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Modular invariants for genus 3 hyperelliptic curves

Research output: Contribution to journalArticle

  • Sorina Ionica
  • Pınar Kılıçer
  • Kristin Lauter
  • Elisa Lorenzo García
  • Adelina Mânzăţeanu
  • Maike Massierer
  • Christelle Vincent
Original languageEnglish
Article number9
Number of pages22
JournalResearch in Number Theory
Volume5
Issue number1
DOIs
DateAccepted/In press - 6 Dec 2018
DatePublished (current) - 2 Jan 2019

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.

    Research areas

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

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    Rights statement: This is the final published version of the article (version of record). It first appeared online via SpringerOpen at https://resnumtheor.springeropen.com/articles/10.1007/s40993-018-0146-6 . Please refer to any applicable terms of use of the publisher.

    Final published version, 564 KB, PDF document

    Licence: CC BY

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