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 t 0 ∈ P 1 (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. Recall that the latter is independent of the choice of P (up to k p -isomorphism). 1 2 JOACHIM KÖNIG, FRANÇ OIS LEGRAND, AND DANNY NEFTINExamples of Grunwald problems (G, (L (p) /k p ) p∈S ) with no solution L/k occur already for cyclic groups G, when S contains a prime of k lying over 2 [Wan48]. However, it is expected [Har07, §1] that, for solvable groups G, every Grunwald problem (G, (L (p) /k p ) p∈S ) has a solution, provided S is disjoint from some finite set S exc of "exceptional" primes of k, depending only on G and k. This is known when (1) G is abelian, and S exc is the set of primes of k dividing 2 [NSW08, (9.2.8)]; (2) G is an iterated semidirect product A 1 ⋊ (A 2 ⋊ · · · ⋊ A n ) of finite abelian groups, and S exc is the set of primes of k dividing |G|; see [Har07, Théorème 1] and [DLAN17, Theorem 1.1]; (3) G is solvable of order prime to the number of roots of unity in k, and S exc = ∅ [NSW08, (9.5.5)]; and (4) there exists a generic extension for G over k, and S exc = ∅ [Sal82, Theorem 5.9]. Among the above, the latter is the only method which applies to non-solvable groups. However, the family of non-solvable groups for which a generic extension is known is quite restricted, e.g., it is unknown whether the alternating group A n has a generic extension for n ≥ 6. See [JLY02] for an overview on generic extensions.The main source of realizations of non-solvable groups G over k is via k-regular Gextensions, that is, via G-extensions E/k(T ), where T is an indeterminate over k and k is algebraically closed in E. Indeed, by Hilbert's irreducibility theorem, every non-trivial k-regular G-extension E/k(T ) has infinitely many linearly disjoint specializations E t 0 /k, t 0 ∈ P 1 (k), with Galois group G. Many groups have been realized by this method; see, e.g., [MM99], and references within, as well as [Zyw14] for more recent examples.This specialization process provides a natural way to attack Grunwald problems for k-regular Galois groups, that is, for finite groups G admitting a k-regular G-extension of k(T ). Namely, given such an extension E/k(T ), it is natural to ask for the local behaviour of specializations E t 0 /k, t 0 ∈ P 1 (k). That is, which local extensions L (p) /k p , which local Galois groups Gal(L (p) /k p ), and which local degrees [L (p) : k p ] arise by completing the specialization E t 0 /k at primes p of k, when t 0 runs over P 1 (k)? For points t 0 ∈ P 1 (k) which are p-adically far from branch points of E/k(T ), this approach was deeply investigated by Dèbes ...
We propose an approach for the computation of multi-parameter families of Galois extensions with prescribed ramification type. More precisely, we combine existing deformation and interpolation techniques with recently developed strong tools for the computation of 3-point covers. To demonstrate the applicability of our method in relatively large degrees, we compute several families of polynomials with symplectic Galois groups, in particular obtaining the first totally real polynomials with Galois group PSp 6 (2).Note: Supplementary data are contained in an extra file, available at:https://arxiv.org/src/1803.08778/anc/anc.txt
Given a number field k, we show that, for many finite groups G, all the Galois extensions of k with Galois group G cannot be obtained by specializing any given finitely many Galois extensions E/k(T ) with Galois group G and E/k regular. Our examples include abelian groups, dihedral groups, symmetric groups, general linear groups over finite fields, etc. We also provide a similar conclusion while specializing any given infinitely many Galois extensions E/k(T ) with Galois group G and E/k regular of a certain type, under a conjectural "uniform Faltings' theorem".1 4 In a recent paper of Neftin and the two authors, a completely different method is used to show that A n (n ≥ 4) has no finite 1-parametric set over k; see [KLN17, Corollary 7.3]. We also mention this weaker result (which can be used with simple or non-simple groups): any given non-trivial regular Galois group G over k is the Galois group of a k-regular Galois extension of k(T ) that is not parametric over k; see [Leg16a, Theorem 1.3] and [Kön17, Theorem 2.2]. 5 i.e., if there exists no k-regular Galois extension of k(T ) with Galois group G. 6 Replace T − t 0 by 1/T if t 0 = ∞. 7 One has r = 0 if and only if G is trivial.
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