A conjecture of Manin predicts the asymptotic distribution of rational points of bounded height on Fano varieties. In this paper we use conic bundles to obtain correct lower bounds for a wide class of surfaces over number fields for which the conjecture is still far from being proved. For example, we obtain the conjectured lower bound of Manin's conjecture for any del Pezzo surface whose Picard rank is sufficiently large, or for arbitrary del Pezzo surfaces after possibly an extension of the ground field of small degree.

We provide an asymptotic estimate for the number of rational points of bounded height on a non-singular conic over Q. The estimate is uniform in the coefficients of the underlying quadratic form.

We study almost prime solutions of systems of Diophantine equations in the Birch setting. Previous work shows that there exist integer solutions of size B with each component having no prime divisors below B 1{u , where u " c0n 3{2 , n is the number of variables and c0 is a constant depending on the degree and the number of equations. We improve the polynomial growth n 3{2 to the logarithmic plog nqplog log nq´1. Our main new ingredients are the generalisation of the Brüdern-Fouvry vector sieve in any dimension and the incorporation of smooth weights into the Davenport-Birch version of the circle method.

Estimating averages of Dirichlet convolutions 1˚χ, for some real Dirichlet character χ of fixed modulus, over the sparse set of values of binary forms defined over Z has been the focus of extensive investigations in recent years, with spectacular applications to Manin's conjecture for Châtelet surfaces. We introduce a far-reaching generalization of this problem, in particular replacing χ by Jacobi symbols with both arguments having varying size, possibly tending to infinity. The main results of this paper provide asymptotic estimates and lower bounds of the expected order of magnitude for the corresponding averages. All of this is performed over arbitrary number fields by adapting a technique of Daniel specific to 1˚1. This is the first time that divisor sums over values of binary forms are asymptotically evaluated over any number field other than Q. Our work is a key step in the proof, given in subsequent work, of the lower bound predicted by Manin's conjecture for all del Pezzo surfaces over all number fields, under mild assumptions on the Picard number.

Abstract. -Let π : X Ñ P 1 Q be a non-singular conic bundle over Q having n non-split fibres and denote by N pπ, Bq the cardinality of the fibres of Weil height at most B that possess a rational point. Serre showed in 1990 that a direct application of the large sieve yieldsand raised the problem of proving that this is the true order of magnitude of N pπ, Bq under the necessary assumption that there exists at least one smooth fibre with a rational point. We solve this problem for all non-singular conic bundles of rank at most 3. Our method comprises the use of Hooley neutralisers, estimating divisor sums over values of binary forms, and an application of the Rosser-Iwaniec sieve.