Rational and Integral Points via Analytic and Geometric Methods (18w5012)
Cecília Salgado (Universidade Federal do Rio de Janeiro)
Ulrich Derenthal (Leibniz Universität Hannover)
Tim Browning (University of Bristol)
-Hasse principle and weak approximation on varieties. A deep conjecture of Colliot-Thélène predicts that the Brauer-Manin obstruction provides an algorithm for deciding whether or not the Hasse principle or weak approximation hold for the class of smooth projective rationally connected varieties over any number field. This is a cohomological obstruction that was put forward by Manin in 1970 and undoubtedly lies very deep. Colliot-Thélène's conjecture is still wide open but there are several recent examples where tools from analytic number theory have been successfully paired with algebro-geometric methods to handle important special cases of the conjecture.
-The density of rational (or integral) points on varieties; and Given a higher-dimensional variety with rational points, the next natural question is to ask about the distribution of its rational points. For example, we may wonder whether all of its rational points lie on one algebraic curve, or more generally on a finite number of lower-dimensional subvarieties. More precisely, the question is whether the rational points are dense on the variety in the Zariski topology. By Colliot-Thélène's conjecture, this is expected to be true for smooth projective rationally connected varieties over all global fields, but Zariski density is still open even for some rational surfaces. At the other extreme, Lang's conjecture predicts that rational points are not Zariski dense on varieties of general type.
-The behaviour of rational points in families. Another exciting new development is the combination of the two main topics presented above: can one quantify how often a point exists or weak approximation fails in a given family of varieties. Such questions have been recently considered for example by Bhargava, Poonen and Stoll in the case of families of curves, and by Loughran for families of Brauer-Severi varieties parameterized by toric varieties. There is great potential to consider similar questions for other families. Again, progress on this topic appears to require a genuine combination of geometric, cohomological and analytic tools.
These are all fundamental and extensively studied topics. We shall focus on surfaces and higher-dimensional varieties, approaching them from both a quantitative and qualitative point of view.