Manin's conjecture on a nonsingular quartic del Pezzo surface

Abstract

Given a nonsingular quartic del Pezzo surface, Manin's conjecture predicts the density of rational points on the open subset of the surface formed by deleting the lines. We prove that this prediction is of the correct order of magnitude for a particular surface.

Quantitative Arithmetic of Projective Varieties

“…Leung [89] has revisited Salberger's proof of (2.17), ultimately replacing the B ε by a small power of log B under the same hypotheses. It sometimes happens in analytic number theory that a method of attack that yields a good upper bound can be pushed into giving the full asymptotic formula if one is prepared to work hard enough.…”

Linear growth for Châtelet surfaces

“…To overcome this, we will gain significant extra leverage by restricting the summation to only those p ∈ P 1 (Q) of small height that produce isotropic conics π −1 (p). It seems likely that this innovation could also be put to use in the analogous situation studied by Leung [10].…”

“…For such surfaces an upper bound O ε (B 1+ε ) is achieved for the corresponding counting function by taking advantage of the morphism π : X → P 1 in order to count rational points of bounded height on the conics π −1 (p), uniformly for points p ∈ P 1 (Q) of small height. In subsequent work Leung [10] has refined this argument, replacing B ε by (log B) A for a certain integer A 5. However, the value of A is often bigger than the exponent predicted by Manin.…”

Counting rational points on quartic del Pezzo surfaces with a rational conic

“…For comparison, Leung's work [21,Chapter 4] establishes an upper bound for NpBq with the potentially larger exponent 1`δ 1 . This exponent agrees with the Batyrev-Manin conjecture if and only if X Ñ P 1 is a conic bundle with a section over Q, a hypothesis that our main result avoids.…”