Chapter 5 is devoted to giving a detailed proof of Faltings’s theorem (Theorem 5.1), asserting that "any algebraic curve of genus at least two over a number field has only finitely many rational points." We begin by recalling the proof of Liouville’s inequality in classical Diophantine approximation, and compare it with the proof of Faltings’s theorem, that is, Bombieri's version of Vojta's proof. We then explain how to reduce Faltings’s theorem to Vojta’s inequality (Theorem 5.2) and to Theorems 5.4, 5.5, and 5.6, which respectively assert the existence of sections of Vojta divisors having small heights, an upper bound for the indices of the small sections, and a lower bound for the indices of the small sections. We give proofs of Theorems 5.4, 5.5, and 5.6, which completes the proof of Faltings’s theorem. Finally, we give an easy application of Faltings’s theorem to Fermat curves.

Chapter 3 is devoted to the theory of heights, which is fundamental in Diophantine geometry. We explain archimedean and nonarchimedean absolute values on a number field, and prove the product formula. We define the absolute (logarithmic) Weil height. We explain heights associated to line bundles, and prove Northcott’s finiteness theorem. In the latter part of Chapter 3, we briefly introduce abelian varieties and some properties of line bundles on abelian varieties, such as the seesaw theorem and the theorem of cube. We define Neron–Tate height pairings on abelian varieties. We introduce Jacobian varieties and the Abel–Jacobi maps. We prove the Hermite–Minkowski theorem and the Mordell–Weil theorem.

Chapter 4 is devoted to several fundamental results of Diophantine geometry such as Siegel's lemma (Lemma 4.1 and Proposition 4.3) and Roth's lemma (Theorem 4.20). Besides them, we also introduce Guass’s lemma, the Mahler measure, the height of a polynomial, Gelfond’s inequality, the index with respect to a weight, the Wronskian, the norm of an invertible sheaf, the height of a norm and the local Eisenstein theorem. We will use them in Chapter 5. Because our purpose is to give a proof of Faltings's theorem in not too many pages, we touch on only the essential results of Diophantine geometry.

Chapter 1 is a gentle introduction of the Mordell conjecture for beginners ofDiophantine geometry. We explain what the Mordell conjecture is, its brief history and its importance in current mathematics.

In Chapter 2, we explain some of the basics of algebraic number theory, which we will need in Chapter 3 to introduce the theory of heights and to give a proof of the Mordell-Weil theorem. We begin by introducing the trace and the norm of an element of a finite extention field. We show the existence of an integral basis for a ring of integers and define the discriminant of a number field. After showing the existence of a prime factorization of a fractional ideal of a ring of integers (Theorems 2.5 and 2.6), we prove Minkowski's convex body theorem (Theorem 2.9) and Minkowski's discriminant theorem (Theorem 2.13). Finally, we introduce the notions of the ramification index and the residue degree at a prime ideal of an extension field. We define the difference of a number field, and explain several results relating the discriminant, the difference, and the ramifications of prime ideals (Lemma 2.17 and Theorem 2.18).