MATH Seminar

Title: Integral points on groupic varieties (work of Yang Cao and Fei Xu)
Seminar: Algebra
Speaker: J.L. Colliot-Thelene of CNRS et Universite Paris-Sud
Contact: David Zureick-Brown, dab@mathcs.emory.edu
Date: 2016-03-15 at 4:00PM
Venue: W304
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Abstract: Summary : Let G be a connected linear algebraic group over a field k. By definition, a groupic G-variety X over k is a smooth (left) G-variety with a dense open set isomorphic to G with its (left) action on itself. Let X be a groupic G-variety over a number field. Under a suitable noncompactness hypothesis for the simple factors of the semisimple part of G at the archimedean places, Cao and Xu show that the Brauer-Manin obstruction is the only obstruction to strong approximation for X off the archimedean places. The proof builds upon the case X=G (handled in earlier papers by Xu and the speaker, Harari, Demarche). The toric case (G is a torus) was already handled in a previous paper by Cao and Xu. For X projective, the statement is a weak approximation result and the theorem has been known for a long time (Sansuc).. The proof of the strong approximation result for an arbitrary groupic G-variety X involves novel arguments, both geometric and arithmetic.