Non-cuspidal Hida theory for Siegel modular forms and trivial zeros of $p$-adic $L$-functions

TL;DR

This paper develops a non-cuspidal Hida theory for Siegel modular forms to analyze the derivatives of $p$-adic $L$-functions at trivial zeros, advancing understanding of Greenberg's conjecture.

Contribution

It introduces a new Hida theory framework for non-cuspidal Siegel modular forms and applies it to study trivial zeros of $p$-adic $L$-functions.

Findings

Proves part of Greenberg's conjecture on trivial zeros.

Constructs an improved $p$-adic $L$-function using the developed theory.

Analyzes derivatives of $p$-adic $L$-functions at semi-stable trivial zeros.

Abstract

We study the derivative of the standard $p$-adic $L$-function associated with a $P$-ordinary Siegel modular form (for $P$ a parabolic subgroup of $\mathrm{GL}(n)$) when it presents a semi-stable trivial zero. This implies part of Greenberg's conjecture on the order and leading coefficient of $p$-adic $L$-functions at such trivial zero. We use the method of Greenberg-Stevens. For the construction of the improved $p$-adic $L$-function we develop Hida theory for non-cuspidal Siegel modular forms.