The problem
Claim: a smooth projective curve of genus g ≥ 2 over a number field k has finitely many k-rational points. (Equivalently via Weil: the set C(k) is finite for g ≥ 2.)
Mordell's Conjecture (Faltings' Theorem)
historicReferences
History & significance
Mordell stated it in his 1922 Cambridge paper proving the genus-1 case (finite generation of elliptic curve groups). Siegel's 1929 theorem handled integral points; Shafarevich's 1962 finiteness of good-reduction curves reframed the landscape; Parshin and Arakelov built the reduction bridges that made the height argument inevitable.
The resolution (human proof)
Proven by Gerd Faltings, 1983 (Annals), via heights on Jacobians and Semistable Reduction — Fields Medal 1986. The proof is INFECTIVE: it shows the rational points are finite but gives no bound, no method to find them. Effectiveness remains the live frontier: Chabauty–Coleman techniques, Kim's Selmer varieties, and the Mordell–Weil sieves attack individual curves (and Magellan-grade computations settle families), while a general effective Faltings is arguably the biggest outstanding gap in arithmetic geometry after BSD.