Determining the primes of bad reduction of cm curves of genus 3-Reference-Cited by-同舟云学术

Determining the primes of bad reduction of cm curves of genus 3

Published:2024-03-01 Issue:1 Volume:75 Page:239-276
ISSN:0033-5606
Container-title:The Quarterly Journal of Mathematics
language:en
Short-container-title:

Author:

Ionica Sorina¹²³⁴⁵⁶⁷,KılıÇer Pınar¹²³⁴⁵⁶⁷,Lauter Kristin¹²³⁴⁵⁶⁷,Lorenzo GarcíA Elisa¹²³⁴⁵⁸⁷^ORCID,MÂnzĂŢeanu Adelina¹²³⁴⁵⁸⁷,Vincent Christelle¹²³⁴⁵⁶⁷

Affiliation:

1. Laboratoire MIS, Université de Picardie Jules Verne , 33 Rue Saint Leu, Amiens 80039, France

2. Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence , 9747 AG Groningen, The Netherlands

3. Facebook AI Research, Meta, Seattle, WA, USA

4. Institut de Mathématiques, Université de Neuchâtel , Rue Emile-Argand 11, 2000, Neuchâtel, Switzerland

5. Laboratoire IRMAR, Office 602, Université de Rennes 1 , Campus de Beaulieu, 35042, Rennes Cedex, France

6. Mathematics Institut, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands

7. Department of Mathematics and Statistics, University of Vermont , 82 University Place, Burlington VT 054015, Vermont

8. Mathematics Institut , Niels Bohrweg 1, 2333 CA Leiden, The Netherlands

Abstract

Abstract In this paper, we introduce a new problem called the Isogenous Embedding Problem (IEP) and relate the existence of solutions to this problem to the primes of bad reduction of complex multiplication curves of genus 3. More precisely, the absence of solutions to IEP implies potentially good reduction. We propose an algorithm for computing the solutions to the IEP and run the algorithm through different families of curves. Using this algorithm, we were able to prove the reduction type of some particular curves at certain primes that were open cases in R. Lercier, Q. Liu, E. Lorenzo García and C. Ritzenthaler, Reduction type of smooth plane quartics, Algebra Number Theory 15 no. 6 2021, 1429–1468.

Publisher

Oxford University Press (OUP)

Link

https://academic.oup.com/qjmath/article-pdf/75/1/239/58704998/haae005.pdf

Reference45 articles.

1. Constructing Picard curves with complex multiplication using the Chinese remainder theorem;Arora,2019

2. Constructing genus-3 hyperelliptic Jacobians with CM;Balakrishnan;LMS J. Comput. Math.,2016

3. Constructing elliptic curves with prescribed embedding degrees;Barreto,2003

4. Computing Hilbert Class Polynomials

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