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 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, 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)