Abstract

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 $n \geq 3$ 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 $GL_n$. I will explain what a $p$-adic $L$-function is, state the conjecture more precisely, and then report on ongoing joint work with David Loeffler, in which we prove this conjecture for $n=3$ (without any transfer or self-duality assumptions)