The Equation ax ² + by ² + cz ² = dxyz over Quadratic Imaginary Fields

Abstract

The diophantine equation ax ²+by ²+cz ² = dxyz with a, b, c, d ∈ ℤ{0} and a, b, c ¦ d has been studied in connection with discrete subgroups of PSL(2, ℝ) ([R2, KR, K, Sch]). Rosenberger and Kern-Isberner have determined the complete set of integral solutions. Further Silverman ([S]) gave a description of the solutions of the equation x ² + y ² + z ² = dxyz, |d| ≥ 3, over orders in quadratic imaginary fields, whereas Bowditch, Maclachlan and Reid studied the equation x ² + y ² + z ² = xyz in order to describe the arithmetic once-punctered torus bundles ([BMR]). In this paper we give a survey of the set of solutions over the ring of integers O _k in quadratic imaginary fields, $\mathbb{Q}\left( {\sqrt { - k} } \right)$, k > 0 squarefree, of the equation ax ² + by ² + cz ² = dxyz, where the coefficients are chosen as follows:

$$\bullet a,b,c,d \in O_k \backslash \{ 0\} ,\left. {a,b,c} \right|d,\left| {\tfrac{d} {{\sqrt {abc} }}} \right| \geqslant 3$$

$$\bullet a,b,c,d \in \mathbb{Z}\backslash \{ 0\} ,\left. {a,b,c} \right|d,1 \leqslant \left| {\tfrac{d} {{\sqrt {abc} }}} \right| < 3, resp. a = b = c = 1, d \in O_k \backslash \{ 0\} ,\left| d \right| < 3.$$