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

Abstract

In this paper, we propose and explore a new connection in the study of $p$-adic $L$-functions and eigenvarieties. We use it to prove 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 non-vanishing of a (standard) $p$-adic $L$-function implies smoothness of the eigenvariety at such points. Key to our methods are a family of distribution-valued functionals on (parahoric) overconvergent cohomology groups, which we construct via $p$-adic interpolation of classical representation-theoretic branching laws for ${\mathrm{GL}}_n\times {\mathrm{GL}}_n\subset {\mathrm{GL}}_{2n}$. More precisely, we use our functionals to 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. Our new parahoric distribution coefficients allow us to obtain optimal non-critical slope and growth bounds for this construction. When $\pi$ has regular weight and the corresponding $p$-adic Galois representation is irreducible, we exploit non-vanishing of our functionals to show that the parabolic eigenvariety for ${\mathrm{GL}}_{2n}/F$ is étale at $\tilde{\pi }$ over an $([F:\mathbb{Q}]+1)$-dimensional weight space and contains a dense set of classical points admitting Shalika models. 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.

69 pages (inc. glossary of notation). Accepted version, now published in Advances in Mathematics