Applications of additive combinatorics to Diophantine equations
Applications of additive combinatorics to Diophantine equations

Alexei Skorobogatov , Imperial College London
IAS Room S101
The work of Green, Tao and Ziegler can be used to prove existence and approximation properties for rational solutions of the Diophantine equations that describe representations of a product of norm forms by a product of linear polynomials. One can also prove that the BrauerManin obstruction precisely describes the closure of rational points in the adelic points for pencils of conics and quadrics over Q when the degenerate fibres are all defined over Q. In this setting the result of Green, Tao and Ziegler replaces Schinzel's Hypothesis (H) used in earlier papers of ColliotThélène and Sansuc. I will give an overview of recent work in this direction due to L. Matthiesen, T. Browning, Y. Harpaz, O. Wittenberg and the speaker.