Mahler’s conjecture and other transcendence Results

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 ω₁/ω₂ 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.