It has already been noted in the past that there is a deep connection between number theory, algebraic geometry and algebraic topology. An example of this was Grothendieck’s proof of the rationality part of the Weil conjectures, where he provided an étale cohomological interpretation of the Hasse-Weil zeta function for “nice” varieties over finite fields.

The goal of this talk is to follow the work of Lars Hesselholt and extend this result to the realm of homotopy theory, providing a cohomological interpretation of the Hasse-Weil zeta function using the cohomology associated to a certain spectrum, namely the Topological Periodic Cyclic Homology spectrum.