Coverings of curves of genus 2

We shall discuss the idea of finding all rational points on a curve C by first finding an associated collection of curves whose rational points cover those of C. This classical technique has recently been given a new lease of life by being combined with descent techniques on Jacobians of curves, Chabauty techniques, and the increased power of software to perform algebraic number theory. We shall survey recent applications during the last 5 years which have used Chabauty techniques and covering collections of curves of genus 2 obtained from pullbacks along isogenies on their Jacobians

Experiences of post-qualifying study in social work

This article is based on a research project to explore the experiences of past and current candidates for post-qualifying awards in social work in England. Also included in the study are the Leads of the post-qualifying consortia in England. The study used questionnaire survey and nominal group techniques to gather data, which were coded and categorised into themes. The main findings relate to the perceived purposes of post-qualifying study, motivations for undertaking post-qualifying study, the factors that sustain and hinder study, the advice that those who have or who are experiencing post-qualifying study would give to those about to start and future plans and hopes in this area.Post-qualifying study is generally valued, especially in relation to the opportunities it provides for professional development. The support of a mentor who has direct experience of the candidate's programme is highly prized, as are clear and consistent guidance from the programme and meaningful study time and workload relief from employers. There are also frustrations for some candidates who do not feel that their post-qualifying study has stretched them beyond qualifying standards or who experience the teaching as divorced from the realities of daily practice. The appetite for a wider choice of post-qualifying modules suggests that providers of post-qualifying study will need to collaborate within and across regions in order to achieve a critical mass of candidates for more specialist or focused learning. The study suggests a need for further research to understand the impact of post-qualifying study on candidates' social work practice.The article concludes with two checklists of questions, one for individual candidates and another for agencies and programmes. These questions arise from the findings in the research

Covering collections and a challenge problem of Serre

We answer a challenge of Serre by showing that every rational point on the projective curve X$^4$ + Y$^4$ = 17 Z$^4$ is of the form ($\pm$1, $\pm$2, 1) or ($\pm$2, $\pm$1, 1). Our approach builds on recent ideas from both Nils Bruin and the authors on the application of covering collections and Chabauty arguments to curves of high rank. This is the only value of c$\le$81 for which the Fermat quartic X$^4$ + Y$^4$ = c Z$^4$ cannot be solved trivially, either by local considerations or maps to elliptic curves of rank 0, and it seems likely that our approach should give a method of attack for other nontrivial values of c

The arithmetic of hyperelliptic curves

We summarise recent advances in techniques for solving Diophantine problems on hyperelliptic curves; in particular, those for finding the rank of the Jacobian, and the set of rational points on the curve

On a theorem of Coleman

A simplified method of descent via isogeny is given for Jacobians of curves of genus 2. This method is then used to give applications of a theorem of Coleman for computing all of the rational points on certain curves of genus 2

The effect of the Solar motion on the flux of long-period comets

The long-term dynamics of Oort cloud comets are studied under the influence of both the radial and the vertical components of the Galactic tidal field. Sporadic dynamical perturbation processes are ignored, such as passing stars, since we aim to study the influence of just the axisymmetric Galactic tidal field on the cometary motion and how it changes in time. We use a model of the Galaxy with a disc, bulge and dark halo, and a local disc density, and disc scale length constrained to fit the best available observational constraints. By integrating a few million of cometary orbits over 1 Gyr, we calculate the time variable flux of Oort cloud comets that enter the inner Solar System, for the cases of a constant Galactic tidal field, and a realistically varying tidal field which is a function of the Sun's orbit. The applied method calculates the evolution of the comets by using first-order averaged mean elements. We find that the periodicity in the cometary flux is complicated and quasi-periodic. The amplitude of the variations in the flux are of order 30%. The radial motion of the Sun is the chief cause of this behaviour, and should be taken into account when the Galactic influence on the Oort cloud comets is studied.Comment: 8 pages, 9 figures, 3 tables, Accepted MNRA

On the problems of measuring transient temperature in cryogenic fluids

Cryogenic sensor errors in measuring transient temperature in cryogenic fluid

Exhibiting Sha[2] on hyperelliptic jacobians

We discuss approaches to computing in the Shafarevich-Tate group of Jacobians of higher genus curves, with an emphasis on the theory and practice of visualisation. Especially for hyperelliptic curves, this often enables the computation of ranks of Jacobians, even when the 2-Selmer bound does not bound the rank sharply. This was previously only possible for a few special cases. For curves of genus 2, we also demonstrate a connection with degree 4 del Pezzo surfaces, and show how the Brauer-Manin obstruction on these surfaces can be used to compute members of the Shafarevich-Tate group of Jacobians. We derive an explicit parametrised infinite family of genus 2 curves whose Jacobians have nontrivial members of the Sharevich-Tate group. Finally we prove that under certain conditions, the visualisation dimension for order 2 cocycles of Jacobians of certain genus 2 curves is 4 rather than the general bound of 32

Descent via (3,3)-isogeny on Jacobians of genus 2 curves

We give parametrisation of curves C of genus 2 with a maximal isotropic (ZZ/3)^2 in J[3], where J is the Jacobian variety of C, and develop the theory required to perform descent via (3,3)-isogeny. We apply this to several examples, where it can shown that non-reducible Jacobians have nontrivial 3-part of the Tate-Shafarevich group.Comment: 17 page