The topic of investigation is cubic forms F over Z in n variables that are representable as a sum L 3 1 +L 3 2 of two cubes of linear forms with algebraic coefficients. If Z 2 (n, X ) denotes the number of such forms F, the main result, stated as Theorem 1.3, gives its order of magnitude as Z 2 (n, X ) ร X 2nร3 , with the implied constants depending on n only. The gist of our method consists of the analysis of the p-adic conditions for the coefficients of the linear forms L 1 and L 2 which stem from the fact that F is defined over Z. This leads to results concerning local lattices and their connection to global lattices that seem of interest even beyond the treated problem, and which are therefore stated with some more generality in Theorems 4.1 and 4.8. The combination of a fact from the Geometry of Numbers with the above then leads to the main theorem. Even though these steps are applied to changes of variables leading to the special diagonal form X 3 +Y 3 , they may be applicable to more general situations, the final goal being the treatment of forms that can be transformed into an arbitrary given form f (X, Y) by a suitable linear, algebraic change of variables. Another, probably difficult generalisation consists in increasing the number of variables to deal with forms X 3 1 + } } } +X 3 k for k 3. 1998 Academic Press 1. SUMS OF 2 CUBES OF LINEAR FORMS WITH ALGEBRAIC COEFFICIENTS 1.1. Notations and Problem Setting. As usual we denote by v C the complex number field, v Q the rational number field, v Z the rational integers, v K an algebraic number field, v O K its ring of integers.
With F we denote cubic forms in n variables with coefficients from one of the rings mentioned above, whereas L is reserved for linear forms.