Rocky Mountain Journal of Mathematics

On high rank $\pi/3$ and $2\pi/3$-congruent number elliptic curves

A.S. Janfada, S. Salami, A. Dujella, and J.C. Peral

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Consider the elliptic curves given by \[ \ent : y^2=x^3+2s n x^2-(r^2-s^2) n^2 x \] where $0 \lt \ta\lt \pi$, $\cos(\ta)=s/r$ is rational with $0\leq |s| \lt r$ and $\gcd (r,s)=1$. These elliptic curves are related to the $\ta$-congruent number problem as a generalization of the congruent number problem. For fixed $\ta$, this family corresponds to the quadratic twist by $n$ of the curve $\ttt: y^2=x^3+2s x^2-(r^2-s^2) x$. We study two special cases: $\ta=\pi/3$ and $\ta=2\pi/3$. We have found a subfamily of $n=n(w)$ having rank at least $3$ over $\Q(w)$ and a subfamily with rank~$4$ parametrized by points of an elliptic curve with positive rank. We also found examples of $n$ such that $E_{n, \ta}$ has rank up to $7$ over $\Q$ in both cases.

Article information

Source
Rocky Mountain J. Math., Volume 44, Number 6 (2014), 1867-1880.

Dates
First available in Project Euclid: 2 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1422885098

Digital Object Identifier
doi:10.1216/RMJ-2014-44-6-1867

Mathematical Reviews number (MathSciNet)
MR3310952

Zentralblatt MATH identifier
1321.11062

Subjects
Primary: 11G05: Elliptic curves over global fields [See also 14H52]

Keywords
$\theta$-congruent number elliptic curve Mordell-Weil rank

Citation

Janfada, A.S.; Salami, S.; Dujella, A.; Peral, J.C. On high rank $\pi/3$ and $2\pi/3$-congruent number elliptic curves. Rocky Mountain J. Math. 44 (2014), no. 6, 1867--1880. doi:10.1216/RMJ-2014-44-6-1867. https://projecteuclid.org/euclid.rmjm/1422885098


Export citation

References

  • B.J. Birch and H.P.F. Swinnerton-Dyer, Notes on elliptic curves, II, J. reine angew. Math. 218 (1965), 79–108.
  • W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, J. Symb. Comp. 24 (1997), 235–265.
  • G. Campbell, Finding elliptic curves and families of elliptic curves over $\Q$ of large rank, Ph.D. thesis, Rutgers University, 1999.
  • J. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, 1997.
  • A. Dujella, High rank elliptic curves with prescribed torsion, http:// www.maths.hr/~duje/tors.html.
  • A. Dujella, A.S. Janfada and S. Salami, A search for high rank congruent number elliptic curves, J. Int. Seq. 12 (2009), 09.5.8.
  • A. Dujella and M. Jukić Bokun, On the rank of elliptic curves over $\Q(i)$ with torsion group $\Z/4\Z \times \Z/4\Z$, Proc. Japan Acad. Math. Sci. 86 (2010), 93–96.
  • N.D. Elkies, Three lectures on elliptic surfaces and curves of high rank, Lecture notes, Oberwolfach, 2007, arXiv:0709.2908.
  • M. Fujiwara, $\ta$-congruent numbers, in Number theory, K. Győry, A. Pethő and V. Sós, eds., de Gruyter, Berlin, 1997.
  • M. Fujiwara, Some properties of $\ta$-congruent numbers, Natural Science Report, Ochanomizu University, 118 (2001), 1–8.
  • F. Gouvêa and B. Mazur, The square-free sieve and the rank of elliptic curves, J. Amer. Math. Soc. 4 (1991), 1–23.
  • M. Kan, $\ta$-congruent numbers and elliptic curves, Acta Arith. 94 (2000), 153–160.
  • N. Koblitz, Introduction to elliptic curves and modular forms, Grad. Texts Math. 97, 2nd edition, Springer-Verlag, Berlin, 1993.
  • J.-F. Mestre, Construction de courbes elliptiques sur $\Q$ de rang $\geq 12$, C.R. Acad. Sci. Paris 295 (1982), 643–644.
  • ––––, Formules explicites et minorations de conducteurs de variétés algébriques, Comp. Math. 58 (1986), 209–232.
  • ––––, Rang de certaines familles de courbes elliptiques d' invariant donné, C.R. Acad. Sci. Paris 327 (1998), 763-764.
  • K. Nagao, An example of elliptic curve over $\Q$ with rank $\geq 21$, Proc. Japan Acad. Math. Sci. 70 (1994), 104–105.
  • PARI/GP, version 2.3.3, Bordeaux, 2008, http://pari.math.u-bordeaux.fr
  • N. Rogers, Rank computations for the congruent number elliptic curves, Exp. Math. 9 (2000), 591–594.
  • ––––, Elliptic curves $x^3+y^3=k$ with high rank, Ph.D. thesis, Harvard University, Cambridge, 2004.
  • K. Rubin and A. Silverberg, Twists of elliptic curves of rank at least four, in Ranks of elliptic curves and random matrix theory, Cambridge University Press, 2007.
  • ––––, Rank frequencies for quadratic twists of elliptic curves, Exp. Math. 10 (2001), 559–569.