Preprints/Publications

We show that the density of abelian extensions satisfying the Hasse norm principle (every element which is a local norm everywhere is a global norm) exists. Our results strengthen previous work of Frei--Loughran--Newton using a completely distinct method.

Given an extension of fields K/Q with abelian galois group G, we study weak approximation on the affine torus defined by the equation N_{K/Q}(\x) =1. For any fixed abelian G, we develop an asymptotic formula for the number of extensions with Galois group G of discriminant at most X such that weak approximation holds on the associated torus.

We prove an asymptotic formula for how often the equations aX^2 + bY^2 + cZ^2 = 0 have solutions as you vary a,b,c in a box of sidelength B. This improves upper bounds of Serre and lower bounds of Guo and Hooley. We then use this asymptotic formula as the jumping off point to predict the growth of a very general variation on this problem.

The rational points of a variety naturally embed into the set of adelic points, understanding the image of this embedding is a central theme in Diophantine geometry. We say a projective variety satisfies weak approximation if this image is dense. In other words, for any finite collection (x_p)_{p \in S} of local points, you can find a rational point x which is arbitrarily "p-adically close" to each x_p.

We completely determine whether or not this property holds for a particular class of varieties. The varieties we study all admit a map to P^2 such that the fibres are 2-dimensional quadrics. The reason for studying this class is that their Brauer groups have already been investigated (over C) in recent breakthrough work of Hassett-Pirutka-Tschinkel on classical rationality problems in algebraic geometry.

Given a polynomial f(n) without any obvious local obstructions, the Bateman-Horn conjecture predicts that as n varies from 1 to X, f(n) is a prime number roughly C_f X/(log X) times. In this paper, we prove a new large sieve inequality for r-th order Dirichlet characters which we use to show that 100% of polynomials of the form f(n) = n^r + k satisfy the Bateman-Horn conjecture.

This result is then used, along with a variation of the fibration method, to study varieties defined by equations of the form X^2 - d Y^2 = T^r + k. Under certain technical restrictions on d, we show that 100% of such varieties have a Q point, and under further assumptions on d, 100% of such varieties have a Z point.

Point sets in Euclidean space can act as mathematical models for the atomic structure of materials in the real world. One way to study such atomic structures is to look at the diffraction pattern produced when a light wave passes through the atom. Clear patterns with visible symmetries are produced when the atom has a crystalline or quasi-crystalline structure. There is a mathematical formalism describing when a point set has a so called "pure point" diffraction pattern, as well as notions of crystalline and quasicrystalline point sets.

However some point sets of number theoretic origin exhibit pure point diffraction but are neither crystalline or quasi-crystalline! We investigate one such set, the set of k-free numbers, and using tools from multiplicative number theory give an asymptotic formula for the scaling factor of the diffraction measure of this set.

For every natural number e there is a number N_e such that N_e points in general position in the plane lie on finitely many degree e rational curves over the complex numbers. Our motivating question is: if the co-ordinates of these points are all rational numbers then how many of the resulting curves are defined over the rationals? And what is the Galois group of the smallest field over which all the curves are defined?

We find that 0% of such collections of points result in even a single curve being defined over Q. Moreover we conjecture that 100% of the time the Galois group is as large as it could be (i.e. S_{N_e}), which we prove in the case e=3.

Next we replace the plane by a Fano hypersurface and play the same game. The key ingredient in the quantitative part of the problem is a new large sieve for rational points on hypersurfaces of low degree.

We study smooth projective models of the varieties defined by the equation

X^2 + Y^2 = (aT^2 + b)(cT^2 + d)=/=0,

and ask how often they satisfy the Hasse principle. By producing asymptotic formulae for the number of locally soluble/globally soluble surfaces in this family, it is shown that if one imposes the condition that ad-bc = \pm 1 then a positive proportion fail the Hasse principle.

If K/Q is a number field then there is an associated field norm N_{K/Q} from K^* to Q^*. This map induces norm maps on all the completions K_{mu}/Q_{nu} and hence one can ask the local-global question: if an element in Q^* is in the image of all the local norms, must it also be a global norm?

The answer is: not always! We give an explicit asymptotic formula for the number of biquadratic fields Q(sqrt(a), sqrt(b)) for which the local-global principle (known as the Hasse norm principle) fails.