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

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

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

On the representation of integers by quadratic forms

Let Q be a non-singular quadratic form with integer coefficients. When Q is indefinite we provide new upper bounds for the least non-trivial integral solution to the equation Q=0. When Q is positive definite we provide improved upper bounds for the least positive integer k such that the equation Q=k is insoluble in integers, despite being soluble modulo every prime power.Comment: 33 page

Integral points on cubic hypersurfaces

Let g be a cubic polynomial with integer coefficients and n>9 variables, and assume that the congruence g=0 modulo p^k is soluble for all prime powers p^k. We show that the equation g=0 has infinitely many integer solutions when the cubic part of g defines a projective hypersurface with singular locus of dimension <n-10. The proof is based on the Hardy-Littlewood circle method.Comment: 18 page

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