Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Elliptic curves with nonsplit mod $11$ representations

Authors: Imin Chen and Chris Cummins
Journal: Math. Comp. 73 (2004), 869-880
MSC (2000): Primary 11G05; Secondary 14G05
Published electronically: June 17, 2003
MathSciNet review: 2031412
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We calculate explicitly the $j$-invariants of the elliptic curves corresponding to rational points on the modular curve $X_{ns}^+(11)$ by giving an expression defined over $\mathbb{Q} $ of the $j$-function in terms of the function field generators $X$ and $Y$of the elliptic curve $X_{ns}^+(11)$. As a result we exhibit infinitely many elliptic curves over $\mathbb{Q} $ with nonsplit mod $11$representations.

References [Enhancements On Off] (What's this?)

  • 1. B.J. Birch and W. Kuyk, editors.
    Modular Functions of One Variable IV, number 476 in Lecture Notes in Mathematics. Springer-Verlag, 1972. MR 51:12708
  • 2. I. Chen.
    The Jacobian of Modular Curves Associated to Cartan Subgroups.
    PhD thesis, University of Oxford, 1996.
  • 3. I. Chen.
    Surjectivity of mod $\ell$ representations attached to elliptic curves and congruence primes.
    To appear in the Canadian Mathematical Bulletin, 2002.
  • 4. P. Deligne and M. Rapoport.
    Les schémas de modules de courbes elliptiques.
    In P. Deligne and W. Kuyk, editors, Modular Functions of One Variable II, number 349 in Lecture Notes in Mathematics, pages 143-316. Springer-Verlag, 1972. MR 49:2762
  • 5. N. Katz and B. Mazur.
    Arithmetic Moduli of Elliptic Curves.
    Number 108 in Annals of Mathematics Studies. Princeton University Press, 1985. MR 86i:11024
  • 6. M.A. Kenku.
    A note on the integral points of a modular curve of level 7.
    Mathematika, 32:45-48, 1985. MR 87d:11040
  • 7. D. Kubert and S. Lang.
    Modular Units, volume 244 of Grundlehren der mathematischen Wissenschaften.
    Springer-Verlag, 1981. MR 84h:12009
  • 8. Daniel Kubert.
    Quadratic relations for generators of units in the modular function field.
    Math. Ann., 225(1):1-20, 1977. MR 55:5536
  • 9. G. Ligozat.
    Courbes modulaires de niveau 11.
    In J.P. Serre and D.B. Zagier, editors, Modular Functions of One Variable V, number 601 in Lecture Notes in Mathematics, pages 149-237. Springer-Verlag, 1977. MR 57:3079
  • 10. B. Mazur.
    Rational isogenies of prime degree.
    Inventiones mathematicae, 44:129-162, 1978. MR 80h:14022
  • 11. J.P. Serre.
    Propriétés galoisiennes des points d'ordre fini des courbes elliptiques.
    Inventiones Mathematicae, 15:259-331, 1972. MR 52:8126
  • 12. J.P. Serre.
    Lectures on the Mordell-Weil Theorem.
    Number E15 in Aspects of Mathematics. Friedr, Vieweg & Sohn, Braunschweig, 1989. MR 90e:11086
  • 13. C.L. Siegel.
    Zum Beweise des Starkschen Satzes.
    Inventiones Mathematicae, 5:180-191, 1968. MR 37:4045

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 11G05, 14G05

Retrieve articles in all journals with MSC (2000): 11G05, 14G05

Additional Information

Imin Chen
Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada, V5A 1S6

Chris Cummins
Affiliation: Department of Mathematics and Statistics, Concordia University, Montreal, Quebec, Canada, H3G 1M8

Received by editor(s): May 2, 2002
Received by editor(s) in revised form: September 11, 2002
Published electronically: June 17, 2003
Additional Notes: Research supported by NSERC
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society