On singular moduli of Drinfeld modules in characteristic two (Q1267294)

Let \(E\) be an elliptic curve over \(\mathbb C\) with complex multiplication. Then the absolute norm of \(j(E)\) is a highly composite integer. More generally, \textit{B. Gross} and \textit{D. Zagier} [J. Reine Angew. Math 355, 191-220 (1985; Zbl 0545.10015)] gave factorization formulae for the norm of the difference of singular moduli. \textit{D. Dorman} [Compos. Math. 80, 235-256 (1991; Zbl 0744.11032)] took up the corresponding problem for Drinfeld modules. Let \(\phi\) be an \(\mathbb F_q[T]\)-Drinfeld module of rank 2 with complex multiplication by the maximal order of an ``imaginary'' quadratic extension of \(\mathbb F_q(T)\). He determined the prime decomposition of the \(\mathbb F_q(T)\)-norm of \(j(\phi)\) in case the characteristic of \(\mathbb F_q\) is odd. The paper under review settles the characteristic 2 case, which needs separate treatment because the explicit descriptions of the quadratic extensions and quaternion algebras involved look quite different. A factorization formula for the norm of \(j(\phi)\) and also some examples are given.