Minhyong Kim's reply to a question John Baez once asked about the analogy between $\text{Spec } \mathbb{Z}$ and 3-manifolds contains the following snippet:

Finally, regarding the field with one element. I'm all for general theory building, but I think this is one area where having some definite problems in mind might help to focus ideas better. From this perspective, there are two things to look for in the theory of $\mathbb{F}_1$.1) A theory of differentiation with respect to the ground field. A well-known consequence of such a theory could include an array of effective theorems in Diophantine geometry, like an effective Mordell conjecture or the ABC conjecture. Over function fields, the ability to differentiate with respect to the field of constants is responsible for the

considerably stronger theorems of Mordell conjecture type[emphasis mine], and makes the ABC conjecture trivial.

What is the strongest such theorem? Does anyone have a reference? (A preliminary search led me to results that are too general for me to understand them. I'd prefer to just see effective bounds on the number of rational points on a curve of genus greater than 1 in the function field case.)