Arithmetic Lifting of Dihedral Extensions

Transcription

1 JOURNAL OF ALGEBRA 203, ARTICLE NO JA Arithmetic Lifting of Dihedral Extensions Elena V Black* Department of Mathematics, Uniersity of Maryland, College Park, Maryland Communicated by Mel Hochster Received March 28, 995 INTRODUCTION The Inverse Galois problem remains a fascinating yet unanswered uestion The standard approach through algebraic geometry is to construct a Galois branched covering of the projective line over the rationals with a desired group G Then one invokes the Hilbert Irreducibility Theorem to construct a G-Galois extension of If every Galois extension of number fields is the specialization of a Galois branched covering with the same group, this approach is very logical Of course, the uestion whether every Galois extension of number fields is the specialization of a branched covering is of interest in its own right Answered affirmatively, it would put all Galois extensions of a given number field in families It is conjectured that the answer to this uestion is yes Beckmann addressed this problem in her paper Is every extension of the specialization of a branched covering? She answers it affirmatively when G is either a symmetric or a finite abelian group We will take one step further and prove this conjecture when G is a dihedral group Dn with n odd However, before proving this result, we will reexamine the case of an abelian group In Section I, we reinterpret the work of Beckmann and Saltman in terms of etale cohomology In Section II, we generalize to the case of a dihedral group *Present address: Department of Mathematics, University of Oklahoma, 60 Elm Avenue, Room 423, Norman, OK eblack@mathouedu $2500 Copyright 998 by Academic Press All rights of reproduction in any form reserved 2

2 ARITHMETIC LIFTING OF DIHEDRAL EXTENSIONS 3 NOTATION AND DEFINITIONS Let X be a complex, irreducible, smooth, projective algebraic curve Let X be a finite dominant morphism; ie, a nonconstant rational map We call X a branched coering If the corresponding function field extension Ž X Ž t is Galois with GalŽŽ X Ž t G, then it is called G a G-Galois branched coering and sometimes denoted X G G K K Let K be a number field We say X has a model X over K if the following holds: Ž i XK is a connected, complete, smooth curve over K, such that XKSpec K Spec X; ii the maps XK K and X are compatible; Ž iii the function field extension KŽ X KŽ t K is Galois with group G We will also say in this case that XK K is a G-Galois branched covering defined over K A field extension LKŽ t is called regular if K is algebraically closed in L 8 Note, in particular, that an extension LKŽ t corresponds to the function field extension of some branched covering XK K defined over K if and only if it is regular f K K be a branched covering defined over K, and let y K Let X Ž be a K-rational point If K t denotes the function field of K, let t a correspond to the point y We have a commutative diagram, G X f K K X f y yspec K, where X X y K K Spec K is the fiber of f over y Since f is finite, so is f Thus X y Spec A for some K-algebra A finitely generated as a K- f module 2 The specialization of XK K at t a is defined to be the extension of K-algebras K A It is clear that if X K is a G-Galois branched covering and A is a field, then the field extension AK is Galois with group G Let LK be a G-Galois extension of fields View Spec K as a point of K Then by a lifting of LK we mean a G-Galois branched covering G XK K, together with K-rational point y K, such that the fiber over

3 4 ELENA V BLACK y is isomorphic to Spec L In other words, XK K specializes to LK at some K-rational point y K Two branched coverings X and Y are called disjoint if whenever there is a commutative diagram, X Y, Z G then Z is an isomorphism In the rest of the paper we use the following notation: Let LK denote a Galois extension of number fields Let n denote the group of nth roots of unity, with fixed generator n Its dual n HomŽ, n is generated by, ie, Ž n n We denote GalŽKŽ K n by nif n is cyclic, we denote its generator by and its order by s Let K be an algebraic closure of K If Mis a GalŽ KK -mod- i ule, we write H Ž K, M for the ith Galois cohomology group H i ŽGalŽ KK, M Let G be a group acting on n and n and let M be a G-module We define M to be the G-module M n By using a fixed choice of a generator for we identify MŽ n and M as abelian groups via m m, but we have different actions of G SECTION 0: TECHNICAL PRELIMINARIES Let p d be a prime power Let R be a discrete valuation ring, with a fixed uniformizer, field of fractions K, and residue field F Let GalŽKŽ K and GalŽFŽ F Note that via decomposition group we may consider as a subgroup of Let Q, and let the order of Q be m Let S be an integral closure of R in KŽ Let M be a -torsion abelian group In what follows, H i Ž A, M et for any ring A is the ith etale cohomology group of Spec A of the sheaf defined by a group scheme M We denote Galois cohomology groups by H i Note that if A is a field, the groups H i Ž A, M and H i Ž A, M are isomorphic 5, p 53 et PROPOSITION With notation as aboe, there is a commutatie diagram of spectral seuences: i H j, H Ž S, M ij H Ž R, M et et Ž Ž i j ij H, H F, M H F, M

4 ARITHMETIC LIFTING OF DIHEDRAL EXTENSIONS 5 Moreoer, if is cyclic, M is a sheaf with triial action, and the integers m and are relatiely prime, then the maps gien by the left ertical arrow for j 0 are isomorphisms for all i Proof We have a commutative diagram of rings: mod S FR S R mod F By taking spectra we obtain a corresponding commutative diagram of schemes Note that F S FŽ Q Using the Hochschild-Serre spectral seuences 5, p 05 and the fact that spectral seuences are natural under pullbacks, we obtain the following commutative diagram: Ž i j H, Het S, M Het R, M ij Ž i j ij H, Het FS, M Het F, M Using the Hochschild-Serre seuence again, we have a spectral seuence: ž Ž / Ž H k Q, H i, H j Ž F S, M H ki, H j Ž F S, M Ž We work with the left-hand side of with finite direct sums, we have Ž Since cohomology commutes Since the Q-module Ž H k ŽQ, H i, H j Ž F Ž, M Ž et Q H k ŽQ, H i, H j ŽFŽ, M et Q Q Ž Ž H k Q, H i, H j ŽFŽ, M Ž 2 H i Ž, H j Ž F Ž, M Q is induced, and therefore cohomologically trivial, Ž 2 is eual to zero if k0 and is eual to H i Ž, H j ŽFŽ, M if k 0

5 6 ELENA V BLACK We thus have an isomorphism ž Ž H i, H j F S, M H i, H j F, M et Therefore we get a commutative diagram: Ž i j H, Het S, M Het R, M ij Ž i j ij H, Het FS, M H F, M Ž Ž i j ij H, H F, M H F, M We next assume that is cyclic, generated by, and that the integers m and are relatively prime Then H 0 Ž S, M M and H 0 Ž FS, M et et m M, since M is a trivial sheaf The map H 0 Ž S, M H 0 Ž F S, M et is the diagonal map We thus have a short exact seuence of -modules 0 M MA0 diag m Note that the action of on the first term of the above seuence is trivial, while permutes the factors of the second term To prove our claim, we only need to show that H i Ž, A 0 for all i Since is cyclic by assumption, the Tate cohomology groups depend only on the parity of i 7 They can be calculated as follows: Hˆi, A A NA, i 0 mod 2 et Hˆi, A Ker N DA, i mod 2 Here N Ýi0 s i is the norm of the action on A and D To see the action of on A we identify A MdiagŽ M m with m M via the isomorphism that sends the element Ž a,,a to Ž m a2 a,,a a Then the action of takes Ž a a,,a a m 2 m to Ž a a,,a a,a a 3 2 m 2 2 It is easy to check that NA 0 and hence KerŽ N A The kernel of the map of A to itself induced by D is eual to A To see that A is m-torsion, we set Ž a a,,a a Ž 2 m a3 a,,a a,a a That easily leads to the conclusion that ma Ž 2 m 2 2 i

6 ARITHMETIC LIFTING OF DIHEDRAL EXTENSIONS 7 a 0 for all i 2,,m, and hence A is indeed m-torsion Since A is also -torsion, and m and are relatively prime by assumption, A is a trivial group Therefore DA A, and the proposition is proved SECTION I: ABELIAN GROUPS The result presented in this section is not new As mentioned in the introduction, Beckmann has proved that every finite abelian extension of a number field K is a specialization of a Galois branched covering defined over K with the same group Here we reinterpret her work Žand that of Saltman 6 in terms of etale cohomology In the next section, this method is generalized to the case of dihedral groups Let K be a number field and GalŽKŽ K n n Define a homomor- phism : Gal K Ž K Ž n n n Žg that satisfies g n n, for any g n Now let p d be a prime power Let LK be a cyclic Galois extension of number fields with GalŽ LK By Galois theory, the extension of number fields LK is classified by an element Ž a H K, Hom Gal KK, Note that a represents a surjective homomorphism from GalŽ KK to This follows because L is the fixed field of the kernel of a From the HochschildSerre spectral seuence, we have the following exact seuence Ž also called the Inflation-Restriction seuence : Inf Res 2 0H, H K, H K, H, Since a is surjective, it is of order and is not contained in the image of H Ž, Therefore, we can view the field extension LK as one of the extensions given by the element defined as Res a H K, and 0

7 8 ELENA V BLACK View M KŽ K as a -module with the usual Galois action We can identify MŽ with M as an abelian group; however, the action has been twisted Let and m MŽ Then Ž m Ž m Ž Ž m Ž Ž Ž m So the norm of the action of on MŽ is N Ý Žg g n g Observe that there is a natural isomorphism as -modules H K Ž, H K Ž, M MŽ We therefore view as an element of KŽ K such that MŽ and such that 0 We now turn our attention to cyclic Galois branched coverings of the Ž projective line defined over K Let K t be the function field of K Note that GalŽKŽ K GalŽKŽ, t KŽ t,so acts on KŽ, t in the usual way Let M t denote the -module K, t K, t Thus we have the -module t M K,t K,t d Ž LEMMA Let p be a prime power Let K t be a function field of Let K Ž Ž t K,t K,t Ž Ž Ž t be such that t M, t is not a pth power in K t and t Ž Ž 0 Then Ž t gies a regular -Galois extension LKŽ t, corre- sponding to a regular -Galois branched coering XK K defined oer K Proof To find a -Galois branched covering X defined over K, it suffices to construct a regular -Galois extension of fields LKŽ t To ensure regularity, we find a -Galois extension of fields L KŽ t, corresponding to a branched covering of complex curves X, which descends to KŽ t It is sufficient to produce a regular cover over KŽ, and then descend it to K As before, we have an exact seuence Ž Inf Res 2 Ž Ž 0H, H K t, H K,t, H,

8 ARITHMETIC LIFTING OF DIHEDRAL EXTENSIONS 9 If Ž t H ŽKŽ,t, is such that t Ž 0 then it lifts to an element in H ŽKŽ t,, defining a -Galois extension of KŽ t This extension is regular if we assume in addition that Ž t is not a pth power in KŽ t This is because the extension x x Ž t K t over KŽ t is a -Galois extension of fields 3, p 33, which descends without change to a -Galois field extension over KŽ, t Since H KŽ,t xž x Ž t Ž K, t, M M Ž t t as -modules, we can view Ž t as an element of M Ž t This concludes the proof of Lemma We intend to specialize at t 0 Hence we also reuire that Ž t gives a covering unramified at the point t 0 To ensure this, we pick Ž t in H Ž S,, where S is an integral closure of R Kt in KŽ, t et Žt We apply Proposition to this situation to obtain the following result COROLLARY Let be a positie integer, let K be a number field, and GalŽKŽ K Then there is an exact commutatie diagram Inf Res 2 0H Ž, H Ž K, H K, H Ž, t0 Inf Res 2 et et 0H Ž, H Ž R, H Ž S, H Ž, We are now in the position to prove the following theorem: THEOREM Let p d be a prime power Let LK be a Galois extension of number fields with GalŽ LK Then there exists a regular -Galois branched coering XK K defined oer K and specializing to LK at t0 Moreoer, there are infinitely many such coerings, each specializing to LK att0, disjoint from one another Proof The extension LK is determined by an element a H Ž K, This element a is one of the lifts of an element KŽ, such that MŽ t0

9 20 ELENA V BLACK By Lemma, in order to construct the desired -Galois branched Ž covering, we need an element t S KŽ, t, such that Ž t M Ž We also need Ž t to satisfy the following reuirements: Ž i Ž t is not a pth power in KŽ t ; Ž ii t Ž 0; Ž iii Ž 0 The commutative diagram in Corollary allows us to conclude that Ž iii implies Ž ii, and hence we only need to ensure Ž i and Ž iii The same diagram shows that the lift of Ž t to H Ž R, et can be chosen so that its specialization is precisely an element a, which determines our extension LK We construct the reuired element Ž t as follows: Choose KŽ such that g for all g id, that is has maximal orbit under the Galois action of GalŽKŽ K Let Ž t NŽ t N Here, recall that N stands for the norm of the action of on M Ž t, ie, N Ý Žg g Hence, we can expand the above expression as g Ž g Ž t Ł Ž g t Ł Ž g g g g Ž It is clear that t t Ž g t g Ł ž / g is such that Ž t MtŽ, and Ž 0 Because of the maximal orbit condition in our choice of, t Ž is not a pth power in KŽ t Finally, observe that we should have written Ž t to stress the fact that the resulting branched covering depends on the choice of In fact, the set of branch points of this covering is either gž, g 4 Ž when is not trivial or, 4 Ž if is trivial Different Ž subject to the maximal orbit condition give rise to disjoint -branched coverings of each K

10 ARITHMETIC LIFTING OF DIHEDRAL EXTENSIONS 2 specializing to LK at t 0 This follows because there is no nontrivial branched covering of the projective line with at most one branch point For a detailed proof of the above the reader is referred to This completes the proof of the Theorem Remark Let LK be a finite abelian extension of number fields with Galois group A Use the Fundamental Theorem of the Structure of Abelian Groups to write A as a direct sum n, where the i are prime powers Use Theorem and fibered products to construct an A-Galois branched covering XK K defined over K and specializing to LK For details, see, cor 24 SECTION II: DIHEDRAL GROUPS We fix more notation for this case We still let LK be a Galois extension of number fields, with Galois group GalŽ LK Dn n 2 We assume throughout that n is odd Let K E L be the uniue uadratic subextension; ie, E is the fixed field of n Write E KŽ ' r for some r K We denote the generator of GalŽ EK 2 by, ie, Ž ' r ' r As before, denotes GalŽKŽ K n n We let denote GalŽEŽ K and GalŽEŽ E n n n Note that n is a subgroup of and is an extension of by ² : n n As before we have a homomor- Žg phism : n n defined by g n n for g n Let be such that E Let p d be an odd prime power Then both and are cyclic with generators and respectively We denote the order of by s Next, we consider two cases, depending on whether E and KŽ are disjoint over K or not In the disjoint case, E KŽ K, and ² : is the trivial extension In the nondisjoint case, E is the uniue uadratic subfield of KŽ over K, so E KŽ,, and is the uniue subgroup of index 2 in Thus, 2 and Let M EŽ E be a -module with the usual Galois action As in the abelian case, we write M for M and identify it with M as an abelian group with the twisted action of We reserve the notation for a cyclic group of order whenever the action of on it is trivial However, when we refer to the cyclic group of order considered as a ² :-module with the action of taking the generator to its inverse, we denote this group as C We write M for M C and identify it with M as abelian group We make M into a -module by twisting the action of Finally, we denote M C by M Ž, identify it with M as an abelian group, and view it as a -module with action of twisted The

11 22 ELENA V BLACK norm of this -action is denoted by N The action of on M Ž has a norm N Ý Žh h h, as in abelian case The action of on M Ž is different depending on whether we have the disjoint or nondisjoint case In the disjoint case, the norm of this action is and so, N NN In the nondisjoint case, N Ž s i Ž i i, so the norm of twisted action of is N Ý LEMMA 2 i0 With notation as aboe, there is an exact seuence 2 Ž Ž Ž 0H,C H E,C M H,C Proof We use the HochschildSerre spectral seuence to see that 2 0 H, C H E,C H E,C H,C is exact Denote the uotient H Ž E, C H Ž, C by A for simplicity We have another exact seuence Ž ² : 0H,C H E,C A H, H,C 0 Ž 3 Ž 4 The last term in Ž 4 is zero by torsion considerations So rewriting Ž 3 and taking invariants under the action of ² : we get an exact seuence: Res, 2 0 A H EŽ,C H,C Ž 5 Ž Ž Put together seuences 5 and 4 to get Res, 2 0 H, C H E,C H E,C H,C As in the abelian case, there is a canonical -module isomorphism H EŽ, C H EŽ, C H EŽ, C This concludes the proof of Lemma 2 E E C M

12 ARITHMETIC LIFTING OF DIHEDRAL EXTENSIONS 23 LEMMA 3 Let p d be an odd prime power Let LK bead-galois extension of number fields with uadratic subfield EK Then the fields EŽ and L are disjoint oer E Proof Let F denote the intersection of L and EŽ over E Then EŽ is an abelian extension of K since it is a compositum of two abelian extensions KŽ and E over K Therefore, FK is an abelian extension, containing the subextension EK IfF:E, then it is a power of p The Galois group of EK acts on GalŽ FE by taking its generator to its inverse, which contradicts the fact that FK is an abelian extension Therefore, F : E must be and the fields L and EŽ are disjoint over E Remark In particular, it follows from this lemma that if L L E Ž E, then L E y y for some EŽ, which is not a pth power in EŽ We next note that LK is uniuely determined by an element ž / ah E,C Hom Gal KE, C, where acts on Gal KE by conjugation and acts on C by sending In other words, dihedral extensions of K with fixed uadratic subfield EK are classified by H Ž E,C This can be seen from Galois theory and the fact that L is a fixed field of a surjective homomorphism GalŽ KE Define an element as Res a M Recall that N denotes the norm of the twisted action of on M Ž We let N be the submodule of M Ž consisting of norms N for EŽ LEMMA 4 Let p d be an odd prime power Let LK bead-galois extension of number fields with uadratic subfield EK Let E KŽ ' r Ž i If KŽ and E are disjoint, then there is a KŽ, such that ' r ž ' r / Ž i s ' i N Ž r Ł i0 Ž ii If K E KŽ, then there exists KŽ such that s i Ž i i N Ł Ž i0

13 24 ELENA V BLACK Proof We first show there exist a short exact seuence 0 H, C H E,C N 0, where N N Ž, EŽ 4 Using a result of Saltman 6, Th 23Ž a, b we conclude that the kernel of in HochschildSerre seuence Ž 3 is precisely N Therefore we have a short exact seuence: 0 H Ž, C H Ž E, C N 0 ² : Taking invariants under the appropriate action of 2 on it, we obtain another exact seuence: Ž ² : 0H,C H E,C N H, H,C 0 The last term in the above seuence is zero by torsion considerations So is an element of N Since N is -torsion, and has order 2, the Tate cohomology module Hˆ0Ž² :, N vanishes We therefore can write N for some element E In case Ž ii we simply let, since EŽ KŽ in this case In case Ž i we proceed to show that can be written as ' r This is because EŽ KŽ Ž' r Write in general form as b0 b ' r, where b KŽ i Note that b 0 since, clearly, Then Ł Ž g ' r Ł g ž r/ ž / ' Ž g g g g Above, we let b b This concludes the proof of Lemma 4 0 We now turn our attention to the construction of a D-Galois branched covering X with a D-Galois model XK K over K That amounts to constructing a regular D -Galois field extension LKŽ t Again we intend to specialize at a point t 0, so we need this extension to be unramified at Ž t We start by fixing a uadratic subextension We take Ž Ž 2 K t x x rt KŽ x with GalŽKŽ x KŽ t 2² : This is clearly regular and unramified at Ž t Note that our choice of uadratic subextension is only limited by regularity, since there is no obstruction to the embedding into the dihedral extension Once again, in our construction of the desired branched covering we need to make sure that the point Ž 2 t0 is not a branch point In K x the ideal x r is lying over Ž t in Kt For the local ring R in Proposition, we take Kx 2 Ž x r As before the ring S is an integral closure of R in KŽ, x

14 ARITHMETIC LIFTING OF DIHEDRAL EXTENSIONS 25 We observe that here the picture is similar to that of the case of disjoint KŽ and E in Lemma 4, with playing the role of and Gal K Ž K Gal K Ž xž KŽ x GalŽ SR Here by GalŽ SR we understand the group AutŽ Spec SSpec R We denote GalŽKŽ x, KŽ t by In other words, ² : Using considerations similar to those in Lemma 2, we obtain an exact seuence: 2 Ž et Ž x Ž 0H,C H R,C M H,C Thus, a D -extension of KŽ t with uadratic subextension KŽ x, unram- Ž ified at t, is given by a lift of an element x S KŽ, x such that x Mx Ž and x 0 Here Mx S S, viewed as -module with the usual Galois action, and M M C We identify M Ž x x x with Mx as abelian groups and twist the action of The norm of that action is Ý Ý i i N g g i0 g Moreover, we need Ž x not be a pth power in KŽ x, so that it defines an extension of fields As in Lemma and for the same reasons, these conditions on Ž x ensure regularity of the resulting extension and enables us to descend to K We thus have: d LEMMA 5 Let p be a prime power Let KŽ t be a function field of Let KŽ x be uadratic extension of KŽ t defined as aboe Let Ž x K K, x be such that x Mx, Ž x is not a pth power, and Ž Ž x 0 Then Ž x gies a regular D-Galois extension LKŽ t, corresponding to a regular D -Galois branched D coering XK K defined oer K and unramified at t 0 We can apply Proposition, with R and S as defined, fracž R KŽ x and F E, GalŽ SR and GalŽEŽ E We act by 2 ina comparable fashion on both rows of the resulting commutative diagram

15 26 ELENA V BLACK We then use Proposition, Lemma 2, and Lemma 5 to obtain the following: COROLLARY 2 diagram: With the notation as aboe there is an exact commutatie Inf Res 2 0H,C H E,C H E,C H,C t0 Inf Res 2 et et 0H,C H R,C H S,C H,C We now combine the Lemmas to prove the main result of this section THEOREM 2 Let p d be an odd prime power Let LK be a Galois extension of number fields with GalŽ LK D Let E be the uniue uadratic subfield of L oer K Assume that E KŽ K Then there exists a regular D-Galois branched coering XK K defined oer K and specializing at t 0 to LK Proof Given an extension LK we have a corresponding element t0 Ž By Lemma 5, in order to construct the desired covering we need an element x S such that N M M p x Mx D -Galois branched We need Ž x to satisfy the following reuirements: Ž i Ž x is not a pth power in KŽ x ; Ž ii Ž Ž x 0; Ž iii Ž ' r By Corollary 2 we only need to ensure Ž i and Ž iii By Lemma 4, there exists an element KŽ, such that N Ž ' r We let / Ž g x Ž x N Ž x Ý g ž x g It is obvious that Ž x specializes to at the point corresponding to Ž 2 x r We need to show that Ž x is not a pth power in KŽ x Since gž x x for all g the rational function Ž x KŽ, x is of the form Ž x a Ž xau Ž x p, Ž Ž xb Ž xb u

16 ARITHMETIC LIFTING OF DIHEDRAL EXTENSIONS 27 for some integers and u and elements a, b KŽ i i for i,,u Therefore, if Ž x is a pth power in KŽ x it is also a pth power in KŽ Ž x But then it specializes to a pth power in KŽ at any element this field But that contradicts that is not a pth power in KŽ Ž Lemma 3 This concludes the proof of Theorem 2 THEOREM 3 Let p d be an odd prime power Let LK be a Galois extension of number fields with GalŽ LK D Assume that the uadratic Galois subextension EK is such that K E K Ž Then there is a regular D -Galois branched coering X specializing to LK att0 K K Proof By Lemma 4, LK is given by s i Ž i i Ł Ž, i0 for some KŽ Recall that is a generator and s is the order of Let KŽ be defined by It is clear that and Let ž / / Ž i s ' i r x ' Ł ž ' x N r x r x 6 As in Theorem 2, Ž x is not a pth power in KŽ x because of its form and the fact that it specializes to which is not a pth power To see that Ž ' r, we rewrite expression Ž 6 The reason we can t simply put in Ž 6 ' r for x to obtain Ž ' r is because Ž ' r ' r while Ž x x In fact, the reason specialization at t 0 gives the right answer is precisely because acts on KŽ ' r the same way acts on KŽ x i i ' r x For simplicity we denote ž / by FŽ x i ' r x Then, i0 s ' r x Ł i Ł i Ł i0 ieven iodd ž ' r x/ Ž i ' r x Ł i Ł ieven iodd ž ' r x/ Ž i ' r x i Ł i Ł ieven iodd ž ' r x/ Ž x FŽ x FŽ x i F x i F x Ž i

17 28 ELENA V BLACK ' At x r, and, we have ž / iodd ž / ' i Ž r Ł Ł ieven Ł i Ž Ž i i Ł i Ž i i Ž i ieven iodd This concludes the proof of the Theorem 3 THEOREM 4 Let LK bead-galois n extension of number fields, with n odd Then there exists a regular Dn-Galois branched coering XK K, specializing to LK att0 Proof A dihedral group is a semidirect product of a cyclic group of order n and a cyclic group of order 2, ie, Dn n 2 By the Chinese Remainder Theorem, n, where p d i m i i odd prime powers Let EK as before denote the uadratic Galois subextension of LK; in other words, let E be a fixed field of n The field L is a compositum of the fields L i, where LiE is a Galois field extension with Galois group i The group 2 acts on the group n, by sending Ž,, Ž,, ; we have m 2 acting on each i by sending Hence each LiK is a D i -Galois extension of number fields with uadratic subextension EK for all i By Theorem 2 and Theorem 3 for each i we have a D i -Galois branched covering X over the complex numbers, with D -Galois i model Xi, K K over K, specializing to LiK In each case, the branched covering X factors through the same curve Y In our case, Y is a i i i complex projective line Hence, for each i we have Xi Y, with model over K, such that the 2-Galois branched covering Y over K specializes to EK at t 0 We construct the reuired D -Galois n branched covering by taking the fibered product over Y of X i s We view the resulting product as a covering of via the map X X X X Y Y 2 Y Y m This clearly gives a Dn-Galois branched covering X On the function field level, the fibered product XiY Xj Y corresponds to the field Ž X Ž X This follows from the disjointness of Ž X and Ž X i Ž x j i j over Ž x which in turn follows from relatively prime degrees Ž X : Ž x i and Ž X : Ž x j Hence XiY Xj Y is indeed an irreducible branched covering By induction X Y is an irreducible branched covering K K

18 ARITHMETIC LIFTING OF DIHEDRAL EXTENSIONS 29 Finally, we note that since Xi, K YK specializes to LiE at t 0, the constructed Dn branched covering X has a model over K specializ- ing to LK at t 0 This concludes the proof of Theorem 4 ACKNOWLEDGMENTS I acknowledge helpful conversations with Henri Darmon, Kevin Coombes and Larry Washington concerning material in this paper REFERENCES S Beckmann, Is every extension of the specialization of a branched covering?, J Algebra 65 Ž 994, R Hartshorne, Algebraic Geometry, Springer-Verlag, BerlinHeidelbergNew York, S Lang, Algebra, AddisonWesley, Reading, MA, S Lang, Fundamentals of Diophantine Geometry, Springer-Verlag, BerlinHeidelberg New York, Tokyo, J S Milne, Etale cohomology, Princeton Univ Press, Princeton, NJ, D Saltman, Generic Galois extensions and problems in field theory, Ad Math 43 Ž 982, J P Serre, Local Fields, Springer-Verlag, BerlinHeidelbergNew York, J P Serre, Topics in Galois Theory, Jones and Bartlett Publishers, BostonLondon, 992