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2016 Galois-generic points on Shimura varieties
Anna Cadoret, Arno Kret
Algebra Number Theory 10(9): 1893-1934 (2016). DOI: 10.2140/ant.2016.10.1893

Abstract

We discuss existence and abundance of Galois-generic points for adelic representations attached to Shimura varieties. First, we show that, for Shimura varieties of abelian type, -Galois-generic points are Galois-generic; in particular, adelic representations attached to such Shimura varieties admit (“lots of”) closed Galois-generic points. Next, we investigate further the distribution of Galois-generic points and show the André–Pink conjecture for them: if S is a connected Shimura variety associated to a -simple reductive group, then every infinite subset of the generalized Hecke orbit of a Galois-generic point is Zariski-dense in S. Our proof follows the approach of Pink for Siegel Shimura varieties. Our main contribution consists in showing that there are only finitely many Hecke operators of bounded degree on (adelic and connected) Shimura varieties. Compared with other approaches of this result, our proof, which relies on Bruhat–Tits theory, is effective and works for arbitrary Shimura varieties.

Citation

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Anna Cadoret. Arno Kret. "Galois-generic points on Shimura varieties." Algebra Number Theory 10 (9) 1893 - 1934, 2016. https://doi.org/10.2140/ant.2016.10.1893

Information

Received: 13 September 2015; Revised: 29 June 2016; Accepted: 12 August 2016; Published: 2016
First available in Project Euclid: 16 November 2017

zbMATH: 06657570
MathSciNet: MR3576114
Digital Object Identifier: 10.2140/ant.2016.10.1893

Subjects:
Primary: 11G18
Secondary: 14F20, 20G35

Rights: Copyright © 2016 Mathematical Sciences Publishers

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Vol.10 • No. 9 • 2016
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