An introduction to p-adic L-functions

This is an expository introduction to p-adic L-functions and the foundations of Iwasawa theory.We focus on Kubota-Leopoldt's p-adic analogue of the Riemann zeta function, which we describe in three different ways.We first present a measure-theoretic (analytic) p-adic interpolation of special values of the Riemann zeta function.Next, we describe Coleman's (arithmetic) construction via cyclotomic units.Finally, we examine Iwasawa's (algebraic) construction via Galois modules over the Iwasawa algebra.The Iwasawa Main Conjecture, now a theorem due to Mazur and Wiles, says that these constructions agree.We will state the conjecture precisely, and give a proof when p is a Vandiver prime (which conjecturally covers every prime).Throughout, we discuss generalisations of these constructions and their connections to modern research directions in number theory. 1. Introduction 103 1.1.What do we cover in these notes?103 2. What is a p-adic L-function, and what should it do?105 2.1.Classical L-functions 105 2.2.Motivating questions for Iwasawa theory 107 2.3.The Riemann zeta function 110 2.4.p-adic L-functions 112 Part I: The Kubota-Leopoldt p-adic L-function 3. Measures and Iwasawa algebras 117 3.1.Preliminaries on p-adic Banach spaces 118 3.2.p-adic measures 119 3.3.The Iwasawa algebra 121 3.4.p-adic analysis and Mahler transforms 123 3.5.A measure-theoretic toolbox 126 3.6.Pseudomeasures 129 3.7.Locally analytic functions and distributions 132 3.8.Further remarks 134 4. The Kubota-Leopoldt p-adic L-function 136 4.1.The measure