Affine congruences and rational points on a certain cubic surface

Abstract

We establish estimates for the number of solutions of certain affine congruences. These estimates are then used to prove Manin's conjecture for a cubic surface split over Q and whose singularity type is D_4. This improves on a result of Browning and answers a problem posed by Tschinkel

Linear Growth for Certain Elliptic Fibrations

“…Moreover, since e 1 , e 2 , and e 3 are distinct, there is at most one such solution n∈ 4) and the conditions gcd(d i , a i b i ) = 1 for i ∈ {1, 2, 3}, we deduce that gcd(…”

“…Indeed, this strategy has been successful to establish Manin's conjecture for several examples of singular del Pezzo surfaces of low degree (see [1,4] for the most striking results). Hausen and Süss [3,…”

Density of Châtelet surfaces failing the Hasse principle

“…Indeed, according to [3, Lemme 3.1.2], the set W (Q v ) has dense image in X a,b,c,d (Q v ) for the topology defined by the natural topology of Q v . In this way we see that it suffices to examine the local solubility of the affine equation 8) where Q 1 , Q 2 are as in (2.5). Our first order of business is a precise characterisation of when an element can locally be written as the sum of two squares.…”

“…We close this section with a modified result of Le Boudec [8,Lemma 2], concerning pairs of integers constrained to satisfy congruences modulo q and modulo r, with gcd(q, r) = 1. We are interested in the cardinality…”

Quadratic congruences on average and rational points on cubic surfaces

“…-For the degree 2 del Pezzo surface of type E 7 [BB13], the first two summations over η 11−d , η 12−d are treated simultaneously. Furthermore, Manin's conjecture is known for some smooth and singular del Pezzo surfaces of degree greater or equal to 3 for which the factor N (a, q) does not appear, in particular for certain singular cubic surfaces of types 2A 2 + A 1 [LB12] and D 4 [LB14].…”