Algebra & Number Theory

Tubular approaches to Baker's method for curves and varieties

Samuel Le Fourn

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Baker’s method, relying on estimates on linear forms in logarithms of algebraic numbers, allows one to prove in several situations the effective finiteness of integral points on varieties. In this article, we generalize results of Levin regarding Baker’s method for varieties, and explain how, quite surprisingly, it mixes (under additional hypotheses) with Runge’s method to improve some known estimates in the case of curves by bypassing (or more generally reducing) the need for linear forms in p-adic logarithms. We then use these ideas to improve known estimates on solutions of S-unit equations. Finally, we explain how a finer analysis and formalism can improve upon the conditions given, and give some applications to the Siegel modular variety A2(2).

Article information

Algebra Number Theory, Volume 14, Number 3 (2020), 763-785.

Received: 18 February 2019
Revised: 20 August 2019
Accepted: 7 October 2019
First available in Project Euclid: 2 July 2020

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 11G35: Varieties over global fields [See also 14G25]
Secondary: 11J86: Linear forms in logarithms; Baker's method

integral points Baker's method Runge's method $S$-unit equation


Le Fourn, Samuel. Tubular approaches to Baker's method for curves and varieties. Algebra Number Theory 14 (2020), no. 3, 763--785. doi:10.2140/ant.2020.14.763.

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