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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: N.P.Smart@ukc.ac.uk
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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

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