Math 390C Algebraic Combinatorics
Fall 2008
Instructor: Geir Helleloid
Homework 5 Solutions
1. Let G be a tree on n + 1 vertices with one vertex identied as the sink. What is the
chip-ring group of G (that is, what are the stable recurrent chip conguratio
Elliptic Curves
Notes taken by James Jones for Dr. Voloch
March 4, 2010
Let k be a eld containing a primitive N th root of unity so that
char(k ) = 0 or char(k ) = p N . Let E/k (t) be an elliptic curve with j+
invariant t. Recall FN = k (t)(E [N ]) and F
Elliptic Curves
Felipe Voloch
March 30, 2010
X1 (N ) X0 (N )
We consider here
X1 (N ) X0 (N ),
which we may choose to think of as
(E, P ) (E, P ) or alternatively, X (N )/H1 X (N )/H2
where
H1 =
1b
01
H2 =
ab
0d
SL2 (Z/N Z)/ 1
: ad = 1
SL2 (Z/N Z)/ 1
No
ELLIPTIC CURVES, MODULAR CURVES, AND MODULAR FORMS
4/6/10
We begin with a computation involving the Hecke operator Tp , p a prime. We have, for
an elliptic curve E (suppressing the level structure from the notation),
Tp Tp (E ) = Tp
E/C
=
(E/C )/C .
C E/C
15
an
ns
an e2in
L(E/Q, s) =
n=1
f ( ) =
n=1
(s) := (2 ) (s)L(E/Q, s)
(s) = N 1s (2 s)
f ( ) = N 1 2 f (1/N )
s
0 1
N0
1
N
GL2 \SL2
i
(i )s f ( )d /
(s) =
0
X0 (N ) = cfw_(E, C ) | C E, C N
(Ei , C ) j (E, C )
X0 ( N ) P 1
j
w N : X0 ( N ) X 0 ( N
1
Notes 02/25/10
Corrections: Corrections from previous notes.
1. Y (2) = A1 \ cfw_0, 1 = P1 \ cfw_0, 1,
2. Y0 (N ) is never a ne moduli space.
We state a few result whose proof will be given later.
Theorem. Let E be an elliptic curve over K (t) (t a var
MODULAR CURVES
FELIPE VOLOCH
FEB 23, 2010
(NOTES BY YUAN YAO)
1. Level structure
We want to study the moduli space of elliptic curves together with some
specic structure related to their torsion. These structures are called level
structure, and have the f
ELLIPTIC CURVES, MODULAR CUVERS, AND MODULAR FORMS,
WEEK 1
Denition 1. Let K be a eld. An elliptic curve E/K is a smooth projective curve of
genus 1 over K , together with a point O E (K ).
We will prove that an an elliptic curve has an equation of the fo
1
The Group Law on an Elliptic Curve
January 26, 2010
We assume temporarily that K is algebraically closed. We x a smooth
projective curve C/k . A divisor on C is a formal linear combination of the
points of C with coecients in Z. We denote the collection
ELLIPTIC CURVES
WEEK 3
2-2-10
First, we recall what we covered last time about isogenies.
Denition 1. An isogeny is a (nonconstant) map f : E E between elliptic
curves such that f (O) = O .
Ex: [n] : E E , P nP .
Theorem 2. An isogeny is a homomorphism.
T
ELLIPTIC CURVES OVER C (CONTINUED)
FELIPE VOLOCH
FEB 18, 2010
(NOTES BY YUAN YAO)
1. Isogenies
Proposition 1.1. Let E1 , E2 be two elliptic curves over C, 1 , 2 the
corresponding lattices. Then f : E1 E2 is an isogeny if and only if there
is a complex num
Proposition 1. Let E/K be an elliptic curve.
(1) If char K = 0 or if char K = p, p not dividing n, then
E [n]
Z/nZ Z/nZ
(2) If char K = p > 0, then
E [pr ]
Z/pr Z or 0
If E is dened over K then E [n] is dened over its algebraic closure K , and we
can cons
Math 390C Algebraic Combinatorics
Fall 2008
Instructor: Geir Helleloid
Homework 1 Solutions
1. (Easy Question) Use generating functions to evaluate
(a)
n
k=0
(1)k
m
k
m
nk
Solution. We nd that
n
m
k
(1)k
n=0
k=0
(1)
=
mn
x
n
n
n=0
m
= (1 x) (1 + x)
= (1 x
Math 390C Algebraic Combinatorics
Fall 2008
Instructor: Geir Helleloid
Homework 2 Solutions
1. Prove that the number of set partitions of [n] in which you cannot nd i < j < k < l
such that i and k are in the same block, j and l are in the same block, and
Math 390C Algebraic Combinatorics
Fall 2008
Instructor: Geir Helleloid
Homework 3 Solutions
1. A threshold graph is a simple graph (no loops or multiple edges) which may be dened
inductively as follows:
The graph with one vertex is a threshold graph.
If
Math 390C Algebraic Combinatorics
Fall 2008
Instructor: Geir Helleloid
Homework 4 Solutions
1. Let be the hypercube graph on 2n vertices, that is, the vertices are the binary strings
of length n and there is an edge between two vertices if they dier in ex
1
Elliptic Curves Notes May 4, 2010
Lemma 1. Let E/Q be semistable. Let l and p be distinct primes with
l = 2, 3. Suppose E has multiplicative reduction at l and
vl () 0
mod p
Then Q(E [p])/Q is unramied at l.
Proof. Let E be given by y 2 = x3 + ax + b. =