In 2009, Jean-Pierre Serre asked whether an action of a finite -group on an affine space over a field of characteristic different from always has a rational fixed point. Serre proved this result when the field is finite or algebraically closed, but the question turns out to be quite difficult over other base fields: for instance, Serre pointed out that it is not known whether an involution on affine 3-space over the field of rational numbers always has a rational fixed point. The problem is that the automorphism group of affine space is poorly understood in dimension >2.
　  In my talk, I will present a joint work with Hélène Esnault (FU Berlin) where we give a positive answer to Serre's question when the base field is a henselian discretely valued field with algebraically closed residue field of characteristic different from . For instance, our result applies to the field of complex Laurent series. The first part of the talk will be an introduction to the study of rational points in algebraic geometry.