arXiv subjects
Publications and source records attributed to T. D. Browning.
An asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain Zariski dense subset of the biprojective hypersurface $x_1y_1^2+\dots+x_4y_4^2=0$ in $\mathbb{P}^3\times\mathbb{P}^3$. This confirms the modified Manin conjecture for this variety, in which the removal of a thin set of rational points is allowed.
An asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain Zariski open subset of an arbitrary smooth biquadratic hypersurface in sufficiently many variables. The proof uses the Hardy--Littlewood circle method.
We give an upper bound for the number of rational points of height at most $B$, lying on a surface defined by a quadratic form $Q$. The bound shows an explicit dependence on $Q$. It is optimal with respect to $B$, and is also optimal for typical forms $Q$.
Upper and lower bounds, of the expected order of magnitude, are obtained for the number of rational points of bounded height on any quartic del Pezzo surface over $\mathbb{Q}$ that contains a conic defined over $\mathbb{Q}$.
For a general class of non-negative functions defined on integral ideals of number fields, upper bounds are established for their average over the values of certain principal ideals that are associated to irreducible binary forms with integer coefficients.
Building on recent work of Bhargava--Elkies--Schnidman and Kriz--Li, we produce infinitely many smooth cubic surfaces defined over the field of rational numbers that contain rational points.
We show that a twisted variant of Linnik's conjecture on sums of Kloosterman sums leads to an optimal covering exponent for $S^3$.
We investigate Fano varieties defined over a number field that contain subvarieties whose number of rational points of bounded height is comparable to the total number on the variety.
Given a symmetric variety Y defined over the rationals and a non-zero polynomial with integer coefficients, we use techniques from homogeneous dynamics to establish conditions under which the polynomial can be made r-free for a Zariski dense set of integral points on Y. We also establish an asymptotic counting formula for this set. In the special case that Y is a quadric hypersurface, we give explicit bounds on the size of r by combining the argument with a uniform upper bound for the density of integral points on general affine quadrics.
Given a family of varieties $X\to \mathbb{P}^n$ over a number field $k$, we determine conditions under which there is a Brauer-Manin obstruction to weak approximation for $100\%$ of the fibres which are everywhere locally soluble.
We generalise Birch's seminal work on forms in many variables to handle a system of forms in which the degrees need not all be the same. This allows us to prove the Hasse principle, weak approximation, and the Manin-Peyre conjecture for a smooth and geometrically integral projective variety, provided only that its dimension is large enough in terms of its degree.
We show that a non-singular integral form of degree d is soluble non-trivially over the integers if and only if it is soluble non-trivially over the reals and the p-adic numbers, provided that the form has at least (d-\sqrt{d}/2)2^d variables. This improves on a longstanding result of Birch.
For any number field k, upper bounds are established for the number of k-rational points of bounded height on non-singular del Pezzo surfaces defined over k, which are equipped with suitable conic bundle structures over k.
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 investigate the Hasse principle for complete intersections cut out by a quadric and cubic hypersurface defined over the rational numbers.
A new polynomial sieve is presented and used to show that almost all integers have at most one representation as a sum of two values of a given polynomial of degree at least 3.
For suitable pairs of diagonal quadratic forms in 8 variables we use the circle method to investigate the density of simultaneous integer solutions and relate this to the problem of estimating linear correlations among sums of two squares.
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.