L -functions of GL 2 n : p -adic properties and non-vanishing of twists

Dimitrov, Mladen; Januszewski, Fabian; Raghuram, A.

Document type :

Compte-rendu et recension critique d'ouvrage

DOI :

10.1112/S0010437X20007551

Title :

L -functions of GL 2 n : p -adic properties and non-vanishing of twists

Author(s) :

Dimitrov, Mladen [Auteur]

Université de Lille
Januszewski, Fabian [Auteur]
University of Karlsruhe [TH]
Raghuram, A. [Auteur]
Indian Institute of Science Education and Research Pune [IISER Pune]

Journal title :

Compositio Mathematica

Pages :

2437-2468

Publisher :

Foundation Compositio Mathematica

Publication date :

2020-12

ISSN :

0010-437X

HAL domain(s) :

Mathématiques [math]/Théorie des nombres [math.NT]

English abstract : [en]

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 ...
Show more >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.Show less >

Language :

Anglais

Popular science :

Non

ANR Project :

Représentations galoisiennes, formes automorphes et leurs fonctions L

Collections :

Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL

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