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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.

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This chapter gives a sample of work done on prime gaps spanning more than half of the twentieth century, showing the importance of the problem and giving a context for what was to follow. Erdős proved in 1940 that there are an inﬁnite number of consecutive primes strictly less than the average gap. His proof is given in Section 3.3. Section 3.4 gives the theorem of Bombieri and Davenport showing that there are an infinite number of primes less than half the average gap. Their proof uses many advanced methods and tools with a fundamental lemma of 10 steps. Section 3.5 gives Granville and Soundararajan’s version of the work of Maier, which has applications to primes in intervals as well as prime gaps.

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The Mordell Conjecture

3 - Early Work

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The Mordell Conjecture

3 - Early Work

Summary