Anticyclotomic 𝑝-adic 𝐿-functions and the exceptional zero phenomenon

Abstract

Let A A be a modular elliptic curve over a totally real field F F , and let K / F K/F be a totally imaginary quadratic extension. In the event of exceptional zero phenomenon, we prove a formula for the derivative of the multivariable anticyclotomic p p -adic L L -function attached to ( A , K ) (A,K) , in terms of the Hasse-Weil L L -function and certain p p -adic periods attached to the respective automorphic forms. Our methods are based on a new construction of the anticyclotomic p p -adic L L -function by means of the corresponding automorphic representation.