Subrings of p-power index in endomorphism rings of simple abelian varieties over finite fields

JOURNAL OF NUMBER THEORY

卷:141

Article

关键词: Abelian variety;    Finite field;    Endomorphism ring;    p-rank;   

DOI  :  10.1016/j.jnt.2014.01.029

来源: Elsevier

【 摘 要 】

Text. In this paper we look at endomorphisms of simple abelian varieties defined over a finite field k = F-p(n). with End(k)(A) commutative. We give a new proof of a formula that connects the p-rank r(A) with the splitting behavior of p in E = Q(pi), where pi is a root of the characteristic polynomial of the Frobenius endomorphism. We then prove that p does not divide [O-E : Z[pi, pi]] when p >= 3 and A is an absolutely simple abelian surface. It then follows that the endomorphism ring of an absolutely simple abelian surface is maximal at p when p >= 3. When A is not a surface, we derive a criterion that gives cases where p divides [OE : Z[pi, pi]]. Video. For a video summary of this paper, please click here or visit http://youtu.be/tgQMp-MLnwM. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free