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Abelian surfaces over finite fields as Jacobians. With an appendix by Everett W. Howe. (English) Zbl 1101.14056
Summary: For any finite field \(k=\mathbb{F}_q\), 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 \(t^4+(1-2q)t^2+q^2\) \(q\) 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.

MSC:
14K15 Arithmetic ground fields for abelian varieties
11G10 Abelian varieties of dimension \(> 1\)
11G25 Varieties over finite and local fields
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