Parahoric level $p$-adic $L$-functions for automorphic representations of $\operatorname{GL}_{2n}$ with Shalika models

TL;DR

This paper constructs $p$-adic $L$-functions for certain automorphic representations of $ ext{GL}_{2n}$ with Shalika models over totally real fields, and proves their properties in families, advancing the understanding of automorphic $L$-functions in the $p$-adic setting.

Contribution

It introduces new local tools like improved Shalika functionals and extends Shahidi's theory to Shalika models, enabling the construction of $p$-adic $L$-functions in a broader context.

Findings

Construction of $p$-adic $L$-functions for automorphic representations with Shalika models.

Proof of etaleness of the parabolic eigenvariety at relevant points.

Development of new local ingredients and factorization formulas for local coefficients.

Abstract

We construct $p$-adic $L$-functions for regularly refined cuspidal automorphic representations of symplectic type on $\operatorname{GL}_{2n}$ over totally real fields, which are parahoric spherical at every finite place. Furthermore, we prove etaleness of the parabolic eigenvariety at such points and construct $p$-adic $L$-functions in families. The novel local ingredients are the construction of improved Ash--Ginzburg Shalika functionals and production of Friedberg--Jacquet test vectors relating local zeta integrals to automorphic $L$-functions beyond the spherical level. Our proofs rely on a generalization of Shahidi's theory of local coefficients to Shalika models, for which we establish a general factorization formula related to the exterior square automorphic $L$-function.