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

TL;DR

This paper explores how representation theory and explicit units can be used to relate Leopoldt's conjecture across intermediate fields in Galois extensions, providing new infinite families of fields where the conjecture holds.

Contribution

It establishes conditions under which Leopoldt's conjecture for intermediate fields implies it for larger fields, and constructs infinite families of totally real extensions satisfying the conjecture at multiple primes.

Findings

Leopoldt's conjecture at $p$ for intermediate fields implies it for the whole extension under certain conditions.

Existence of infinite families of totally real $S_3$-extensions where Leopoldt's conjecture holds at specified primes.

Relations between Leopoldt defects of intermediate extensions.

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 $p \in \mathcal{P}$.