Infinite families of elliptic curves with high rank and prescribed torsion

New version of this tables

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.

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    T         G(T)>=             Author(s)
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    0           18        Elkies (2006) 
   Z/2Z          9        Kihara (2001)
   Z/3Z          6        Kulesz (1998), Kihara (2000)
   Z/4Z          5        Kihara (2004)		 
   Z/5Z          3        Lecacheux (2001)
   Z/6Z          3        Lecacheux (2001), Kihara (2006) 	 	 
   Z/7Z          1        Kulesz (1998), Lecacheux (2003)
   Z/8Z          1        Kulesz (1998), Lecacheux (2002) 
   Z/9Z          0        Kubert (1976)
   Z/10Z         0        Kubert (1976)  
   Z/12Z         0        Kubert (1976)
Z/2Z × Z/2Z      6        Kihara (2004)
Z/2Z × Z/4Z      3        Lecacheux (2001)
Z/2Z × Z/6Z      1        Kulesz (1998), Campbell (1999), Lecacheux (2002), Dujella (2007)
Z/2Z × Z/8Z      0        Kubert (1976) 
__________________________________________________________________________________________



_________________________________________________________________________________________________________

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

    0           19        Elkies (2006) 
   Z/2Z          9        Kihara (1997)
   Z/3Z          6        Kulesz (1998), Kihara (2000)
   Z/4Z          5        Kihara (2004) 		 
   Z/5Z          3        Lecacheux (2001)
   Z/6Z          4        Kihara (2006)	 	 
   Z/7Z          2        Lecacheux (2003), Elkies (2006)
   Z/8Z          2        Lecacheux (2002) 
   Z/9Z          1        Atkin - Morain (1993), Kulesz (1998)
   Z/10Z         1        Atkin - Morain (1993), Kulesz (1998)
   Z/12Z         1        Suyama (1985), Kulesz (1998)
Z/2Z × Z/2Z      6        Kihara (2004)
Z/2Z × Z/4Z      3        Lecacheux (2001), Adelstein - Christiansen (2006)
Z/2Z × Z/6Z      2        Lecacheux (2002)
Z/2Z × Z/8Z      1        Atkin - Morain (1993), Kulesz (1998), Lecacheux (2002), Campbell - Goins (2003) 	 	
_________________________________________________________________________________________________________

References:

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

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

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

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

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10. N. D. Elkies, Personal communication, 2006.

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22. 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.

23. 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.

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

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

26. 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.

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

28. 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.

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

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

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

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

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

High rank elliptic curves with prescribed torsion

History of elliptic curves rank records