Rational Points and Algebraic Geometry
September 26 – 30, 2016
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
arithmetic and algebraic geometry is the theory of rationally connected vari- eties over various ground fields. Other such examples include the recent proof of unirationality of all del Pezzo surfaces of degree 2 over arbitrary fields, ex- plicit construction of Brauer classes, applications of derived categories of coherent sheaves, Cox rings and their applications to universal torsors.

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.
-Brauer  groups of varieties  and  of their  function  fields, structure of the
Brauer-Manin  set.
-Existence of points on homogeneous spaces of algebraic groups.
-Links to abelian varieties, consequences of Tate conjecture.

Between  16 and  18 one-hour talks will be scheduled  (free afternoon on
Wednesday, and at most one talk on Friday afternoon).

Scientific Committee

Tim Browning (University of Bristol)
David Harari (Université Paris-Sud)
Alexei Skorobogatov (Imperial College London)

Organizing Committee

David Harari (Université Paris-Sud)
Alexei Skorobogatov (Imperial College London)


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