A fundamental question in arithmetic geometry consists of studying rational and integral points on algebraic varieties. A fruitful approach to rational points is based on the local to global principle. More precise variants of the Hasse principle and weak approximation can be conveniently formulated in terms of the Brauer-Manin obstruction and its generalizations. The area has recently seen much progress due to applications of powerful results from additive combinatorics of Green and Tao, and results from analytic number theory. These works motivated a revision of the classical methods of fibration and descent that produced a host of new results for both rational points and zero-cycles. Other ideas and methods stem from Grothendieck section conjecture that exploits the rich structure of the étale algebraic fundamental group of hyperbolic varieties.
Much recent work has been devoted to rational points on homogeneous spaces of algebraic groups, where cohomological methods are very efficient. Another promising area is the arithmetically crucial class of K3 surfaces that occupy the middle ground between rational varieties and the varieties of gen- eral type. In the case of Kummer K3 surfaces a method of Swinnerton-Dyer relates the existence of rational points to the variation of the Selmer group in a family of twists of the associated abelian variety. In view of exciting recent results in the direction of the Birch and Swinnerton-Dyer conjecture (Bhar- gava and others) it is a good moment to explore their possible applications to rational points on Kummer surfaces.
A better understanding of K3 surfaces and their Brauer groups was made possible by the recent proof of the Tate conjecture for these surfaces in pos- itive characteristic. Another example of the dynamic interaction between
We plan to bring together experts in arithmetic geometry in a broad sense, including experts in algebraic geometry over non-closed fields, whose work has direct or indirect applications to rational points. We would like to summarize a very active period that this area has seen in 2012-2015 and discuss new avenues for future research. Participation of leading experts from different areas will be important for fostering a fruitful exchange of ideas and forming new collaborations.
Among others, the following topics will be considered :
-Hasse principle and weak approximation on higher dimensional varieties, descent and fibration method.
Between 16 and 18 one-hour talks will be scheduled (free afternoon on
Tim Browning (University of Bristol)
David Harari (Université Paris-Sud)
Spaces of rational curves on Fano hypersurfaces
Orbifoldes, aspects géométriques, hyperboliques et arithmétiques
Rational points on curves under field extensions
Second 2-descent and rational points on Kummer surfaces
Pseudo-split varieties and arithmetic surjectivity
Correlations of multiplicative functions
Heuristics for boundedness of ranks of elliptic curves
Representations of integers by systems of three quadratic forms
The generalized Fermat equation x2 + y3 = z11 (pdf)
Geometry of Severi–Brauer varieties (after Kollár)
Complex multiplication of K3 surfaces
Weak approximation for cubics over global function fields
Differential descent obstructions
Sur la conjecture de Hodge entière pour les solides réels
Compatible systems of l-adic representations arising from abelian varieties