On the Galois structure of equivariant line bundles on curves

Abstract. Let k be a finite field, and let X be a smooth, projective curve over k with structure sheaf O. Let G be a finite group, and write Cl (O[G]) for the reduced Grothendieck group of the category of O[G]-vector bundles. In this paper we describe explicitly the subgroup of Cl (O[G]) which is generated by the classes arising from G-stable invertible sheaves on tame Galois covers of X which have Galois group G. Introduction. Let k be a finite field of characteristic p, and let G be a finite abelian group. Suppose that f: Y,! X is a tamely ramified Galois covering of smooth projective curves over k, with Galois group G. (We shall refer to such coverings as “tame G-covers ” of X). In this paper we study the structure of G-stable line bundles on such curves Y