Hecke characters of singular Drinfeld modules (Q1893294)

Let \(k= \mathbb{F}_q (T)\), \(K\) an imaginary quadratic extension of \(k\), and \(H\) the Hilbert class field of \(K\). The author uses \(\mathbb{F}_q [T ]\)- Drinfeld modules \(\varphi\) of rank 2 admitting complex multiplication by an order in \(K\) to construct Hecke characters \(\chi_\varphi\) (of weight one) for extension fields of \(K\). These Hecke characters are then used to describe \(H\)-isogeny classes and \(H\)-isomorphism classes of Drinfeld modules having complex multiplication by the maximal order of \(K\). The proofs are analogous to those in the classical setting [\textit{S. Lang}, Elliptic functions (1987; Zbl 0615.14018), chapter 10 \S 4, and \textit{B. Gross}, Arithmetic of elliptic curves with complex multiplication, Lect. Notes Math. 776, 20-31 (1980; Zbl 0433.14032)]. It should be noted however that the statements in section 4 should be slightly modified in case \(K/k\) is inseparable, because then \(H= k(j (\varphi))\).