The Mordell conjecture (proved by Faltings (2x), Vojta and Bombieri).
The Weil conjectures (proved by Deligne (2x)).
The Theorem of Roth (proved by Roth and Faltings).
References:
Faltings, Gerd (1983). "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern" [Finiteness theorems for abelian varieties over number fields]. Inventiones Mathematicae (in German) 73 (3): 349–366. doi:10.1007/BF01388432. MR 0718935.
Faltings, Gerd (1984). "Erratum: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern". Inventiones Mathematicae (in German) 75 (2): 381. doi:10.1007/BF01388572. MR 0732554.
Faltings, Gerd (1991). "Diophantine approximation on abelian varieties". Ann. of Math. 133 (3): 549–576. doi:10.2307/2944319. MR 1109353.
Faltings, Gerd (1994). "The general case of S. Lang's conjecture". In Cristante, Valentino; Messing, William. Barsotti Symposium in Algebraic Geometry. Papers from the symposium held in Abano Terme, June 24–27, 1991. Perspectives in Mathematics. San Diego, CA: Academic Press, Inc. ISBN 0-12-197270-4. MR 1307396.
Bombieri, Enrico (1990). "The Mordell conjecture revisited". Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17 (4): 615–640. MR 1093712.
Vojta, Paul (1991). "Siegel's theorem in the compact case". Ann. of Math. 133 (3): 509–548. doi:10.2307/2944318. MR 1109352.
Deligne, Pierre (1974), "La conjecture de Weil. I", Publications Mathématiques de l'IHÉS (43): 273–307, ISSN 1618-1913, MR 0340258
Deligne, Pierre (1980), "La conjecture de Weil. II", Publications Mathématiques de l'IHÉS (52): 137–252, ISSN 1618-1913, MR 601520
Bombieri, Gubler, Heights in Diophantine Geometry, http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511542879
Faltings, Wüstholz, Diophantine approximations on projective spaces, http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002111748