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

Let ${\mathfrak{V}\mathfrak{a}\mathfrak{r}}_k^G$ 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 ${\mathfrak{V}\mathfrak{a}\mathfrak{r}}_k^G$ 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 ${\sum }_{n=0}^{\mathrm{\infty }}[{\text{Sym}}^n(X,\sigma )]t^n$ 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.