X’morphisms & Projective Geometric J. Liu

Outline  Homomorphisms 1.Coset 2.Normal subgrups 3.Factor groups 4.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.

G G/H f H H 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 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 1.x N +y N = 1 (solutions of rational number) 2.X N +Y N = Z N (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 P 2 = {[a, b, c]: a, b, c are not all zero}/~  General define: P n = {[a 0, a 1,…, a n ]: a 0, a 1,…, a n are not all zero}/~

Algebraic  As we see above, P 2 is a factor group by normal subgroup L, which is a line go through (0,0,0).  It is easy to see P 2 with dim 2.  P 2 exclude the triple [0, 0, 0]   X +  Y +  Z = 0 is a line on P 2 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 P 2.

Definition of projective plane  Affine plane (Euclidean plane)  A 2 = {(x,y) : x and y any numbers}  P 2 = A 2  {the set of directions in A 2 } = A 2  P 1  P 2 has no parallel lines at all !  Two definitions are equivalence (Isomorphic).

Maps between them

Curves in the projective plane  Define projective curve C in P 2 in three variables as F(X, Y, Z) = 0, that is C = {(a, b, c): F(a, b, c) = 0, where [a, b, c]  P 2 }  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) = Y 2 Z-X 3 +XZ 2 = 0 with degree 3.

Affine part  As we know, P 2 = A 2  P 1, C  A 2 is the affine part of C, C  P 1 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) = x 2 +xy+x 2 y 2 +y 3 F(X, Y, Z) = X 2 Z 2 +XYZ 2 +X 2 Y 2 +Y 3 Z  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]  P 2 : 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 C 0 (Z) with rational curve as {(r,s)  A 2 : 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. F p )  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)