Siegel's Theorem and the Shafarevich Conjecture

TL;DR

This paper explores the relationship between the effective Shafarevich conjecture for hyperelliptic Jacobians and Siegel's theorem on integral points, highlighting implications for number theory and algebraic geometry.

Contribution

It demonstrates that extending the effective Shafarevich conjecture to Jacobians would lead to an effective version of Siegel's theorem for hyperelliptic curves.

Findings

Effective computation of hyperelliptic curves with good reduction outside S

Extension of the Shafarevich conjecture implies an effective Siegel's theorem

Connections between hyperelliptic Jacobians and integral points

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.