$p$-adic L-functions of automorphic forms and exceptional zeros

Abstract

We construct $p$-adic L-functions for automorphic representations of $\mathrm{GL}$_2 of a number field $F$ , and show that the corresponding $p$-adic L-function of a modular elliptic curve $E$ over $F$ has an extra zero at the central point for each prime above $p$ at which $E$ has split multiplicative reduction, a part of the exceptional zero conjecture.