Variations on the Six Exponentials Theorem

Abstract

According to the Four Exponentials Conjecture, a 2 × 2 matrix whose entries λ _ij (i = 1,2, j = 1,2) are logarithms of algebraic numbers is regular, as soon as the two rows as well as the two columns are linearly independent over the field ℚ of rational numbers. The question we address is as follows: are the numbers

$${\lambda _{12}} - \left( {{\lambda _{11}}{\lambda _{22}}/{\lambda _{21}}} \right),\quad \left( {{\lambda _{11}}{\lambda _{22}}} \right)/\left( {{\lambda _{12}}{\lambda _{21}}} \right),\quad \left( {{\lambda _{12}}/{\lambda _{11}}} \right) - \left( {{\lambda _{22}}/{\lambda _{21}}} \right)$$

and

$${\lambda _{11}}{\lambda _{22}} - {\lambda _{21}}{\lambda _{12}}$$

transcendental?

Denote by $\tilde{\mathcal{L}}$ the set of linear combination, with algebraic coefficients, of 1 and logarithms of algebraic numbers. A strong form of the Four Exponentials Conjecture states that a 2 × 2 matrix whose entries are in $\tilde{\mathcal{L}}$ is regular, as soon as the two rows as well as the two columns are linearly independent over the field $\bar{\mathbb{Q}}$ of algebraic numbers. Prom this conjecture follows a positive answer (apart from trivial cases) to the previous question for the first three numbers: not only they are transcendental, but, even more, they are not in the set $\tilde{\mathcal{L}}$. This Strong Four Exponentials Conjecture does not seem sufficient to settle the question for the last number, which amounts to prove that a 3 × 3 matrix is regular; the Conjecture of algebraic independence of logarithms of algebraic numbers provides the answer.

The first goal of this paper is to give the state of the art on these questions: we replace the Strong Four Exponentials Conjecture by the Strong Six Exponentials Theorem of D. Roy; we deduce that in a set of 2 numbers, at least one element is not in $\tilde{\mathcal{L}}$ (and therefore is transcendental). The second goal is to replace the Conjecture of algebraic independence of logarithms by the Linear Subgroup Theorem; we obtain partial results on the non existence of quadratic relations among logarithms of algebraic numbers. The third and last goal is to consider elliptic analogs of these statements.

An appendix by Hironori Shiga provides a link with periods of K3-surfaces.