Derivative at s = 1 of the p-adic L-function of the symmetric square of a Hilbert modular form

TL;DR

This paper derives a formula for the derivative at s=1 of a p-adic L-function associated with the symmetric square of a Hilbert modular form, addressing trivial zeros and generalizing prior unpublished results.

Contribution

It provides a new explicit formula for the p-adic derivative at s=1 of the symmetric square L-function, extending Greenberg and Tilouine's work to a broader setting.

Findings

Derived a formula for the p-adic derivative at s=1.

Proved a case of Greenberg's conjecture on trivial zeros.

Generalized previous unpublished results to new cases.

Abstract

Let p be an odd prime and F a totally real number field. Let f be a Hilbert cuspidal eigenform of parallel weight 2, trivial Nebentypus and ordinary at p. It is possible to construct a p-adic L-function which interpolates the complex L-function associated to the symmetric square representation of f. This p-adic L-function vanishes at s=1 even if the complex L-function does not. Assuming p inert and f Steinberg at p, we give a formula for the p-adic derivative at s=1 of this p-adic L-function, generalizing unpublished work of Greenberg and Tilouine. Under some hypotheses on the conductor of f we prove a particular case of a conjecture of Greenberg on trivial zeros.