Applications of additive combinatorics to Diophantine equations

Applications of additive combinatorics to Diophantine equations

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Alexei Skorobogatov , Imperial College London
IAS Room S-101

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 Brauer-Manin 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 Colliot-Thélène 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.