Rational fixed points for finite group actions, and a question of Serre

Abstract:  Serre asked if the action of a finite $\ell$-group on an affine space over a field of characteristic different from $\ell$ always admits a rational fixed point. I will give a brief survey of known results and techniques, and show how one can use Loeser and Sebag's motivic Serre invariant to answer the question affirmatively if the base field is a henselian discretely valued field of characteristic zero with algebraically closed residue field of characteristic different from $\ell$. This is joint work with Hélène Esnault.