Algebraic and Arithmetic Geometry Seminar

# On reduction steps for Leopoldt's conjecture

## by Fabio Ferri (Università di Exeter)

Europe/Rome
Aula Magna (Department of Mathematics)

### Aula Magna

#### Department of Mathematics

Description

Let $p$ be a rational prime and let $L/K$ be a Galois extension of number fields with Galois group $G$. Under some hypotheses, we show that Leopoldt's conjecture at $p$ for certain proper intermediate fields of $L/K$ implies Leopoldt's conjecture at $p$ for $L$; a crucial tool will be the theory of idempotent relations in $\mathbb Q[G]$. We also consider relations between the Leopoldt defects at $p$ for intermediate extensions of $L/K$. We finally show that our results combined with some techniques introduced by Buchmann and Sands allow us to find infinite families of nonabelian totally real Galois extension of $\mathbb Q$ satisfying Leopoldt's conjecture for certain primes. This is joint work with Henri Johnston.