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Automorphic vector bundles with global sections on $G\text{-}\mathtt{Zip}^{{\mathcal{Z}}}$-schemes

Part of: Discontinuous groups and automorphic forms Algebraic groups

Published online by Cambridge University Press:  31 October 2018

Wushi Goldring  and
Jean-Stefan Koskivirta
Wushi Goldring
Affiliation:
Department of Mathematics, Stockholm University, Stockholm SE-10691, Sweden email wushijig@gmail.com
Jean-Stefan Koskivirta
Affiliation:
Department of Mathematics, South Kensington Campus, Imperial College London, London SW7 2AZ, UK email jeanstefan.koskivirta@gmail.com
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Abstract

A general conjecture is stated on the cone of automorphic vector bundles admitting nonzero global sections on schemes endowed with a smooth, surjective morphism to a stack of $G$-zips of connected Hodge type; such schemes should include all Hodge-type Shimura varieties with hyperspecial level. We prove our conjecture for groups of type $A_{1}^{n}$, $C_{2}$, and $\mathbf{F}_{p}$-split groups of type $A_{2}$ (this includes all Hilbert–Blumenthal varieties and should also apply to Siegel modular $3$-folds and Picard modular surfaces). An example is given to show that our conjecture can fail for zip data not of connected Hodge type.

Keywords

G-zipsHasse invariantsShimura varietiesautomorphic vector bundles

MSC classification

Primary: 11F33: Congruences for modular and $p$-adic modular forms 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms 14L30: Group actions on varieties or schemes (quotients)
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 12 , December 2018 , pp. 2586 - 2605
Copyright
© The Authors 2018 

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