Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4593950 | Journal of Number Theory | 2014 | 25 Pages |
•We examine when an endomorphism ring is maximal at p.•We give a new proof of a formula computing p-rank from global data.•We prove that absolutely simple abelian surfaces are maximal at p.•We give explicit structure of subrings of p-power index for abelian 3-folds.
TextIn this paper we look at endomorphisms of simple abelian varieties defined over a finite field k=Fpnk=Fpn with Endk(A)Endk(A) commutative. We give a new proof of a formula that connects the p -rank r(A)r(A) with the splitting behavior of p in E=Q(π)E=Q(π), where π is a root of the characteristic polynomial of the Frobenius endomorphism. We then prove that p does not divide [OE:Z[π,π¯]] when p⩾3p⩾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⩾3p⩾3. When A is not a surface, we derive a criterion that gives cases where p divides [OE:Z[π,π¯]].VideoFor a video summary of this paper, please click here or visit http://youtu.be/tgQMp-MLnwM.