Premium
S ‐Unit Equations, Binary Forms and Curves of Genus 2
Author(s) -
Smart NP
Publication year - 1997
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s002461159700035x
Subject(s) - mathematics , discriminant , jacobian matrix and determinant , degree (music) , genus , exposition (narrative) , binary number , pure mathematics , equivalence relation , table (database) , discrete mathematics , arithmetic , physics , botany , artificial intelligence , computer science , acoustics , biology , art , literature , data mining
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.