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The Mordell conjecture (Faltings's theorem) is one of the most important achievements in Diophantine geometry, stating that an algebraic curve of genus at least two has only finitely many rational points. This book provides a self-contained and detailed proof of the Mordell conjecture following the papers of Bombieri and Vojta. Also acting as a concise introduction to Diophantine geometry, the text starts from basics of algebraic number theory, touches on several important theorems and techniques (including the theory of heights, the Mordell-Weil theorem, Siegel's lemma and Roth's lemma) from Diophantine geometry, and culminates in the proof of the Mordell conjecture. Based on the authors' own teaching experience, it will be of great value to advanced undergraduate and graduate students in algebraic geometry and number theory, as well as researchers interested in Diophantine geometry as a whole.

Mordell conjecture.

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Model theory --- Arithmetical algebraic geometry --- Mordell conjecture --- Algebraic geometry [Arithmetical] --- Arithmetic algebraic geometry --- Conjecture, Mordell --- Diophantine geometry --- Geometry [Arithmetical algebraic] --- Geometry [Diophantine] --- Géométrie algébrique arithmétique --- Modeles [Theorie des ] --- Modellentheorie --- Mordell's conjecture --- Rekenkundige algebraïsche geometrie --- Rekenkundige algebraïsche meetkunde --- 512.75 --- 51 --- Curves, Algebraic --- Logic, Symbolic and mathematical --- Algebraic geometry, Arithmetical --- Geometry, Arithmetical algebraic --- Geometry, Diophantine --- Number theory --- 51 Mathematics --- Mathematics --- 512.75 Arithmetic problems of algebraic varieties. Rationality questions. Zeta-functions --- Arithmetic problems of algebraic varieties. Rationality questions. Zeta-functions --- Model theory. --- Arithmetical algebraic geometry. --- Mordell conjecture. --- Théorie des modèles --- 51 Wiskunde. Mathematiek --- Wiskunde. Mathematiek --- Algebraic geometry. --- Mathematical logic. --- Number theory. --- Algebraic Geometry. --- Mathematical Logic and Foundations. --- Number Theory. --- Number study --- Numbers, Theory of --- Algebra --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Algebraic geometry --- Geometry