Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-19T10:10:17.082Z Has data issue: false hasContentIssue false
  • English
  • Français

S-Unit equations, binary forms and curves of genus 2

Published online by Cambridge University Press:  01 September 1997

NP Smart
NP Smart
Affiliation:
University of Kent, Canterbury, UK Present address: Mathematics Group, Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol BS12 6QZ, UK. E-mail: [email protected]
Get access

Abstract

In this paper a practical algorithm is given to find all binary forms with rational coefficients of given degree with discriminant divisible by a given finite set of rational primes, up to an obvious equivalence relation.

This is done by adapting the finiteness result of Evertse and Gy\H{o}ry. A technical assumption that all fields used have class number $1$ is made to aid the exposition.

All binary forms of degree less than or equal to $6$ with $2$-power discriminant are then calculated. This is then used to complete the table of curves of genus $2$ which had previously been computed by Merriman and the author. Some ranks of the associated Jacobian varieties are also computed.

1991 Mathematics Subject Classification: primary 11E76, 11G30; secondary 11Y40, 11D61.

Keywords

binary forms$S$-unit equationscurves of genus 2computational number theory
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 75 , Issue 2 , September 1997 , pp. 271 - 307
Copyright
London Mathematical Society 1997

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)