10 octobre 2025 (salle 16-26-113)

14h: Melvyn El Kamel Meyrigne (Ecole Polytechnique).

Tate-Shafarevich group of a constant group of multiplicative type over a finite extension of Q((x_1,...,x_n)).

Let K be a finite extension of Q((x_1,...,x_n)) and let R be the integral closure of Q[[x_1,...,x_n]] in K with residue field k. Consider a group of multiplicative type G defined over K. We define the Tate-Shafarevich group with respect to the points of codimension 1 of Spec(R). We show the finiteness of a quotient of the Tate-Shafarevich group when G comes from a group of multiplicative type G_k defined over k provided that a geometrical hypothesis on the special fiber of a regular model of Spec(R) is satisfied. We then prove that the Tate-Shafarevich group is trivial when the ring of integers R is regular. Finally, we show how the arguments used can be adapted to work over other kinds of fields.

15h30: Harry Shaw (University of Bath).

The existence of quartic del Pezzo surfaces over global function fields which do not admit a quadratic point.

Creutz and Viray have recently proven the existence of quartic del Pezzo surfaces over Q which do not admit a quadratic point, despite every such surface admitting a quadratic point over any local field. We prove the analogous result over global function fields. This is joint work with Giorgio Navone, Katerina Santicola and Haowen Zhang. 

7 novembre 2025

M. Archita (Université d'Anvers).

Zhenghui Li (Sorbonne Université).

5 décembre 2025

Sho Tanimoto (Université de Nagoya).

Lucas Lagarde (Université Sorbonne Paris Nord).