(Both the published vesion and the arXiv version are available online.)
We determine the isogeny classes of abelian surfaces over Fq whose group of Fq -rational points has order divisible by q2. We also solve the same problem for Jacobians of genus-2 curves. The latter result generalizes a recent result of Ravnshøj, who treated the case that q is prime, the endomorphism ring of the Jacobian is the ring of integers in a primitive quartic CM-field, and the Frobenius endomorphism of the Jacobian has a certain special form.
Our proof of the result for abelian surfaces uses this Magma program.
Addendum: after the referee made the entirely valid complaint that the initial version of this paper was unreadably terse, I expanded the paper by a factor of four. However, the editor's remark about my original version was not quite correct (although correct in spirit): the editor suggested that I ``was trying to write a 2 page paper'', when in fact my goal was to write a paper which would be announced in the next daily arXiv mailing after the one which contained the paper by Ravnshøj that had originally led me to consider this question. I don't claim that this was a meritorious goal, or that it was good for me to submit an unreadable paper to either the arXiv or a journal. My only defense is that I was unemployed, many years past my Ph.D., and I had very few papers, so I was trying to build up my publication record as quickly as possible in order to give myself a chance to get an academic position.
Second addendum: I'm very sad to see that the paper of Ravnshøj which introduced me to this topic was not published. It certainly was not my intent to prevent Ravnshøj from publishing on this topic. In fact, I suggested that we combine our papers, which he agreed would be best, but he declined since he had already submitted his. In hindsight, I wish I had explictly told him that my offer to combine our papers would remain open until one of the papers was published. I'll be more careful in the future when faced this sort of situation.
Michael Zieve: home page publication list