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Brauer group and integral points of two families of affine cubic surfaces. (Groupe de Brauer et points entiers de deux familles de surfaces cubiques affines.) (French) Zbl 1295.14020
The main objects of the paper under review are two families of affine cubic surfaces: \(x^3+y^3+z^3=a\) and \(x^3+y^3+2z^3=a\). The authors are interested in the existence of integer points and thus assume that for the first family \(a\) is not of the form \(9n\pm 4\) (otherwise there are no points modulo 9). In the spirit of the papers by J.-L. Colliot-Thélène and F. Xu [Compos. Math. 145, No. 2, 309–363 (2009; Zbl 1190.11036)] and A. Kresch and Y. Tschinkel [Bull. Lond. Math. Soc. 40, No. 6, 995–1001 (2008; Zbl 1161.14019)], they consider the integral Brauer-Manin obstruction to strong approximation. The main result of the paper (Theorem 4.1) states that under the assumption on \(a\) mentioned above, there is no integral Brauer-Manin obstruction to the existence of integer points.
Apart from this statement, experts will find many auxiliary results which are interesting by their own right. Such are all lemmas and propositions of Section 5.1, allowing one to compare, in a much more general set-up, the Brauer groups of an affine variety and of its projective closure. Section 5.2 contains many interesting examples, among which one can mention Remark 5.7 with an interpretation, in terms of the Brauer-Manin obstruction, of a computation by [J. W. S. Cassels, Math. Comput. 44, 265–266 (1985; Zbl 0556.10007)].

MSC:
14F22 Brauer groups of schemes
11D25 Cubic and quartic Diophantine equations
11G05 Elliptic curves over global fields
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