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L-functions of GL2n: p-adic properties and non-vanishing of twists

Part of: Discontinuous groups and automorphic forms Algebraic number theory: local and $p$-adic fields

Published online by Cambridge University Press:  08 January 2021

Mladen Dimitrov ,
Fabian Januszewski  and
A. Raghuram
Mladen Dimitrov
Affiliation:
Laboratoire Paul Painlevé, CNRS, UMR 8524, University of Lille, 59000Lille, Francemladen.dimitrov@univ-lille.fr
Fabian Januszewski
Affiliation:
Institut für Mathematik, EIM, Paderborn University, Warburger Str. 100, 33098Paderborn, Germanyfabian.januszewski@math.upb.de
A. Raghuram
Affiliation:
Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pashan, Pune411021, Indiaraghuram@iiserpune.ac.in
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Abstract

The principal aim of this article is to attach and study $p$-adic $L$-functions to cohomological cuspidal automorphic representations $\Pi$ of $\operatorname {GL}_{2n}$ 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 because we draw heavily upon the methods used in the recent and separate works of all three authors. By construction, our $p$-adic $L$-functions are distributions on the Galois group of the maximal abelian extension of $F$ unramified outside $p\infty$. Moreover, we work under a weaker Panchishkine-type condition on $\Pi _p$ rather than the full ordinariness condition. Finally, we prove the so-called Manin relations between the $p$-adic $L$-functions at all critical points. This has the striking consequence that, given a unitary $\Pi$ whose standard $L$-function admits at least two critical points, and given a prime $p$ such that $\Pi _p$ is ordinary, the central critical value $L(\frac {1}{2}, \Pi \otimes \chi )$ is non-zero for all except finitely many Dirichlet characters $\chi$ of $p$-power conductor.

Keywords

p-adic L-functionsnon-vanishing of L-functionsautomorphic forms on GL(2n)

MSC classification

Primary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols
Secondary: 11S40: Zeta functions and $L$-functions 11F55: Other groups and their modular and automorphic forms (several variables) 11F70: Representation-theoretic methods; automorphic representations over local and global fields
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 12 , December 2020 , pp. 2437 - 2468
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Copyright
© The Author(s) 2021

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