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Time: Tuesday, September 20, 2011, 17h30-18h30

Location: salle W

Arithmetic of algebraic groups in positive characteristic

Brian Conrad (Stanford University - Personal website)

Let X be a projective variety over a global field k. Consider the set of projective varieties X' that become isomorphic to X over every completion of k. It is natural to wonder if the set of such X', taken up to k-isomorphism, is finite. Mazur proved such a finiteness result conditional on the Tate--Shafarevich conjecture when k is a number field and the component group of the automorphism scheme of X satisfies some group-theoretic finiteness properties. When k is a global function field, several new difficulties arise. We explain a bit about the structure theory of pseudo-reductive groups and how (together with strong approximation) it helps to overcome these problems.


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