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Modular invariants for genus 3 hyperelliptic curves

  • Sorina Ionica
  • Pınar Kılıçer
  • Kristin Lauter
  • Elisa Lorenzo García
  • Adelina Mânzăţeanu
  • Maike Massierer
  • Christelle Vincent
Research
  • 20 Downloads

Abstract

In this article we prove an analogue of a theorem of Lachaud, Ritzenthaler, and Zykin, which allows us to connect invariants of binary octics to Siegel modular forms of genus 3. We use this connection to show that certain modular functions, when restricted to the hyperelliptic locus, assume values whose denominators are products of powers of primes of bad reduction for the associated hyperelliptic curves. We illustrate our theorem with explicit computations. This work is motivated by the study of the values of these modular functions at CM points of the Siegel upper half-space, which, if their denominators are known, can be used to effectively compute models of (hyperelliptic, in our case) curves with CM.

Keywords

Hyperelliptic curve Invariant of curve Bad reduction Siegel modular form Complex multiplication Theta constant 

Mathematics Subject Classification

14H42 11F46 11G15 

Notes

Acknowlegements

The authors would like to thank the Lorentz Center in Leiden for hosting the Women in Numbers Europe 2 workshop and providing a productive and enjoyable environment for our initial work on this project. We are grateful to the organizers of WIN-E2, Irene Bouw, Rachel Newton and Ekin Ozman, for making this conference and this collaboration possible. We thank Irene Bouw and Christophe Ritzenhaler for helpful discussions. Ionica acknowledges support from the Thomas Jefferson Fund of the Embassy of France in the United States and the FACE Foundation. Most of Kılıçer’s work was carried out during her stay in Universiteit Leiden and Carl von Ossietzky Universität Oldenburg. Massierer was supported by the Australian Research Council (DP150101689). Vincent is supported by the National Science Foundation under Grant No. DMS-1802323 and by the Thomas Jefferson Fund of the Embassy of France in the United States and the FACE Foundation.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Sorina Ionica
    • 1
  • Pınar Kılıçer
    • 2
  • Kristin Lauter
    • 3
  • Elisa Lorenzo García
    • 4
  • Adelina Mânzăţeanu
    • 5
  • Maike Massierer
    • 6
  • Christelle Vincent
    • 7
  1. 1.Laboratoire MISUniversité de Picardie Jules VerneAmiensFrance
  2. 2.Johann Bernoulli Instituut voor Wiskunde en InformaticaRijksuniversiteit GroningenGroningenThe Netherlands
  3. 3.Microsoft ResearchRedmondUSA
  4. 4.Laboratoire IRMARUniversité de Rennes 1Rennes CedexFrance
  5. 5.Institute of Science and Technology Austria, School of MathematicsUniversity of BristolBristolUK
  6. 6.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia
  7. 7.Department of Mathematics and StatisticsUniversity of VermontBurlingtonUSA

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