Derived equivalence, Albanese varieties, and the zeta functions of 3-dimensional varieties

TL;DR

This paper proves that derived equivalent smooth, projective three-dimensional varieties over finite fields have identical zeta functions, extending known results about Albanese torsors and derived equivalences.

Contribution

It establishes the equality of zeta functions for derived equivalent 3-folds over finite fields, extending complex field results to finite fields.

Findings

Derived equivalent 3-folds over finite fields have equal zeta functions.

Extension of Albanese torsor isogeny results from complex to finite fields.

Connection between derived equivalence and arithmetic invariants established.

Abstract

We show that any derived equivalent smooth, projective varieties of dimension 3 over a finite field $\mathbb{F}_q$ have equal zeta functions. This result is an application of the extension to smooth, projective varieties over any field of Popa and Schnell's proof that derived equivalent smooth, projective varieties over $\mathbb{C}$ have isogenous Albanese torsors; this result is proven in an appendix by Achter, Casalaina-Martin, Honigs and Vial.