On a problem of Hasse and Ramachandra

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


Introduction
Let K be an imaginary quadratic eld with ring of integers O K , and let E be an elliptic curve with complex multiplication by O K . When E is given by the a ne model where ∆ = g − g and ℘(z) = ℘(z; O K ). This map gives rise to an isomorphism of E/Aut(E) onto P (C) ( [8,Theorem 7 in Chapter 1]).
Let f be a proper nontrivial ideal of O K . We denote by H the Hilbert class eld of K, and by K f the ray class eld of K modulo f. As a consequence of the main theorem of the theory of complex multiplication, Hasse proved in [4] that See also [8,Chapter 10]. In his letter to Hecke, Hasse further asked whether K f can be generated by a single value of h without the j-invariant ([3, p. 91]), and Ramachandra also mentioned this problem later in [10]. It was Sugawara who rst gave a partial answer to this question ( [12] and [13]), however, it still remains an open question.
In this paper, through careful understanding about the characters on class groups and the second Kronecker limit formula, we shall eventually resolve Hasse-Ramachandra's problem for f = (N) with any positive integer N excluding , , and (Theorem 5.1).

The second Kronecker limit formula
For v = r r ∈ (Q \ Z) , we de ne the ( rst) Fricke function fv(τ) on the upper half-plane H by where  Let K be an imaginary quadratic eld, let f be a proper nontrivial ideal of O K and let N (> ) be the smallest positive integer in f. We denote by Cl(f) the ray class group of K modulo f.
We de ne the Fricke invariant f f (C) and the Siegel-Ramachandra invariant g f (C) by

Proposition 2.2.
The invariants f f (C) and g f (C) belong to K f . Furthermore, they satisfy Proof. See [6, Theorem 1.1 in Chapter 11].
Let χ be a nonprincipal character of Cl(f). We de ne the Stickelberger element S(χ) = S f (χ) by and the L-function L f (s, χ) by where a runs over all nontrivial ideals of O K prime to f and [a] stands for the class in Cl(f) containing the ideal a. We shall denote by fχ the conductor of the character χ.

Proposition 2.3. Let χ be the primitive character of χ on Cl(fχ). If fχ ≠ O K , then we obtain the relation
Proof

Di erences of Weber functions
For an imaginary quadratic eld K, x an element and by the de nitions (1) and (3). Let H N be the ring class eld of the order of conductor N in K. Then we have a tower of elds For an integer t prime to N, by C t = C N, t we mean the class in the ray class group Cl(N) of K modulo (N) containing the ideal (t). Note that C is the identity element of Cl(N).

Lemma 3.1. If t is an integer prime to N, then we get
Proof. Since we deduce the lemma by the de nition (4).
For an intermediate eld F of the extension K (N) /K, we shall denote by Cl(K (N) /F) the subgroup of Cl(N) corresponding to Gal(K (N) /F).

Lemma 3.2. We have
Let t be an integer such that gcd(N, t) = and t ≢ ± (mod N).
Note that such an integer t always exists except for the four cases N = , , , . Express (t + )/N and (t − )/N as where n+, N+, n−, N− are integers such that N+, N− > and gcd(n+, N+) = gcd(n−, N−) = . Observe that the condition t ≢ ± (mod N) is equivalent to saying that neither N+ nor N− is equal to . Now, we de ne Furthermore, for a character χ of Cl(N) we denote by

Lemma 3.3. If χ is nontrivial on Cl(K (N) /H), then we obtain
Proof. We derive that by the de nition (6) (2) by the assumption that χ is nontrivial on Cl(K (N) /H) and the de nition (5).

Lemmas on characters of class groups
If we set then we obtain by (2) In this section, we shall prove the existence of certain characters of class groups under the assumption that F is properly contained in K (N) .
We then achieve that Then, there exists an integer t satisfying the following properties: Proof. Let be an integer such that > and gcd( , ) = . One can take t as listed in Table 1.   (13) = K(h( /N)) by the rst case of the theorem.
This completes the proof.