Groupe de Brauer et points entiers de deux familles de surfaces cubiques affines

TL;DR

This paper investigates the Brauer-Manin obstruction to the existence of integer solutions for specific cubic surface equations, showing that under certain conditions, no such obstruction prevents solutions.

Contribution

It provides new results on the absence of Brauer-Manin obstructions for particular families of affine cubic surfaces related to sums of cubes.

Findings

No Brauer-Manin obstruction when a ≠ 4,5 mod 9 for x^3+y^3+z^3=a

No Brauer-Manin obstruction for x^3+y^3+2z^3=a under the same conditions

Results clarify when integer solutions are not obstructed by Brauer-Manin

Abstract

Let a be a nonzero integer. If a is not congruent to 4 or 5 modulo 9 then there is no Brauer-Manin obstruction to the existence of integers x, y, z such that x^3+y^3+z^3=a. In addition, there is no Brauer-Manin obstruction to the existence of integers x, y, z such that x^3+y^3+2z^3=a.