Modularity and effective Mordell I

TL;DR

This paper provides an effective proof of Faltings' theorem for certain curves over odd-degree totally real fields, using new methods related to the Shafarevich conjecture for abelian varieties of -type, leading to explicit height bounds.

Contribution

It introduces an effective proof approach for Faltings' theorem in the context of Hilbert modular stacks over odd-degree totally real fields, with applications to explicit height bounds.

Findings

Effective height bounds for specific curves over totally real fields

Proof of the Shafarevich conjecture for -type abelian varieties over such fields

Extension of hyperbolic hyperelliptic curve properties over algebraic closures

Abstract

We give an effective proof of Faltings' theorem for curves mapping to Hilbert modular stacks over odd-degree totally real fields. We do this by giving an effective proof of the Shafarevich conjecture for abelian varieties of $\mathrm{GL}_2$-type over an odd-degree totally real field. We deduce for example an effective height bound for $K$-points on the curves $C_a : x^6 + 4y^3 = a^2$ ($a\in K^\times$) when $K$ is odd-degree totally real. (Over $\overline{\mathbb{Q}}$ all hyperbolic hyperelliptic curves admit an \'{e}tale cover dominating $C_1$.)