An introduction to $p$-adic $L$-functions

TL;DR

This paper introduces $p$-adic $L$-functions, focusing on their various constructions and the proof of the Iwasawa Main conjecture, highlighting their significance in modern number theory.

Contribution

It provides an expository overview of different constructions of $p$-adic $L$-functions and discusses the proof of the Iwasawa Main conjecture for Vandiver primes.

Findings

The constructions of $p$-adic $L$-functions via measure, cyclotomic units, and Galois modules are equivalent.

The Iwasawa Main conjecture is proven for Vandiver primes.

Connections to modern research in number theory are explored.

Abstract

These expository notes introduce $p$-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.