Everett W. Howe, Daniel Maisner, Enric Nart, and Christophe Ritzenthaler: Principally polarizable isogeny classes of abelian surfaces over finite fields, Math. Res. Lett. 15 (2008) 121–127, MR 2006g:11125, Zbl 1145.11045.
(A preprint and an official version are available online.)
Let A be an isogeny class of abelian surfaces over Fq with Weil polynomial x4 + ax3 + bx2 + aqx + q2. We show that A does not contain a surface that has a principal polarization if and only if a2 - b = q and b < 0 and all prime divisors of b are congruent to 1 modulo 3. We use this result in a forthcoming paper in which we determine which isogeny classes of abelian surfaces over finite fields contain Jacobians.