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

  • M. A. Kenku (a1)

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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  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
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