On the singular values of the Drinfeld modular function \(\mu\) (Q5960977)

Let \(k= \mathbb F_q(T)\) be the rational function field over the finite field \(\mathbb F_q\), \(A = \mathbb F_q[T]\) its integers, and \(X_0(n)\) the Drinfeld modular curve of Hecke type and conductor \(n \in A\). If \(\deg n \leq 2\), \(X_0(n)\) is a rational curve with a distinguished uniformizer \(\mu\), whose conductor is concentrated in the cusps of \(X_0(n)\). Hence \(\mu\) may be used in a similar fashion like the classical elliptic \(j\)-invariant or the Drinfeld \(j\)-invariant to work out the abelian class field theory of imaginary quadratic extensions \(K\) of \(k\). The authors start such a program in the case where \(\deg n = 1\) (without restriction: \(n = T\)). They obtain a description of the Hilbert class field and the \(T\)-th ring class field of \(K\) as the field generated over \(K\) by \(\mu(\omega)\), where \(\omega \in K\) is suitably chosen.