Rationality of motivic zeta-functions for curves with finite abelian group actions

TL;DR

This paper proves that the motivic zeta-function for algebraic curves with finite abelian group actions is rational, extending previous conjectures and providing new insights into the structure of varieties with symmetries.

Contribution

It introduces a Grothendieck ring for varieties with group actions and demonstrates the rationality of their motivic zeta-functions for curves with finite abelian groups.

Findings

Motivic zeta-function for curves with group actions is rational.

Defines Grothendieck ring for varieties with group actions.

Generalizes aspects of Weil's First Conjecture.

Abstract

Let $\mathfrak{Var}_k^G$ denote the category of pairs $(X,\sigma)$, where $X$ is a variety over $k$ and $\sigma$ is a group action on $X$. We define the Grothendieck ring for varieties with group actions as the free abelian group of isomorphism classes in the category $\mathfrak{Var}_k^G$ modulo a cutting and pasting relation. The multiplication in this ring is defined by the fiber product of varieties. This allows for motivic zeta-functions for varieties with group actions to be defined. This is a formal power series $\sum_{n=0}^{\infty}[\text{Sym}^n (X,\sigma)]t^n$ with coefficients in the Grothendieck ring. The main result of this paper asserts that the motivic zeta-function for an algebraic curve with a finite abelian group action is rational. This is a partial generalization of Weil's First Conjecture.