Generation of class fields by using the Weber function

Abstract

Let K be an imaginary quadratic field and O-K be its ring of integers. Let h(E) be the Weber function on a certain elliptic curve E with complex multiplication by O-K. We show that if N (> 1) is an integer prime to 6, then the function h(E) alone generates the ray class field modulo NOK over K when evaluated at some N-torsion point of E, which would be a partial answer to the question mentioned in [10, p. 105]. (C) 2016 Elsevier Inc. All rights reserved.