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

TL;DR

This work establishes vanishing results for the coherent cohomology of good reductions of Siegel varieties with automorphic vector bundles in positive characteristic. It introduces a general vanishing propagation framework via a weight-encoding function $g_{I_0,e}$, built on a combination of $G$-Zip stratifications, generalized Hasse invariants, and a logarithmic Kodaira–Nakano vanishing theorem, and then leverages $ abla$-filtrations of tensor products to deduce new vanishing from known cases. A key strength is extending access beyond regular and $p$-small weights by incorporating orbitally $p$-close conditions, enabling explicit computations via a Sage algorithm and yielding concrete vanishing results in low-genus cases (notably $g=2,3$). The results provide a practical toolkit for deriving vanishing patterns for a wide class of automorphic bundles on Siegel varieties, with potential arithmetic consequences for automorphic representations in characteristic $p$. The combination of theoretical framework and computational implementation makes the approach usable for explicit weight-by-weight vanishing analysis and for probing the structure of coherent cohomology 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 $λ$ 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.

Figures (9)

• Figure 1: $g = 2$, $p = 5$.
• Figure 2: $E_2$-page of the spectral sequence.
• Figure 3: $E_2$-page of the spectral sequence when $e = 0$
• Figure 4: $E_2$-page of the spectral sequence when $e = 1$
• Figure 5: $g = 2, p = 5$. The weights $\lambda = ( k_1 \geq k_2)$ such that the cohomology is concentrated in degree $0$ contains in particular the positive parallel weights $(k,k)$ below $(-4,-4)$. The vanishing results in the region located near the positive parallel line comes from the degeneration with $I = \{(1,-1)\}$ and the rest corresponds to the degeneration with $I = \emptyset$.

Theorems & Definitions (149)

• Theorem : Theorem \ref{['th1']}
• Definition 2.1: MR2015057
• Proposition 2.2: MR2015057
• Remark 2.3
• Definition 2.4: MR2015057
• Proposition 2.5: MR2015057
• proof
• Definition 2.6
• Remark 2.7
• Lemma 2.8