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

Alexandre, Thibault

doi:10.2140/ant.2025.19.143

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

Thibault Alexandre

Vol. 19 (2025), No. 1, 143–193

DOI: 10.2140/ant.2025.19.143

Abstract

We prove vanishing results for the coherent cohomology of the good reduction modulo $p$ of the Siegel modular variety with coefficients in some automorphic bundles. We show that for an automorphic bundle with highest weight $λ$ near the walls of the antidominant 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 SageMath that computes explicitly the vanishing results.

Keywords

coherent cohomology, automorphic vector bundles, vanishing results, Siegel variety, Siegel modular variety, positive characteristic, highest weight, Weyl chamber, SageMath, Shimura varieties, automorphic representations, abelian schemes, g-zip theory.

Mathematical Subject Classification

Primary: 14G35

Milestones

Received: 12 February 2023

Revised: 20 December 2023

Accepted: 13 February 2024

Published: 4 December 2024

Authors

Thibault Alexandre
Institut de Mathématiques de Jussieu-Paris Rive Gauche Sorbonne Université Paris France

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Vol. 19, No. 1, 2025