On the non-existence of certain curves of genus two

Abstract

We prove that if q is a power of an odd prime, then there is no genus-2 curve over $\mathbf{F}_q$ whose Jacobian has characteristic polynomial of Frobenius equal to $x^4 + (2 - 2q)x^2 + q^2$. Our proof uses the Brauer relations in a biquadratic extension of $\mathbb{Q}$ to show that every principally polarized abelian surface over $\mathbf{F}_q$ with the given characteristic polynomial splits over $\mathbf{F}_{q^2}$ as a product of polarized elliptic curves.