Minimal models for Drinfeld modules of rank 2 with complex multiplication (Q1977797)

This paper continues the first author's paper [Pac. J. Math. 167, 215-230 (1995; Zbl 0826.11025)] on Drinfeld modules with complex multiplication and Hecke characters. Let \(\phi\) be a rank 2 Drinfeld \({\mathbb F}_q[T]\)-module with CM by the maximal order of a separable, imaginary quadratic extension \(K\) of \(k={\mathbb F}_q(T)\), and suppose that \(\phi\) is defined over the Hilbert class field \(H\) of \(K\). In analogy with \(\mathbb Q\)-curves in the theory of elliptic curves, such a Drinfeld module is called a \(k\)-module if it is \(H\)-isogenous to all its \(Gal(H/k)\)-conjugates. The main result is that certain \(k\)-modules \(\phi\) have a global minimal model over \(k(j(\phi))\). For the corresponding result on \(\mathbb Q\)-curves see \textit{B. Gross} [Compos. Math. 45, 155-164 (1982; Zbl 0541.14010)].