X’mo rphisms & Projective Geometric

X’morphisms & Projective Geometric J. Liu

Outline • Homomorphisms • Coset • Normal subgrups • Factor groups • Canonical homomorphisms • Isomomorphisms • Automomorphisms • Endomorphisms

Homomorphisms • f: GG’ is a map having the following property  x, y G, we have f(xy) = 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: GG’, 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 xH 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 aHbH = . • 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 xG 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’xS and x’S} • {aHaG, H is normal} is a group, denote by G/H and called it factor groups of G. • A mapping f: GG/H is a homomorphism, and call it canonical homomorphism.

f aH aH H H G/H G

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 plane • Curves 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 • xN+yN = 1 (solutions of rational number) • XN+YN= ZN (solutions of integer number) • 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. • [0, 0, 0] … • [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 • P2has 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, CA2 is the affine part of C, CP1 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+Y3Z • Classic 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, cQ} • 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, sZ } • For a project curve C(Q) = C(Z). • It’s also possible to look at polynomial equations 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

Sharp point P (singular point) of a curve: if • Singular Curve • In projective plane can change coordinates for … • To be continuous… (this Friday)

X’mo rphisms & Projective Geometric