### EPSRC Grant Number GR/R93155/01

### 08/2002 -- 08/2005

Number Theory Group, Mathematical Institute

This was a three year EPSRC research project, with principal investigator Professor Heath-Brown and postdoctoral research assistant Dr Browning.

### Abstract

The project aims to investigate the density of rational points on algebraic varieties from the perspective of an analytic number theorist. A great many problems in number theory refer to (or can be reduced to) questions involving rational or integral solutions of Diophantine equations. Understanding the distribution of solutions of Diophantine equations has therefore always been a crucial part of the subject. Indeed the modern field of Arithmetic Geometry is ultimately concerned with the equivalent issue of the distribution of integral and rational points on algebraic varieties.

For many algebraic varieties we expect there to be infinitely many integral points, and for many others we expect, but cannot prove, that there should be only finitely many such points. In both these cases it is therefore of considerable interest to establish upper bounds for the counting function for integral points of bounded height. To be precise, if *F( x)* is an absolutely irreducible form of degree

*d*, where

**x**= (

*x*

_{1},...,

*x*), and if

_{n}*B*is an arbitrary real number bigger than 1, then we may define the counting function

*N*(*F;B*) = #{**x** \in **Z*** ^{n}* : max(|

*x*

_{1}|,...,|

*x*|) <

_{n}*B*,

*F*(

**x**) = 0}.

The most basic questions concern the behaviour of *N*(*F;B*) as *B* tends to infinity, and the following pages give a more detailed synopsis of the project:

**Examples:** gives an illustration of the diverse types of problems that are covered by the formulation above.

**Background:** gives an idea of the current state of knowledge concerning the above counting function.

### Publications

- T.D. Browning.

Counting rational points on del Pezzo surfaces of degree five,*Proceedings of the session in analytic number theory and Diophantine equations*, Bonner Math. Schriften 360 (2003). - T.D. Browning.

The density of rational points on a certain singular cubic surface,*submitted*, 2004. - T.D. Browning and D.R. Heath-Brown.

Equal sums of three powers,*Invent. Math.*, 157 (2004), 553--573. - T.D. Browning and D.R. Heath-Brown.

Plane curves in boxes and equal sums of two powers,*Math. Zeit.*, 251 (2005), 233--247. - T.D. Browning and D.R. Heath-Brown.

Counting rational points on hypersurfaces,*Journal fï¿½r die reine und angewandte Mathematik*, 584 (2005), 83-115. - T.D. Browning and D.R. Heath-Brown.

The density of rational points on non-singular hypersurfaces, I,*Bull. London Math. Soc.*, 38 (2006), 401-410. - T.D. Browning, D.R. Heath-Brown and P. Salberger.

Counting rational points on algebraic varieties,*Duke Math J.*, 132 (2006), 545-578. - T.D. Browning and D.R. Heath-Brown.

The density of rational points on non-singular hypersurfaces, II,*Proc. London Math. Soc.*(3), 93 (2006), 273-303. - D.R. Heath-Brown.

The density of rational points on curves and surfaces,*Annals of Math.*, 155 (2002), 553-595. - D.R. Heath-Brown.

The density of rational points on Cayley's cubic curface,*Proceedings of the session in analytic number theory and Diophantine equations*, Bonner Math. Schriften 360 (2003). - D.R. Heath-Brown.

Linear relations amongst sums of two squares,*Number theory and algebraic geometry --- to Peter Swinnerton-Dyer on his 75th birthday*, CUP (2003). - D.R. Heath-Brown.

Counting rational points on algebraic varieties,*C.I.M.E. lecture notes*, Springer Lecture Notes Vol. 1891.

### Conferences

- Diophantine geometry, Pisa, 04/05--07/05.
- Diophantine geometry, Goettingen, 6/04.
- Rational and integral points on higher-dimensional varieties, Palo Alto, 12/02.

(See Swinnerton-Dyer's report 'Diophantine equations: progress and problems') - Analytic number theory summer school, Cetraro, 07/02.
- Special activity in analytic number theory and Diophantine equations, Bonn, 06/02.