nt.number theory - What does Faltings' theorem look like over function fields?

The "function field analogues" of Faltings' theorem were proved by Manin, Grauert and Samuel: see

29/PMIHES_1966_29_55_0/PMIHES_1966_29_55_0.pdf">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://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.