12,060 research outputs found
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 jacobian and formal group of a curve of genus 2 over an arbitrary ground field
We present an explicit model of the Jacobian variety, and give a set of quadratic defining equations. We develop constructively the theory of formal groups for genus 2, including an explicit pair of local parameters which induce a formal group law defined over the same ring as the coefficients of the original curve
Solving Diophantine problems on curves via descent on the jacobian
We suggest that the following plan will provide a powerful tool for trying to find the set of Q-rational points C(Q) on a curve C of genus>1:\ud
(1) Attempt to find J(Q)/2J(Q) via descent on J, the Jacobian of C.\ud
(2) Deduce generators for J(Q) via an explicit theory of heights.\ud
(3). Apply local techniques to try to deduce C(Q) via an embedding of C(Q) inside J(Q).\ud
We describe work just completed, which gives versions of (1),(2),(3) which are often workable in practice for genus 2, and outline the potential for a computationally viable generalisation. We note that (1),(2),(3) (quite aside from being part of this plan) have their own independent applications to other branches of the Mathematics of Computation, such as the search for large rank, the higher dimensional testing of well known conjectures, and algorithms for symbolic integration
On Q-derived polynomials
It is known that Q-derived univariate polynomials (polynomials defined over Q, with the property that they and all their derivatives have all their roots in Q) can be completely classified subject to two conjectures: that no quartic with four distinct roots is Q-derived, and that no quintic with a triple root and two other distinct roots is Q-derived. We prove the second of these conjectures
Sequences of rational torsions on abelian varieties
We address the question of how fast the available rational torsion on abelian varieties over Q increases with dimension. The emphasis will be on the derivation of sequences of torsion divisors on hyperelliptic curves. Work of Hellegouarch and Lozach (and Klein) may be made explicit to provide sequences of curves with rational torsion divisors of orders increasing linearly with respect to genus. The main results are applications of a new technique which provide sequences of hyperelliptic curves for all torsions in an interval where is quadratic in g and is linear in g. As well as providing an improvement from linear to quadratic, these results provide a wide selection of torsion orders for potential use by those involved in computer integration. We conclude by considering possible techniques for divisors of non-hyperelliptic curves, and for general abelian varieties
A flexible method for applying Chabauty's Theorem
A strategy is proposed for applying Chabauty's Theorem to hyperelliptic curves of genus>1. In the genus 2 case, it shown how recent developments on the formal group of the Jacobian can be used to give a flexible and computationally viable method for applying this strategy. The details are described for a general curve of genus 2, and are then applied to find C(Q) for a selection of curves. A fringe benefit is a more explicit proof of a result of Coleman
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
Descent Via Isogeny on Elliptic Curves with Large Rational Torsion Subgroups.
We outline PARI programs which assist with various algorithms related to descent via isogeny on elliptic curves. We describe, in this context, variations of standard inequalities which aid the computation of members of the Tate-Shafarevich group. We apply these techniques to several examples: in one case we use descent via 9-isogeny to determine the rank of an elliptic curve; in another case we find nontrivial members of the 9-part of the Tate-Shafarevich group, and in a further case, nontrivial members of the 13-part of the Tate-Shafarevich group
Finite Weil restriction of curves
Given number fields , smooth projective curves defined over
and defined over , and a non-constant -morphism ,we consider the curve defined over whose -rational points
parametrize the -rational points on whose images under are defined
over . Our construction provides a framework which includes as a special
case that used in Elliptic Curve Chabauty techniques and their higher genus
versions. The set can be infinite only when has genus at most 1;
we analyze completely the case when has genus 1.Comment: Comments are welcome
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