Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



Elliptic curves over the rationals with bad reduction at only one prime

Authors: Bas Edixhoven, Arnold de Groot and Jaap Top
Journal: Math. Comp. 54 (1990), 413-419
MSC: Primary 11G05; Secondary 11D25
MathSciNet review: 995209
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A list is given of elliptic curves over Q having additive reduction at exactly one prime. It is also proved that for primes congruent to 5 modulo 12, no such curves having potentially good reduction exist. This enables one to find in a number of cases a complete list of all elliptic curves with bad reduction at only one prime.

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

  • [1] B. J. Birch and W. Kuyk (eds.), Modular functions of one variable. IV, Lecture Notes in Math., vol. 476, Springer-Verlag, 1975. MR 0376533 (51:12708)
  • [2] A. Brumer and K. Kramer, The rank of elliptic curves, Duke Math. J. 44 (1977), 715-743. MR 0457453 (56:15658)
  • [3] S. J. Edixhoven, Stable models of modular curves and applications, Ph.D. Thesis, Math. Inst., Univ. of Utrecht, 1989.
  • [4] B. H. Gross, Arithmetic on elliptic curves with complex multiplication, Lecture Notes in Math., vol. 776, Springer-Verlag, 1980. MR 563921 (81f:10041)
  • [5] B. Mazur, Rational isogenies of prime degree, Invent. Math. 44 (1978), 129-162. MR 482230 (80h:14022)
  • [6] J.-F. Mestre, La méthode des graphes. Exemples et applications, Proc. Internat. Conf. on Class Numbers and Fundamental Units of Algebraic Number Fields (Nagoya Univ., Nagoya, 1986), Katata, Japan, 1986, pp. 217-242. MR 891898 (88e:11025)
  • [7] T. Nagel, Über die rationalen Punkte auf einigen kubischen Kurven, Tôhoku Math. J. 24 (1925), 48-53.
  • [8] M. T. Nagell, L'analyse indéterminée de degré supérieur, Mémorial des Sciences Mathématiques 39 (1929).
  • [9] B. Setzer, Elliptic curves of prime conductor, J. London Math. Soc. 10 (1975), 367-378. MR 0371904 (51:8121)
  • [10] J. H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Math., vol. 106, Springer-Verlag, 1986. MR 817210 (87g:11070)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 11G05, 11D25

Retrieve articles in all journals with MSC: 11G05, 11D25

Additional Information

Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society