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.
Categories: math.NT math.AG