Zeta functions and Fourier-Mukai transforms

ongoing

Arithmetic geometry and the study of derived categories of coherent sheaves are two central areas of research in algebraic geometry. Despite their many points of contact, they have until recently remained largely disjoint. The zeta function of an algebraic variety over a finite field is one of the most studied invariants in arithmetic geometry, and a conjecture of Orlov predicts that this invariant can be detected by the derived category of coherent sheaves on the variety. In this project, I will prove this for large classes of varieties. To achieve this, I will enrich a wide range of techniques from arithmetic geometry with ideas that have classically been used in the study of derived categories. In this way, this project will also serve as a catalyst for further interaction between arithmetic geometry and derived categories.