Klein forms and the generalized superelliptic equation

• 成果类型: Article
• 署名作者: Bennett, Michael A.; Dahmen, Sander R.
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2013.177.1.4
• 发表日期: 2013
• 页码: 171-239
• 关键词: cubic thue inequalities; elliptic-curves; REPRESENTATIONS; number; families; points; FIELDS

摘要

If F(x, y) is an element of Z[x, y] is an irreducible binary form of degree k >= 3, then a theorem of Darmon and Granville implies that the generalized superelliptic equation F(x, y) = z(l) has, given an integer l >= max{2, 7 - k}, at most finitely many solutions in coprime integers x, y and z. In this paper, for large classes of forms of degree k = 3, 4, 6 and 12 (including, heuristically, most cubic forms), we extend this to prove a like result, where the parameter l is now taken to be variable. In the case of irreducible cubic forms, this provides the first examples where such a conclusion has been proven. The method of proof combines classical invariant theory, modular Galois representations, and properties of elliptic curves with isomorphic mod n Galois representations.