Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-24T13:21:16.654Z Has data issue: false hasContentIssue false
  • English
  • Français

High-rank elliptic curves with torsion $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/4\mathbb{Z}$ induced by Diophantine triples

Part of: Arithmetic algebraic geometry

Published online by Cambridge University Press:  01 June 2014

Andrej Dujella  and
Juan Carlos Peral
Andrej Dujella
Affiliation:
Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia email [email protected]
Juan Carlos Peral
Affiliation:
Departamento de Matemáticas, Universidad del País Vasco, Aptdo. 644, 48080 Bilbao, Spain email [email protected]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We construct an elliptic curve over the field of rational functions with torsion group $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/4\mathbb{Z}$ and rank equal to four, and an elliptic curve over $\mathbb{Q}$ with the same torsion group and rank nine. Both results improve previous records for ranks of curves of this torsion group. They are obtained by considering elliptic curves induced by Diophantine triples.

MSC classification

Secondary: 11G05: Elliptic curves over global fields
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 17 , Issue 1 , 2014 , pp. 282 - 288
Copyright
© The Author(s) 2014 

References

Aguirre, J., Dujella, A. and Peral, J. C., ‘On the rank of elliptic curves coming from rational Diophantine triples’, Rocky Mountain J. Math. 42 (2012) 1759–1776.Google Scholar
Campbell, G. and Goins, E. H., ‘Heron triangles, Diophantine problems and elliptic curves’, Preprint,http://www.swarthmore.edu/NatSci/gcampbe1/papers/heron-Campbell-Goins.pdf.Google Scholar
Carmichael, R. D., Diophantine analysis (Dover, New York, 1959).Google Scholar
Connell, I., Elliptic curve handbook (McGill University, 1999).Google Scholar
Cremona, J., Algorithms for modular elliptic curves (Cambridge University Press, Cambridge, 1997).Google Scholar
Dujella, A., ‘Diophantine triples and construction of high-rank elliptic curves over ℚ with three non-trivial 2-torsion points’, Rocky Mountain J. Math. 30 (2000) 157–164.CrossRefGoogle Scholar
Dujella, A., ‘On Mordell–Weil groups of elliptic curves induced by Diophantine triples’, Glas. Mat. Ser. III 42 (2007) 3–18.CrossRefGoogle Scholar
Dujella, A., ‘High rank elliptic curves with prescribed torsion’,http://web.math.pmf.unizg.hr/∼duje/tors/tors.html.Google Scholar
Dujella, A. and Peral, J. C., ‘Elliptic curves coming from Heron triangles’, Rocky Mountain J. Math., to appear.Google Scholar
Elkies, N. D., ‘ E (ℚ) = (ℤ∕2ℤ) × (ℤ∕4ℤ) ×ℤ8 ’, Number Theory Listserver (2005), https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0506&L=NMBRTHRY&P=R194.Google Scholar
Elkies, N. D., ‘Three lectures on elliptic surfaces and curves of high rank’, Lecture notes, Oberwolfach (2007) , arXiv:0709.2908.Google Scholar
Gusić, I. and Tadić, P., ‘A remark on the injectivity of the specialization homomorphism’, Glas. Mat. Ser. III 47 (2012) 265–275.Google Scholar
Gusić, I. and Tadić, P., ‘Injectivity of the specialization homomorphism of elliptic curves’, Preprint, 2012, arXiv:1211.3851.Google Scholar
Knapp, A., Elliptic curves (Princeton University Press, Princeton, NJ, 1992).Google Scholar
Lecacheux, O., ‘Rang de courbes elliptiques avec groupe de torsion non trivial’, J. Théor. Nombres Bordeaux 15 (2003) 231–247.CrossRefGoogle Scholar
Mestre, J.-F., ‘Formules explicites et minorations de conducteurs de variétés algébriques’, Compositio Math. 58 (1986) 209–232.Google Scholar
Nagao, K., ‘An example of elliptic curve over ℚ with rank ≥ 20’, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993) 291–293.Google Scholar
PARI/GP, version 2.4.0, Bordeaux, 2008, http://pari.math.u-bordeaux.fr.Google Scholar
Silverman, J. H., Advanced topics in the arithmetic of elliptic curves (Springer, New York, 1994).Google Scholar