Minimal Fields of Definition for Galois Action

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1 Minimal Fields of Definition for Galois Action Hilaf Hasson Department of Mathematics, Stanford University, Palo Alto, CA 94305, USA Abstract 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 f Ks is Galois. Assuming f admits a Galois K-form f : X Y, we use it to analyze fields of definition over 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 f L does not descend to a cover of Y. 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 [CH85] 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 Y Ks with Galois group G, and let M be its associated field of moduli. Under the additional assumption that Y (K), [CH85, Proposition 2.5] shows that there is a descent to a cover of Y M which is possibly not Galois; and [CH85, 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 Y F. 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 f Ks 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 K s over K so that X E Y E 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, under the assumption that there exists some Galois K-form X Y. 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 f L 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. address: hilaf@stanford.edu (Hilaf Hasson) Preprint submitted to Elsevier February 29, 2016

2 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 f Ks is Galois. Since X Y is étale, the sheaf Aut X/Y is representable by a group scheme over Y (also denoted Aut X/Y ). Clearly, X Y is a left Aut X/Y -torsor, and (Aut X/Y ) YKs = Aut XKs /Y Ks = G opp is a constant Y Ks -group. Let the continuous homomorphism ρ : Gal(K s /K) = Gal(K s (Y )/K(Y )) Aut(G opp ) be the Galois action induced by Aut X/Y as a Y Ks /Y -form of G opp. Then, clearly, the splitting field E := (K s ) ker(ρ) of ρ is the unique minimal subfield of K s containing K for which the base change X E Y E is Galois. We remark that E/K is Galois, and its Galois group is canonically isomorphic to a subgroup of Aut(G opp ). Theorem 2.2. With the notation as above, assume that there exists some Galois K-form X Y of f, and let P be a point such that X P (K). Let T be the P -fiber X P 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 X Y has a K-rational point if and only if XP and X Q are isomorphic as right G-torsors. Remark 2.4. A variant of Theorem 2.3 has been known before, and goes by the name the Twisting Lemma ([Sko01, 2.2]; [DG12, Section 2]; see also Lemma 3.3). It has been applied in Galois- Theoretic contexts, most notably by Pierre Dèbes ([Dèb99], [DG12], [DL12], [DL13], [Dèb14]). 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. 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 X P (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

3 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 Isom Y,G (T Y, Z) over Y, classifying G-equivariant isomorphisms over Y -schemes. For any right G-torsor T over Spec(K), the map X T Y is a (not necessarily Galois) K-form of f. 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 Z F over Y F. 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. 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 (The Twisting Lemma) Let T Spec(K) be a right G-torsor. Then, in the notation of Theorem 2.2, the fiber over P in X is isomorphic as a right G-torsor to T if and only if the twisted cover X T Y has a K-rational point above P. 2. In this situation, the number of K-rational points of XT in the fiber over P is the size of the centralizer C G opp(gal(f/k)) of Gal(F/K) in G opp = Aut G (T Ks ), where F is the splitting field of T. Proof. The formation of XT is compatible with base-change on Y over K, so in particular the P - fiber of X T is the twist of X P against T. Therefore the existence of a K-point in the P -fiber of X T is equivalent to the existence of an isomorphism between T and the fiber in X over P. The second assertion follows from the fact that Aut G (T ) = C G opp(gal(f/k)). (Indeed, by Galois descent, the elements of Aut G (T F ) = G opp that descend to K are exactly those that commute with the action of Gal(F/K).) Proposition 3.4. Let f : X Y be a Galois K-form of f. Then for every K-form X Y of f, there exists a right G-torsor T over Spec(K) so that X T is isomorphic to X over Y. Proof. We will make the following two identifications for the (étale) Čech cohomology set Ȟ1 (K, G). On the one hand, since the Galois group for X over Y is naturally identified with G, this set classifies isomorphism 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 via the above identifications to the K-form X Y of X Y. We shall now check that X T is isomorphic to X over Y. It is easy to see that the right G Y -torsor T Y represents the Isom-functor Isom Y (X, X) equipped with its natural right G Y -action through X, and so evaluation at points of X defines an evident map of functors X Isom Y,G (T Y, X). But working over an étale cover of Y splitting these finite étale covers shows that this latter map is an isomorphism. The target is X T by definition, so we are done. Now we can finally prove Theorems 2.2, 2.3 and 2.5: 3

4 Proof. By Proposition 3.4, there exists a right G-torsor T so that X T is isomorphic to X over Y. Note that X P = ( X P ) T = Isom G (T, X P ), so T X P =: T as G-torsors since X P (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(G opp ) is a subgroup of C G opp(h) for any subgroup H of G opp. It remains to prove Theorem 2.5(1). Let x be a point in X P (K), and x a point in X P (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 such existence and uniqueness when K = K s, as we now assume. In particular, now X = X is a G-torsor 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. 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 Y Ks that descends to a cover of Y there exists a finite extension L/K and an L-descent X Y L 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 K s having a Galois group that is isomorphic to a subgroup of Aut(G opp ). The field W is clearly Galois over K, and is not equal to K s. Let L be a nontrivial finite field extension of W. By Weissauer s Theorem ([Wei82, Satz 9.7]; see also [FJ05, Theorem ]), the field L is Hilbertian. Since, by the arguments in the beginning of Section 2, every descent of E Y Ks to a cover of Y becomes Galois over some Galois extension of K with Galois group a subgroup of Aut(G opp ), they all become Galois over L. In particular, there exists a G-Galois L -descent X Y L of E Y Ks. 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 C G opp(aut(f/l )) = Z(G opp ), where F is the splitting field of T. Since by assumption Z(G opp ) is a proper subgroup of G opp, it follows that X T is not Galois over Y L, 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 Y L descends. 5. Acknowledgements The author would like to thank the anonymous reviewer for comments that were very helpful in improving this paper. References [CH85] Kevin Coombes and David Harbater, Hurwitz families and arithmetic Galois groups, Duke Math. J. 52 (1985), no. 4, [Dèb99] Pierre Dèbes, Galois covers with prescribed fibers: the Beckmann-Black problem, Ann. Scuola Norm. Sup. Pisa, Cl. Sci. 4 (1999), no. 28,

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