Abelian surfaces over finite fields as Jacobians

For any finite field k = Fq, we explicitly describe the k-isogeny classes of abelian surfaces defined over k and their behavior under finite field extension. In particular, we determine the absolutely simple abelian surfaces. Then, we analyze numerically what surfaces are k-isogenous to the Jacobian of a smooth projective curve of genus 2 defined over k. We prove some partial results suggested by these numerical data. For instance, we show that every absolutely simple abelian surface is fc-isogenous to a Jacobian. Other facts suggested by these numerical computations are that the polynomials t4 + (1 - 2q)t2 +q2 (for all q) and t4 + (2 — 2q)t2 + q2 (for q odd) are never the characteristic polynomial of Frobenius of a Jacobian. These statements have been proved by E. Howe. The proof for the first polynomial is attached in an appendix. © A K Peters, Ltd.