Non-parametric sets of regular realizations over number fields

König, Joachim; Legrand, François

doi:10.1016/j.jalgebra.2017.11.023

Cited by 10 publications

(13 citation statements)

References 22 publications

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“…Non-existence of finite parametric sets. As in [KL18], given a finite group G, we call a set S of k-regular G-extensions of k(T ) parametric if every G-extension of k occurs as a specialization of some extension E/k(T ) in S; see Definition 7.1. If S consists of a single extension E/k(T ), the extension E/k(T ) is called parametric.…”

Section: 32mentioning

confidence: 99%

On the Local Behavior of Specializations of Function Field Extensions

König

Legrand

Neftin

2018

International Mathematics Research Notices

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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 ...

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Section: 32mentioning

confidence: 99%

On the Local Behavior of Specializations of Function Field Extensions

König

Legrand

Neftin

2018

International Mathematics Research Notices

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“…(b) By Corollary 3.5(c), if q 7 is a prime number, then no regular Z/qZ-extension of Q(T ) is parametric, under the abc-conjecture. The interest of this remark is that none of the methods from [26] and [27, § 7] applies to finite groups of prime order. 6 Indeed, at least one such branch point must exist since the inertia groups at branch points generate More generally, by the above, no regular G-extension of Q(T ) with r 7 branch points is parametric, under the abc-conjecture and, possibly, the lower bound (1.2).…”

Section: Explicit Examplesmentioning

confidence: 99%

“…, g s ). By (a) and as the uniformity conjecture holds, one may apply [26,Proposition 2.5] to get that there exists a positive constant B = B(|G/H |, g 0 ) such that, for each i ∈ {1, . .…”

Section: Variantsmentioning

confidence: 99%

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Density Results for Specialization Sets of Galois covers

König¹,

Legrand²

2019

J. Inst. Math. Jussieu

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We provide evidence for this conclusion: given a finite Galois cover f : X → P 1 Q of group G, almost all (in a density sense) realizations of G over Q do not occur as specializations of f . We show that this holds if the number of branch points of f is sufficiently large, under the abc-conjecture and, possibly, the lower bound predicted by the Malle conjecture for the number of Galois extensions of Q of given group and bounded discriminant. This widely extends a result of Granville on the lack of Q-rational points on quadratic twists of hyperelliptic curves over Q with large genus, under the abc-conjecture (a diophantine reformulation of the case G = Z/2Z of our result). As a further evidence, we exhibit a few finite groups G for which the above conclusion holds unconditionally for almost all covers of P 1 Q of group G. We also introduce a local-global principle for specializations of Galois covers f : X → P 1 Q and show that it often fails if f has abelian Galois group and sufficiently many branch points, under the abc-conjecture. On the one hand, such a local-global conclusion underscores the "smallness" of the specialization set of a Galois cover of P 1 Q . On the other hand, it allows to generate conditionally "many" curves over Q failing the Hasse principle, thus generalizing a recent result of Clark and Watson devoted to the hyperelliptic case.

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“…Yet, no such group was known before [KL18,KLN19], which provide many examples. But the question remains of the classification of all the finite groups G with a k-parametric polynomial…”

Section: Introductionmentioning

confidence: 99%

On parametric and generic polynomials with one parameter

Dèbes¹,

König²,

Legrand³

et al. 2021

Preprint

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Given fields k ⊆ L, our results concern one parameter L-parametric polynomials over k, and their relation to generic polynomials. The former are polynomialsof group G which parametrize all Galois extensions of L of group G via specialization of T in L, and the latter are those which are L-parametric for every field L ⊇ k. We show, for example, that being L-parametric with L taken to be the single field C((V ))(U ) is in fact sufficient for a polynomial P (T, Y ) ∈ C[T ][Y ] to be generic. As a corollary, we obtain a complete list of one parameter generic polynomials over a given field of characteristic 0, complementing the classical literature on the topic. Our approach also applies to an old problem of Schinzel: subject to the Birch and Swinnerton-Dyer conjecture, we provide one parameter families of affine curves over number fields, all with a rational point, but with no rational generic point.1 There are PAC fields k with R G (k) = ∅ for every finite group G, and so for which the k-parametricity property is not trivial as it is for algebraically closed fields. A concrete example (due to Pop) is Q tr ( √ −1).2. Preliminaries §2.1 and §2.2 are devoted to terminology and notation. In §2.3, we review the classification of all the finite groups with a one parameter generic polynomial with coefficients in a given field of characteristic zero (see Theorem 2.5). As said in §1, Corollary 3.6 will more precisely give the list of corresponding polynomials. Finally, in §2.4, we briefly discuss étale algebras, which will be used to prove Theorem 1.2 and its generalizations.2.1. Basic terminology. Let k be a field of characteristic zero and F/k(T ) a finite Galois extension. Say that F/k(T ) is k-regular if F ∩ k = k. By the branch point set of F/k(T ), we mean the finite set t of points t ∈ P 1 (k) such that the associated discrete va-

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Non-parametric sets of regular realizations over number fields

Cited by 10 publications

References 22 publications

On the Local Behavior of Specializations of Function Field Extensions

On the Local Behavior of Specializations of Function Field Extensions

Density Results for Specialization Sets of Galois covers

On parametric and generic polynomials with one parameter

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