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An Elementary Formal Proof of the Group Law on Weierstrass Elliptic Curves in Any Characteristic

Authors David Kurniadi Angdinata , Junyan Xu

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Author Details

David Kurniadi Angdinata
  • London School of Geometry and Number Theory, UK
Junyan Xu
  • Cancer Data Science Laboratory, National Cancer Institute, Bethesda, MD, USA

Funding

This work was supported by the Engineering and Physical Sciences Research Council [EP/S021590/1], EPSRC Centre for Doctoral Training in Geometry and Number Theory (London School of Geometry and Number Theory), University College London. This research was supported in part by the Intramural Research Program of the Center for Cancer Research, National Cancer Institute, NIH.

Acknowledgements

We thank the Lean community for their continual support. We thank the mathlib contributors, especially Anne Baanen, for developing libraries this work depends on. We thank Marc Masdeu and Michael Stoll for proposing alternative proofs. DKA would like to thank Kevin Buzzard for his guidance and Mel Levin for suggesting the formalisation in the first place.

Cite As Get BibTex

David Kurniadi Angdinata and Junyan Xu. An Elementary Formal Proof of the Group Law on Weierstrass Elliptic Curves in Any Characteristic. In 14th International Conference on Interactive Theorem Proving (ITP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 268, pp. 6:1-6:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ITP.2023.6

Abstract

Elliptic curves are fundamental objects in number theory and algebraic geometry, whose points over a field form an abelian group under a geometric addition law. Any elliptic curve over a field admits a Weierstrass model, but prior formal proofs that the addition law is associative in this model involve either advanced algebraic geometry or tedious computation, especially in characteristic two. We formalise in the Lean theorem prover, the type of nonsingular points of a Weierstrass curve over a field of any characteristic and a purely algebraic proof that it forms an abelian group.

Subject Classification

ACM Subject Classification
  • Theory of computation → Interactive proof systems
  • Security and privacy → Logic and verification
  • Mathematics of computing → Mathematical software
Keywords
  • formal math
  • algebraic geometry
  • elliptic curve
  • group law
  • Lean
  • mathlib

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Supplementary Materials

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