E. E. Eischen | Searcharxiv

arXiv2021

An introduction to Eisenstein measures

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.

math.NT↗

arXiv2020

Applications of nonarchimedean developments to archimedean nonvanishing results for twisted $L$-functions

We prove the nonvanishing of the twisted central critical values of a class of automorphic $L$-functions for twists by all but finitely many unitary characters in particular infinite families. While this paper focuses on $L$-functions associated to certain automorphic representations of unitary groups, it illustrates how decades-old nonarchimedean methods from Iwasawa theory can be combined with the output of new machinery to achieve broader nonvanishing results. In an appendix, which concerns an intermediate step, we also outline how to extend relevant prior computations for $p$-adic Eisenstein series and $L$-functions on unitary groups to the case where primes dividing $p$ merely needs to be unramified (whereas prior constructions required $p$ to split completely) in the associated reflex field.

math.NT↗