Rend. Sem. Mat. Univ. Poi. Torino Voi. 53, 4(1995) . Number Theory

A. Mori*

POWER SERIES EXPANSIONS OF MODULAR FORMS AT GM POINTS

Abstract. This paper is a survey of previous work of the author, and includes also

a list of open questions. Let / be a modular form of arbitrary weight with respect

to To(N). It is shown that the power series expansion of / at a CM point is a

good substitute for the more usuai Fourier expansion. An advantage of considering

expansions at CM points is that this method can be naturally employed also when / is

modular with respect to anisotropie quaternion groups, where Fourier expansions do

not exist. In particular, it is possible to characterize explicitly the jR-rational structure

of the space of cuspforms of quaternionic type, at least when R is a subfìeld of C.

No other explicit characterization of the jR-rational structure was known before.

Introduction

Modular forms arise classically in Number Theory as generating functions for the

number of representations of integers by a fi xed quadrati e forni, [18]. Over the last few

decades mathematical work of Hecke, Weil, Shimura, Eichler, Langlands, Serre, Deligne,

Katz, Hida, Wiles and many others, has increased enormously the scope and the range of

applications of the theory of modular and, more generally, automorphic forms. Even a very

concise outline of the state of the art of the theory of modular and automorphic forms is

beyond the goals of the present paper. The reader should consult [1], [4], [13] and the

references listed therein.

A basic arithmetical property of the spaces of modular forms with respect to SL2CZ)

and its congruence subgroups, is that they can be endowed with a canonical /2-structure, for

any not too small subring R C C. This can be done naively, following Shimura, in terms

of the Fourier coefficients, or else, as in [2] and [5], viewing modular forms as global

sections of a suitable line bundle defined over a modular curve. These two approaches

define actually the same 72-structure. This is a fact known as the q-expansion principle. It

*The author is member of the GNSAGA of CNR.

362 A. Mori

is indeed a key foundational result in the p-adic theory of modular forms and the study of congruences.

Consideration of the Shimura curves associated with indefinite quaternion di vision algebras over Q leads to a defìnition of canonical substructures also for the corresponding spaces of modular forms. Since a group of elements of norm l in an indefinite quaternion division algebra over Q contains no parabolic elements, Fourier analysis is not available and the substructures remain implicit and not well suited for applications.

In [16] the author developed an alternative method to characterize the ^-integrai structure of the spaces of modular forms with respect to subgroups of SL2CZ). Precisely, let ók = - ^ |2z*^- + M be the classical Maass operator, [11], tfjp = ók+2n-2 o • • - o 6k

and r a point in the complex upper halfplane such that Q(r) is a quadratic imaginary field. Then, with some technical restrictions, a modular form / ofweight k is ^-integrai if and only if ali the numbers <!>["' ( / ) ( T ) are, up to some well specifìed transcendental factor, v-adic integers and satisfy the Kummer-Serre congruences.

In [17] the author has extended the methods of [16] to modular forms associated with an indefinite quaternion division algebra. In particular a characterization of the j^-rational subspaces of these forms (F field) is now available.

The aim of this note is to survey the papers [16] and [17] and give a list of a few open problems and questions. The first two sections are devoted to a brief review of the fundamentals of the theory. In particular we recali the algebraic defìnition of the space of modular forms and give a statement of the #-expansion principle.

The results obtained by the author are presented ih Sections 3 and 4. We keep the case of SL^i^) and the case of a di vision quaternion algebra separate, although the methods used are quite similar and actually rest on the same idea. For the sake of economy in exposition, we only deal with the group To(A/r) in Section 3 and with an Eichler order of level N in Section 4, but the results can be proved for more general congruence groups. Also, we nave left out the modular forms with non trivial nebentypus: this is truly necessary only in Section 4 because we are unable to deal with forms of odd weight (see also (5.2)).

The final section lists a few open questions, some of which may be relevant for applications. For reasons of space we keep from going further into details. More on these topics will appear in forthcoming papers.

Notations and Conventions. The symbols Z, (Q), M and C denote respectively, as usuai, the integers, the rational, the real and the complex numbers. We fix once for ali an embedding Q ^ C , i.e. we think of any algebraic extension of Q as a subfìeld of C.

If p is a rational prime, Zp, Qp denote the p-adic integers and the p-adic numbers respectively. If L is a number field and v a non-archimedean place of L, the symbols Lv

Power serìes expansions of modular forms at CM points 363

and Rv denote the v-adic completion of L and its ring of integers, respectively. In some context Rv may denote the ring of v-adic integers in L (no confusion should arise). The completion of the maximal unramifìed extension of Lv will be denoted LJJr.

If V is a vector space, Sym2(V) denotes the symmetric 2-fold product of V by itself. If i;, w E V, let v o w = |(u<g)u;-f-ii>®i;). We shall use the same symbol ism for vector bundles and their sections.

1. Quaternion algebras and Sili mura ciirves

Let D be a quaternion algebra over Q. It is called split at a rational place p if D <g) Qp ~ M2(Qp). Let ED denote the set of places not splitting D. We say that D is split if ED = 0, and w^ s/?/*Y otherwise. It is well known that, [21] Ch.3,

1) ED is a finite set containing an even number of places.

2) Given a finite set E consisting of an even number of places of Q, there exists a quaternion algebra D such that E = ED-

3) The isomorphism class of D is completely determined by ED-

In particular, D = M2(Q) is the only split quaternion algebra up to isomorphism. The discriminant AD of D is the product of the finite primes in ED or A D = 1 if D is split. Throughout this paper we will assume that D is indefinite, i.e. it is split at oo, and we shall fix an isomorphism iTO:Z>® M^il//2(M). Fix also a positive involution b\-+b* (the.n 6* = /-16tf for some t £ D such that /2 is a negative integer, by the Skolem-Noether Theorem).

Let N be a naturai number prime to A#. An Eichler order of level N in D is a subring O which is a free Z-module of rank 4 and satisfies the following conditions.

1) For each prime P\AD, O ® Xp is the unique discrete valuation ring of the division algebra £>®Qp.

2) Ifpn||iV, there is an isomorphism ip:.D 0 QP^M2(QP) such that ip(0 <g> Zp) =

VPnZp ZpA •3) For a prime p not dividing JV, there is an isomorphism ip: D ® Qp-^M2(QP) such

that Ap ((9 <8> Zp ) = M2(Zp).

Given an Eichler order (9, consider the multiplicative group

r = T(0) = {T G O such that Nr(7) = 77 = 1} .

If D is split the group T is, up to an inner automorphism, the usuai congruence subgroup

r = T0{N) =\[c ^ ) € SL2(Z) such that e = 0( mod i\T)|.

364 A. Morì

In either case we let T act on the upper halfplane % = {z = x + yi £ C | y > 0} via the isomorphism L^ and the usuai action of SL2(M.), namely

(1.1) T ^ = , , ' ^ o o ( T ) = ( r

Assume that T acts without fixed points, which is always the case if N > 3.

The geometry of the Riemann surface Yp = T\H depends strongly on the type of D. If D is split, Y0(N) = YTO(AT) is not compact. It can be reinterpreted as the set of complex points of the scheme yo(N) which represents the functor Fw:Schemes —»• Sets such that

. . _ f Isomorphism classes of elliptic curves E — E/T ì \ with a cyclic isogeny E -* E' of order N ) '

The complex elliptic curve with a cyclic isogeny of order N corresponding tozG YQ(N) is ^ = C/(Z 0 Zz) with isogeny Ez -* Ez/C where C is the cyclic subgroup of Ez

consisting of the points a/N modulo TL^TLz for a = 0 , 1 , . . . , N ~ 1. The scheme 3^0(N) is defined over 7L and it is known to be smooth over 1[1/N]. The reduction modulo a prime dividing the leve! has been fully analyzed in [9]. Consider the universal elliptic curve with a level N structure, SN-^y^N) and let LO = ^*(^eN/y0(N))' The Kodaira-Spencer map gives rise to an isomorphism

(1.2) K-.^^n^y,

see [7] §1.0 where the isomorphism is constructed for the more general moduli spaces of Hilbert-Blumenthal abelian varieties of any genus. The curve YQ(N) can be compactifìed by adding the finite set consisting of the T-orbits in P1(Q)1 [20] §1.3. Analogously, the scheme yo[N) can be compactifìed considering families of generalized elliptic curves, [2]. The resulting curve XQ(N) (or scheme XQ(N)) is called the modular curve of level N. The invertible sheaf LO extends uniquely to the compactification and (1.2) extends to an isomorphism o;®2-^fì1(logC) where C is the divisor of the cusps.

If D is not split, the curve Yp is compact, [20] Proposition 9.3, and we let Xp = Yp. It can be reinterpreted as the set of complex points of the scheme Xp = yp which represents the functor Fa • Schemes —»> Sets such that

( . f Isomorphism classes of abelian surfaces A/T ì *°^ ' ~ \ with an embedding O ^ End(A) J *

The elements in Fo(T) are called QM abelian surfaces. The complex QM abelian surface corresponding to z e Yp is Az = C2 /t>oo(0)(\)., [19]. The complex torus Az admits a unique (up to a scalar) Riemann form whose corresponding Rosati involution is precisely d H-> d*, [12]. By choosing / in a suitable way, [3] [15], we may assume that the polarization is principal. The scheme yp is defined over % and it is known to be smooth over Z[ì/NA].

Power series expansions of modular forms at CM points 365

Consider the universal family ^QM-^3^r and the sheaf u = ̂ ( ^ Q M / A , ). Now the fìbers of the universal family over the curve Yr are of dimension 2, so the sheaf u> cannot be an invertible sheaf. The Kodaira-Spencer map gives rise only to a surjective map

(1.3) A':Sym2w —• fì^.

2. Modular forms and the (/-expansion principle

Let D and T be as in Section 1. For g = ( , ) E SLn(M) and a function <j> on

H let j(g, z) — cz + d and 0|À?[J/](2) = </<>(# ' Z)J(.9Ì z)~k- A modular form ofweight k G 2

with respect to T is a holomorphic function / on H satisfying the functional relation

(2-1) /IfcMW = /(*), V 7 € r ,

and a certain growth condition as z —> s for each cusp s. The growth condition can be rephrased as follows. For a given cusp s choose an element a E 5Xo(M) such that

a • s = oo. Then, since ( n j G ro(A^) the function /U[0"-1] is invariant under the

transformation z *-*-z + l. We can thus write

(2.2) /l»[tf-1](*) = /.(«) = S < 4 ' ) ( / ) 9 B . where<, = e 2 ' " . n

The expression (2.2) is independent of a and is called the Fourier or 5- expansion of f at s. The growth condition alluded to above is equivalent to the condition «n (/) = 0 for ali n < 0 and for ali cusps s (we say that / is holomorphic at the cusps). If furthermore «o (/) = 0 for ali cusps, f is called a cuspform.

As recalled in Section 1, if D is not split T has no cusps and so the Fourier expansions (2.2) are not available. Also, the distinction between cuspforms and modular fórms has no meaning.

We shall denote M*. = Mk(T) the complex vector space of modular forms of weight k with respect to T, and Sk — Sk(T) the subspace of cuspforms. The notion of C°° modular and cuspform is obtained in the same way starting with a C°° function / . The bigger spaces of C°° modular and cuspforms will be denoted by M%° and 5£°, respectively. Since - 1 E T, the relation (2.1) forces Mjf° = (0) for odd k. When D is split, the assignment f(z) •-* f(z)(2iridu)®k sets an isomorphism

(2.3) Mk^UH\Xr,(^k),

which combined with the Kodaira-Spencer isomorphism gives rise to an isomorphism

(2.4) M2k-^H\Xv,{Sll{\ogC)fk).

366 A. Morì

In the latter identification S2k ^ H°(Xr, {tii)®k). The identification (2.3) shows that the spaces Mk are finite dimensionai in general and trivial for k < 0.

Let R be any Z-algebra. Using the modular inteipretation of Xo(N) Katz gave in [5] an algebraic definition of a space Mk{R) = Mk(To(N),R,) of modular forms defined over R as functions of triples (E/R,UJ,P) where EJR is an elliptic curve over R, LO an invariant differential on E/R and P a point of exact order N in E/R, satisfying certain formai axioms translating (2.1). Any / E Mk can be viewed as an element in Mk(C): its tf-expansions can be reconstructed as the value al. the Tate curve Gm(Z((^)) )/q& endowed with its canonical invariant form and the various cyclic subgroups of order N. Moreover,

Mk = {feMk(C)\fs(q)e£[[q}],\/S}.

We can thus define, for U C C a canonical sub i2-module Mk(R) = Mk Ci Mk(R) of the space of classical modular forms. It turns out that, at least when N is invertible in R, the construction of Mk{R) is compatible with the isomorphism (2.4) and the Kodaira-Spencer isomorphism:

(2.5) M2k(R)-^H°(MM) <8> Rt (u>f2k)-^H°(XQ(N) <g> R, (tt\logC)fk).

This shows that iVfk(R) is a functorial .ft-structure for Mk, i.e. Mk{R) ®R£ = Mk. This canonical i^-structure is made explicit by the

THEOREM 1 (</-expansion principle): / E Mk is defined over R if and only if

fs(<l) € R[[(l}] for at least one cusp s.

A similar algebraic construction can be carried out also in the case of D not split, using QM abelian surfaces instead of elliptic curves. In particular, the identification (2.4) holds as well. Obviously, the problem is that this i2-structure remains implicit and so useless for applications such as the foundation of a p-adic theory and the study of congruences.

The naturai goal is to develop an alternative method to test i?-rationality which works independently of the type of D, We will attack and solve this problem in a special case (R — fìeld) in the next sections.

3. An alternative method: the split case

In this section D is always split. In [5], Theorem 1 is proved combining the following two faets:

(3.1) The g-expansion (2.2) of the form / is the jet of / at the cusp s.

(3.2) Let £ be a line bundle on a connected curve C, both defined over a ring R. Let S be an i2-algebra. Then a section / of C is S'-rational if and only if its jet at an ,S-rational point of C is 5-rational.

Power series expansions of modular forms al CM points 367

Note that the existence of a canonica! differential invariant form on the Tate curve allows to trivialize canonically the bundle of weight k modular forms in a neighborhood of the cusp. Then, Using the canonical locai parameter q = e2vtz at ?'oo, there is no ambiguity in identifying the jet of / with a power series as in (3.i).

Our pian is to follow this very same course of action, but considering the jet of / at a finite point instead of a cusp and use (3.2). In order to get an explicit expansion we need to:

1) Pick a specific point x £ yo(N) and trivialize the bundle of modular forms around it.

2) Choose a locai parameter at x and identify the jet of a form / at x with a power series.

3) Find a way to compute explicitly in terms of / the coefficients of this power series. It turns out that we cannot pick just any point in y$(N) but we have to proceed as follows. Let K be a quadratic imaginary field and consider embeddings i: K c-^ D which are normalized in the sense of [20], (4.4.5). These embeddings are in one to one correspondence with the r G H such that Q(r) = K. The correspondence is realized as

. / n o r m a ^ z e ^ embeddings ì ( r E'H such that ( ^ 1 r.K^D J ' K t f x y = {7 e GL+(Q|7 • r = r)

The elliptic curve E - ET with r as in (3.3) is a CM curve with End(£) <g> Q = K. The points of Xo(N) constructed in this way are called CM points.

Let p be a prime number split in K. AIso, assume that p does not divide N. Let a? be a CM point defined over a number field I containing K and pick a place v of L dividi ng p such that the CM curve E corresponding to x has good reduction E^ modulo v. Because of our choice of p, E = E^ 0 kv is ordinary.

Let R be an Artin ring with residue field kv and let E fu be a deformation of E over R. Since elliptic curves are self-dual, a choice of a non-zero P E TP(É) determines a trivialization of E (i.e. an isomorphism E-*Gm), hence an invariant form LO E — ^E(P) by pulling back the canonical invariant form on Gm, [8] §3.3. This construction is functorial and so the various LUE can be simultaneously lifted to a single form u on the universal formai deformation £ of E. The form LO trivializes the bundle UJ in the formai neighborhood of x over the ring R"1'.

Because of ordinarity, the field K of complex multiplications can be embedded in the p-adic field Qp. Of the two possible embeddings, choose the one that makes /J e End(^) C ^p act on TP(E) as multiplication by /./. When (j, e End(#) is a p-adic unit, its action on E lifts to an action on the formai neighborhood (this action obviously does not preserve the fìbers). A computation shows that the behavior of the Serre-Tate

368 A. Mori

parameter q under this action is precisely q i-s> qPff*. By evaluating the Serre-Tate parameter at the universal formai deformation £ we can construct a (formai) locai parameter at x, namely Q = log q, which is defined over L"r and is eigen for the action of the complex multiplications with character x(fJ>) = fili1-

The action of Kx on the analytical neighborhood of x can be computed over C using the explicit embedding i and (1.1). A straightforward calculation shows that Kx acts on the holomorphic cotangent space T* ~ T* via the character %. By solving successive linear equations, we can lift any non-zero cotangent vector to an analytic formai locai x~ eigen locai parameter Ux at x. The actual computation using the coordinate z — T in Ti shows that in fact Ux £ mx -mi C Ox.

It is possible to give a third interpretation of this action of K x in terms of algebraic correspondences on the modular curve X0(N). It can be used to show that the action preserves the L-rational structure of Ox and in fact that a suitable multiple of Ux (or Q) is defined over L.

Comparison of these different descriptions of the locai parameter yields the following numerical characterization of the ^-integrai jets at x:

THEO REM 2 The re exists an L-rational locai parameter Ux E Ox which is a fi//i-eigenparameter for the action of p E Kx. Moreover, a jet X ^ o anU£/n\ is v-integrai if and only if an E Rv far ali n > 0 and the an satisfy the Kummer-Serre congruences.

The Kummer-Serre congruences alluded to in the previous statement are the congruences characterizing the image of Rv [[X]] in Lv [[Y]] under the formai substitution X = eV = Zn>0Y

n/nl

Let / be a modular form of weight k with respect to To(A^), and let /* be the corresponding global section of ujk as in (2.3). We write

(3.4) j e t ( r ) = |gMZlQ.. ,)02®i-. \n=Q U' I

The coeffìcient bn(f) can be computed using [8], Theorems 3.7.1 and 4.3.1. Namely,

(3.5) 0Ì^(f)(x) = bn(f)^+2n

In the formula (3.5), ux is the 1-form on E obtained by restricting Q to the centrai fiber (i.e. putting Q — 0) and 0JT; = 9k+2n-2,P ° • • •°^,p where 0k)P:uk —* cuk+2 is the differential operator defined as follows. Let Ti1 = TC^R(£/'Xo(N)) be the relative de Rham bundle associated with the universal family. Over a suffìciently large p-adic ring the bundle Ti1

admits a decomposition under the action of the Frobenius morphism, the unit root space decomposition

(3.6) Hl=H1>°eU

Power series expansions of modular forms at CM poìnts 369

(note that 'H1'0 ~ u>). Then 9k)P is the composition

(3.7) w®* c-> Symk{Hl)^^mk{Til) ® fì1-^

- ^ S y m * ^ 1 ) <8> io®2^u®k ® u>®2 - u;®fc+2,

where V denotes the Gauss-Manin connection [10] and PFI- the projection defìned by (3.6). The following remark is cruciai

(3.8) For CM curves, the unit root decomposition (3.6) coincides with the Hodge decomposition. In fact they both coincide with the eigenspace decomposition of Hj)K under the action of the complex multiplications, [6] Lemma 4.0.7 and Lemma 8.0.13.

The remark implies that the formula (3.5) holds unchanged when we replace the operator 9k)P with the operator 9k defìned as the composition

(3.9) u>®k e - S y m ^ ^ S y m ^ f t 1 ) ® « 1 J ^

where PR is the projection defìned by the Hodge decomposition

(3.10) H^R^H^QK0'1.

Note that since the decomposition (3.10) is not holomorphic, the form 0*. (/) 6 Mg+2n

is not holomorphic.

The advantage of the operator 9^ is that it.can be easily computed in terms of the z-coordinate in Ti. Let V be the 1-dimensionai representation of Cx given by the character Pk(s) = sk. The line bundle u®k is holomorphically equivalent to the bundle Tk quotient of the trivial G-bundle V x Ti on Ti with action

g-(viZ) = (pk{j{giZ))v,g-Z).

The isomorphism T^^*u®h depends on the choice of a Constant section v of V x Ti. For a, say C°°, function <p on Ti let

6k(<p) = -~r ( 2i— + - ) ((p), where z = x + «'?/• 47r \ dz y J

Following the step-by-step defìnition (3.9) of the operator 9k, we eventually get a commutative diagram

Tk -^ w®*

ih [9k

Tkvi - ^ w®*+1

370 A. Mori

where Sk(fv) = Sk(f)v. This identification can be used to show that when Q and u in (3.5) are replaced by an L-rational multiple, the coefficients bn(f) are replaced by the values

*(/)=(^ff">(/)(r) where fì„ is a suitable period of E, defined up to a u-adic unit. Thus, applying (3.2) to the present situation yields the

THEOREM 3 Let f be a modular forni of weight k with respect to TQ(N). Let r E% be such that K = Q(r) is a quadratic imaginary field in which the prime p splits (assume that p does not divide N). Let L be a number field containing K such that the elliptic curve E = C/Z 0 Zr is defined over L and has good ordinary reduction modulo a prime v over p. Then f is defined over the ring Rv if and only if cn(f) G Rv, for ali n — 0 ,1 , . . . , and satisfy the Kummer-Serre congruences.

If p divides the level TV, the above result does not hold any longer because the components of the modular curve modulo p are not reduced. A necessary condition for the u-integrality of a modular form of level prN (with (p,N) = 1) is obtained in [14] using the results of [9]. In particular, when r = 1 the components of the modular curve modulo p are reduced and meet each other only at the supersingular points, so a characterization very similar to Theorem 2 can be obtained.

4. An alternative method: the non-split case

In this section D is always non-split. The ideas explained in the previous section can be adapted to this case to obtain a result analogous to Theorem 3 for modular forms with respect to T.

Let K be a quadratic imaginary field. Embeddings K C D exist if and only if K splits D, i.e. D (g) K ~ Mo(K). In the same way as before, the normalized embeddings defìne the CM points of Yp via the bijection

( . ( normalized embeddings ì J re Ti such that L00(L(KX)) = ì ( > ) l i.K^D } M l { T G ^ o o p x n G L + ( Q ) ) | T - r = r}J

The QM abelian surface corresponding to the CM point x G Xr defined by an embedding of K is isogenous to the twofold product of a CM curve E with End(^) ® Q = K. The choice of a representati ve r E7i of x determines uniquely an isogeny

, * C C A—••

Let $i be the composition of $ with the projection on the i-th factor.

Let p be a prime number split in K. Then D is split at p. Fix a CM point x and let L be a number field such that

Power series expansions of modular forms at CM points 371

(4.2) K C L;

(4.3) The QM-abelian variety A = Axt its isogeny factor E and the isogeny $ are defined over L.

In particular (4.2) implies that L splits D. Pick a place i> of Z such that A has a good reduction A^ modulo v. Because of our choice of p, A = A(v) <g> kv is ordinary.

Let M be the general formai moduli space corresponding to A, and let MCiM be the formai neighborhood of x in X?. Forgetting the QM-structure gives rise to a cartesian diagram of schemes and universal deformations

^ M — • A

i i •

where the horizontal maps are closed embeddings. Recali that the QM-abelian surfaces can be assumed principally polarized, hence self-dual. A choice of a pair of Zp-independent elements {Pi, /^} in TP(A) defìnes, as in the previous section, invariant forms £>i and w2

on A. Their restrictions to ̂ 4^M trivialize the bundle a; over MCiM.

Attached to the pair {Pi^2} we also have the Serre-Tate parameters qn,qi2,Q22 which generate the ring of functions of M. The pair {Pi, P2) can be chosen so that the following two conditions are simultaneously satisfied:

(4.4) The image of M^ in M is the intersection of the "surfaces" qn — 1 and #22 = 1.

(4.5) The symmetric 2-form Q\ o a)2 restricted to the centrai fìber belongs to the 1-dimensional subspace $l(H°(Et QX

E)) o ̂ *2(H°(E, Q^)) of the fiber of Sym2(a;).

It follows from (4.4) that the Serre-Tate parameter q = qi2 can be naturally seen as a parameter on M^M. The action of the complex multiplication p on it can be calculated and turns out to be q \-> q^^ again.

An analysis of the Kx -action from the complex analytical and the algebraic point of view as in the previous section, leads to the conclusion that Theorem 2 holds unchanged also in the present situation.

The difficulty that now arises is that there is no naturai way to attach a global section of a power of u to a modular form, since LO is not a line bundle. The remark (4.5) and the general shape of the computations of the Kodaira-Spencer map over M suggest the following approach. The bundle Sym2(io) is homogeneous, i.e. it supports an action of the group GL^(M) which extends the obvious action of T. Given a homogeneous bundle V we can define a line subbundle C CV simply by picking a point x and choosing a line (equivalently a non-zero vector) Cx in the fìber Vx. The line bundle C is defined over each

372 A. Mori

fiber by translating with the group action the elements of Cx. It can be characterized as the unique homogeneous line subbundle of V whose fiber at x is exactly Cx. Following this procedure, let C = CTìv be the line subbundle of Sym2(o;) obtained as the quotient of the unique homogeneous line subbundle CT C Sym2(u>) such that

(4.6) (CT)T = **1(H0(E,a1

E))oVt(H0(E;a)s)).

The bundle C depends on r, but we can prove that

(4.7) For each r the Kodaira-Spencer map restricts to an isomorphism K-.C^Ù^ of complex line bundles.

(4.8) For each r the bundle C (and so its tensor powers) is subject to an algebraic description that endows it with a naturai L-rational structure.

These two facts combined with the observation that the Kodaira-Spencer map is algebraic show that the L-rational structure of C is independent of r. Hence, we can identify

(4.9) M2k(T)-^H°(Xr,C®k) via / >-> f = fs®k

where s is a suitable Constant section of C over Ti. In fact we pick s such that K(s) = 27ridZ. So we can now write

(4.10) jet(/*)= ( f ; ^ Q n ) ® u ^ , Q = Q1oQ2. Vn=0 H- )

Following the same procedure explained in the second naif of Section 3, we can now defìne numbers

(_4ff)fc+n-CnU) = nk+n % » ( T )

for a suitable transcendental period &A, and prove the

THEOREM 4 Let f be a modular forni of weight k with respect to T. Let r £% be such that K — Q(r) is a quadratic imaginary jìeld. Let L be a numberfield satisfying (4.2-3). Then f is defined over afìeld F D L if and only ifcn(f) £ F, for ali n — 0 ,1 , . . . .

5. Open problems

The methods and the results of Sections 3 and 4 give rìse to several questions and problems that we discuss very briefly fiere. We shall freely use the notations and terminology introduced in the previous sections.

(5.1) The result obtained for modular forms with respect to non split T is weaker. It would be interesting to have also in this case a characterization of the v-integral structure, at least for primes of good reduction. The problem is that although each line bundle C can be endowed with a canonical v-integral structure, the map

Power series expansions of modular forms al CM points 373

need not be an isomorphism over Rv in general. In particular the map of global sections

H°(Xr ® Rv,C®k) —> H°(Xr <g> Rv, (fì1)®*)

is not surjective in general. Since our methods allow to characterize the ^-integrai sections of C®k, this seems to suggest that the "correct" defìnition of the space Mk{Rv) is via the direct identifìcation M2k ^ H°(C®k). Then a problem of defìnition arises.

(5.2) Over the modular curve X0(N) the sheaf Q^logC) has a canonical "square root" provided by the Kodaira-Spencer isomorphism. As remarked many times, this is not so for Xr with T not split. A consequence of this is that modular forms of odd weight with non-trivial Nebentypus become intractable with our methods. To include these, some new approach must be devised.

(5.3) With each space Mk(T) is associated the Hecke algebra Hk(T). It is the subalgebra of End(Mfc) generated by the basic Hecke operators Ti and Titi for a prime number L Each basic operator is defìned in terms of doublé cosets of the form TaT with a £ D, det(a) = £, [20] Chapter 3. Equivalently Ti can be described modularly in terms of isogenies of degree L When T — To(N) the action of the Hecke algebra on the g-expansion at oo is easily calculated. Namely,

,_ , , , ,«**(/) + ««/<(/W if( ' , iv) = i • an\J-lt) — \

and

an(Tltlf) = r2an(f\ae) if(.£,AT)

0 ìf£\N

fi-1 0\ where ai e TQ(N) is such that ai = ( . 1 mod N and with the convention that

am = 0 if m ^ Z. This explicit knowledge of the action on (/-expansions allows to prove some remarkable properties of the Hecke algebras, such as duality theorems.

It would be quite interesting to know the action of the Hecke operators on the expansions (3.4), (4.10) or, equivalently, on the numbers cn(f). On one side this would permit to prove directly faets about Hk(T) for D split. On another side, it would give a formula linking the Hecke eigenvalues to the values at CM points of the Maass operators. (5.4) Let / e Mk(T0(N), Rv) and r a CM point as in Section 3. The formai power series J] n(c n( / ) /n!)T n G £JP1] defìnes an Iwasawa function, hence a measure fi = fiftT

such that Jz xn dj.i = cn(f). Is the Mellin transform of \i related to the standard p-adic L-function of / ?

374 A. Mori

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Andrea MORI Dipartimento di Matematica Università di Torino via Carlo Alberto 10 10123 Torino, Italy.