Rank–conductor conjecture for elliptic curves over global fields

Let $K$ be a global field with $\operatorname{char}(K)\neq 2,3$, let $E$ be an elliptic curve over $K$, and let $\mathscr{C}_E$ be its conductor. Write $\mathrm{N}(\mathscr{C}_E)$ for the norm of the conductor.

Rank–conductor conjecture. One has

$$\operatorname{rank} E=o(\log \mathrm{N}(\mathscr{C}_E))\quad\text{as }\mathrm{N}(\mathscr{C}_E)\to\infty.$$

This conjecture supplies a sublogarithmic upper bound for elliptic-curve ranks in terms of conductor norms and is used in the paper to derive consequences for upper bounds in Manin's conjecture for del Pezzo surfaces. The source states that it follows from the Birch and Swinnerton-Dyer conjecture, that 2-descent gives the weaker bound $\operatorname{rank} E=O(\log \mathrm{N}(\mathscr{C}_E))$, and that it is proven in characteristic greater than three.