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

Abstract

Let p ≥ 3 be a 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 with 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and f Steinberg at p, we give a formula for the p-adic derivative at s = 1 of this p-adic L-function, generalizing unpublished work of Greenberg and Tilouine.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.

Derivative of symmetric square p-adic L-functions via pull-back formula

“…That's why in [Ros13a,Ros13b] and in this article we can deal only with forms which are Steinberg at p.…”

“…This method has been used successfully many other times [Mok09,Ros13a,Ros13b]. It is very robust and easily adaptable to many situations in which the expected order of the trivial zero is one.…”

“…For example, in [Gre94,Mok09] we have s(κ) = 1 2 Log p (π(κ)) and in [Ros13a,Ros13b] we have s(κ) = Log p (π(κ)) − 1. In the first case the line corresponds to the vanishing on the central critical line which is a consequence of the fact that the ε-factor is constant in the family.…”

“…This is why in [Mok09,Ros13a] the authors deal only with forms of parallel weight. It seems very hard to generalize the method of Greenberg and Stevens to higher order derivatives without new ideas.…”

“…The aim of this section is to recall the method of Greenberg and Stevens [GS93] to calculate analytic L-invariant. This method has been used successfully many other times [Mok09,Ros13a,Ros13b]. It is very robust and easily adaptable to many situations in which the expected order of the trivial zero is one.…”