4 - Preliminaries for the Proof of Faltings’s Theorem

Summary

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.