(An official version and a preprint version are available online.)

We prove another conjecture of Maisner and Nart: namely, that if *k*
is a finite field with *q* elements, and *q* is odd,
then there is no
curve of genus 2 over *k* whose Jacobian has characteristic
polynomial of Frobenius
*x*^{4} + (2 − 2*q*) *x*^{2} + *q*^{2}.
The proof uses the Brauer relations in a biquadratic extension of **Q**
to show that every principally-polarized abelian surface over *k*
with the given characteristic polynomial splits over the quadratic
extension of *k* as a product of polarized elliptic curves.