Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-24T09:58:40.013Z Has data issue: false hasContentIssue false
  • English
  • Français

Zeta Functions of Supersingular Curves of Genus 2

Published online by Cambridge University Press:  20 November 2018

Daniel Maisner  and
Enric Nart
Daniel Maisner
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C, 08193 Bellaterra, Barcelona, Spainemail: [email protected], [email protected]
Enric Nart
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C, 08193 Bellaterra, Barcelona, Spainemail: [email protected], [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We determine which isogeny classes of supersingular abelian surfaces over a finite field $k$ of characteristic 2 contain jacobians. We deal with this problem in a direct way by computing explicitly the zeta function of all supersingular curves of genus 2. Our procedure is constructive, so that we are able to exhibit curves with prescribed zeta function and find formulas for the number of curves, up to $k$-isomorphism, leading to the same zeta function.

Keywords

11G2014G1511G10
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 59 , Issue 2 , 01 April 2007 , pp. 372 - 392
Copyright
Copyright © Canadian Mathematical Society 2007

References

[CNP] Cardona, G., Nart, E., and Pujolàs, J., Curves of genus two over fields of even characteristic. Math. Z. 250(2005), no. 1, 177–201.Google Scholar
[MN] Maisner, D. and Nart, E., Abelian surfaces over finite fields as Jacobians. With an appendix by Everett W. Howe. Experimental Math. 11(2003), no. 3, 321–337.Google Scholar
[VV1] van der Geer, G. and van der Vlugt, M., Reed-Muller codes and supersingular curves. I. Compositio Math. 84(1992), no. 3, 333–367.Google Scholar
[VV2] van der Geer, G. and van der Vlugt, M., Supersingular curves of genus 2 over finite fields of characteristic 2. Math. Nachr. 159(1992), 73–81.Google Scholar
[Wat] Waterhouse, W. C., Abelian varieties over finite fields. Ann. Sci. école Norm. Sup. (4) 2(1969), 521–560.Google Scholar