- Email: [email protected]
Minimal fields of definition for Galois action
Journal of Pure and Applied Algebra 220 (2016) 3327–3331
Contents lists available at ScienceDirect
Journal of Pure and Applied Algebra www.elsevier.com/locate/jpaa
Minimal fields of definition for Galois action Hilaf Hasson Department of Mathematics, Stanford University, Palo Alto, CA 94305, USA
a r t i c l e
i n f o
Article history: Received 18 November 2014 Received in revised form 19 February 2016 Available online 9 March 2016 Communicated by R. Vakil
a b s t r a c t Let K be a field, and let f : X → Y be a finite étale cover between reduced and geometrically irreducible K-schemes of finite type such that fKs is Galois. Assuming ¯ → Y , we use it to analyze fields of definition over f admits a Galois K-form f¯ : X K for the Galois property of f and the presence of K-points in general K-forms f : X → Y over Y (K). Additionally, we show that if K is Hilbertian, the group G is non-abelian, and the base variety is rational, then there are finite separable extensions L/K such that some L-form of fL does not descend to a cover of Y . © 2016 Elsevier B.V. All rights reserved.
1. Introduction The focus of this paper is the descent of G-Galois covers. For a finite group G, a map of varieties (over a fixed field) is said to be a G-Galois cover if it is finite, étale, and G acts freely and transitively on its geometric fibers. (See Definition 2.1.) David Harbater and Kevin Coombes have made several observations in [1] about the Galois property of descents. Let K be a number field, let Y be a reduced geometrically irreducible K-scheme of finite type, fix a Galois finite étale surjection over YKs with Galois group G, and let M be its associated “field of moduli”. Under the additional assumption that Y (K) = ∅, [1, Proposition 2.5] shows that there is a descent to a cover of YM which is possibly not Galois; and [1, Proposition 2.7] shows that M is the intersection of the number fields F for which there is a descent to a Galois cover of YF . The setup of this paper will be slightly different. Let K be any field, and let f : X → Y be a finite étale surjection between reduced and geometrically irreducible K-schemes of finite type such that fKs is Galois with Galois group G. It is easy to show (see the beginning of Section 2) that there exists a unique minimal subfield E ⊂ Ks over K so that XE → YE is Galois. (Henceforth, this is called the minimal field of Galois action of X → Y .) The main theorems (Theorems 2.2, 2.3 and 2.5), shed light on how E is determined, ¯ →Y. under the assumption that there exists some Galois K-form X E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jpaa.2016.02.017 0022-4049/© 2016 Elsevier B.V. All rights reserved.
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In Proposition 4.1 we show that if K is Hilbertian, the group G is non-abelian, and the base variety is rational, then there exist finite separable extensions L/K such that some L-form of fL does not descend to a cover of Y . 2. Main theorems In this section, we state our main results, proved at the end of Section 3. Definition 2.1. A map f : X → Y between integral noetherian schemes is called a cover if it is a finite, étale surjection. Letting G := Aut(X/Y )opp , the fiber-degree is a constant d > 0 with |G| ≤ d, and |G| = d if and only if X is a right G-torsor over Y . In such cases we say that f is Galois (and G is naturally identified with the Galois group of the extension of function fields). For a finite group Γ, we say that f is a Γ-cover if it is Galois and an isomorphism Γ ∼ = Aut(X/Y )opp is specified. The notion of isomorphism for covers and Γ-covers is defined in the obvious manner. In what follows, we fix a reduced and geometrically irreducible K-scheme Y of finite type and a cover f : X → Y such that X is geometrically irreducible over K and fKs is Galois. Since X → Y is étale, the sheaf AutX/Y is representable by a group scheme over Y (also denoted AutX/Y ). Clearly, X → Y is a left AutX/Y -torsor, and (AutX/Y )YKs = AutXKs /YKs ∼ = Gopp is a constant YKs -group. Let the continuous homomorphism ρ : Gal(Ks /K) = Gal(Ks (Y )/K(Y )) → Aut(Gopp ) be the Galois action induced by AutX/Y as a YKs /Y -form of Gopp . Then, clearly, the splitting field E := (Ks )ker(ρ) of ρ is the unique minimal subfield of Ks containing K for which the base change XE → YE is Galois. We remark that E/K is Galois, and its Galois group is canonically isomorphic to a subgroup of Aut(Gopp ). ¯ → Y of f , and Theorem 2.2. With the notation as above, assume that there exists some Galois K-form X ¯ let P be a point such that XP (K) = ∅. Let T be the P -fiber XP viewed as a right G-torsor over Spec(K). Then the minimal field of Galois action for X → Y is contained in the splitting field of T . We may conclude from Theorem 2.2 that the minimal field of Galois action for X → Y is contained in the specialization at P of all Galois K-forms of f . The effect of changing the point P in Theorem 2.2 is expressed by the following result: Theorem 2.3. With the notation of Theorem 2.2, let Q be another point in Y (K). Then the fiber over Q in ¯ P and X ¯ Q are isomorphic as right G-torsors. X → Y has a K-rational point if and only if X Remark 2.4. A variant of Theorem 2.3 has been known before, and goes by the name the “Twisting Lemma” ([8, 2.2]; [4, Section 2]; see also Lemma 3.3). It has been applied in Galois-Theoretic contexts, most notably by Pierre Dèbes [2–6]. However, the Twisting Lemma is a bit weaker since it merely says that if f is Galois then it admits some K-form X → Y such that K-points in fibers over Y (K) can be detected by fibers of f as in Theorem 2.3. Finally, I will prove the following.
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Theorem 2.5. With the notation of Theorem 2.3, the following hold: 1. If X → Y is a K-form of f such that XP (K) = ∅, then it is isomorphic to X over Y . 2. The number of K-rational points in X lying above P is divisible by |Z(G)| and divides |G|. 3. Twisted covers Throughout this section, we continue to use the setup of Theorem 2.2. Definition 3.1. If Z → Y is a right G-torsor, and T → Spec(K) is a right G-torsor, then we define the right G-torsor Z T → Y as the finite étale Isom-scheme IsomY,G (TY , Z) over Y , classifying G-equivariant isomorphisms over Y -schemes. ¯ T → Y is a (not necessarily Galois) K-form of f . For any right G-torsor T over Spec(K), the map X More precisely: Lemma 3.2. With the notation of Definition 3.1, let F/K be the splitting field of T . Then (Z T )F is isomorphic to ZF over YF . Proof. The twisting construction is compatible with extension of the ground field, so we may rename F as K; i.e., we may assume T is the trivial G-torsor over K. But then it is obvious from the definition of Z T as an Isom-scheme that it is Y -isomorphic to Z. 2 The interest in twisting G-Galois covers by a right G-torsor arises from a property that the twisted cover satisfies, namely the first assertion in the following lemma. Lemma 3.3. 1. (The Twisting Lemma) Let T → Spec(K) be a right G-torsor. Then, in the notation of Theorem 2.2, ¯ is isomorphic as a right G-torsor to T if and only if the twisted cover X ¯T → Y the fiber over P in X has a K-rational point above P . ¯ T in the fiber over P is the size of the centralizer 2. In this situation, the number of K-rational points of X CGopp (Gal(F/K)) of Gal(F/K) in Gopp = AutG (TKs ), where F is the splitting field of T . ¯ T is compatible with base-change on Y over K, so in particular the P -fiber of Proof. The formation of X T ¯ ¯ ¯ T is equivalent to X is the twist of XP against T . Therefore the existence of a K-point in the P -fiber of X ¯ over P . The second assertion follows from the existence of an isomorphism between T and the fiber in X ∼ opp the fact that AutG (T ) = CG (Gal(F/K)). (Indeed, by Galois descent, the elements of AutG (TF ) ∼ = Gopp that descend to K are exactly those that commute with the action of Gal(F/K).) 2 ¯ → Y be a Galois K-form of f . Then for every K-form X → Y of f , there exists Proposition 3.4. Let f¯ : X ¯ T is isomorphic to X over Y . a right G-torsor T over Spec(K) so that X ˇ 1 (K, G). On the Proof. We will make the following two identifications for the (étale) Čech cohomology set H ¯ over Y is naturally identified with G, this set classifies isomorphism one hand, since the Galois group for X ¯ classes of K-forms of X → Y (since every descent datum is effective). On the other hand, as is well-known, this set also classifies isomorphism classes of G-torsors over K. Let T be the G-torsor over K that corresponds ¯ → Y . We shall now check that X ¯ T is isomorphic via the above identifications to the K-form X → Y of X to X over Y .
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¯ equipped with its It is easy to see that the right GY -torsor TY represents the Isom-functor IsomY (X , X) ¯ natural right GY -action through X, and so “evaluation” at points of X defines an evident map of functors ¯ But working over an étale cover of Y splitting these finite étale covers shows that X → IsomY,G (TY , X). ¯ T by definition, so we are done. 2 this latter map is an isomorphism. The target is X Now we can finally prove Theorems 2.2, 2.3 and 2.5: ¯ T is Proof of Theorems 2.2, 2.3 and 2.5. By Proposition 3.4, there exists a right G-torsor T so that X ¯ P )T = Isom (T , X ¯ P =: T as G-torsors since ¯ P ), so T X isomorphic to X over Y . Note that XP = (X G XP (K) is non-empty by hypothesis. Then Theorem 2.2 follows immediately from Lemma 3.2. Theorem 2.3 follows immediately from the Twisting Lemma (Lemma 3.3(1)) to the fiber at Q, and Theorem 2.5(2) follows immediately from Lemma 3.3(2), using that Z(Gopp ) is a subgroup of CGopp (H) for any subgroup H of Gopp . It remains to prove Theorem 2.5(1). Let x be a point in XP (K), and x a point in XP (K). In this ∼ situation, if there exists a Y -isomorphism X → X carrying x to x , then it is unique since X is irreducible. By the uniqueness and geometric irreducibility of X over K, Galois descent reduces our task to proving ¯ is a G-torsor such existence and uniqueness when K = Ks , as we now assume. In particular, now X = X over Y , so it is Galois. Our task is to prove that there is a unique Y -automorphism of X swapping a chosen pair of rational points in the fiber of P . That in turn is obvious since X is Galois over Y . 2 4. An obstruction to the descent of forms Proposition 4.1. Let K be a Hilbertian field, let G be a non-abelian finite group, and let Y be a rational variety over K. For any geometrically irreducible G-Galois cover E of YKs that descends to a cover of Y there exists a finite extension L/K and an L-descent X → YL of that cover such that it does not descend to a cover of Y . Proof. Let W be the compositum of all of the Galois field extensions of K in Ks having a Galois group that is isomorphic to a subgroup of Aut(Gopp ). The field W is clearly Galois over K, and is not equal to Ks . Let L be a nontrivial finite field extension of W . By Weissauer’s Theorem ([9, Satz 9.7]; see also [7, Theorem 13.9.1]), the field L is Hilbertian. Since, by the arguments in the beginning of Section 2, every descent of E → YKs to a cover of Y becomes Galois over some Galois extension of K with Galois group a subgroup of Aut(Gopp ), they all become Galois over L . In particular, there exists a G-Galois L -descent X → YL of E → YKs . Since L is Hilbertian and Y is rational, there exists a connected right G-torsor T over L achieved by specializing X at some L -point P . By Lemma 3.3, the number of L -rational points of X T in the fiber of P is |CGopp (Aut(F/L ))| = |Z(Gopp )|, where F is the splitting field of T . Since by assumption Z(Gopp ) is a proper subgroup of Gopp , it follows that X T is not Galois over YL , and therefore has no descent to a cover of Y . Finally, let L be some subfield of L , finite over K, to which X T → YL descends. 2 Acknowledgement The author would like to thank the anonymous reviewer for comments that were very helpful in improving this paper. References [1] Kevin Coombes, David Harbater, Hurwitz families and arithmetic Galois groups, Duke Math. J. 52 (4) (1985) 821–839. [2] Pierre Dèbes, Galois covers with prescribed fibers: the Beckmann–Black problem, Ann. Sc. Norm. Super. Pisa, Cl. Sci. 4 (28) (1999) 273–286.
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[3] Pierre Dèbes, On the Malle conjecture and the self-twisted cover, arXiv:1404.4074 [math.NT], 2014. [4] Pierre Dèbes, Nour Ghazi, Galois covers and the Hilbert–Grunwald property, Ann. Inst. Fourier (Grenoble) 62 (3) (2012) 989–1013. [5] Pierre Dèbes, François Legrand, Twisted covers and specializations, in: Galois–Teichmüller Theory and Arithmetic Geometry, in: Adv. Stud. Pure Math., vol. 63, Math. Soc. Japan, Tokyo, 2012, pp. 141–162. [6] Pierre Dèbes, François Legrand, Specialization results in Galois theory, Trans. Am. Math. Soc. 365 (10) (2013) 5259–5275. [7] Michael Fried, Moshe Jarden, Field Arithmetic, third ed., Ergeb. Math. Ser., vol. 11, Springer-Verlag, 2005. [8] Alexei Skorobogatov, Torsors and Rational Points, Cambridge Tracts Math., Cambridge University Press, Cambridge, 2001. [9] Rainer Weissauer, Der Hilbertsche Irreduzibilitätssatz, J. Reine Angew. Math. 334 (1982) 203–220.
Contents lists available at ScienceDirect
Journal of Pure and Applied Algebra www.elsevier.com/locate/jpaa
Minimal fields of definition for Galois action Hilaf Hasson Department of Mathematics, Stanford University, Palo Alto, CA 94305, USA
a r t i c l e
i n f o
Article history: Received 18 November 2014 Received in revised form 19 February 2016 Available online 9 March 2016 Communicated by R. Vakil
a b s t r a c t Let K be a field, and let f : X → Y be a finite étale cover between reduced and geometrically irreducible K-schemes of finite type such that fKs is Galois. Assuming ¯ → Y , we use it to analyze fields of definition over f admits a Galois K-form f¯ : X K for the Galois property of f and the presence of K-points in general K-forms f : X → Y over Y (K). Additionally, we show that if K is Hilbertian, the group G is non-abelian, and the base variety is rational, then there are finite separable extensions L/K such that some L-form of fL does not descend to a cover of Y . © 2016 Elsevier B.V. All rights reserved.
1. Introduction The focus of this paper is the descent of G-Galois covers. For a finite group G, a map of varieties (over a fixed field) is said to be a G-Galois cover if it is finite, étale, and G acts freely and transitively on its geometric fibers. (See Definition 2.1.) David Harbater and Kevin Coombes have made several observations in [1] about the Galois property of descents. Let K be a number field, let Y be a reduced geometrically irreducible K-scheme of finite type, fix a Galois finite étale surjection over YKs with Galois group G, and let M be its associated “field of moduli”. Under the additional assumption that Y (K) = ∅, [1, Proposition 2.5] shows that there is a descent to a cover of YM which is possibly not Galois; and [1, Proposition 2.7] shows that M is the intersection of the number fields F for which there is a descent to a Galois cover of YF . The setup of this paper will be slightly different. Let K be any field, and let f : X → Y be a finite étale surjection between reduced and geometrically irreducible K-schemes of finite type such that fKs is Galois with Galois group G. It is easy to show (see the beginning of Section 2) that there exists a unique minimal subfield E ⊂ Ks over K so that XE → YE is Galois. (Henceforth, this is called the minimal field of Galois action of X → Y .) The main theorems (Theorems 2.2, 2.3 and 2.5), shed light on how E is determined, ¯ →Y. under the assumption that there exists some Galois K-form X E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jpaa.2016.02.017 0022-4049/© 2016 Elsevier B.V. All rights reserved.
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In Proposition 4.1 we show that if K is Hilbertian, the group G is non-abelian, and the base variety is rational, then there exist finite separable extensions L/K such that some L-form of fL does not descend to a cover of Y . 2. Main theorems In this section, we state our main results, proved at the end of Section 3. Definition 2.1. A map f : X → Y between integral noetherian schemes is called a cover if it is a finite, étale surjection. Letting G := Aut(X/Y )opp , the fiber-degree is a constant d > 0 with |G| ≤ d, and |G| = d if and only if X is a right G-torsor over Y . In such cases we say that f is Galois (and G is naturally identified with the Galois group of the extension of function fields). For a finite group Γ, we say that f is a Γ-cover if it is Galois and an isomorphism Γ ∼ = Aut(X/Y )opp is specified. The notion of isomorphism for covers and Γ-covers is defined in the obvious manner. In what follows, we fix a reduced and geometrically irreducible K-scheme Y of finite type and a cover f : X → Y such that X is geometrically irreducible over K and fKs is Galois. Since X → Y is étale, the sheaf AutX/Y is representable by a group scheme over Y (also denoted AutX/Y ). Clearly, X → Y is a left AutX/Y -torsor, and (AutX/Y )YKs = AutXKs /YKs ∼ = Gopp is a constant YKs -group. Let the continuous homomorphism ρ : Gal(Ks /K) = Gal(Ks (Y )/K(Y )) → Aut(Gopp ) be the Galois action induced by AutX/Y as a YKs /Y -form of Gopp . Then, clearly, the splitting field E := (Ks )ker(ρ) of ρ is the unique minimal subfield of Ks containing K for which the base change XE → YE is Galois. We remark that E/K is Galois, and its Galois group is canonically isomorphic to a subgroup of Aut(Gopp ). ¯ → Y of f , and Theorem 2.2. With the notation as above, assume that there exists some Galois K-form X ¯ let P be a point such that XP (K) = ∅. Let T be the P -fiber XP viewed as a right G-torsor over Spec(K). Then the minimal field of Galois action for X → Y is contained in the splitting field of T . We may conclude from Theorem 2.2 that the minimal field of Galois action for X → Y is contained in the specialization at P of all Galois K-forms of f . The effect of changing the point P in Theorem 2.2 is expressed by the following result: Theorem 2.3. With the notation of Theorem 2.2, let Q be another point in Y (K). Then the fiber over Q in ¯ P and X ¯ Q are isomorphic as right G-torsors. X → Y has a K-rational point if and only if X Remark 2.4. A variant of Theorem 2.3 has been known before, and goes by the name the “Twisting Lemma” ([8, 2.2]; [4, Section 2]; see also Lemma 3.3). It has been applied in Galois-Theoretic contexts, most notably by Pierre Dèbes [2–6]. However, the Twisting Lemma is a bit weaker since it merely says that if f is Galois then it admits some K-form X → Y such that K-points in fibers over Y (K) can be detected by fibers of f as in Theorem 2.3. Finally, I will prove the following.
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Theorem 2.5. With the notation of Theorem 2.3, the following hold: 1. If X → Y is a K-form of f such that XP (K) = ∅, then it is isomorphic to X over Y . 2. The number of K-rational points in X lying above P is divisible by |Z(G)| and divides |G|. 3. Twisted covers Throughout this section, we continue to use the setup of Theorem 2.2. Definition 3.1. If Z → Y is a right G-torsor, and T → Spec(K) is a right G-torsor, then we define the right G-torsor Z T → Y as the finite étale Isom-scheme IsomY,G (TY , Z) over Y , classifying G-equivariant isomorphisms over Y -schemes. ¯ T → Y is a (not necessarily Galois) K-form of f . For any right G-torsor T over Spec(K), the map X More precisely: Lemma 3.2. With the notation of Definition 3.1, let F/K be the splitting field of T . Then (Z T )F is isomorphic to ZF over YF . Proof. The twisting construction is compatible with extension of the ground field, so we may rename F as K; i.e., we may assume T is the trivial G-torsor over K. But then it is obvious from the definition of Z T as an Isom-scheme that it is Y -isomorphic to Z. 2 The interest in twisting G-Galois covers by a right G-torsor arises from a property that the twisted cover satisfies, namely the first assertion in the following lemma. Lemma 3.3. 1. (The Twisting Lemma) Let T → Spec(K) be a right G-torsor. Then, in the notation of Theorem 2.2, ¯ is isomorphic as a right G-torsor to T if and only if the twisted cover X ¯T → Y the fiber over P in X has a K-rational point above P . ¯ T in the fiber over P is the size of the centralizer 2. In this situation, the number of K-rational points of X CGopp (Gal(F/K)) of Gal(F/K) in Gopp = AutG (TKs ), where F is the splitting field of T . ¯ T is compatible with base-change on Y over K, so in particular the P -fiber of Proof. The formation of X T ¯ ¯ ¯ T is equivalent to X is the twist of XP against T . Therefore the existence of a K-point in the P -fiber of X ¯ over P . The second assertion follows from the existence of an isomorphism between T and the fiber in X ∼ opp the fact that AutG (T ) = CG (Gal(F/K)). (Indeed, by Galois descent, the elements of AutG (TF ) ∼ = Gopp that descend to K are exactly those that commute with the action of Gal(F/K).) 2 ¯ → Y be a Galois K-form of f . Then for every K-form X → Y of f , there exists Proposition 3.4. Let f¯ : X ¯ T is isomorphic to X over Y . a right G-torsor T over Spec(K) so that X ˇ 1 (K, G). On the Proof. We will make the following two identifications for the (étale) Čech cohomology set H ¯ over Y is naturally identified with G, this set classifies isomorphism one hand, since the Galois group for X ¯ classes of K-forms of X → Y (since every descent datum is effective). On the other hand, as is well-known, this set also classifies isomorphism classes of G-torsors over K. Let T be the G-torsor over K that corresponds ¯ → Y . We shall now check that X ¯ T is isomorphic via the above identifications to the K-form X → Y of X to X over Y .
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¯ equipped with its It is easy to see that the right GY -torsor TY represents the Isom-functor IsomY (X , X) ¯ natural right GY -action through X, and so “evaluation” at points of X defines an evident map of functors ¯ But working over an étale cover of Y splitting these finite étale covers shows that X → IsomY,G (TY , X). ¯ T by definition, so we are done. 2 this latter map is an isomorphism. The target is X Now we can finally prove Theorems 2.2, 2.3 and 2.5: ¯ T is Proof of Theorems 2.2, 2.3 and 2.5. By Proposition 3.4, there exists a right G-torsor T so that X ¯ P )T = Isom (T , X ¯ P =: T as G-torsors since ¯ P ), so T X isomorphic to X over Y . Note that XP = (X G XP (K) is non-empty by hypothesis. Then Theorem 2.2 follows immediately from Lemma 3.2. Theorem 2.3 follows immediately from the Twisting Lemma (Lemma 3.3(1)) to the fiber at Q, and Theorem 2.5(2) follows immediately from Lemma 3.3(2), using that Z(Gopp ) is a subgroup of CGopp (H) for any subgroup H of Gopp . It remains to prove Theorem 2.5(1). Let x be a point in XP (K), and x a point in XP (K). In this ∼ situation, if there exists a Y -isomorphism X → X carrying x to x , then it is unique since X is irreducible. By the uniqueness and geometric irreducibility of X over K, Galois descent reduces our task to proving ¯ is a G-torsor such existence and uniqueness when K = Ks , as we now assume. In particular, now X = X over Y , so it is Galois. Our task is to prove that there is a unique Y -automorphism of X swapping a chosen pair of rational points in the fiber of P . That in turn is obvious since X is Galois over Y . 2 4. An obstruction to the descent of forms Proposition 4.1. Let K be a Hilbertian field, let G be a non-abelian finite group, and let Y be a rational variety over K. For any geometrically irreducible G-Galois cover E of YKs that descends to a cover of Y there exists a finite extension L/K and an L-descent X → YL of that cover such that it does not descend to a cover of Y . Proof. Let W be the compositum of all of the Galois field extensions of K in Ks having a Galois group that is isomorphic to a subgroup of Aut(Gopp ). The field W is clearly Galois over K, and is not equal to Ks . Let L be a nontrivial finite field extension of W . By Weissauer’s Theorem ([9, Satz 9.7]; see also [7, Theorem 13.9.1]), the field L is Hilbertian. Since, by the arguments in the beginning of Section 2, every descent of E → YKs to a cover of Y becomes Galois over some Galois extension of K with Galois group a subgroup of Aut(Gopp ), they all become Galois over L . In particular, there exists a G-Galois L -descent X → YL of E → YKs . Since L is Hilbertian and Y is rational, there exists a connected right G-torsor T over L achieved by specializing X at some L -point P . By Lemma 3.3, the number of L -rational points of X T in the fiber of P is |CGopp (Aut(F/L ))| = |Z(Gopp )|, where F is the splitting field of T . Since by assumption Z(Gopp ) is a proper subgroup of Gopp , it follows that X T is not Galois over YL , and therefore has no descent to a cover of Y . Finally, let L be some subfield of L , finite over K, to which X T → YL descends. 2 Acknowledgement The author would like to thank the anonymous reviewer for comments that were very helpful in improving this paper. References [1] Kevin Coombes, David Harbater, Hurwitz families and arithmetic Galois groups, Duke Math. J. 52 (4) (1985) 821–839. [2] Pierre Dèbes, Galois covers with prescribed fibers: the Beckmann–Black problem, Ann. Sc. Norm. Super. Pisa, Cl. Sci. 4 (28) (1999) 273–286.
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[3] Pierre Dèbes, On the Malle conjecture and the self-twisted cover, arXiv:1404.4074 [math.NT], 2014. [4] Pierre Dèbes, Nour Ghazi, Galois covers and the Hilbert–Grunwald property, Ann. Inst. Fourier (Grenoble) 62 (3) (2012) 989–1013. [5] Pierre Dèbes, François Legrand, Twisted covers and specializations, in: Galois–Teichmüller Theory and Arithmetic Geometry, in: Adv. Stud. Pure Math., vol. 63, Math. Soc. Japan, Tokyo, 2012, pp. 141–162. [6] Pierre Dèbes, François Legrand, Specialization results in Galois theory, Trans. Am. Math. Soc. 365 (10) (2013) 5259–5275. [7] Michael Fried, Moshe Jarden, Field Arithmetic, third ed., Ergeb. Math. Ser., vol. 11, Springer-Verlag, 2005. [8] Alexei Skorobogatov, Torsors and Rational Points, Cambridge Tracts Math., Cambridge University Press, Cambridge, 2001. [9] Rainer Weissauer, Der Hilbertsche Irreduzibilitätssatz, J. Reine Angew. Math. 334 (1982) 203–220.