One of the most famous results of 20th century mathematics is Wiles' proof of Fermat's Last Theorem, which states that the only rational solutions to xⁿ+yⁿ=zⁿ for n>2 satisfy xyz=0. Thus there are at most 3 points on such a curve in ℙⁿ_ℚ. In 1922, Mordell conjectured that a similar result holds in general: if we take a complete smooth curve C of genus g>2 over a finite extension K of ℚ, then the set C(K) of K−rational points is finite.

We will cover Gerd Faltings' proof of Mordell's conjecture, now known as Faltings' Theorem. In broad strokes, the K−rational points on the curve C correspond to abelian varieties over K of specified dimension and bad reduction. We discuss how to show only finitely many isomorphism classes of these abelian varieties exist; this generalizes a known result for global fields of prime characteristic to characteristic zero.