$L$-functions of ${\mathrm{GL}}(2n):$ $p$-adic properties and nonvanishing of twists

Mladen Dimitrov, Fabian Januszewski, A. Raghuram

Published 2018-02-27Version 1

The principal aim of this article is to attach and study $p$-adic $L$-functions to cohomological cuspidal automorphic representations $\Pi$ of ${\mathrm{GL}}_{2n}$ over a totally real field $F$ admitting a Shalika model. We use a modular symbol approach, along the global lines of the work of Ash and Ginzburg, but our results are more definitive since we draw heavily upon the methods used in the recent, and separate, works of all the three authors. Our $p$-adic $L$-functions are by construction distributions on the Galois group of the maximal abelian extension of $F$ unramified outside $p\infty.$ Moreover, we prove the so-called Manin relations between the $p$-adic $L$-functions at {\it all} critical points. This has the striking consequence that, given a $\Pi$ admitting at least two critical points, and given a prime $p$ such that $\Pi_p$ is ordinary, the central critical value $L(\tfrac12, \Pi\otimes\chi)$ is not zero for all except finitely many Dirichlet characters $\chi$ of finite order which are unramified outside $p\infty$.