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Brauer--Manin obstruction and integral points

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For a scheme over the integers, the Brauer-Manin set consists of the integral adèles orthogonal to the Brauer group of the variety over the rationals attached to that scheme. One then has two basic questions: If the Brauer-Manin set is not empty, do there exist integral points? If so, are the integral points dense in the Brauer-Manin set?

For homogeneous spaces of connected linear algebraic groups, fairly general results have been obtained Xu and the speaker, Harari, Xu and Wei, Borovoi and Demarche.

At least two more cases will be discussed: representation of an integer as a sum of three cubes (Wittenberg and the speaker), integral points on curves (Harari and Voloch).

This talk is part of the Number Theory Seminar series.

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