Location of Repository

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

Topics:
Mathematics - Algebraic Geometry, 14G10

Year: 2011

OAI identifier:
oai:arXiv.org:1103.2160

Provided by:
arXiv.org e-Print Archive

Download PDF:To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.