Generalised Divisor Sums of Binary Forms Over Number Fields

Abstract

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.

Rational points of bounded height on general conic bundle surfaces

“…As was proved in [, Theorem 1.1], the lower bound conjecture holds for systems of forms of complexity at most 3. From Theorem , this allows us to obtain the following unconditional result.…”

“…Theorem implies Theorem , since the complexity (as defined in ) of is exactly the complexity of the conic bundle , as will follow from Lemma .…”

“…In the companion paper , the first and last named authors introduce certain generalised divisor sums over the values of binary forms and state a lower bound conjecture for such divisor sums. We construct now an instance of [, Conjecture 1], whose validity will imply the validity of Theorem for the particular hypersurface of bidegree in as above.…”

“…Assume that there is , such that the fibre of above is smooth and has rational points, and define the ideal . If the lower bound conjecture for divisor sums [, Conjecture 1] holds for , then holds for every non‐empty open subset of . The implicit constant may depend on every parameter except .…”

“…The present paper's central result obtains sharp lower bounds for Manin's conjecture for conic bundle surfaces, conditional on this conjecture. Theorem Let and assume the validity of [, Conjecture 1] for systems of binary forms of complexity . Let be a conic bundle of complexity with a rational point lying on a smooth fibre.…”