This project concerns rational points, which are essentially whole-number solutions to equations, a fundamental object of interest in mathematics that has been studied since ancient times but is even now understood only to a very limited extent. The modern point of view is that the equations define a geometric object, called a variety, such as a curve or a surface.

To give a basic example, consider the equation x^2+y^2=1, which describes the unit circle in the sense that the points (x , y) in the xy -plane that satisfy this equation make up this circle. The points (4/5, 3/5) and (5/13, 12/13) on this circle have coordinates that are fractions, also called rational numbers; these points are called rational points. To Pythagoras, and even in ancient Babylonia and India, it was known that this circle contains infinitely many such rational points, though they did not phrase it in these geometric terms. In contrast to the case of the circle, Fermat proved in the 17th century that the only rational points on the curve given by x^4 + y^4 = 1 are (\pm1, 0) and (0, \pm1).

infinitely many rational points
only finitely many rational points

By viewing whole-number solutions as rational points on the variety described by the equations, we open the door to tools from algebraic geometry. Since the 20th century, this geometric point of view has led to major advances. Moreover, it turns out that the geometry of a variety determines to an amazing extent the qualitative and quantitative behaviour of its rational points.

In the case of curves, which have dimension 1, we can associate with each curve a geometric invariant, called its genus; a celebrated theorem by Faltings (1983) states that if the genus is at least 2, then there are only finitely many rational points on the curve. This is indeed consistent with the examples above, as the circle has genus 0, while the second curve has genus 3.

For varieties of higher dimension, it is generally believed that their geometry still governs the distribution of rational points. The geometry, however, is much more subtle. There are long-standing conjectures, such as those by Colliot-Thélène (1980), Batyrev–Manin (1989) and Bombieri–Lang (1974), that formulate in a precise way how the behaviour of rational points is affected by the geometry. The study of rational points, particularly on surfaces and higher-dimensional objects, has flourished in recent years.

A cubic surface with rational points and lines

The time is right to develop both our theoretical and computational understanding of rational points on surfaces and higher-dimensional varieties, as well as their applications, to the same degree of sophistication as the study of rational points on curves achieved in the 20th century.

In this project we have identified a number of challenges, grouped under three interrelated themes, that will act as stepping-stones towards this goal. The first theme, from curves to surfaces and beyond, will take emerging techniques from the study of rational points on curves and apply and extend them to surfaces and higher-dimensional settings. The second theme, from characteristic zero to characteristic p and back, will investigate rational points using tools from geometry over finite fields and local-global methods. The third theme, from rational points to Campana points, will explore the exciting new territory of Campana points, a generalisation of rational points.

Through this project, postdocs and PhD students will collaborate with our team of established experts to attack long-standing challenges in the area by devising, extending and applying techniques from our complementary areas of expertise. We will have monthly joint seminars, frequently with international guests, and organise two international meetings.