arXiv subjects
Publications and source records attributed to R. de la Bretèche.
For integer $n\geqslant 1$ and real $u$, let $Δ(n,u):=|\{d:d\mid n,\,{\rm e}^u<d\leqslant {\rm e}^{u+1}\}|$. The Erdős--Hooley Delta-function is then defined by $Δ(n):=\max_{u\in{\mathbb R}}Δ(n,u).$ We provide new upper bounds for weighted real moments of this function.
Châtelet surfaces provide a rich source of geometrically rational surfaces which do not always satisfy the Hasse principle. Restricting attention to a special class of Châtelet surfaces, we investigate the frequency that such counter-examples arise over the rationals.
We use an explicit description of the Brauer-Manin obstruction, recently obtained by Colliot-Thélène, to study the density of varieties in a certain family which do not satisfy the Hasse principle.
The representation of integral binary forms as sums of two squares is discussed and applied to establish the Manin conjecture for certain Châtelet surfaces defined over the rationals.
An asymptotic formula is established for the number of rational points of bounded height on a non-singular quartic del Pezzo surface with a conic bundle structure.
The Manin conjecture is established for Châtelet surfaces over Q arising as minimal proper smooth models of the surface Y^2+Z^2=f(X) where f is a totally reducible polynomial of degree 3 without repeated roots. These surfaces do not satisfy weak approximation.
We study the average order of the divisor function, as it ranges over the values of binary quartic forms that are reducible over the rationals.