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Pure Anderson motives over finite fields. (English) Zbl 1227.11076
Authors’ 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 latter. 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 Zeta function and 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.”
Reviewer’s addition: Note that the references might be completed: the authors [Math. Z. 268, No. 1–2, 67–100 (2011; Zbl 1227.11077); J. Number Theory 129, No. 2, 247–283 (2009; Zbl 1227.11076) and U. Hartl, Ann. Math. (2) 173, No. 3, 1241–1358 (2011; Zbl 1304.11050)].

##### MSC:
 11G09 Drinfel’d modules; higher-dimensional motives, etc. 14L05 Formal groups, $$p$$-divisible groups 13A35 Characteristic $$p$$ methods (Frobenius endomorphism) and reduction to characteristic $$p$$; tight closure 16K20 Finite-dimensional division rings
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