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

TL;DR

This paper establishes vanishing results for the coherent cohomology of automorphic vector bundles over Siegel varieties in positive characteristic, providing new bounds and an explicit computational algorithm.

Contribution

It proves new vanishing theorems for automorphic bundles with near-wall weights and offers an explicit Sage algorithm for computing these results.

Findings

Cohomology is concentrated in degrees [0, e] for certain weights.

Vanishing results apply to non-regular and non-$p$-small weights.

Provides an explicit computational tool for the results.

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.