Daniel Maisner and Enric Nart with an appendix by Everett W. Howe: Abelian surfaces over finite fields as Jacobians, Experiment. Math. 11 (2002) 321–337.
(An unofficial electronic version is available online.)
In the appendix to this paper I prove a conjecture of Maisner and Nart: namely, that if k is a finite field with q elements, then there is no curve of genus 2 over k whose Jacobian has characteristic polynomial of Frobenius x4 + (1 − 2q) x2 + q2.