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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.

Let 𝕀 denote an imaginary quadratic field or the field ℚ of rational numbers and let ℤ𝕀 denote its ring of integers. We shall prove a new explicit Baker-type lower bound for a ℤ𝕀-linear form in the numbers 1, eα1 , . . . , eαm, m ⩾ 2, where α0 = 0, α1, . . . , αm are m + 1 different numbers from the field 𝕀. Our work gives substantial improvements on the existing explicit versions of Baker’s work about exponential values at rational points. In particular, dependencies on m are improved.

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4 - Preliminaries for the Proof of Faltings’s Theorem

Summary

An explicit Baker-type lower bound of exponential values

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2 results

4 - Preliminaries for the Proof of Faltings’s Theorem

Summary

An explicit Baker-type lower bound of exponential values