Let $K$ be a quadratic imaginary number field of discriminant $d$. For $t\in \mathbb{N}$ let ${𝔒}_t$ denote the order of conductor $t$ in $K$ and $j\left({𝔒}_t\right)$ its modular invariant which is known to generate the ring class field modulo $t$ over $K$. The coefficients of the minimal equation of $j\left({𝔒}_t\right)$ being quite large Weber considered in [We] the functions $f,f_1,f_2,{\gamma }_2,{\gamma }_3$ defined below and thereby obtained simpler generators of the ring class fields. Later on the singular values of these functions played a crucial role in Heegner’s solution [He] of the class...

Let K be a number field. Assume that the 2-rank of the ideal class group of K is equal to the 2-rank of the narrow ideal class group of K. Moreover, assume K has a unique dyadic prime and the class of is a square in the ideal class group of K. We prove that if ₁,...,ₙ are finite primes of K such that ∙ the class of ${}_i$ is a square in the ideal class group of K for every i ∈ 1,...,n, ∙ -1 is a local square at ${}_i$ for every nondyadic ${}_i\in ₁,...,ₙ$, then ₁,...,ₙ is the wild set of some self-equivalence of the field...