Let T be one of 26 admissible torsion groups for
an elliptic curve over a quadratic field [KM,Ka].
We define B(T) as the supremum of the ranks
of elliptic curves defined over any quadratic field and having torsion group T
(for the trivial torsion group we put T = 0).
In the following table, in the second column we give the best known
lower bounds for B(T),
while the third column gives
the values d (smallest in absolute value) of the corresponding quadratic number fields
Values in the brackets in the second column are conditional, assuming
the Parity Conjecture.
________________________________________________________________________________________________________________
T B(T)>= d References
________________________________________________________________________________________________________________
0 34 -3 [El5]
Z/2Z 28 -1 [Wa,ACP,ADJBP]
Z/3Z 22 -3 [Na,ER]
Z/4Z 16 605113 [Vo]
Z/5Z 13 367327 [EK1,Vo]
Z/6Z 15 330374 [EK1,Vo]
Z/7Z 10 30781 [DK,Vo]
Z/8Z 10 1680281 [DP1,Vo]
Z/9Z 8 408729 [Du,Vo]
Z/10Z 8 234649 [Ra2,Vo]
Z/11Z 3 12601 [Vo]
Z/12Z 7 2014 [ADJBP]
Z/13Z 2 193 [Ra1]
Z/14Z 3 430 [HH]
Z/15Z 2 190 [Vo]
Z/16Z 4 -183443183 [Vo]
Z/18Z 2 9049 [BBDN], [Vo]
Z/2Z × Z/2Z 20 (21) d37 [EK,Vo]
Z/2Z × Z/4Z 13 (14) -83201 (186503453333275772907657409) [ADJBP], [EK,Vo]
Z/2Z × Z/6Z 10 624341 [ADJBP]
Z/2Z × Z/8Z 8 31230597 [ADJBP]
Z/2Z × Z/10Z 4 (5) 4063 (2740223761) [BBDN], [ADJBP], [Vo]
Z/2Z × Z/12Z 5 1098305 [Vo]
Z/3Z × Z/3Z 8 -3 [Du,Pe]
Z/3Z × Z/6Z 6 -3 [JB], [At,Vo], [At,Pe,Vo]
Z/4Z × Z/4Z 8 -1 [Pe]
________________________________________________________________________________________________________________
d37 = 3466878976650056958751531193538841866
[ADJBP] J. Aguirre, A. Dujella, M. Jukic Bokun, J. C. Peral, High rank elliptic curves with prescribed torsion group over quadratic fields, Period. Math. Hungar. 68 (2014), 222-230.
[At] Y. AttarBashi, Personal communication, 2021, 2022.
[BBDN] J. Bosman, P. Bruin, A. Dujella and F. Najman, Ranks of elliptic curves with prescribed torsion over number fields, Int. Math. Res. Not. IMRN 2014 (11) (2014), 2885-2923.
[Du] A. Dujella, Personal communication, 2001, 2022.
[DJB] 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.
[DJBS] A. Dujella, M. Jukic Bokun and I. Soldo, On the torsion group of elliptic curves induced by Diophantine triples over quadratic fields, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM 111 (2017), 1177-1185.
[DK] A. Dujella and L. Kulesz, Torsion group Z/7Z, rank = 5, (2001)
[DL] A. Dujella and O. Lecacheux, Torsion group Z/7Z, rank = 5, (2009)
[DN] 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.
[DP1] A. Dujella and J. C. Peral, Torsion group Z/8Z, rank = 5, (2012)
[DP2] A. Dujella and J. C. Peral, Construction of high rank elliptic curves, J. Geom. Anal. 31 (2021), 6698-6724.
[El1] N. D. Elkies, Z28 in E(Q), etc., Number Theory Listserver, May 2006.
[El2] N. D. Elkies, Personal communication, 2006, 2009.
[El3] N. D. Elkies, Three lectures on elliptic surfaces and curves of high rank, Lecture notes, Oberwolfach, 2007, arXiv:0709.2908.
[El4] N. D. Elkies, j = 0; rank 15; also 3-rank 6 and 7 in real and imaginary quadratic fields, Number Theory Listserver, Dec 2009.
[El5] N. D. Elkies, Rank of an elliptic curve and 3-rank of a quadratic field via the Burgess bounds, Proceedings of ANTS-16, (2024).
[EK1] N. D. Elkies and Z. Klagsbrun, New rank records for elliptic curves having rational torsion, Proceedings of the Fourteenth Algorithmic Number Theory Symposium, Mathematical Sciences Publishers, Berkeley, 2020, pp. 233-250.
[EK2] N. D. Elkies and Z. Klagsbrun, Z29 in E(Q), Number Theory Listserver, Aug 2024.
[ER] 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.
[Er] Y. G. Eroshkin, Personal communication, 2009, 2011.
[Fi] T. A. Fisher, Personal communication, 2009.
[HH] L. Halbeisen and N. Hungerbühler, Personal communication, 2020.
[HHVZ] L. Halbeisen, N. Hungerbühler, A. Shamsi Zargar, 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.
[JB] M. Jukic Bokun, On the rank of elliptic curves over Q(√-3) with torsion groups Z/3Z × Z/3Z and Z/3Z × Z/6Z, Proc. Japan Acad. Ser. A Math. Sci. 87 (2011), 61-64.
[JB2] M. Jukic Bokun, Elliptic curves over quadratic fields with fixed torsion subgroup and positive rank, Glas. Mat. Ser. III 47 (2012), 277-284.
[Ka] S. Kamienny, Torsion points on elliptic curves and q-coefficients of modular forms, Invent. Math. 109 (1992), 221-229.
[KM] M. A. Kenku and F. Momose, Torsion points on elliptic curves defined over quadratic fields, Nagoya Math. J. 109 (1988), 125-149.
[Kl] Z. Klagsbrun, Personal communication, 2020.
[Ku] L. Kulesz, Arithmetique des courbes algebriques de genre au moins deux, These de doctorat, Universite Paris 7, 1998.
[Le1] 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.
[Le2] O. Lecacheux, Rang de courbes elliptiques dont le groupe de torsion est non trivial, Ann. Sci. Math. Quebec 28 (2004), 145-151.
[Me] J.-F. Mestre, Rang des courbes elliptiques d'invariant donné, C. R. Acad. Sci. Paris 314 (1992), 919-922.
[Na] F. Najman, Some rank records for elliptic curves with prescribed torsion over quadratic fields, An. Stiint. Univ. "Ovidius" Constanta Ser. Mat. 22 (2014), 215-220.
[Pe] V. Petricevic, Personal communication, 2022.
[Ra1] F. P. Rabarison, Structure de torsion des courbes elliptiques sur les corps quadratiques, Acta Arith. 144 (2010), 17-52.
[Ra2] R. Rathbun, Torsion group Z/10Z, rank = 3, (2003)
[SZ] 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.
[Vo] M. Voznyy, Personal communication, 2021, 2022, 2023, 2024, 2025.
[Wa] M. Watkins, Personal communication, 2005.
High rank elliptic curves with prescribed torsion
Infinite families of elliptic curves with high rank and prescribed torsion
History of elliptic curves rank records