On the non-existence of certain curves of genus two

We prove that if q is a power of an odd prime, then there is no genus-2 curve over 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 Q to show that every principally polarized abelian surface over F-q with the given characteristic polynomial splits over F-q 2 as a product of polarized elliptic curves.