Variétés rationnelles
Horaires : Le mardi 20 septembre 2011, 17h30-18h30
Lieu : salle W
Arithmetic of algebraic groups in positive characteristic
Brian Conrad (Stanford University - page personnelle)
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.