The Manin conjecture in dimension 2

These lecture notes describe the current state of affairs for Manin's conjecture in the context of del Pezzo surfaces.Comment: 57 pages. These are a preliminary version of lecture notes for the "School and conference on analytic number theory", ICTP, Trieste, 23/04/07-11/05/0

Many cubic surfaces contain rational points

Building on recent work of Bhargava--Elkies--Schnidman and Kriz--Li, we produce infinitely many smooth cubic surfaces defined over the field of rational numbers that contain rational points.Comment: 23 pages; minor edits and added new remark (Remark 2.1) following an argument of Jahne

Density of integer solutions to diagonal quadratic forms

Let Q be a non-singular diagonal quadratic form in at least four variables. We provide upper bounds for the number of integer solutions to the equation Q=0, which lie in a box with sides of length 2B, as B tends to infinity. The estimates obtained are completely uniform in the coefficients of the form, and become sharper as they grow larger in modulus.Comment: 23 page

Equal sums of like polynomials

Let $f$ be a polynomial of degree $d>6$, with integer coefficients. Then the paucity of non-trivial positive integer solutions to the equation $f(a)+f(b)=f(c)+f(d)$ is established. The corresponding situation for equal sums of three like polynomials is also investigated.Comment: 8 pages; to appear in Bull. London Math. So

Varieties with too many rational points

We investigate Fano varieties defined over a number field that contain subvarieties whose number of rational points of bounded height is comparable to the total number on the variety.Comment: 23 page

Power-free values of polynomials on symmetric varieties

Given a symmetric variety Y defined over the rationals and a non-zero polynomial with integer coefficients, we use techniques from homogeneous dynamics to establish conditions under which the polynomial can be made r-free for a Zariski dense set of integral points on Y. We also establish an asymptotic counting formula for this set. In the special case that Y is a quadric hypersurface, we give explicit bounds on the size of r by combining the argument with a uniform upper bound for the density of integral points on general affine quadrics.Comment: 47 pages; accepted versio