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The cone conjecture for vector-valued Siegel automorphic forms

Jean-Stefan Koskivirta

Abstract

We prove that the effective cone of automorphic vector bundles on the Siegel modular variety of rank $n$ in characteristic $p$ at a place of good reduction is encoded by the stack of $G$-zips of Pink--Wedhorn--Ziegler. Specifically, we show that the degree zero cohomology groups of automorphic vector bundles always vanish outside of the zip cone. This result is a special case of a general conjecture formulated by the Goldring and the author for all Hodge-type Shimura varieties of good reduction. In the case $n=3$, we give explicit conditions for the vanishing of the $0$-th cohomology group. Finally, in the course of the proof we define the notion of automorphic forms of trivial-type and study their properties.

The cone conjecture for vector-valued Siegel automorphic forms

Abstract

We prove that the effective cone of automorphic vector bundles on the Siegel modular variety of rank in characteristic at a place of good reduction is encoded by the stack of -zips of Pink--Wedhorn--Ziegler. Specifically, we show that the degree zero cohomology groups of automorphic vector bundles always vanish outside of the zip cone. This result is a special case of a general conjecture formulated by the Goldring and the author for all Hodge-type Shimura varieties of good reduction. In the case , we give explicit conditions for the vanishing of the -th cohomology group. Finally, in the course of the proof we define the notion of automorphic forms of trivial-type and study their properties.
Paper Structure (35 sections, 25 theorems, 134 equations)

This paper contains 35 sections, 25 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Vector bundles and quotient stacks
  3. Associated sheaf of a representation
  4. Torsors and global sections
  5. Finite étale group schemes
  6. Symmetric transforms
  7. Preliminaries
  8. Notation
  9. Parametrization of strata
  10. The ordinary locus
  11. Siegel and Hilbert--Blumenthal Shimura varieties
  12. Siegel modular varieties
  13. Hodge-type Shimura varieties
  14. Hilbert--Blumenthal Shimura varieties
  15. Toroidal compactification
  16. ...and 20 more sections

Key Result

Theorem 3.3.1

Theorems & Definitions (43)

  • Theorem 3.3.1: Pink-Wedhorn-Ziegler-zip-data
  • Theorem 3.6.1: Koskivirta-automforms-GZip, Imai-Koskivirta-vector-bundles
  • Proposition 4.4.1
  • proof
  • Theorem 4.5.1: Lan-Stroh-stratifications-compactifications
  • Lemma 4.6.1
  • proof
  • Corollary 4.6.2
  • Theorem 4.7.1: van-hoften-ordinary-hecke
  • Theorem 4.9.1
  • ...and 33 more