This website is the home site for the IHP thematic period on
rational points which will take place at Henri Poincaré Institute
from April to July 2019.
Description
Rational points on algebraic varieties represent a modern
way of thinking about one of the oldest problems in mathematics:
integral and rational solutions of Diophantine equations. In
arithmetic geometry one views rational points in the context of
geometric properties of underlying algebraic varieties. In analytic
number theory many different analytic techniques are used to count
the number of rational or integral points, and so understand their
“average” behaviour. In logic, rational points feature
prominently in the work on Hilbert’s Tenth Problem over Q,
which asks for an algorithm to decide the existence of rational
solutions to all Diophantine equations. Here one searches for examples
of “weird” or “far from average” behaviour
of rational points.
There is a large body of conjectures that describe the
behaviour of rational points. They include various versions of
Mazur’s conjectures on the real topological closure of the set
of rational points. A related circle of conjectures deals with
the Brauer–Manin obstruction designed to describe the closure of
the set of rational points inside the topological space of adelic
points. It is conjectured that the Brauer–Manin obstruction should
exactly describe this closure for certain classes of algebraic varieties
such as rationally connected varieties, K3 surfaces and algebraic curves.
Supportive heuristic and theoretical evidence for these difficult
conjectures is slowly emerging from the work of many people. The
Batyrev–Manin conjectures on the growth of rational points of bounded
height have received much attention by analytic number theorists.
New techniques that have revolutionised analytic number theory,
such as additive combinatorics (Green, Tao, Ziegler) or arithmetic
invariant theory (Bhargava, Gross), have made it possible to solve
some of the long standing problems in arithmetic geometry. A new
feature in recent years has been an increased interaction between
the analytic and geometric thinking: questions motivated by various
counting problems give rise to novel geometric ideas, whereas
conjectures coming from geometry open up new fields of investigation
for analytic number theorists.
The aim of this thematic period is to bring together
senior and junior mathematicians from the various domains related
to rational points to foster new interactions and new research.
