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Modular invariants for genus 3 hyperelliptic curves

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.

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Notes

  1. Here by denominator we mean the least common multiple of the (rational) denominators that appear in an algebraic number’s monic minimal polynomial.

  2. Apart from curves (1) and (6), we could recognize these values as algebraic numbers with 15,000 bits of precision; for curve (1), we needed 30,000 bits of precision. In fact, for CM field (6), the theta constants obtained using the Magma implementation [16] for high precision (i.e. \(\ge 30,000\) bits) were not conclusive. We therefore ran an improved implementation of the naive method to get these values up to 30,000 bits of precision, and recognized the invariants as algebraic numbers after multiplying by the expected denominators.

  3. An absolute invariant is a ratio of homogeneous invariants of the same degree.

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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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Correspondence to Sorina Ionica.

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Ionica, S., Kılıçer, P., Lauter, K. et al. Modular invariants for genus 3 hyperelliptic curves. Res. number theory 5, 9 (2019). https://doi.org/10.1007/s40993-018-0146-6

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  • DOI: https://doi.org/10.1007/s40993-018-0146-6

Keywords

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

Mathematics Subject Classification

  • 14H42
  • 11F46
  • 11G15