Generation of ray class fields modulo 2, 3, 4 or 6 by using the Weber function

Abstract

Let K be an imaginary quadratic field with ring of integers O-K. Let E be an elliptic curve with complex multiplication by O-K, and let h(E) be the Weber function on E. Let N is an element of {2, 3, 4, 6}. We show that h(E) alone when evaluated at a certain N-torsion point on E generates the ray class field of K modulo NOK. This would be a partial answer to the question raised by Hasse and Ramachandra.