Siegel's Theorem and the Shafarevich Conjecture

2011/09/28 by Aaron Levin, Levin, Aaron
Mathematics · #11G30 (Primary) 14K15 #14H40 (Secondary) #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1109.6070

openalex publication_date 2011/09/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is known that in the case of hyperelliptic curves the Shafarevich conjecture can be made effective, i.e., for any number field k and any finite set of places S of k, one can effectively compute the set of isomorphism classes of hyperelliptic curves over k with good reduction outside S. We show here that an extension of this result to an effective Shafarevich conjecture for Jacobians of hyperelliptic curves of genus g would imply an effective version of Siegel's theorem for integral points on hyperelliptic curves of genus g.

Citations

• Good Reduction of Abelian Varieties

Related

• An effective proof of the hyperelliptic Shafarevich conjecture
• Fiber products of hyperelliptic curves
• Uniformly effective Shafarevich Conjecture on families of hyperbolic curves over a curve with prescribed degeneracy locus
• Bogomolov's Conjecture for Hyperelliptic Curves over Function Fields
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