Derivative of the standard $p$-adic $L$-function associated with a Siegel form

TL;DR

This paper constructs a two-variable p-adic L-function for Siegel forms, analyzes its derivative at trivial zeros, and confirms predictions of Greenberg's conjecture in this context.

Contribution

It develops a new two-variable p-adic L-function for Siegel forms and computes its derivative at trivial zeros, extending previous one-variable methods.

Findings

Constructed a two-variable p-adic L-function for Siegel forms.

Calculated the first derivative at trivial zeros using Greenberg--Stevens method.

Confirmed Greenberg's conjecture on trivial zeros for this setting.

Abstract

In this paper we construct a two variables $p$-adic $L$-function for the standard representation associated with a Hida family of parallel weight genus $g$ Siegel forms, using a method previously developed by B\"ocherer--Schmidt in one variable. When a form of weight $g+1$ is Steinberg at $p$, a trivial zero appears and, using the method of Greenberg--Stevens, we calculate the first derivative of this $p$-adic $L$-function and show that it has the form predicted by a conjecture of Greenberg on trivial zeros.