Pure Anderson Motives over Finite Fields

TL;DR

This paper explores the arithmetic properties of pure t-motives over finite fields, focusing on their semisimplicity, endomorphism rings, and isogenies, drawing parallels to Tate's results for abelian varieties.

Contribution

It provides new insights into the semisimplicity and endomorphism structures of pure t-motives over finite fields, extending Tate's theorems to this setting.

Findings

Semisimplicity of pure t-motives over finite fields is characterized by endomorphism algebra properties.

Examples of non-semisimple pure t-motives are constructed.

Criteria for the existence of isogenies between pure t-motives are established.

Abstract

In the arithmetic of function fields Drinfeld modules play the role that elliptic curves take on in the arithmetic of number fields. As higher dimensional generalizations of Drinfeld modules, and as the appropriate analogues of abelian varieties, G. Anderson introduced pure t-motives. In this article we study the arithmetic of the later. We investigate which pure t-motives are semisimple, that is, isogenous to direct sums of simple ones. We give examples for pure t-motives which are not semisimple. Over finite fields the semisimplicity is equivalent to the semisimplicity of the endomorphism algebra, but also this fails over infinite fields. Still over finite fields we study the endomorphism rings of pure t-motives and criteria for the existence of isogenies. We obtain answers which are similar to Tate's famous results for abelian varieties.