MODULAR SYMBOLS AND L-FUNCTIONS
BRANDON LEVIN
1.
Introduction
As the rst non-overview lecture in this seminar, we will be setting up a lot of notation, getting
comfortable working with modular symbols, and then hopefully discussing some of the major input
LECTURE 1: GENERALITIES ON ENDOMORPHISMS (2/15/12)
NOTES BY I. BOREICO
In this lecture and the two that follow, we will summarize many facts about abelian varieties and
their endomorphisms, with an emphasis on making statements over a general ground eld (
TWO-VARIABLE p-ADIC L-FUNCTIONS
PAYMAN L KASSAEI
1. Introduction
This is a write-up of my talk in the Stanford reading group on the work of BertoliniDarmon. The objective of my talk is to present a construction a two-variable p-adic Lfunction attached to
p-adic L-function of an ordinary weight 2 cusp form
The following exposition essentially summarizes one section of the excellent article [MS]. For any
xed cusp form S2 (0 (N ), Q), we will associate a p-adic modular form on weight space.
The idea is to pr
p-adic L-functions for Dirichlet characters
Rebecca Bellovin
1
Notation and conventions
Before we begin, we x a bit of notation.
We make the following convention: for a xed prime p, we set q = p if p is
odd, and we set q = 4 if p = 2.
We will always view
HIDA THEORY
CAMERON FRANC
Abstract.
Contents
1. Introduction
1.1. The Eisenstein family
1.2. Some congruences between cusp forms
1.3. Hida families
1.4. Outline of proof of Theorem 7
1.5. Periods of Hida families
2. Existence of Hida families
2.1. Tower o
LECTURE 2: CM TYPES, REFLEX FIELDS, AND MAIN THEOREM (2/22/12)
NOTES BY I. BOREICO
1. The CM type in characteristic 0.
Recall the setting at the end of the previous lecture: A is an abelian variety over a eld F of
characteristic zero, with dimension g > 0
Algebraic Hecke Characters
Motivation
This motivation was inspired by the excellent article [Serre-Tate, 7].
Our goal is to prove the main theorem of complex multiplication. The Galois theoretic
formulation is as follows:
Theorem (Main Theorem of Complex
Lecture 2: Abelian varieties
The subject of abelian varieties is vast. In these notes we will hit some highlights of the
theory, stressing examples and intuition rather than proofs (due to lack of time, among other
reasons). We will note analogies with th
SHIMURATANIYAMA FORMULA
BRIAN CONRAD
As we have seen earlier in the seminar in the talk of Tong Liu, if K is a CM eld and (A, i) is a complex
torus with CM by K , then A is necessarily an abelian variety and moreover the pair (A, i) descends to a
number e
FINAL LECTURE: THE PROOF OF THE GALOIS TWISTING FORM AND
THE ALGEBRAIC FORM OF THE MAIN THEOREM OF CM
(05/24/12-05/31/12)
Recall the premise of the previous lecture. We considered (A, i, ) an abelian variety with CM
structure, dened over Q, which (with al
CONSTRUCTIONS WITH FRACTIONAL IDEALS
JEREMY BOOHER
In preparation for the proof of the Main Theorem of Complex Multiplication, for an Abelian
variety A0 over K with CM type (, L) we need to construct an isomorphism so that
,P
/ A
A0 NN
O0
NNN
NNN
NNN
N&
N
ALGEBRAIC FORMULATION OF THE MAIN THEOREM OF CM
BRANDON LEVIN
1. Introduction
The goal of this talk is to introduce the algebraic form of the Main Thm of CM and show that
it implies the "down to earth" form of the main thm stated in Mikes lecture. Along t
THE SHIMURA-TANIYAMA FORMULA AND p-DIVISIBLE GROUPS
DANIEL LITT
1. Notation and Introduction
Let us x the following notation:
K is a number eld;
L is a CM eld with totally real subeld L+ ;
(A, i) is an Abelian variety/K with CM i : L End0 (A) (remark: