Principally polarizable isogeny classes of abelian surfaces over finite fields
Everett W. Howe, Daniel Maisner, Enric Nart, Christophe Ritzenthaler · Mathematical Research Letters · 2008
Let A be an isogeny class of abelian surfaces over Fq with Weil polynomial x 4 +ax 3 +bx 2 +aqx+q 2 .We show that A does not contain a surface that has a principal polarization if and only if a 2 -b = q and b < 0 and all prime divisors of b are congruent to 1 modulo 3.