On the GL(2n) eigenvariety: branching laws, Shalika families and $p$-adic $L$-functions

Salazar, Daniel Barrera; Dimitrov, Mladen; Graham, Andrew; Jorza, Andrei; Williams, Chris

Document type :

Pré-publication ou Document de travail

Title :

On the GL(2n) eigenvariety: branching laws, Shalika families and $p$-adic $L$-functions

Author(s) :

Salazar, Daniel Barrera [Auteur]
Universidad de Santiago de Chile [Santiago] [USACH]
Dimitrov, Mladen [Auteur]

Université de Lille
Graham, Andrew [Auteur]
Laboratoire de Mathématiques d'Orsay [LMO]
Jorza, Andrei [Auteur]
University of Notre Dame [Indiana] [UND]
Williams, Chris [Auteur]
University of Nottingham, UK [UON]

Publication date :

2022-11-15

HAL domain(s) :

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

English abstract : [en]

In this paper, we prove that a GL(2n)-eigenvariety is etale over the (pure) weight space at non-critical Shalika points, and construct multi-variabled $p$-adic $L$-functions varying over the resulting Shalika components. ...
Show more >In this paper, we prove that a GL(2n)-eigenvariety is etale over the (pure) weight space at non-critical Shalika points, and construct multi-variabled $p$-adic $L$-functions varying over the resulting Shalika components. Our constructions hold in tame level 1 and Iwahori level at $p$, and give $p$-adic variation of $L$-values (of regular algebraic cuspidal automorphic representations, or RACARs, of GL(2n) admitting Shalika models) over the whole pure weight space. In the case of GL(4), these results have been used by Loeffler and Zerbes to prove cases of the Bloch--Kato conjecture for GSp(4). Our main innovations are: a) the introduction and systematic study of `Shalika refinements' of local representations of GL(2n), evaluating their attached local twisted zeta integrals; and b) the $p$-adic interpolation of representation-theoretic branching laws for GL(n)$\times$GL(n) inside GL(2n). Using (b), we give a construction of many-variabled $p$-adic functionals on the overconvergent cohomology groups for GL(2n), interpolating the zeta integrals of (a). We exploit the resulting non-vanishing of these functionals to prove our main arithmetic applications.Show less >

Language :

Anglais

ANR Project :

Représentations galoisiennes, formes automorphes et leurs fonctions L

Comment :

57 pages. Comments welcome

Collections :

Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL

Files

2211.08126
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