An introduction to Eisenstein measures

TL;DR

This paper introduces Eisenstein measures as a key tool for constructing $p$-adic $L$-functions, tracing their development from classical cases to more complex settings, and discussing ongoing challenges.

Contribution

It provides a comprehensive introduction to Eisenstein measures, highlighting their role in extending Kummer congruences to broader $L$-functions and discussing unresolved issues.

Findings

Eisenstein measures facilitate the construction of $p$-adic $L$-functions.

They extend classical Kummer congruences to new contexts.

Challenges remain in generalizing Eisenstein measures to all settings.

Abstract

This paper provides an introduction to Eisenstein measures, a powerful tool for constructing certain $p$-adic $L$-functions. First seen in Serre's realization of $p$-adic Dedekind zeta functions associated to totally real fields, Eisenstein measures provide a way to extend the style of congruences Kummer observed for values of the Riemann zeta function (so-called {\em Kummer congruences}) to certain other $L$-functions. In addition to tracing key developments, we discuss some challenges that arise in more general settings, concluding with some that remain open.