X’morphisms & Projective Geometric

J. Liu

Outline

Homomorphisms1.Coset2.Normal subgrups3.Factor groups4.Canonical homomorphismsIsomomorphismsAutomomorphisms Endomorphisms

Homomorphisms

f: GG’ is a map having the following property

x, y G, we have f(xy) = f(x)f(y).

Where “” is the operator of G,

and “”is the operator of G’.

Some properties of homomorphism

f(e) = e’ f(x-1) = f(x)-1

f: GG’, g: G’ G” are both homomorphisms, then fg is homomorphism form G to G”

Kernel If ker(f) = {e’} then f is injective Image of f is a subgroup of G’

The group of homomorphisms

A, B are abelian groups then, Hom(A,B) denote the set of homomorphisms of A into B. Hom(A,B) is a group with operation + define as follow.

(f+g)(x) = f(x)+g(x)

Cosets

G is a group, and H is a subgroup of G. Let a be an element of G. the set of all elements ax with xH is called a coset of H in G, denote by aH. (left or right)

aH and bH be coset of H in the group G. Then aH = bH or aHbH = .

Cosets can (class) G.

Lagrange’s theorem

Index of H: is the number of the cosets of H in group G.

order(G) = index(H)*order(H) Index(H) = order(image(f))

Normal subgroup

H is normal

for all xG such that xH = Hx

H is the kernel of some homomorphism of G into some geoup

Factor group

The product of two sets is define as follow

SS’ = {xx’xS and x’S}{aHaG, H is normal} is a group, denote

by G/H and called it factor groups of G.A mapping f: GG/H is a homomorphism,

and call it canonical homomorphism.

G G/H

f

H H

aH aH

Isomomorphisms

If f is a group homomorphism and f is 1-1 and onto then f is a isomomorphism

Automorphisms

If f is a isomorphism from G to G then f is a automorphism

The set of all automorphism of a group G is a group denote by Aut (G)

Endomorphisms

The ring of endomorphisms. Let A be an abelian group. End(A) denote the set of all homomorphisms of A into itself. We call End(A) the set of endomorphism of A.

Thus End (A) = Hom (A, A).

Projective Algebraic Geometry

Rational Points on Elliptic Curves

Joseph H. Silverman & John Tate

Outline

General philosophy : Think Geometrically, Prove Algebraically.

Projective plane V.S. Affine planeCurves in the projective plane

Projective plane V.S. Affine plane

Fermat equations Homogenous coordinates Two constructions of projective plane Algebraic (factor group) Geometric (geometric postulate) Affine plane Directions Points at infinite

Fermat equations

1. xN+yN = 1 (solutions of rational number)

2. XN+YN= ZN (solutions of integer number)

3. If (a/c, b/c) is a solution for 1 is then [a, b, c] is a solution for 2. Conversely, it is not true when c = 0.

4. [0, 0, 0] …

5. [1, -1, 0] when N is odd

Homogenous coordinates

[ta, tb, tc] is homogenous coordinates with [a, b, c] for non-zero t.

Define ~ as a relation with homogenous coordinates

Define: projective plane P2 = {[a, b, c]: a, b, c are not all zero}/~

General define: Pn = {[a0, a1,…, an]: a0, a1,…, an are not all zero}/~

Algebraic

As we see above, P2 is a factor group by normal subgroup L, which is a line go through (0,0,0).

It is easy to see P2 with dim 2.P2 exclude the triple [0, 0, 0]X + Y + Z = 0 is a line on P2 with

points [a, b, c].

Geometry

It is well-know that two points in the usual plane determine a unique line.

Similarly, two lines in the plane determine a unique point, unless parallel lines.

From both an aesthetic and a practical viewpoint, it would be nice to provide these poor parallel lines with an intersection point of their own.

Only one point at infinity?

No, there is a line at infinity in P2.

Definition of projective plane

Affine plane (Euclidean plane)A2 = {(x,y) : x and y any numbers}P2 = A2 {the set of directions in A2}

= A2 P1 P2 has no parallel lines at all ! Two definitions are equivalence (Isomorphic).

Maps between them

Curves in the projective plane

Define projective curve C in P2 in three variables as F(X, Y, Z) = 0, that is C = {(a, b, c): F(a, b, c) = 0, where [a, b, c] P2 }

As we seen below, (a, b, c) is equivalent to it’s homogenous coordinator (ta, tb, tc), that is, F is a homogenous polynomial.

EX: F(X, Y, Z) = Y2Z-X3+XZ2 = 0 with degree 3.

Affine part

As we know, P2 = A2 P1, CA2 is the affine part of C, CP1 are the infinity points of C.

Affine part: affine curve

C’ = f(x, y) = F(X, Y, 1)Points at infinity: limiting tangent directions

of the affine part.(通常是漸進線的斜率 , 取 Z = 0)

Homogenization & Dehomogenization

Dehomogenization: f(x, y) = F(X, Y, 1)Homogenization:

EX: f(x, y) = x2+xy+x2y2+y3

F(X, Y, Z) = X2 Z2+XYZ2+X2Y2+Y3ZClassic algebraic geometry: complex

solutions, but here concerned non-algebraically closed fields like Q, or even in rings like Z.

Rational curve

A curve C is rational, if all coefficient of F is rational. (non-standard in A.G)

F() = 0 is the same with cF() = 0. (intger curve)

The set of ration points on C: C(Q) = {[a,b,c]P2: F(a, b, c) = 0 and a, b, cQ}

Note, if P(a, b, c)C(Q) then a, b, c is not necessary be rational. (homo. c.)

We define the set of integer points C0(Z) with rational curve as

{(r,s)A2 : f(r, s) = 0, r, sZ }For a project curve C(Q) = C(Z). It’s also possible to look at polynomial equati

ons and sol in rings and fields other than Z or Q or R or C.(EX. Fp)

The tangent line to C at P is

0))(,())(,(

sysry

frxsr

x

f

Sharp point P (singular point) of a curve: if

Singular Curve In projective plane can change coordinates

for …

To be continuous… (this Friday)

0)()(

Py

fP

x

f