Finiteness of certain products of algebraic groups over a finite field (Algebraic Number Theory and Related Topics 2012)

Abstract

Let G_{1}, ..., G_{n} be smooth connected and commutative algebraic groups over a finite field F. We show the finiteness of the tensor product (G_{1}Motimescdots Motimes G_{n})(Spec F) of G_{1}, ..., G_{n} in the category of Mackey functors and also Ivorra-Rülling s K-group T(G_{1}, ..., G_{n})(Spec F) associated with those algebraic groups as reciprocity functors. We apply this to prove that, for a product of open curves, the finiteness of the relative Chow group and an abelian fundamental group which classies abelian coverings with bounded ramication along the boundary.