$p$-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.