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
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.
Authors:
Stéphane R. Louboutin (1)
Author institution:(1) Aix Marseille Université, CNRS, Centrale Marseille, I2M, Marseille, France
Summary:
Keywords: Units, exceptional units, cubic number fields, unit index, Thue's lemma, The abc conjecture.
Contents
Full-Text PDF