Location of Repository

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

Topics:
Algebraic geometry, Number theory

Publisher: Springer

Year: 2000

OAI identifier:
oai:generic.eprints.org:261/core69

Provided by:
Mathematical Institute Eprints Archive

Downloaded from
http://eprints.maths.ox.ac.uk/261/1/art18.pdf

- (1999). The Diophantine equations
- Chabauty methods using covers on curves of genus 2.
- KASH-based program for performing 2-descent on elliptic curves over number fields. http://www.math.uu.nl/people/bruin/ell.shar
- When Newton met Diophantus: A study of rational-derived polynomials and their extension to quadratic fields.
- Local Fields. LMS–ST 3.
- (1996). Prolegomena to a Middlebrow Arithmetic of
- Sur les points rationnels des varie´te´s alge´briques dont l’irre´gularite´ est supe´rieure a` la dimension.
- (1985). Effective Chabauty,
- (1999). Computing the p-Selmer group of an elliptic curve. Manuscript
- (1997). A flexible method for applying chabauty’s theorem.
- (1997). Canonical heights on the Jacobians of curves of genus 2 and the infinite descent.
- (1997). Cycles of quadratic polynomials and rational points on a genus-two curve.
- (1999). Finding Rational Points on Bielliptic Genus 2 Curves.
- On Q-Derived Polynomials.
- (2001). Covering Collections and a Challenge Problem of Serre. Acta Arithmetica XCVIII.2:197–205,
- (1994). On the method of Coleman and Chabauty.
- (1998). Arithmetic properties of periodic points of quadratic maps,
- (1998). Computing a Selmer group of a Jacobian using functions on the curve.
- Lectures on the Mordell-Weil Theorem Transl. and ed. by Martin Brown. From notes by Michel Waldschmidt.
- (1982). Books IV to VII of Diophantus’ Arithmetica in the Arabic Translation attributed to Qusta ibn Luqa.
- (1995). Infinite descent on elliptic curves.
- The Arithmetic of Elliptic Curves.
- On the height constant for curves of genus two.
- Implementing 2-descent for Jacobians of hyperelliptic curves,
- (2000). On the height constant for curves of genus two,
- (1997). Bounding the Number of Rational Points on Certain Curves of High Rank.
- (1924). On the Solution of a Pair of Simultaneous Diophantine Equations Connected with the Nuptial Numbers of Plato.

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.