The Mathematical Institute, University of Oxford, Eprints Archive

Solving Diophantine problems on curves via descent on the jacobian

Flynn, E. V. (1995) Solving Diophantine problems on curves via descent on the jacobian. In: Number Theoretic and Algebraic Methods in Computer Science. World Scientific Publishing, River Edge, USA, pp. 65-68. ISBN 9-8102-2334-X

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Abstract

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:
(1) Attempt to find J(Q)/2J(Q) via descent on J, the Jacobian of C.
(2) Deduce generators for J(Q) via an explicit theory of heights.
(3). Apply local techniques to try to deduce C(Q) via an embedding of C(Q) inside J(Q).
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.

Item Type:Book Section
Uncontrolled Keywords:Descent; Jacobians; heights; local techniques.
Subjects:A - C > Algebraic geometry
H - N > Number theory
Research Groups:Number Theory Group
ID Code:268
Deposited By:E. Victor Flynn
Deposited On:12 Jul 2006
Last Modified:08 Oct 2012 14:34

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