Everett W. Howe: On the nonexistence of certain curves of genus two, Compositio Math. 140 (2004) 581–592, MR 2005a:11088, Zbl 1067.11035.

(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 x4 + (2 − 2q) x2 + q2. 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.