Galois Covers and the Hilbert-Grunwald Property

Abstract

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Specialization results and ramification conditions

“…In this unramified context, similar more precise conclusions are given in the two papers [DG12] and [DG11] of Dèbes and Ghazi: they have some additional control on the decomposition groups. As shown in §3.3, it is in fact possible to conjoin their statement and theorem 1 to obtain, for any finite group G which occurs as the Galois group of a Galois extension E/Q(T ) with E/Q regular, a general existence result of Galois extensions of Q of group G with specified local behavior (ramified or unramified).…”

“…This has been already investigated in [Dèb99], [DG12], [DG11] (and also in [DL13] and [DL12] in the non Galois case) for any base field k and positive answers have been given over various fields such as PAC fields, finite fields or complete valued fields. Recall for example that, given a PAC 2 field k, any Galois extension F/k of group G is the specialization E t 0 /k at t 0 of any Galois extension E/k(T ) with group G and E/k regular for infinitely many distinct points t 0 ∈ P 1 (k).…”

“…As already noted in remark 3.2, theorem 3.1 also includes trivial ramification. Previous works, namely [DG11] and [DG12], are concerned with this kind of conclusions: Dèbes and Ghazi show that, for each finite group G, any Galois extension E/Q(T ) of group G such that E/Q is regular has specializations with the same group which each is unramified at any finitely many prescribed large enough primes and such that the associated Frobenius at each such prime is in any specified conjugacy class of G.…”

“…Proof of theorem 3.8. We first recall how [DG12] handles condition (3). Fix a prime p ∈ S ur and an element g p ∈ C p .…”

“…Let ϕ : G Qp → g p be the corresponding epimorphism. Since p ≥ r 2 |G| 2 and p is a good prime for E/Q(T ), [DG12] provides some integer θ p such that, for each integer t ≡ θ p mod p, t is not a branch point and the specialization (EQ p ) t /Q p corresponds to the epimorphism ϕ.…”