---
_id: '10874'
abstract:
- lang: eng
  text: 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.
acknowledgement: "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\r\nthank 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. "
article_number: '9'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Sorina
  full_name: Ionica, Sorina
  last_name: Ionica
- first_name: Pınar
  full_name: Kılıçer, Pınar
  last_name: Kılıçer
- first_name: Kristin
  full_name: Lauter, Kristin
  last_name: Lauter
- first_name: Elisa
  full_name: Lorenzo García, Elisa
  last_name: Lorenzo García
- first_name: Maria-Adelina
  full_name: Manzateanu, Maria-Adelina
  id: be8d652e-a908-11ec-82a4-e2867729459c
  last_name: Manzateanu
- first_name: Maike
  full_name: Massierer, Maike
  last_name: Massierer
- first_name: Christelle
  full_name: Vincent, Christelle
  last_name: Vincent
citation:
  ama: Ionica S, Kılıçer P, Lauter K, et al. Modular invariants for genus 3 hyperelliptic
    curves. <i>Research in Number Theory</i>. 2019;5. doi:<a href="https://doi.org/10.1007/s40993-018-0146-6">10.1007/s40993-018-0146-6</a>
  apa: Ionica, S., Kılıçer, P., Lauter, K., Lorenzo García, E., Manzateanu, M.-A.,
    Massierer, M., &#38; Vincent, C. (2019). Modular invariants for genus 3 hyperelliptic
    curves. <i>Research in Number Theory</i>. Springer Nature. <a href="https://doi.org/10.1007/s40993-018-0146-6">https://doi.org/10.1007/s40993-018-0146-6</a>
  chicago: Ionica, Sorina, Pınar Kılıçer, Kristin Lauter, Elisa Lorenzo García, Maria-Adelina
    Manzateanu, Maike Massierer, and Christelle Vincent. “Modular Invariants for Genus
    3 Hyperelliptic Curves.” <i>Research in Number Theory</i>. Springer Nature, 2019.
    <a href="https://doi.org/10.1007/s40993-018-0146-6">https://doi.org/10.1007/s40993-018-0146-6</a>.
  ieee: S. Ionica <i>et al.</i>, “Modular invariants for genus 3 hyperelliptic curves,”
    <i>Research in Number Theory</i>, vol. 5. Springer Nature, 2019.
  ista: Ionica S, Kılıçer P, Lauter K, Lorenzo García E, Manzateanu M-A, Massierer
    M, Vincent C. 2019. Modular invariants for genus 3 hyperelliptic curves. Research
    in Number Theory. 5, 9.
  mla: Ionica, Sorina, et al. “Modular Invariants for Genus 3 Hyperelliptic Curves.”
    <i>Research in Number Theory</i>, vol. 5, 9, Springer Nature, 2019, doi:<a href="https://doi.org/10.1007/s40993-018-0146-6">10.1007/s40993-018-0146-6</a>.
  short: S. Ionica, P. Kılıçer, K. Lauter, E. Lorenzo García, M.-A. Manzateanu, M.
    Massierer, C. Vincent, Research in Number Theory 5 (2019).
date_created: 2022-03-18T12:09:48Z
date_published: 2019-01-02T00:00:00Z
date_updated: 2023-09-05T15:39:31Z
day: '02'
department:
- _id: TiBr
doi: 10.1007/s40993-018-0146-6
external_id:
  arxiv:
  - '1807.08986'
intvolume: '         5'
keyword:
- Algebra and Number Theory
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1807.08986
month: '01'
oa: 1
oa_version: Preprint
publication: Research in Number Theory
publication_identifier:
  eissn:
  - 2363-9555
  issn:
  - 2522-0160
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Modular invariants for genus 3 hyperelliptic curves
type: journal_article
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 5
year: '2019'
...