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Rational Points on Genus 3 Real Hyperelliptic Curves
Faculté Des Sciences et Techniques, Université Marien Ngouabi, Brazzaville, Congo
Faculté Des Sciences et Techniques, Université Marien Ngouabi, Brazzaville, Congo
Faculté Des Sciences et Techniques, Université Marien Ngouabi, Brazzaville, Congo
- 1 Faculté Des Sciences et Techniques, Université Marien Ngouabi, Brazzaville, Congo
- 2 Faculté Des Sciences et Techniques, Université Marien Ngouabi, Brazzaville, Congo
- 3 Faculté Des Sciences et Techniques, Université Marien Ngouabi, Brazzaville, Congo
Open Journal of Discrete Mathematics·Volume 11 (2021)·Pages 103–113·Published 19 October 2021·DOI10.4236/ojdm.2021.114008
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Abstract
We compute rational points on real hyperelliptic curves of genus 3 defined on whose Jacobian have Mordell-Weil rank r=0 . We present an implementation in sagemath of an algorithm which describes the birational transformation of real hyperelliptic curves into imaginary hyperelliptic curves and the Chabauty-Coleman method to find C ( ) . We run the algorithms in Sage on 47 real hyperelliptic curves of genus 3.
KeywordsHyperelliptic CurveJacobianColeman IntegrationRational Point
- Coleman, R.F. (1985) Effective Chabauty. Duke Mathematical Journal, 52, 765-770. https://doi.org/10.1215/S0012-7094-85-05240-8
- Balakrishnan, J.S., Bianchi, F., Cantoral-Farfán, V., Çiperiani, M. and Etropolski, A. (2018) Chabauty-Coleman Experiments for Genus 3 Hyperelliptic Curves. In: Balakrishnan, J., Folsom, A., Lalín, M. and Manes, M., Eds., Research Directions in Number Theory, Springer, Cham, 67-90. https://doi.org/10.1007/978-3-030-19478-9_3
- Balakrishnan, J.S. (2015) Coleman Integration for Even-Degree Models of Hyperelliptic Curves. LMS Journal of Computation and Mathematics, 18, 258-265. https://doi.org/10.1112/S1461157015000029
- Balakrishnan, J.S., Bradshaw, R.W. and Kedlaya, K.S. (2010) Explicit Coleman Integration for Hyperelliptic Curves. In: Hanrot, G., Morain, F. and Thomé, E., Eds., Algorithmic Number Theory, ANTS 2010. Lecture Notes in Computer Science. Springer, Berlin, 16-31. https://doi.org/10.1007/978-3-642-14518-6_6
- de Frutos Fernaandez, M.I. and Hashimoto, S. (2019) Computing Rational Points on Rank 0 Genus 3 Hyperelliptic Curves. arXiv, 190904808v2, 1-10.
- de Frutos Fernaandez, M.I. and Hashimoto, S. (2019) Sage Code. https://github.com/sachihashimoto/rational-points-hyperelliptic
- Booker, A., Platt, D., Sijsling, J. and Sutherland, A. (2018) Genus 3 Hyperelliptic Curves. 1-4. http://math.mit.edu/~drew/lmfdb_genus3_hyperelliptic.txt
- Jacobson, M.J., Scheidler, R. and Stein, A. (2009) Cryptographic Aspects of Real Hyperelliptic Curves. Cryptology ePrint Archive, 1-28.
- Stoll, M. (2014) Arithmetic of Hyperelliptic Curves. Screen Version of August 1, 2014, 1, 1-35.
- McCallum, W. and Poonen, B. (2012) The Method of Chabauty and Coleman, Explicit Methods in Number Theory. In: Panoramas et Synthèses, Société Mathématique de France, Paris, 36, 99-117.
- Balakrishnan, J.S. (2011) Coleman Integration for Hyperelliptic Curves: Algorithms and Applications. Thesis, Massachusetts Institute of Technology, Cambridge, MA, USA.
- Zimmermann, P. (2017) The Sage Developers Sagemath, the Sage Mathematics Software System (Version 9.1).