Title :

On $p$-adic $L$-functions for $GL_{2n}$ in finite slope Shalika families

Author(s) :

Salazar, Daniel Barrera [Auteur]
Universidad de Chile = University of Chile [Santiago] [UCHILE]
Dimitrov, Mladen [Auteur]
Université de Lille
Williams, Chris [Auteur]
University of Nottingham, UK [UON]

HAL domain(s) :

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

English abstract : [en]

In this paper, we prove new results on the geometry of the cuspidal eigenvariety for $\mathrm{GL}_{2n}$ over a totally real number field $F$ at classical points admitting Shalika models. We also construct $p$-adic $L$-functions ...
Show more >In this paper, we prove new results on the geometry of the cuspidal eigenvariety for $\mathrm{GL}_{2n}$ over a totally real number field $F$ at classical points admitting Shalika models. We also construct $p$-adic $L$-functions over the eigenvariety around these points. Our proofs proceed in the opposite direction to established methods: rather than using the geometry of eigenvarieties to deduce results about $p$-adic $L$-functions, we instead show that -- via evaluation maps on parahoric overconvergent cohomology groups -- non-vanishing of a $p$-adic $L$-function implies smoothness of the eigenvariety at such points. More precisely, we attach a $p$-adic $L$-function to a non-critical refinement $\tilde\pi$ of a regular algebraic cuspidal automorphic representation $\pi$ of $\mathrm{GL}_{2n}/F$ which is spherical at $p$ and admits a Shalika model. This gives the first construction of $p$-adic $L$-functions in this generality beyond the $p$-ordinary setting. Further, when $\pi$ has regular weight and the corresponding $p$-adic Galois representation is irreducible, we show that the parabolic eigenvariety for $\mathrm{GL}_{2n}/F$ is \'etale at $\tilde\pi$ over an $([F:\mathbb{Q}]+1)$-dimensional weight space and contains a dense set of classical points admitting Shalika models. Finally, under a hypothesis on the local Shalika models at bad places which is empty for $\pi$ of level 1, we construct a $p$-adic $L$-function for the family.Show less >

Language :

Anglais

ANR Project :

Représentations galoisiennes, formes automorphes et leurs fonctions L
ULNE

Comment :

101 pages (inc. glossary of notation), comments welcome! Changes: v4: greatly expanded exposition and reformatted to improve readability. Rewrote material on Shalika new vectors. Other minor corrections. The main results are unchanged. v2,v3: minor corrections and expositional improvements

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL