On the torsion group of elliptic curves induced by D(4)-triples

Dujella, Andrej; Mikić, Miljen

doi:10.2478/auom-2014-0033

Abstract

A D(4)-m-tuple is a set of m integers such that the product of any two of them increased by 4 is a perfect square. A problem of extendibility of D(4)-m-tuples is closely connected with the properties of elliptic curves associated with them. In this paper we prove that the torsion group of an elliptic curve associated with a D(4)-triple can be either ℤ/2ℤ × ℤ/2ℤ or ℤ/2ℤ × ℤ/6ℤ, except for the D(4)-triple {−1, 3, 4} when the torsion group is ℤ/2ℤ × ℤ/4ℤ.

References

[1] L. E. Dickson, A history of the Theory of numbers, Vol. 2, Chelsea, New York, 1966., pp. 513–520.
Search in Google Scholar
[2] Diophantus of Alexandria, Arithmetics and the Book of Polygonal Numbers (I. G. Bashmakova, Ed.), Nauka, 1974, (in Russian), pp. 103–104, 232.
Search in Google Scholar
[3] A. Dujella, A parametric family of elliptic curves, Acta Arith. 94 (2000), 87–101.10.4064/aa-94-1-87-101
Search in Google Scholar
[4] A. Dujella, Diophantine m-tuples and elliptic curves, J. Theor. Nombres Bordeaux 13 (2001), 111–124.10.5802/jtnb.308
Search in Google Scholar
[5] A. Dujella, On the size of Diophantine m-tuples, Math. Proc. Cambridge Philos. Soc. 132 (2002), 23–33.10.1017/S0305004101005515
Search in Google Scholar
[6] A. Dujella, There are only finitely many Diophantine quintuples, J. Reine Angew. Math. 566 (2004), 183–214.
Search in Google Scholar
[7] A. Dujella, On Mordell-Weil groups of elliptic curves induced by Diophantine triples, Glas. Mat. Ser. III 42 (2007), 3–18.10.3336/gm.42.1.01
Search in Google Scholar
[8] A. Dujella and A. Pethő, Integer points on a family of elliptic curves, Publ. Math. Debrecen 56 (2000), 321–335.10.5486/PMD.2000.2192
Search in Google Scholar
[9] A. Dujella and A. M. S. Ramasamy, Fibonacci numbers and sets with the property D(4), Simon Stevin 12 (2005), 401–412.
Search in Google Scholar
[10] A. Filipin, There does not exist a D(4)-sextuple, J. Number Theory 128 (2008), 1555–1565.10.1016/j.jnt.2007.07.012
Search in Google Scholar
[11] A. Filipin, There are only finitely many D(4)-quintuples, Rocky Mountain J. Math. 41 (2011), 1847–1859.10.1216/RMJ-2011-41-6-1847
Search in Google Scholar
[12] A. Filipin, An irregular D(4)-quadruple cannot be extended to a quintuple, Acta Arith. 136 (2009), 167–176.10.4064/aa136-2-5
Search in Google Scholar
[13] A. Filipin, Bo He and A. Togbé, On a family of two-parametric D(4)-triples, Glas. Mat. Ser. III 47 (2012), 31–51.10.3336/gm.47.1.03
Search in Google Scholar
[14] K. S. Kedlaya, Solving constrained Pell equations, Math. Comp. 67 (1998), 833–842.10.1090/S0025-5718-98-00918-1
Search in Google Scholar
[15] A. Knapp, Elliptic Curves, Princeton Univ. Press, 1992.
Search in Google Scholar
[16] B. Mazur, Rational isogenies of prime degree, Invent. Math. 44 (1978), 129–162.10.1007/BF01390348
Search in Google Scholar
[17] S. P. Mohanty and A. M. S. Ramasamy, The characteristic number of two simultaneous Pell’s equations and its applications, Simon Stevin 59 (1985), 203–214.
Search in Google Scholar
[18] F. Najman, Integer points on two families of elliptic curves, Publ. Math. Debrecen 75 (2009), 401–418.10.5486/PMD.2009.4413
Search in Google Scholar

On the torsion group of elliptic curves induced by D(4)-triples

Abstract

Paradigm

My account