A dynamical approach to generalized Weil’s Riemann hypothesis and semisimplicity

In a letter to Weil, on November 9, 1959, Serre sketched an elegant proof of a Kähler analog of Weil’s Riemann hypothesis. The positive-characteristic analog of Serre’s result is thus called (by us) generalized Weil’s Riemann hypothesis and semisimplicity. In this talk, I shall discuss how a dynamical approach can be applied towards these questions. In particular, I will describe the relationships between the above questions, the standard conjectures, and our dynamical conjectures. In the end, some applications to abelian varieties and Kummer surfaces are given.