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.)
Best Answer
The "function field analogues" of Faltings' theorem were proved by Manin, Grauert and Samuel: see
http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1966__29_/PMIHES_1966__29__55_0/PMIHES_1966__29__55_0.pdf
especially Theorem 4. (The quotation marks above are because all of this function field work came first: the above link is to Samuel's 1966 paper, whereas Faltings' theorem was proved circa 1982.)
The statement is the same as the Mordell Conjecture, except that there is an extra hypothesis on "nonisotriviality", i.e., one does not want the curve have constant moduli. For some discussion on why this hypothesis is necessary, see e.g. p. 7 of
http://alpha.math.uga.edu/~pete/hassebjornv2.pdf
An effective height bound in the function field case is given in Corollaire 2, Section 8 of
Szpiro, L.(F-PARIS6-G) Discriminant et conducteur des courbes elliptiques. (French) [Discriminant and conductor of elliptic curves] Séminaire sur les Pinceaux de Courbes Elliptiques (Paris, 1988). Astérisque No. 183 (1990), 7--18.
Note that effectivity on the height is much better than effectivity on the number of rational points (Faltings' proof does give the latter). This is not to be confused with uniform bounds on the number of rational points, for which I believe there are only conditional results known in any case.