A starting point of arithmetic is to solve diophantine equations over the field Q of rational numbers. More generally, given an algebraic variety over Q, 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. As an example, a geometric consequence of this conjecture in characteristic p, is that families of Severi-Brauer varieties parametrised by the projective line should be unirational.

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. This boils down to the study of explicit moduli spaces of curves on surfaces over finite fields.