Motivic zeta-functions and the Grothendieck ring for varieties with group actions

Let VarGk 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 VarGk 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=0infinity [Symn(X, sigma)] tn with coefficients in the Grothendieck ring. The main result of this paper asserts that when 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.;To prove this we study the Picard bundle Symn X over Picn X, a fiber bundle with a fiber preserving G action on it. Studying the vector bundle En whose projectivization is Symn X, we find that En can be broken up into a product in the Grothendieck ring of a smaller bundle Em and an affine space with a regular G action. This is used to construct a formula for Symn X in the Grothendieck ring which can be used to prove the rationality of the power series.