Let \(\pi\) be a p-ordinary cohomological cuspidal automorphic representation of \(GL_n(\mathbb A_{\mathbb Q})\). A conjecture of Coates--Perrin-Riou predicts that the (twisted) critical values of its L-function \(L(\pi \times \chi,s)\), for Dirichlet characters \(\chi\) of p-power conductor, satisfy systematic congruence properties modulo powers of p, captured in the existence of a p-adic L-function. For \(n = 1,2\) this conjecture has been known for decades, but for \(n\) at least 3 it is known only in special cases, e.g. symmetric squares of modular forms; and in all known cases, \(\pi\) is a functorial transfer from a proper subgroup of \(GL_n\). In this talk, I will give a gentle introduction to p-adic L-functions and describe some of their applications. I will then state the conjecture more precisely, and describe recent joint work with David Loeffler, in which we prove this conjecture for \(n=3\) (without any transfer or self-duality assumptions). 

At the end, I will describe a cute application of our construction. In the non-trivial weight setting, our p-adic L-function satisfies the so-called 'Manin relations', i.e. interpolates twisted L-values at all critical integers \(j\). By comparing with the existing constructions for symmetric square p-adic L-functions, we obtain a refinement of a theorem of Manhkopf on the algebraicity of L-values for \(GL(3)\). More precisely, we show that his archimedean periods, which are non-vanishing by work of Kasten--Schmidt, satisfy the compatibility in j expected by Coates--Perrin-Riou.