An introduction to Eisenstein measures

Eischen, Ellen

Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 3.1, pp. 779-808.

Cet article fournit une introduction aux mesures d’Eisenstein, un outil puissant pour construire certaines fonctions $L$ $p$-adiques. Vues pour la première fois dans la réalisation par Serre des fonctions zêta de Dedekind $p$-adiques associées aux corps totalement réels, les mesures d’Eisenstein fournissent un moyen d’étendre les congruences de style kummerien, observées par Kummer pour les valeurs de la fonction zêta de Riemann (dites congruences de Kummer) à certaines autres fonctions $L$. En plus de retracer les développements clés, nous discutons certains défis qui se posent dans des contextes plus généraux, en concluant par certains qui restent ouverts.

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 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.

Classification : 11F33, 11F85, 11S40, 11F30, 11F03, 11R23
Mots clés : Eisenstein measures, $p$-adic families of modular forms, $p$-adic modular forms, $p$-adic $L$-functions, $p$-adic measures

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