On certain cohomology groups attached to $\mathfrak{p}^{\infty}$-towers of quaternionic Hilbert modular varieties

TL;DR

This paper constructs specific sheaf cohomology groups related to quaternionic Hilbert modular varieties, enabling the study of infinitesimal p-adic deformations of associated Galois representations in a new geometric framework.

Contribution

It introduces novel sheaf cohomology groups for quaternionic Hilbert modular varieties that connect to p-adic deformation theory of automorphic and Galois representations.

Findings

Defines cohomology groups intertwining p-infinity towers and sheaves.

Provides a geometric construction for infinitesimal Galois deformations.

Links automorphic deformations to geometric cohomology frameworks.

Abstract

For a totally real number field $F$ and a nonarchimedean prime $\mathfrak{p}$ of $F$ lying above a prime number $p$ we introduce certain sheaf cohomology groups that intertwine the $\mathfrak{p}^{\infty}$-tower of a quaternionic Hilbert modular variety associated to a quaternion algebra $D$ over $F$ that is split at $\mathfrak{p}$ and a $p$-adically admissible representation of $\mbox{PGL}_2(F_{\mathfrak{p}})$. Applied to infinitesimal $p$-adic deformations of the local factor at $\mathfrak{p}$ of a cuspidal automorphic representation $\pi$ of $D^*(\mathbb{A})$ this yields a natural construction of infinitesimal deformations of the Galois representation attached to $\pi$.