Abelian Surfaces over Finite Fields as Jacobians

Abstract

For any finite field k = F _q , 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 k-isogenous to a Jacobian. Other facts suggested by these numerical computations are that the polynomials t ⁴ + (1 – 2q)t ² + q ² (for all q) and t ⁴ + (2 – 2q)t ² + q ² (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.

Keywords:

• Abelian surface
• zeta function
• finite field
• Jacobian variety