Abstract
Let $\calA$ be an isogeny class of abelian surfaces over $\fq$ with Weil polynomial $x^4 + ax^3 + bx^2 + aqx + q^2$. We show that $\calA$ 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~$3$.
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