Results
author = "Flynn, E."
Coverings of curves of genus 2
Author(s) :
Flynn, E. V.
Description :
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, Cha...
Repository :
Oxford University Research Archive (ORA)
Language(s) :
Undetermined
Sequences of rational torsions on abelian varieties
Author(s) :
Flynn, E. V.
Description :
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 sequen...
Repository :
Oxford University Research Archive (ORA)
Language(s) :
Undetermined
The jacobian and formal group of a curve of genus 2 over an arbitrary ground field
Author(s) :
Flynn, E. V.
Description :
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 th...
Repository :
Oxford University Research Archive (ORA)
Language(s) :
Undetermined
A flexible method for applying Chabauty's Theorem
Author(s) :
Flynn, E. V.
Description :
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 ...
Repository :
Oxford University Research Archive (ORA)
Language(s) :
Undetermined
On a theorem of Coleman
Author(s) :
Flynn, E. V.
Description :
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.
Repository :
Oxford University Research Archive (ORA)
Language(s) :
Undetermined
Homogeneous Spaces and Degree 4 del Pezzo Surfaces.
Author(s) :
Flynn, E. V.
Description :
It is known that, given a genus 2 curve C : y^2 = f(x), where f(x) is quintic and defined over a field K, of characteristic different from 2, and given a homogeneous space H_delta for complete 2-descent on the Jacobian of C, there is a V_delta (which we shall describe), which is a degree 4 del Pezzo...
Repository :
Oxford University Research Archive (ORA)
Language(s) :
Undetermined
On Q-derived polynomials
Author(s) :
Flynn, E. V.
Description :
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 t...
Repository :
Oxford University Research Archive (ORA)
Language(s) :
Undetermined
Large rational torsion on abelian varieties
Author(s) :
Flynn, E. V.
Description :
A method of searching for large rational torsion on Abelian varieties is described. A few explicit applications of this method over Q give rational 11- and 13-torsion in dimension 2, and rational 29-torsion in dimension 4.
Repository :
Oxford University Research Archive (ORA)
Language(s) :
Undetermined
An explicit theory of heights
Author(s) :
Flynn, E. V.
Description :
We consider the problem of explicitly determining the naive height constants for Jacobians of hyperelliptic curves. For genus>1, it is impractical to apply Hilbert's Nullstellensatz directly to the defining equations of the duplication law; we indicate how this technical difficulty can be overcome b...
Repository :
Oxford University Research Archive (ORA)
Language(s) :
Undetermined
The group law on the jacobian of a curve of genus 2
Author(s) :
Flynn, E. V.
Description :
An explicit description is given of the group law on the Jacobian of a curve C of genus 2. The Kummer surface provides a useful intermediary stage; bilinear forms relating to the Kummer surface imply that the global group law may be given projectively by biquadratic forms defined over the same ring ...
Repository :
Oxford University Research Archive (ORA)
Language(s) :
Undetermined