High rank elliptic curves with prescribed torsion

Maintained by Andrej Dujella, University of Zagreb

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

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

The conjecture is that B(T) is unbounded for all T. In the following table we give the best known lower bounds for B(T). 

______________________________________________________________________________________________________

    T         B(T)>=	              Author(s)
______________________________________________________________________________________________________

    0           28        Elkies (2006)

   Z/2Z         19        Elkies (2009)	

   Z/3Z         13        Eroshkin (2007,2008,2009)

   Z/4Z         12        Elkies (2006)  

   Z/5Z          8        Dujella - Lecacheux (2009), Eroshkin (2009) 

   Z/6Z          8        Eroshkin (2008), Dujella - Eroshkin (2008), Elkies (2008), Dujella (2008) 	

   Z/7Z          5        Dujella - Kulesz (2001), Elkies (2006), Eroshkin (2009), 
                          Dujella - Lecacheux (2009), Dujella - Eroshkin (2009)

   Z/8Z          6        Elkies (2006) 

   Z/9Z          4        Fisher (2009) 

   Z/10Z         4        Dujella (2005,2008), Elkies (2006)    

   Z/12Z         4        Fisher (2008) 

Z/2Z × Z/2Z     15        Elkies (2009)

Z/2Z × Z/4Z      8        Elkies (2005), Eroshkin (2008), Dujella - Eroshkin (2008)

Z/2Z × Z/6Z      6        Elkies (2006)

Z/2Z × Z/8Z      3        Connell (2000), Dujella (2000,2001,2006,2008), Campbell - Goins (2003), 
                          Rathbun (2003,2006), Dujella - Rathbun (2006), 
                          Flores - Jones - Rollick - Weigandt - Rathbun (2007), Fisher (2009)
                                                   
______________________________________________________________________________________________________

Click on rank r to see the corresponding "record" curve(s) with torsion points and independent points P1, P2, ... , Pr of infinite order.

References:

  1. J. Aguirre, F. Castaneda, J. C. Peral, High rank elliptic curves with torsion group Z/(2Z), Math. Comp. 73 (2004), 323-331.

  2. J. Aguirre, A. Dujella and J. C. Peral, On the rank of elliptic curves coming from rational Diophantine triples, Rocky Mountain J. Math., to appear.

  3. J. Aguirre, J. C. Peral, Personal communication, 2009.

  4. K. P. Ansaldi, A. R. Ford, J. L. George, K. M. Mugo, C. E. Phifer, In search of an 8: Rank computations on a family of quartic curves, The Journal of the SUMSRI, Summer 2005.

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

  6. T. D. Brooks, E. A. Fowler, K. C. Hastings, D. L. Hiance, M. A. Zimmerman, Elliptic curves with torsion subgroup Z/2Z × Z/8Z: does a rank 4 curve exist?, The Journal of the SUMSRI, Summer 2006.

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

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

  9. I. Connell, APECS, ftp://ftp.math.mcgill.ca/pub/apecs/

  10. A. Dujella, Number Theory Listserver, Apr 2000, May 2000, Mar 2001, Apr 2001, Sep 2002, Dec 2005.

  11. 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.

  12. A. Dujella, Irregular Diophantine m-tuples and elliptic curves of high rank, Proc. Japan Acad. Ser. A Math. Sci. 76 (2000), 66-67.

  13. A. Dujella, An example of elliptic curve over Q with rank equal to 15, Proc. Japan Acad. Ser. A Math. Sci. 78 (2002), 109-111.

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

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

  16. A. Dujella, 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., to appear.

  17. N. D. Elkies, Algorithmic Number Theory: Tables and Links, http://www.math.harvard.edu/~elkies/compnt.html

  18. N. D. Elkies, E(Q) = (Z/2Z) * (Z/4Z) * Z8, Number Theory Listserver, Jun 2005.

  19. N. D. Elkies, E(Q) = (Z/4Z) * Z11 [also (Z/2Z)2 * Z11], Number Theory Listserver, Jun 2005.

  20. N. D. Elkies, E(Q) = (Z/2Z) * Z17, Number Theory Listserver, Jun 2005.

  21. N. D. Elkies, E(Q) = (Z/2Z)2 * Z14, Number Theory Listserver, Dec 2005.

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

  23. N. D. Elkies, Some more rank records: E(Q) = (Z/2Z) * Z18, (Z/4Z) * Z12, (Z/8Z) * Z6, (Z/2Z) * (Z/6Z) * Z6, Number Theory Listserver, Jun 2006.

  24. N. D. Elkies, Personal communication, 2006, 2008, 2009.

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

  26. N. D. Elkies and N. F. Rogers, New rank records for x3 + y3 = k, Number Theory Listserver, May 2003, Jul 2003, Oct 2003.

  27. N. D. Elkies and N. F. Rogers, Elliptic curves x3 + y3 = k of high rank, Proceedings of ANTS-6 (D. Buell, ed.), Lecture Notes in Comput. Sci. 3076 (2004), 184-193.

  28. Y. G. Eroshkin, Personal communication, 2006, 2007, 2008, 2009, 2010.

  29. S. Fermigier, Un exemple de courbe elliptique definie sur Q de rang ≥ 19, C. R. Acad. Sci. Paris Ser. I 315 (1992), 719-722.

  30. 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 322 (1996), 949-952.

  31. S. Fermigier, Une courbe elliptique definie sur Q(t) de rang ≥ 22, Acta Arith. (1997), 359-363.

  32. T. A. Fisher, Personal communication, 2008, 2009.

  33. J. Flores, K. Jones, A. Rollick, J. Weigandt, A statistical analysis of 2-Selmer groups for elliptic curves with torsion subgroup Z2 × Z8, The Journal of the SUMSRI, Summer 2007.

  34. S. Ivy, B. Jefferson, M. Josey, C. Outing, C. Taylor, S. White, 4-covering maps on elliptic curves with torsion subgroup Z2 × Z8, The Journal of the SUMSRI, Summer 2008.

  35. T.J. Kretschmer, Construction of elliptic curves with large rank, Math. Comp. 46 (1986), 627-635.

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

  37. L. Kulesz and C. Stahlke, Elliptic curves of high rank with nontrivial torsion group over Q, Experiment. Math. 10 (2001), 475-480.

  38. 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.

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

  40. A. MacLeod, Personal communication, 2004.

  41. R. Martin and W. McMillen, An elliptic curve over Q with rank at least 23, Number Theory Listserver, March 1998.

  42. R. Martin and W. McMillen, An elliptic curve over Q with rank at least 24, Number Theory Listserver, May 2000.

  43. J.-F. Mestre, Construction de courbes elliptiques sur Q de rang ≥ 12, C. R. Acad. Sci. Paris Ser I Math. 295 (1982), 643-644.

  44. J.-F. Mestre, Un exemple de courbe elliptique sur Q de rang ≥ 15, C. R. Acad. Sci. Paris Ser I Math. 314 (1992), 453-455.

  45. P.L. Montgomery, Speeding the Polard and elliptic curve methods of factorization, Math. Comp. 48 (1987), 243-264.

  46. K. Nagao, Examples of elliptic curves over Q with rank ≥ 17, Proc. Japan Acad. Ser. A Math. Sci. 68 (1992), 287-289.

  47. K. Nagao, An example of elliptic curve over Q with rank ≥ 20, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), 291-293.

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

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

  50. R. L. Rathbun, Number Theory Listserver, Oct 2003.

  51. R. L. Rathbun, Personal communication, 2006.

  52. U. Schneiders and H.G. Zimmer, The rank of elliptic curves upon quadratic extensions, in: Computational Number Theory (A. Petho, H.C. Williams, H.G. Zimmer, eds.), de Gruyter, Berlin, 1991, pp. 239-260.

  53. M. Watkins, Personal communication, 2005.

  54. T. Womack, Curves with moderate rank and interesing torsion group, http://www.tom.womack.net/maths/torsion.htm

Infinite families of elliptic curves with high rank and prescribed torsion

History of elliptic curves rank records