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

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 à la Pilloni, constructing, for $0\le q\le [F:\mathbb{Q}]$, some modules $M^q$ which $p$-adically interpolate the ordinary part of the cohomology groups $H^q(X,\underline{\ddot{I}}{‰}^{\hat{I}}º)$, varying the weight $\hat{I}º$ of the automorphic sheaf.