Infinite families of elliptic curves with high rank and prescribed torsion

Maintained by Andrej Dujella, University of Zagreb

Let T be an admissible torsion group for an elliptic curve over the rationals. Define

G(T) = sup {rank E(Q(t)) : torsion group of elliptic curve E over Q(t) is T},

C(T) = lim sup {rank E(Q) : torsion group of elliptic curve E over Q is T}.

In the following two tables the best known lower bounds for G(T) and C(T) are given. If C(T) > G(T), it means that the current record for C(T) comes from a parametrization by rational points of some elliptic curves with positive rank. 

___________________________________________________________________________________________________

    T         G(T)>=             Author(s)
___________________________________________________________________________________________________

    0           18        Elkies (2006) 
   Z/2Z         11        Elkies (2009), Dujella - Peral (2023)
   Z/3Z          7        Elkies (2007), Eroshkin (2023)
   Z/4Z          6        Dujella - Peral (2022) 		 
   Z/5Z          4        Eroshkin (2020)
   Z/6Z          3        Lecacheux (2001), Kihara (2006), Eroshkin (2008), Woo (2008), 
                          Dujella - Peral (2012,2020), MacLeod (2014,2015), Voznyy (2021)	 	 
   Z/7Z          1        Kulesz (1998), Lecacheux (2003), Rabarison (2008), 
                          Harrache (2009), MacLeod (2014)
   Z/8Z          2        Dujella - Peral (2012), MacLeod (2013), 
                          Dujella - Kazalicki - Peral (2021) 
   Z/9Z          0        Kubert (1976)
   Z/10Z         0        Kubert (1976)  
   Z/12Z         0        Kubert (1976)
Z/2Z × Z/2Z      7        Elkies (2007)
Z/2Z × Z/4Z      4        Dujella - Peral (2012)
Z/2Z × Z/6Z      2        Dujella - Peral (2012,2015,2017), MacLeod (2013), 
                          Dujella - Kazalicki - Peral (2021) 
Z/2Z × Z/8Z      0        Kubert (1976) 
___________________________________________________________________________________________________



___________________________________________________________________________________________________

    T         C(T)>=             Author(s)
___________________________________________________________________________________________________

    0           19        Elkies (2006) 
   Z/2Z         11        Elkies (2007,2009), Dujella - Peral (2023)
   Z/3Z          8        Eroshkin (2023)
   Z/4Z          6        Elkies (2007), Dujella - Peral (2021,2022)		 
   Z/5Z          4        Eroshkin (2009)
   Z/6Z          5        Eroshkin (2009)	 	 
   Z/7Z          2        Lecacheux (2003), Elkies (2006), Rabarison (2008), Harrache (2009), 
                          Voznyy (2022), Youmbai (2024)
   Z/8Z          3        Dujella - Peral (2012), Dujella - Kazalicki - Peral (2021) 
   Z/9Z          1        Atkin - Morain (1993), Kulesz (1998), Rabarison (2008), 
                          Gasull - Manosa - Xarles (2010), Youmbai (2024)
   Z/10Z         1        Atkin - Morain (1993), Kulesz (1998), Rabarison (2008), Voznyy (2021)
   Z/12Z         1        Suyama (1985), Kulesz (1998), Rabarison (2008), 
			  Halbeisen - Hungerbühler - Shamsi Zargar - Voznyy (2021)
Z/2Z × Z/2Z      8        Elkies (2007)
Z/2Z × Z/4Z      5        Eroshkin (2009)
Z/2Z × Z/6Z      3        Dujella - Peral (2013), Dujella - Kazalicki - Peral (2021)
Z/2Z × Z/8Z      1        Atkin - Morain (1993), Kulesz (1998), Lecacheux (2002), 
                          Campbell - Goins (2003), Rabarison (2008) 	 	
___________________________________________________________________________________________________

References:

  1. I. Adelstein and E. Christiansen, Personal communication, 2006.

  2. J. Aguirre, A. Dujella and J. C. Peral, On the rank of elliptic curves coming from rational Diophantine triples, Rocky Mountain J. Math. 42 (2012), 1759-1776.

  3. A. O. L. Atkin and F. Morain, Finding suitable curves for the elliptic curve method of factorization, Math. Comp. 60 (1993), 399-405.

  4. G. Campbell, Finding Elliptic Curves and Families of Elliptic Curves over Q of Large Rank, Dissertation, Rutgers University, 1999.

  5. G. Campbell, On the rank of elliptic curves with a rational point of order 4, preprint.

  6. G. Campbell and E. H. Goins, Heron triangles, Diophantine problems and elliptic curves, preprint.

  7. A. Dujella, 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.

  8. A. Dujella, On Mordell-Weil groups of elliptic curves induced by Diophantine triples, Glas. Mat. Ser. III 42 (2007), 3-18.

  9. A. Dujella, Diophantine m-tuples. Connections with elliptic curves, http://web.math.hr/~duje/coell.html

  10. A. Dujella, High-rank elliptic curves with given torsion group and some applications, Proceedings of NuTMiC 2022, Poznań, Banach Center Publications, Warszawa, to appear.

  11. A Dujella, I. Gusic and P. Tadic, 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.

  12. A. Dujella and M. Jukic Bokun, On the rank of elliptic curves over Q(i) with torsion group Z/4Z × Z/4Z, Proc. Japan Acad. Ser. A Math. Sci. 86 (2010), 93-96.

  13. A. Dujella, M. Kazalicki and J. C. Peral, Elliptic curves with torsion groups Z/8Z and Z/2Z × Z/6Z, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM 115 (2021), Article 169, (24pp)

  14. A. Dujella and F. Najman, Elliptic curves with large torsion and positive rank over number fields of small degree and ECM factorization, Period. Math. Hungar. 65 (2012), 193-203.

  15. 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.

  16. A. Dujella and 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.

  17. A. Dujella and J. C. Peral, Elliptic curves induced by Diophantine triples, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM 113 (2019), 791-806.

  18. A. Dujella and J. C. Peral, Construction of high rank elliptic curves, J. Geom. Anal. 31 (2021), 6698-6724.

  19. A. Dujella and J. C. Peral, High rank elliptic curves induced by rational Diophantine triples, Glas. Mat. Ser. III 55 (2020), 237-252.

  20. A. Dujella and J. C. Peral, An elliptic curve over Q(u) with torsion Z/4Z and rank 6, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 28 (2024), 185-192.

  21. A. Dujella, J. C. Peral and P. Tadic, Elliptic curves with torsion group Z/6Z, Glas. Mat. Ser. III 51 (2016), 321-333.

  22. N. D. Elkies, Z28 in E(Q), etc., Number Theory Listserver, May 2006.

  23. N. D. Elkies, Personal communication, 2006, 2009.

  24. N. D. Elkies, Three lectures on elliptic surfaces and curves of high rank, Lecture notes, Oberwolfach, 2007, arXiv:0709.2908.

  25. Y. G. Eroshkin, Personal communication, 2008, 2009, 2020, 2023.

  26. S. Fermigier, Exemples de courbes elliptiques de grand rang sur Q(t) et sur Q possedant des points d'ordre 2, C. R. Acad. Sci. Paris Ser. I 332 (1996), 949-952.

  27. A. Gasull, V. Manosa and X. Xarles, Rational periodic sequences for the Lyness recurrence, Discrete Contin. Dyn. Syst. 32 (2012), 587-604.

  28. L. Halbeisen, N. Hungerbühler, A. Shamsi Zargar and M. Voznyy, A geometric approach to elliptic curves with torsion groups Z/10Z, Z/12Z, Z/14Z, and Z/16Z, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 27 (2023), 87-109.

  29. T. Harrache, Etude des firations elliptiques d'une surface K3, These de doctorat, Universite Paris 6, 2009.

  30. F. Khoshnam and D. Moody, High rank elliptic curves with torsion Z/4Z induced by Kihara's elliptic curves, Integers 16 (2016), #A70, 1-12.

  31. S. Kihara, On an infinite family of elliptic curves with rank ≥ 14 over Q, Proc. Japan Acad. Ser A Math. Sci. 73 (1997), 32.

  32. S. Kihara, On the rank of elliptic curves with three rational points of order 2, Proc. Japan Acad. Ser A Math. Sci. 73 (1997), 77-78.

  33. S. Kihara, On the rank of elliptic curves with three rational points of order 2, II, Proc. Japan Acad. Ser A Math. Sci. 73 (1997), 151.

  34. S. Kihara, Construction of high-rank elliptic curves with a non-trivial rational point of order 2, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), 165.

  35. S. Kihara, On the rank of elliptic curves with a rational point of order 3, Proc. Japan Acad. Ser. A Math. Sci. 76 (2000), 126-127.

  36. S. Kihara, On an elliptic curve over Q(t) of rank ≥ 9 with a non-trivial 2-torsion point, Proc. Japan Acad. Ser. A Math. Sci. 77 (2001), 11-12.

  37. S. Kihara, On an elliptic curve over Q(t) of rank ≥ 14, Proc. Japan. Acad. Ser. A Math. Sci. 77 (2001), 50-51.

  38. S. Kihara, On the rank of elliptic curves with three rational points of order 2, III, Proc. Japan Acad. Ser A Math. Sci. 80 (2004), 13-14.

  39. S. Kihara, On the rank of elliptic curves with a rational point of order 4, Proc. Japan Acad. Ser A Math. Sci. 80 (2004), 26-27.

  40. S. Kihara, On the rank of elliptic curves with a rational point of order 4, II, Proc. Japan Acad. Ser A Math. Sci. 80 (2004), 158-159.

  41. 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.

  42. D. S. Kubert, Universal bounds on the torsion of elliptic curves, Proc. London Math. Soc. 33 (1976), 193-237.

  43. L. Kulesz, Arithmetique des courbes algebriques de genre au moins deux, These de doctorat, Universite Paris 7, 1998.

  44. L. Kulesz, Courbes elliptiques de rang ≥ 5 sur Q(t) avec un groupe de torsion isomorphe a Z/2Z × Z/2Z, C. R. Acad. Sci. Paris Ser. I Math. 329 (1999), 503-506.

  45. L. Kulesz, Families of elliptic curves of high rank with nontrivial torsion group over Q, Acta Arith. 108 (2003), 339-356.

  46. O. Lecacheux, Rang de courbes elliptiques sur Q avec un groupe de torsion isomorphe a Z/5Z, C. R. Acad. Sci. Paris Ser. I Math. 332 (2001), 1-6.

  47. O. Lecacheux, Rang de familles de courbes elliptiques, Acta Arith. 109 (2003), 131-142.

  48. O. Lecacheux, Rang de courbes elliptiques avec groupe de torsion non trivial, J. Theor. Nombres Bordeaux 15 (2003), 231-247.

  49. O. Lecacheux, Rang de courbes elliptiques dont le groupe de torsion est non trivial, Ann. Sci. Math. Quebec 28 (2004), 145-151.

  50. A. MacLeod, Personal communication, 2013, 2014.

  51. A. MacLeod, A simple method for high-rank families of elliptic curves with specified torsion, preprint, 2014.

  52. J.-F. Mestre, Courbes elliptiques de rang ≥ 11 sur Q(t), C. R. Acad. Sci. Paris Ser. I 313 (1991), 139-142.

  53. J.-F. Mestre, Courbes elliptiques de rang ≥ 12 sur Q(t), C. R. Acad. Sci. Paris Ser. I 313 (1991), 171-174.

  54. J.-F. Mestre, Construction polynomiales et theorie de Galois, Proc. International Congress Mathematicians, Zurich 1994, Birkhauser, 1995, pp. 318-323.

  55. K. Nagao, An example of elliptic curve over Q(T) with rank ≥ 13, Proc. Japan Acad. Ser. A Math. Sci. 70 (1994), 152-153.

  56. K. Nagao, Construction of high-rank elliptic curves, Kobe J. Math. 11 (1994), 211-219.

  57. K. Nagao, Construction of high-rank elliptic curves with a nontrivial torsion point, Math. Comp. 66 (1997), 411-415.

  58. A. Néron, Problèmes arithmétiques et géométriques rattachés à la notion de rang d'une courbe algébrique dans un corps, Bull. Soc. Math. France 80 (1952), 101-166.

  59. F. P. Rabarison, Torsion et rang des courbes elliptiques définies sur les corps de nombres algébriques, These de doctorat, Université de Caen, 2008.

  60. F. P. Rabarison, Construction of infinite family of elliptic curves with non trivial torsion group and high rank, internal report 2009-17-LMNO - CNRS UMR 6139.

  61. F. P. Rabarison, Structure de torsion des courbes elliptiques sur les corps quadratiques, Acta Arith. 144 (2010), 17-52.

  62. H. Suyama, Informal preliminary report (8), October 1985.

  63. M. Voznyy, Personal communication, 2021, 2022.

  64. J. Woo, Arithmetic of Elliptic Curves and Surfaces. Descent and Quadratic sections, PhD Thesis, Harvard University, 2010.

  65. A. E. A. Youmbai, Personal communication, 2024.

Old version of this tables (2006)

High rank elliptic curves with prescribed torsion

History of elliptic curves rank records

High rank elliptic curves with prescribed torsion over quadratic fields