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

TL;DR

This paper establishes a formula for the derivative of multivariable anticyclotomic p-adic L-functions associated with modular elliptic curves over totally real fields and imaginary quadratic extensions, addressing the exceptional zero phenomenon.

Contribution

It introduces a new construction of the anticyclotomic p-adic L-function using automorphic representations, linking derivatives to classical L-values and p-adic periods.

Findings

Derived a formula relating derivatives of p-adic L-functions to classical L-values.

Developed a novel automorphic representation-based construction of p-adic L-functions.

Addressed the exceptional zero phenomenon in the anticyclotomic setting.

Abstract

Let A be a modular elliptic curve over a totally real field F, and let E/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-adic L-function attached to (A,E), in terms of the Hasse-Weil L-function and certain p-adic periods attached to the respective automorphic forms. Our methods are based on a new construction of the anticyclotomic p-adic L-function by means of the corresponding automorphic representation.