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

Abstract

Let VarkG denote the category of pairs (X,σ), where X is a variety over k and σ 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 VarkG 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 ∑n=0∞[Symn(X,σ)]tn 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.