$p$-adic $L$-functions of Hilbert cusp forms and the trivial zero conjecture

TL;DR

This paper proves a strong form of the trivial zero conjecture for $p$-adic $L$-functions associated with certain automorphic representations over totally real fields, using novel methods for higher order zeros.

Contribution

It introduces new techniques for analyzing higher order zeros of $p$-adic $L$-functions, extending previous approaches to a broader class of automorphic forms.

Findings

Proved the trivial zero conjecture at the central point for specific $p$-adic $L$-functions.

Developed a new approach for higher order zeros based on root number variation.

Constructed improved $p$-adic $L$-functions using automorphic symbols and overconvergent cohomology.

Abstract

We prove a strong form of the trivial zero conjecture at the central point for the $p$-adic $L$-function of a non-critically refined self-dual cohomological cuspidal automorphic representation of $\mathrm{GL}_2$ over a totally real field, which is Iwahori spherical at places above $p$. In the case of a simple zero we adapt the approach of Greenberg and Stevens, based on the functional equation for the $p$-adic $L$-function of a nearly finite slope family and on improved $p$-adic $L$-functions that we construct using automorphic symbols and overconvergent cohomology. For higher order zeros we develop a conceptually new approach studying the variation of the root number in partial families and establishing the vanishing of many Taylor coefficients of the $p$-adic $L$-function of the family.