We use a function field version of the circle method to prove that a positive proportion of elements in 𝔽_q[t] are representable as a sum of three cubes of minimal degree from 𝔽_q[t], assuming a suitable form of the Ratios Conjecture and that the characteristic is greater than 3. The analogue of this conjecture for quadratic Dirichlet L-functions is known for large fixed q, via recent developments in homological stability.

We introduce a matrix divisor function τ_n(T,M), counting factorisations AB=M for n× n integer matrices A,B of height at most T. For a fixed non-singular M, or for the zero matrix M=O_n, we prove an asymptotic formula for τ_n(T,M), as T→∞, using lattice point counting. We also prove an essentially sharp uniform upper bound for τ_n(T,M), for an arbitrary non-singular matrix M.

We show how the circle method can be used to study rational points on a certain cubic fourfold, going beyond the square-root barrier.

The Davenport-Heilbronn method is a version of the circle method that was developed for studying Diophantine inequalities in the paper (Davenport and Heilbronn, J. Lond. Math. Soc. (1) 21 (1946), 185-193). We discuss the main ideas in the paper, together with an account of the development of the subject in the intervening 80 years.

Let p,q,r≥ 2 and consider the Campana orbifold ( ℙ^1, (1-1p)[0] +(1-1q)[1] +(1-1r)[∞] ). Primitive positive Campana points on this orbifold correspond to solutions of a+b=c in which a, b, and c are respectively p-full, q-full, and r-full. We establish upper bounds for the number of such points of bounded height in a broad range of exponents, with a power-saving over the trivial bound. The main analytic input is an estimate for primitive integral points in lopsided boxes on generalized Fermat surfaces a_1x^p+a_2y^q+a_3z^r=0, which is uniform in the coefficients.

We extend work of Heath-Brown and Salberger, based on the determinant method, to provide a uniform upper bound for the number of integral points of bounded height on an affine surface, which are subject to a polynomial congruence condition. This is applied to get a new uniform bound for points on diagonal quadric surfaces, and to a problem about the representation of integers as a sum of four unlike powers.

We use the circle method to prove that a density 1 of elements in 𝔽_q[t] are representable as a sum of three cubes of essentially minimal degree from 𝔽_q[t] , assuming the Ratios Conjecture and that char (𝔽_q)>3 . Roughly speaking, to do so, we upgrade an order of magnitude result to a full asymptotic formula that was conjectured by Hooley in the number field setting.

We use function field analytic number theory to establish the irreducibility and dimension of the moduli space that parameterises morphisms of fixed degree from ℙ^2 to an arbitrary smooth hypersurface of sufficiently small degree.

Among the set of hypersurfaces of degree d and dimension ℓ defined by the vanishing of a homogeneous polynomial with coefficients ± 1, we investigate the probability that a hypersurface contains a rational point as d and ℓ tend to infinity.

The large sieve is used to estimate the density of quadratic polynomials Q E Z[x], such that there exists an odd degree polynomial defined over Z which has resultant +1 with Q. Given a monic polynomial R E Z[x] of odd degree, this is used to show that for almost all quadratic polynomials Q E Z[x], there exists a prime p such that Q and R share a common root in Fp. Using recent work of Landesman, an application to the average size of the odd part of the class group of quadratic number fields is also given.

We prove upper and lower bounds on the number of pairs of commuting n× n matrices with integer entries in [-T,T] , as T→∞ . Our work uses Fourier analysis and leads to an analysis of exponential sums involving matrices over finite fields. These are bounded by combining a stratification result of Fouvry and Katz with a new result about the flatness of the commutator Lie bracket.

For a given elliptic curve E in short Weierstrass form, we show that almost all quadratic twists E_D have no integral points, as D ranges over square-free integers ordered by size. Our result is conditional on a weak form of the Hall-Lang conjecture in the case that E has partial 2-torsion. The proof uses a correspondence of Mordell and the reduction theory of binary quartic forms in order to transfer the problem to counting rational points of bounded height on a certain singular cubic surface, together with extensive use of cancellation in character sum estimates, drawn from Heath-Brown's analysis of Selmer group statistics for the congruent number curve.

Given a family of varieties over the projective line, we study the density of fibres that are everywhere locally soluble in the case that components of higher multiplicity are allowed. We use log geometry to formulate a new sparsity criterion for the existence of everywhere locally soluble fibres and formulate new conjectures that generalise previous work of Loughran-Smeets. These conjectures involve geometric invariants of the associated multiplicity orbifolds on the base of the fibration in the spirit of Campana. We give evidence for the conjectures using Chebotarev's theorem and sieve methods.

We develop a heuristic for the density of integer points on affine cubic surfaces. Our heuristic applies to smooth surfaces defined by cubic polynomials that are log K3, but it can also be adjusted to handle singular cubic surfaces. We compare our heuristic to Heath-Brown's prediction for sums of three cubes, as well as to asymptotic formulae in the literature around Zagier's work on the Markoff cubic surface, and work of Baragar and Umeda on further surfaces of Markoff-type. We also test our heuristic against numerical data for several families of cubic surfaces.

The circle method has been successfully used over the last century to study rational points on hypersurfaces. More recently, a version of the method over function fields, combined with spreading out techniques, has led to a range of results about moduli spaces of rational curves on hypersurfaces. In this paper a version of the circle method is implemented in the setting of the Grothendieck ring of varieties. This allows us to approximate the classes of these moduli spaces directly, without relying on point counting, and leads to a deeper understanding of their geometry.

The question of whether or not a given integral polynomial takes infinitely many square-free values has only been addressed unconditionally for polynomials of degree at most 3. We address this question, on average, for polynomials of arbitrary degree.

We study the geometry of the space of rational curves on smooth complete intersections of low degree, which pass through a given set of points on the variety. The argument uses spreading out to a finite field, together with an adaptation to function fields of positive characteristic of work by Rydin Myerson on the circle method. Our work also allows us to handle weak approximation for such varieties.

This paper corrects an error in an earlier work of the author.

We investigate strong divisibility sequences and produce lower and upper bounds for the density of integers in the sequence that only have (somewhat) large prime factors. We focus on the special cases of Fibonacci numbers and elliptic divisibility sequences, discussing the limitations of our methods. At the end of the paper, there is an appendix by Sandro Bettin on divisor closed sets that we use to study the density of prime terms that appear in strong divisibility sequences.

We prove the Manin-Peyre conjecture for the number of rational points of bounded height outside of a thin subset on a family of Fano threefolds of bidegree (1,2). The proof uses a mixture of the circle method and techniques from the geometry of numbers.