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Mahler’s conjecture and other transcendence Results

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1752)

Abstract

The first transcendental result about values of the modular invariant j (“if τ is algebraic but not imaginary quadratic then j(τ) is transcendental”) has been proved in 1937 by Th. Schneider, by means of elliptic functions. To be precise, let ℘ be a Weierstrass elliptic function with algebraic invariants; if (ω1ω2) is a basis of periods then ω12 is either quadratic or transcendental. This is transcribed in the above mentioned result on j. There are other translations of this kind; for example :

  • We know that ω1/π is transcendental (Th. Schneider); this gives a modular consequence : if j(τ) ∉ {0, 1728} then j (τ) and j’(τ) /π are not simultaneously algebraic.

  • We know that ω1, is transcendental (Th. Schneider); consequently if τ is quadratic with j (τ) ∉ {0, 1728} then j’(τ) is transcendental.

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© 2001 Springer-Verlag Berlin Heidelberg

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(2001). Mahler’s conjecture and other transcendence Results. In: Nesterenko, Y.V., Philippon, P. (eds) Introduction to Algebraic Independence Theory. Lecture Notes in Mathematics, vol 1752. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44550-1_2

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  • DOI: https://doi.org/10.1007/3-540-44550-1_2

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-41496-4

  • Online ISBN: 978-3-540-44550-0

  • eBook Packages: Springer Book Archive