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Arithmetic on curves. (English) Zbl 0593.14021

The aim of this survey article is to discuss the recent progress made in the arithmetic of curves by Faltings. The ideas involved are explained without requiring substantial background in algebra, number theory, or algebraic geometry. In contrast to most other survey articles on the subject this one provides an extended introduction for non-experts, starting with an account of the attempts so far for a systematic study of Diophantine problems. This goes on with a discussion of properties of algebraic curves leading to Faltings’ theorem that Mordell’s conjecture is true. The next section treats Jacobians and Abelian varieties. In the following section on ”Classification of families with bounded bad reduction” Shafarevich’s conjecture is discussed and how the latter implies Mordell’s conjecture via Parshin’s construction. Finally the important role of height of abelian varieties is explained in particular how they allow to make ”finiteness statements” leading to the proof of the conjectures of Tate, Shafarevich and Mordell.
Reviewer: C.-G.Schmidt

MSC:

14H25 Arithmetic ground fields for curves
14K15 Arithmetic ground fields for abelian varieties
01A65 Development of contemporary mathematics
14H52 Elliptic curves
11D99 Diophantine equations
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