De Franchis + Kodaira-Parshin + Shafarevich ===> Mordell

I will explain and sketch the proof of a trick due to Kodaira-Parshin which, roughly speaking, takes as input sections of a family of curves, and gives as output ramified coverings of the generic fiber of that family. Then I will prove a classical theorem of De Franchis to the effect that a curve of genus at least 2 admits only finitely many dominant maps from a fixed source curve. Finally, I'll explain how these facts combine to prove the implication "Shafarevich's conjecture implies Mordell's conjecture".