Modularity and effective Mordell I

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 $C_1$.)