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A vanishing theorem for vector-valued Siegel automorphic forms in characteristic $p$

Jean-Stefan Koskivirta

Abstract

We show that the space of vector-valued Siegel automorphic forms in characteristic $p$ is zero when the weight is outside of an explicit locus. This result is a special case of a general conjecture about Hodge-type Shimura varieties formulated in previous work with W. Goldring.

A vanishing theorem for vector-valued Siegel automorphic forms in characteristic $p$

Abstract

We show that the space of vector-valued Siegel automorphic forms in characteristic is zero when the weight is outside of an explicit locus. This result is a special case of a general conjecture about Hodge-type Shimura varieties formulated in previous work with W. Goldring.
Paper Structure (24 sections, 17 theorems, 125 equations)

This paper contains 24 sections, 17 theorems, 125 equations.

Table of Contents

  1. Shimura varieties
  2. Siegel-type Shimura varieties
  3. Hodge-type Shimura varieties
  4. Hecke operators
  5. Hilbert--Blumenthal Shimura varieties
  6. Definition
  7. Parametrization of strata
  8. The flag space
  9. Flag strata
  10. Functoriality
  11. Automorphic vector bundles
  12. Extension to the toroidal compactification
  13. Finite etale maps
  14. Stable base locus
  15. Hecke-equivariance and automorphic vector bundles
  16. ...and 9 more sections

Key Result

Theorem 1.3.1

Assume that $\mathbf{E}_{{\mathfrak p}}=\mathbb{Q}_p$. For any point $x\in S_K^{\mathop{\mathrm{ord}}\nolimits}$, the set ${\mathcal{H}}^p(x)$ is Zariski dense in $S_{K}$.

Theorems & Definitions (33)

  • Theorem 1.3.1
  • Theorem 2.2.1: Pink-Wedhorn-Ziegler-zip-data
  • Proposition 2.4.1
  • proof
  • Theorem 3.3.1: Lan-Stroh-stratifications-compactifications
  • Proposition 3.5.1
  • proof
  • Definition 3.6.1
  • Theorem 3.6.2
  • proof
  • ...and 23 more