arXiv subjects
Publications and source records attributed to T. D. Browning.
Given a projective intersection of two quadrics X in at least 9 variables, the quantitative behaviour of the rational points on X is investigated under the assumption that X contains a pair of conjugate singular points defined over the Gaussians.
For given non-zero integers a,b,q we investigate the density of integer solutions (x,y) to the binary cubic congruence ax^2+by^3=0 (mod q). We use this to establish the Manin conjecture for a singular del Pezzo surface of degree 2 defined over the rationals and to examine the distribution of elliptic curves with square-free discriminant.
The Hasse principle and weak approximation is established for equations of the shape P(t)=N(x_1,x_2,x_3,x_4), where P is an irreducible quadratic polynomial in one variable and N is a norm form associated to a quartic extension of the rationals containing the roots of P. The proof uses analytic methods.
We investigate the frequency of positive squareful numbers x,y,z for which x+y=z.
We investigate the density of integer solutions to certain binary inhomogeneous quadratic congruences and use this information to detect almost primes on a singular del Pezzo surface of degree 6.
We investigate the first and second moments of shifted convolutions of the generalised divisor function $d_3(n)$.
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.
We investigate the average order of the divisor function at values of totally reducible binary cubic forms and discuss some applications.
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.
An effective search bound is established for the least non-trivial integer zero of an arbitrary integral cubic form in at least 17 variables.
An upper bound of the expected order of magnitude is established for the number of rational points of bounded height on Châtelet surfaces defined over the rationals.
The Manin conjecture is established for a split singular del Pezzo surface of degree four, with singularity type A_4.
The Manin conjecture is established for a split singular cubic surface in P^3, with singularity type D_5.
Let X be a projective cubic hypersurface of dimension 11 or more, which is defined over the rationals. In this paper it is shown that X contains rational points provided that the cubic form defining X can be written as the sum of two forms that share no common variables. ----- This paper features an appendix "Groupe de Brauer non ramifié des hypersurfaces cubiques singulières (d'après P. Salberger)", by J.-L. Colliot-Thĺène.
The arithmetic of ternary diagonal equation is considered for degree d >1, with the outcome that the set of coefficients for which the equation admits a non-zero integer solution is shown to have density zero.
Let X be a geometrically integral projective cubic hypersurface defined over the rationals, with dimension D and singular locus of dimension at most D-4. For any ε>0, we show that X contains O(B^{D+ε}) rational points of height at most B. The implied constant in this estimate depends upon the choice of εand the coefficients of the cubic form defining X.