École polytechnique  – Centre de Mathématiques Laurent Schwartz

11h: Haowen Zhang (Sorbonne Université et École Polytechnique).

"Weak approximation for homogeneous spaces over C((X,Y)) or C((X))(Y)"

We study obstructions to weak approximation for connected linear groups and homogeneous spaces with connected or abelian stabilizers over finite extensions of C((X, Y)) or function fields of curves over C((X)). We show that for connected linear groups, the usual Brauer-Manin obstruction works as in the case of tori, using some dévissage argument. However, this Brauer-Manin obstruction is not enough for homogeneous spaces, and this leads us to somehow combine the Brauer-Manin obstruction with descent obstructions along torsors under quasi-trivial tori. This is another natural tool used in the study of such questions, as done by Izquierdo and Lucchini Arteche for the study of obstructions to the existence of rational points.

14h: Giulio Bresciani (Scuola Normale Superiore di Pisa).

"Fields of moduli and arithmetic of quotient singularities"

(Joint work with A. Vistoli). Let k be a field with algebraic closure K, assume char k = 0 for simplicity. Given a variety X over K with additional structure (such as marked points, or a polarization), via Galois theory one can define the field of moduli X: loosely speaking, it is the smallest extension of k on which we can hope that X is defined.

Generalizing a result of Dèbes and Emsalem, we give a condition which ensures that X is defined over its field of moduli. We give two applications: we show that every curve with n>0 marked points is defined over its field of moduli, and we generalize a theorem of Shimura stating that a generic principally polarized abelian variety of odd dimension is defined over its field of moduli.

In dimensions higher than 1, the problem of fields of moduli versus field of definition is tightly connected to what we call arithmetic of quotient singularities, i.e. the study of whether a rational point of a variety with quotient singularities lifts to a resolution. We prove several results concerning this question.

15h30: Charles De Clercq (Université Sorbonne Paris Nord).

"Motifs et traces de Tate des variétés projectives homogènes"

On établira la classification complète des motifs de variétés projectives homogènes sous l'action d'un groupe semisimple, à coefficients finis. Le critère obtenu met en jeu une nouvelle famille d'invariants motiviques, les traces de Tate. Nous déduirons ensuite de ce résultat une généralisation de la caractérisation de l'équivalence motivique des groupes semisimples en fonction de leurs p-indices de Tits supérieurs.