On Faltings' Proof of the Mordell Conjecture

Abstract

In this paper we give an exposition of Gerd Faltings' 1983 proof of the Tate and Shafarevich conjectures for abelian varieties over a number field, along with their corollary, Mordell's conjecture. This is already accomplished in the volume of Silverman-Cornell. However, the author finds this text to be opaque in some places for (1) intuition for theorems/propositions and (2) the actual arguments taking place in Faltings' proof. Working out these details is undoubtedly a worthy exercise for the AG-inclined graduate student, but we wish to make the content within approachable for undergraduates. Our focus is on the theory of abelian varieties and group schemes, with approachable references (in the author's opinion) for the black boxes of moduli stacks, étale cohomology, and the Hodge-Tate decomposition.