The fibration method: a local-global principle in family

par Elyes Boughattas

jeudi 5 mars 2026, 14:00 → 15:00 Europe/Paris

M7-411 (ENS Lyon)

A starting point of arithmetic is to solve diophantine equations over the field Q of rational numbers. More generally, given an algebraic variety, can we say whether it has a rational point? Local-global principles are a well-known strategy to tackle this question. For example, is the existence of solutions over Q implied by the existence of solutions over all completions of Q? The first non-examples to this question have been found in the 1960's. Then Manin introduced in 1970 a cohomological obstruction, called the "Brauer-Manin obstruction" which is conjectured to explain the lack of rational points for the nice family of rationally connected varieties.

The fibration method is a conjecture which predicts that the Brauer-Manin obstruction behaves well for families of varieties parametrised by the projective line. During the last ten years, this conjecture has known a new impetus, after the foundational works of Harpaz and Wittenberg.

During this talk, I will present a work in progress where I prove an analogue of the fibration method over function fields of curves over finite fields. To complete the picture, I will mention two joint works in progress towards both quantitative and qualitative aspects of the fibration method over Q.