Rad HAZU, Matematičke znanosti, Vol. 24 (2020), 29-37.

RANK ZERO ELLIPTIC CURVES INDUCED BY RATIONAL DIOPHANTINE TRIPLES

Andrej Dujella and Miljen Mikić

Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia
e-mail: duje@math.hr

Kumičićeva 20, 51000 Rijeka, Croatia
e-mail: miljen.mikic@gmail.com

Abstract.   Rational Diophantine triples, i.e. rationals a, b, c with the property that ab + 1, ac + 1, bc + 1 are perfect squares, are often used in the construction of elliptic curves with high rank. In this paper, we consider the opposite problem and ask how small can be the rank of elliptic curves induced by rational Diophantine triples. It is easy to find rational Diophantine triples with elements with mixed signs which induce elliptic curves with rank 0. However, the problem of finding such examples of rational Diophantine triples with positive elements is much more challenging, and we will provide the first such known example.

2020 Mathematics Subject Classification.   11G05, 11D09.

Key words and phrases.   Elliptic curves, Diophantine triples, rank, torsion group.

Full text (PDF) (free access)

DOI: https://doi.org/10.21857/m8vqrtq4j9

References:

  1. J. Aguirre, A. Dujella and J. C. Peral, On the rank of elliptic curves coming from rational Diophantine triples, Rocky Mountain J. Math. 42 (2012), 1759-1776.
    MathSciNet     CrossRef

  2. A. Baker and H. Davenport, The equations 3x2 - 2 = y2 and 8x2 - 7 = z2, Quart. J. Math. Oxford Ser. (2) 20 (1969), 129-137.
    MathSciNet     CrossRef

  3. W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I. The user language, J. Symb. Comp. 24 (1997), 235-265.
    MathSciNet     CrossRef

  4. G. Campbell and E. H. Goins, Heron triangles, Diophantine problems and elliptic curves, preprint, http://www.swarthmore.edu/NatSci/gcampbe1/papers/heron-Campbell-Goins.pdf

  5. I. Connell, Elliptic Curve Handbook, McGill University, 1999.

  6. J. Cremona, Algorithms for Modular Elliptic Curves, Cambridge University Press, Cambridge, 1997.
    MathSciNet

  7. 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.
    MathSciNet     CrossRef

  8. A. Dujella, Diophantine m-tuples and elliptic curves, J. Théor. Nombres Bordeaux 13 (2001), 111-124.
    MathSciNet

  9. A. Dujella, There are only finitely many Diophantine quintuples, J. Reine Angew. Math. 566 (2004), 183-214.
    MathSciNet     CrossRef

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

  11. A. Dujella, What is ... a Diophantine m-tuple?, Notices Amer. Math. Soc. 63 (2016), 772-774.
    MathSciNet     CrossRef

  12. A. Dujella, M. Jukić 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.
    MathSciNet     CrossRef

  13. A. Dujella and M. Kazalicki, More on Diophantine sextuples, in: Number Theory - Diophantine problems, uniform distribution and applications, Festschrift in honour of Robert F. Tichy’s 60th birthday (C. Elsholtz, P. Grabner, Eds.), Springer, Cham, 2017, pp. 227-235.
    MathSciNet

  14. A. Dujella, M. Kazalicki, M. Mikić and M. Szikszai, There are infinitely many rational Diophantine sextuples, Int. Math. Res. Not. IMRN 2017 (2) (2017), 490-508.
    MathSciNet     CrossRef

  15. A. Dujella, M. Kazalicki and V. Petričević, There are infinitely many rational Diophantine sextuples with square denominators, J. Number Theory 205 (2019), 340-346.
    MathSciNet     CrossRef

  16. A. Dujella, M. Kazalicki and V. Petričević, Rational Diophantine sextuples containing two regular quadruples and one regular quintuple, Acta Mathematica Spalatensia, to appear.

  17. A. Dujella and M. Mikić, On the torsion group of elliptic curves induced by D(4)-triples, An. Stiint. Univ. "Ovidius" Constanta Ser. Mat. 22 (2014), 79-90.
    MathSciNet     CrossRef

  18. A. Dujella and J. C. Peral, High rank elliptic curves with torsion Z/2Z × Z/4Z induced by Diophantine triples, LMS J. Comput. Math. 17 (2014), 282-288.
    MathSciNet     CrossRef

  19. A. Dujella and J. C. Peral, Elliptic curves induced by Diophantine triples, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM 113 (2019), 791-806.
    MathSciNet     CrossRef

  20. A. Dujella and J. C. Peral, Construction of high rank elliptic curves, J. Geom. Anal. (2020), https://doi.org/10.1007/s12220-020-00373-7
    CrossRef

  21. A. Dujella and J. C. Peral, High rank elliptic curves induced by rational Diophantine triples, Glas. Mat. Ser. III, to appear.

  22. T. Fisher, Higher descents on an elliptic curve with a rational 2-torsion point, Math. Comp. 86 (2017), 2493-2518.
    MathSciNet     CrossRef

  23. P. Gibbs, Some rational Diophantine sextuples, Glas. Mat. Ser. III 41 (2006), 195-203.
    MathSciNet     CrossRef

  24. B. He, A. Togbé and V. Ziegler, There is no Diophantine quintuple, Trans. Amer. Math. Soc. 371 (2019), 6665-6709.
    MathSciNet     CrossRef

  25. B. Mazur, Rational isogenies of prime degree, Invent. Math. 44 (1978), 129-162.
    MathSciNet     CrossRef

  26. T. Skolem, Diophantische Gleichungen, Chelsea, 1950.

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