Pure Anderson Motives and Abelian \tau-Sheaves

TL;DR

This paper explores the relationship between pure t-motives and abelian au-sheaves, establishing an equivalence of their quasi-isogeny categories and developing foundational theories for both structures.

Contribution

It clarifies the connection between pure t-motives and abelian au-sheaves and develops their elementary theory including morphisms, isogenies, and Tate modules.

Findings

Established an equivalence of quasi-isogeny categories

Developed the theory of morphisms and isogenies for both structures

Analyzed Tate modules and local shtukas as analogs of p-divisible groups

Abstract

Pure t-motives were introduced by G. Anderson as higher dimensional generalizations of Drinfeld modules, and as the appropriate analogs of abelian varieties in the arithmetic of function fields. In order to construct moduli spaces for pure t-motives the second author has previously introduced the concept of abelian \tau-sheaf. In this article we clarify the relation between pure t-motives and abelian \tau-sheaves. We obtain an equivalence of the respective quasi-isogeny categories. Furthermore, we develop the elementary theory of both structures regarding morphisms, isogenies, Tate modules, and local shtukas. The later are the analogs of p-divisible groups.