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Chabauty–Coleman Experiments for Genus 3 Hyperelliptic Curves

  • Jennifer S. Balakrishnan
  • Francesca Bianchi
  • Victoria Cantoral-Farfán
  • Mirela ÇiperianiEmail author
  • Anastassia Etropolski
Conference paper
Part of the Association for Women in Mathematics Series book series (AWMS, volume 19)

Abstract

We describe a computation of rational points on genus 3 hyperelliptic curves C defined over \(\mathbb {Q}\) whose Jacobians have Mordell–Weil rank 1. Using the method of Chabauty and Coleman, we present and implement an algorithm in SageMath to compute the zero locus of two Coleman integrals and analyze the finite set of points cut out by the vanishing of these integrals. We run the algorithm on approximately 17,000 curves from a forthcoming database of genus 3 hyperelliptic curves and discuss some interesting examples where the zero set includes global points not found in \(C(\mathbb {Q})\).

Keywords

Chabauty-Coleman method Rational points Hyperelliptic curves 

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Copyright information

© The Author(s) and The Association for Women in Mathematics 2019

Authors and Affiliations

  • Jennifer S. Balakrishnan
    • 1
  • Francesca Bianchi
    • 2
  • Victoria Cantoral-Farfán
    • 3
  • Mirela Çiperiani
    • 4
    Email author
  • Anastassia Etropolski
    • 5
  1. 1.Department of Mathematics and StatisticsBoston UniversityBostonUSA
  2. 2.Mathematical InstituteUniversity of OxfordOxfordUK
  3. 3.The Abdus Salam International Center for Theoretical Physics, Mathematics SectionTriesteItaly
  4. 4.Department of MathematicsThe University of Texas at AustinAustinUSA
  5. 5.Department of MathematicsRice University MS 136HoustonUSA

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