The arithmetic of L-functions has long been a topic of intense interest in number theory. Via the Bloch–Kato Conjecture and its p-adic avatars, special values of L-functions are expected to carry deep algebraic data, and good understanding of p-adic L-functions, eigenvarieties, and p-adic L-functions over eigenvarieties have been instrumental in most recent progress towards these conjectures. 

In this talk, we will focus on the construction of p-adic L-functions for regular algebraic symplectic cuspidal automorphic representation of GL(2n,Q) which are everywhere spherical, and sketch possible generalisations. The proofs will partly proceed in opposite direction to established methods: rather than using the geometry of eigenvarieties to deduce results about p-adic L-functions in families, we instead show how non-vanishing of a (standard) p-adic L-function implies smoothness of the eigenvariety at such points.