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On subgroups of semi-abelian varieties defined by difference equations


Authors: Zoé Chatzidakis and Ehud Hrushovski
Journal: Trans. Amer. Math. Soc. 369 (2017), 3673-3705
MSC (2010): Primary 03C60, 03C98, 12H10
DOI: https://doi.org/10.1090/tran/6924
Published electronically: December 30, 2016
MathSciNet review: 3605984
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Abstract: We study the induced structure on definable groups in existentially closed difference fields. If $ G$ is a definable subgroup of a semi-abelian variety, orthogonal to every definable field, we show that $ G$ is stable and stably embedded; every definable subset of $ G^n$ is a Boolean combination of cosets of definable subgroups of $ G^n$, and $ G^n$ has at most countably many definable subgroups. This generalises to positive characteristic earlier results of the authors.


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Additional Information

Zoé Chatzidakis
Affiliation: Département de Mathématiques et Applications (UMR 8553), Ecole Normale Supérieure, 45 rue d’Ulm, 75230 Paris Cedex 05, France
Email: zchatzid@dma.ens.fr

Ehud Hrushovski
Affiliation: Institute of Mathematics, The Hebrew University, Givat Ram, 91904 Jerusalem, Israel
Email: ehud@math.huji.ac.il

DOI: https://doi.org/10.1090/tran/6924
Received by editor(s): December 25, 2011
Received by editor(s) in revised form: March 30, 2015, and February 19, 2016
Published electronically: December 30, 2016
Additional Notes: The first author was partially supported by PITN-2009-238381 and by ANR-06-BLAN-0183, ANR-09-BLAN-0047, ANR-13-BS01-0006.
The research of the second author leading to these results received funding from the European Research Council under the European Unions Seventh Framework Programme (FP7/2007- 2013)/ERC Grant Agreement No. 291111, as well as the Israel Science Foundation 1048/07.
Article copyright: © Copyright 2016 American Mathematical Society

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