The Mordell conjecture or Falting’s theorem is a statement about the finiteness of rational points on an algebraic curve over a number field of genus $g \gt 1$.
Its statement motivated the introduction of anabelian geometry (Grothendieck).
The Mordell conjecture completed a classification of the behavior of rational points on curves over $\mathbb{Q}$. For genus 0 one has no points, or something isomorphic to $\mathbb{P}^1$ and hence infinitely many. For genus 1, the Mordell-Weil theorem? tells us it is either empty or a finitely generated abelian group (sometimes finite, sometimes infinite). To finish off this type of classification the Mordell conjecture shows that only the finite case can occur for higher genus.
This result also implies many non-trivial results. For example, fix a finite set of primes $S$, a dimension $n$, and a polarization degree? $d$. There are only finitely many abelian varieties of dimension $n$ and polarization degree $d$ with bad reduction? inside $S$. This result is so interesting that a whole industry popped up asking what other varieties this type of behavior occurs for. It seems to work for K3s.
The Mordell conjecture is implied by the abc conjecture. (See there.)
The Mordell conjecture implies Tate's isogeny theorem?.
See also Vojta's conjecture.
The Mordell comnjecture originates in
It was proven in
Reviews include
Barry Mazur, Abelian varieties and the Mordell-Lang conjecture (pdf)
Enrico Bombieri, The Mordell conjecture revisited, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Sér. 4, 17 no. 4 (1990), p. 615-640 (Numdam)
Encyclopedia entries are in
Encyclopedia of Mathematics, Mordell conjecture
Wikipedia, Faltings’ theorem
The relation to anabelian geometry originates in