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2002 Abelian Surfaces over Finite Fields as Jacobians
Enric Hart, Daniel Maisner
Experiment. Math. 11(3): 321-337 (2002).

Abstract

For any finite field {\small $k=\fq$}, 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 k-isogenous to a Jacobian. Other facts suggested by these numerical computations are that the polynomials {\small $t^4+(1-2q)t^2+q^2$} (for all q) and {\small $t^4+(2-2q)t^2+q^2$} (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.

Citation

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Enric Hart. Daniel Maisner. "Abelian Surfaces over Finite Fields as Jacobians." Experiment. Math. 11 (3) 321 - 337, 2002.

Information

Published: 2002
First available in Project Euclid: 9 July 2003

zbMATH: 1101.14056
MathSciNet: MR1959745

Subjects:
Primary: 11G20 , 14G15
Secondary: 11G10

Keywords: abelian surface , finite field , Jacobian variety , zeta function

Rights: Copyright © 2002 A K Peters, Ltd.

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Vol.11 • No. 3 • 2002
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