Counting rational points over imaginary quadratic number fields on a singular cubic surface

Let [Formula: see text] be an imaginary quadratic number field. Consider the singular cubic surface [Formula: see text] defined by [Formula: see text] in [Formula: see text] We denote by U the complement of the three lines [Formula: see text] [Formula: see text] in [Formula: see text] and denote by [Formula: see text] the set of [Formula: see text]-rational points on U. For any [Formula: see text] let [Formula: see text] denote the counting function of rational points in [Formula: see text] of Weil height [Formula: see text] In this paper, we shall prove the following asymptotic formula: [Formula: see text] where [Formula: see text] is a polynomial in u of degree [Formula: see text] Our method is an application of the multi-variable version of the well-known Perron’s formula.