# Moscow Mathematical Journal

Volume 21, Issue 4, October–December 2021  pp. 789–805.

On Ennola's Conjecture on Non-Galois Cubic Number Fields with Exceptional Units

Authors:  Stéphane R. Louboutin (1)
Author institution:(1) Aix Marseille Université, CNRS, Centrale Marseille, I2M, Marseille, France

Summary:

Let $\varepsilon$ be a non-Galois totally real cubic special unit, i.e., a unit such that $\varepsilon-1$ is also a unit. Then $\varepsilon$ and $\varepsilon-1$ are multiplicatively independent and the unit index $j_\varepsilon$ of the groups of units generated by $-1$, $\varepsilon$ and $\varepsilon-1$ in the group of units of the ring of algebraic integers of ${\mathbb Q}(\varepsilon)$ is finite. It is known that $\{\varepsilon,\varepsilon-1\}$ is a system of fundamental units of the cubic order ${\mathbb Z}[\varepsilon]$. V. Ennola conjectured that $\{\varepsilon,\varepsilon-1\}$ is always a system of fundamental units of the maximal order of ${\mathbb Q}(\varepsilon)$, i.e., that $j_\varepsilon$ is always equal to $1$. Fix an algebraic closure of ${\mathbb Q}$. We prove that for any given prime $p$ there are only finitely many cases for which $p$ divides $j_\varepsilon$. We explain how this result makes Ennola's conjecture very reasonable for its possible exceptions would be few and far between. Our proof is conditional: we conjecture that the degrees of some explicit rational fractions that clearly are Laurent polynomials are always negative and given by conjectured explicit formulas. These degrees being easy to compute by using any formal language for algebraic computation, we checked enough of them to obtain that for any given prime $p\leq 1875$ there are only finitely many cases for which $p$ divides $j_\varepsilon$. We also prove that under the assumption of the ABC conjecture there are only finitely many exceptions to Ennola's conjecture.

2020 Math. Subj. Class. 11R16, 11R27.

Keywords: Units, exceptional units, cubic number fields, unit index, Thue's lemma, The abc conjecture.