Finiteness Problems in Diophantine Geometry

TL;DR

This survey discusses Faltings' proof of key conjectures in Diophantine geometry, highlighting ideas, results, and historical context related to finiteness problems in the field.

Contribution

It provides an exposition of Faltings' proof and related results, offering historical insights and clarifications on foundational finiteness problems in Diophantine geometry.

Findings

Exposition of Faltings' proof of Shafarevich, Tate, and Mordell conjectures.

Historical overview of the development of finiteness results.

Clarifications and updates on the original exposition.

Abstract

This survey contains an exposition of ideas and results related to Faltings' proof of the conjectures of Shafarevich, Tate and Mordell. This paper originally appeared in 1986 as an Appendix to the Russian translation of Serge Lang, "Fundamentals of Diophantine Geometry" (Springer Verlag, 1983) published by "Mir", Moscow (MR0854670, 88a:11054). A history of the publication of the Appendix is briefly described by Lang in Section 4 of his paper "Mordell's review, Siegel's letter to Mordell, Diophantine geometry, and 20th century mathematics" that was published (in 1995) simultaneously in Notices of the AMS and Gazette des Math\'ematiciens (SMF) (MR1316025, 96g:11002a; MR1316133, 96g:11002b) http://smf.emath.fr/Publications/Gazette/1995/63/smf_gazette_63_17-36.pdf . Later an expanded version of the Appendix was translated into English by Neal Koblitz and published in 1989 by the American Mathematical Society as part of the collection "Eight papers translated from the Russian", AMS Translations, Series 2, Vol. 143 http://www.ams.org/bookstore-getitem/item=TRANS2-143 (MR1008476, 90b:00011). We put this paper on the arXiv with the kind permission of the American Mathematical Society. For this version we slightly updated the bibliography and added a few short notes (marked as "Added in December 2009"). We also corrected inaccuracies that were kindly pointed out to us by J.-P. Serre - one of the few people to read this paper (in English) twenty years ago.