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Genus 2 Curves with (3, 3)-Split Jacobian and Large Automorphism Group

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Algorithmic Number Theory (ANTS 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2369))

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Abstract

Let \( \mathcal{C} \) be a genus 2 curve defined over k, char(k) = 0. If \( \mathcal{C} \) has a (3, 3)-split Jacobian then we show that the automorphism group Aut(\( \mathcal{C} \) ) is isomorphic to one of the following: ℤ2, V 4, D 8, or D 12. There are exactly six ℂ-isomorphism classes of genus two curves \( \mathcal{C} \) with Aut(\( \mathcal{C} \) ) isomorphic to D 8 (resp., D 12) and with (3, 3)-split Jacobian. We show that exactly four (resp., three) of these classes with group D 8 (resp., D 12) have representatives defined over ℚ. We discuss some of these curves in detail and find their rational points.

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Author information

Authors and Affiliations

  1. University of California at Irvine, Irvine, CA, 92697

    Tony Shaska

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  1. Tony Shaska
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Editor information

Editors and Affiliations

  1. School of Mathematics and Statistics, F07, University of Sydney, Sydney, NSW, 2006, Australia

    Claus Fieker  & David R. Kohel  & 

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Shaska, T. (2002). Genus 2 Curves with (3, 3)-Split Jacobian and Large Automorphism Group. In: Fieker, C., Kohel, D.R. (eds) Algorithmic Number Theory. ANTS 2002. Lecture Notes in Computer Science, vol 2369. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45455-1_17

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  • DOI: https://doi.org/10.1007/3-540-45455-1_17

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-43863-2

  • Online ISBN: 978-3-540-45455-7

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