Applications of representation theory and of explicit units to Leopoldt's conjecture

Abstract

Let $L/K$ be a Galois extension of number fields and let $G=\mathrm{Gal}(L/K)$. We show that under certain hypotheses on $G$, for a fixed prime number $p$, Leopoldt's conjecture at $p$ for certain proper intermediate fields of $L/K$ implies Leopoldt's conjecture at $p$ for $L$. We also obtain relations between the Leopoldt defects of intermediate extensions of $L/K$. By applying a result of Buchmann and Sands together with an explicit description of units and a special case of the above results, we show that given any finite set of prime numbers $\mathcal{P}$, there exists an infinite family $\mathcal{F}$ of totally real $S_{3}$-extensions of $\mathbb{Q}$ such that Leopoldt's conjecture for $F$ at $p$ holds for every $F \in \mathcal{F}$ and $p \in \mathcal{P}$.