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

•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.