Invariants and the theory of numbers DISCKSONtomlr.free.fr/Math%E9matiques/Fichiers%20Claude/... ·...

On Invariants and the Theory

of Numbers

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[THE MADISON COLLOQUIUM, 1913, PART I]

On Invariants and the Theory

of Numbers

b Leonard Eugene Dickson

DOVER PUBLICATIONS, INC., NEW YORK

-- -. .-__--_~ ,---.-..---- ___-

Published in Canada by General Publishing Company, Ltd., 30 Lesmill Road, Don Mills, Toronto, Ontario.

Published in the United Kingdom by Constable and Company, Ltd., 16 Orange Street, .London WC 2.

This Dover edition, first published in 1966, is an unabridged and unaltered republication of the work originally published by the American Mathe- matical Society in 1914 as Part I of The Madison Colloquium (1913), Volume IV of the Colloquium Lectures.

Part II of The Madison Colloquium is reprinted separately by Dover Publications under the title Topics in the Theory of Several Complex Variables, by William Fogg Osgood.

This edition is published by special arrangement with the American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02964.

Library of Congress Catalog Card Number: 66-23743

Manufactured in the United States of America Dover Publications, Inc.

180 Varick Street New York, N. Y. 16614

CONTENTS

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

LECTURE I

A !hiEORY OF h’hRIlLNT s APPLICABLE TO ALGEBRAIC AND

MODTJLAR FORMS

l-3. Introduction to the algebraic side of the theory by means of the example of an algebraic quadratic form in m variables. . . . . . . . . . . . . . . . . . . . . . . . .

47. Introduction to the number theory side of the theory of invariants by means of the example of a modular quadratic form.. . . . . . . . . . . . . . . . . . .

8-Q. Modular invariants are rational and integral. . . . . . 10. Characteristic modular invariants. . . . . . . . . . . . . . . 11. Number of linearly independent modular invariants 12. Fundamental system of modular invariants. . . . . . . 13. Minor r81e of modular covariants . . . . . . . . . . . . . . . 14. References to further developments. . . . . . . . . . . . .

4

6 12 13 13 14 15 15

LECTURE II

SEMINV -s OF ALGEBRAIC AND MODULAR BINARY FORMS

l-6. Introductory example of the binary quartic form. . 16 7-10. Fundamental system of modular seminvariants of a

binary n-ic derived by induction from n - 1 to n 21 11. Explicit fundamental system when the modulus p

exceeds n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

. . . Vlll CONTENTS.

12. Anothermethodforthecasep> It . . . . . . . . . . . . . . 27 13. Number of linearly independent seminvariants. . . . 28

14-15. Derivation of modular invariants from seminvariants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

LECTURE III

INVARIANTS OF A MODULAR GROUP. FORMAL INVARIANTS AND COVARIANTS OF MODULAR FORMS. APPLICATIONS

l-4. Invariants of certain modular groups; problem of Hurwitz.................................... 33

5-11. Formal invariants and seminvariants of binary modular forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

12. Theorem of Miss Sanderson. . . . . . . . . . . . . . . . . . . . 54 13. Fundamental systems of modular covariants. . . . . . 55 14. Form problem for the total binary modular group 58 15. Invariantive classification of forms. . . . . . . . . . . . . . 61

LECTURE IV

MODULAR GEOMETRY AND COVARIANTIVE THEORY OF A QUAD- RATIC FORM IN m VARIABLES MODULO 2

1-2. Introduction. The polar locus. . . . . . . . . . . . . . . . . 3. Odd number of variables; apex; linear tangential

4. 5. 6. 7. 8.

9-32. 33.

equation................................... 66 Covariant line of a conic. , . . . . . . . . . . . . . . . . . . . . . 69 Even number of variables. . . . . . . . . . . . . . . . . . . . . . 70 Covariant plane of a degenerate quadric surface . . 71 A configuration defined by the quinary surface. . . . 72 Certain formal and modular covariants of a conic 73 Fundamental system of covariants of a conic. . . . . 76 References on modular geometry.. , . . . . . . . . . . . . . 98

65

CONTENTS. ix

LECTURE V

A THEORY OF PIANE CUBIC CURVES NITH A REAL INFLEXION POINT VALID IN ORDINARY AND IN MODULAR GEOMETRY

1. Normal form of a ternary cubic. . . . . . . . . . . . . . . . . 99 2. The invariants s and t. . . . . . . . . . . . . . . . . . . . . . . . 99 3. The four inflexion triangles. . . . . . . . . . . . . . . . . . . . . 100 4. The parameter 6 in the normal form. . . . . . . . . . . . . 101

5-9. Criteria for 9,3 or 1 real inflexion points; sub-cases 101

__I______ -__ -___-... _-

INTRODUCTION

A simple theory of invariants for the modular forms and linear transformations employed in the theory of numbers should be of an importance commensurate with that of the theory of invariants in modern algebra and analytic projective geometry, and should have the advantage of introducing into the theory of numbers methods uniform with those of algebra and geometry.

In considering the invariants of a modular form (a homogeneous polynomial with integral coefficients taken modulo p, where p is a prime), we see at once that the rational integral invariants of the corresponding algebraic form with arbitrary variables as coefficients give rise to as many modular invariants of the modular form, and that there are numerous additional invariants peculiar to the case of the theory of numbers. More- over, nearly all of the processes of the theory of algebraic invariants, whether symbolic or not, either fail for modular invariants or else become so complicated as to be useless. For instance, the annihilators are no longer linear differential oper- ators. The attempt to construct a simple theory of modular invariants from the standpoints in vogue in the algebraic theory was a failure, although useful special results were obtained in this laborious way. Later I discovered a new standpoint which led to a remarkably simple theory of modular invariants. This standpoint is of function-theoretic character, employing the

1

2 THE MADISON COLLOQUIUM.

values of the invariant, and using linear transformations only in the preliminary problem of separating into classes the particular forms obtained by assigning special values to the coefficients of the ground form. As to the practical value of the new theory as a working tool, it may be observed that the problem to find a fundamental system of modular seminvariants of a binary form is from the new standpoint a much simpler problem than the corresponding one in the algebraic case; indeed, we shall exhibit explicitly the fundamental system of modular seminvariants for a binary form of general degree.

It will now be clear why these Lectures make no use of the technical theories of algebraic invariants. On the contrary, they afford an introduction to that subject from a new standpoint and, in particular, throw considerable new light on the relations between the subjects of rational integral invariants and transcendental invariants of algebraic forms and the corresponding questions for seminvariants. Again, I shall make no use of technical theory of numbers, presupposing merely the concepts of congruence and primitive roots, Fermat’s theorem, and (in Lectures III and V) the concept of quadratic residues.

The developments given in these Lectures are new, with exceptions in the case of Lecture I, which presents an introduction to the theory, and in the case of the earlier and final sections of Lecture III. But in these cases the exposition is considerably simpler and more elementary than that in my published papers on the same topics. The contacts with the work of other writers will be indicated at the appropriate places. Much light is thrown upon the unsolved problem of Hurwitz concerning formal invariants.

In many parts of these Lectures, I have not aimed at complete generality and exhaustiveness, but rather at an illumination of typical and suggestive topics, treated by that particular method which I have found to be the best of various possible methods. Surely in a new subject in which most of the possible methods are very complex, it is desirable to put on record an account of the

INVARLWTS AND NUMBER THEORY. 3

simple successful methods. Finally, it may be remarked that the present theory is equally simple when the coefficients of the forms and linear transformations are not integers, but are elements of any Snite field.

I am much indebted to Dr. Sanderson and Professors Cole and Glenn for reading the proof sheets.

LECTURE I

A THEORY OF INVARIANTS APPLICABLE TO ALGEBRAIC AND MODULAR FORMS

INTRODUCTION TO THE ALGEBRAIC SIDE OF THE THEORY BY

MEANS OF THE EXAMPLE OF AN ALGEBRAIC QUADRATIC

FORM IN m VARIABLES, $0 l-3

1. Classes of Algebraic Quadratic Forma.-Let the coefficients of

be ordinary real or complex numbers. Let the determinant

(2) D = l&l (i,j = 1, s-s, m)

of a particular form qm be of rank r (r > 0); then every minor of order exceeding t is zero, while at least one minor of order r is not zero. There exists a linear transformation of determinant unity which replaces this qm by a form*

(3) W12 + ***+cY$.x:r2 (a1$:0,“‘,a,*0).

Indeed, if 811 =l= 0, we obtain a form lacking 21x2, . . ., ~1% by substituting

Xl - 811~v312s + * * * + PlA)

for XI. If BII = 0, Bi6 * 0, we substitute xc for x1 and - 21 for xi; while, if every /3kk = 0, and 812 =I= 0, we substitute x2 + x1 for x2; in either case we obtain a form in which the coefficient of xl2 is not zero. We now have ~1x12 + 4, where crl =j= 0 and Cp involves only x2, . . . , G. Proceeding similarly with 4, we ultimately obtain a form (3).

Now (3) is replaced by a similar form having al = 1 by the * Note for later use that each ar and each coeflicient of the transformation

ia a rational function of the B’s with integral coefficients. 4

IN-VARIANTS AND NUMBER THEORY. 5

transformation .*

51 = cQ-*xl’, xm = a,+&‘, xj = Xi’ (i=2, **.,m-1)

of determinant unity. Hence there exists a linear transformation with complex coefficients of determinant unity which replaces qni by

(4) Xl2 + ’ - * + xi-1 + Da;n2, Xl2 + * * * + x,2,

according as r = m or r < m. In the first case, the final coefficient is D since the determinant (2) of a form q,,, equals that of the form derived from q,,, by any linear transformation of determinant unity. Hence all quadratic forms (1) may be separated into the classes

(5) C =,D, C, (DPO,r=O,L--.,m-l),

where, for a particular number D =j= 0, the class C,,,, D is composed of all forms qn, of determinant D, each being transformable into (41); while, for 0 < r < m, the class C, is composed of all forms of rank r, each being transformable into (42) ; and, finally, the class Co is composed of the single form with every coefficient zero. In the last case, the determinant D is said to be of rank zero. Using also the fact that the rank of the determinant of a quadratic form is not altered by linear transformation, we conclude that two quadratic forms are transformable into each other by linear transformations of akterminant unity if and only if they belong to the same &a.? (5).

2. Single-palued InvaGz& of q,,,.-Using the term function in Dirichlet’s sense of correspondence, we shall say that a single- valued function I$ of the undetermined coefficients /9ij of the general quadratic form qfii is an invariunt of q,,, if C#I has the same value for all sets pi,, /3$, l * * of coefficients of forms q:, qz, - - l

belonging to the same class.* The values v,,,,=, v, of I$ for the various classes (5) are in general different. For example, the determinant D is an invariant; likewise the single-valued func-

* Briefly, if + has the same value for all fonm in any class.


tion T of the undetermined coefficients p<j which specifies the rank Of lhjl-

Each consistent set of values of D and T uniquely determines a class (5) and, by definition, each class uniquely determines a value of C/L Hence C$ is a single valued function of D and r.

Every single-valued invariant of a system of fomns & a single- td.d function of the &variants (D and r in our example) which

completely characterize the classes.

3. R&onul Integral Invariants of q,,,.-If the invariant C#J is a rational integral function of the coefficients @ii, it equals a rational integral function of D. For, if the /3’s have any values such that D =I= 0, + has the same value for the form (1) as for the particular form (4J of the same class. Hence C$ = P(D), where P(D) is a polynomial in D with numerical coefficients. Since this equation holds for all sets of B’s whose determinant is not zero, it is an identity.

INTRODUCTION TO THE NUMBER THEORY SIDE OF THE

THEORY OF INVARIANTS BY MEANS OF THE EXAMPLE OF A MODULAR QUADRATIC FORM, $0 4-7

4. Classes of Modular Quadratic Forms q,,,.-Let xl, . * ., x,,, be indeterminates in the sense of Kronecker. Let each &j be an integer taken modulo p, where p is an odd prime. Then the expression (1) is called a modular quadratic form. By 0 1, there exists a linear transformation, whose coe5cients are integers* taken modulo p and whose determinant is congruent to unity, which replaces qm by a quadratic form (3) in which each Cyk is an integer not divisible by p. Thust each (Yk is congruent to a power of a primitive root p of p. After applying a linear transformation of determinant unity which permutes s12, . . ., x,2, we may assume that crl, . . . , CY* are even powers of p and that %+1, . . ., CY~ are odd powers of p. The transformation which

* Perhaps initially of the form a/b, where a and b are integers, b not divisible by p. But there exists an integral solution q of qb = a (mod p).

t For p = 5, p = 2, 1 = 2: 2 = 2l, 3 = 2: 4 = 2% (mod 5).

INVARIANTS AND NUMBER THEORY. 7

multiplies a particular xi (; < m) by pk and k by pmk is of determinant unity.

First, let r < m. Applying transformations of the last type to (3), we obtain

(6) Xl2 + * * * + X82 + pxf,, + * * * + px2.

Under the transformation of determinant unity

Xj = O!Xj + @Xj, Xj = - flXi + dj, Gn = (2 + PYxm,

xi2 + xj” becomes (a2 + p2) (Xi2 + Xiz>. Choose* integers cy, /3 so that

(7) Pw+P2) = 1 (mod PI.

Hence the sum of two terms of (6) with the coefficient p can be transformed into a sum of two squares. Thus by means of a linear transformation, with integral coefficients of determinant unity, q,,, can be reduced to one of the forms

(8) a2+ - - - + XT-, + x,2, xl2 + . - - +xE-,+pxr2 (O<r<m).

Next, let r = m. We obtain initially

Xl2 + - * * + x2 + PX?,, + * * * + P&4 + m?L2,

in which u need not equal p as in (6). If there be an even number of terms with the coefficient p, we obtain as above a form of type (41). In the contrary case, we get

f = x12+ * * * + xi-2 + #ox:-1 + p-‘Dxm2.

If D = p2z+1 (mod p), f is transformed into (41) by

xm-l = - plXm, xm = p-lx,*.

But if D = p2z, f is reduced to (41) by the transformation

xp+l = ax-1 + 6p2”lXln, xm = - ax,-, + a!pXm,

p (a2 + p2vi2) = 1, *If p = 5, p = 2, we may take Q = fi = 2. For any p, either there is an

integer 1 such that 1% = - 1 (mod p) and we may take p(a + Z@) = 1, a-~=l;orelsez*+1takee1+(p - 1)/2 incongruent values modulo p, no one divisible by p, when x ranges over the integers 0, 1, . . *, p - 1, so that z2 + 1 takes at lea& one value of the form p**-l. In the latter event, a = p-*, fi = za satisfy (7).


of determinant unity. The final condition is of the form (7) with@= pW and hence has integral solutions (Y, 6.

Hence the classes of modular quadratic forms are

(9) c na, D, c,, 1, CT. -1, co

(D = 1, . * *, p - 1; r = 1, * * *, 712 - l),

where Cm, D is composed of all modular quadratic forms whose determinant is a given integer D not divisible by p, each being transformable into (41), where C,, 1 and C,, -1 are composed of all forms transformable into (81) and (&) respectively, and Co is composed of the form all of whose coefficients are zero.

Two modular quadratic forms are transformable into each other by linear transformations with integral coejicienta of determinant unity module p if and only if thy belong to the same class (9).

Indeed, since D and T are invariants,* it remains only to show that the two forms (8) are not transformable into each 0ther.t But if a linear transformation

(i = 1, ..., m)

replaces f = xl2 + - - - + xr2 by F = Xl2 + . . . + XT-, + pXr2, then, for j > r,

g= ~~x'+g= 0, .g= cvij =0 (iSr,j>r), i L-1 i i i

(i = 1, ’ * *, 9).

Hence under this partial transformation on 51, . * . , z,., we would have f = F. Thus the determinant of F would equal [aij12 times the determinant unity off and hence equal an even power of p. But the determinant of F is actually p.

*tjBnowthe maximm order of a minor not divisible by p. t An immediate proof follows from the values taken by the invariant A,

given below. But aa the neceseity of constructing A, is based upon the fact that the forma (8) do not belong to the same &as, it seem8 preferable to prove the last fact without the use of A,.

INVARIANTS AND hTUMBER THEORY. 9

The invariants D and T therefore do not completely characterize the classes of modular quadratic forms, a result in contrast to that for algebraic quadratic forms. We shall give ‘a criterion to decide whether a given form of rank r (0 < r < m) is of class C,, 1 or of class C,, --1 and later deduce an invariantive criterion.

5. t%terion for chaaea C,, Ir.-Such a criterion may be obtained from Kronecker’s elegant theory of quadratic forms.* We shall make use of the theorem that a symmetrical determinant of rank T (r > 0) has a non-vanishing principal minor M of order r, i. e., one whose diagonal elements lie in the main diagonal of the given determinantt After an evident linear transformation of determinant unity, we may set

w M = l&l + 0 (mod P> (i,j = 1, * * *, T).

In the present problem, r < m. To q,,, apply the transformation

Xj = Xj + CiXm (i = 1, * . *, r),

Xj = Xj (i = T+ 1, . . ..rn)

of determinant unity in which the ci are integers. We get

where

Bjm = $, Pijci + Pjm (j = 1, * * *, m).

In view of (10) there are unique values of cl, -6 ., G such that

Bjm z 0 (mod p) (j = 1, a. a, r).

But the determinant of the coefficients of cl, * * ., cr, 1 in

B lm9 Bzm, - - - , Bm, &cm (T < k 2 m)

* Kronecker, Werke, vol. 1, p. 166, p. 357; cf. Gundelfinger, C’relb, vol. 91 (1881), p. 221; B8cher, Introduction to Higher Algebra, p. 58, p. 139.

t The most elementary proof ie that by Dickson, Annals of Mathematics, ser. 2, vol. 15 (1913), pp. 27, 28. For other short proofs, see Wedderburn, ibid., p. 29, and Kowalewski, Determinantentheorie, pp. 122-124.

-_-. .-. .-._- ._- ~.. ._ .~ . . .“._ ._. .._-.-.--. ~.. -.-.... .._


is the minor of /31rn in the determinant

Ii&l (i, j = 1, * * *, r, k, m)

and hence is zero, being of order r + 1. Hence Bktn = 0. Thus q,,, has been transformed into

n-1

‘ZZl Bijxi;yj*

After repetitions of this process, qm is transformed into*

(11) 4iil BijXiXj* 1-

This form, of determinant M, can be reduced (0 4) to

x12+ --- + x3-1 + Jfxr2

by a linear transformation on x1, . . . , xr with integral coefficients of determinant unity modulo p. Express M as a power p2’+’ (e = 0 or 1) of a primitive root. Since r < m, we may replace xr by p-‘xr and G by p’x,,, and obtain (81) or (82) according as e=Oore= 1. Now ~(=-l)‘~ is not congruent to umty, but its square is congruent to unity modulo p, by Fermat’s theorem; hence it is = - 1. Thus, in the respective cases,

p-1 MT=+1 or -1 (mod P>-

Hence if a form is of rank r and if M is any chosen r-rowed principal minor not divisible by p, the form is of class C,, 1 or c r, -I according as the first or second alternative (12) holds.

6. Invariantive Criterion for Cbses CG, *I.-A function which has the value + 1 for any form of class C,, +I, the value - 1 for any form of class C;, - 1, and the value zero for the remaining classes C,,,, =, Co, CL, *I (k =I= r), is an invariant (0 2). This functiont is

* This proof and the results in $8 4-13 am due to Dickson, Transactions of the American Mahma.ticuZ Society, vol. 10 (1909), pp. 123433.

t Constructed synthetically in the paper last cited.

____l__-_ -- _-- ____-.-- .-.-. -I

INVARIANTS AN’D NUMBER THEORY. 11

p-1

A, = (13)

@I!&” + l!fsp+ (1 - MP-1) + * * * p-1

+ J&7(1 - MrP-‘) -** (1 - M;::)]II(l - @-I),

where Ml, 0 s . , M,, denote the principal minors of order 7 taken in any sequence, and d ranges over the principal minors of orders exceeding r. For, if any d + 0, the rank exceeds r and A, = 0 by Fermat’s theorem. Next, let every d = 0, so that the rank is r or less, and the final factor in (13) is congruent to unity. Then, if every Mg = 0, the rank is less than T and A, = 0. But, if Mr + 0,

P--l

A, s >I,-% f 1 (mod ~1,

by (12), the sign being the same as in Cr, *I. If Ml 3 0, Mz $ 0, p-1

A,~&~sdzl (mod PI,

etc. Note for later use that P--l

(14) A,,, = D”.

7. Ri&md Integral Invariunh of q,,,.-The function

(15) l()=II(l-+;;‘) (&j= l,*.*,m;iSj)

has the value 1 for the form (of class Co) all of whose coefficients are zero and the value 0 for all remaining forms qm, and hence is an invariant of qm. We now have rational integral invariants

(16) D, AI, ---> A nSl1 IO

which completely characterize the classes (9). Hence, by the general theorem in 0 12, any rational integral invariant of the modular form q,,, is a rational integral function of the invariants (16) with integral coefficients. In other words, invariants (16) form a fundamental system of rational integral invariants of qm.

If we employ not merely, as before, linear transformations with integral coefficients of determinant unity modulo p, but those of all determinants, we obtain at once the classes

c co r, *I, (r = 1, . . ., m),

. .-. -- -..-.__.


and see that these are characterized by AX, . . *, A,,,, IO. The latter therefore form a fundamental system of rational integral absolute invariants. But D is a relative invariant.

GENERAL THEORY OF MODULAR INVARIANTS, $0 8-14

8. De$nitions.-Let S be any system of forms in x1, l . . , x,,, with undetermined integral coefficients taken modulo p, a prime. Let G be any group of linear transformations on xl, 0 . . , x,,, with integral coefficients taken modulo p. The particular systems S’, S”, * - * , obtained from S by assigning to the coefficients particular sets of integral values modulo p, may be separated into classes CO, Cl, - - -, C-1 such that two systems belong to the same class if and only if they are transformable into each other by transformations of G.

A single-valued function 4 of the coefficients of the forms in the system S is called an invariant of S under G if, for i = 0, 1, . . . ,n - 1, the function 4 has the same value vi for all systems of forms in the class Ci.

In case the values taken by 4 are integers which may be reduced at will modulo p and congruent values be identified, the invariant is called modular. Since this reduction can be effected on each coefficient of the modular forms comprising our system S, any rational integral invariant of S is a modular invariant.

An example of a non-modular invariant is the transcendental function r defining the rank of the determinant of the modular quadratic form qm. The values of r are evidently not to be identified when merely congruent modulo p. However, the residue of r modulo p is a modular invariant, since

(17) r = Al2 + 2A22 -I- . . . + mAm2 (mod P>.

9. Modular Invariant-s are Rational and Integral.-Any modular invariant 4 of a system S of modular forms can be identified with a rational integral function (with integral coefficients) of the coefficients cl, . . ., cs appearing in the forms of the system S.

INVARIANTS AND NUMRER THEORY. 13

For, if

t$=V dir . ..* % when cl= el, l *s,c.=e, (mod P>,

then + is identically congruent (as to cl, . . . , c,) to

(18)

p--l

C q, -, e,=O

vel, . . . . #, fj I1 - (ci - eP),

as shown by Fermat’s theorem.

10. Characteristic Modular Invatiants.-The characteristic invariant Ik of the class Ck is defined to be that modular invariant which has the value unity for systems of fornk of the class Ck and the value zero for any of the remaining classes.

For example, for a single quadratic form q,,,, I,, is given by (15), while,the characteristic invariants for the classes C,, 1 and c ,, -I are (19) I,. 1 = +(A,2 + A,), IT,-1 = 6(A,2 - A,).

For any system of forms with the coefficients cl, . . ., c8, we have

(20) I,=Cl&l-(Ci- , Ci(Jd)P-l)

where the sum extends over all sets of coefficients ~(~1, . . a, cl@) of the various systems of forms of class Ck. In particular, in accord with (15),

(21) IO = *g (1 - v-9.

11. Number of Linearly Independent Modular Invariants.- Since any modular invariant I takes certain values vo, s . . , v,+l for the respective classes CO, . . *, C-1, we have

cm I = VOIO + VA + -. ’ + v,Jn--l.

Hence any modular invariant can be expressed in one and but one way as a linear homogeneous function of the characteristic invariants. Moreover, the number of linearly independent modular invariant8 equals the number of classes.


For example, using (19), we see that a complete set of linearly independent modular invariants of the quadratic form q,,, module p (p > 2) is given by

(23) IO, A,, A,2 (T = 1, . . ., m - l), Dk (k = 1, . a ., p-l).

12. Fundamental Systems of Modular Invariants.--While, by (22), the characteristic invariants IO, . * . , 1,1 form a fundamental system of modular invariants of a system S of modular forms, it is usually much easier to find another fundamental system. In fact, certain invariants are usually known in advance, e. g., the invariants of the corresponding system of algebraic forms. M’e shall prove the following fundamental theorem:

If the modular invariants A, B, . . ., L completely characterize the classes, they form a fundamental system of modular invariants.

For example, 10, . . . , 1-1 evidently completely characterize the classes and were seen to form a fundamental system.

Letc1, -6.) c8 be the coefficients of the forms in the system S. Let each cd take the values 0, 1, . . . , p - 1. For the resulting p” sets of values of the c’s, let the rational integral functions A, B, es., Lof cl, . . . . c8 take the distinct sets of values

A;, Bi, ..., Li (i = 0, ..., n - 1).

Thus there are n classes of systems S and by hypothesis the ith class is uniquely defined by the values A.;, . . . , L; of our invariants. A rational integral invariant +(cr, . a s, c,) takes the same value for all systems of forms in the ith class, so that this value may be designated by &. Now the polynomial

n-1

P(A, B, . . ., L) = gO&{l - (A- Ai)P’J...(l- (L-L@‘+)

is congruent to +i when A = Ai, . . . , L = Li (mod p). Hence

d&9 - * -, c,) = P(A,B, se., L) (mod P) for all sets of integral values of cl, . + . , cg. In view of Fermat’s theorem, we may assume that each exponent in r$(cr, . . . , c,) is less than p. If we replace A, . . ., L by their expressions in


terms of the c’s, P (A, * * *, L) becomes a polynomial, which, after exponents are reduced below p, will be designated by # (cl, . . . , c,). Then 4 and + are identically congruent in cl, - . . , c., that is, corresponding coefficients are congruent modulo p. In fact, a polynomial of type 4 is uniquely determined by its values for the p* sets of values of cl, . . . , cs, each chosen from 0, 1, . . ., p - 1 (0 9). Hence 4 can be expressed as a polynomial in A, * . . , L with integral coefficients.*

13. Minor R61e of Modular Covariank-In contrast with the case of algebraic forms, the classes of modular forms are completely characterized by rational integral invariants. Such invariants therefore suffice to express all invariantive properties of a system of modular forms. In this respect, modular covariants play a superfluous Ale. For instance, a projective property of a system of algebraic forms is often expressed by the identical vanishing of a covariant. But if C is a modular covariant with the coefficients cl, . . ., c8, then 10 given by (21) is a modular invariant of C and hence of the initial system of forms. We have C = 0 or C + 0 (mod p) identically, according as 10 = 1 or 10 = 0.

14. References to Further Developments.-This general theory of modular invariants has been applied by me to determine a complete set of linearly independent modular invariants of Q linear forms on m variables,t and a fundamental system of modular invariants of a pair of binary quadratic forms and of a pair of binary forms, one quadratic and the other 1inear.S

The theory has been extended to combinants and applied to a pair of binary quadratic forms.5

* Thii correct theorem for any finite field cannot be extended at once to any field es stated by me in American Journul of Mathematics, vol. 31 (lQOQ), top of p. 338.

t Proceedings of the Lo&m Ma&matical Society, ser. 2, vol. 7 (lQOQ), pp. 430444.

$ American Journal of Mathematice, vol. 31 (lQOQ), pp. 343-354; cf. pp. 103-146, where a less effective method ia used.

0 Dickson, Quarterly Journal of Mathematics, vol. 40 (lQOQ), pp. 349-366.

LECTURE II

SEMINVARIANTS OF ALGEBRAIC AND MODULAR BINARY FORMS

INTRODUCTORY EXAMPLE OFTHE BINARYQUARTICFORM, $8 l-6

1. Comparative View.-Let the forms

f = ad + %x% + 6a&y2 + hq? + ad,

with real or complex coefficients, be separated into classes such that two forms f are transformable into one another by a transformation of type

(1) x = 2’ + ty’, y = y’,

if and only if they belong to the same class. Then a single- valued function S(ao, . * *, ad) is called a seminvariant of f if it has the same value for all of the forms in any class.

By the repeated application of this definition and without the aid of new principles, we shall obtain a fundamental system of rational integral seminvariants of f, then on the one hand the additional single-valued seminvariant needed to form with these a fundamental system of single-valued seminvarints, and on the other hand the additional rational integral modular seminvariants needed to form with them a fundamental system of modular seminvariants off. It is such a comparative view that we desire to emphasize here. In later sections, we shall show that it is usually much simpler to treat the modular case independently and in particular without introducing all of the algebraic seminvariants, which become very numerous and most unwieldy for forms of high degree. The rational integral seminvariants S of an algebraic form are of special importance since each is the leading coefficient of one and but one covariant, which can be found from S by a process of differentiation. For example, the seminvariant a0 is the leading coefficient of the covariant f.

16


2. The Classes of Algebraic Quurtic Forma.-Consider a quartic form f in which a, is the first non-vanishing coefficient. Apply transformation (1) with

(2)

We obtain a form having zero in place of the former awl. Drop- ping the accents on xl, y’, we obtain, for k = 0, 1, 2, 3, the respective forms

(3) a0+0: aox + 6ao%dy2 + 4aom2&x$ + ao-SS&,

(4) a0 = 0, al=/=O: 4ady + al3hqP + al-2S1&,

(5) ao=al=O, a2*0: 6ady2 + Qa2+&4$,

(6) a0 = al = a2 = 0, a3 =j= 0: kq?,

(7) ao=al=a2=a~=O: 4,

no transformation having been made in the last case. Here

(8) S2 = adz2 - a12, SS = ao2a3 - 3aoala2 + 2a1*,1

(9) S4 = aosa4 - 4a02a1a3 + 6aoa12a2 - 3a14,

(10) &a = 4a1a3 - 3a22, S14 = a12a4 - 2w2a3 + a2s,

St4 = 3a2a4 - 2as2.

If we apply to one of the forms (3)-(6) a transformation (1) with t + 0, we obtain a form having an additional (second) term. Hence no two of the forms (3)-(7) can be transformed into each other by a transformation (l), so that each represents a class of forms. For example, there is a class (5) for each set of values of the parameters a2 and 6’24 (a2 + 0).

3. Rational Integral Seminvariants of an Algebraic Qua&.- First, a0 is a seminvariant since it has a definite value =j= 0 for any form in any class (3) and the value zero for any form in any class (4)-(7). Next, S2, &, S4 are seminvariants, since they have constant values

(11) S2 = - a12, S3 = 2a13, S4 = - 3a14 (if a0 = 0)


for any form in any class (4)-(7), and constant values for any form of a definite class (3), for which therefore a0 has a definite value =j= 0 and UO-%‘~, . * . , and hence each Si, has a definite value. Moreover, these seminvarianta ao, Sz, &, Sr completely churacterize the claw38 (3).

Consider a quartic form f in which a~, al, a~, US, u4 are arbitrary, except that a0 =+ 0. Any rational integral seminvariant Sk-Jo, -**9 ad) has the same value for f as for the particular form (3) in the same class as f. Hence

s2 ss s4 S=S a0,0,- 2,~~ ( > +(a0, 82, SS, 841

a0' a0 =

aoj ,

where + is a rational integral function of its arguments. We therefore seek such functions 4 as are divisible by a power of ao, and hence by (11) in which the terms involving only aI cancel. The function of lowest degree is evidently

(12) S4 + 3~92~ = ao21, I = aoa4 - 4ala8 + 3e2.

The next lowest degree is 6 and the function is

d&i& + e8S2 + (3d + 4e)&3s

The coefficient of d is ao21Ss, that of e is

(13) (D z a&Q

S2 -I- 4st8 = ao2D

- 6aoamaa + 4aoa2 + 4aPar - 3a12az2).

Henceford= l,e= - 1, the function is the product of ao2 and

(14) I&-D=aoJ, J=aaoa2a4-a,,aa2+2ala2a~-a~2a4--3.

We do not retain D since it is expressible in terms of the other functions. Eliminating D between (13) and (14), we get

(15) 8a2 + 482’ - ao21S2 + ao3J = 0.

Now I and J are aeminvariank Indeed, if a0 + 0, they are expressible in terms of the parameters ao, SC in (3) and hence each has the same value for any form in a class (3) ; while

(16) I = - 513, J = - 514 (if a0 = O),


so that each has the same value for any form in a class (4); finally, (17) I = 3at2, J = - a33 (if a0 = al = 0),

so that each has the same value for any form in a class (5)-(7). From 4 we eliminate S* by means of (12) and then the second

and higher powers of S3 by means of (15). Thus S equals N/aok, where N is a rational integral function of

(18) ao, &, 83, I, J,

of degree 0 or 1 in S3. If k > 0, we may evidently assume that not every term of the polynomial N in the arguments (18) has the factor a~. Let P(&, 6’3, I, J) denote the aggregate of the terms of N not involving a0 explicitly. We shall prove that, if k > 0, N/aok isthen not a rational integral function of a~, * * a, ~24. For, if it be, P vanishes when a0 = 0. By (11) and (16), the terms independent of a0 in J involve a4, while those in I, SH, S’s do not. Hence J does not occur in P. Then, by (11) and the term 3as2 in I, we conclude that I does not occur in P. Thus P is a polynomial in & and S3 of degree 0 or 1 in SS and is not identically zero. By (ll), it cannot vanish for a0 = 0.

Under the initial assumption that a0 =j= 0, we have now proved that any rational integral seminvariant S equals a polynomial in the functions (18). The resulting equality is therefore an identity.

Th-e seminvariank, (18) form a ficndamental system of rational integral seminvariants of the algebraic quurtic form.*

They are connected by the relation, or syzygy, (15).

4. Invariantive Characterization of the Classes.-By 0 3, the classes (3) are completely characterized by the seminvariants a~, S’s, S3, I. These with J characterize the classes (4) having a0 = 0, al + 0. For, by (ll), AS’S and S3 determine al; while, by (16), I and J determine the remaining parameters in (4).

* The above proof differs from that by Cayley in minor details and in the method of obtaining the functions (18) and the vetication that they are seminvariants (the present method being based upon the classes).


The parameter ~42 (a2 =I= 0) in (5) is determined by I and J, in view of (17).

We have now gone as far as is possible in the characterization of the classes by means of rational integral seminvariants S, since the parameters S 24, u3, a4 in (5)-(7) cannot be determined by such seminvariants. Indeed,* for a0 = al = 0, we have I!32 = ss = 0 by (1 l), while I and J reduce to powers of a2 by (17).

5. Single+&& Semin~uriunts.-We may, however, construct a single-valued seminvariant which shall determine these outstanding parameters 6’24, a3, ~4. To this end consider the single- valued function V defined as follows by its values in the sense of Dir&let. We take V = 0 if a0 =l= 0 or if al $: 0, and v = S24, us, u4 in the respective cases (5), (6), (7). Since V has the same value for all forms in any class, it is a seminvariant. The seminvariants (18) and V completely characterize the classes (3)-(7) and hence, by 0 2 of Lecture I, form a fundamental system of single-valued seminvariants of the algebraic binary quartic form.

6. Seminvuriants of a Modular Qua&c Form.-Passing to the number theory case, let the coefficients of the quartic form f be integers taken modulo p, where p is a prime exceeding 3. The denominator in (2) is then not divisible by p, so that the classes are again (3)-(7).

By the general theory in Lecture I, it is possible to characterize all of the classes by means of rational integral seminvariants, and the latter will then form a fundamental system. In particular, we do not now require the use of such a bizarre function as that used in 0 5.

* A proof of this fact, not bssed upon the f&l theorem of 0 3, would afford a better insight into the nature of the last steps in $3 and explain, in particular, why we stopped with I and J and did not consider combinations of the Si of higher than the sixth degree in the u’s. To this end, let S be a seminvariant homogeneous of total degree i, in the a’s, and isobaric, of constant weight W. As well known, 4i 2 220. Thus S cannot have a term u8f or a& and cannot reduce, when a0 = al = 0, to U&S’LP (m > 0), of degree 2 + 2m and weight 21+&n.


We shall make frequent use of the abbreviation

(19) P, = (1 - a$-‘)(1 - UP--‘) * ** (1 - UP-‘).

Then PI&, Psa, and Ptal are seminvariants* since each takes the same value for all forms in any class. For the classes (5), (6), (7), their values are S24, a3 and a4, respectively. Hence the jive seminvarianti (18) together with P&J, Ptaa and Psar

completely ch4zracterize the classes and therefore form a fundamental system of rational integral seminvariank, of the qua& form f with integral coefients taken module p, p > 3.

SEMINVARIANTS OF A MODULAR BINARY [FORM OF ORDER n,

7. Fundamental System of Moclulur Seminvariants Derived

by Induction from n - 1 to n.-It is necessary to distinguish the case in which the modulus p is prime to n from the case in which p divides n. Binomial coefficients for the form are not per- missible in the second case and often not in the first case (for example, if n = 4, p = 3, since (3 is then divisible by p). Denote the form by

(20) F, = AG” + A#-ly + . . . + A,y”.

First, let p be prime to n. For A0 =+ 0, we can transform F,, into a form lacking the second term and having as coefficients the quotients of

(21) g2 = nAoA2 - *(n - l)A?,

a3=n2A>A3-(n-2)n&4A++(n-l)(n-2)Ar3, ***

by powers of nA0. These may also be obtained from (8) by identifying F, with

(22) fn = aoxn + nalxn-ly + n(n - 1)

2 azxn-2y~+ . . . .

* The first is one-half the discriminant of the semicovariant

PIfly4 = Pl(6w9 + da&q/ + a@) (mod P),

and the last two are the seminvariants of PzT11/s = P2(4asl: + a& (mod p).

1 - . . . I ._ _ . . . I . _- .__-_-_ -_ - . - . . ‘ . - I - . - - - i._.._. . - . ” - __. . “~. . - _ - . . , - . . -


For p prime to n, a fundamental system of seminvariants of F, is given by Ao, ~2, . * . , u, together with a fundamental system of the particular form of order n - 1

(23) FL = PoF&

E P&x”-‘+ P~A~x”-~Y+ m - a +PoA,y-’ (mod p>,

where PO = 1 - A#‘+.

Indeed, Ao, ~2, . . . , u,, completely characterize the classes of forms F, with A0 + 0. Since yF,+< = F, identically, when A0 = 0, the classes of forms F, with A0 = 0 are completely characterized by the seminvariants of the fundamental system for F-1’.

For example, A0 and PoAl form a fundamental system of modular seminvariants of A@ + Aly (since these characterize the classes represented by A@ and Aly). The corresponding functions for

are PoAl and Fl’ = PO&X + PoA2y

{ 1 - (P0A3p+}PoA2 = (1 - A1+)PoA2 = P1A2 (mod P).

Hence the theorem shows that, if p > 2,

(24) Ao, 202 = 4Ad2 - A12, PO&, PA2

form a fundamental system of modular aeminvariants of F2. For f2, these are

(249 2a0, S2 = adz2 - a12, POal, Plan.

8. Order a Multiple of the Modulus.-Next, let n = pg. By Fermat’s theorem, xP - xyp’ and hence

(25) C#B = AO(xp - zyr’)*

is unaltered modulo p by any transformation (1). Hence if, for each value of the seminvariant Ao, we separate the forms

(26)


into classes under (l), multiply each form by y and add 9, we obtain the classes of forms F, for this value of Ao. Hence, iif n & o?iv&ible by p a fundamental system of modular seminvariants

of F, is given by A0 and a fundamental system for F-1. For example, if n = p = 2,

- FI = (AI + Adz + A2y

can be transformed into z or A2y by (l), according as Ao + AI = 1 or 0 (mod 2). Adding C$ = A&? - xy) to xy and Azy2, we obtain representatives of the classes of forms F2. Hence the 6 classes are completely characterized by the seminvariants A0 and those (0 7) of Fl, and hence by

(27) Ao, AI, J = (1 + Ao + A&&.

9. Seminvarianf.s of the Binary Cub&~ Form.-The classes of algebraic forms f3 are

(2% a& + 3a0C1S2a$ + a0-2S3y3,

(29) 3ady+ ia14S13ti, 3a2xy2, a&,

where the S’s are given by (8) and (101). The discriminant D of fa is given by (13). As in 0 3, ao, S2, &, D form a fundamental system of seminvariants of fa; they are connected by the syzygy

(13). Henceforth, let the coefficients of fa be integers taken modulo p,

the excluded case p = 3 being treated in 0 15. If p > 3, the classes are again (28) and (29), and a fundamental system of seminvariants is given by

(30) ao, 82, 83, D, Pm, Pm.

It is instructive to compare this result with that obtained by the method of 0 7. Forming the functions (24) for

f2’ = Pofs/2/ = 3po%z + 3P!lwY + Poa3Y2 (mod PI,

and deleting the factor 3 from the first and second, we get*

P0a1, 6 = Po(4ala8 - 3a22) = h&3, Pla2, P2a3.

* They characterize the classes (29) of ja with a0 = 0 and may be so derived.


Hence, if p > 3, these four functions and a~, St, Sa form a fundamental system of modular seminvariants of fa. We may drop POUI since

P--s (31) PQ&~ Sa = f 2Pgz$ = f 2Poal (mod PI. Hence a fundamental system of seminvariants of $3 for p > 3 is

(32) a0, SZ, SS, 6 = PO&~, Pla2, P2a3.

It is easy to deduce 6 from the old set (30), and D from this new set.*

Finally, let p = 2. By 4 7, a fundamental system of seminvariants fort8 is given by ao, Sz, S3 and a fundamental system for f2’. The latter system is derived from (27) by replacing Ao, Al, A2 by Poa~, Poaz, Pea,, and hence is

(I+ a0h (1 + a0la2, (1 + a0>(1 + al + aph.

We may drop (1 + UO)UI = (1 + a&!!&.

10. The Binary Qua& Form. For p = 2, we have

Fs = Ad + (Ao + A2)+/ + Asq.i2 + A&,

whose seminvariants are obtained from those offs at the end of Q 9. They with A0 give a fundamental system of seminvariants of Fd:

Ao, AI, AlAa + Ao + A2, Cl+ AM,

AlA, + AlAs(Ao + Az), K = Cl+ A)(1 + Ao + A2 + AaM-

An equivalent fundamental system is?

(33) Ao, AI, A2 + At, (1 + AI) At,

Au44 + AoA2 + AA, K.

* D = a~“-‘(& + 4W) - 6Sa (mod PI.

For, if a0 ( 0, then 6 E 0 and this relation follows from (13); while, if uo = 0, D = alWl~ = a13 = - S& Conversely, 6 can be expressed in terms of the functions (30). The above relation givea s&. The product of this by SP iscongruentto6if&+O. Also~=Oifa~+O. Thereremainsthecaee in which SI = 0, a0 = 0, whence al = 0, 1 = - 3a2 = - 3(Plar)*.

t Annals of Mat-h, ser. 2, vol. 15, March, 1914. I there give also a complete set of linearly independent invaxiants and of linear ~~variants


For p > 3, fs’ is obtained from f3 by replacing ao, al, a2, as by

4alP0, &PO, &P0, a4P0,

respectively. Making this replacement in the second set of seminvariants of f3 in Q 9, we obtain P@I, which may be dropped in view of (31), and the last five functions (34). Hence,for p > 3, a fundamental system of modular seminvariants of f4 ti given by

(34) a0, 82, Sat 84, P&13, PO&~, P824, P2aa, Paa4.

Here the three i!?<j are given by (10). Since the functions (34) completely characterize the classes (3)-(7), we have a new proof that they form a fundamental system.

11. Explicit Fundamental System when p > n.-Instead of employing the above step by step process, we can obtain directly a fundamental system of modular seminvariants of fn when the modulus p exceeds the order n of the binary form (22). Consider a particular f,, in which ok is the first non-vanishing coefficient:

II n =(.> 2=x 2

ap?&-' i Y (ak $: 0).

To this we apply transformation (1) and obtain

where we have replaced j by I- i and set

Take k < n and give to t the value (2). Thus

n ak

0 ckl

(35) Akz = ((k + ~)a,)zwky


In particular,

gkk = 1, uk,,+l = 0, (roz = g (- 1)“” (r) aO+zl%i,

the last being the algebraic seminvariant designated earlier by SZ. It is obtained from the expansion of (a0 - al)’ by replacing a single a0 in each term by ai. Except for a numerical factor not divisible by p, (TkZ (for 0 < k < I - 1) equals the SkZ in (10) and in (38) below.

The classes ck of forms f,, in which ak is the first non-vanishing a are distinguished from each other by the value of a, if k = n, and if k < n by the values of the parameters ak, UkZ (I! = k -i- 2,

* * *9 n). Employing the notation (19), we shall verify that Pk-lak and PklCTkZ are modular seminvariants of f,,. They vanish for a form Cj (j 2 k - 1) since then 1 - a,“’ = 0. For ck, they reduce to the parameters ak and rkZ of that class. For a0 = 0, se., ok = 0, the first is zero and the second is the expression for (Tkl when ak = 0, whose non-vanishing terms (given by i = k and i = k + 1) are constant multiples of a&:; but uk+l is constant for any class Cj (j > k).

It foilows also that the parameter awl in a class ck+l is determined by the seminvariants PklakZ (I = k -I- 2, k -I- 3), provided k + 3 s n. But u,~ and a,, not so determined, are found from Pklak (k = n - 1, n). Hence a fundamental system of modular seminvariunts of fn, for p > n, is given by

(36)

a0, Qoz (I = 2, - - -, n),

PJ+lukZ (k = 1, .*., n - 2; I = k-t- 2, a.., n),

Pn-2%-l, Pn-l&b.

For n = 2, 3, 4, these are (24’), (32), (34), respectively, except for the difference of notation indicated above. For n = 5, we see that a fundumental system of modular seminvwiants of fat

for p > 5, ti

(37) a0, 82, 83, S4, SK, PO&, PO&~, PO&S,

pls24, PlSPE, p2&51 P$4, p4a5,

. ..---~--._---- -----.- -. _..” -..-” .-...... ..- ----.----__^_I ----_ --.-- --- - -~- J


in which the symbols are defined by (@-(lo), (19) and

SS I= ao4a3 - 5agsa1a4 + 10a02a12as - lOadQa2 + 4a16,

&S = 16a13as - (33) s

40a12a2a4 + 40ala22a3 - 15h4,

25 = 27a32a3 - 45a2a3ad + 20a33,

S 35 = 8a3a5 - 5aa2.

12. Another Methodfor the Case p > n.-We may formulate the method of 0 7 so that it shall be free from the induction process. The classes of forms (23) with P,& + 0, and hence the classes of forms F, with A,, = 0, AI + 0, are characterized by the seminvariants given by the products of PO by the functions Q’, * . . obtained from ~2, aa, . . ., u-1 by increasing the subscript of each Ai by unity and replacing n by n - 1; indeed, P,2 = PI (mod p). When the process of deriving (23) from (20) is applied to (23), we get

r:-, = 11 - (PoA1)~‘lF:-,/y = (1 - Ar1)P8,/y2

(39) = P1F,/y2 = P~A~x”-~ + P1A3x”~y

+ - - . + P~A,Y”~ (mod PI.

The class of forms (39) with PlA3 + 0, and hence the classes of forms F, with A0 = AI = 0, A3 + 0, are characterized by the seminvariants given by the products of PI by the functions ua”, . . . obtained from IJ~‘, . . . , u-2’ by increasing the subscript of each Ai by unity and replacing n by n - 1. Finally, we obtain P,3A,,-lx + P,3A,y, characterized by the seminvariants Pa-2A ,+r and P-IA,. The earlier P+IAs may be dropped (0 11).

For example, if n = 3, p > 3, the fundamental system of F3 is

Ao, ~2, ~3, Pouz’ = Po(4AA2 - A22), Pd2, P2Aa.

Changing the notation from F3 to f3, we see that ~2’ becomes 3(4ala3 - 3az2), so that the resulting seminvariants are (32). We may of course apply the method directly to fa; in S2 we replace a~, al, a2 by 3aI, #a2, a3 and obtain $(4ala3 - 3az2).


Again, to find a fundamental system of f4 for p > 3, we take ao, SZ, &, Sq and the products of PO by the functions $S’lz and 16S14 obtained from Sz and S’s by replacing ao, al, a2, aa by 4al, 6 . 6a2, Q * 4a3, ad; then the product of PI by the function 2524 obtained from Sz by replacing a~, al, a2 by 6~2, 4 * 4as, ah; then P2a.3 and Paah, to characterize P2(4asx + say). We again have (34).

13. Number of Linearly Independent Seminvariants.-Let p > n and employ the notations of 0 11. In the classes C’k (k < n). Akk = ak (;> has p - 1 values, f&+1 = 0, while &k+?, * * a, Akn take independently the values 0, 1, . . . , p - 1. In the classes C,, a, has p values. Hence there are

r-1

p+k~O(p-l)Pn-~l=P+P”-l

distinct classes of forms f,,. Thus by 0 11 of Lecture I, there are exactly p” -I- p - 1 linearly independent modular seminvariants of

f,whenp>n.

I)ERIVATION* OF MODULAR IN-VARIANTS FROM SEMINVARIANTS,

$5 14-15 14. Invarianti of the Binary Q?.idratic Form.-First, let p=2.

Any polynomial in the seminvariants (27) is a linear function of

1, Ao, A, Aoh, J, AoJ = AoAIAz,

since (A0 + A1)J = 0. Since there were six classes, these six seminvariants form a co.mplete set of linearly independent seminvariants. Now a seminvariant is an invariant if and only if it is symmetrical in A0 and AZ. But

I=(l-Ao)(l-Ar)(l-A+(l-Ao)(J+l+Al) (mod 2).

Thus 1, AI, AoJ and I are invariants. By subtracting constant --___ *While this method is usually longer than the method of Lecture I, it

requires no artificea and makes no use of the technical theory of numbers. Moreover, it leads to the actual expressions of the invariants in terms of the semiuvariants of a fundamental system, thus yielding material of value in the construction of covariants.

- -_ ” ) - - - - ~ . . - . - . - - . - - - - “ - l _ -~ - . ~~.-.


multiples of these four, any seminvariant can be reduced to cAO + &A,& which is an invariant only when identically zero. Hence 1, AI, A&AZ and I form a complete set of linearly huh- pendent invariants of Fz m&do 2.

Next, let p > 2. The discriminant of f2 is D = 5’2. Any polynomial in the four fundamental seminvariants (24’) is a linear function of

a2Dj, Poa14, Plazi (i, j = 0, 1, . . ., p - l),

since the product of Peal or Plas by a0 is zero, that of PlaP by Pea, or D is zero, while DPoal = - Pals. Further,

PO = 1 - aoF’, Po[Dj - (- a12)j] = 0,

Pl = PO - P&-I, aoPIDi s Di _ (- l)ipoa12i,

modulo p. Hence any seminvariant is a linear function of P-l

(40) ao , aoiDj (i = 0, 1, . . ., p - 2; j = 0, 1, . . ., p - l),

Pod, Plazk (k= 1, ..*,p-1).

The number of these is p2 + p - 1. Hence (0 13) they form a complete set of linearly independent modular seminvariunt8 of f2 for p > 2.

The invariant A = A1 in Q 6 of Lecture I becomes for two variables

(41) A= {ao“+a~“(l- ao@)) (l- DP-I) = ao’(l- Dp1)+PIa2”,

where p = (p - 1)/2. By the expansion of Dpl, we get*

(4) A = (sop + as“) ( 1 - 2 ao(,2;a12@-2i> . t=o

* Transactions of the American Mathematical Society, vol. 10 (1909), p. 132. To give a direct proof of the identity of the final expression (41) and (42), note that the product of the ikal factor in (42) by D equala a&t - (a&r+1 algebraically, 80 that the product AD is divisible by p. But the product of (41) by D is evidently divisible by p. It therefore remains only to treat the case D = 0. Replacing aI* by acat, we see that the Cnal factor in (42) becomes 1 - G( + l)ao”aac. Hence (41) and (42) are now identical if

aOfiaP (ao* - azw) = 0 (mod PI.

But, if aDal + 0, aOpaylr = a+ = 1, aOp = a2c = * 1.


Since (42) is therefore a seminvariant and is symmetrical in uo and a2 and since the weight of every term is divisible by P - 1, A is an absolute invariant. By (41),

(43) A2 = up (1 - 0”1) + P&2P, (1 - a#--1)~“1 = PO@-‘,

LP+ LP-l- l= - IO, IO= (l-ua,p-‘)(l-ualq(l-uP).

Hence also IO is an absolute invariant. Subtracting multiples of

lo=l-u~~-‘-~~oa~~‘-P~u2~‘, A, Dj o’=O, 1, . . . . p-l),

we may reduce any seminvariant to a linear function of the expressions (40) other than Pru$‘+, P&‘, Dj (j = 0, * * *, p - 1). The resulting linear function L is not an invariant. For example, ifp= 3, it is

L=u4z02+bu0+cu0D+du~2+ePoul+fP~~2 (a, . . ., f constants).

Interchange a0 and 1x2, and change the sign of al. We get

UU~~ -I- but + cu2D + du2D2 + (1 - a~~)(ful - euJ.

This is to be identically congruent to the invariant L. Taking a2 = 0, we see that e = f = a = b = 0, c = d. Then L

= cuoaz(uo + u2) + CCZ~~U~~U~ is not symmetric in a0 and ~22. Hence L = 0. For any p, a like result may be proved by considering separately the terms of L of constant weights module P - 1. Hence in accord with 0 11 of Lecture I, a complete set

of linearly independent invariants of f2, for p > 2, i-9 given by IO, A and the powers of D. In place of DO = 1, we may use A2, in view of (43).

15. Invariants of th Binary Cubic Modulo 3.-A fundamental system of seminvariants of F3 modulo 3 is given by A0 and a fundamental system of

F2 = Ad + (Ao + A2)zy + A3y2.

Hence, by (24), a fundamental system for F3 is given by

Ao, AI, t = AlA3 - (Ao + A212, (1 - A12>(Ao + Az),

D = (1 - A12)[1 - (Ao + A2j2]A3.

INV.QRIANTS AND NUMBER THEORY. 31

In place of the fourth and third we may evidently use

x= (1 - h2M2, u= &43+&A*- A?A*2 = t + A02 + x2.

Here u is the discriminant of Fa for p = 3. By 0 13 there are 11 classes of forms i;~. Hence, by Q 8, there are 3 * 11 classes of forms R. Thus there are exactly 33 linearly independent seminvariants of Fa. Since

AJ = A@ = 0, ax = AOF, p(u + AI)*) = 0,

p(x + Ao) = 0, (1 - &*)a = A&

modulo 3, any polynomial in the seminvariants Ao, Al, u, X, p of the fundamental system is congruent to a linear function of

(44) Ao’Alj, Aoiak, Ao’Ad, Ao’Xk, A&” (i, j= 0, 1, 2; k= 1, 2).

Hence these 33 functions form a complete set of linearly independent seminvariants of Fa. The seminvariants

P = 1 - A? - X2 = (1 - A12)(1 - As*),

(45) IO = (I - Ao*)(P - 4) = & (1 - Ai%

E = A&(u - a*) + Aop = A,,As(A,,Az. - AlAa+ AI*- AZ*)

are seen to be invariants as follows.* The weights of the terms of each are all even or all odd. Moreover, under the substitution (A&(AIA2), induced upon the coefficients of Fs by the interchange of z and y, the functions u, P and 10 are unaltered, while E is changed in sign. Hence u, P, 10 are absolute invariants, while E is an invariant of index unity. We now have 7 linearly independent invariants

(46) IO, E, E, us 9, P, 1. Noting that

(47) E = Ao2p2 + A,,*(u - f + X2) - A&

*Or by general theorems, Transactions of the Ammixn Mathematicd Society, vol. 8 (X107), pp. 206-207. Note that E ia the eliminant of Fa = 0, 2’ = 2,2/’ 5 g (mod 3).


we may employ the functions (46) to delete from (44)

4, AOP, A 2 2 OPJ QJ o-2, x2, 1

in turn (no one of these terms being reintroduced at a later stage). There remain 11 seminvariants of odd weight

(48) A&Al, AOiAlU, AoiAd, p, A&

and 15 of even weight

(i = 0: 1, 2),

(49) A,,, Ao2, AoiA12, Aou, Ad, A,,%, A,,W,AoIX, AoX2, Ao2xz, Ati2.

Now the weight and index of a seminvariant of F3 modulo 3 are both even or both odd.* A linear combination of the functions (48) which is changed in sign by the substitution (AoAs) (AIA2) is seen to be identically zero (it suffices to set Aa = 0, A2 = 0 in turn). A linear combination of the functions (49) which is unaltered by that substitution is seen similarly to be identically zero. Hencet a complete set of hearty independent invariants of Fa modulo 3 is gigen by (46).

* When the sign of y is changed, a seminvariant is unaltered or changed in sign according 88 ita weight is even or odd.

t Another proof, using the cl- of Fa under the group of all binary linear transformations of determinant unity module 3, and involving a use of more technical theory of numbers, is given in Tranaadims of the Ammicun Mathe ctz&kd Society, vol. 10 (lQOQ), pp. 144-154. The case of any modulus p ia there treated.

LECTURE III

INVARIANTS OF A MODULAR GROUP. FORMAL INVARIANTS AND COVARIANTS OF MODULAR FORMS. APPLICATIONS

INVARIANTS OF CERTAIN MODULAR GROUPS, $0 14

1. Introduction.-Let G be any given group of g linear homogeneous transformations on the indeterminates x1, . . ., x,,, with integral coefficients taken modulo p, a prime. Hurwitz* raised the question of the existence of a finite fundamental system of invariants of G. For the relatively unimportant case in which g is not divisible by p, he readily obtained an affirmative answer by use of Hilbert’s well known theorem on a set of homogeneous functions, but emphasized the difficulty of the problem in the general case.

In 0 5 I shall consider the relation of this question to that of modular covariants and formal invariants of a system of forms and incidentally answer the above question for special groups of orders divisible by p.

I shall, however, first present a simplification of my own work on the total group. Its invariants are universal covariants, i. e., covariants of any system of modular forms (§ 13). It was from the latter standpoint that I was led to the subject of invariants of a modular group independently of Hurwitz’s paper, in the title of which the word invariant does not occur.

2. Invariants of the Total Binary Group.-Consider the group G of all modular linear homogeneous transformations with integral coefhcients of determinant unity:

(1) X’ = by + dy, 9’ = cx + ey, be - cd = 1 (mod P>.

The term point will be used in the sense of homogeneous coordinates, so that (2, y) = (ax, ay), while (0, 0) is excluded.

* Archiv elm Mathematik und Phytik, (3), vol. 5 (1903), p. 25. 33


We do not restrict the coordinates to be integers, but permit their ratio to be a root of any congruence with integral coefficients modulo p. A point is called real if the ratio of its coordinates is rational.

A point (x, y) is invariant under a transformation (1) if 5’ = px, y’ = py, or

(2) (b - p)x + cly = 0, cx + (e - p)y = 0 (mod PI.

If these congruences hold identically as to x, y, then

dscco, bzes+l (mod P)

and the transformation is one of the transformations

(3) 2’ G f 2, y' E f y (mod P),

which leave every point invariant. A special point is one invariant under at least one trans-

formation (1) not of the form (3). There are p(p2 - 1) transformations (1). We shall assume in the text that p > 2 (rele- gating to foot-notes the modifications to be made when p = 2). Then there are two transformations (3). Hence any non-special point is one of exactly*

(4) w = &p(p” - 1)

conjugate points under the group G, while a special point is one of fewer than o conjugates.

Let (x, y) be a special point and let (1) be a transformation, not of the form (3), which leaves it invariant. Thus the congruences (2) are not both identities. The determinant of their coefficients must therefore be divisible by p. Hence p is a root of the characteristic congruence (in which (Y = b + e)

(5) P2 - 0p+1=0 (mod PI.

First, suppose that (5) has an integral root p. For this value of p, one of the congruences (2) is a consequence of the other, and the ratio x : y is uniquely determined as an integer modulo p.

* For p = 2, o is to be replaced by 2(2* - 1) = 6.


Hence only real special points are invariant under a transformation [other than (3)] whose characteristic congruence has an integral root. Moreover, all real points are conjugate under the group G. Indeed,

x’ = bx, y’ = x + b-ly, and x’ = - y, y’ = x

replace (1, 0) by (b, 1) and (0, 1) respectively. Hence if an invariant of G vanishes for one of the real points, it vanishes for all and has the factor

p--l (6) L = Ygcx - uy) = xpy - xyp (mod PI,

the congruence following from Fermat’s theorem. Obviously, any transformation of G replaces a real point by a real point, and therefore L by kL. The constant k is in fact unity and L is an invariant of G. Indeed, for

(7) x=aX+bY, y=cX+dY (mod P>,

where a, se., d are integers of determinant A = ad - bc,

Next, suppose that (5) has no integral root and therefore two Galois imaginary roots. By (2), each root p uniquely determines a point (x, y) with y =!= 0. We may therefore take y = 1, whence cx = p - e. The resulting two special points are therefore imaginary points of the form (rp + 8, l), where r and a are integers modulo p, and r is not divisible by p. The imaginaries introduced* by new transformations are expressible linearly in terms of this p. Indeed, (2~ - a)” = A, where A = (y2 - 4 is a quadratic non-residue of p (i. e., is not the re- mainder when the square of any integer is divided by p). Thus A = a2v, where v is a fixed non-residue of p. Hence the roots of all congruences (5) having no integral roots are expressible in the form k + 16, where k and 1 are integers.

* There tie no new onea if p = 2, since a = 0 (mod 2).


Hence the special points invariant under transformations whose characteristic congruences have no integral roots are all of the form (rp + S, l), where T and a are integers, r not divisible by p, while p is a fixed root of a particular one of these congruences (5).

We next show that these p2 - p imaginary special points are all conjugate under the group G. It suffices to prove that they are all conjugate with (p, I), which is invariant under

2’ s cyx - y, y’ = 5.

Now transformation (1) replaces (p, 1) by (R, l), where

We are to prove that there exist integers b, c, d, e satisfying

(9) be - cd= 1 (mod P>,

such that R = rp + s, where r and s are any assigned integers for which r is not divisible by p. Denote the second root of (5) by p’ and multiply the numerator and denominator of R by cp’ + e. Using (9), we get

j&X!? Q ’

N = bc + de + &a, q = 8 -I- ace -I- e2.

We first show* that we can choose integers c and e such that q E i (mod p), where i is any assigned integer not divisible by p.

If i is a quadratic residue of p, we may take c = 0. Next, let i be a quadratic non-residue of p. Taking c + 0, e = kc, we have

q = St(k), f(k) = 1 + ak + k2.

Now f(k) 3 f(K) if and only if K = k or K = - a! - k. Hence the p - 1 values of k other than - 42 give by pairs the same valueoff( Thusfork = 0, . . ..p - l,f(k) takes 1 + *(p-l) incongruent values, no one a multiple of p [since (5) has no

* If p = 2, then a = 0; taking c = 1, e = 0, we have p = 1 = i (mod 2).


integral root], and consequently a value which is a quadratic non-residue of p. Then, by choice of c, q can be made congruent to any assigned non-residue.

Having made q = i (mod p) by choice of c and e, we proceed to choose integral solutions b and d of (9) such that N will be congruent to any assigned integer j. If c = 0, so that e + 0, we take d = j/e. If c + 0, we eliminate d from N by use of (9) and obtain

N+bq- e-m), q=c2+ace+e2.

Since q $ 0, we may make N = j by choice of b. We have therefore proved that there are exactly p2 - p

imaginary special points, viz., (rp + s, l), r + 0, and that they are all conjugate under the group G. Hence any invariant of G which vanishes for an imaginary special point has the factor

(10)

Indeed, the numerator of the first fraction vanishes for x= rp+s, y = 1, since

(rp + 8)PZ = rppD + 8, p+ = p (mod ~1,

the last congruence* being a case of Galois’s generalization of Fermat’s theorem. We have divided out L, which vanishes for the real points (8, 1) and (1, 0). Since any transformation of G replaces one of our imaginary points by another, it replaces Q by kQ. The constant k is in fact unity and Q is an invariant of G. Indeed, (8) holds if .we replace the exponents p by p2. Hence the quotient Q is invariantt under all transformations (7).

* It may be proved by noting that (5) implies

(d - ap + 1)p = p*’ - apP + 1 = 0 (mod PI,

80 that pp ia the second root of (5). By the same argument, (pp)p is a root, distinct from p*, and hence identical with p.

t I gave the notation & to the invariant (10) since it is the product of all of the binary quadratic forma z* + +. . which are irreducible modulo p. Indeed, the latter vanishes for two point8 of the form (rp + s, 1) and (rp’ + s, l), where p and p’ are the roots of (5) and T, s are integers, r + 0, and conversely.


We are now ready to prove that any rational integral invariant I, with integ+ral coejbients, of the group G ti a rational integral function of L and Q with integral coe+nb.

After removing possible factors L and Q, we may assume that I vanishes for no special point. If I is not a constant, it vanishes at a point (c, d) and hence at the w distinct points conjugate with (c, d) under the group G. The invariants*

(11) P+l tip--l)

q=QT, l=Lz

are of degree o. The constant 7, determined by

q(c, d) + 7 * Z(c, d) = 0 (mod PI,

is a root of a congruence of a certain degree t with integral coefficients and irreducible modulo p. Now q + 71 is a factor of I. Since q, I and I have integral coefficients, I has also the factors

(12) q+rp1, q+&, ‘-a) q+Fl.

For, by Galois’s theorem mentioned above,

7, 7P* TP’ f . . . TP” ,

are the roots of our irreducible congruence of degree t. Since the conditions which imply that q + zl shall be a factor of I are congruences satisfied when z = r, they are satisfied when z = r@. Hence if we multiply q + 71 by the product of the invariants (12), we obtain an invariant T with integral coefficients module p. Since L and Q have no common factor, no two of the functions q + rl and (12) have a common factor. Hence T is a factor of I. Proceeding in like manner with I/T, we arrive finally at the truth of the theorem.?

3. Invarianti of Smaller Binary Groups.-We shall later need the theorem that a fun&amental system of rational integral invarianta

* If p = 2, we omit the divisor 2 in the exponents. t Proved less simply in !&ansactions of the Avnericun Mathmatkal Society,

vol. 12 (1911), p. 1. Still simpler ia the proof that various coe&icients of an invariant are zero, QuuT~~~ Journd of Mathematic+ 1911, p. 158.


of the group composed of the p powers of tk tran$oormation

(13) 2’ = 2 + y, y’ = y (mod P> . G given by y and X, where

(14) x = x(x+y)(x+ay)* * ’ (x+p-ly) =xp-qpl (mod PI-

Now (1, 0) is the only special point, being the only point unaltered by (13) or its kth power, k < p. Hence an invariant not having a factor y or X vanishes at imaginary points falling into sets each of p points conjugate under our group. As at the end of 9 2, the invariant is a product of factors yp + A so related that the product equals a polynomial in yP and X with integral coefficients.

Other results will be merely stated, since they are not presupposed in what follows. Within the group G of all transformations (l), any subgroup of order a multiple of p is conjugate with one containing (13) and transformations exclusively of the form

(15) 2’ = tx + ly, y’ = t-ly (mod P),

and having y and X as a fundamental system of invariants.* The invariants of any subgroup whose order is prime to p have been f0und.t

4. Invariants of the Total Group on m Variables.-The functions

W-3 L =

QP=+-l . . . g&P=-’

xlP--” . . . &&PC’

. . . . . .

xlP . . . &P

Xl a-* &

QP” . . . &P”

. . . . .

p*+l %??+’

9 Qm = :;p-, : : : hp.-, . . . . .

2 . . . %I

are seen, by a generalization of (8), to be invariants of index 1 and 0 respectively of the group I?, of all linear homogeneous transformations on x1, 9 . . , G with integral coefficients modulo p.

+ BuL?e.t~n of the America n Mathematical Society, vol. 20 (1913), pp. 132-4. t American Journal of Mathentatics, vol. 33 (1911), p. 175.


Since L,,, is an invariant of I?,,, and has the factor 51, it follows from an examination of its diagonal term that*

(17) L = ii 2 (ac + %+1ac+1+ - * * + G&J k=l y=o

(mod P>,

in which occurs one of each set of proportional linear forms modulo p. A like proof shows that the numerator of Q,,,* is divisible by each of the linear functions (17) and hence by L,, modulo p.

Making use of the theorem in 0 2, I have proved by inductiont that the m invariants L,,,, &,,,I, + . ., &,,,,,,-I are independent and form a fundamental system of rational integral invariants of I?,.

A fundamental system of invariants of the group of all modular linear transformations on two sets of two cogredient variables has been obtained very recently by Dr. W. C. Krathwohl in his Chicago dissertati0n.S

FORMAL INVARIANTS AND SEMINVARIANTS OF MODULAR FORMS,

00 5-13

5. Formal Modular Invariants.-Consider a binary form

f(x, Y) = a0f + as?% + - - - + c~ry:

in which x, y, ao, * a 0, a, are arbitrary variables. The transformation (7) with integral coefficients, whose determinant A is not divisible by the prime p, replaces f by a form

c$(X, Y) = AoX’ + AlX’-‘Y + * * - + k&Y’, in which

(18) Aa = f(a, c), A1 = Pa)-%a0 + . . a, . . ., A, = f(b, d).

A polynomial P(ao, * . *, a;) with integral coefficients is called a formal invariant modulo p of index X off under the transforma-

* E. H. Moore, BdZetin of the Americu n Mathematical Society, vol. 2 (1896), p. 189. His proofs do not use the iuvariantive property. A like remark ia true of the proof that the product (17), iu the case zn = 1, is congruent to a determimmt of order m - 1, then obviously equal to L,,,, by R. Levavasseur, M.z%wirea de l’Acm%mie des Seienees ok Twbuse, ser. 10, vol. 3 (1903), pp. 39-48; compka R-endw, 135 (1902), p. 949.

t Transadms of the America n Mdhenatical Society, vol. 12 (1911), p. 75. $ American Journal of Mat~ics, October, 1914.

TNVe4RIANTS AND NUMBER THEORY. 41

tion (7) if

(19) P(Ao, Al, * * -9 A,) = A’P(ao, al, . . . , a,) (mod P>,

identically as to ao, . + . , a,, after the A’s have been replaced by their values (18) in terms of the ui. If P is invariant modulo p under all transformations (7), it is called a formal invariant modulo p off.

The term formal is here used in connection with a formf whose coefficients are arbitrary variables in contrast to the case, treated in the earlier Lectures, in which the coefficients are undeter-

mined integers taken modulo p. In the latter case, (19) neces- sarily becomes an identical congruence in the u’s only after the exponent of each a is reduced to a value less than p by means of Fermat’s theorem UP = a (mod p).

The functions (18) are linear in a~, . * a, a,. It is customary to say that relations (18) define a linear transformation on uo, “.,a, which is induced by the binary transformation (7). Let r be the group of all of the transformations (18) induced by the group of all of the binary transformations (7). Making no further

use of the form f, we may state the above problem of the determination of the formal invariants off in the following terms. We desire a fundamental system of invariants of group r. This

problem is of the type proposed in $1; the group r is a special group of order a multiple of p. Here and below the term invariant is restricted to rational integral functions of uo, . . ., a,.

A theory of formal invariants has not been found. For no form f has a fundamental system of formal invariants been published. Some light is thrown upon this interesting but

di5cult problem by the following complete treatment of a binary quadratic form, first for the exceptional case p = 2 and next for the case p > 2, and preliminary treatment of a binary cubic form.

6. Formal Invarianti Moddo 2 of a Binary &uudrutic Fornz.- Write

(20) f = ad + bxy + cg2,


where a, b, c are arbitrary variables. Under the transformation

(21) x = 2’ + y’, y = y’,

f becomes f’, in which the coefficients are

cm a’ e a, b’ = b, c’ = a + b + c (mod 2).

By $3, the only invariants under d’ = d, c’ = c + d, modulo 2, are the polynomials in d and c(c + d). Take d = a + b. Hence the only seminvariunh off are the polynomials in a, b and

(23) a = c(c + a + b).

Such a polynomial is an invariant of f if and only if it is unaltered by the substitution (ac) induced by (q). Thus

(24) b, k = as, q = b(a + c) -I- a2 + UC + c2 = s + ab + a2

are invariants off. Introducing q in place of s, we see that any seminvariant is a polynomial in a, b, q. Consider an invariant of this type. Since its terms free of a are invariants, the sum of its terms involving a is an invariant with the factor a and hence also the factors c and a + b + c, the last by (22). Hence this sum has the factor k, and its quotient by k is an invariant. By induction we have the theorem:

Any rational integral formal invariant of f equal a rational iniegral function* of b, q, k.

7. Formal Seminvarianh of a Binary Quadratic Form for p > 2. Write

(25) f = d + 2bxy + cy”,

where a, b, o are arbitrary variables. Under the transformation (21), f becomes?, whose coefficients are

(2’3 a’ = a, b’ = a + b, c’ = a + 2b + c.

* Replace 01, zg ZS, of 0 4 by a, b, c ; then

La - bkk(k + W, &:a = b’ + bk + P, &a~ = Pq’ + bq7c + b% + V.


Evident formal seminvariants are a, A = be - ac, and

P-1 (27) f9=p+b)“bp-baPl (mod P>,

(28)

Indeed, the linear function under the product sign in (28) is transformed by (26) into the function derived from it by replacing t by t + 1. As in (27),

(29) [-ykJa=cl = cp - a-1 (mod PI.

Let S(a, b, c) be a homogeneous rational integral seminvariant with integral coefficients. Then, by (26),

S(0, b, c) = S(0, b, 2b + c) (mod PI.

Thus, by 0 3, S(0, b, c) equals a polynomial in b, CC’ - cP. Hence, by (29),

S(a, b, 4 = da, h 4 + $4, ok) (mod P),

where u and tj are polynomials in their arguments. Now

Hence b2’ = A’ + a( ), bpf2’ = @A; + a( ).

(30) S = ah(a, b, c) + #(&A, Yk) + ‘p~~pdgb2i+1ykeJ,

where X and $ are polynomials in their arguments, and di is an integer.

When y is multiplied by a primitive root p of p, a, b, c are multiplied by 1, p, p2, respectively. Hence p is multiplied by p,

while, by (29), Tk and A are multiplied by p2. If therefore we attribute the weights 0, 1,2 to a, b, c, respectively, and the weight 8 + 2t to a’b*c’, we see that the weight of every term of rk is congruent to 2 modulo p - 1.

We can now prove that every di is divisible by p. For, if not, the seminvariant S - # h as a term of odd weight, so that every


term of X is of odd weight and hence has the factor b. Thus S - $ has the factor b and therefore the factor p, so that its terms free of a have the factor bp. But this is impossible, since 2i + 1 < p and (29) does not have the factor b.

Hence S - $I has the factor a and the quotient is a seminvariant of the form ax’ + #‘. Proceeding in this way, we obtain the theorem :

Any seminvariant & a polynomial in a, A, fi and any single Yk.

Of these, @ alone is of odd weight. Hence any seminvariant is a polynomial in a, A, Yk, @* or the product of such a polynomial by ,f3. But

P-1 (31) p2= aPyo+A(A T - az’12 > (mod PI.

To prove this, it suffices to show that the second member is divisible by b and hence by & and being of even weight therefore by p2, and to remark that each member of (31) reduces to b2p for a = 0. Now

hola=o = ‘Q (t’a + c) = c ( “g (Pa + c) )”

p-1 p--l s ~(0” - (- a)“12 (mod P>,

aP[yO]b,O = ac( (- ac)’ - aP1j2 (mod P>.

But A reduces to - ac for b = 0. Hence the second member of (31) has the factor b. We therefore have the theorem:

For p > 2, any formal seminvariant of a binary quadratic form

is a polynomial in a, A, y. or the product of such a polynomial by /3.

8. Formal Invariants of a Binary Quadratic Form for p > 2.

The product

(32) r = UYk (k ranging over the quadratic non-residues of p) k

is an absolute invariant of f under the group G of all binary transformations with integral coefficients taken module p of

INVARL4NTS AND NUMBER THEORY. 45

determinant unity. It suffices to prove that this seminvariant is unaltered by the sub&it ution

(33) a’ = c, c’ = a, b’ = - b,

induced by the transformation x = y’, y = - x’. Under (33), the general factor in (28) is replaced by

where (t”-k){(T2-K)a+2Tb+c},

T=& k

K = (p - k)2’

Hence K is quadratic non-residue of p when k is. Also,

p@~-k)=-k(i~j$k-t~)}2--k(;‘-l)2=-4k (mod P>

if k is a non-residue. To show that the product of the resulting numbers - 4k is congruent to unity, we set x = 0 in

(34) p-1

v(x- k) = x2+ 1 (modp),

and note that 2P-l = 1. Hence (32) is unaltered by (33) and is an absolute invariant off under G.

It is very easy to verify that

(35) J = aTo

is unaltered by (33), so that J is an invariant of f under G. If an invariant has the factor /3, it has the factor

(36) B= B&r ( r ranging over the quadratic residues of p).

For, under the substitution (33), b+Ta (T/O) becomes T(c-b/T). By choice of r, we reach c + 2tb, where t is any assigned integer not divisible by p. This is a factor of Yk where k = P.

The fact that B is an invariant may be verified as in the case of (32) or deduced from the fact that

P-1 @nYk = am - 131’

k=O


is an invariant, being the product of all non-proportional linear functions of a, b, c with integral coefficients modulo p.

Hence any invariant is the product of a power of B by an invariant which is a polynomial P in a, A, +yo.

Since rk is a seminvariant not divisible by /3, it equals a polynomial in a, A, r. (9 7). But if a = 0, 7k = y. (mod p), by (29), and A = b2 is free of c, so that ok is not a polynomial in a and A only. Hence

(37) rrn = YO + gda, A) (mod PI.

For p = 3, the polynomial P therefore equals a polynomial in a, A, 7~ = T. Now an invariant 4(a, A, I’) differs from the invariant +(O, A, I’) by an invariant with the factor a and hence the factor (35). Treating the quotient similarly, we ultimately obtain the following theorem for the case p = 3:

A fundumental system of formal invarianti of the binary quadratic form f modulo p, p > 2, is given by the discriminani A and I’, J, B, defined by (32), (35), (36). The product of the last three is congruent module p to the product of all the non-proportional linear function8 of th4 coe~ienh off.

To prove the theorem for p > 3, note first, by (37), that I’, given by (32), differs from ~0% by a polynomial in 70, a, A of degree n - linyo,wheren = (p - 1) 12. Hence a polynomial in a, A, 70 equals a polynomial in a, A, 70, I’ of degree at most n - 1 in yo. Subtract from each the terms of the latter involving only the invariants A, I’. We have therefore to investigate invariants of the type

s-1 n-1 C CCYO’P~@, r) + C y0Vi(a, A, r),

in which the ci are integers, while Pi and t#~i are polynomials in their arguments, and +i has the factor a. If every cl = 0, the invariant has the factor a and hence the factor aye = J, and the quotient by J is an invariant which may be treated similarly. The theorem will therefore follow if we show that a contradiction

INVARIANTS ANB NUMBER THEORY. 47

is involved in the assumption that a certain cj is not divisible by p. First, the remaining ci are divisible by p. For if also ci + 0, let &A’PJ be the term of Pi of highest degree in A. Since +yo and I? are of degrees p and np, and of weights = 2 and 0 bc+- l), y&Pi is of degree P; + 2ri + s;np and of weight = 2i+ 21-i (mod p - 1). But p = 1 (mod n). Hence

i + 2Ti s j + 2Tjp 2i + 2ri s 2j + 2rj (mod 4,

so that i = j (mod n). But i and j are positive integers < n. Hence i = j. Multiplying our invariant by a suitably chosen integer, we have the invariant

(39) rojPj(A, I’> + EJ~o~&(u, A, I’), Pj = A’I” + *. l .

Now - (c - ka)b- is the term of highest degree in b in Tk. Hence

(40) Yo = -cbpl+ **a, r = apb-O+ . . . .

(41) u = ${- (c - ka)) = (- c)” + (- a)” (mod p),

where k ranges over the non-residues of p, the last following from (34) for 2 = c/u. Since 70 and I’ are of even weights, only even powers of b enter (39). Hence an invariant (39) is symmetrical in a and c. We shall prove that this is not the case for the terms of highest degree in b. For TaPj this term is

(42) (- c)ju*b8, /I = j(p - 1) + 2r + sn(p - 1).

Let C&Af~P be one of the terms of & in which the exponent of b is a maximum. Then in y&i the highest power of b occurs in the terms

(43) C,xf~( - c) Wbb i 2 Pi= vi + g;n(p - 1) + i(P - 1).

Since the weight and degree is the same as for (42),

2i + fli s 2j + /3 (mod P - 11,

ei + i + gsn + pi = j + 8?2 + 8.


First, let /3i = /3. Then i = j, ei = 0 (mod n), whence i = j. Thus the exponent of a in any term (42) or (43) is divisible by n, while the exponent of c is not, being congruent to j modulo n. Hence the coefficient of bS in the sum of (42) and the various terms (43), with i = j, is not symmetrical in a and c, unless identically zero. But (43) has the factor a while (42) does not. Hence the greatest & exceeds 8.

Next, consider a set of terms (43) and a set of terms of like form with i replaced by k, all being of equal degree in b. Then fli = /3k. By (441), 2i + Pi = 2k + &, i = k. Consider finally terms (43) with /3i constant. In them the residue modulo n of ei is a constant + i. For, if ei = i, then 2i + & = j + @ (mod n) by (44,), so that j = 0 (mod n) by (441). Hence these terms (43) are not symmetric in a and c and yet do not cancel.*

Our fundamental invariants are connected by a syzygy; for P = 3, (45) B2 = A312 + J(J - A2)2.

9. Formal Invariants of a Binary Cubic Form for p + 3.- We have seen that the theory of formal invariants of a binary quadratic form is dominated by the invariantive products of linear functions of the coefficients. While these products de- pended upon the classification of integers into the quadratic residues and the non-residues of p, we shall find that for a cubic form it is a question not merely of cubic residues and non-residues of p, but of the larger classes of reducible and irreducible congruences. Write

f = ax3 + Sba+y + 3cxy2 + dy3,

thus taking p + 3. Under transformation (21), f becomes f’, whose coefficients are given by (26) and

(46) d’ = a + 3b + 3c + d.

* If two are of lie degree in c, their g’s are equal and hence their fs are equal; then, if of like degree in a, their e’s are equal. But then we have the same term of +i.


Hence a, /3 and ok, given by (27) and (28), are again seminvariants; also,

p-1 (47) si,=~{(tS-3kt--~a+3(P-k)b+3tc+a~

(j, k = 0, - * -, p - 1).

Indeed, if Ft(a, b, c, d) is the function in brackets,

Ft(a’, b’, G’, 8) = Fwl(a, b, c, 4.

Any invariant with the factor a has the factor

(48) a&o = a’5 (Pa + 3t2b + 3tc + d) = f(1, 0)&t, l), t=o

whose vanishing is the condition that one of the points (z, 9) represented by f = 0 shall be one of the existing p + 1 real points (1, 0), (t, 1) of the modular line. To verify algebraically that the seminvariant (48) is an invariant,* note that it is unaltered modulo p by the substitution

(49) a’ = - a, dt = a, b’ = C, d = - b,

which is induced on the coefficients of f by x = y’, y = - x’. The product P of the ajh in which j and k are such that

x = t3 -3kt-j

is irreducible modulo p is a formal invariant. The substitution (49) replaces the general factor of (47) by

- a + 3tb - 3(P - k)c + Xd

= X((T3 - 3KT - J)a + 3(T2 - K)b + 3Tc + d}, where

T-k-t2 -- x , K=;, J=;, g = l2 + k2a + tj,

h= - 2k3 + 6k2t2 + 3ktj + t3j + Jo.

*For any form, e.ee Transactions of tk American Mathemuticd Society, vol. 8 (X307), pp. 207-208.


We are to show that there is no integral solution z of

x3-3Kx- J=O (mod P>.

Multiply this by X3 and set kr = y. Then

y3 - 3gy - h = 0 (mod P).

But the negative of the left member is the result of substituting

?-+a= - t, rs = - y - 2k

in the expansion of the product

(T3 - 3kr - 31 (s3 - 3ks - j).

The latter is congruent to zero modulo p for no values of r and s which are integers or the roots of an irreducible quadratic congruence with the integral coefficients t, - y - 2k.

For p = 2, P = 611. For p = 5, P is the product of two invariants*

(50) b1822832~41, &3624&4643,

neither of which is a product of invariants. The last property is true also of the following invariants:

(51) 71603, 74602, y2~04&2~30~20~42,

~3~01'%0~2S~33~40, @~0&4~21831~44.

The product of these seven invariants and a&o equals the product of all the linear functions of a, b, c, d, not proportional module 5.

For p = 2, each of the 15 linear functions is a factor of just one of the following invariants (no one with an invariant factor) :

(52) USOO, 611, ProSo~, K = b + c, (a + b + c)&o.

For any p + 3, the cubic form has the formal invariant

(53) G = 3(bcp - bpc) - (a&’ - cd),

* In those linear factors of the limt which lack c, the product of the coefficients of a and b is a quadratic non-residue of 5; in those of the second invariant, a quadratic residue.

INVARIANTB AND NUMBER THEORY. 51

and an absolute formal invariant* K of degree p - 1. For

P = 5,

(54) K = b4 + c4 - b2d2 - a2c2 - bc2d - ab2c + ad2 + a2bd.

Thus, for p = 5, K and the discriminant D are invariants of degree 4, and weights = 0, 2 (mod 4), while a&o and G are of degree 6 and weight = 3 (mod 4). It follows from 0 10 that there are no further invariants of degree less than 8. Now the first and second invariants (51) are of degree 10 and weight = 1 (mod 4). Hence if either is expressible as a polynomial in invariants of lower degrees, it must be the product of D by a linear function of a&o and G. This is seen to be impossible either by a consideration of the terms of degree 2 5 in d or by noting that D has no linear factor. Thus y1& or cy,& occurs in a fundamental system of invariants.

Invariantive products of linear functions of the coefficients of the cubic form therefore play an important r81e in the theory of its formal invariants. Whether or not they play as dominant a r81e as in the case of the quadratic form is not discussed here. We shall however treat more completely the seminvariants.

10. Formal Seminnariants of a Binary Cubic for p > 3.-We shall first determine the character of the function to which any seminvariant S(a, b, c, d) reduces when a = 0. Set A = 3b, 2B = 3c, C = d. Then (26) and (46) give

A’ = A, B’ = A + B, C’ = A + 2B + C (when a = 0).

Any function unaltered by this transformation is (0 7) a polynomial in A, B2 - AC, TO’, or the product of such a polynomial by /3’, where 70’ and /3’ are the functions yo and @ written in capitals. But

p-1 ~0’ = 9 (3cb + 3~ + a) = [~jo]o=o,

* Tranaactiona of the America n Mahvnutieal Society, vol. 8 (1907), p. 221; vol. 10 (lQOQ), p. 154, foot-note. Bulletin of the American Ma#mnutical Society, vol. 14 (1908), p. 316. Cf. Humitz, 1. c.

52 THE MADISON COLLOQUITJM.

modulo p. Hence

(55) S = m4a, b, c, 4 + ykV(b q, Sj0) (e = 0 or l),

where k, j may be given any assigned integral values and

(56) q = c2 - +ba, - wq = [D]a=o,

D being the discriminant off. We use the seminvariants (II, 0 2)

(57) S2 = - b2 + ac, 6’3 = 2b3 + a(ad - 3bc).

First, let p = 5. Then q = 8 + 2bd. We have the formal seminvariants*

u3 = bq - abb + 24,

u4= K-S522=q2+a(abd-2ac2+b2c+c#),

us= bg+a(-aaa- bc@ + 3&i + ab$ - 2b3c + a9b),

us = qa + a(ad4 - 2bcd3 - CW + absa - ab%d + a3bd + 2~4

(58) - b2cS - 2aV- + ab’),

a7 = no + a(2(b2 - ac)d4 + a2bd3 - b&P - 2c4d2 + 2a2cW

- 2ac(b2 - ac)# - (b2 - ac)2d2 - 2a4d2 + 2abc3d

+ 2a3bcd + 2ab4c + 3(b2 - ac)c4 - a4b2 + 2aV),

while 2G differs from bya by a multiple of a. By (55)-(58), S differs from a polynomial in the seminvariants

(59) a, D, S2, Ss, ~a, K, us, us, ~7, G, YO, 600

by a function aX + pb&, + uq6&, in which p and u are constants at least one of which is zero (in view of the degree of the terms). But the increment to b&, under transformation (26), (46), is

* Aa the terma with the factor a were taken all of the proper degree and weight; then a term common to a combination of the seminvarianta (59) was deleted. Finally the coefficients were found by a proceaa equivalent to the use of a (non-linear) annihilator, Tranmctima of the Am&can Mathematical Society, vol. 8 (1907), p. 205. Expansions were made in powers of d and the terma involving d rechecked. As each remaining term in~olvw a new coefficient, there is no doubt ae to the existence of covariante of type US, ~6, 01, though the terms free of d were not rechecked.

1 .


CL& with the term acZ50, while d does not occur to this power in the increment to a function X of degree 5g. Again, the increment to qS$ has the term ~cu-P+~~, while the increment to a function X of degree 5h + 1 is of smaller degree in d. Hence p = u = 0. Then in ah, X is a seminvariant which may be treated as was the initial S.

A fundamental system of formal eeminvariante of the binary cubic form module 5 is given by the functions (59).

11. For p = 2, the method of 0 10 fails. In place of c we now introduce the seminvariant K = b + c. Then the transformation (26), (46), becomes

(60) a’ = a, K’=K, b’=a+b, d’=a+K+d.

By 0 3, any seminvariant S(a, K, b, d) becomes for a = 0 a polynomial in K, b, d(K + d), In place of the last we may use 600. Hence

S = a~ + M, K, 6001, 600 = d(a + K + 4.

We make use of the seminvariants

(61) A = ad + bc = 600 + 601, p = b2 + ub,

/Y+A=bK+a(b+d).

Hence S differs from a polynomial in K, 600, A, /3 by a function up + bT(P, 600). Let (60) replace p by p’. Then p + p’ = r (mod 2). Take a = K = 0; then (60) is the identity and 0 = T(b2, d2) identically in b, d. Hence the function ~(8, S) is identically zero. Thus up and hence p is a seminvariant. Hence a, K, 600, A, /3 form a fundamental system of formal eemin- variants of th cubic module 2.

Note that A2 is the discriminant, so that A is an invariant. The invariants (52) may be expressed in terms of our seminvariants:

(62) 811 = I + A, PYOSOI = B@ + K2 + UK) (A + So,),

(a+KNo= (a+K>(a2+I)=a600+KI,

where I = a2 + aK + 600 is an invariant.


12. Miss Sanderson’s Theorem.*-Given a modular invariant i of a system of forms under any modular group G, we can construct a formal modular invariant I of the system of forms under G such that I = i (mod p) for all integral values of the coefficients of the forms. As the proof does not give a simple method of actually constructing I from i, it is in place here to give a very interesting illustration of the theorem with independent veri- fication. Take as i the fundamental seminvariant (- l)=P,+ra, of a binary form f (Lecture II). Then 1 is the quotient L&L,,,, where Lkr is given by (16) or (17) with 21, . . . , Z~ replaced by the first m coefficients a0, al, . . . , a-l of the binary form f. Now z = 2’ + y’, y = y’, replaces f(z, y) by a form in which the coefficient aj’ is a linear function of ao, . * a, aj. Hence Lj is a formal seminvariant of f modulo p. First,

La sop alp -=

I I Ll a0 al i a0 = aOP1al - alp

is a formal seminvariant which reduces to - Peal for integral values of ao, al, where PO = 1 - aopl. Compare (27). Next,

aOps a/ a#

La = a# alp a.2P ,

a0 al a2

C = L3/L2 = a# - a$& -I- a&P

where, as in (lo),

(mod PI,

Q = aoPsal - L2 aoa1p4 = $ ao’(p+Ul’j (8 = p - 1).

For integral values of the a’s, we have

L2 = 0, Q = a$ + al* + (p - l)aoaa~* = 1 - PI,

PI = (1 - a#)(1 - al’),

module p, since each term of Q, with j + 0, j =l= p, is congruent * Transadms of the Americun Mathemdicd Socktg, vol. 14 (1913), p. 499.


to ao8ala. Hence C = PIaz. Similarly,

L4fL1= - mp’ + a/Q32 - apQ31 + diP1 (mod PI,

where the Q’s are defined by (16) and are congruent to*

&al = Q(L&P-1 + Lpgp, &a = (L/L2P-’ + Q”,

with Q as above. Hence for integral values of the a’s,

Qal G (1 - P1)Plap+= 0, &a2 = 1 - PI (1 - a$‘-‘) = 1 - P2,

L4/L3 = - P2a,,

13. Modular Covarianta.-Extending the usual definition of a covariant of an algebraic form f to the case in which the group is the set of all linear transformations with integral coefficients taken modulo p, we obtain the concepts modular covariants or formal modular covariants according as the coefficients of f are integers taken modulo p or are indeterminates. The contrast is the same as in 0 5. The universal covariants obtained in Q 2 and 0 4 do not involve the coefficients off and hence are forma1 covariants.

I have recently proved? that all rational integral modular covariante of any system of modular forms are rational integral functions of a finite number of these covariants. In the same paper I proved that a fundamental system of modular covarianta of the binary quadratic form (25) modulo 3 is given by the form f it&f, its diwriminunt A, the m&versa1 covariants L and Q, together u&!hS

q= (a+c)(b2+ac-l), f4=&+bz3y+bzy3+cy4,

(63) Cl = (a2b - b3)z2 + 2(b2 + ac) (c - a)zy + (b3 - b$)y2,

CZ = (A + a”)9 - 2b(a -I- c)zy + (A + 8)y”.

Here f4 is a formal covariant, which is congruent to f for integral

* T~ama&nw of the Americun Mathemakd 8ocietg, vol. 12 (IgIl), p. 77. t !i?a?hsa&ons of the American Mathematical Society, vol. 14 (1913), pp.

299-310. The extension to cogredient seta of variablea has since been made by Professor F. B. Wiley, and will be published in hi Chicago dissertation.

$ No one of the eight is a rational integral function of the remaining seven even in the case of integral coefficients a, b, c taken modulo 3.


values of z, y. Also C; and (as here written) Cr are formal covariants. Note that - q is the invariant (42) of Lecture II. When q is made homogeneous by replacing - a - c by - a* - cs, we obtain the formal invariant I? = 79, given by (32). The resulting eight formal covariants off do not form a fundamental system of formal covariants; not all the formal invariants are polynomials in A and I’ (0 8). No instance of a fundamental system of formal covariants has yet been published.

The method of proof will be here illustrated by the new and simpler case of a binary quadratic form (20) with integral coefficients modulo 2. By 0 6 any invariant off is a polynomial in

(247 b, ah q= (b+l)(a+c)+w

to which the formal invariants (24) reduce modulo 2. Such a polynomial is congruent to a linear function of these three and unity, since

bq=abc (mod .2).

Further, any seminvariant is a polynomial in a, b and q (0 S), and hence is a linear function of 1, a, b, ab, q, abc. For,

aq=a+ab+abc (mod 2).

These results are in accord with those obtained otherwise in 0 14 of Lecture II. We shall now prove the following theorem:

Every rational integral covariant K of th.e binary quudratic form f modulo 2 6 a rational integral function off, its invariants b and q, the universal covarianta

Q = 9 + q/ + y”, L = 2~ + XI/“,

and the linear covariant

l= (a+b)x+ (b+c)y, P=f+bQ (mod 2).

The leading coefficient 8 of K is a seminvariant and hence is of the form I + ra + sab, where r and s are constants, and I is an invariant, a linear combination of the invariants (24’) and unity.


First, let K be of even order 2n. Then

K1 = K - IQ” - rf” - 8bs(

is a covariant in which the coefficient of x2” is zero and hence has the factor y. Thus K1 has the factor L and the quotient is a covariant of order 2n - 3 to which the next argument applies.

Next, let K be of odd order:

K = i3~9”+~+ S#‘y+ sm..

After subtracting from K constant multiples of IQ” and bl&“, in which the coefficients of $‘+I are a + b and ab + b, respectively, we may assume that S is an invariant. After also subtracting from K a constant multiple of IL&“-l, where I is a linear combination of the invariants (24’) and unity, we may assume that S1 = /3ra + &c, where the S’s are functions of b only. Then the covariance of K with respect to the transformation (21) gives

sx’2m+‘+$~x’my’+ . . . GKE Sx@‘+l+ (S+&)x’2ny’+ . . . (mod 2),

where Sr’ denotes the function Sr formed for the new coefficients (22). Hence

Sl’ - SI = Btb + b)

must equal the invariant S. Since &b is a function of the invariant b, &a must be an invariant, so that & = 0. Thus S = 0 and K has the factor L as before. Hence the theorem is true for covariants of order w if true for those of order w - 3. But it was proved true for those of order zero.

By a similar method I obtain the following theorem: A fundamental system of covariants of the binary quadratic form

f, given by (20), and the linear form X = a2x + sly module 2 is

given by f, X, I, h = (aa2 + j>z + (cm + j>y,

Q, L and f/w invuriunte b, q, (al - l)(a2 - 1) and

j = (a + b)al + (b + 4~2.

58 THE MADISON COLLOQUIUM. I

Since aI and a2 are cogredient with x and y, the function j obtained from the covariant t off is an invariant off and X.

The reverse of the last process is important. If we adjoin to a system of binary forms in the variables z’ and y’ the linear form yx’ - xy’, any modular invariant of the enlarged system, formal as to x, y, is a modular covariant of the given system with CC’, y’ replaced by x, y. The theorem of 0 12 therefore proves the existence of certain formal covariants. *

APPLICATIONS OF INVARIANTS OF A MODULAR GROUP, 50 14,15

14. Form Problem for the Total Binary Modutur Group I?.- This group is composed of all binary linear transformations (7) with integral coefficients taken modulo p whose determinant A is not divisible by p. By (S),

034) Lb, Y) =AL(Z Y>, &(x9 Y> = &(X9 y> (mod PI,

so that L@ and Q are absolute invariants of I’. Hence, of the functions (ll), Q is invariant under I?, while t is unaltered by certain transformations and changed in sign by others. Thus a homogeneous function of q and t having a term which is a power of q is a relative invariant of I’ only when an absolute invariant. Hence if p > 2, it involves only even powers of t, and by the homogeneity, only even powers of q. Hence any absolute invariant of r is a product of powers of LPI and Q by a polynomial in q7, t7, where 7 = 1 if p = 2, 7 = 2 if p > 2.

In particular, LPI and Q form a fundamental system of absolute invariants of r. The so-called form problem for the group I’ requires the determination of all pairs of values of the variables x and y for which Lpl and Q are congruent modulo p to assigned values X and p, either integers or imaginary roots of congruences modulo p. We have therefore to solve the system of congruences

(65) IL&, Y>P-’ = A, Qb, Y) = P (mod p>.

*After these lecturea were delivered, I saw a manuscript by Professor 0. E. Glenn, containing tables of formal concomitants for forms of low ordera and moduli 2 and 3. He employs transvection between the form and the covariant L of 0 2.

___-. .__ ~~-,.-__“---__--____.-“.__- .,... --._-_-.-- __^____.. -.. I

INVARIANTS AND NUbfBER THEORY. 59

First, let X + 0. For z = x or z = y, we have

2P’ gP2 ZP’

0 s XP yp ZP = LZP” - QLzp + LPZ (mod ~1.

x Y z

Hence x and y are roots of

(66) F(z) = zp* - pzp + AZ = 0 (mod P>.

Having no double root, this congruence has p2 distinct integral or imaginary roots. These roots are

(67) eX + fY (e,f = 0, 1, . . ., p - l),

where X and Y are particular roots linearly independent modulo p. For, (68) F(eX + fY) = eF(X) + fW7.

Hence any pair of solutions x, y of (65) is of the form (7), where a, a--, d are integers, whose determinant A is not divisible by p, in view of (641) and X + 0.

Conversely, if X and Y are fixed linearly independent solutions of (66), any pair of linear functions of X and Y with integral coefficients, whose determinant is not divisible by p, gives a solution of (65). Indeed, by (68), x and y are solutions of (66). From the two resulting identities, we eliminate X and p in turn and get

P = Q(x, Y), IUx, ~11” = Wx, Y).

Since X and Y are linearly independent modulo p, L(X, Y) is not divisible by p [cf. (6)]. Thus L(x, y) + 0 by (64). Hence (65) hold.

Hence, for X + 0, the form problem has been reduced to the solution of congruence (66). The latter will be discussed here in the simple but typical* case in which X and p are integers. Now the problem to find the real and imaginary roots of a con-

* For the general case, see Tran.mtims of the Am&can Mathmatieal Society, vol. 12 (1911), p. 87.


gruence with integral coefficients is at bottom the problem to factor it into irreducible congruences with integral coefficients.

When v is an integer, zp - vz is a factor of (66) if and only if v is a root of the characteristic* congruence

(6% u2 -p+x=o (mod P).

Such a binomial is a prod&t of binomials zd - 6, irreducible modulo p, whose degree d is the exponent to which the integer v belongs modulo p. Since 2p - 1 < p”, the function (66) has an irreducible factor d(z) of degree D > 1, not of the preceding type ti - 6, and hence with a root T such that P/T is not congruent to an integer. Thus every root of (66) is of the form C~T+C~P, where the c’s are integers. The irreducible factors of (66) are of degree D except those, occurring only when (69) ha.s an integral

root, of the for?72 Zd - 6, where d is a ditior of D.

To find D, note that by raising (66) to the powers p, p2, . . . , we can express ZP” as a linear function It of ZP and z. Now D is the least value of t for which 4 = z. But the coefficients of Ir are the elements of the first row of the matrix of S”, where

* Note the analogy of (66) with the linear differential equation

having the solution z = ert if u is a root of u2 - PU + X = 0. Also, (68) holds. Make dz/dt correspond to zp and hence d%/dP to (zp)p. Thus the differential equation corresponds to (66), and the integral z = ear (viz., dz/dt = vz) to 2.p = uz.

t Let f(z) be an irreducible factor of degree d. Its roots are

r, 7-p = u-l-, ,.P= E &., . . ., Ted-’ s ,,d-IT,

where ud = 1, vz + 1, 0 < 2 < d. Thus d is a divisor of p - 1. Hence

p-1 - v = ZP-1 - $-P-l

has the factor zd - rd. The latter has a root T in common with f(z). But

(,d)z+ c vd E 1.

Thus 6 = rd is an integer. Hence f(z) = zd - 6.

~----~-_-I I -_ - -_ - - _-__--


But ID = z implies that &,+I = sp. The condition for the latter is therefore SD = 1. Hence D is the period of S. But (69) is the characteristic determinant of S. According as it has distinct roots rl and 2)2 or equal roots v = 8~ = X*, a linear substitution of matrix S can be transformed linearly into one of matrix*

According m the characteristic congruence (69) has distinct (real or imaginary) roots or a double root, D is the least common multiple of the exponente to which the &tin& roots belong moddo p, or ia p timea the exponent to which thfs double root be,?onga.

Finally, let X = 0. By (6), either y = 0 or x - uy = 0 (mod p), where a is an integer. In the fist case,

Q = xPLP, xPa - *P = 0.

If j,L = 0, then x = y = 0. If p =I= 0, the roots x are equal in sets of p and hence are cx1 (c = 0, 1, . . ., p - l), where x1’ is a particular root not divisible by p. In the second case x - uy = 0, we take x - uy as a new variable X and conclude from the absolute invariance of Q that

Q(x, Y) = &to, Y) = ypLp.

We thus have the first case with y in place of x. Using similar methods, I have solved the form problem for

the total group of modular linear transformations on m variableat

15. Invariantive Ck&$cation of Forms.-Let

(70) 4(x, $4 = p+ *** (m > 1)

be a binary form irreducible modulo p and having unity as the coefficient of the highest power of x. Let G be the group of all modular binary linear transformations (1) with integral coef-

* In the second case we use the new variables x and x - uy. t Tranaudons of the American Mathem&& Society, vol. 12 (1911), pp.

84-92.


ficients of determinant unity. Let ~$1 = 4, &, . . . , 4k: denote all the forms of type (70) which can be transformed into constant multiples of C#J by transformations of G. Evidently their product P = 4142 - - * C$k is transformed into c,P by any transformation t of G. The constant ct is easily seen* to be congruent to unity. Hence P is an absolute invariant of G. If m > 2, no C& vanishes for a special point. We now apply the theorem in the first part of 0 14. Hence, if m > 2, the absolute invariant P is an integral function with integral coemnte of the invariants q, 1, each exponent of q and 1 be&g even ij p > 2. In view of the definition of the &, this function of q and 1 is an irreducible function of those arguments modulo p.

Two binary forms shall be said to be equivalent if and only if one of them can be transformed into a constant multiple of the other by a transformation of G. A set of all forms equivalent to a given one shall be called a genus. Thus &, . . . , $k form a genus. All of the irreducible forms (70) separate into a finite number f of distinct genera; let PI, . . ., Pf denote the products of the forms in the respective genera. Thus ?r, = PI . . * Pf is the product of all of the binary forms z”’ + 0 . . irreducible modulo p. Hence ?r, is a polynomial in q, 1 with integral coefficients. Hence the f genera of irreducible binary forms of degree m > 2 are characterized invariantively by the f irreducible factors

P&7, 0 of dq, 0 moddo p. We shall see that ?r,,,(q, I) is easily computed. By finding its

factors irreducible modulo p in the arguments q, Z, we shall have invariantive criteria for the equivalence of two irreducible binary forms of degree m. For example, we shall prove that r3 = q - 1 if p = 2, so that all irreducible binary cubic forms module 2 are equivalent. Further, 1r3 = q2 - F if p > 2, so that the irreducible cubic factors of q - I are all equivalent, also those of q + 1, while no factor of the former is equivalent to one of the latter.

* T~an.w&ms of the Am&cm Mathematical Society, vol. 12 (1911), p. 3, Q 4. The present section is an account of the simpler topics there treated at length.


In general, let m be a product of powers of the distinct prime numbers q1, . . *, q,,, and set

Ft = (zq/ - qp”)/L.

From the expression for ?r, due to Galois we readily obtain

in which the first product in the numerator extends over the +I-& - 1) combinations of ql, . . . , q,, two at a time, and similarly for the remaining products. By the first theorem of this section, and (1 1), n,,, is a polynomial in

Jzqy= QP+‘, K=~Y= Lpcpl) (+y = 1 if p=2, y=2 if p>2).

We readily verify the recursion formula

Ft = QF:-, - KF;:, (mod P>,

since F1 = 1, Fs = Q. In particular,

Fs = J - K, Fq = Q(F3p - KJp’).

NOW IQ = Fa, ~4 = Fd/Q. Hence

a3 = J - K, T4 s Jp - Kp - KJp-1 (mod PI.

The first of these results was discussed above. Next, for P = 2, ~4 is the irreducible quadratic form q2 - P - lq, so that all quartic forms irreducible modulo 2 are equivalent. For p > 2, 7r4 vanishes for K = pJ, where

pP=l-p (mod P>.

Except for p = 9, p is a quadratic Galois imaginary since

pPs E 1 - pP s p (mod P>.

Thus 7r4 is a product of J - 2K and +(p - 1) irreducible quadratic forms in J, K. Some of the latter yield a quartic in q and 1 which is irreducible; others yield a quartic which is a product of two irreducible quadratics modulo p. A simple discussion shows


that the number of irreducible factors of T*(Q, I) is 6k + t + 1 if p = 8k + t (t = * 1 or - 3), but is 6k + 2 if p = 8k + 3. We have therefore the number f of genera of irreducible quart& modulo p. For quintics and septics, the analogous discussion is simple, for sextics laborious.

We may utilize similarly the invariants (16) of the group on m variables, obtain expressions in terms of them of the product of all forms in m variables of specified types (as quadratic forms transformable into an irreducible binary form, non-vanishing ternary forms, nondegenerate ternary quadratic forms, etc.), and hence draw conclusions as to the equivalence of forms of the specified type. *

* Tranmctions of the A merican i!hdmatti 8ociety, vol. 12 (lQll), pp. 92-98.

LECTURE IV

MODULAR GEOMETRY AND COVARIANTIVE THEORY OF A QUADRATIC FORM IN m VARIABLES MODULO 2

1. Introduction.-The modular form that has been most used in geometry and the theory of functions is the quadratic form

(1) Qm (X) = ZCijXiXj + ZbiXi2 (i, j = 1, * * *, m; i < 3)

with integral coefficients taken modulo 2. In accord with Lecture III, we shall use the term point to denote a set of m ordered elements, not all zero, of the infinite field Fa composed of the roots of all congruences module 2 with integral coefficients. We shall identify such a point (x1, * * ., z,,J with (~21, * * a, ~z,,J where p is any element not zero in Fz. The point is called real if the ratios of the x’s are congruent to integers modulo 2. Let the cij and bi in (1) be elements not all zero of the field Fs. Then the aggregate of the points (x) = (x1, * * 0, z,,J for which ~~(2) = 0 (mod 2) shall be called a quadric locus, in particular, a conic if m = 3. The locus is thus composed of an infinitude of points, a finite number of which are real.

While our results are purely arithmetical, we shall find that the employment of the terminology and methods of analytic projective geometry is of great help in the investigation. Usually the proofs are given initially in an essentially arithmetical form. In case a preliminary argument is based upon geometrical intuition, a purely algebraic proof is given later. The geometry brings out naturally the existence of a linear covariant, which is important in the problem of the determination of a fundamental system of covariants.

2. Tlw Polar Locus.-The point (KYI -j- &I, - - a, KY,,, + AZ,,,)

is on q(x) = 0 if

(2) +7(Y) + Khp(Y, 2) + h*Q(z) = 0 (mod 3, 65


where (3) P(y, 2) = ZCij(y$j + YjZi) (i, j = 1, ’ ’ ‘) m; i < j).

If (y) is a tlxed point, all points (8) for which P(y, z) = 0 (mod 2) are said to form the polar* locus of (y). For (2) = (y), each summand in (3) is congruent to zero modulo 2. Hence the polar of (y) passes through (y). If (z) is on the polar of (y), (2) has a double root K : X and the line joining (y) and (z) is tangent to q = 0.

We may write (3) in the form

(3’) where

P(y,z) = u1y1+ *** + urnynl,

Ul = c12zz + k&3 + Cl424 + * * - + Qdm,

242 = c1221 + c2323 + c2d4 + " - + cZm%,

(4) u3 = cl321 + c23z2 + c34z4 + --- + c37&n,

urn = cl&l+ C2&2 + c3l&+ .** + G&-1+1.

There is a striking difference between the cases m odd and m even.

3. Odd Number of Variables; Apex; Linear Tangential Equation. Let m be odd. Then the determinant of the coefficients in (4) is congruent modulo 2 to a skew-symmetric determinant of odd order and hence is identically congruent to zero. Hence we can find values of 21, * . . , z,,, not all congruent to zero such that

Ul, . . ., u, are all zero modulo 2. Thus the polars of all points (y) have at least one point in common.

We shall limit attention to the case in which the pfaffians

(5) Cl= [23 - * - m], CZ= 1134 - - - m], - - -, C,= [12 - - - m-l]

are not all congruent to zero. The point (C,, . . ., C,) shall be *Take I = 1 and let (z) be a point not on p(z) = 0. Then (2) is a quad-

ratic congruence in A with coefficients in Fa and hence haa two roots hl and X2 in that field. Now the points (u) and (z) are separated harmonically by (y + Alz) and (1/ + Ati) if and only if X1 = - AZ, that is, if A1 = An (mod 2). But the condition for a double root of (2) is P E 0 (mod 2).


called the apex* of the locus a(r) = 0. Now each zbi = 0 if 21 = Cl, - - -, z,,, = C,,,. Hence, for m odd, the polars of all points pass through the apex.

If (y) is any point not the apex, the line joining (y) to the apex is tangent to q(x) = 0 (0 2). Thus any Zim through the apex is

tangent to q(x) = 0. For m = 3, it is true conversely that, if the line

(6) ZUiXi G 0 (mod 2)

is tangent to q(x) = 0, it passes through the apex, so that

(7) K = I;C;Ui

is zero modulo 2. Taking, for example, u3 =l= 0, we obtain by eliminating x3 from (6) and q(x) = 0 a quadratic equation in x1 and x2 whose left member is the square of a linear function module 2 if and only if the coefficient of ~1x2 is congruent to zero. But this coefficient is the product of K by a power of u3. Thus

K = 0 is the tangential equation of q(x) = 0. The last result is true for any odd m. The spread (6) is said

to be tangent to q(x) = 0 if the locus of their intersections is degenerate. Taking u, =l= 0, and eliminating G between (6) and q(x) = 0, we obtain a quadratic form whose discriminant, defined by (24), equals a product of K by a power of u,,,, and hence is degenerate if and only if K = 0.

We thus have geometrical evidence that K ti a formal contravariant of q(x), i. e., an invariant of q(x) and Zuixi.

To give an algebraic proof, note that K is unaltered when xi and xj are interchanged, while

(8) x1=21’+52’, 52=x2’, -a-, %?a = %I&'

replaces q(x) by q’(x’) in which the altered coefficients are

(9) bz’ = bz + h + ~12, C;j = C2j + Clj (i = 3, - - *, m).

*After these lectures were delivered, I learned that Professor U. G. Mitchell had obtained, independently of me, the notion apex (“ outside point “) for the case m = 3, Princeton dissertation, 1910, printed privately, 1913.


The pfaffians Cz, . . *, C,,, are unaltered modulo 2, while

(10) Cl’= c;+ cz, T.&Z’= %+?&I, z&i’= UC (i* 2) (mod 2).

Hence K is unaltered modulo 2. Note that

0 Cl2 Cl3 ” - CM Ul

Cl2 0 C23 * ’ - C2?n ua (11) $G . . . . . . . . . (mod 2).

Clm C2m C3m .‘. 0 u,

Ul u2 243 a** u, 0

We saw that Cl, . * ., C, are cogredient with ~1, * -0, z~. Tbis is evident from the fact that the apex is covariantively related to q(z). Hence if we substitute Cl for 21, . l a, C,,, for z,,, in (l), we obtain the formal invariant

(12)

If this invariant vanishes, the apex is on the locus, which is then a cone. Indeed, by (2), every point on the line joining (C) to a point on q(z) = 0 lies on the latter. Hence q(x) can be transformed into a form in m - 1 variables and hence has the discriminant zero. To argue algebraically, let new variables be chosen so that the apex becomes (0, . l *, 0,l). The polar of any point (3) passes through the apex. Taking z1 = 0, l * ., z,,,-1 = 0, &I& = 1 in (4), we see that the polar (3’) becomes clmyr + . - l

+ c,,+l,y,l, which must vanish for arbitrary y’s. Hence b,,,~~ is the only term of (1) involving z,,,. But the apex is on the locus. Hence b,,, = 0 and q(z) is free of G. The converse is obvious from (5).

Whether m is odd or even, q(z) has the invariant

(13) A,,, = lI(Cij+ 1) (i, j= 1, ***,m;i<$.

This is evidently true by (9) or as follows. If A, = 1 (mod 2), every cij s 0 and q = (2%&2; while if A, = 0, at least one cij is not congruent to zero, and q is not a double line.

Hence the product A,q(z) is a covariant; in fact, the square


of the linear covariant A,G%pi. We shall see however that there exists a more fundamental linear covariant.

4. Cmariant fine of a Conic.-Since we shall later treat in detail the case m = 3, we shall replace (1) by the simpler notation

(14) F(z) = alx2a + a2xla + wax2 + by? + b2~2~ + f~4~.

Its apex is (at a2, a2). Its discriminant (12) is

(15) A = F(ar, a2, as) = alazaa + a12bl + az*bz + as2ba.

The invariant (13) becomes

(16) A = crlct~a!~ (cyj = ai + 1).

Consider a form (14) with integral coefficients and not the square of a linear function. Then not every ai is congruent to zero modulo 2. By an interchange of variables we may set a3 = 1. Replace 21 by X1 + alza and 52 by Xz + a2x8. We get

X,X, + blX12 + b2Xz2 + Axa2.

Let A = 1. Replace x3 by X8 + blXl + b2X2. We get

(17) fp = x,x* + X82.

The only real points on t$ = 0 (mod 2) are (1, 1, l), (1, 0, 0), (0, l,O). In addition to these and the apex (0, 0, l), the only real points in the plane are (1, 1, 0), (0, 1, l), (1, 0, 1). These lie on the straight line

(18) Xl + x2 + x* = 0 (mod 2).

Hence with every nondegenerate conic modulo 2 is associated covariantly a straight line.

The inverse of the transformation used above is

Xl = xl + alx8, X2 = x2 + 62x8,

&I = blxl + btm + (I+ a&l + azbdxa.

It must therefore replace 4 by the general form (14) having


as = A = 1. It actually replaces (18) by

(bl+ ur1+ (bz + 1)x2 + (a, + ata2 + 0%

in which we have added A + 1 = 0 to the initial coefficient of x8. Guided by symmetry, we restore terms which become zero for a3 = 1 and get

Making the terms homogeneous we obtain the formal covariant (20) L = Blzl+ B2s2 + &a,

B1 = b12 + ~42~3 + a22 + aa2, B2 = bs2 + alu3 + al2 + aa2, (21)

Bs = ba2 + ala2 + a? + az2.

Under the substitution (aiaj)(bibj) induced upon the coefficients of F by (z,zi), we see that Bi and Bj are interchanged. Under cv, viz.,

(2.3 b2’ = b2 + bl + a3, al’ = al + a2 (mod 21,

there results

(23) B1’ = B1, B2’ = B2 + B1, B,’ = Ba (mod 2).

Hence (20) is a formal covariani of F. For other interpretations of L see 0 8.

5. Even Number of Variables.-The determinant of the coefficients in (4) is congruent modulo 2 to the square of the pfaffian

(24) & = [123 --- m].

This is in fact the discriminant of q,,,, which is degenerate if and only if Is, = 0 (mod 2). I have elsewhere* discussed at length the invariants of qm.

* Tran.sa&kns of the Anwican Mathemuticd Society, vol. 8 (1907), p. 213 (case m = 2); vol. 10 (1909), pp. 133-149; American Journal of Mathematics, vol. 30 (1908), p. 263; Procdings of the London Mathe?tuzttical Society, (2), vol. 5 (1907), p. 301.


If G, + 0 (mod 2), we can solve equations (4) for the z’s. Substituting the resulting values into q(z), we obtain the tangential equation U, = 0 of a(z) = 0. For m = 2 and m = 4, we get

Bordering the algebraic discriminant of (l), we find that

Cl2 2bz ~23 0 - - cz,,, u2 (26) 2u,,,z . . . . . . . . .

Cl?78 C2m em -- - 2b,,, u,,,

,241 u2 U) -* * um 0

(mod 4).

Finally, let & = 0 (mod 2). Then all of the first minors of the matrix of the coefficients in (4) are zero modulo 2. Hence the polars of all points have in common the points of a straight line S. Since its discriminant vanishes, q(z) can be transformed linearly into a quadratic form in zl, . . . , -1, which therefore represents a cone with the vertex (0, . . ., 0, 1). Let (z) be the vertex of the initial cone q(z) = 0. If (x) is any point on the cone, (Z + xz> is on the cone, and, by (2), P(z, z) is congruent to zero identically in ~1, . . ., z~. Hence the linear functions (4) all vanish. Thus the line S meets the cone in its vertex, and h2 is the discriminant of Q+~(x), while xi2 is obtained from that discriminant by interchanging m and i. For example, if m=4,

z42 = cl2cl3c23 + b& + &A + b&, ** -,

Z12 = c23c24c34 + b& + btda + ba&.

The product of the general form (1) by 6 = & + 1 is a quadratic form whose discriminant is zero modulo 2 and hence has the vertex (6~1, . . ., Sz,J, where z? has the value just given. Hence 6z12, * . . , Sk2 are cogredient with xl, . . . , x,,,.

6. Covariant Plane of a Degenerate Qua&G Surface.-The product of q4 by 6 = [1234] + 1 is a quaternary form f whose


discriminant is zero and hence can be transformed into a form (14) free of x4. With this cone F = 0 is associated covariantively the plane I = 0, where I is the ternary covariant (19). Hence f has a linear covariant L which reduces to 1 when b4 = 0, ci4 2 0 (i = 1, 2, 3). Relying upon symmetry and the presence of the factor 6, we are led to conjecture that

(27) L = 6{h + 1 + (CIZ + 1)hs + l>(c14+ l>h + -a -

+ 6Iba + 1 + (‘314 + l)(c24 + 1>(‘34 + 01x4.

It is readily verified algebraically tbat L is a covariant of 44. There is a simple interpretation of L. If [1234] + 0 (mod 2),

then 8 = 0 and L is identically zero. If [1234] = 0, q4 is degenerate and can be transformed into C# = x1x2 + x8* or a form involving only x1 and x2, In the former case, L = x1 + x2 + x8. Of the 15 real points in space, the seven (100x), (010x), (111x) and (0001) are on the cone t$ = 0, the two (001x) are on the invariant line S through the vertex (0001) of the cone and the apex (0010) of the conic cut out by 24 = 0, while the remaining six (Lola), (011x), (110x) lie on the plane L = 0. Hence with a degenerate quadric surface, not a pair of planes, is associated covariantively a plane, just as a line (19) is associated with a non-degenerate conic (14).

Every linear covariant is of the form IL, where I is an invariant. Every quadratic covariant is a linear combination of the ILP and Iq4.

7. A Cmfiguratiun Defined by the Q&nary Surface.-A q6 whose discriminant is not zero modulo 2 can be transformed into

F = x1x2 + 52x4 + xs2.

The 15 real points on F = 0 (mod 2) are given in the last column of the table below. In addition to these and the apex (00001) of F, there are just 15 real points in space:

l= (00011), 2= (OlOOl), 3= (OlOll), 4= (OOlOl), 5= (OllOl),

S=(OOllO), 7= (OlllO), 8= (lOOOl), 9= (lOOll), a= (lOlOl),

b= (lOllO), c= (llOOO), d= (llOlO), e= (lllOO), f= (11111).

.--.._ ._ ~- ~_..-..-


These lie by threes in exactly 20 straight lines, which occur in the columns of the table, with the heading “Sides.” With these lines we can form exactly 15 complete quadrilaterals, the three diagonals of each of which intersect* in a point on F = 0, given in the last column. The columns, with the heading “Plane,” give the equations defining the plane of the quadrilateral. In each case, the two equations of the plane have in common with F = 0 a single real point, the intersection of the diagonals. Thus the real points on F = 0 are its points of contact with these tangent planes.

Sides Diagonals Plane

146 157 356 347 146 lab 49b 69a 146 lef 4df 6de 157 lab 5~4 7bc 157 lef 58e 78f lab lef 2ae 2bf 28~ 29d 38d 39c 28c 2ae 5ac 58e 28c 2bf 78f 7bc 29d 2ae 69a 6de 29d 2bf 49b 4df 347 38d 4df 78f 347 39c 49b 7bc 356 38d 58e 6de 356 39c 5ac 69a

13 45 67 z1=0, zr+zd+zs=O 19 4a 6b x,=0, za+zt+zs=O Id 4e 6f cl=zs=z:+z,+z~ lc 5b 7a zl+s~+~~==q+z~+z~=(l 18 5f 7e zt=zs, z~=q+z~+zs 12 af be m=za, ZZ=Z~+Z,+Z~ 23 89 cd x*=0, x~=x2+xs 25 8a ce x,=0, xl=xt+xa 27 8b cf x8=x+ z1=zp+zp+z~ 26 9e ad xl=xa+xc=x2+xa 24 9f bd x1=x+ xa=g+x,+x~ 3f 48 7d x,=x4, x~=x~+x,+x~ 3b 4c 79 x,=x~+x~=x~+x~ 3e 5d 68 xz=xa+xc, zl=xe+xa 3a 59 6c x~=x~+x~=x~+x~

.-

I

Iuter- section

01000 10000 11001 11011 10010 01010 00010 00100 00111 01111 01100 10100 11101 10111 11110

8. Certain Formal and Modular Covariants of a Conk-For conic (14), the polar form is

(28)

al a2 a3

Yl !I2 Y3 l

21 z2 83

Hence if two sets of variables yi and zi be transformed cogredi- ently with the set zi, this polar form (28) is a covariant of F and the two points (y), (z), in an extended sense of the term

* The dual of the theorem of Veblen and Bueaey, ” Fiuite projective geometries,” Transa&ma of the Amenkan Mathematical Society, vol. 7 (1906), p. 245.

74 THE MADISON COLLOQTJIUM.

covariant. In particular, if we take (y) = (z), (II) = (.2”), we obtain a covariant of Fin the narrow sense used in these lectures. In particular,

‘al a2 a3 al a9 a3

(29) K = Xl z2 23 , M = Xl x2 Z3

Xl2 x22 x32 Xl" x24 sf

are formal covariants of F. While the discriminant A, given by (15), is a formal invariant, (16) is not. But

(30) A+A+l=a! (mod 3,

(31) ff = E&i+ 2d+ am + ala3 + w3,

a! being a formal invariant of F. By (23), the B’s are contragredient to the x’s and hence to the a’s, so that

(32) Al = ZaiBi = E&b? + ZU& + ama

is a formal invariant. For integral values of aij bi,

(33) Al s A s Zai(& + 1) (mod 2).

Any form with undetermined integral coefficients cl, CZ, . . ., taken module 2, has, by (21) of Lecture I, the invariant (cl+ U(c2 + 1) **-* Thus (16) is an invariant of (7) and hence of F. Likewise from (19) and F itself, we obtain the invariants

(34) J = &&,&, AJ = An(bi + 1).

In (6) we made use geometrically of

(35) x = u1x1+ %X2 + u3x3.

Now F + tx2 is congruent module 2 to the quadratic form derived from F by replacing each bi by bi + tuz. Making this replacement in A, we see that the coefficient of t is congruent to Ke, where

(36) K = am + a2u2 + a3ut

is therefore a formal invariant* of F and X. Making the same

* Since (36) is a contravariant of F, za&X/axi) is a covariant of F if C is. Taking &s, &I, L as C, we get K, M, A, respectively.

INVARIANTS AND NUMBER THEORY. 75 I

replacement in J and taking t and ug to be integers, we obtain as the coefficient of t = t2

(37) w = p&ZuS + 8$3u2 + t&33% + hu2uS

+ &ulu3 + 83ulu2 + UlU2U3,

a modular invariant of F and X. By the theorem used above,

(38) u = cm+ l>(u2 + wur+ 11

is an invariant of A. In w + u + 1, we replace /% by the congruent value Bi + 1, and render the expression homogeneous in the u’s and B’s separately. We get

(39) w = Z(BIB2 + B12 + B3’)~3~ + ZB12U3~3,

a formal invariant of F, X. For, it is unaltered by the substitution

induced by (xizj), and by the substitution (23) and (10) induced by (8). Let the coefficients of F be integers not all even. Then (39) becomes (39’) wbB2 + 1>u32 + m&+ 1hu3.

Its covariant I, is identically zero. Hence, by the table in 0 9, if w is not identically zero it can be transformed into ut + us2 + 241~2 and hence vanishes for a single set of integral values of ~1, 212, 7.43. These are seen to be ui = pi + 1. Hence* the line L = 0 i.e the only line with integral line coordinates on the line locus (39).

The invariant A for (39) is J (its discriminant is zero, as just seen). Thus a knowledge of any one of the concomitants L, J, w implies that of the other two.

The covariance of K in (29) implies that

* Also thus: just EILI the point conic F = 0 determines ita line equation (36) and hence ita apex (a), so the covariant line conic (39) determinea the point equation ZBh = 0, which is the line L = 0 for integral values of the coefficients.

“-- ----.--_l--.~ -... -“- “._ _.-.._ --.______ ----- -.-l__;_l_--..- - ---- - --- ~.


are contragredient with al, clz, a3 and hence with xl, x2, x3, and therefore cogredient with ~1, u2, u3. Thus (39) yields the formal covariant (41’) w’ = x(&B2 + IL2 + B22)+f32 + =lhE,.

From this or (39’), we obtain the modular covariant

(41) w = %3482 + l>ta2 + z@l + l)bb.

In these notations (29) become

(42) K = ZOJ&~, M = ZU&(Q~ + x2x3 + ~3~).

Finally, by (16) of Lecture III, we have the universal covariants

4 x2 x3

(43) La = x12 x22 x32 ,

&I = ZX~~X~~ + Zx14x2x3 + x12x22x3:32,

x4 224 x34 Q2 = 2x1~ + 2~1~x2~ + XIX~X~ZX~.

The covariant line .L = 0 of a non-degenerate conic F = 0 is determined by the three (collinear) diagonal points of the complete quadrangle having as its vertices the apex (a) and the three intersections of F = 0 with its covariant cubic curve K = 0.

FUNDAMENTALSYSTEMOFCOVARIANTSOFTHETERNARY FORM F,

00 9-32 9. Invariants of F.-A fundamental system of invariants of F

is given by A, A, J. It suffices to prove that they completely characterize the classes of forms F under the group of all ternary linear transformations with integral coefficients modulo 2. This is evident from the following table

Class A

x1x2 + x32 1

21x2 + Xl2 + x22 0

x1x2 0

Xl2 0

0 0

,

A J -- 0 0

0 1

0 0

1 0

1 1

. L

a+ x2 + x3

0

Xl + x2

Xl

0


As to the classes, we saw in 0 4 that, if F is not the square of a linear function (i. e., not reducible to xl2 or 0), it can be transformed into 51x2 + bl~1~ + 132x2~ + Axs2 and hence into one of the f&t three classes of the table. By means of the relations

(44) AA=O, AJ=O, A2=A, A2=A, P=J (mod 3,

any polynomial in A, A, J equals a linear function of

(455) 1, A, A, J, AJ.

These are linearly independent sihce there are five classes.

10. Leader of a &variant of F.-Let S be the coefficient of 58”’ in a covariant of order w of F. Writing (14) in the form

(46) F = f+ lx:,+ bm2, f = bm*+ wm+ b2xz2, I= a2x1+am,

we see that the leader S is a function of bs and the invariants of the pair of forms f and I under the linear group on x1, x2.

In the modular covariants forming a fundamental system for f (Q 13 of Lecture III), we replace x1 by al and 22 by a2 and obtain a fundamental system of modular invariants of the pair f and I:

(47) a3, ff1a2, q= bd2+(bl+bdw, j= (bl+adal+(b2+ada2,

where CQ = ai + 1. By means of the relations

(48) a~cu2j = 0, qj = j + a3j (mod 2),

any polynomial in the four functions (47) can be reduced to a linear combination of

(49) 1, ~3, q, a3q, aa2, w2a3, wx2q, w2a3q, j, a3j.

These form a complete set of linearly independent* invariants off, 1.

*Instead of verifying as usual that these 10 functions are linearly independent, we may deduce that result from the fact that there are 10 clssstx:

I = 21, f = uslw2 + asxa* or QXl” + wwr + wf,

2 = 0, f = 22 + 21x2 + x2, x1x2, x1* or 0.

Since (47) characterize the classes, they form a fundamental system.

- --. ---._II_I_I.-..--____ -- _._. .___ _.,.-_ - ,_._-.= . _.._..


Hence S is a linear combination of the functions (49) and their products by bs. Moreover, S must remain unaltered modulo 2 when US and bl are replaced by

(50) aa’ s US + al, bl’ = bl + bs + az,

which are the only altered coefficients of the form obtained from F by the transformation

(51) x1 = Xl’, x2 = x2’, x3 = x3’ + 21’ (mod 2).

Both requirements are evidently met by the functions

(52) 1, a1~12, ba, b3am

and any invariant of F. We find that

A = ~@$!(@ + l), A = crm’2U3 + j + aaba + as,

(53) J = a~cuz(m + l)@s + 1) + btj + c&j + b3q + mcuzq,

AJ = cw2(a3 + Wa + l)(q + lb

From these and their products by b3, we see that

(54) AJ, baJ, J, W, kb 4 A

contain the respective terms

bww~q, hamq, cumq, hj, bma3, j, cmm,

while no one involves an earlier one of these terms. Hence any linear combination of the functions (49) and their products by ba is a linear combination of the functions (52), (54) and

(55) as, ba, q, bag, w, hq, a3.i bmj, mww.

A linear combination of the latter is of the form

u = ma3 + m2q + m3a39 + m4a3j + mw2a39,

where ml, . - a, m4 are linear functions of bs, while m is a constant. The coefficient of a,bl is seen to be

p = m2 + m&t + ma + mbm(R + b3 + 11,

_I -... ----_I_


where R = ba + u2 is the increment to bl in (50). Set

(56) u=p&+ra~+&+t (p, *es, t independent of as, bl).

Let the substitution (50) replace u by Q’. Then

(57) u’ - Q = pRas + palbl + palR + rul + sR.

This is zero for every ~3, bl if and only if

(53) pR = 0, pal = 0, ral = sR (mod 2).

For p= mn+ a.., pal = 0 gives m3 = 0, m4 z m. Then pR = 0 gives m2 = 0, rnba = 0, whence m = 0. Thus u = mlas, so that ml = 0. Hence the leader of a covariant of F has the form

(59) I + Ml + cala2 + okw&a,

where I and II are invariants, c and d are constants.

COVARIANTS WHOSE LEADERS ARE NOT ZERO, @ll-19

11. Consider a covariant of odd order w :

(60) c = Sxs” + s1x3w-1x1 + s2x3o-2x12 + * * *.

If S1’ is derived from S1 by the substitution (50), then, by (51),

(61) fiy = s1+ us = i&-j- s (mod 2).

Give S1 the notation (56). Then S is given by (57) and has no term with the factor atbl. Now a3bl enters no term of (59) except J and AJ of I and* b,J of bsI1, and in these is multiplied by

(62) bm + WQ, wa(b2 + l>(h + 11, bala2,

respectively. Since the latter are linearly independent, neither J nor AJ occurs in the I, II of the leader (59). Also, A and crlcv2 occur only in the combinations A + 1, cvlcvz + 1, since (57) has no constant term. The coefficients of xao in L”, AL*, (A + A)L” are respectively

(63) ba + am2 + 1, Ah, A + Ah +bma2,

* AJ is not retained in II, since b.dJ = 0, AJ being (34).

80 THE MADISON COLLOQTJIUM.

where L is the linear covariant (19). After subtracting from C a linear combination of these three covariants, we may set

S = mdA + 1) + m2A + m&a + mb8ala2.

Since /3&xrul(rs = 0, AJ = 0, the leader of the covariant JC is

JS = mlAJ + md + m3b3J.

Hence ml = ma = 0. The coefficient of a8 in S is now m2(a1a2 + b3) and must vanish for b3 = a2 since it is of the form pR by (57). Hence m2 = 0. Thus S = rnbgxlas. For w > 1, mFLue2 has this same leader. For u = 1,

C = m(bma223 + b1a2wa + b2ww2),

which satisfies (61) only when m = 0. Hence eeery linear covariand is a linear function of L, AL, AL; evey covariant of odd oxler w > 1 difers from a linear combination of L”, AL”, AL”, FLWe2 by a covariant whose leader G zero.

12. In the covariants of order 4n

(64) I&$‘, IP”, L4”, F2”-‘L2 (I an invariant),

the coefficients of xz4” are respectively

I, &I, bt + cw~2 + 1, kwxz.

Linear combinations of these give every leader (59). Hence every covariunt of OTO%T 4n dij’era from a linear combination of the covariants (64) by a covariant whose leader is zero.

13. In the covariants of order w = 4n + 2

(65) IQt”F, Q2”L2, AQ2”L2 (I an invariant),

the coefficients of xs* are respectively

baI, bs + WYZ + 1, A + bs(A + alaza3).

The sum of the third function and bt(A + A) is A -I- bgxlctrz. Hence any covariant C is of the form P + C’, where P is a linear

-.- ____----I . . -.--- _____ l__l_--.--. . .-~ _ J

INVARIANTO AND NUMBER THEORY. 81

combination of the covariants (65), while C’ is a covariant whose leader is an invariant. For w = 2,

C’ = Sx? + 51X3X1 + 5x12 + x29.

This is transformed by (51) into a function having Sr as the coefficient of 21’~. Since S is an invariant, Sr = S. Thus every coefficient of C’ equals S. Then (51) transforms C’ into a function in which the coefficient of ~1~x2’ is zero, so that S = 0. Hence every quadratic covaria& is a linear fwhon of

(66) F, AF, AF, JF, L2, AL2.

14. There remains the more difhcult case of covariants (60) of order w = 4n + 2 > 2. If Si’ is the function obtained from Si by the substitution (50), then

(67) 231’ = Sl, S2’ = s + 6% + s2.

Now S1 is unaltered also by the substitutions (22) and

um as’ = ag + a2, b2’ = b2 + ba + al (mod 21,

induced on the coefficients of F by the transformations (8) and

(69) x1 = Xl’, x2 = x2’, 53 = 23’ + x2’.

15. A fun&mental system of invariants of F, under t/w group I’ generated by the transformutiona (8), (51) and (69), is given by A, A, J, a2, h, am2 and

(70) P = Wa + a2>.

It suffices to prove that these seven functions, which are evidently invariant under I’, completely characterize the classes of forms F under I’. There are six cases.

(i) bt=a2= 1. Replacing x1 by x1 + ~1x2 and x8 by x3 + w2, we get

F = ,dx12 + Axz2 + xa2 + ~1x3.

(ii) b, = 1, a2 = 0, all = 1. Replacing x3 by z3 + asxr, we get

F = AXI’ + b2xz2 + ~3~ + ~2x3.


If A = 0, then bz = J. If A = 1, we replace z1 by x1 + b2zz and get

a2 + a2 + ~2%

(iii) bs = 1, a2 = ala2 = 0. Replacing 23 by za + bm + b2x2, we get

xa2 +&x2.

(iv) bs = 0, a2 = 1. After replacing xS by x3 + u8x2, we obtain a form with also aa E 0. Taking this as F, and replacing xl by x1 + alxz, we get

haa2 + ha2 + ~1~s.

Replacing xa by x8 + blxr, we get Ax? + ~1x1. (v) bs 5 az = 0, alcu, = 1. Replacing x3 by ZQ + ~3x1 + b2x2,

we get px12+ X2X8.

(vi) bs = a+? E am = 0. Then F is the binary formf in (46). The effective part of I’ is now the subgroup rl generated by (8). Now

@ = bt A + 1 = aa, J = B + (bl + l)cra, B = br(bl + era).

These seminvariants bl, ea, B of f completely characterize the classes of formsf under I’l. For, if aa = bt

f = blxr2 + Bxz2 + blxlq;

while if a8 = bl + 1, we replace x1 by x1 + b2x2 and get

bm2+ (bl+ 1)x152.

16. The number of classes of forms F in the respective cases (i)-(vi) is 4, 3, 2, 2, 6. Hence there are exactly 19 linearly independent invariants of F under the group l?. As these we may take

(71)

1, a?, alcrz, A, bar b,az, bsam, bsA,

A= bm+ *es, ad= blalazf ---,

B = bdbs + a~>, azP = Msm,

----..--.---J

INVARIANTS ANB NUMBER THEORY. 83

(71) &=b~(ba+l)A, bA=b&m+ - - -, ahA=b&w2+ - - -,

J- blbtbs+ -se, asJ= blbzb3az+ s-e,

bsJ = b,b,bs(ala, + al + ar) + l l a, AJ = blbzbaA + - - a.

These are linearly independent since the first eight do not involve bl, while all the terms with the factor bl in the next seven are given explicitly, likewise all with the factor blbzbs in the last four. Hence tha 19 functions (71) form a complete set of linearly independent invariants of F under the group I?.

17. Hence, in 0 14, Sr is a linear combination of the functions (71). By 03721, S + S 1 is of the form (57) if S2 be denoted by (56). Now aabl occurs in J, AJ, biJ, a2J, A/3, but in no further function (71). In the first three, asbl is multiplied by the linearly independent functions (62), respectively; in the last two by belat and alcu2(b3 + l), whose sum is congruent to the first function (62). Hence the part of S + Sr involving J, - - -, A/3 is a linear combination of

(72) (b3 + adJ = b&&w2 + b&wwa,

(73) J + baJ + 43 = @a + lNdtcw~ + bzA + A).

But bl occurs in just six of the functions (71) other than the five just considered. Thus the factor pal of bl in (57) is a linear combination of the coefficients of bl in (72), (73), S, a@, A, QA, bb, azb&. Now al is a factor of the coefficients of bl in all except the second, third and fourth, while in these the coefficients are

(bs + Www2, bs+e+ 1, 43

and are linearly independent. Hence (73), 8, a@ do not occur in S + I!&. By (57), the latter has no constant term and hence involves 1, A only in the combination A + 1. This cannot occur since the total coefficient of a3 must be of the form pR and hence vanish for b3 = ~22. At the same time we see that the sum of the constant multipliers of A, aA, bb, Rbd is zero modulo 2. Hence S + SI is a linear combination of the functions


~2, 58, bsscz, ala2, and the last six in (74) below. Lie (57), this combination must vanish for al = 0, ba = a. Since all but the first three of the ten functions then vanish, the sum of the multipliers of these three must be zero module 2. Hence S + S1 is a linear combination of

b8 + a2,

(74) at(b, + 11, a1a2, bw2, h-4

fh2, NJ, + 11, Na2h + 11, @a + az>J.

18. Without altering the invariant S, we may simplify S1 by subtracting from C constant multiples of L4*+ K and its product by A, where K is given by (29), and hence delete a2(ba + 1) and A(a&s + 1) from the terms (74) of &. Then

s1 = S + mh + mA(bs + 1) + m&s + az)J

+ Mb3 + 4 + m4am + m5baalcuz + m&A.

The coefficient T of q”+ x2 in C is obtained from S1 by applying the substitution (a&(b~b2) induced by (51~2). In view of the transformation (8), we see that T’ = T + 6’1, where T’ is derived from T by (22). Hence

S = (m + ml)A + mdb + m&J

+ (m4 + m&)(a1a2 + a+ 4 + m&3 + m&A.

Let L: be the sum of the second member and the function obtained from it by the substitution (ma3)(b2ba). Thus E = 0. Taking bs = b2, we get m4 = ma = 0. Then

Z = (b2 + bt)I, I = m9 + m2J + ma + meA.

Applying to Z the substitution (68), we get (b2 + al)1 = 0. Applying (alas)(b~bg) to the latter, we get (b2 + aa)1 = 0. Adding, we get (al + a3)I = 0. Applying (50), we see that aJ = 0. Then each ail= 0, so that I = gA, where g is a constant. By 2: = 0, g = 0. Thus ml, m2, ma, me are zero. Hence S = mA, ~‘31 = mAa2. But

(75) E=F(L4+AF2)+(A+A)L8=Ax~6+~~~.

1 -


Hence 12 - &2”+E has the leader zero. Any covariant of order

&j = 4.n + 2 > 2 diJ$ers fTOV2 a linear co?nbination of the co-

variante (65) and Qz “+E by a covmiant whose leader is zero.

19. &?gukM and Irregular copa&&; Rank.-A covariant shall be called regular or irregular according as it has not or has the factor La, given by (43). The quotient of an irregular covariant by Ls is a covariant. Hence the determination of all irregular covariants reduces to that of the regular covariants. If a covariant has a linear factor it has as a factor each of the seven ternary linear functions incongruent modulo 2, whose product is LS. Hence a. regular covariant has a non-vanishing component involving only 21, 53. In a regular covariant C without terms z$’ (i. e., with leader zero), this component has the factors zl, 5 and (by the covariant property) also Al+ x3. The product of these three linear factors was denoted by & in (40). Let tSm be the highest power of .$ which is a factor of the component and let n be the degree of the quotient in the x’s, Then C may be given the notation

(76) &a,* = &fit? + xlx2x3:ad,

where, if n = 0, fz is a function of the a’s and b’s not identically zero, while, if n > 0, f2 is a function also of XI, x3 in which the coefficients of xrn and 23” are not zero; fi is a function of x2, x3; f3 of Xl, 52.

The regular covariant (76) shall be said to be of rank m. In an irregular covariant the component free of x2 is zero and hence is divisible by an arbitrary power of &; it is proper and convenient to say that an irregular covariant is of infinite rank.

Any covariant of rank zero differs from one of rank greater than zero by a polynomial in the known covariants

(77) A, A, J, F, 4 Q2.

This is a consequence of the theorems in Q§ 11-18, where the polynomial is given explicitly. Any product, of order w in the


z’s, of powers of the covariants (77) can be reduced by means of the syzygies

JL = 0, AL2 = AF, (A + A + J + l)(FL + K) = 0,

(78) AK = 0, FL2 + (A + A)L4 + AF2 + AQ, = LK,

FS + QzF = L3K + (A + JW + (A + 1)LG + (-4 + l)Q,,

to a sum of covariants of order w given in 00 11-18 and a linear function, with covariant coefficients, of K, &I and

G = Q2L -I- L6 = x2[@3@1+ 1)2s2 + (h/33 + 1)x351

(79) + PlcB3 + US121 + w253[(Pl+ P2 + P3 + 1)

x (51x2 + 21x3 + z2:253) + z@i + lh?].

Here G and K, given by (42), are of rank 1, while &I= tt2+x2 ( ) is of rank 2. As this theorem is not presupposed in what follows, its proof is omitted. However, it led naturally to the important relations (75) and (79) and showed that no new combinations of the covariants (77) of rank zero yield covariants of rank > 0, a fact used as a guide in the investigation of the latter covariants.

REGULAR COVARIANTS RI,,,,, 00 20-22 20. A separate treatment is necessary for covariants (76)

withn= 0. Then each fi is a function of the coefficients ui, bi. Since the factor E3’” of the partf3t3(11 of &o free of 23 is unaltered by every linear transformation on 21 and ~2, f3 is a linear combination of the functions (49) and their products by b3. Also, fa must be unaltered by

(80) zl = a$ + ~3’: al’ = al + ~3, ba’ = b3 + bl + ~2.

Both conditions are evidently satisfied by the ternary invariants and by us and q, in (47). In view of (53), we may employ

AJ, J, 4, A, USJ, q4 A to replace in turn

b3wza3q, b 3ala2a39 &j# j, a3b3% ala2a3qt wff2h

- INVARIANTS AND NUMBER THEORY. 87

since a term previously replaced is not introduced later. Thus fs is a linear combination of these seven functions, aa, q, asq, and

wa2, ww, h, hat, be, b3ala2, kww, b3j, bmj.

Give to any linear function m~cv~acz + * - - of these the notation

Call e the increment bl + a2 to ba in (80) and employ e to eliminate bl. Then u is unaltered by (80) if and only if

cue = 0, aa3 = 0, /?u3 = ye (mod 2).

Since ba does not occur in q or j, nor al in q, we have

(Y = mscw2 + m?ar2q + m8(e + u2 + a3) + ms3(e + a2 + as).

Thus eye = 0 gives me = mr = 0, ms = me. Then aas = 0 gives m = 0. Now

B = mla2 + m2a2q, y = ma + m4a3 + m5q,

and flag = ‘ye readily gives Q = 0. Any function of b3 and the invarianta (49) off and 1, which i.~ unaltered by (80), is a linear combination of the ternary invariant8 (45) and a3, q, a3q, aa, asJ, qA.

21. For n = 0 and m even, there exists a covariant (76) in which fs is any function specified in the preceding theorem. For, if I is any ternary invariant, I&r”/” has f3 = I. By (42) and (41), I-P and PI2 are of the form (76) with fa = a3 and /392 + 1, respectively; they may be multiplied by any invariant. By (19) and (47), we have

(81) /W2 + 1 = q + ad + A + 1, eq = aA + qA + a3J.

Hence we obtain q, then qA, qA, and therefore arq. Any covariunt with n = 0, m even, differ8 by an irregular covariant from a linear function of

IQf’$I#“, MP’~2 (I=l,A, A, J, AJ; L=l,A, J; &=l,A, A).


22. For n = 0 and m odd, we may delete the terms aJ1 from f* by use of IlK”. First, let m = 1 and apply transformation (51) ; we get

(82)

41 = tl'+ 4st, 52 = (2', h = s,

R'=fl&'+fib' + dfi +fdEa'+ (zlk2'd+ d2z2')+.

Thus + = 0. Since fs = I + Lq, condition fi + fa = fs’ gives

I = I2(& + a2b2 + a&8 + ws + alad.

Add to this the relation obtained by permuting the subscripts 1, 2. Thus

0 = I2@1 + b2 + a2m + am).

The increment under (22) is I& + aa + a2cx8) = 0. Now I2 is of the form z + PA + z,A, where z, y, 2: are constants. From the terms in b&, we get y = 0. Then z = z = 0. The only oova~nte are therefore LK.

Second, let m > 1. Then KW(-1)/2 is of the form (76) with fs = arq + aa, by (811). Hence we may set

fs=I+cq+&A (c, d constants).

In R given by (76), let g denote the coefficient of

(83) 3w2:22a * xpxp@+.

In the function derived from R by the transformation (51), the term corresponding to (83) has the coefficient g + fi, since by (82) the & parts contribute only one such term, that from f G”-lEs’. Now

fi = I + Cq‘ + 41’4 Q’ = b2b3 + @2 + hi)W.

When g is given the notation (56), g’ - g = fi is the function (57). But a& occurs in fi only in J and AJ and in them with the linearly independent multipliers (62). Hence

I = nl(A + 1) + mA.

The coefficient of aa in fi is now

nlala2 + n&w2 + bi3) + dQIcw2 = p(h + az>.

.l_l_-^_____ll--l_____ - J .-

INVARIANTS AND NUMBER THEORT. 89

Taking ba = as, we see that nl = nz = d = 0. Thus fr = cq’. By (57) for al = 0, bs = a2, we get c E 0. Any copariant wit6 ta = 0 and m oo?d di$ers by an irregular covariant from a linear fun&m of Km, AK@‘, JK” and, if m > 1, KW(-l) D.

COVARIANTS OF RANK UNITY, $5 23-26

23. Henceforth let m > 0, n > 0 in (76) and set

(84) fi = Szg + 512pz1+ s$lpi$ + * * l (S 9 0) .

Since S is unaltered by the group I? of 0 15, it is a linear combination of the functions (71). We may omit the functions az(ba + 1) and Aas(ba + l), since K”L” is of the form (76) with S = az(bs + 1). Thus

(85) S = I+ a& -5 bJ2 + hala2 + k2bsalcuz+kB+Sa28+lcsAB,

where I is any invariant, II a linear function of 1, A, J; I2 one of 1, A, A, J;ewhile /3 = bl(bs + ~2~).

First, let m = 1. If T and B are the coefficients of x$’ in f3 and fi, transformation (51) replaces the covariant (76) by a function in which, by (82), the coefficient of 21’x~~~+~ is

036) T + B = T’,

where T’ is derived from T by the induced substitution (50). But T is obtained from S by the interchange [23] of subscripts, and B from T by [13]. We thus find by (86) that

I = b& + (kl + k&t)(al + am)

+ k&&l + ah + a.& + am + am>

+ k&&h + aabs + alas + aza3).

Let Z be the sum of the second member and the function obtained by applying (u2aa)(b2bs) to it. .InX=O,setb2= ba;weget

{kl+k(+b3(k2+lea))(a2+as)crl=O, h=kl, k4=k2.

Then Z = 0 may be written in the form

(bz + b& = 0, X = 12 + kz(A + A + 1).


As in 0 18, X = 0. Thus Is and I are the products of A + A + 1 by ks, kl, so that

(87) S= (h+Ws)@+A + 1) +a2Il+k1(b1ba+blcvz+a1cun)

+ knbdazh + ~2) + kd(& + bl).

For n odd, S is the increment to SI under (50) and hence has no term containing a&l. If t is the coefficient of J in II, a& occurs in (87) only in tazJ and in the final part, being multiplied by fa?cu& and kgxla2(ba + l), respectively. Hence t = ka = 0. Since S is of the form (57), the coefficient of by must vanish if a1 = 0. Thus

kl(ba + 4 + kzbrae = 0, kl= k2 = 0.

Now S = a2Il = as(u + VA) must vanish for al = 0, bs = az by (57) ; then A = az(b2 + us), so that u = v = 0, S = 0. Any covariant with m = 1 and n odd &few from on% of rank > 1 by a linear function of KL”, AKL”.

24. For m = 1, n = 4v, we may delete ~11 from (85) by use of 11KQ2”. Set fi = Bs” + - l l + &,~a”. Then (51) replaces (76) by

R’ = ~2[S&’ + S~X~‘-~XI + (SI + S2‘3)~3”-~ct5~ + - . -1 + &fa

+ (cl+ Esmd~a"+ av-%'+ -*->

+ Bn-lz2(ar1+ ar2cc1+ ***>I + cw2nl+ Z123J2w.

Since S1 is the increment of SZ, it is a linear combination of the functions (74). By use of Lw8Q1, LnssKa and their products by A and A, we may, without disturbing S, delete from SI

ba+alcurr+l, Ah, A+bd+hwz, a&+1), wWa+l).

Hence we may set

SI = t&s + 4 + Mwm + t&a + t& + 4J. Applying (arap)(blb2) to S and &, we obtain B, and B-I.

Let 1 be the coefficient of x2xan+ in c$, By the coefhcient of

IN-VARIANTS AND NTJMBER THEORY. 91

x~x~q - x2xP in R’, we have

B,+ B,r+l= 1’.

For 1 given by (56), B, + B ,+I is given by (57). By the coefficient of a&, we get ta 5 0. The coefficient of a3 must vanish for ba = a2. Hence

km + (k2 + t2)wzz + kgzlcuzbz = 0, kl= ks = 0, ts = ks,

S = kzW + A + 1 + a1cuz + usbd.

The coefficient of kz equals that of 52x3” in

GFQ$‘-’ + AKL” + AKQ2y.

Any covariant with m = 1, n = 4v, di$ers from one widh m 2 2 by a linear fun&m of KL”, AKL”, IKQZ”, GFQ2”l (I = 1, A, J).

25. For m = 1, n = 4v + 2, we may delete a211 from S, given by (87), by use of IlQ2’M. The coefficient of f2xan in Q2”G is

d = B&-h + 1) = A + (bl + Ww2 + ba) + haam.

The coefficient of kl in S equals d + Q,A + az(ba + l), the final term of which was reached in 0 23, and a& above. The coefficients of ks and kz in S equal Ad and

bM~l+az)+bd~zbz+ ws+azaa+az) =Ad+dJ+l)+az(ba+l),

respectively. Any copariunt with m = 1, n = 4v + 2, d$ers from one with m 2 2 by a linear function of KL”, AKL”, IQS’G, I1Q2”M (I = 1, A, A; II = 1, A, J).

For use in 0 26, we replace QsVM by Qz’FK, noting that

@a M = (F + L2)K

and that QvL2K ditfers from KL” by a covariant of rank 2.

26. By the last four theorems, any covariant of rank 1 differs from one of rank 2 2 by CK + DG, where C and D are known covariants of rank zero. Taking as CI and D1 arbitrary func-


tions of the proper degree in the z’s, of the generators (77) of covariants of rank zero, I found the syzygies needed to reduce CrK + DIG to an expression differing from the above CK + DG by a covariant of rank 2 2, in which those of rank 2 are linear combinations of lP, KG, 62, W, Q1 and the new one

(89) V = GP + AQzG + (A + J + 1)QzFK

where + ALsKz + ALa& = &2x&% + * - -,

(90) v = a2 + b3(1 + UlcY2).

The only new syzygies needed for this reduction are

LG=QzL2+L6= W, FLK=AW+AQl+(J+l)P,

(91) (P + L4 + QX = (A + W3,

(A + l>(FG + a4 + KQd + JKQz = AL&l + wL3,

in which w is an invariant not computed. Proof need not be given of these facts since we presuppose below merely the existence of relation (89) which may be verified independently. Of course, the fact that V is the only new covariant of rank 2 was a guide in the later investigation.

COVAHIANTS OF EVEN RANK m = 2~ > 0, $527-29

27. First, let n be odd. In the covariant (76) replace 58 by x3 + XI. In view of (82), we get

R’ = fi’fz” + f353’” + fi’(t? + t32>” + (x1x223 + x12x2)4’.

Using the notation (84) for f2, we have Sr’ = Sr + S in f2’. Thus, as in 5 17, S is a linear combination of the functions (74). Now QrPLn and its products by A and A + A are covariants (76) with S given by (63). Using also K”L”, in which S = ~3@3+l),

and its product by A, we may set

8 = kl(b3 + a2) + k2b3ala2 + k&z + kr(bs + an)J.

In x1x2x3$, let g be the coefficient of

X1X2x3 * X22p-1X34w+n’2 = (X22X34) “X3”+xl.


Such a term occurs in neither of the first two parts of R’, since they are functions of only two variables. To obtain such a term from the third part of R’, we must omit terms with the factor Es2 (and hence s12) and take (~2s3~)~“ in [12”, so as not to make the degree in x2 too high. Hence if T be the coefficient of zan in jr, g’ = g + T. Now (ala2)(bJ4 replaces S by T. The resulting T must be of the form (57). By the coefficient of U&I, kq = 0; cf. (72). By the coefficient ktcxlbs of as, ks = 0. Since T = 0 for al = 0, b3 = a2, we get kl= k2. Hence S = kg, where v is given by (90).

For n = 1, f2 = 8x3 + &Xl. Thus S1 = klv’, where v’ is derived from v by interchanging the subscripts 1 and 3. Then SI’ = S1 + S gives kl 3 0.

For n 2 3, Q1@-1Ln-3V is of the form (76) with S = v, since #f&v = 0.

Any covariant with n odd, m = 2p > 0, differs from one of rank > m by a linear combination of IQlwLn (I = 1, A, A), K”L”, AKmLn and, if n > 1, Q1u-lLn-sV.

28. For m = 2~ > 0, n = 4v > 0, the coefficients of ~Z*XS” in

(92) QI~&P~, Km&z", &?I-“, Q,‘L”, K”‘L”,

Qf-1Q2-1G2, K-2Q$‘-W

are respectively

1, a2, bt, Pa + 1, a2@3 + 11, d = P3@3 + 11, a2d.

These may be multiplied by any invariant. Now

133 + 1 + a2 + b3 = w2,

AU% + 1) + (A + A + l)b3 + ha2 + A = b3ala2,

d + A + 83 + a2@ + b3) = br(b3 + ~2) = 8,

ad + a2b3 = nzblbo = a& Ad = Abl(b3 + 1) = A@.

Hence we have a covariant (76) in which the coefficient of [2"%3*

is any linear combination of the functions (71). Hence ths

94 THE MADISON COILOQUIUM.

covariant differs from ooze of rank > m by a linear function of the covariante (92), the producte of the first three by any invarfuti except 1, the products of the fourth and jifth by A and the product of the eixth by A.

29. For m = 2~ > 0, n = 4v -I- 2, the coefficients of &“x$’ in

(93) MK”‘+Qs’, K”‘L”, GK-lQ$‘, Fn’2Q2’, L”QP

are respectively

a, az(ba + 11, aMb1 + 11, h, b3 + am + 1.

Linear combinations of products of these by invariants give*

a2, aA azJ, apba, Aazb,, add,, Iba, am, A + bma.

Since S and SI are unaltered by the group I’ of 0 15, they are linear combinations of the functions (71). Deleting the above functions a2, a4, - - - from S, we have

S = I + c/3 + e&3, B = bdbs + m),

where c and e are constants, and I is an invariant. Set

fl = BZ~” + BlX‘pX~ + * - * + Bn-lx*x$--l + Bnx3n,

and call u the coefficient of .

w X1X2& * x~4*+~~xp-1 = (xgxpxpxl

in x~x~x&. The coefficientt of (94) in R’ of 0 27 is BI + u. Hence

u’ - u = BI,

if (50) replaces u by u’. Thus B1 must be of the form (57). For n = 2, Sz is derived from S by applying (alaJ(blba).

Then (672) gives &. Applying (am)(bdd to Sl, we get

B, = I + c(b&s + bzcvl + bed + eA(bh + ba + U.

* For the last two, use the first two of the four equations in Q 28. t The first part of R’ is free of 21, the second of z8, while in the third psrt

@ hag the factor ~9, and in fl’@r there is a single term (94) and it has the caefficient BI.


Since this must be of the form (57), we get I = 0, c = e = 0. A covatiant with m = 2p, n = 2, di$ers from ooze of rank > m by a linear function of

iMK”‘+, K”L2, AK”L2, GK”-‘, IF&l“, L2Qic, AL2Q1”

(i = 1, A, J; I = 1, A, A, J).

For n > 2, we may delete A from the part I of S by use of E&I~&~-~, where E is given by (75). Without disturbing S we may delete as(ba + 1) and its product by A from &‘I by use of K!+tlLn+, since the term of t2”“f2 with the coefficient S1 is the term of highest degree in za in f22P+1(S15P + - * -). Since S + Sl is a linear combination of the functions (74),

(95) Sl = S + tdba + ad + tzam + t&ma2 + t&d + t&-a

+ tA(ba + 1) + to@3 + 4J.

Apply (alazas)(blb&~) to Br, of the form (57). Hence

(96) $1 = pm + pa&2 + pazp + m + SP, P = bl + e.

Now a&s occurs in S only in the terms J, AJ of I and in the part of (95) after S only in the last term, given by (72). In these the factors of a&t are linearly independent. Hence to = 0, I = s(A + 1). The coefficient of al in Sr must vanish for br = as, and 6’1 itself if also ua = 0. Hence

c= t2 = 2, t1 = ta = t* = t, ta = x + t,

Sl = %(A + 1+ Ma + ha2 + ala2 + Aad + eAbl(ba + 1)

+ t(b3 + aft + hala + &A + bb + a&.

Call E the coefficient in z1x2z8~ of

XlZ2X.9 * xpx$‘+“-a = (xgxa4) kT1x$ITp.

In R’ of 0 27, the coefficient of this product is e + B-1. Hence L&-r is of the form (57). Interchanging the subscripts 1 and 2 in B-1, we get Sr. Thus the coefficient of aa in Sl vanishes for bs = al. Hence S = Sl = 0. Any covariant with n > 2 difTer8


from one of rank > m by a linear combination of

iMKnrlQ~“, jKmLn, GKhlQzy, IFn12QlM, jiY&l’, EQ1@&2)‘1

(i = 1,. A, J; j = 1, A; I = 1, A, A, J).

COVARIANTS OF ODD RANK m = 2~ + 1 > 1, $6 30-31

30. Replacing 23 by 23 + 51 in the covariant (76), we get

R’ = fa’[zm + f3tsrn + fi’(& + Es>” + (x1x2x3 + x12x2)4’.

In 2122x34, let g be the coefficient of (xlx22)(x22x3)“‘+x2n. The coefficient of the corresponding term of R’ is g’ = g + B, where B is that of ~2” in fi. Hence B is of the form (57).

First, let n be odd. Then &‘I’ = S1 + S under (50), so that S is a linear combination of functions (74) with a2(b3 + 1) and its product by A deleted (0 23). Thus S is the sum of the terms (95) after the first. Applying (ala2as)(blb2b3) to B, of the form (57), we see that S is of the form (96). By these two results,

S = t(b3 + a2 + baalcvz + &A + b3L\ + ad).

If I is the coe5cient of (z2z#%!3’L-1z1 in z1%2z3+, that in R’ is I?’ = l+nB,. Hence, for n odd, B, is of the form (57). Inter- changing the subscripts 1,2 in B,, we get S. Thus the coefficient of aa in S vanishes for bs = al, so that t = 0. Any cmariand with m and n odd di$eTs from one of rank > m by a linear function of K*P and AKmLn .

31. Finally, let m be odd and n even. According as n = 4u or 4~ + 2, KmQ2” or K”‘+MQ2’ is of the form (76) with a2 as the coe5cient of fzrnx3*. Hence we may delete the terms a211 in (85) and hence the terms aJr in B of $! 23. But (0 30), B is of the form (57). Now a3bl occurs in J and AJ of I and in b2J of b212, having in these linearly independent multipliers. Hence

I =x(A+l)+yA, I,=e+fA+gA.

Since the coefficient of a3 in B shall vanish for b3 = a2, and B itself if also al = 0, we get kl = x = y = k3, k2 = f = g = 8.


Thus

(97) S = x(A + 1 + A + alcvz + b&3 + l&g) + kraibrba

+g(A+l+A+ adba + kdh(be + 1).

First, let n = 4~ + 2 and write 2~ + 1 for m. Then

GQl'Qc', IPGQQ’-~Q~’

have d = @a(& + 1) and atd as the coefficients of &?x$‘. As in 0 25, the coefficients of x, kq, g, k5 in (97) equal respectively

d + a@ + b3 + l), a2d + azba, Ad + a& + a2J, Ad.

The terms not containing d are combinations of the above a211 and a2(b3 + 1) of 5 23. Any covuriunt with m = 2p -I- 1 > 1, n=4v+2,d$ f i ers ram one of rank > m by a linear function of

iK”L”, 11Km-IMQ2’, IGQl’Q$‘, K2GQPQ2”

(i = 1, A; 11 = 1, A, J; I = 1, A, A).

Next, let n = 4~ > 0. In the last two covariants of the theorem below, the coefficients of f22P+1x+” are a&& + 1) and a = b&@r + 1). We had reached covariants in which the corresponding coefficients are ~221 and c&b8 + 1)I. Thus we obtain the coefficient of kq in (97) and 6 + Aa2bs + ablb3, which equals the coefficient of g. We may therefore set kq = g = 0. Subtracting covariants of the fourth and fifth types in the theorem, we may take as S1 the function in 0 24, without disturbing S. Applying (a&(blb2) to S and S1, we get B, and B ,,-I. If b is the coefficient of x~x~~+~x~“+” in xrxsx&, its coefhcient in R’ of 0 30 is 1’ = I+ B, + B-1. Thus B, + B,,-1 is of the form (57). By the coefficient of a3bl, t4=0. Since the coefficient of a3 is zero for b3 = a2, we get z = k5 = ta = 0. Thus S= 0. Any covariant zoith m = 2p + 1 > 1, n = 4v > 0, di$ere from one of rank > m by a linear function of

IPL”, AK”‘L”, IK*Qz”, iLLR3Q1@+I, iL-3K2r+2, G2KQ2-1Ql”--l,

FGQ2-1Q~U (i = 1, -4, A; I = 1, A, J).


32. We have now completed the proof of the theorem: As a fundamental system of modular eovariants of the ternary

quadratic form F with integral coejbienls mod& 2, we may take

F, its invariants A, A, J, its linear covariant L, 6% “polar ” cubti covariant K, and the universal covarianti Qr, Q2, La.

Incidentally, we have obtained a complete set of linearly independent covariants of each order and rank. We might then find a complete set of independent syzygies. Syzygies whose members are covariants of low rank are given in (78), (88), (91).

33. References on Modular Geometry.-Gther aspects of the modular geometry of quadratic forms modulo 2 and, in particular, applications to theta functions have been considered by Cable.* For a treatment of non-homogeneous quadratic forms in x, y modulo p (p an odd prime), analogous to that of conies in elementary analytic geometry, but employing only real points on the modular locus, see G. AFnoux, Essai de GBom&rie analytique modulaire, Paris, 1911. The earlier paper by Veblen and Bussey was cited in 0 7. The paper by Mitchell was cited in 8 3. Appli- cations of modular geometries have been made by Conwe1l.t

The problem of coloring a map has been treated from the standpoint of modular geometry by Veb1en.S

* Transactions of Ihe Ammimn Ma&m&cd Society, vol. 14 (1913), pp. 241-276.

t Ann& of Mathematics, ser. 2, vol. 11 (lQlO), pp. 66-76. $ Ann& of Mathematics, ser. 2, vol. 14 (1912), pp. 86-94.

LECTURE V

A THEORY OF PLANE CUBIC CURVES WITH A REAL INFLEXION POINT VALID IN ORDINARY AND IN MODULAR GEOMETRY

1. Normal Form of Cubic.-Consider a ternary cubic form C(Z, y, z) with coefficients in a field F not having modulus 2 or 3. After applying a linear transformation with coefficients in F and of determinant unity, we may assume that (1, 0, 0) is an inflexion point. In particular, C lacks the term z3. If it lacks also $2 and $2, its first partial derivatives vanish for y = z = 0. But we shall assume that the discriminant of C is not zero. Hence the coefficient of 9 may be taken as the new variable y. At the inflexion point (1, 0, 0) the tangent y = 0 is to be an inflexion tangent, i. e., meet the cubic in a single point. Hence C lacks the term ~9. Thus

c = x3/ + way2 + By4 + 4(y, 2).

Replacing z by z - cuy - pz, we see that x!$ is now the only term involving Z. If y were a factor, the discriminant would be zero. Hence the term z3 occurs. Adding a suitable multiple of y to z, we get (1) C = x2y + gy3 + hy2z + 6z3 (6 =I= 0).

2. The Invariants s and L-The Hessian of (1) is

II= - 36Sz - h2y3 + 9SgyZz + 36hyz2.

The sides of an inflexion triangle form a degenerate cubic belonging to the pencil of cubits kC + H. The latter has the factor z only when k = h = 0 and the factor y - lz only when kl = 36 (as shown by the terms in $), where k is a root of

P + 186hk2 + 108cS2gk - 27a2h2 = 0.

Before considering the factors involving x, we note that the 99


coefficients of this quartic equation are the values which relative invariants of a general cubic assume for the case of our cubic (1). Indeed, a linear transformation of determinant unity which replaces C by a cubic C’ must replace H by the Hessian H’ of C’, and hence replace the inflexion triangle of C given by a root k of the qua&c by that inflexion triangle of C’ which is given by the same number k. We denote the invariants by*

(2) S= - 36/z, t = - 108Pg.

The above quark now becomes

(3) k4 - 6sk2 - tk - 3f = 0 .

The discriminant A of C is such that

(4) 27A = P - 64~~.

There are four distinct roots of (3) since its discriminant is - 273A2 .

Our earlier results are that kC + H has the factor z only when k = s = 0 and the factor y - 36k% if k is a root =l= 0 of (3). It has the factor x - ry - pz if and only if

3p2 = k, 9a2kr2 = a2 + tk/12, kp2 - 66~7 = s,

66kpr - 962r2 - sk - t/4 = 0.

These conditions are satisfied if and only if k is a root of (3) and

p = k = 0, 36J2r2 = - t (k = 01,

3p2 = k, 6Skr = p(p - 38) (k + 0).

3. The Four Injiexion Triangles.-First, let a = 0. Then t+Oby(4). Th e root lc = 0 gives the inflexion triangle with the sides

(5) z = 0, 2 = f 71y (366e,,2 = - t). .-

* Wehaves = - 345, t = - 3eT, where S and T, given in Salmon’s Higher Plane Curves, p. 189, are the invarianta of the general cubic with multinomial coeflkients.

INVARIANTS AND NUMBER THEORY.

Each root of k3 = t gives an inflexion triangle

101

Y’$, x= ‘=n(Yf$) (3.366W=i),

Next, let a =I= 0. Each root of (3) gives an inflexion triangle

(7) x=f k

4 - ‘+ k2 - 38

3 66k

4. The Parameter &-If we multiply x, y, z by p, p+, p, we obtain from (1) a cubic with 6 replaced by 6p3. If F is the field of all complex numbers, the field of all real numbers, or the finite field of the residues of integers modulo 3j + 2, a prime, every element is the cube of an element of the field [in the third case, e 5 (e-J)3], so that the parameter 6 may be taken to be unity. Although we do not use the fact below, it is in place to state here that for all further fields a new invariant is needed to distinguish the classes of cubits (1). Indeed, two cubits (l), with coefficients in F and with the same invariants a and t and discriminants not zero, are equivalent under a linear transformation with coefficients in F and having determinant unity if and only if the ratio of their 6’s is the cube of an element of F.

CRITERIA FOR 9, 3 OR 1 REAL INFLEXION POINTS, $0 5-9

5. Infition Points when a = O.-Let K be a fixed root of k3 = t. Let ~1 and 72 be fixed roots of t.he equations at the end of (5) and (6). Then

(71/72)2 = - 3 = (1 + 2@)2, 3 + w + 1 = 0.

Choose w so that 71/72 = 1 + 2~. Denote the lines 2: = 0, x = 71y, x = - rly in (5) by L1, L2, La. For each value of i = 0, 1, 2, denote the three lines (6) with k = uoi by LG, Ls, Lsi, that with the lower sign being L3i. Then the 9 inflexion points and the subscripts of the 4 inflexion lines through each are given in the following table:

11

12

THE MADISON COLLOQUIUM.

(n, 1, 0)

1

20

21

22

(--72, 1, 0) (n,l, $)I( -d$) 1 2 3

30 li li

31 2,i- 1 2,i-2

32 3,i-2 3,i--l

In the last two columns, i has the values 0, 1, 2; whilei - 1 or . z - 2 is to be replaced by the number 0, 1, 2 to which it is congruent module 3.

When F is the field of all real numbers, K may be taken to be real, while just one of the numbers ~1 and 72 is real. Hence 3 and only 3 of the 9 inflexion points are real. The same result is true if F is the field of the p residues of integers modulo p, where p is a prime 3j + 2 > 2. For, K may be taken to be integral (0 4), while w is imaginary and hence - 3 is a quadratic non-residue of p. If - t is a quadratic residue, 71 is real and 72 imaginary. If - t is a non-residue, the reverse is true.

Next, let p = 3j + 1, so that w is real and hence - 3 a quadratic residue. By (5) and (6), 71 and 72 are both real or both imaginary according as - t is a quadratic residue or non-residue of p. Hence all 9 inflexion points are real if and only if - t is both a square and a cube and hence a 6th power module p. If - t is a square but not a cube, only the first 3 inflexion points are real. If - t is a quadratic non-residue, (1, 0, 0) is the only real inflexion point.

A cubic with integral coeficients taken modulo p, a prime > 3, with at least one real inflexion point and with invariant e = 0 and inva.riant t + 0, haa 9 real injlexion points if p = 3j + 1 and - t ti a sixth power module p, a &gle real in$exion point if P = 3j + 1 and - t is a quadratic non-residue of p, and exactly 3 real inJEexion points in all of the remaining cases.

For example, if p = 7 and s = 0, t + 0, there are 9 real inflexion points only when t = - 1. Taking 6 = 3, rl= - 2,


l,K=- 1, we get o = 2. Thus a+$ - yJ + 328 = 0 has the 9 inflexion points (1, 0, 0), (1, 1, 0), (- 1, 1, 0), (-2,1, 3 * 23, (2, 1, 3 * 2i) (; = 0, 1, 2).

6. Inj?.eti Points when s =I= 0, A $: O.-These are (1, 0, 0) and

(9)

where k ranges over the roots of the qua&c (3). We seek the number of real roots k for which m is real. In order that the left member of (3) shall have the factors

(10) k2+wk+l, k2-wk+m,

it is necessary and sufficient that

(11) 1+ m - w2 = - 68, (I - m)w = t, lm = - 3a2.

Let t + 0 (for t = Osees9). Thenw+Oand

(13) 21= w2 - 6s + t/w, 2m = w2 - 69 - t/w.

Inserting these values into (ll~), we get

(13) W6 - 12sw4 + 4892w2 - t2 = 0.

Set w2 = y + 4s. Then

(14) y= = t2 - 64sa = 27A.

For the rest of this section, let the field be that of the residues of integers module p, where p is an odd prime 3j + 2. Since any integer e has a unique cube root e-j modulo p, there is a single real root y of (14).

First, let y + 49 be a quadratic residue of p. Then w is real and hence also 1 and m. The product of the discriminants of the quadratic functions (10) is seen by (111) and (11~) to equal

(15) (d - 4Z)(w2 - 4m) = - 3(~? - 48)2 = - 3y2

and hence is a quadratic non-residue of p. Thus a single one of the quadratics (lo), say the first, has a disc&&ant which is a

-- -.---- -----.- _.._. --_-_.c^__-.“._“_ _“..-l-,-“-._~ ___. ---- ~..- ~- -.


quadratic residue and hence has real roots. By (121),

41(ti - 4&t+ = - 2we - 6w3t + 36sw4 - 4t2 + 48dw - 144.#w2.

Adding the vanishing quantity (13), we see that

(16) 4l(w” - 4l)w2 = - 3(w3 - &SW + ty.

Since w2 - 41 is a quadratic residue and - 3 is a non-residue of p, it follows that 1 is a non-residue. Hence a single one of the roots of the first quadratic (lo), and hence a single one of the roots of the qua&c (3), is the negative of a quadratic residue. Thus just two of the inflexion points (9) are real.

Next, let y + 49 be a quadratic non-residue of p. Then there is no factorizat.ion of the quartic (3) into real quadratic factors. Nor is there a real linear factor k - r and a real irreducible cubic factor. For, if so, the roots of the latter are of the form X, Xp, X** (cf. the first foot-note p. 37). Then

(r-X)(r-X*)(r-Xp’), P=(x-XP)(xP--XP”)(XP*-X)~Ppp (modp)

are real, so that the discriminant of (3) is a quadratic residue. But this discriminant was seen to be - 3(81A)2, and - 3*is a non-residue. Hence (3) is irreducible modulo p. Thus (1, 0, 0) is the only real inflexion point.

For p = 3j + 2 > 2, a cubic (1) with stA =I= 0, )LLY exactly three real &flex&m points or a single one according aa the real number 3A4 -I- 49 W a quadratic residue or non-residue of p.

7. C&ic with &A =I= 0, p = 3j + l.-Now - 3 is a quadratic residue of p and there are three real cube roots 1, o, w2 of unity modulo p.

In this section we shall assume that A is a cube modulo p. Then there are three real roots yi of (14). At least one of the yi + 49 is a quadratic residue of p since

a (yi + 48) = J/l3 + MS3 = t2.

If yr + 49 is a quadratic residue, while y2 + 49 and y3 + 48


are non-residues, there is a single factorization of quark (3) into real quadratics (10) and hence certainly not four real roots. The product (15) of the discriminants of the real quadratic factors is now a quadratic residue of p. If each were a residue, there would be four real roots. Hence each is a non-residue and there is no real root. There is a single real in$exion point if p = 3j -I- 1, &A + 0, A is a cube, and if the three values of 3A*$49 are not all quadratic residues of p.

Next, let each yi + 4.9 be a quadratic residue of p. Then there are three ways of factoring quartic (3) into real quadratics (10). But a root common to two distinct real quadratics is real. Hence all four roots are real. The discriminant of each quadratic (10) is therefore a quadratic residue of p. Hence, by (IS), I is a quadratic residue of p; similarly for the constant term of each quadratic factor. Thus the negatives of the four roots are all quadratic residues or all non-residues.

To decide between these alternatives, we need the actual roots. In z0,2 = yi + 43, let the signs of the wi be chosen so that

k2 -Wik+??Zi= 0

have a common root. As in (12),

2mi = Wz - 68 - t/Wi.

(i = 1, 2, 3)

For e =l= 1, we find by subtraction and cancellation of WI - we that

2k = Wl + we + tl(wJc>.

Comparing the results for e = 2 and e = 3, we get

(17) WlW2WQ = t.

Hence* the roots of (3) are

(18) 3(wl+ w2 + w3), 6<w1- w2 - w3),

3(- w+ w2 - WI, SC-- Wl - w2 -I- w3).

The product of the first and (i + 1)th roots is seen to equal mi

* In particular, we have deduced Euler’s solution by the method of Descartes.

- 106 THE MADISON COLLOQUIUM.

and hence is a quadratic residue. For given values of p, 8, t,

we can readily find by a table of indices the real values of the wi

and thus a real root and hence decide whether or not it (and hence each of the four roots) is the negative of a quadratic residue,

However, changing our standpoint, we shall make an explicit determination of all sets a, t for which the quartic (3) has four real roots each the negative of a quadratic residue of p.

By the definition of the w?, or direct from (13),

(19) zW12 = 128, ~W&U~~ = 4882, W12W22W32 = t?

Let w be a tied integral root of o2 + w + 1 E 0 (mod p). Then

o = (128)2 - 3(4882) = 2w14 - Zw?w22

= (WI2 + ww22 + dwa2) (WI2 + fJ?w22 + wws2>.

Interchanging w2 and wa, if necessary, we have

(20~ WI2 + cdw22 + w2w32 3 0 (mod P>.

Conversely, if the w? are any quadratic residues satisfying (20) and if we define 9 and t by (19J and (17), we obtain a qua&c (3) with the four real roots (18). If we permute ~01, wa, 208 cyclically we obtain solutions of (20) leading to the same 8 and t and to the same four roots (18).

Our first problem is therefore to find all sets of solutions of (20). To this end it is necessary to treat separately the cases - 1 a quadratic residue and - 1 a non-residue; viz., p = 12q+ 1 and p = 12q + 7 (since already p = 3j + 1).

First, let p = 12q + 1. Then - 1s zY (mod p), where i is an integer. Set

2p = 201 - iwwa, 2c = WI+- iuw3.

Then (20) becomes

4pu = - cow22 = (i&w#,

so that pu must be a quadratic residue. Hence we may take


u = pF, where p and 1 are integers not divisible by p. Then

(21) Wl = p(l + P), wq = 2iwpz, wg = ?Yp(l - I”).

We must exclude the values of I which lead to equal values of two of the wi2, and hence to equal yi)s, since the roots of (14) are incongruent. Now if any two of the w? in (20) are congruent, all three are congruent. But wr2 = w22 implies

1+12= *2iwZ, (Z=~=iw)~=w~, I= +b+eew2 (8s 1).

The values 12 = 0, f 1 make one of the wi = 0. Hence we must exclude the 9 incongruent integral values of I:

(22) 1 = 0, f 1, f ;., w2 f iw, - 02 f iw.

Using the values (21), we get

(23) 129 = p2{(1 - w)(l + Z4) - 6w2P}, t = 2p*Z(Z4 - l),

(24) Hwl+w2+w) = $Pu+ioa)(l+$&)p.

To make the negative of the last a. square, we must take

(25) p = - 2(1 + S)r2 (r + 0).

Now s, given by (23), is zero only when

W-9 l=wzt.2, -wzt&$.

The desired sets s, t are given by (23) and (25), where r is any integer not &tXble by p, while 1 i.9 any one of the p - 13 positive

integers < p not congruent m0h.410 p to one of tke 13 incongrwmt integers (22), (26). The minimum p is 37.

Second, let p = 12q + 7. Then X2 = - 1 (mod p) is irreducible. Its roots i and - i = ip are Galois imaginaries. Set

(27) 7r=p+1, o=p-1.

There exists a linear function R of i with integral coefficients such that R”” = 1, whiIe no lower power of R is unity. Any function of i is zero or a power of R and any integer is a power of


R”, a primitive root of p. Hence we may set

&‘wt = R=“, WI+ ww3i = R”, WI - cowsi = Rps,

where 0 =< q < u, 0 2 e < wc. Then (20) is equivalent to

R”” + R2=9 = 0 > re = 2rq + +m (mod KU).

The last condition is equivalent to

(2% e = 27 + a/2 +ju (0 2 j < T). We have

w2 = wR”“, 2wl= Re+Rpe, 2w3 = -id’(R”- Rpa),

2w2Zw1 = 2Rn”+ (co2 - iw)R”+ (u2+iu)R~*,

(29) (co” - ;w)(w” + 720) = - 1,

(cl.9 - &,,)= = - 1, u2 - iw = R/c’/2 (f odd), 2&zwI = ~R”‘I + &+f~ I2 _ R~doI2

= R~bYf+l~l2l(R ~-3-pWkl) 12 + Rm-d-W-1) /2)2.

The last binomial is its own pth power and hence is real. We desire that the root @wI shall be the negative of a quadratic residue and hence a non-residue. Since R” is a primitive root of p, the condition is that j - cf + 1)/2 shall be odd:

(30) f = 21- 1, j - I= odd.

We must exclude the values making w12 = wz2:

0 = 2R~12(wl =,= w2) = R2l)-hfjo =,= &,fp~k-~P - R%v-j~,

the second term having been simplified by use of

R-12 = - 1, RP* = &-=.

Completing the square of the first two terms, we get

(Rvl-~Wl)/2 F uRm--oj/2)2 = (o2 + l)R2m-~j.

NowW2+1=-U= (c&o~)~, where c = 1 or - 1. Hence

Rv+‘(i+l)/2 = (& w + gu2)Rm-ffi/2.

INVARIANTS AND NUMRER THEORY. 109

But

(31) (u + ;w”)(u - iu2) = - 1, w + ,iw2 = R”“‘2,

a-h2= -R-““/2 (v odd).

Hence we must exclude the four cases in which

(32) ?J = j + 2p= v + l), j + $(f v + 7r + 1) (mod 4,

these four values being incongruent. No one of the w’s in (29) is zero, since e is odd by (28), so that

e + 0, a/2 (mod n). By (191) and (17),

(33) 489 = (1 - w)(R2” + R2p”) + 6W2R2=“,

4t = _ iR=l)(R26 _ PP”).

Finally, we must here exclude the cases in which a = 0. Combining Zwr2 = 0 with (20), we obtain the necessary and sufficient condition w12 = owa2 for s = 0. But w1 = f w2w3, in connection with (29), gives

R”(1 f ;w) = R=‘“(- 1 f ;w), R”(w f 3)” = Rpb.

Thus, by (31), the condition is that e f vu = pe (mod nu) or e = f v (mod x). Then, by (28), q is congruent modulo ?r to one of the values (32) decreased by x/4. Hence the desired

sets s, t are given by (33), subject to (28), in which the 8 incongruent $8 given by (32) and those values decreased by 7r/4 are excluded. In particular, p > 7.

For p = 19, the only admissible pairs are

s = 2 - 22z, t = 6(- 2)3z (1 = 0, 1, . . ., 8).

For any 2, the negatives of the roots of quartic (3) are the products of - 3 = 42, 4, 7 = g2, - 8 = 72 by (- 2) z and hence are quadratic residues of 19 since - 2 = 62.

For p = 31, the only pairs are

CT= 321, t=5(-3)3l; s=-3321, t=13(-3)3’ (I = 0, * * a, 15),

the negatives of the roots of (3) being the products of 7, - 11, - 12, - 15 and - 3, 5, 9, - 11, respectively, by (- 3) z, and hence are quadratic residues of 31.

110 THE MADISON COLLOQUITJM.

8. Casep= 3j + 1, &A $1 0, A not a Cube.-The roots of (14) are now Galois imaginaries y, yp, 9~‘. As at the beginning of 0 7,

t2 = (y + 48)(yp + +f8)(yp* + 48) = (y + 48)““p”.

Raise each member to the power (p - 1)/2. We see that y + 48 is the square of an element, say w, of the Galois field of order ~3. The 6rst root (18) is $(w + wp -I- wP7 and equals its own pth power, and hence is real. This is not true of the remaining roots (18), since WP =I= w, or since a real quadratic factor would imply that w is real. Hence the quart& haa a single real root.

For p = 7, the only cases in which the negative of the single real root is a quadratic residue are t = - 1 or 3, a = - 1, - 2,3; t = 2, a arbitrary =/= 0. For p = 13, the only cases are

+t= 4, 5, 6; a = - 1, - 3, 4 (83 = - 1);

*t= 1, 5, 6; s = - 2, - 5, - 6 (8’ = 5);

and f t = 3, - s equals one of the preceding six values of 8.

9. Cubic z&h t = 0, s =i= O.-In this case, (3) becomes

(P - 38)2 = 1282.

If there be a real root k, 3 is a quadratic residue of p, and

P=ls, 1=3*2&a

First, let p = 3j + 2, so that - 3 is a quadratic non-residue of p. Then - 1 must be a non-residue of p and hence p = 12~ + 11. The product of the two Z’s is - 3, so that a single value of P is a quadratic residue. Since the two real k’s are of opposite sign, there is a single real root k whose negative is a quadratic residue. For t = 0, s + 0, and p = 12r + 5, there is a tingle real in.xion

point; for p = 12~ + 11, thme are just three real infiti points.

Finally, let p = 3j + 1, so that - 3 is a quadratic residue of p. If p = 12r + 7, then 3 is a non-residue, and there is no real k

and hence a single real inflexion point. If p = 127 + 1, the four roots k are all real or all imaginary. For p = 13, h? z - 28 or - 53, and - k is a quadratic residue if and only if k6 = 1, 83 G 8, s = 2, 5, 6. For p = 37, k2 = - 49 or loS, and - k is a residue if and only if s* = 1.

The end

CATALOGUE OF DOVER BOOKS

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MATHEMATICS-INTERMEDIATE TO ADVANCED

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INTRODUCTION TO APPLIED MATHEMATICS, Francis 0. Murnaghan. A practical and thoroughly sound introduction to a number of advanced branches of higher mathematics. Among the selected topics covered in detail are: vector and matrix analysis, partial and differential eauations. integral eauations. calculus of variations. Laolace transform theorv. the vector triple product, hear vector functions quadratic and ‘bilinear forms Fourier se&s, spherical harmonics, Bessel functions, the Heiviside expansion formula, and many others. Extremely USefUf book for graduate students in physics, engineering, chemistry, and mathematics. Index. 111 study exercises with answers. 41 illustrations. ix + 389pp. 5% x 8%.

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OPERATIONAL METHODS IN APPLIED MATHEMATICS. H. S. Carslaw and 1. C. Jaeger. Exolana- tion Of the application of the Laplace Transformation to differential equations, g simpli -and effective substitute for more difficult and obscure operational methods. Of great practical value to engineers and to all workers in applied mathematics. Chapters on: Ordinary Linear Differential Equations with Constant Coefficients:: Electric Circuit Theorv: Dvnamical ADD& cations; The Inversion Theorem for the Laplace’iransformation; Conduciion of Heat; Vibra- tions of Continuous Mechanical Svstems: Hvdrodvnamics: lmnulsive Functions: Chains of Differential Equations: and other answers. 22 figures. xvi -t 359pp.

r*atated ’ maiters-’ . ,5ys x ax?.

appendices. -..- -..-_..- _.

153 problems, many with SlOll Paperbound $2.25

APPLIED MATHEMATICS FOR RADIO AND COMMUNICATIONS ENGINEERS. C. E. Smith. No extraneous material here!-only the theories, equations, and operations essential and im- mediately useful for radio work. Can be used as refresher, as handbook of applications and tables, or as full home-study course. Ranges from simplest arithmetic through calculus? series, and wave forms! hyperbolic trigonometry, simultaneous equations in mesh circuds, etc. Supplies applicahons right along with each math topic discussed. 22 useful tables of functions, formulas, logs, etc. Index. 166 exercises, 140 examples, all with answers. 95 diagrams. Bibliography. x + 336~~. 5% x 8. S141 Paperbound $1.75

Algebra, group theory, determinants, sets, matrix theory

ALGEBRAS AND THEIR ARITHMETICS, 1. E. Dickson. Provides the foundation and background necessary to any advanced undergraduate or graduate student studying abstract algebra. Begins with elementary introduction to linear transformations, matrices, field of complex numbers; proceeds to order, basal units, modulus, quaternions, etc.; develops calculus of linears sets, describes various examples of algebras including invariant, difference, nilpotent, semi-simple. “Makes the reader marvel at his genius for clear and profound analysis,” Amer. Mathematical Monthly. Index. xii + 241~~. 5% x 8. S616 Paperbound $1.50

THE THEORY OF EGUATIONS WITH AH INTRODUCTION TO THE THEORY OF BINARY ALGEBRAIC FORMS, W. S. Burnside and A. W. Panton. Extremely thorough and concrete discussion of the theory of equations, with extensive detailed treatment of many topics curtailed in later texts. Covers theory of algebraic equations, properties of polynomials, symmetric functions, derived functions, Homer’s process, complex numbers and the complex variable, determinants and methods of elimination, invariant theory (nearly 100 pages), transformations, introduction to Galois theory, Abelian equations, and much more. Invaluable supplementary work for modern students and teachers. 759 examples and exercises. Index in each volume. TWO volume set. Total of xxiv + 604~~. 5% x 8. S714 Vol I Paperbound 1.85

S715 Vol II Paperbound 1.85 The set f 3.70

COMPUTATIONAL METHODS OF LINEAR ALGEBRA, V. N. Faddaeva, translated by C. 0. Benster. First English translation of a unique and valuable work, the only work in English present- ing a systematic exposition of the most important methods of linear algebra-classical and contemporary. Shows in detail how to derive numerical solutions of problems in mathematical physics which are frequently connected with those of linear algebra. Theory as well as individual practice. Part I surveys the mathematical background that Is indispensable to what follows. Parts II and Ill, the conclusion, set forth the most important methods of solution, for both exact and iterative groups. One of the most outstanding and valuable features of this work is the 23 tables, double and triple checked for accuracy. These tables will not be found elsewhere. Author’s preface. Translator’s note. New bibliography and index. x + 252~~. 5% x 8. S424 Paperbound $1.95

ALGEBRAIC EGUATIONS,, E. Oehn. Careful and complete presentation of Galois’ theory of algebraic equations; theorres of Lagrange and Galois developed in logical rather than historical form, with a more thorough exposition than in most modern books. Many concrete applications and fully-worked-out examples. Discusses basic theory (very clear exposition of the symmetric group); isomorphic, transitive, and Abelian groups; applications of Lagrange’s and Galois’ theories; and much more. Newly revised by the author. Index. List of Theorems. xi + 208~~. 5% x 8. S697 Paperbound $1.45


ALGEBRAIC THEORIES. 1. E. Dickson. Best thorough introduction to classical topics in higher algebra develops theories centering around matrices, invariant!, groups. Higher algebra, Galois theory, finite linear groups, Klein’s icosahedron, algebraic invariants, linear transformations, elementary divisors, Invariant factors; quadratic, bi-linear, Hermitian forms, singly and in pairs. Proofs rigorous, detailed; topics developed lucidly, in close connection with their most frequent mathematical applications. Formerly “Modern Algebraic Theories.” 155 problems. Bibliography. 2 indexes. 285~~. 5% x 8. S547 Paperbound $1.50

LECTURES ON THE ICOSAHEDRON AND THE SOLUTION OF EQUATIONS OF THE FIFTH DEGREE. Felix Klein. The solution of quintics in terms of rotation of a regular icosahedron around its axes of symmetry. A classic & indispensable source for those interested in higher algebra, geometry, crystallography. Considerable explanatory material included. 230 footnotes, mostly bibliographic. 2nd edition, xvi + 289pp. 5% x 8. S314 Paperbound $2.25

LINEAR GROUPS, WITH AN EXPOSITION OF THE GALOIS FIELD THEORY, 1. E. Dickson. The classic exposition of the theory of groups, well within the range of the graduate student. Part I contains the most extensive and thorough presentation of the theory of Galois Fields available, with a wealth of examples and theorems. Part II is a full discussion of linear groups of finite order. Much material in this work is based on Dickson’s own contributions. Also Includes expositions of Jordan, Lie, Abel, Betti-Mathieu, Hermite, etc. “A milestone in the development of modern algebra,” edition. Index. xv + 312~~. 5% x 8.

W. Magnus, in his historical introduction to this S482 Paperbound $l.SS

INTRODUCTION TO THE THEORY OF GROUPS OF FINITE ORDER, R. Carmichael. Examines fundamental theorems and their application. Beginning with sets, systems, permutations, etc., it progresses in easy stages through important types of groups: Abelian, prime power, permutation, etc. Except 1 chapter where matrices are desirable, no higher math needed. 783 exercises, problems. Index. xvi + 447~~. 5% x 8. S3DO Paperbound $2.25

THEORY OF GROUPS OF FINITE ORDER, W. Burnside. First published some 40 years ago, this is still one of the clearest introductory texts. Partial contents: permutations, groups independent of representation, composition series of a group, isomorphism of a group with itself, Abelian groups, prime power groups, permutation groups, invariants of groups of linear substitution, graphical representation, etc. 45pp. of notes. Indexes. xxiv + 512~~. 5H x 8.


CONTINUOUS GROUPS OF TRANSFORMATIONS, 1. P. Eisenhart. Intensive study of the theory and geometrical applications of continuous groups of transformations; a standard work on the subject, called forth by the revolution in physics in the 1920’s. Covers tensor analysis, Riemannian geometry, canonical parameters, transitivity, imprimitivity, differential invariants, the algebra of constants of structure, differential geometry, contact transformations,, etc. “Likely to remain one of the standard works on the subject for many years . . . prmcipel theorems are proved clearly and concisely, and the arrangement of the whole is coherent,” MATHEMATICAL GAZETTE. Index. 72-item bibliography. 185 exercises. ix + 301~11. 5% x 8.


THE THEORY OF GROUPS AND PUANTUM MECHANICS, H. Weyl. Discussions of Schroedinger’s wave equation, de Broglie’s waves of a particle, Jordan-Hoelder theorem, Lie’s continuous groups of transformations, Pauli exclusion principle, quantization of Maxwell-Dirac field equations, etc. Unitary geometry, quantum theory, groups, application of groups to quantum mechanics, symmetry permutation group, algebra of symmetric transformation, etc. ?nd revised edition. Bibliography. Index. xxii + 422~~. 5% x 8. S269 Paperbound $2.35

APPLIED GROUP-THEORETIC AND MATRIX METHODS, Bryan H&man. The first systematic treatment of group and matrix theory for the physical scientist. Contains a comprehensive, easily-followed exposition of the basic ideas of group theory (realized through matrices) and its applications in the various areas of physics and chem,stry: tensor analysis, relativity, quantum theory, molecular structure and spectra, and Eddington’s quantum relativity. Includes rigorous proofs available only in works of a far more advanced character. 34 figures, numerous tables. Bibliography. Index. xiii + 454~~. 53/e x 8%


THE THEORY OF GROUP REPRESENTATIONS, Francis D. Murnaghan. A comprehensive introduction to the theory of group representations. Particular attention is devoted to those groups-mainly the symmetric and rotation groups-which have proved to be of fundamental significance for quantum mechanics (esp. nuclear physics). Also a valuable contribution to the literature on matrices, since the usual representations of groups are groups of matrices. Covers the theory of group integration (as developed by Schur and Weyl), the theory of 2-valued or spin representations, the representations of the symmetric group, the crystallographic groups, the Lorentz group, reducibility (Schur’s lemma, Burnside’s Theorem, etc.), the alternating group, linear groups, the orthogonal group, etc. Index. List of references. xi + 369pp. 5% x 8%. S1112 Paperbound $2.35

THEORY OF SETS, E. Kamke. Clearest, amplest introduction in English, well suited for independent study. Subdivision of main theory, such as theory of sets of points, are discussed, but emphasis is on general theory. Partial contents: rudiments of set theory, arbitrary Sat.5 and their cgrdinal numbers, ordered sets and their order types, well-ordered sets and thelr cardinal numbers. Bibliography. Key to symbols. Index. vii + 144~~. 5% x 8.

S141 Paperbound $135

THE THEORY OF DETERMINANTS, MATRICES., AND INVARIANTS, H. W. Tumbull. Important study includes all salient features and maJor theories. 7 chapters on determinants and matrices cover fundamental properties, Laplace identities, multiplication, linear equations, rank and differentiation, etc. Sections on invariants gives general properties, symbolic and direct methods of reduction, binary and polar forms, general linear transformation, first fundamental theorem, multilinear forms. Following chapters study development and proof of Hilbert’s Basis Theorem, Gordan-Hilbert Finiteness Theorem, Clebsch’s Theorem, and include discussions of apolarity, canonical forms, geometrical Interpretations of algebraic forms, complete system of the general quadric, etc. New preface and appendix. Biblio raphy. xviii + 374pp. 5% x 8. S699 Paperboun f $2.25

AN INTRODUCTION TO THE THEORY OF CANONICAL MATRICES, H. W. Turnbull and A. C. Aitken. All principal aspects of the theory of canonical matrices, from definitions and fundamental

1 . roperties of matrices to the practical applications of their reduction to canonical form. egrnnmg with matrix multiplications, reciprocals, and partitioned matrices, the authors go

on to elementary transformations and bilinear and quadratic forms. Also covers such topics as a rational canonical form for the collineatory group, congruent and conjunctive transformation for quadratic and hermitian forms, unitary and orthogonal transformations, canonical reduction of pencils of matrices, etc. Index. Appendix. Historical notes at chapter ends. Bibliographies. 275 problems. xiv + 200~~. 5% x 8. S177 Paperbound $1.55

A TREATISE ON THE THEORY OF DETERMINANTS,,T. Muir. Unequalled as an exhaustive compila- tion of nearly all the known facts about determmants up to the early 1930’s. Covers notation and general properties, row and column transformation, symmetry, compound determinants, adjugates, rectangular arrays and matrices, linear dependence, gradrents, Jacobians, Hessians, Wronskians, and much more. Invaluable for libraries of industrial and research organizations as well as for student, teacher, and mathematician: very useful in the field of computing machines. Revised and enlarged by W. H. Metzler. Index. 485 problems and scores of numerical examples. iv + 766~~. 5% x 8. S670 Paperbound $3.00

THEORY OF DETERMINANTS IN THE HISTORICAL ORDER OF DEVELOPMENT, Slr Thomas Muir. Unabridged reprinting of this complete study of 1,859 papers on determmant theory written between 1693 and 1900. Most important and orrginal sections reproduced, valuable com- mentary on each. No other work is necessary for determinant research: all types are covered- each subdivision of the theory treated separately; all papers dealing with each type are covered; you are told exactly what each paper is about and how important its contribution is. Each result, theory, extension, or modification is assigned its own identifying numeral so that the full history may be more easily followed. Includes papers on determinants in general, determinants and linear equations, symmetric determinants, alternants, recurrents, determinants having invariant factors, and all other major types. “A model of what such histories ought to be,” NATURE. “Mathematicians must ever be grateful to Sir Thomas for his monu- mental work,” AMERICAN MATH MONTHLY. Four volumes bound as two. Indices. Bibliop raphies. Total of lxxxiv + 1977pp. 53,‘s x 8. S672-3 The set, Clothbound $12.50

Calculus and function theory, Fourier theory, infinite series, calculus of variations, real and complex functions

FIVE VOLUME “THEORY OF FUNCTIONS’ SET BY KONRAO KNOPP

This five-volume set, prepared by Konrad Knopp, provides a complete and readily followed account of theory of functions. Proofs are given concisely, yet without sacrifice of complete- ness or rigor. These volumes are used as texts by such umversities as M.I.T., University of Chicago, N. Y. City College, and many others. “Excellent introduction remarkably readable, concise, clear, rigorous,” JOURNAL 0~ THE AMERICAN STATISTICAL ASSOCIATION.

ELEMENTS OF THE THEORY OF FUNCTIONS, Konrad Knopp. This book provides the student with background for further volumes in this set, or texts on a similar level. Partial contents: foundations, system of complex numbers and the Gaussian plane of numbers, Riemann sphere of numbers, mapping by linear functions, normal forms, the logarithm, the cyclometric functions and binomial series. “Not only for the young student, but also for the student who knows all about what is in it,” 5% x 8.

MATHEMATICAL JOURNAL. Bibliography. Index. 140~~. S154 Paperbound $1.35


THEORY OF FUNCTIONS, PART II, Konrad Knopp. Application and further development of general theory, special topics. Single valued functions, entire Weierstrass Meromorphtc functions. Riemann surfaces. Algebraic functions. Analytical con/iguration, Riimann surface. Bibliography. Index. X -I 150PP. 5% x 8. S157 Paperbound $1.35

PROBLEM BOOK IN THE THEORY OF FUNCTIONS, VOLUME 1, Konrad Knopp. Problems In elementary theory, for ,use with Knopp’s THEORY OF FUNCTIONS, or any other text, arranged according to rncreasmg difficulty. Fundamental concepts, sequences of numbers and infinite series, complex variable, integral theorems, development in series, conformal mapping. 182 problems. Answers. viii + 126~~. 5% x 8. S158 Paperbound $1.35

PROBLEM BOOK IN THE THEORY OF FUNCTIONS, VOLUME 2, Konrad Knopp. Advanced theory of functions, to be used either with Knopp’s THEORY OF FUNCTIONS, or any other com- parable text. Singularities, entire & meromorphic functions, periodic, analytic, continuation, multiple-valued functions, Riemann surfaces, conformal mapping. Includes a section of additional elementary problems. “ The difficult task of selecting from the immense material of the modern theory of functions the problems just within the reach of the beginner is here masterfully accomplished,” AM. MATH. SOC. Answers. 138~~. 5% x 8. S159 Paperbound $1.35

Vol. 2 part 2 S556 Paperbound $1.85 3 vol. set $8.26

MODERN THEORIES OF iNTEGRATION, H. Kestelman. Connected and concrete coverage, with fully-worked-out proofs for every step. Ranges from elementary definitions through theory of aggregates, sets of points, Rremann and Lebesgue integration, and much more. This new revised and enlarged edrtion contains a new chapter on Riemann-Stieltjes integration, as well as a supplementary section of 186 exercises. Ideal for the mathematician, student, teacher, or self-studier. Index of Definitions and Symbols. General Index. Bibliography. x + 310~~. 5% x 8%. S572 Paperbound $2.25

THEORY OF MAXIMA AND MINIMA, H. Hanoook. Fullest treatment ever written; only work in English with extended discussion of maxima and minima for functions of l! 2, or n variables, problems with subsidiary constraints, and relevant quadratic forms. Detarled proof of each important theorem. Covers the Scheeffer and von Dantscher theories, homogeneous quadratic forms, reversion of series, fallacious establishment of maxima and minima etc. Unsurpassed treatise for advanced students of calculus, mathematicians, economists, siatisticians. Index. 24 diagrams. 39 problems, many examples. 193pp. 5% x 8. S665 Paperbound $1.50

AN ELEMENTARY TREATISE ON ELLIPTIC FUNCTIONS, A. Cayley. Still the fullest and clearest text on the theories of Jacobi and Legendre for the advanced student (and an excellent supplement for the beginner). A masterpiece of exposition by the great 19th century British mathematician (creator of the theory of matrices and abstract geometry), it covers the addition-theory, Landen’s theorem, the 3 kinds of elliptic integrals, transformations, the q-functions, reduction of a differential expression, and much more. Index. xii + 386~~. 5% x 8.


THE APPLICATIONS OF ELLIPTIC FUNCTIONS, A. G. Greenhill. Modern books forego detail for sake of brevity-this book offers complete exposition necessary for proper understanding, use of elliptic integrals. Formulas developed from definite physical, geometric problems; examples representative enough to offer basic information in widely useable form. Elliptic integrals, addition theorem, algebraical form of addition theorem, elliptic integrals of 2nd, 3rd kind, double periodicity, resolution into factors, series, transformation, etc. Introduction. Index. 25 illus. xi + 357~~. 5% x 8. S603 Paperbound $1.75

THE THEORY OF FUNCTIONS OF REAL VARIABLES, James Pierpont. A 2-volume authoritative exposition, by one of the foremost mathematicians of his time. Each theorem stated with all conditions, then followed by proof. No need to go through complicated reasoning to dis- cover conditions added without specific mention. Includes a particularly complete, rigorous presentation of theory of measure; and Pierpont’s own work on a theory of Lebesgue integrals, and treatment of area of a curved surface. Partial contents, Vol. 1: rational numbers, exponentials, logarithms, point aggregates, maxima, minima, proper integrals, improper integrals, multiple proper integrals, continuity, discontinuity, indeterminate forms. Vol. 2: point sets, proper integrals, series, power series, aggregates, ordinal numbers, discontinuous functions, sub-, infra-unrform convergence, much more. Index. 95 illustrations. 1229pp. 5% x 8. S558-9, 2 volume set, paperbound $5.20


ELEMENTS OF THE THEORY OF REAL FUNCTIONS, 1. E. Littlewood. Based on lectures given at Trinity College, Cambridge, this book has proved to be extremely successful In introaucing graduate students to the modern theory of functions. It offers a full and concise covera e of classes and cardinal numbers, well-ordered series, other types of series, and elemen s ‘i of the theory of sets of points. 3rd revised edition. vi1 + 71pp. 5% x 8.

S171 Clothbound S172 Paperbound f

2.85 1.25

TRANSCENDENTAL AND ALGEBRAIC NUMBERS, A. 0. ttelfond. First English translation of work by leading Soviet mathematician. Thue-Siegel theorem, its p-adic analogue, on approximation of algebraic numbers by numbers in fixed algebraic field; Hermite-Lindemann theorem on transcendency of Bessel functions, solutions of other differential equations; Gelfond-Schneider theorem on transcendency of alpha to power beta: Schneider’s work on elliptic functions, ;&hxmgethod developed by Gelfond. Translated by L. F. Boron. Index. Bibliography. 200~~.


ELLIPTIC INTEGRALS, N. Hancock. Invaluable in work involving differential equations containing cubits or quartrcs under the root sign, where elementary calculus methods are inade. quate. Practical solutions to problems that occur in mathematics, engineering, physics: differential equations requiring integration of Lame’s, Briot’s, or Bouquet’s equations; determination of arc of ellipse, hyperbola, lemniscate; solutions of problems in elastica; motion of a projectile under resistance varying as the cube of the velocity; pendulums; many others. Exposition is in accordance with Legendre-Jacobi theory and includes rigorous dfs- cussion of Legendre transformations. 20 figures. 5 place table. Index. 104~~. 5% x 8.


LECTURES ON THE THEORY OF ELLIPTIC FUNCTIONS, H. Hancock. Reissue of the only book in English with so extensive a coverage, especially of Abel, Jacobi, Legendre, Weierstrasse, Hermite, Liouville, and Riemann. Unusual fullness of treatment, plus applications as well as theory, in discussing elliptic function (the universe of elliptic integrals originating in works of Abel and Jacobi), their existence, and ultimate meaning. Use is made of Riemann to provide the most general theory. 40 page table of formulas. 76 figures. xxiii + 498pp.


ALMOST PERIODIC FUNCTIONS, A. S. Besicovitch. This unique and important summary by a well-known mathematician covers in detail the two stages of development in Bohr’s theory of almost periodic functions: (1) as a generalization of pure periodicity, with results and proofs; (2) the work done by Stepanoff, Wiener, Weyl, and Bohr in generalizing the theory. Bibliography. xi + 180~~. 5% x 8. S18 Paperbound $1.75

THE ANALYTICAL THEORY OF HEAT, Joseph Fourier. This book, which revolutionized mathematical physics, is listed in the Great Books program, and many other listings of great books. It has been used with profit by generations of mathematicians and physicists who are interested in either heat or in the application of the Fourier integral. Covers cause and reflection of rays of heat, radiant heating, heating of closed spaces, use of trigonometric series in the theory of heat, Fourier integral, etc. Translated by Alexander Freeman. 20 figures. xxii + 466~~. 5% x 8. S93 Paperbound $2.50

AN INTRODUCTION TO FOURIER METHODS AND THE LAPLACE TRANSFORMATION, Philip Franklin. Concentrates upon essentials, enabling the reader with only a working knowledge of calculus to gain an understanding of Fourier methods in a broad sense, suitable for most applications. This work covers complex qualities with methods of computing elementary functions for complex values of the argument and finding approximations by the use of charts; Fourier series and integrals with half-range and complex Fourier series; harmonic analysis; Fourier and Laplace transformations, etc.; partial differential equations with applications to transmission of electricity; etc. The methods developed are related to physical problems of heat flow, vibrations, electrical transmission, electromagnetic radiation, etc. 828 problems with answers. Formerly entitled “Fourier Methods.” Bibliography. Index. x + 289pp. 5% x 8.


THE FOURIER INTEGRAL AND CERTAIN OF ITS APPLICATIONS, Norbert Wiener. The only book- length study of the Fourier integral as link between pure and applied math. An expansion of lectures given at Cambridge. Partial contents: Plancherel’s theorem, general Tauberian theorem, special Tauberian theorems, generalized harmonic analysis. Bibliography. viii -I- 201~1. 5% x 8. S272 Paperbound $1.50


INTRODUCTION TO THE THEORY OF FOURIER’S SERIES AND INTEGRALS, Ii. S. Carrlaw. 3rd revised edition. This excellent introduction is an outgrowth of the author’s courses at Cambrid e. Historical introduction, rational and irrational numbers, infinite sequences and series, unctrons of a single variable, definite integral, Fourier series, Fourier integrals, and f . similar topics. Appendixes discuss ractical harmonic analysis, periodogram analysis. Lebes- gue’s theory. Indexes. 84 examples, fi rbllography. xii + 368~~. 5% x 8. S48 Paperbound $2.25

FOURIER’S SERIES AND SPHERICAL HARMONICS, W. E. Byerly. Continues to be recognized as one of most practical, useful expositions. Functions, series, and their differential equations are concretely explained in great detail; theory is applied constantly to practical problems, which are fully and lucidly worked out. Appendix includes 6 tables of surface zonal harmonics, hyperbolic functions, Bessel’s functions. Bibliography. 190 problems, a proximately half with answers. ix + 287~~. 5% x 8. S536 Paper ound 1 $1.75

INFINITE SEGUENCES AND SERIES, Konrad Knopp. First publication in any language! Excel- lent introduction to 2 topics of modern mathematics, designed to give the student background to penetrate farther by himself. Sequences & sets, real & complex numbers, etc. Functions of a real & complex variable. Sequences G series. Infinite series. Convergent power series. Expansion of elementary functions. Numerical evaluation of series. Bibliog- raphy. v + 186~~. 5% x 8. S153 Paperbound $1.75

TRI6ONOMETRICAL SERIES, Antoni Zygmund. Unique in any language on modern advanced level. Contains carefully organized analyses of trigonometric, orthogonal, Fourier systems of functions, with clear adequate descriptions of summability of Fourier series, proximation theory, conjugate series, convergence, divergence of Fourier series. Especially valuable for Russian, Eastern European coverage. Bibliography. 329pp. 5% x 8. S290 Paperbound $2.00

DICTIONARY OF CONFORMAL REPRESENTATIONS, H. Kober. La solved in this unique book developed by the British P

lace’s equation in 2 dimensions Admira ty. Scores of geometrical forms

(P their transformations for electrical engineers, Joukowski aerofoil for aerodynamists. Schwarz-Christoffel transformations for hydrodynamics, transcendental functions. Contents classified according to analytical functions describing transformation. Twin diagrams show curves of most transformations with corresponding regions. Glossary. Topological index. 447 dlagrams. 244~~. 61/s x 9%. S160 Paperbound $2.00

CALCULUS OF VARIATIONS, A. R. Forsyth. Methods, solutions, rather than determination of weakest valid hypotheses. Over 150 examples completely worked-out show use of Euler, Legendre, Jacobi, Weierstrass tests for maxima, minima. Integrals with one ori inal pendent variable: with derivatives of 2nd order; two dependent variables, one f

de- in ependent

variable; double Integrals involving 1 dependent variable, 2 first derivatives; double integrals involving partial derivatives of 2nd order; triple integrals; much more. 50 diagrams. 678~~. 5% x 8%. S622 Paperbound $2.95

LECTURES ON THE CALCULUS OF VARIATIONS, 0. Bolu. Analyzes in detail the fundamental concepts of the calculus of variations, as developed from Euler to Bilbert, with sharp formu- lations of the problems and rigorous demonstrations of their solutions. More than a score of solved examples; systematic references for each theorem. Covers the necessary and sufficient conditions; the contributions made by Euler, Du Bois-Reymond, Hilbert, Weierstrass, Legendre, Jacobi, Erdmann, Kneser, and Gauss; and much more. Index. Bibliography. xi + 271~~. 5% x 8. S218 Paperbound $1.65

A TREATISE ON THE CALCULUS OF FINITE DIFFERENCES, 6. Boole. A classic in the literature of the calculus. Thorough, clear discussion of basic principles, theorems, methods. Covers MacLaurin’s and Herschel’s theorems, mechanical quadrature, factorials, periodical constants, Bernoulli’s numbers, difference-equations (linear, mixed, and partial), etc. Stresses anal- ogies with differential calculus. 236 problems, answers to the numerical ones. viii + 336~~. 5% x a. S695 Paperbound $1.55

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