Fedor Bogomolov, Jean-Louis Colliot-Thélène, Bjorn Poonen, Alice Silverberg, Yuri Tschinkel

The theory of diophantine equations is one of the oldest branches of mathematics. Already around 250 A.D., Diophantus discovered a connection between geometry and algebra: the rational points on the circle x2 + y2 = 1 can be parametrized by drawing a line of varying rational slope through (-1, 0). In general, for each system of polynomial equations, there is a variety defined by the same equations over C. One of the greatest successes of 20th century mathematics has been a qualitative understanding of rational and integral points on curves (1-dimensional varieties), through theorems of Mordell, Weil, Siegel, and Faltings. Our focus will be rational and integral points on varieties of dimension > 1. Recently it has become clear that many branches of mathematics can be brought to bear on problems in the area: complex algebraic geometry, Galois and étale cohomology, transcendence theory and diophantine approximation, harmonic analysis, automorphic forms, and analytic number theory. Sometimes it is only by combining techniques that progress is made. We will bring together researchers from these various fields who have an interest in arithmetic applications, as well as specialists in arithmetic geometry itself. The semester-long program will include the following areas:

• Cohomological approaches to existence and density of rational points (e.g., obstructions from Brauer groups and descent, torsors)
• Applications of complex geometry to arithmetic (e.g., Zariski density of rational points, deformation techniques and rationally connected varieties, analogies between diophantine approximation and Nevanlinna theory)
• Counting points of bounded height (e.g., asymptotics for Fano varieties, the circle method, arithmetic applications of automorphic forms and harmonic analysis)
• Algebraic geometry over Qbar and height functions (e.g., points of small height)
• Families of abelian varieties (e.g., variation of the Mordell-Weil rank and Tate-Shafarevich group in families)
• Diophantine undecidability (e.g., Hilbert's Tenth Problem over Q, Mazur's conjectures on topology of rational points, and other connections with logic).

Rational points on a K3 surface, created by Ronald van Luijk.