No abstract

We first generate ray class fields over imaginary quadratic fields in terms of Siegel-Ramachandra invariants, which would be an extension of Schertz's result [12]. And, by making use of quotients of Siegel-Ramachandra invariants we also construct ray class invariants over imaginary quadratic fields whose minimal polynomials have relatively small coefficients, from which we are able to solve certain quadratic Diophantine equations.

Let K be an imaginary quadratic field, and let f be a nontrivial integral ideal of K. Hasse and Ramachandra asked whether the ray class field of K modulo f can be generated by a single value of the Weber function. We completely resolve this question when f = (N ) for an integer N > 1.

Let K be an imaginary quadratic field and O K be its ring of integers. Let h E be the Weber function on 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 N O K over K when evaluated at some N -torsion point of E, which would be a partial answer to the question mentioned in [13, p.134].