The Thue-Siegel-Roth Theorem

Serge Lang
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Serge Lang: Yale University, Department of Mathematics

Chapter Chapter 7 in Fundamentals of Diophantine Geometry, 1983, pp 158-187 from Springer

Abstract: Abstract We shall give an exposition of this important theorem under axioms which are valid in number fields and function fields, namely, the product formula and a weak Riemann-Roch condition. We also need characteristic 0. In §1 we state the theorem. In §2 we reformulate it by a formal argument in a more manageable form (because we consider approximations at several absolute values). This should not be confused with the deep problem of simultaneous approximations at one absolute value solved by Schmidt. In §3 we give the proof, basing it on two propositions. We then prove Proposition 3.1 in §4 and §5, and Proposition 3.2 in §6, §7, §8 and §9. Finally, in §10, we give a geometric formulation adapted to the applications to algebraic geometry and possible generalizations to varieties of arbitrary dimensions.

Date: 1983
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DOI: 10.1007/978-1-4757-1810-2_7

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