The four colour theorem states that the vertices of every planar graph can be coloured with at most four colours so that no two adjacent vertices receive the same colour. This theorem is famous for many reasons, including the fact that its original 1977 proof includes a non-trivial computer verification. Recently, a formal proof of the theorem was obtained with the equational logic program Coq [G. Gonthier, ‘Formal proof–the four color theorem’, Notices of Amer. Math. Soc. 55 (2008) no. 11, 1382–1393]. In this paper we describe an implementation of the computational method introduced by C. S. Calude and co-workers [Evaluating the complexity of mathematical problems. Part 1’, Complex Systems 18 (2009) 267–285; A new measure of the difficulty of problems’, J. Mult. Valued Logic Soft Comput. 12 (2006) 285–307] to evaluate the complexity of the four colour theorem. Our method uses a Diophantine equational representation of the theorem. We show that the four colour theorem is in the complexity class ℭU,4. For comparison, the Riemann hypothesis is in class ℭU,3 while Fermat’s last theorem is in class ℭU,1.

We study the action of the Hecke operators Un on the set of hypergeometric functions, as well as on formal power series. We show that the spectrum of these operators on the set of hypergeometric functions is the set {na:n∈ℕ,a∈ℤ}, and that the polylogarithms play an important role in the study of the eigenfunctions of the Hecke operators Un on the set of hypergeometric functions. As a corollary of our results on simultaneous eigenfunctions, we also obtain an apparently unrelated result regarding the behavior of completely multiplicative hypergeometric coefficients.

We discuss the Mordell–Weil sieve as a general technique for proving results concerning rational points on a given curve. In the special case of curves of genus 2, we describe quite explicitly how the relevant local information can be obtained if one does not want to restrict to mod p information at primes of good reduction. We describe our implementation of the Mordell–Weil sieve algorithm and discuss its efficiency.

Supplementary materials are available with this article.

Rosen gave a determinant formula for relative class numbers for cyclotomic function fields, which may be regarded as an analogue of the classical Maillet determinant. In this paper, we give a determinant formula for relative congruence zeta functions for cyclotomic function fields. Our formula may be regarded as a generalization of the determinant formula for the relative class number.

We exhibit a strong link between the Hall algebra HX of an elliptic curve X defined over a finite field 𝔽l (or, more precisely, its spherical subalgebra U+X) and Cherednik’s double affine Hecke algebras of type GLn, for all n. This allows us to obtain a geometric construction of the Macdonald polynomials Pλ(q,t−1) in terms of certain functions (Eisenstein series) on the moduli space of semistable vector bundles on the elliptic curve X.

This paper studies affine Deligne–Lusztig varieties in the affine flag manifold of a split group. Among other things, it proves emptiness for certain of these varieties, relates some of them to those for Levi subgroups, and extends previous conjectures concerning their dimensions. We generalize the superset method, an algorithmic approach to the questions of non-emptiness and dimension. Our non-emptiness results apply equally well to the p-adic context and therefore relate to moduli of p-divisible groups and Shimura varieties with Iwahori level structure.

In this paper, we compute Ford fundamental domains for all genus-zero and genus-one congruence subgroups. This is a continuation of previous work, which found all such groups, including ones that are not subgroups of PSL(2,ℤ). To compute these fundamental domains, an algorithm is given that takes the following as its input: a positive square-free integer f, which determines a maximal discrete subgroup Γ0(f)+ of SL(2,ℝ); a decision procedure to determine whether a given element of Γ0(f)+ is in a subgroup G; and the index of G in Γ0(f)+. The output consists of: a fundamental domain for G, a finite set of bounding isometric circles; the cycles of the vertices of this fundamental domain; and a set of generators of G. The algorithm avoids the use of floating-point approximations. It applies, in principle, to any group commensurable with the modular group. Included as appendices are: MAGMA source code implementing the algorithm; data files, computed in a previous paper, which are used as input to compute the fundamental domains; the data computed by the algorithm for each of the congruence subgroups of genus zero and genus one; and an example, which computes the fundamental domain of a non-congruence subgroup.

Let L/k be a finite Galois extension of number fields with Galois group G. For every odd prime p satisfying certain mild technical hypotheses, we use values of Artin L-functions to construct an element in the centre of the group ring ℤ(p)[G] that annihilates the p-part of the class group of L.

We apply a method of Davenport to improve several estimates for slim exceptional sets associated with the asymptotic formula in Waring’s problem. In particular, we show that the anticipated asymptotic formula in Waring’s problem for sums of seven cubes holds for all but O(N1/3+ε) of the natural numbers not exceeding N.

It is shown that any subset of can be the exceptional set of some transcendental entire function. Furthermore, we give a much more general version of this theorem and present a unified proof.

Let f∈ℚ[X] and let us consider a Diophantine equation z2=f(x)2±f(y)2. In this paper, we continue the study of the existence of integer solutions of the equation, when the degree of f is 2 and if f(x) is a triangular number or a tetrahedral number.

We investigate the distribution of real algebraic numbers of a fixed degree that have a close conjugate number, with the distance between the conjugate numbers being given as a function of their height. The main result establishes the ubiquity of such algebraic numbers in the real line and implies a sharp quantitative bound on their number. Although the main result is rather general, it implies new estimates on the least possible distance between conjugate algebraic numbers, which improve recent bounds obtained by Bugeaud and Mignotte. So far, the results à la Bugeaud and Mignotte have relied on finding explicit families of polynomials with clusters of roots. Here we suggest a different approach in which irreducible polynomials are implicitly tailored so that their derivatives assume certain values. We consider some applications of our main theorem, including generalisations of a theorem of Baker and Schmidt and a theorem of Bernik, Kleinbock and Margulis in the metric theory of Diophantine approximation.

We generalize the main result of the paper by Bennett and Mulholland [‘On the diophantine equation xn+yn=2αpz2’, C. R. Math. Acad. Sci. Soc. R. Can.28 (2006), 6–11] concerning the solubility of the diophantine equation xn+yn=2αpz2. We also demonstrate, by way of examples, that questions about solubility of a class of diophantine equations of type (3,3,p) or (4,2,p) can be reduced, in certain cases, to studying several equations of the type (p,p,2).

Let σA(n)=∣{(a,a′)∈A2:a+a′=n}∣, where n∈ℕ and A is a subset of ℕ. Erdős and Turán con-jectured that for any basis A of ℕ, σA(n) is unbounded. In 1990, Ruzsa constructed a basis A⊂ℕ for which σA(n) is bounded in square mean. Based on Ruzsa’s method, we proved that there exists a basis A of ℕ satisfying ∑ n≤Nσ2A(n)≤1449757928N for large enough N. In this paper, we give a quantitative result for the existence of N, that is, we show that there exists a basis A of ℕ satisfying ∑ n≤Nσ2A(n)≤1069693154N for N≥7.628 517 798×1027, which improves earlier results of the author [‘A note on a result of Ruzsa’, Bull. Aust. Math. Soc.77 (2008), 91–98].

Beilinson [Higher regulators and values of L-functions, Itogi Nauki i Tekhniki Seriya Sovremennye Problemy Matematiki Noveishie Dostizheniya (Current problems in mathematics), vol. 24 (Vserossiisky Institut Nauchnoi i Tekhnicheskoi Informatsii, Moscow, 1984), 181–238] obtained a formula relating the special value of the L-function of H2 of a product of modular curves to the regulator of an element of a motivic cohomology group, thus providing evidence for his general conjectures on special values of L-functions. In this paper we prove a similar formula for the L-function of the product of two Drinfeld modular curves, providing evidence for an analogous conjecture in the case of function fields.

In a paper by Badulescu [Global Jacquet–Langlands correspondence, multiplicity one and classification of automorphic representations, Invent. Math. 172 (2008), 383–438], results on the global Jacquet–Langlands correspondence, (weak and strong) multiplicity-one theorems and the classification of automorphic representations for inner forms of the general linear group over a number field were established, under the assumption that the local inner forms are split at archimedean places. In this paper, we extend the main local results of that article to archimedean places so that the above condition can be removed. Along the way, we collect several results about the unitary dual of general linear groups over ℝ, ℂ or ℍ which are of independent interest.

We show that if G is a group and A⊂G is a finite set with ∣A2∣≤K∣A∣, then there is a symmetric neighbourhood of the identity S such that Sk⊂A2A−2 and ∣S∣≥exp (−KO(k))∣A∣.

We describe the structure of ‘K-approximate subgroups’ of torsion-free nilpotent groups, paying particular attention to Lie groups.

Three other works, by Fisher et al., by Sanders and by Tao, have appeared that independently address related issues. We comment briefly on some of the connections between these papers.

We exhibit central simple algebras over the function field of a diagonal quartic surface over the complex numbers that represent the 2-torsion part of its Brauer group. We investigate whether the 2-primary part of the Brauer group of a diagonal quartic surface over a number field is algebraic and give sufficient conditions for this to be the case. In the last section we give an obstruction to weak approximation due to a transcendental class on a specific diagonal quartic surface, an obstruction which cannot be explained by the algebraic Brauer group which in this case is just the constant algebras.