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.
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
[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
[4]
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
[5]
de Frutos Fernaandez, M.I. and Hashimoto, S. (2019) Computing Rational Points on Rank 0 Genus 3 Hyperelliptic Curves. arXiv, 190904808v2, 1-10.
[6]
de Frutos Fernaandez, M.I. and Hashimoto, S. (2019) Sage Code. https://github.com/sachihashimoto/rational-points-hyperelliptic
[7]
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
[8]
Jacobson, M.J., Scheidler, R. and Stein, A. (2009) Cryptographic Aspects of Real Hyperelliptic Curves. Cryptology ePrint Archive, 1-28.
[9]
Stoll, M. (2014) Arithmetic of Hyperelliptic Curves. Screen Version of August 1, 2014, 1, 1-35.
[10]
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.
[11]
Balakrishnan, J.S. (2011) Coleman Integration for Hyperelliptic Curves: Algorithms and Applications. Thesis, Massachusetts Institute of Technology, Cambridge, MA, USA.
[12]
Zimmermann, P. (2017) The Sage Developers Sagemath, the Sage Mathematics Software System (Version 9.1).