Modularity and effective Mordell I

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 étale cover dominating $C_1$.)

~20 page main body, ~5 page (superfluous) appendix. Comments (and especially complaints) always welcome! Enjoy. [v2: added a citation and fixed some typos.]

References in corpus (2)

• On two dimensional weight two odd representations of totally real fields
• Finite descent obstruction for Hilbert modular varieties