## High rank elliptic curves with prescribed torsion over quadratic fields

Maintained by Andrej Dujella, University of Zagreb

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 K = Q(√d).
Values in the brackets in the second column are conditional, assuming the Parity Conjecture.

```________________________________________________________________________________________________________________

T          B(T)>=	                 d              			References
________________________________________________________________________________________________________________

0            30                     -3					[Na,El4], [El1,El3,Me]

Z/3Z          22		        -3					[Na,ER]

Z/4Z          15 (16)            -25689 (407589695068990858395)   		[ADJBP], [Na], [EK,Me], [El2,Vo]

Z/5Z          11        	         *					[Kl,Me]

Z/6Z          15                 330374    					[EK,Vo]

Z/7Z           8        	         *					[EK,Me]

Z/8Z           9 (10)              -227 (68434553555382)			[ADJBP], [DP,Vo]

Z/9Z           8        	    408729 					[Du,Vo]

Z/11Z          3		     12601				        [Vo]

Z/13Z          2		       193					[Ra]

Z/14Z          3		       430					[HH]

Z/15Z          2		       190					[Vo]

Z/16Z          3 (4)                205 (17381446)				[Na], [Vo], [HHVZ]

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 (55325286553)			[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

The marks * in the table refer to the curves obtained by an application of Mestre's general construction
(he proved that for any elliptic curve E over Q there exist infinitely many quadratic twists with rank ≥ 2)
which produces quadratic fields with huge discriminants.

### References:

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

[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.

[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.

[DP]   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. Elkies, j = 0; rank 15; also 3-rank 6 and 7 in real and imaginary quadratic fields, Number Theory Listserver, Dec 2009.

[EK]   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.

[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.. to appear.

[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.

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

[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.

[Wa]   M. Watkins, Personal communication, 2005.