p-adic L-functions provide a beautiful tool for making progress on important, yet extremely difficult, conjectures such as Birch–Swinnerton-Dyer and Bloch-Kato.

We expect them to exist in very wide generality, but in practice they are hard to construct, and even the fundamental case of GL(n) is poorly understood for n at least 3. In this talk, I will give an exposition of a method for constructing p-adic L-functions via overconvergent cohomology, which has been exploited by a number of people to give some of the most general available constructions for GL(2). This conceptually slick method was inspired by Stevens’ theory of overconvergent modular symbols, requires no ordinarity condition and varies very naturally in p-adic families. At the end of the talk I will describe ongoing joint work with Barrera and Dimitrov, sketching the application of this method to the case of self-dual automorphic representions of GL(2n), allowing an extension of previous results of Gehrmann/Dimitrov–Januszewski–Raghuram to the non-ordinary setting and allowing variation in families.