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Galois-generic points
on Shimura varieties
Anna Cadoret and Arno Kret |
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Vol. 10 (2016), No. 9, 1893–1934
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DOI: 10.2140/ant.2016.10.1893
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Abstract |
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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 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 . 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. |
Keywords
Shimura varieties, Hecke orbits, Adelic representations of
étale fundamental group, Galois generic points
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Mathematical Subject Classification 2010
Primary: 11G18
Secondary: 20G35, 14F20
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Milestones
Received: 13 September 2015
Revised: 29 June 2016
Accepted: 12 August 2016
Published: 22 November 2016
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Authors |
| Anna Cadoret | |
| Centre de Mathématiques Laurent
Schwartz Ecole Polytechnique 91128 Palaiseau France |
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| Arno Kret | |
| Faculty of Science Korteweg–de Vries Instituut Postbus 94248 1090 GE Amsterdam Netherlands |
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