The Markoff equation \(X^ 2+Y^ 2+Z^ 2=aXYZ\) over quadratic imaginary fields (Q914725)

It is known that all solutions of the Markov-type equation \[ (*)\quad X^ 2+Y^ 2+Z^ 2=aXYZ \] (0\(\neq a\in {\mathbb{Z}})\) in X,Y,Z\(\in {\mathbb{Z}}\) can be generated from an initial solution by means of simple transformations. The purpose of the paper is to consider (*) in orders R of imaginary quadratic fields. The main theorem gives a similar description of all solutions X,Y,Z\(\in R\) of (*) for any \(a\in R\), \(| a| \geq 3\). It turns out, that solutions exist only if the imaginary quadratic field is \({\mathbb{Q}}(i)\). Moreover, in case \(R={\mathbb{Z}}[i]\), \(a\in {\mathbb{Z}}[i]\) with \(| a| \geq 4\), the author gives an asymptotic formula for the number of solutions X,Y,Z\(\in {\mathbb{Z}}[i]\) of (*) with height \(\leq H\). Such an asymptotic formula was proved in the classical case by \textit{D. Zagier} [Math. Comput. 39, 709-723 (1982; Zbl 0501.10015)].