On $p$-adic $L$-functions for Hilbert modular forms

TL;DR

This paper constructs canonical $p$-adic $L$-functions for Hilbert modular forms over totally real fields, applicable in broad conditions without slope restrictions, advancing the understanding of their $p$-adic properties.

Contribution

It introduces a new canonical construction of $p$-adic $L$-functions that varies in $p$-adic families without slope or non-criticality constraints.

Findings

Constructed $p$-adic $L$-functions for Hilbert modular forms.

Established a canonical map from overconvergent cohomology to distributions.

Proved smoothness of eigenvarieties at critically refined points.

Abstract

We construct $p$-adic $L$-functions associated with $p$-refined cohomological cuspidal Hilbert modular forms over any totally real field under a mild hypothesis. Our construction is canonical, varies naturally in $p$-adic families, and does not require any small slope or non-criticality assumptions on the $p$-refinement. The main new ingredients are an adelic definition of a canonical map from overconvergent cohomology to a space of locally analytic distributions on the relevant Galois group and a smoothness theorem for certain eigenvarieties at critically refined points.