On the local behaviour of specializations of function field extensions

2017/09/10 by König, Joachim, Legrand, François, Neftin, Danny
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1709.03094

Abstract

Given a field k of characteristic zero and an indeterminate T over k, we investigate the local behaviour at primes of k of finite Galois extensions of k arising as specializations of finite Galois extensions E/k(T) (with E/k regular) at points t₀ ∈ ℙ¹(k). We provide a general result about decomposition groups at primes of k in specializations, extending a fundamental result of Beckmann concerning inertia groups. We then apply our result to study crossed products, the Hilbert--Grunwald property, and finite parametric sets.

Related

• Specialization results and ramification conditions
• Hilbert specialization results with local conditions
• The Hilbert-Grunwald specialization property over number fields
• The Grunwald problem and specialization of families of regular Galois extensions
• Specialization results in Galois theory