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On $p$-adic $L$-functions for $GL_{2n}$ 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 $\mathrm{GL}_{2n}$ 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 $\tildeπ$ of a regular algebraic cuspidal automorphic representation $π$ 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 $π$ 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 $\mathrm{GL}_{2n}/F$ is étale at $\tildeπ$ 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 $π$ of level 1, we construct a $p$-adic $L$-function for the family.