Automorphic vector bundles with global sections on $G\text{-}\mathtt{Zip}^{{\mathcal{Z}}}$-schemes. (31st October 2018)

• Record Type: Journal Article
• Authors: Goldring, Wushi; Koskivirta, Jean-Stefan
• Is Part Of: Compositio mathematica. Volume 154:Number 12(2018)
• Page Start: 2586
• Page End: 2605
• Publication Date: 2018-10-31
• Subjects: 11F33, -- 11F46, -- 14L30 (primary); G-zips, -- Hasse invariants, -- Shimura varieties, -- automorphic vector bundles; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X18007467
• Languages: English
• ISSNs: 0010-437X

Abstract

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.