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
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 unram
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 cur
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
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 fro
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
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.
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. Thes
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 pr
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
po
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] :
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
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]
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
(
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
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 follow
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 t
PHYSICS 2 SCIENCE & ENGINEERING PROFESSOR: MAXIM SUMETS
Test #1 :
Question 1
6.67 out of 6.67 points
When gravitational, magnetic and any forces other than static electric forces are not present,
elec