Let V be a nonsingular cubic surface defined over the finite field Fq. It is well known that the number of points on V satisfies #V(Fq) = q2 + nq + 1 where −2 ≤ n ≤ 7 and that n = 6 is impossible; see for example [1], Table 1. Serre has asked if these bounds are best possible for each q. In this paper I shall show that this is so, with three exceptions:

We consider some questions related to the signs of Hecke eigenvalues or Fourier coefficients of classical modular forms. One problem is to determine to what extent those signs, for suitable sets of primes, determine uniquely the modular form, and we give both individual and statistical results. The second problem, which has been considered by a number of authors, is to determine the size, in terms of the conductor and weight, of the first sign-change of Hecke eigenvalues. Here we improve the recent estimate of Iwaniec, Kohnen and Sengupta.

We discuss the question of whether the Brauer–Manin obstruction is the only obstruction to the Hasse principle for integral points on affine hyperbolic curves. In the case of rational curves we conjecture a positive answer, we prove that this conjecture can be given several equivalent formulations and we relate it to an old conjecture of Skolem. Finally, we show that for elliptic curves minus one point a strong version of the question (describing the set of integral points by local conditions) has a negative answer.

We consider a family of dynamical systems (A, α, L) in which α is an endomorphism of a C*-algebra A and L is a transfer operator for α. We extend Exel's construction of a crossed product to cover non-unital algebras A, and show that the C*-algebra of a locally finite graph can be realised as one of these crossed products. When A is commutative, we find criteria for the simplicity of the crossed product, and analyse the ideal structure of the crossed product.

Motivated by a problem from number theory about the relationship between Fermat curves and modular curves, a new class of loops is introduced, the Catalan loops. In the number-theoretic context, these loops turn out to be abelian precisely when the Fermat curves and modular curves coincide. General Catalan loops arise on certain transversals to diagonal subgroups in special linear groups over rings with a topologically nilpotent element. The transversals consist of products of certain affine shears. In a Catalan loop, the multiplication and right division are given by rational functions. The left division is algebraic, corresponding to a quadratic irrationality. The left division embodies generating functions for the Catalan numbers. Structurally, Catalan loops are shown to be residually nilpotent.

In this paper the space of almost commuting elements in a Lie group is studied through a homotopical point of view. In particular a stable splitting after one suspension is derived for these spaces and their quotients under conjugation. A complete description for the stable factors appearing in this splitting is provided for compact connected Lie groups of rank one. By using symmetric products, the colimits Rep(ℤn, SU), Rep(ℤn, U) and Rep(ℤn, Sp) are explicitly described as finite products of Eilenberg–MacLane spaces.

This paper constructs completely integrable convex Hamiltonians on the cotangent bundle of certain k bundles over l. A central role is played by the Lax representation of a Bogoyavlenskij–Toda lattice. The classification of these systems, up to iso-energetic topological conjugacy, is related to the classification of abelian groups of Anosov toral automorphisms by their topological entropy function.

We prove that on a restricted contact type hypersurface the number of leaf-wise intersections is bounded from below by a certain cup-length.

Let K be a compact subset of ℝd and write s for the Hausdorff dimension of K. For a probability measure μ on K, the lower and upper Lq-dimensions of order q ∈ ℝ are defined byIn this paper we study Lq-dimensions of measures that are generic in the sense of prevalence. In particular, we prove that if K satisfies a mild regularity condition, then a prevalent probability measure μ on K satisfies:for all 0 ≤ q ≤ 1, andfor all 1 ≤ q. This result is in sharp contrast to the behaviour of the Lq-dimensions of measures that are generic in the sense of Baire category. Namely, if K satisfies a mild regularity condition, then a probability measure μ on K that is generic in the sense of Baire category satisfies:for all 1 < q.