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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Rouse, Jeremy and Zureick-Brown, David 2015. Elliptic curves over ℚ $\mathbb {Q}$ and 2-adic images of Galois. Research in Number Theory, Vol. 1, Issue. 1,

    Baran, Burcu 2010. Normalizers of non-split Cartan subgroups, modular curves, and the class number one problem. Journal of Number Theory, Vol. 130, Issue. 12, p. 2753.

    Baran, Burcu 2009. A modular curve of level 9 and the class number one problem. Journal of Number Theory, Vol. 129, Issue. 3, p. 715.

    Chen, Imin 1999. On Siegel's Modular Curve of Level 5 and the Class Number One Problem. Journal of Number Theory, Vol. 74, Issue. 2, p. 278.


A note on the integral points of a modular curve of level 7

  • M. A. Kenku (a1)
  • DOI:
  • Published online: 01 February 2010

Let . denote the modular curve associated with the normalizer of a non-split Cartan group of level N., where N. is an arbitrary integer. The curve is denned over Q and the corresponding scheme over ℤ[1/N] is smooth [1]. If N. is a prime, the genus formula for . is given in [5,6]. The curve . has genus 0 if N < 11 and has genus 1. Ligozat [5] has shown that the group of Q-rational points on has rank 1. If the genus g(N). is greater than 1, very little is known about the Q-rational points of . Since under simple conditions imaginary quadratic fields with class number 1 give an integral point on these curves, Serre and others have asked whether all integral points are obtained in this way [8].

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1.P. Deligne and M. Rapoport . Schémas de modules des courbes elliptiques, Vol. II of the Proceedings of the International Summer School on Modular Functions, Antwerp. (1972). Lecture Notes in Mathematics., 349 (Springer, Berlin, 1973).

3.F. Klein . Gesammelte mathematische Abhandlungen, Vol. 3. (Springer, Berlin, 1923).

7.T. Nagell . Sur un type particuliér d'unites algébriques. Arkiv für Mat., 8 (1969), 163184.

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