The modular degree conjecture over totally real fields

Let $F$ be a totally real number field of degree $n$ over $\mathbb{Q}$, and let $E/F$ be an elliptic curve modular of Shimura level $1$. Let $\delta_E$ be the relevant degree of a modular parametrization, let $d_F$ denote the discriminant of $F$, and let $\mathfrak{N}$ be the conductor ideal of $E$. Modular degree conjecture over totally real fields. There is a constant $\kappa$ depending only on $n$ such that

$$\log\delta_E<\kappa\cdot\log\bigl(d_F\operatorname{N}(\mathfrak{N})\bigr).$$

The conjecture is motivated by the preceding theorem bounding the Faltings height in terms of the modular degree. The paper establishes related unconditional estimates but leaves this asymptotic modular-degree bound open.