describe a part of mathematics linking Number Theory and Geometry. Historically this is a question of solving polynomial equations in rational numbers (or integers). Polynomial equations define an algebraic variety. The fundamental problem is to discover links between the geometry of the algebraic variety (say its space of complex points) and the repartition of its rational (or integral) points. 

There is a full dictionary in the case of curves or more generally subvarieties of abelian varieties. For general varieties this problem is wide open. 

For varieties with "lots of rational points", there is a conjecture of Manin, proven in only a few cases.

All these theorems and conjecture point towards a general principle : «varieties with many holomorphic differential forms have few rational points and vice versa».

We will describe this field, giving definitions and examples, and its many open problems.