S -Unit Equations, Binary Forms and Curves of Genus 2
NP Smart · Proceedings of the London Mathematical Society · 1997
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 Györy. 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.