Higher Hida theory for Hilbert modular varieties in the totally split case

TL;DR

This paper develops a higher Hida theory for Hilbert modular varieties over totally real fields where the prime p is totally split, constructing modules that interpolate cohomology groups across weights.

Contribution

It extends higher Hida theory to the setting of Hilbert modular varieties with totally split primes, constructing interpolating modules for cohomology groups.

Findings

Constructed modules $M^q$ interpolating cohomology across weights

Extended Hida theory to totally split prime case

Provides p-adic interpolation of automorphic sheaf cohomology

Abstract

We study $p$-adic properties of the coherent cohomology of some automorphic sheaves on the Hilbert modular variety $X$ for a totally real field $F$ in the case where the prime $p$ is totally split in $F$. More precisely, we develop higher Hida theory \`{a} la Pilloni, constructing, for $0\leq q\leq [F:\mathbb{Q}]$, some modules $M^q$ which $p$-adically interpolate the ordinary part of the cohomology groups $H^q(X, \underline{\omega}^{\kappa})$, varying the weight $\kappa$ of the automorphic sheaf.