### Rational points of bounded height on general conic bundle surfaces

Authors: Christopher Frei, Daniel Loughran, Efthymios Sofos

Journal: Proceedings of the London Mathematical Society

Publication Date: 03 April, 2018

Department of: Mathematics

### In Abstract

**Counting integer solutions of equations**

The great Indian mathematician Ramanujan observed that 1729 is the smallest natural number that can be written in two different ways as the sum of two cubes,

1729=1^{3}+12^{3}=9^{3}+10^{3}.

To study the frequency of such occurrences, one studies integer solutions to the equation

x_{0}^{3}+x_{1}^{3}=x_{2}^{3}+x_{3}^{3},

where one excludes the obvious solutions given by equating variables (or their negatives). The number of such non-trivial solutions in a box is predicted in high generality by a famous conjecture of the Russian-German mathematician Yu. I. Manin. It is one of the core problems of modern number theory and still widely open even in the above case (geometrically, this describes a cubic surface).

This research, conducted by researchers at the University of Manchester and the Max-Planck Institute for Mathematics in Bonn (Germany), confirms the lower bound predicted by the Manin conjecture in high generality, including the cubic surface described above and many other cases in which nothing was previously known. Its innovations, using specialised techniques from analytic number theory in a very general arithmetic-geometric setup, are highly likely to find more applications in the near future. In particular, they will allow researchers to cover more complex cases of the Manin conjecture as the development of analytic methods progresses.

- Integer solutions to polynomial equations (so-called Diophantine equations, after Diophantus of Alexandria) have intrigued mathematicians since antiquity. Tremendous progress was made in the 20th century, building among other things on deep connections to algebraic geometry. Yet many problems remain unsolved.