A Geometric Proof of Mordell’s Conjecture for Function Fields

Let C, C be curves over a base scheme S with g(C) ≥ 2. Then the functor T 7→ {generically smooth T -morphisms T ×S C ′ → T ×S C} from ((Sschemes)) to ((sets)) is represented by a quasi-finite unramified S-scheme. From this one can deduce that for any two integers g ≥ 2 and g, there is an integer M(g, g) such that for any two curves C,C over any field k with g(C) = g, g(C) = g, there are at most M(g, g) separable k-morphisms C → C. It is conjectured that the arithmetic function M(g, g) is bounded by a linear function of g.