Zeta Functions of Curves with no Rational Points

TL;DR

This paper proves that the motivic zeta functions of certain algebraic curves without rational points are rational functions, extending previous results and involving the study of Severi-Brauer schemes in the Grothendieck ring.

Contribution

It establishes the rationality of motivic zeta functions for curves lacking rational points, generalizing earlier known cases.

Findings

Motivic zeta functions of these curves are rational.

Class of Severi-Brauer schemes analyzed in the Grothendieck ring.

Extension of rationality results to broader class of curves.

Abstract

We show that the motivic zeta functions of smooth, geometrically connected curves with no rational points are rational functions. This was previously known only for curves whose smooth projective models have a rational point on each connected component. In the course of the proof we study the class of a Severi-Brauer scheme over a general base in the Grothendieck ring of varieties.