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 ∈ { 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 N O K . This would be a partial answer to the question raised by Hasse and Ramachandra.