On p-adic L-functions for GL2n in finite slope Shalika families

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 GL<inf>2n</inf> 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 GL<inf>n</inf>×GL<inf>n</inf>⊂GL<inf>2n</inf>. More precisely, we use our functionals to attach a p -adic L -function to a non-critical refinement π˜ of a regular algebraic cuspidal automorphic representation π of GL<inf>2n</inf>/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 π 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 GL<inf>2n</inf>/F is étale at π˜ over an ([F: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 π of level 1, we construct a p -adic L -function for the family. © 2025 The Author(s).