Pure Anderson Motives over Finite Fields

Details Pure Anderson Motives over Finite Fields Pure Anderson Motives over Finite Fields

2007-09-18

arxiv.org/abs/0709.2815v2

Mathematics

Number Theory

v2: final version to appear in Journal of Number Theory. Note, this is the second part of the two-in-one article arXiv:math.NT

Scientific paper

10.1016/j.jnt.2008.09.006

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.

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