An effective proof of the hyperelliptic Shafarevich conjecture and applications

Abstract: The purpose of this thesis is to combine the theory of logarithmic forms with geometric tools to deduce new results in Diophantine geometry. Let \(K\) be a number field and let \(S\) be a finite set of places of \(K\).
     We first prove an effective Shafarevich theorem for elliptic curves. It gives an effectively determinable Dedekind domain \(R\subset K\) and an effective constant \(\Omega\), depending only on \(K\) and \(S\), such that for each elliptic curve \(E\) defined over \(K\) with good reduction outside \(S\) there is a globally minimal Weierstrass model of \(E\) over \(\text{Spec}\;(R)\) with height bounded by \(\Omega\). This chapter 1 is joint work with Professor Gisbert Wüstholz and Clemens Fuchs.
     In chapter 2 we introduce a new method to generalize and improve the results of the first chapter. Let \(C\) be an arbitrary hyperelliptic curve of genus \(g\geq 1\) defined over \(K\) with good reduction outside \(S\). We show that \(C\) has a Weierstrass scheme over the ring of integers of \(K\), arising from a hyperelliptic equation for \(C\) with height effectively bounded in terms of \(g\), \(S\) and \(K\). Then we give a new interpretation of this effective Shafarevich theorem for hyperelliptic curves in terms of bad reduction which will be the main tool to deduce the Diophantine applications of the last chapter.
     In chapter 3 we generalize Szpiro’s famous Discriminant Conjecture for elliptic curves over \(K\) to arbitrary hyperelliptic curves \(C\) over \(K\) and we give an effective proof of an exponential version of the generalized conjecture. Then we interpret these results in terms of Arakelov theory and we get also some applications in the theory of geometric Mumford discriminants and minimal regular models respectively. Furthermore, we generalize the Height Conjecture of Frey for elliptic curves to general hyperelliptic curves over \(K\) with a \(K\)-rational Weierstrass point and we prove an effective exponential version of this generalized conjecture. As an application we get that the elliptic \(\mathbb{Q}\)-factors of modular Jacobian’s can be determined effectively.