Here is my Part III essay (supervised by Professor John H. Coates)
Contents
Chapter 1 Introduction
1.1 Kummer's criteria
1.2 p-adic measure theory
1.3 Elliptic curves with complex multiplication
Chapter 2 Grössencharacter
2.1 Commutants
2.2 How CM matters? Defining the Grössencharacter
2.2.1 Factorisation Theorem
2.2.2 Grössencharacter in the sense of Hecke
2.3 Connection with L-values
Chapter 3 Complex Function Theory
3.1 Expressing in terms of Kronecker-Eisenstein series
3.1.1 The cases n=1 and n=2
3.2 Elliptic units
3.3 Proof of the key identity
Chapter 4 p-adic L-function
4.1 Interpretations of the Iwasawa algebra
4.1.1 Algebraic definition
4.1.2 Interpretation as p-adic measures
4.1.3 Interpretation as power series
4.2 Existence of p-adic L-function
4.2.1 Step 1: Searching for a canonical element
4.2.2 Step 2: Relating with complex L-function
4.2.3 Step 3: Getting rid of λ
4.2.4 Step 4: Expressing the integral as a power series
Chapter 5 Iwasawa Main Conjecture
5.1 Kummer Theory for Elliptic Curves
5.2 One Variable Main Conjecture
Chapter 6 On the Birch and Swinnerton-Dyer Conjecture
6.1 Progress to date