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

Joaquín Rodrigues Jacinto, Chris Williams

TL;DR

The notes present a coherent introduction to $p$-adic $L$-functions centered on the Kubota--Leopoldt construction for the Riemann zeta function, illustrating three complementary viewpoints: analytic (measures and pseudo-measures), arithmetic (cyclotomic units and the Coleman map), and algebraic (Galois-module structures over the Iwasawa algebra). They establish a measure-theoretic foundation, prove the existence and interpolation properties of the $p$-adic zeta function, and connect these to Dirichlet $L$-values, culminating in Iwasawa's Main Conjecture in a Vandiver prime case. The exposition then develops the Coleman map as a bridge between cyclotomic units and $p$-adic $L$-functions, leading to an algebraic construction of the same objects and the Main Conjecture, with extensions to Eisenstein series and modular-analytic families. Overall, the work situates $p$-adic $L$-functions at the heart of Iwasawa theory and Beilinson-type conjectures, highlighting their role in linking special $L$-values, Selmer groups, and arithmetic of cyclotomic fields, and outlining avenues for generalization to broader automorphic contexts.

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.

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

TL;DR

The notes present a coherent introduction to -adic -functions centered on the Kubota--Leopoldt construction for the Riemann zeta function, illustrating three complementary viewpoints: analytic (measures and pseudo-measures), arithmetic (cyclotomic units and the Coleman map), and algebraic (Galois-module structures over the Iwasawa algebra). They establish a measure-theoretic foundation, prove the existence and interpolation properties of the -adic zeta function, and connect these to Dirichlet -values, culminating in Iwasawa's Main Conjecture in a Vandiver prime case. The exposition then develops the Coleman map as a bridge between cyclotomic units and -adic -functions, leading to an algebraic construction of the same objects and the Main Conjecture, with extensions to Eisenstein series and modular-analytic families. Overall, the work situates -adic -functions at the heart of Iwasawa theory and Beilinson-type conjectures, highlighting their role in linking special -values, Selmer groups, and arithmetic of cyclotomic fields, and outlining avenues for generalization to broader automorphic contexts.

Abstract

These expository notes introduce -adic -functions and the foundations of Iwasawa theory. We focus on Kubota--Leopoldt's -adic analogue of the Riemann zeta function, which we describe in three different ways. We first present a measure-theoretic (analytic) -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 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.
Paper Structure (83 sections, 111 theorems, 458 equations)

This paper contains 83 sections, 111 theorems, 458 equations.

Table of Contents

  1. Introduction
  2. What do we cover in these notes?
  3. What is a $p$-adic $L$-function, and what should it do?
  4. Classical $L$-functions
  5. Motivating questions for Iwasawa theory
  6. Special values and arithmetic data
  7. Iwasawa Main Conjectures
  8. Applications of $p$-adic methods to classical BSD
  9. The IMC for the Riemann zeta function
  10. The Riemann zeta function
  11. $p$-adic $L$-functions
  12. $p$-adic $L$-functions, a first idea
  13. Ideles, measures and Tate's thesis
  14. $p$-adic $L$-functions via measures
  15. Measures and Iwasawa algebras
  16. ...and 68 more sections

Key Result

theorem 1

Let $F$ be a number field with $r_1$ real embeddings, $r_2$ pairs of complex embeddings, $w$ roots of unity, discriminant $D$, and regulator $R$. The zeta function $\zeta_F$ has a simple pole at $s=1$ with residue where $h_F$ is the class number of $F$.

Theorems & Definitions (279)

  • remark 1
  • theorem 1: Class number formula
  • remark 2
  • theorem 2
  • proof
  • remark 3
  • lemma 1
  • proof
  • lemma 2
  • proof
  • ...and 269 more