A $p$-adic adjoint $L$-function and the ramification locus of the Hilbert modular eigenvariety

TL;DR

This paper investigates the ramification locus of the Hilbert modular eigenvariety over a totally real field, linking it to the $p$-adic properties of adjoint $L$-values through analytic pairings and $L$-ideals.

Contribution

It introduces a new connection between the ramification of eigenvarieties and $p$-adic adjoint $L$-values for totally real fields, extending previous work beyond $Q$.

Findings

Identifies the ramification locus in relation to $p$-adic adjoint $L$-values.

Develops an analytic twisted Poincaré pairing interpolating classical pairings.

Extends the theory of $L$-ideals to totally real fields.

Abstract

Let $F$ be a totally real field and $\mathscr{E}$ the middle-degree eigenvariety for Hilbert modular forms over $F$, constructed by Bergdall--Hansen. We study the ramification locus of $\mathscr{E}$ in relation to the $p$-adic properties of adjoint $L$-values. The connection between the two is made via an analytic twisted Poincar\'e pairing over affinoid weights, which interpolates the classical twisted Poincar\'e pairing for Hilbert modular forms, itself known to be related to adjoint $L$-values by works of Ghate and Dimitrov. The overall strategy connecting the pairings to ramification is based on the theory of $L$-ideals, which was used by Bella\"iche and Kim in the case where $F = \mathbb{Q}$.