Abstract

The sum of three cubes conjecture states that any integer not congruent to 4 or 5 mod 9 can always be represented as the sum of three integer cubes. Geometrically, this means that the affine cubic surfaces corresponding to this family of equations always have an integral point. Colliot-Thélène and Wittenberg showed that these surfaces have no “cohomological obstruction” to the existence of integral points (more formally, there is no integral Brauer-Manin obstruction to the integral Hasse principle). In this talk, we study natural generalizations of these surfaces, namely diagonal affine cubic surfaces, and we will show that, in this more general setting, there are instances of the integral Hasse principle failing. Moreover, we will provide bounds for the frequency of these failures. Time permitting, we will also discuss possible improvements on the results presented in the talk. (Joint work with Julian Lyczak and Vlad Mitankin).