(A preprint and an official version are available online.)

Let **A** be an isogeny class of abelian surfaces over **F**_{q} 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. We use this result in a forthcoming
paper in which we determine which isogeny classes of abelian surfaces over
finite fields contain Jacobians.