Mathematika

Research Article

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

M. A. Kenkua1

a1 Department of Mathematics, University of Lagos, Akoka, Lagos, Nigeria.

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 [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].

(Received January 23 1984)

Key Words:

  • 14K07: ALGEBRAIC GEOMETRY; Special ground fields, arithmetic problems; Elliptic curves