Abelian Sheaves over Finite Fields

Abelian sheaves were introduced by the second author as higher dimensional generalizations of Drinfeld modules and as the appropriate analogues of abelian varieties in the arithmetic of function fields. In this article we devellop their elementary theory regarding morphisms, isogenies, Tate modules, and study their reducibility up to isogeny into direct sums of simple components. Over finite fields we investigate their endomorphism algebras and obtain similar results to Tate’s famous results for abelian varieties. Since abelian sheaves with characteristic different from ∞ are the same as Anderson’s pure t-motives equipped with additional structure at ∞, all our results are also valid for pure t-motives. Mathematics Subject Classification (2000): 11G09, (13A35, 16K20)