Faltings' theorem (Mordell conjecture)

Step 1 (from curves to abelian varieties). Attach to $C$ its Jacobian $J$, an abelian variety of dimension $g$ that contains $C$ (via the Abel–Jacobi embedding, once one rational point is fixed). A rational point of $C$ corresponds to a rational point of $J$, and Parshin's construction (1968) turns a hypothetical infinite sequence of distinct rational points on $C$ into infinitely many pairwise non-isomorphic abelian varieties of dimension $g$ defined over $\mathbb{Q}$, all with good reduction outside one fixed finite set of primes $S$ that depends only on $C$.

Step 2 (Shafarevich's finiteness conjecture). Shafarevich conjectured, for fixed $g$ and fixed finite $S$, that only finitely many isomorphism classes of principally polarized abelian varieties of dimension $g$ over $\mathbb{Q}$ have good reduction outside $S$. Faltings proves this using Arakelov theory: he constructs a height function on the moduli space of such abelian varieties (the Faltings height), shows this height changes in a controlled way under isogeny, and bounds it using the finiteness of number fields unramified outside $S$ together with estimates from the theory of heights and semistable reduction. Bounded height in a fixed-dimensional moduli space forces finiteness.

Step 3 (contradiction closes the argument). Step 1 produced infinitely many non-isomorphic abelian varieties under the false assumption that $C$ has infinitely many rational points; Step 2 shows that only finitely many such abelian varieties can exist. This contradiction is only avoided if the original assumption was wrong, so $C$ has only finitely many rational points: $|C(\mathbb{Q})| < \infty$.