Vanishing results for the coherent cohomology of automorphic vector bundles over the Siegel variety in positive characteristic

Abstract

We prove vanishing results for the coherent cohomology of the good reduction modulo $p$ of the Siegel variety with coefficients in some automorphic bundles. We show that for an automorphic bundle with highest weight $\lambda$ near the walls of the anti-dominant Weyl chamber, there is an integer $e \geq 0$ such that the cohomology is concentrated in degrees $[0, e]$. The accessible weights with our method are not necessarily regular and not necessarily $p$-small. Since our method is technical, we also provide an algorithm written in Sage that computes explicitly the vanishing results.