Transcript Document 7779700

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.
f
aH
H
G
aH
H
G/H
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. 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}
= A 2  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 nonalgebraically 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
f
f
(r , s )( x  r ) 
(r , s )( y  s )  0
x
y
Sharp point P (singular point) of a curve: if
f
f
( P) 
( P)  0
x
y
Singular Curve
In projective plane can change coordinates
for …
To be continuous… (this Friday)