4,087 research outputs found
N-covers of hyperelliptic curves
For a hyperelliptic curve C of genus g with a divisor class of order n=g+1, we shall consider an associated covering collection of curves D, each of genus g. We describe, up to isogeny, the Jacobian of each D via a map from D to C, and two independent maps from D 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 Q-rational points on a curve of genus 2 for which 2-covering techniques would be impractical
Towers of 2-covers of hyperelliptic curves
In this article, we give a way of constructing an unramified Galois cover of a hyperelliptic curve. The geometric Galois-group is an elementary abelian 2-group. The construction does not make use of the embedding of the curve in its Jacobian and it readily displays all subcovers. We show that the cover we construct is isomorphic to the pullback along the multiplication-by-2 map of an embedding of the curve in its Jacobian. We show that the constructed cover has an abundance of elliptic and hyperelliptic subcovers. This makes this cover especially suited for covering techniques employed for determining the rational points on curves. Especially the hyperelliptic subcovers give a chance for applying the method iteratively, thus creating towers of elementary abelian 2-covers of hyperelliptic curves. As an application, we determine the rational points on the genus 2 curve arising from the question whether the sum of the first n fourth powers can ever be a square. For this curve, a simple covering step fails, but a second step succeeds
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
Extending Elliptic Curve Chabauty to higher genus curves
We give a generalization of the method of "Elliptic Curve Chabauty" to higher
genus curves and their Jacobians. This method can sometimes be used in
conjunction with covering techniques and a modified version of the Mordell-Weil
sieve to provide a complete solution to the problem of determining the set of
rational points of an algebraic curve .Comment: 24 page
Arithmetic progressions consisting of unlike powers
In this paper we present some new results about unlike powers in arithmetic
progression. We prove among other things that for given and
there are only finitely many arithmetic progressions of the form
with
gcd and for Furthermore, we
show that, for L=3, the progression is the only such progression
up to sign.Comment: 16 page
The Brauer-Manin Obstruction and Sha[2].
We discuss the Brauer-Manin obstruction on del Pezzo surfaces of degree 4. We outline a detailed algorithm for computing the obstruction and provide associated programs in magma. This is illustrated with the computation of an example with an irreducible cubic factor in the singular locus of the defining pencil of quadrics (in contrast to previous examples, which had at worst quadratic irreducible factors). We exploit the relationship with the Tate-Shafarevich group to give new types of examples of Sha[2], for families of curves of genus 2 of the form y^2 = f(x), where f(x) is a quintic containing an irreducible cubic factor
On finiteness conjectures for modular quaternion algebras
It is conjectured that there exist only finitely many isomorphism classes of endomorphism algebras of abelian varieties of bounded dimension over a number field of bounded degree. We explore this conjecture when restricted to quaternion endomorphism algebras of abelian surfaces of GL-type over Q by giving a moduli interpretation which translates the question into the diophantine arithmetic of Shimura curves embedded in Hilbert surfaces. We address the resulting problems on these curves by local and global methods, including Chabauty techniques on explicit equations of Shimura curves
Chabauty-Coleman experiments for genus 3 hyperelliptic curves
We describe a computation of rational points on genus 3 hyperelliptic curves
defined over whose Jacobians have Mordell-Weil rank 1. Using
the method of Chabauty and Coleman, we present and implement an algorithm in
Sage to compute the zero locus of two Coleman integrals and analyze the finite
set of points cut out by the vanishing of these integrals. We run the algorithm
on approximately 17,000 curves from a forthcoming database of genus 3
hyperelliptic curves and discuss some interesting examples where the zero set
includes global points not found in .Comment: 18 page
Natural equilibrium states for multimodal maps
This paper is devoted to the study of the thermodynamic formalism for a class
of real multimodal maps. This class contains, but it is larger than,
Collet-Eckmann. For a map in this class, we prove existence and uniqueness of
equilibrium states for the geometric potentials , for the largest
possible interval of parameters . We also study the regularity and convexity
properties of the pressure function, completely characterising the first order
phase transitions. Results concerning the existence of absolutely continuous
invariant measures with respect to the Lebesgue measure are also obtained
Development of the sources of work stress inventory.
This article describes the development of the Sources of Work Stress Inventory (SWSI). Factor analyses of the generated items produced (a) a General Work Stress Scale and (b) eight Sources of Work Stress scales, namely Bureaucracy/Autonomy, Relationships, Tools and Equipment, Workload, Role Ambiguity, Work/Home Interface, Job Security and Career Advancement. Rasch rating scale analyses supported the construct validity and reliability of the scales. A multiple regression analysis confirmed the expected strong relationship between the different sources of work stress and the experience of stress in the workplace. It is concluded that the SWSI shows promise as a measure of work stress in the South African context
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