Vol. 10, No. 9, 2016

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Galois-generic points on Shimura varieties

Anna Cadoret and Arno Kret

Vol. 10 (2016), No. 9, 1893–1934
DOI: 10.2140/ant.2016.10.1893

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.

Shimura varieties, Hecke orbits, Adelic representations of étale fundamental group, Galois generic points
Mathematical Subject Classification 2010
Primary: 11G18
Secondary: 20G35, 14F20
Received: 13 September 2015
Revised: 29 June 2016
Accepted: 12 August 2016
Published: 22 November 2016
Anna Cadoret
Centre de Mathématiques Laurent Schwartz
Ecole Polytechnique
91128 Palaiseau
Arno Kret
Faculty of Science
Korteweg–de Vries Instituut
Postbus 94248
1090 GE Amsterdam