For a hyperelliptic curve ${\hbox{\ax C}}$ of genus $g$ with a divisor class of order $n = g + 1$, we shall consider an associated covering collection of curves ${\hbox{\ax D}}_\delta$, each of genus $g^2$. We describe, up to isogeny, the Jacobian of each ${\hbox{\ax D}}_\delta$ via a map from ${\hbox{\ax D}}_\delta$ to ${\hbox{\ax C}}$, and two independent maps from ${\hbox{\ax D}}_\delta$ to a curve of genus $g(g-1)/2$. For some curves, this allows covering techniques that depend on arithmetic data of number fields of smaller degree than standard 2-coverings; we illustrate this by using 3-coverings to find all ${\Bbb Q}$-rational points on a curve of genus 2 for which 2-covering techniques would be impractical.

Let $D\,{=}\,\pi_1\cdots \pi_n$, where $\pi_1,\dots,\pi_n$ are distinct Gaussian odd primes and $n$ is any positive integer. Let $E_{4D^2}$ be the elliptic curve $y^2\,{=}\,x^3-(2D)^2x$. We prove that the value at $s\,{=}\,1$ of the complex $L$-function of $E_{4D^2}$, divided by its natural period(which is $\omega/ \surd{(2D)}$, where $\omega\,{=}\,2{\cdot}6220575\dots$), is always divisible by $2^{n-1}$, and we give a simple combinatorial criterion for it to be exactly divisible by $2^{n-1}$. As a corollary, we construct a series of even non-congruent numbers having an arbitrarily large number of prime factors. Our result is in accord with the predictions of the conjecture of Birch and Swinnerton–Dyer.

Let $M$ be a pure motive over $\Bbb Q$ of weight $w=w(M)$ and of the global conductor $c(M)$. Also given Dirichlet character $\chi$ one can define the twisted motive $M_\chi$; let $L(M_\chi,s)\, (s=\sigma+i t)$ denote the corresponding $L$-function.

Let $(R,M)$ be a Noetherian local domain, $F$ its quotient field and $v$ a valuation of $F$ centred at $R$. In the hope of understanding the valuation $v$, a useful tool is the so-called graded algebra relative to $v$, $gr_{v}R$. Its study can be understood as an attempt to approach the problem of classifying valuations. On the other hand, $gr_{v}R$ is also an object of interest since it can be provided with a geometrical meaning [15].

Partial combinatory algebras (pcas), models for a form of Combinatory Logic with partial application, have been studied for the last thirty years because of their close connection to Intuitionistic Logic (see, for example, [11])

We study compact Riemann surfaces of genus $g\geq2$ having a dihedral group of automorphisms. We find necessary and sufficient conditions on the signature of a Fuchsian group for it to admit a surface kernel epimorphism onto the dihedral group $D_N$. The question of extendability of the action of $D_N$ is considered. We also give an explicit parametrization of the moduli space of Riemann surfaces with maximal dihedral symmetry, showing that it is a one-dimensional complex manifold. Defining equations of all such surfaces and the formulae of their automorphisms are calculated. The locus of this moduli space consisting of those surfaces admitting some real structure is determined.

Let $\Sigma_{g,r}^n$ be a connected orientable surface of genus $g$ with $r$ boundary components and $n$ punctures and let $\Gamma_{g,r}^n$ denote the mapping class group of $\Sigma_{g,r}^n$, namely the group of isotopy classes of orientation-preserving diffeomorphisms of $\Sigma_{g,r}^n$ which are the identity on the boundary and on the punctures. Here, we see the punctures on the surface as distinguished points. The isotopies are required to be the identity on the boundary and on the punctures. If $r$ and/or $n$ is zero, then we omit it from the notation.

We prove that all generalized symmetric spaces of compact simple Lie groups are formal in the sense of Sullivan. Nevertheless, many of them, including all the non-symmetric flag manifolds, do not admit Riemannian metrics for which all products of harmonic forms are harmonic.

We exploit a recent approach to Brascamp–Lieb inequalities, due to Caffarelli [5], and reconsider earlier approaches to establish stochastic domination inequalities between Gaussian variables and random variables with density of the form $g\cdot h, g$ a Gaussian density and $h$ a log-concave or log-convex function. These extend to inequalities on random vectors via a classical result by Prékopa and Leindler and they complement the Brascamp–Lieb moment inequalities. Some applications to a class of Gibbs measures, the anharmonic crystals, are developed.

We consider a variational problem for steady axisymmetric vortex-rings, in which kinetic energy is maximised subject to prescribed impulse, with the ratio of vorticity to axial distance belonging to the class of rearrangements of a prescribed function, as proposed by Benjamin. We prove existence of maximisers in an extended constraint set, allowing some loss of vorticity. We then study a particular family of vortex-rings including Hill's spherical vortex, determining the precise range of impulse values for which the maximiser loses vorticity, and show that the maximisers are spherical when this happens.

It is shown that a spacetime with collisionless matter evolving from data on a compact Cauchy surface with hyperbolic symmetry can be globally covered by compact hypersurfaces on which the mean curvature is constant and by compact hypersurfaces on which the area radius is constant. Results for the related cases of spherical and plane symmetry are reviewed and extended. The prospects of using the global time coordinates obtained in this way to investigate the global geometry of the spacetimes concerned are discussed.

A theorem about local in time existence of spacelike foliations with prescribed mean curvature in cosmological spacetimes will be proved. The time function of the foliation is geometrically defined and fixes the diffeomorphism invariance inherent in general foliations of spacetimes. Moreover, in contrast to the situation of the more special constant mean curvature foliations, which play an important role in global analysis of spacetimes, this theorem overcomes the existence problem arising from topological restrictions for surfaces of constant mean curvature.