Glasnik Matematicki, Vol. 51, No. 2 (2016), 321-333.

ELLIPTIC CURVES WITH TORSION GROUP Z/6Z

Andrej Dujella, Juan Carlos Peral and Petra Tadić

Departamento de Matemáticas, Universidad del País Vasco, 48080 Bilbao, Spain
e-mail: juancarlos.peral@ehu.es

Department of Mathematics, Statistics and Information Science, Juraj Dobrila University of Pula, 52100 Pula, Croatia
e-mail: petra.tadic@unipu.hr

Abstract.   We exhibit several families of elliptic curves with torsion group isomorphic to Z/6Z and generic rank at least 3. Families of this kind have been constructed previously by several authors: Lecacheux, Kihara, Eroshkin and Woo. We mention the details of some of them and we add other examples developed more recently by Dujella and Peral, and MacLeod. Then we apply an algorithm of Gusić and Tadić and we find the exact rank over Q(t) to be 3 and we also determine free generators of the Mordell-Weil group for each family. By suitable specializations, we obtain the known and new examples of curves over Q with torsion Z/6Z and rank 8, which is the current record.

2010 Mathematics Subject Classification.   11G05.

Key words and phrases.   Elliptic curves, torsion, rank.

DOI: 10.3336/gm.51.2.03

References:

1. J. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, 1997.
   MathSciNet    

2. A. Dujella, High rank elliptic curves with prescribed torsion, http://web.math.hr/ duje/tors/tors.html.

3. A. Dujella, I. Gusić and P. Tadić, The rank and generators of Kihara's elliptic curve with torsion Z/4Z over Q(t), Proc. Japan Acad. Ser. A Math. Sci. 91 (2015), 105-109.
   MathSciNet     CrossRef

4. A. Dujella and J. C. Peral, High rank elliptic curves with torsion Z/2Z × Z/4Z induced by Diophantine triples, LMS J. Comput. Math. 17 (2014), 282-288.
   MathSciNet     CrossRef

5. A. Dujella, J.C. Peral, Elliptic curves with torsion group Z/8Z or Z/2Z × Z/6Z, in Trends in Number Theory, Contemp. Math. 649 (2015), 47-62.

6. Y. G. Eroshkin, personal communication, 2008.

7. I. Gusić and P. Tadić, A remark on the injectivity of the specialization homomorphism, Glas. Mat. Ser. III 47 (2012), 265-275.
   MathSciNet     CrossRef

8. I. Gusić and P. Tadić, Injectivity of the specialization homomorphism of elliptic curves, J. Number Theory 148 (2015), 137-152.
   MathSciNet     CrossRef

9. S. Kihara, On the rank of the elliptic curves with a rational point of order 6, Proc. Japan Acad. Ser. A Math. Sci. 82 (2006), 81-82.
   MathSciNet     CrossRef

10. A. Knapp, Elliptic curves, Princeton University Press, 1992.
    MathSciNet    

11. O. Lecacheux, Rang de courbes elliptiques avec groupe de torsion non trivial, J. Théor. Nombres Bordeaux 15 (2003), 231-247.
    MathSciNet     CrossRef

12. A. MacLeod, A simple method for high-rank families of elliptic curves with specified torsion, arXiv:1410.1662.

13. J.-F. Mestre, Construction d'une courbe elliptique de rang ≥ 12, C. R. Acad. Sci. Paris Ser. I 295 (1982) 643-644.
    MathSciNet    

14. The PARI Group, PARI/GP version 2.7.0, Bordeaux, 2014, http://pari.math.u-bordeaux.fr/.

15. K. Nagao, An example of elliptic curve over Q with rank ≥ 21, Proc. Japan Acad. Ser. A Math. Sci. 70 (1994) 104-105.
    MathSciNet     CrossRef

16. J. Woo, Arithmetic of elliptic curves and surfaces descent and quadratic sections, PhD thesis, Harvard University, 2010.
    MathSciNet