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Formalized Class Group Computations and Integral Points on Mordell Elliptic Curves

Authors:

Nirvana Coppola,

Sander R. DahmenAuthors Info & Claims

CPP 2023: Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs

Pages 47 - 62

https://doi.org/10.1145/3573105.3575682

Published: 11 January 2023 Publication History

Abstract

Diophantine equations are a popular and active area of research in number theory. In this paper we consider Mordell equations, which are of the form y²=x³+d, where d is a (given) nonzero integer number and all solutions in integers x and y have to be determined. One non-elementary approach for this problem is the resolution via descent and class groups. Along these lines we formalized in Lean 3 the resolution of Mordell equations for several instances of d<0. In order to achieve this, we needed to formalize several other theories from number theory that are interesting on their own as well, such as ideal norms, quadratic fields and rings, and explicit computations of the class number. Moreover, we introduced new computational tactics in order to carry out efficiently computations in quadratic rings and beyond.

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Cited By

Baanen AChavarri Villarello ADahmen SStark KTimany ABlazy STabareau N(2025)Certifying Rings of Integers in Number FieldsProceedings of the 14th ACM SIGPLAN International Conference on Certified Programs and Proofs10.1145/3703595.3705874(50-66)Online publication date: 10-Jan-2025
https://dl.acm.org/doi/10.1145/3703595.3705874
Giovannini EHaeusler ELassalle Casanave AGiovannini EHaeusler ELassalle Casanave A(2025)De Zolt’s Postulate in Three-DimensionsThe Theory of Plane Area at the Crossroads10.1007/978-3-031-70916-6_4(97-131)Online publication date: 3-Jan-2025
https://doi.org/10.1007/978-3-031-70916-6_4

Index Terms

Formalized Class Group Computations and Integral Points on Mordell Elliptic Curves
1. Mathematics of computing
  1. Mathematical software
2. Security and privacy
  1. Formal methods and theory of security
    1. Logic and verification

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cover image ACM Conferences

CPP 2023: Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs

January 2023

347 pages

ISBN:9798400700262

DOI:10.1145/3573105

General Chairs:
Robbert Krebbers
Radboud University Nijmegen, Netherlands
,
Dmitriy Traytel
University of Copenhagen, Denmark
,
Program Chairs:
Brigitte Pientka
McGill University, Canada
,
Steve Zdancewic
University of Pennsylvania, USA

Copyright © 2023 Owner/Author.

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

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Published: 11 January 2023

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CPP '23: 12th ACM SIGPLAN International Conference on Certified Programs and Proofs

January 16 - 17, 2023

MA, Boston, USA

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Cited By

Baanen AChavarri Villarello ADahmen SStark KTimany ABlazy STabareau N(2025)Certifying Rings of Integers in Number FieldsProceedings of the 14th ACM SIGPLAN International Conference on Certified Programs and Proofs10.1145/3703595.3705874(50-66)Online publication date: 10-Jan-2025
https://dl.acm.org/doi/10.1145/3703595.3705874
Giovannini EHaeusler ELassalle Casanave AGiovannini EHaeusler ELassalle Casanave A(2025)De Zolt’s Postulate in Three-DimensionsThe Theory of Plane Area at the Crossroads10.1007/978-3-031-70916-6_4(97-131)Online publication date: 3-Jan-2025
https://doi.org/10.1007/978-3-031-70916-6_4

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