Compatible systems of symplectic Galois representations and the inverse Galois problem I. Images of projective representations

arXiv:1203.6546v3 [math.NT] 23 Sep 2013 Compatible systems of symplectic Galois representations and the inverse Galois problem I. Images of projective representations. Sara Arias-de-Reyna∗ , Luis Dieulefait† , Gabor Wiese‡ 24th September 2013 Abstract This article is the first part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. In this first part, we determine the smallest field over which the projectivisation of a given symplectic group representation satisfying some natural conditions can be defined. The answer only depends on inner twists. We apply this to the residual representations of a compatible system of symplectic Galois representations satisfying some mild hypothesis and obtain precise information on their projectiveimages for almost all members of the system, under the assumption of huge residual images, by which we mean that a symplectic group of full dimension over the prime field is contained up to conjugation. Finally, we obtain an application to the inverse Galois problem. MSC (2010): 11F80 (Galois representations); 20C25 (Projective representations and multipliers), 12F12 (Inverse Galois theory). 1 Introduction This article is the first part of a series of three articles, about compatible systems of symplectic Galois representations with huge images and applications to the inverse Galois problem. The overall aim is, for given n∈Neven and d∈N, to construct a compatible system of Galois representations over Q such that the projective image of the residual representations at a positive density set of primes ℓis PGSpn(Fℓd)or PSpn(Fℓd), thus realising these groups as Galois groups over Q. Three tasks emerge: ∗Université du Luxembourg, Faculté des Sciences, de la Technologie et de la Communication, 6, rue Richard Coudenhove-Kalergi, L-1359 Luxembourg, Luxembourg, [email protected] †Departament d’Àlgebra i Geometria, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via de les Corts Catalanes, 585, 08007 Barcelona, Spain, [email protected] ‡Université du Luxembourg, Faculté des Sciences, de la Technologie et de la Communication, 6, rue Richard Coudenhove-Kalergi, L-1359 Luxembourg, Luxembourg, gabor[email protected] 1 (1) Given a compatible system of Galois representations, determine the fields of definition of the projective residual representations. For n= 2 and the compatible system arising from a Hecke eigenform f=Pn≥1anqnof level Nand Dirichlet character ψ, these fields are known to be equal to the residue fields of the number field Q(a2 p ψ(p)|p∤N)at almost all places. In this Part I we produce an analog of this number field for an n-dimensional compatible system of Galois representations. We need to make some irreducibility and regularity assumptions, which are automatically satisfied for compatible systems coming from modular forms. (2) Having established the minimal fields of definition for projective representations, the next task is to determine the actual image. In the case n= 2 a well-known classical theorem of Dickson classifies the finite subgroups of PGL2(Fℓ)(up to conjugation) into ‘huge’ ones (i.e. PGL2(Fℓd) or PSL2(Fℓd)for some d∈N), dihedral ones, subgroups of the upper triangular matrices and three exceptional ones; in representation theoretic language, the non-huge and non-exceptional images come from induced or reducible representations. In Part II [AdDW13] we supply an analog of this result for n-dimensional symplectic representations under the assumption of the existence of a nontrivial transvection in the image. As a general strategy, as in [DW11], we will enforce local properties to the compatible system in order to ensure global ones: one local property takes care of the nontrivial transvection, another one (that of a maximally induced place, see below) of absolute irreducibility and in the end huge image and the existence of a suitable subfield of a cyclotomic field in the projective field of definition. (3) The final task is then to construct a compatible system of Galois representations satisfying the specified local requirements. This is achieved through the construction of a suitable automorphic representation in Part III [AdDSW13], which is joint work with Sug Woo Shin. The present Part I is entirely independent of the other two papers. Part II is independent as well, except that in the end the results developed in Part II are combined with Theorem 6.5 from the present Part I in order to prove that the existence of a compatible system of Galois representations with certain specified local properties implies the desired application to the inverse Galois problem. Part III relies on some results of Part I and Part II. Statement of the results We now describe the main results of the present article in slightly simplied forms. Although we are mainly interested in symplectic representations (and compatible systems thereof), we also prove results for general linear representations along the way; some of the assumptions needed are different in that case, making the statements longer; but, we think that including the general linear case is worthwhile. In this overview we will, however, entirely stick to the symplectic case. Our first result determines projective fields of definition for Galois representations. 2 Theorem 1.1. Let L/K be a finite Galois extension of ℓ-adic fields (i.e. finite extensions of Qℓ) or of finite fields and ρ:G→GSpn(L)be a representation of a group Gsuch that its restriction to the subgroup I[ρ]of G(defined in Proposition 2.14; G/I[ρ]is finite abelian) is residually absolutely irreducible. Then there is an extension K[ρ]of Kinside L, which is explicitly described only in terms of inner twists, such that ρproj :Gρ −→ GSpn(L)→PGSpn(L)is conjugate to a representation G→ PGSpn(K[ρ]), and K[ρ]is the smallest field with this property. This is proved as Theorem 2.15 in Section 2. In Section 4 we compute the field K[ρλ]from Theorem 1.1 for almost all residual representations ρλin a compatible system of Galois representations under certain assumptions. The computations are essentially based on relating the inner twists of the residual representations with inner twists of the compatible system. Here we present a simplified form of Theorem 4.6. Theorem 1.2. Let ρ•= (ρλ)λwith ρλ:GQ→GSpn(Lλ)(for a Galois number field L/Q;Lλ denotes an algebraic closure of the completion of Lat the place λ) be a compatible system as in Definition 4.1, consisting of irreducible representations and such that its multiplier is a fixed finite order character times a fixed power of the cyclotomic character. Assume a certain condition at places above ℓ(the rational prime below λ), which is for instance satisfied when the ρλare de Rham with Hodge-Tate weights independent of λ. Then there is a number field Kρ•, which is explicitely determined by the inner twists of the compatible system, such that for all places λof Lat which ρλis residually absolutely irreducible, except possibly finitely many, the projective field of definition of the residual representation ρλis κ((Kρ•)λ), the residue field of Kρ•at λ. We can then already determine the residual projective images under the assumption that almost all of them are ‘huge’ (defined in Definition 5.1). This is done in Corollary 5.8, a version of which we state here. Theorem 1.3. Let ρ•= (ρλ)λbe as in Theorem 1.2. Assume, moreover, that for all but possibly a density zero set of places λof Lthe residual representation ρλhas huge image. Then for all places λof Lwith the possible exception of a density zero set, the image of ρproj λis PGSpn(κ((Kρ•)λ)) or PSpn(κ((Kρ•)λ)). Under these assumptions together with that of the existence of a maximally induced place of a certain order (see Definition 6.1, this goes back to [KLS08] in the context of (n, p)-groups) we can derive the following application to the inverse Galois problem (proved as Theorem 6.5). Theorem 1.4. Let ρ•= (ρλ)λbe as in Theorem 1.3. Assume also that there is a place q∈Ssuch that ρ•is maximally induced at qof order p. Then for any d|p−1 nthere exists a set Ldof rational primes ℓof positive density such that for all ℓ∈ Ldthere is a place λof Labove ℓsatisfying that the image of ρproj λis PGSpn(Fℓd)or PSpn(Fℓd). 3 The maximally induced place of order pserves to produce a certain element ξp∈Q(ζp)of degree p−1 nin Kρ•, generalising the approach of [DW11]. This element guarantees the existence of a subextension of Kρthat is cyclic degree d, from which the existence of a positive density set of primes of residue degree dcan be derived. Acknowledgements S. A.-d.-R. worked on this article as a fellow of the Alexander-von-Humboldt foundation. She thanks the Université du Luxembourg for its hospitality during a long term visit. She was also partially supported by the project MTM2012-33830 of the Ministerio de Economía y Competitividad of Spain. L. V. D. was supported by the project MTM2012-33830 of the Ministerio de Economía y Competitividad of Spain and by an ICREA Academia Research Prize. G. W. was partially supported by the Sonderforschungsbereich TRR 45, the DFG priority program 1489 and by the Université du Luxembourg. 2 Inner twists of a group representation In this section we determine under some mild conditions the smallest field over which a member of the equivalence class of a (projective, symplectic) representation is defined. The main results are Theorem 2.15 and Proposition 2.17. This part owes many of its ideas to Ribet’s paper [Rib85], including Papier’s theorem cited in that article, as well as to other papers of Ribet. A conceptual framework for some of those ideas is provided here as well as a generalisation. We take some time and space to introduce the objects we work with in detail. A guiding example the reader may have in mind is that of a newform f=Pn≥1anqnwith coefficient field Qf= Q(an|n∈N). For a prime ℓtake K=Qℓand Lthe Galois closure over Qℓof the completion of Qf at some place above ℓ(or L=Qℓ). The ρwe will be interested in are the ℓ-adic Galois representation ρfand its mod ℓreduction ρfattached to fwith respect to the chosen place. We are interested in ℓ-adic fields (i.e. finite extensions of Qℓ) as well as finite fields. Since the reasonings are very similar in both cases, we will treat them simultaneously. In what follows, whenwe write that a representation is “(residually) absolutely irreducible” we mean that it is “residually absolutely irreducible” in the case of ℓ-adic fields, and “absolutely irreducible” in the case of finite fields. Whenever we consider a morphism between two topological spaces we assume it to be continuous. In particular, representations and characters will always be continuous. Set-up 2.1. •Kan ℓ-adic field with the ℓ-adic topology or a finite field with the discrete topology. •L/K a Galois extension, Γ := Gal(L/K), endowed with the Krull topology. •Ga finite or a compact topological group. In later sections, Gwill be a Galois group. • E ={ǫ:G→L×character }. 4 Note that the assumptions imply that im(ǫ)lies in a finite extension of K. Note also that Γacts on Efrom the left by composition: γǫ:= γ◦ǫ. •Form the semi-direct product G:= E⋊Γfor the above action. Since we consider the Γ-entry more important than the E-entry, we denote elements of Gas (γ, ǫ)with γ∈Γand ǫ∈ E (instead of the other way around). We concretely have: (γ1, ǫ1)·(γ2, ǫ2) := (γ1γ2,(γ1ǫ2)ǫ1)and (γ, ǫ)−1= (γ−1,γ−1(ǫ−1)). Consequently, we have the exact sequence: 1→ E ι:ǫ7→(1,ǫ) −−−−−→ G π:(γ,ǫ)7→γ −−−−−−→ Γ→1. This sequence is split in the obvious way. Note that the other projection c:G → E,(γ, ǫ)7→ ǫ is a 1-cocycle, when letting Gact on Ethrough its quotient Γ. •Let n∈N. The equivalence class of a representation ρ:G→GLn(L)(for conjugation by an element of GLn(L)) is denoted by [ρ]and equivalence of representations is denoted by ∼. We denote the set of equivalence classes by GL-Repsn(G, L/K). Note ρis defined over a finite extension ˜ K/K inside L, that is ρ:G→GLn(˜ K)nat.emb. −−−−→ GLn(L)(This is a standard fact: the reader can find a proof in e.g. [Ski09], page 244). •Let n∈Nbe even. The symplectic equivalence class of a representation ρ:G→GSpn(L)(for conjugation by an element of GSpn(L)) is also denoted by [ρ]and also symplectic equivalence of representations is denoted by ∼. We denote the set of equivalence classes by GSp-Repsn(G, L/K). Note that ρis defined over a finite extension ˜ K/K inside L, that is ρ:G→GSpn(˜ K)nat.emb. −−−−→ GSpn(L). We make the convention that equivalence of symplectic representations always means symplectic equivalence. Note that the embedding GSpn(L)֒→GLn(L)allows us to see elements of GSp-Repsn(G, L/K)as elements of GL-Repsn(G, L/K). We will call [ρ](absolutely) irreducible, if (any member of) it is (absolutely) irreducible as a linear representation. •Note that Gacts on GL-Repsn(G, L/K)from the left by the following formula: (γ, ǫ).[ρ] := [(γρ)⊗Lǫ−1], where γ∈Γ,ǫ∈ E and [ρ]∈GL-Repsn(G, L/K). In the same way, Gacts on the set GSp-Repsn(G, L/K)from the left. •Let n∈Nand r1, r2:G→PGLn(L)be projective representations. We call r1and r2 equivalent (also denoted r1∼r2) if they are conjugate by the class (modulo scalars) of a matrix in GLn(L). The equivalence class of r1is also denoted [r1]. For ρ:G→GLn(L)we denote by ρproj the composition of ρwith the natural projection GLn(L)→PGLn(L). 5 •Let n∈Nbe even and r1, r2:G→PGSpn(L)be projective symplectic representations. We call r1and r2equivalent (also denoted r1∼r2) if they are conjugate by the class (modulo scalars) of a matrix in GSpn(L). The equivalence class of r1is also denoted [r1]. For ρ:G→ GSpn(L)we denote by ρproj the composition of ρwith the natural projection GSpn(L)→ PGSpn(L). Remark 2.2. Let Kbe an ℓ-adic field and OKits valuation ring. (a) It is well-known that any representation ρ:G→GLn(K)is conjugate in GLn(K)to a representation G→GLn(OK)(see e.g. [Ser98], I-2). We have already used this implicitly when speaking about residual representations. (b) In the symplectic case we assume that ρ:G→GSpn(K)is residually irreducible. One can show that then ρis conjugate in GSpn(K)to a representation G→GSpn(OK). Indeed, if Lis a lattice provided by (a) and the standard symplectic pairing is scaled such that its restriction to L × L is surjective onto OK, then the residual irreducibility together with Nakayama’s lemma implies that the OK[G]-homomorphism Lv7→(w7→hw,vi) −−−−−−−−−→ HomOK(L,OK), where the G-action on the right hand side is twisted by the multiplier of ρ, is an isomorphism. Thus, Ladmits a symplectic OK-basis and the corresponding base change matrix is general symplectic. For the rest of this section we assume that we are in Set-up 2.1. We now make the main definition of this section, which generalises standard definitions for elliptic curves and modular forms. Definition 2.3. Let [ρ]∈GL-Repsn(G, L/K)or [ρ]∈GSp-Repsn(G, L/K). •Define G[ρ]to be the stabiliser group of [ρ]in G(for the G-action on GL-Repsn(G, L/K)or on GSp-Repsn(G, L/K)). Its elements are called inner twists of [ρ]. Explicitly, let (γ, ǫ)∈ G. Then (γ, ǫ)is an inner twist of [ρ]if and only if [ρ] = [(γρ)⊗Lǫ−1], which is the case if and only if [γρ] = [ρ⊗Lǫ]. •Define the groups Γ[ρ]:= π(G[ρ])⊆Γand E[ρ]:= ι−1(G[ρ]) = ι−1(ker(π|G[ρ])). •We say that [ρ]has no complex multiplication if E[ρ]={1}, i.e. if and only if the projection π:G[ρ]→Γ[ρ]is an isomorphism. •Define the field K[ρ]:= LΓ[ρ]. It is called the projective field of definition of [ρ](see Theorem 2.15 for a justification of the terminology). •Define the group ∆[ρ]:= {γ∈Γ[ρ]|(γ, 1) ∈ G[ρ]}. •Define the field E[ρ]:= L∆[ρ]. It is called the field of definition of [ρ](see Proposition 2.17 (b) for a justification of the terminology). In our guiding example of a newform f,ρfhas CM or a nontrivial inner twist if and only if f does. 6 Lemma 2.4. (a) Let [ρ]∈GL-Repsn(G, L/K)or [ρ]∈GSp-Repsn(G, L/K)be absolutely irreducible. Then any M∈GLn(L)commuting with all ρ(g)for g∈Gis a scalar matrix. (b) Let ρ:G→GSpn(L)be absolutely irreducible without complex multiplication, let M∈ GLn(L)and assume that the image of M−1ρM lies in GSpn(L). Then M∈GSpn(L), thus M−1ρM belongs to the symplectic equivalence class of ρ. Proof. (a) This is a form of Schur’s Lemma. See e.g. [CR62], §27 of Chapter IV, Lemma (27.3). (b) Call ρ′=M−1ρM. The symplecticity of ρand ρ′implies for g∈G: ρ(g)trJρ(g) = m(ρ(g))Jand (M−1ρ(g)M)trJ(M−1ρ(g)M) = m(ρ′(g))J, where mis the multiplier and Jis the Gram matrix of the standard symplectic pairing. Combining these two equations, yields (MJ−1MtrJ)−1ρ(g)(MJ−1MtrJ) = m(ρ′(g)) m(ρ(g)) ρ(g). Note that this equation implies that the character m(ρ′(g)) m(ρ(g)) lies in E[ρ]. As we are assuming that ρhas no complex multiplication, this character is trivial. Using (a), we get that MJ−1MtrJis a scalar matrix λidn, showing that M∈GSpn(L), as claimed. One can find examples showing that the conditions in Lemma 2.4 (b) are necessary. On our way to develop the main results of this section, we first include a proposition summarising some representation theory to be used in the sequel. Proposition 2.5. (a) Let [ρ],[ρ′]∈GL-Repsn(G, L/K)be absolutely irreducible. If for all g∈G the equality Tr(ρ(g)) = Tr(ρ(g′)) holds, then [ρ] = [ρ′]. (b) Let [ρ],[ρ′]∈GSp-Repsn(G, L/K)be absolutely irreducible without complex multiplication. If Tr(ρ(g)) = Tr(ρ′(g)) for all g∈G, then [ρ] = [ρ′]. (c) Let [ρ]∈GL-Repsn(G, L/K)be (residually) absolutely irreducible and let K⊆˜ K⊆Lbe a field extension such that Tr(ρ(g)) ∈˜ Kfor all g∈G. Then there is M∈GLn(L)such that M−1ρ(g)M∈GLn(˜ K)for all g∈G. So, the equivalence class [ρ]has a representative with target in GLn(˜ K). (d) Let [ρ]∈GSp-Repsn(G, L/K)be (residually) absolutely irreducible without complex multiplication and let K⊆˜ K⊆Lbe a field extension such that Tr(ρ(g)) ∈˜ Kand m(g)∈˜ K(multiplier map) for all g∈G. Then there is M∈GSpn(L)such that M−1ρ(g)M∈GSpn(˜ K)for all g∈G. So, the equivalence class [ρ]has a representative with target in GSpn(˜ K). Proof. (a) This follows from the Theorem of Brauer-Nesbitt, cf. [CR62], §30 of Chapter V, Theorem (3.15). 7 (b) Applying (a), we obtain that ρand ρ′are equivalent as GLn-representations, meaning that there is some matrix M∈GLn(L)such that ρ′=M−1ρM. Lemma 2.4-(b) shows that M∈GSpn(L), as claimed. (c) For ℓ-adic fields, this is Théorème 2 of [Car94]; see also [Maz97], p. 255. For finite fields the result follows from the triviality of the Brauer group. (d) Applying (c) we obtain a matrix M∈GLn(L)such that M−1ρ(g)M∈GLn(˜ K)for all g∈G. We want to show that there is a matrix N∈GLn(˜ K)such that N−1M−1ρ(g)MN ∈ GSpn(˜ K)for all g∈G. Again denote Jthe Gram matrix of the standard symplectic pairing. Then all M−1ρ(g)Mrespect the symplectic pairing with Gram matrix I:= MtrJM up to the same multiplier m(ρ(g)), i.e. (M−1ρ(g)M)trI(M−1ρ(g)M) = m(ρ(g))I for all g∈G. We claim that I∈GLn(˜ K). Note that conjugating M−1ρ(g)Mby Iyields an element in GLn(˜ K), using here the assumption m(ρ(g)) ∈˜ K. The absolute irreducibility of M−1ρM now implies that conjugating by Ipreserves Mn(˜ K)(see e.g. [Maz97], p. 252). The Skolem-Noether theorem (Corollary 3.63 of [CR81]) consequently shows I∈GLn(˜ K). This now means that Idefines a symplectic pairing on ˜ Kn(the representation space of M−1ρM), which is respected by M−1ρM (up to the multiplier). The claimed existence of the matrix Nis now just the fact that in a symplectic vector space a basis can be chosen such that the symplectic pairing has the standard form, i.e. is given by J. We conclude, using (b), that (MN)−1ρ(MN)is in the equivalence class [ρ]. We now include a word on topology. Lemma 2.6. Let [ρ]∈GL-Repsn(G, L/K)or [ρ]∈GSp-Repsn(G, L/K). Then Γ[ρ]and ∆[ρ]are open subgroups of Γ(and thus of finite index). Proof. There is an open normal subgroup e Γρof Γacting trivially on ρ. Then we have inclusions e Γρ⊆∆[ρ]⊆Γ[ρ]⊆Γ, all of which are of finite index. Fix a system of representatives γ1,...,γmof Γ/e Γρ. Then Ei:= {ǫ∈ E | γiρ∼ρ⊗ǫ}=   ∅or ǫiE[ρ]for some ǫi∈ E. It follows that Γ[ρ]=Sm i=1,Ei6=∅γie Γρand ∆[ρ]=Sm i=1,Ei=E[ρ]γie Γρ, thus they are open subsets of Γ. The following lemma is very useful in our applications of the above theory to compatible systems of Galois representations, where the Frobenius traces lie in some global field. 8 Lemma 2.7. Let [ρ]∈GL-Repsn(G, L/K)or [ρ]∈GSp-Repsn(G, L/K)(in this case also assume that ρhas no complex multiplication) be absolutely irreducible. Then G[ρ]is equal to the set {(γ, ǫ)∈ G | γ(Tr(ρ(g))) = Tr(ρ(g))ǫ(g)∀g∈G}. Proof. Proposition 2.5 (a) and (b). We continue with a useful observation on the values of ǫoccuring for inner twists. Lemma 2.8. Let [ρ]∈GL-Repsn(G, L/K)or [ρ]∈GSp-Repsn(G, L/K). Furthermore, let χ: G→K×be any character and let ψ:G→L×be a character of finite order. Let ǫ∈ E occur in some (γ, ǫ)∈ G[ρ]. (a) If det(ρ) = ψχ, then ǫn=γψ ψand the values of ǫnare contained in the field generated over K by the values of ψ. (b) If [ρ]is symplectic with multiplier map m(ρ) = ψχ, then ǫ2=γψ ψand the values of ǫare contained in the field generated over Kby the values of ψ. Proof. Let (γ, ǫ)∈ G[ρ]. We get: (a) γ(det(ρ)) = χγ(ψ) = ψχǫn, implying ǫn=γψ ψ. The second assertion is now clear. (b) The same calculation with the multiplier map yields ǫ2=γψ ψ. Consequently, the order of ǫ divides 2m, where mis the order of ψ. Let ζbe a primitive m-th root of unity, g∈Gand b, c ∈Z such that ψ(g) = ζb,γ(ζ) = ζc. Then ǫ(g)2=ζbc ζb=ζb(c−1). If mis odd, there exists d∈Zsuch that ζ= (ζ2)d, thus ǫ(g)2=ζ2bd(c−1) and ǫ(g)∈K(ζ). Assume now that mis even. Then c−1 is also even (otherwise cwould be even, and ζcwould not be a primitive m-th root of unity). Hence ǫ(g)2has order dividing m/2, so that ǫ(g)has order dividing m; thus the values of ǫare a subset of the values of ψ. We now study projective representations. Lemma 2.9. (a) For ρ:G→GLn(L)and ǫ∈ E one has ρproj = (ρ⊗ǫ)proj. (b) For [ρ]∈GL-Repsn(G, L/K)or [ρ]∈GSp-Repsn(G, L/K)and ǫ∈ E one has ρproj ∼ (ρ⊗ǫ)proj. (c) Let [ρ1],[ρ2]∈GL-Repsn(G, L/K)or [ρ1],[ρ2]∈GSp-Repsn(G, L/K)such that ρproj 1∼ ρproj 2. Then there is ǫ∈ E such that [ρ1] = [ρ2⊗ǫ]. Proof. (a) and (b) are clear. (c) Denote by sthe injective homomorphism L×→GLn(L)which sends xto the scalar matrix x·idn. Let Mbe a matrix in GLn(L)(or in GSpn(L)) such that ρproj 1=Mρproj 2M−1. Define the character ǫ:G→L×by the formula ǫ(g) := s−1ρ1(g)(Mρ2(g)M−1)−1. Note that ǫis well-defined, is multiplicative and that it obviously satisfies the requirements. 9 (a) ∆ρ•is a normal subgroup of Γρ•and, hence, Eρ•/Kρ•is a Galois extension with Galois group Γρ•/∆ρ•. In particular, the field γ(Eρ•) = Eρ•for all γ∈Γρ•. (b) For each place λof Lsuch that ρλis residually absolutely irreducible (and symplectic if ρ•is a. e. symplectic), the equivalence class [ρλ]contains a representation that can be defined over the field (Eρ•)λand (Eρ•)λis the smallest such field. Proof. (a) First of all, note that, if ρ•is a. e. symplectic, then Eρ•contains the values of ψ. Indeed, for all δ∈∆ρ•, for all places λof Lfixed by δand all places pof Foutside S∪Sℓ(where ℓ is the residue characteristic of λ), we have that Tr(ρλ(Frobp)) = ap=δ(ap) = Tr(δρλ(Frobp)). Here by abuse of notation we denote by δany extension of δto Lλ. Suppose that ρλis absolutely irreducible and symplectic. Then by Remark 4.2 ρλdoes not have complex multiplication. Moreover, by Chebotarev’s Density Theorem and Proposition 2.5 (b), ρλand δρλare equivalent as symplectic representations. Hence δ(m(ρλ)) = m(δρλ) = m(ρλ)for all except possibly finitely many places λ∈L. Since the multiplier of ρ•is ψχa ℓ, we obtain that the values of ψare fixed by δ, thus they lie in Eρ•. This allows us to compute (γǫ)(δ, 1)(γ, ǫ)−1= (γδγ−1,1) ∈ G•as in Proposition 2.17 (a) for all γ∈Γρ•and δ∈∆ρ•, which shows that γδγ−1∈ Eρ•. (b) Let λbe a place of Lsuch that ρλis residually absolutely irreducible (and symplectic if ρ•is a. e. symplectic). For all p6∈ S∪Sℓ(where ℓis the residue characteristic of λ), the trace of ρλ(Frobp) is fixed by all elements of ∆ρ•that fix λ, whence it belongs to (Eρ•)λ, and we saw above that the multiplier of ρλtakes values in (Eρ•)λwhen ρ•is a. e. symplectic. Using Chebotarev’s density theorem and Proposition 2.5 (c) and (d), we see that ρλcan be defined over (Eρ•)λ. Since Eρ•is generated by the {ap|p6∈ S}, it is clear that [ρλ]cannot contain a member which is defined over a subfield smaller than (Eρ•)λ. We will next see that, when ρλis residually absolutely irreducible, the projective field of definition of [ρλ]of Definition 2.3 is just the completion at λof the projective fieldof definition of the compatible system. Lemma 4.4. Let ρ•= (ρλ)λbe an n-dimensional a. e. absolutely irreducible compatible system as in Definition 4.1. Let ψ:GF→L×be a character of finite order. If ρ•is a. e. symplectic, assume that ρ•has no complex multiplication and that the multiplier of ρ•is ψχa ℓ. If ρ•is not a. e. symplectic, assume that Lcontains the n-throots of the values of ψand that the determinant of ρ•is ψχa ℓ. Let λbe a place of Lsuch that ρλis residually absolutely irreducible (and symplectic if ρ•is a. e. symplectic). Then we have Γ[ρλ]= Γλ∩Γρ•, where the intersection is taken inside Γvia the embedding Γλ֒→Γcoming from the embedding L ֒→Lλ. 16 Proof. We use Lemma 2.7. The inclusion ‘⊇’ is clear. Conversely, let γ∈Γ[ρλ]; in particular, γ∈Γλ. There is a character ǫ∈ Eλsuch that (γ, ǫ)∈ G[ρλ], i.e. that γ(ap) = ap·ǫ(Frobp)for all p6∈ S∪Sℓ. Lemma 2.8 implies that ǫcan be lifted to an element of E, i.e. taking values in L×. The equation γ(ap) = ap·ǫ(Frobp)for all p6∈ S∪Sℓis now an equation in the field L, implying (γ, ǫ)∈ Gρ•. The following theorem is our main result about projective fields of definition of compatible systems of ℓ-adic representations. Theorem 4.5. Let ρ•= (ρλ)λbe an n-dimensional a. e. absolutely irreducible compatible system as in Definition 4.1. Let ψ:GF→L×be a character of finite order. If ρ•is a. e. symplectic, assume that ρ•has no complex multiplication and that the multiplier of ρ•is ψχa ℓ. If ρ•is not a. e. symplectic, assume that Lcontains the n-th roots of the values of ψand that the determinant of ρ•is ψχa ℓ. Then for each place λof Lsuch that ρλis residually absolutely irreducible (and symplectic if ρ• is a. e. symplectic), the projective field of definition of [ρλ]is (Kρ•)λ, the completion of Kρ•at the prime lying below λ. Proof. By Lemma 4.4, we have K[ρλ]=LΓ[ρλ] λ=LΓλ∩Γρ• λ= (LΓρ•)λ= (Kρ•)λ. Given a compatible system ρ•= (ρλ)λwe will now consider the residual representations ρλ (see Remark 2.2). If Mis a local field, we denote by κ(M)its residue field. Let λbe a finite place of Land assume ρλis defined over Lλ(which is the case whenever ρλis residually absolutely irreducible (and symplectic without complex multiplication)). Consider Set-up 2.1 for the Galois extension κ(Lλ)/κ(Kλ)with Galois group Γλ, using the notation Eλand Gλ. We apply Definition 2.3 to the equivalence class [ρλ]of the residual representation, using the notation G[ρλ]⊆Gλ,E[ρλ]⊆Eλ and Γ[ρλ]⊆Γλ. We now state and prove our principal result describing the projective fields of definition for almost all residual representations of a compatible system. It may be useful to roughly summarise where the various conditions in the theorem come from. That ψhas finite order is needed to ensure that all ǫ occuring in inner twists are of finite order; the condition on the n-th roots of the values of ψin the non-symplectic case ensures that ǫtakes its values in Eρ•(which is automatic in the symplectic case). The absolute irreducibility and the non-CM condition (in the symplectic case) are needed to ensure that the representations are determined by the characteristic polynomials of Frobenius. The condition on the shape above ℓis needed to exclude that the residual inner twists ramify at ℓ(if infinitely many of them did ramify, then they would not ‘glue’ to an inner twist of the compatible system). Theorem 4.6. Let ρ•= (ρλ)λbe an n-dimensional a. e. absolutely irreducible compatible system as in Definition 4.1. Let ψ:GF→L×be a character of finite order. If ρ•is a. e. symplectic, assume that ρ•has no complex multiplication and that the multiplier of ρ•is ψχa ℓ. If ρ•is not a. e. symplectic, assume that Lcontains the n-th roots of the values of ψand that the determinant of ρ•is ψχa ℓ. Assume that for primes Lof Flying over the residue characteristic of λ, the representation ρλ satisfies Assumption 3.2 (this is automatic if FL=Qℓ). Moreover, assume that there is an integer k, independent of λ, such that the numbers ai,j appearing in Assumption 3.2 are bounded by k. 17 Then for all places λof Lat which ρλis residually absolutely irreducible, except possibly finitely many, the projective field of definition of [ρλ]is κ((Kρ•)λ). Proof. We may restrict to those λof residual characteristic ℓsatisfying ℓ−1 2n> k (cf. Proposition 3.3), and symplectic without complex multiplication if ρ•is a. e. symplectic. We may furthermore limit ourselves to λsuch that λis unramified in L/K. Let (γ, ǫ)∈ G[ρλ]be an inner twist of [ρλ]. We identify ǫwith its lift to an element of Eof the same order (with respect to the fixed L ֒→Lλ) and Γλ= Gal(Lλ/Kλ)with Γλ= Gal(κ(Lλ)/κ(Kλ)), so that G[ρλ]⊆ G[ρλ]. By Lemma 2.8 we know that the order of ǫis bounded independently of λ. Moreover, by Proposition 3.3, ǫis unramified outside S. Let E0be the finite subset of Econsisting of those characters meeting these two requirements. Hence, G0:= E0⋊Γis a finite subgroup of Gand G[ρλ]and G[ρλ]are subgroups. Now consider (γ, ǫ)∈ G0. Assume that (γ, ǫ)∈ G[ρλ]for infinitely many λ. Then for all places p of Fnot in Sand each of these λ(except those above the residue characteristic of p), we have γ(ap)≡ap·ǫ(Frobp) (mod λ). Consequently, we have equality γ(ap) = ap·ǫ(Frobp), whence (γ, ǫ)∈ G[ρλ]. This means that avoiding also those finitely many λsuch that a (γ, ǫ)∈ G0\G[ρλ]is in G[ρλ], we have G[ρλ]=G[ρλ], thus κ(K[ρλ]) = K[ρλ]. Since ρλis residually absolutely irreducible, Theorem 4.5 yields K[ρλ]= (Kρ•)λ, which proves the theorem. 5 Application to compatible systems with huge residual images In this section we make use of Theorem 4.6 to prove a first result that allows us to (almost) determine the projective image of the residual representation ρλ, (except for finitely many λ), for a compatible system ρ•= (ρλ)satisfying suitable conditions. Let Vbe an n-dimensional K-vector space endowed with a symplectic pairing. Recall that an element τ∈GL(V)is a nontrivial transvection if τ−idVhas rank 1, i.e. if τfixes a hyperplane pointwisely and there is a line Usuch that τ(v)−v∈Ufor all v∈V. Any transvection has determinant 1. A symplectic transvection is a transvection in Sp(V). Any symplectic transvection has the form Tv[λ]∈Sp(V) : u7→ u+λhu, viv with direction vector v∈Vand parameter λ∈K(cf. [Art57], p. 137-138). Definition 5.1. Let Lbe an algebraically closed field, and Ga subgroup of GSpn(L). We will say that Gis huge if the subgroup of Ggenerated by the transvections contained in Gis conjugated (in GSpn(L)) to Spn(K)for some subfield K⊆L. Let Kbe an algebraic closure of K. 18 Remark 5.2. When Kis a finite field of characteristic ℓ, a subgroup G⊆GSpn(K)is huge if and only if it contains a subgroup conjugated (in GSpn(K)) to Spn(Fℓ). This result is proved in Part II [AdDW13] of the present series of papers. The main ingredient will be the following group-theoretic result: Proposition 5.3. Let Kbe a finite field of characteristic different from 2, and G⊆GSpn(K)be a group such that the group generated by the transvections in Gis Spn(K). Then G⊆GSpn(K)K×. To prove this proposition, first note the following easy fact: Lemma 5.4. Let G⊆GSpn(K)be a group such that the group generated by the transvections in G is Spn(K). Then Gis contained in the normaliser of Spn(K)in GSpn(K). Proof. Let A∈Gbe any element, say with multiplier α. To see that it belongs to the normaliser NGSpn(K)(Spn(K)), it suffices to see that, for all transvection T=Tv[λ]∈Spn(K),ATA−1∈ Spn(K). An easy computation shows that, for all w∈V, ATv[λ]A−1(w) = A(A−1w+λhA−1w, viAv) = w+λhA−1w, viAv = w+λαhw, AviAv =TAv[λα]. But TAv[λ]is a transvection and belongs to G, so it belongs to Spn(K), as was to be shown. So, in order to obtain Proposition 5.3, all we need is to prove the following proposition. Proposition 5.5. Let Kbe a finite field of characteristic different from 2. Then NGSpn(K)(Spn(K)) ⊆ GSpn(K)K×. We will make use of an auxiliary lemma: Lemma 5.6. Let Kbe field of characteristic different from 2. The group of automorphisms of Spn(K) is generated by two subgroups, as described below: •The group of automorphisms Φsuch that: for all B∈Spn(K),Φ(B) = ABA−1, where A∈GSpn(K)(semi-inner automorphisms). •The group of automorphisms Φsuch that there exists φ:K→Kfield automorphism such that, for all B∈Spn(K),Φ(B)is the matrix with entries obtained by applying φto the entries of B (field automorphisms). Proof. This follows from the main Theorem of [Hua48], page 740. Proof of Proposition 5.5. Denote by Nthe normaliser of Spn(K)in GSpn(K), and consider also the centraliser C=CGSpn(K)Spn(K), that is, the elements A∈GSpn(K)such that, for all B∈ Spn(K),A−1BA =B. 19 First, note that we can view N/C in a natural way as a subgroup of Aut(Spn(K)): namely, given A∈N, it defines an automorphism of Spn(K)by conjugation, and the kernel of the homomorphism N→Aut(Spn(K)) is obviously C. Let us call ˜ Nthe image of Nin the automorphism group. We know that ˜ Nis a subgroup of Aut(Spn(K)). By Lemma 5.6, we know that the group of automorphisms is generated by two subgroups, one consisting of the semi-inner automorphisms and one consisting of the field automorphisms. But no automorphism of ˜ Ncan be a field automorphism. This is proved in [KLS08], page 548, inside of the proof of Corollary 2.6; namely, the reasoning is that any automorphism of the shape B7→ A−1BA must respect the trace. But there is always an element B∈Sp2n(K)such that no field automorphism preserves the trace of B. On the other hand, all semi-inner automorphisms belong to ˜ N. From here we can conclude that the semi-inner automorphisms are precisely those belonging to ˜ N. Indeed, note that the composition of a field automorphism with a semi-inner automorphism coincides with the composition of a semiinner automorphism with a field automorphism, so each automorphism can be written as a product of a semi-inner one and a field one, and then we would get that if ˜ Ncontains an automorphism which is not semi-inner, it contains an automorphism which is a field automorphism. Therefore the group ˜ Ncoincides with the group of semi-inner automorphisms. That is to say, for any matrix A∈N, there exists a matrix A1∈GSpn(K)such that the automorphism B7→ A−1BA coincides with the automorphism A−1 1BA1. That is to say, for all B∈Spn(K),(A1A−1)−1B(A1A−1) = B. Equivalently, A1A−1∈C. But, in our situation, C=K×{Id}. One sees this by considering a basis of Vconsisting of directions of transvections in Spn(K), say Tv1[λ1],...,Tvn[λn]. Assume that B∈C. Then, for all i= 1,...,n,B−1Tvi[λ]B=Tvi[λ]. But B−1Tvi[λ]B=TBvi[λβ−1](where βis the multiplier of B), hence Bmust fix all hvii. Repeating the same reasoning with transvections with directions v1+v2,...,v1+vn, one sees that Bmust be a homothety. Hence N= GSp2n(K)K×. As a consequence of Proposition 5.3, we obtain the following. Corollary 5.7. Let Kbe a finite field of characteristic different from 2, and G⊆GSpn(K)be a group such that the group generated by the transvections in Gis Spn(K). Then the image of Gunder the projection GSpn(K)→PGSpn(K)is either PSpn(K)or PGSpn(K). Corollary 5.8. Let ρ•= (ρλ)λbe an n-dimensional a. e. absolutely irreducible and a. e. symplectic compatible system as in Definition 4.1. Let ψ:GF→L×be a character of finite order. Assume that the multiplier of ρ•is ψχa ℓ. Assume, moreover, that for all but possibly a density zero set of places λof Lthe residual representation ρλhas huge image. Assume that for primes Lof Flying over the residue characteristic of λ, the representation ρλ satisfies Assumption 3.2 (this is automatic if FL=Qℓ). Moreover, assume that there is an integer k, independent of λ, such that the numbers ai,j appearing in Assumption 3.2 are bounded by k. 20 Then for all places λof Lwith the possible exception of a density zero set, the image of ρproj λis PGSpn(κ((Kρ•)λ)) or PSpn(κ((Kρ•)λ)). Proof. Let λbe a place of Lsuch that the image of ρλis symplectic and huge. Then the restriction of ρλto I[ρλ](see Proposition 2.14) is absolutely irreducible and Theorems 2.15 and 4.6 show that ρproj λ can be defined over κ((Kρ•)λ)and that this field is the smallest one with this property. Now, because of Corollary 5.7, we know that there exists a field K′⊆Ksuch that the image of ρproj λis conjugated to PSpn(K′)or PGSpn(K′)whenever ρλhas huge image. In particular ρproj λcan be defined over K′, and this is the smallest field with this property. Hence K′=κ((Kρ•)λ), and the statement follows. 6 Application to the inverse Galois problem In this section we generalise results of [DW11] by proving that compatible systems with certain local properties and the global assumption that all (but a density zero set) residual images are huge lead to the realisation of PSpn(Fℓd)or PGSpn(Fℓd)as Galois groups over Qfor certain fixed dand a positive density set of primes ℓ. In Part II [AdDW13] we replace the residual huge image assumption by weaker local assumptions and in Part III [AdDSW13] we prove the existence of such compatible systems. Some of the ideas in this section are taken from [KLS08], where the terminology of (n, p)- groups is used. Definition 6.1. Let ρ•= (ρλ)λbe an n-dimensional a. e. symplectic compatible system as in Definition 4.1. Let q∈Sbe a place of Fthat is totally split over Qand does not divide 2n. Denote by qthe rational prime under q. Let δ:GQqn→L×be a tame symplectic character of degree nand order 2p as in Section 3.1 of [KLS08] (where it is denoted χ) for an odd prime psuch that the order of qin F× p is equal to n(note that, in particular, n|(p−1)). We say that the compatible system ρ•is maximally induced at qof order pif for each λnot above q the restriction of ρλto a decomposition group at qis equivalent to IndGQq GQqn(δ)⊗α(where we view δ as taking its values in Lλvia L ֒→Lλ) with an unramified character α:GQq→L× λ. The definition is made precisely in such a way in order to ensure that IndGQq GQqn(δ)is symplectic and irreducible. We refer to [KLS08] for more details (see also Part II [AdDW13]). Note moreover that the maximally induced place ensures that ρλis absolutely irreducible for almost all λ. The existence of δis a result of local class field theory. Lemma 6.2. Let pand qbe primes such that the order of qin F× pis equal to n. Let ζp∈Qbe a primitive p-th root of unity. Then the element ξp:= Pn−1 i=0 ζqi phas degree p−1 nover Q. Proof. See the paragraph above Proposition 2.16 of Chapter 2 of [Was82]. Note that the element ξpfrom Lemma 6.2 is the trace of an element in the image of IndGQq GQqn(δ). 21 Lemma 6.3. Let ρ•= (ρλ)λbe an n-dimensional a. e. absolutely irreducible a. e. symplectic compatible system as in Definition 4.1. Let ψ:GF→L×be a character of finite order m. Assume that ρ•has no complex multiplication and that the multiplier of ρ•is ψχa ℓ. Assume also that there is a place q∈Ssuch that ρ•is maximally induced at qof order p. Then ξp:= Pq−1 i=0 ζqi pis an element of Kρ•. Proof. Let λbe a place of Lnot above q(the prime below q), such that ρλis absolutely irreducible and symplectic. Let us take any (γ, ǫ)∈ G[ρλ]so that γρλ∼ρλ⊗ǫ. We first want to prove that γ(ξp) = ξp. There is an element σ0∈Iq(the inertia group at q) such that γρλ(σ0)and (ρλ⊗ǫ)(σ0)are conjugated to       γ(ζp) γ(ζp)q ... γ(ζp)qn−1      and ǫ(σ0)       ζp ζq p... ζqn−1 p       ,(6.2) respectively. Since ζpζq p· · · · · ζqn−1 p= 1, from the equality of the determinants of the matrices in (6.2) we get ǫ(σ0)n= 1. But, ǫ(σ0)is also a p-th root of unity (comparing the eigenvalues of the two matrices), whence ǫ(σ0) = 1, as (n, p) = 1. We then get γ(ζp) = ζqi pfor some 0≤i≤n−1, and therefore γ(ξp) = ζqi p+···+ζqi(n−1) p=ζp+···+ζqn−1 p=ξp. Hence, we have established ξp∈K[ρλ]. But, since this holds for almost all λ, we conclude from Theorem 4.5 (using that, due to the presence of the maximally induced place, the residual representation ρλis absolutely irreducible for almost all λ) that ξplies in Kρ•. Finally we will make use of the following proposition: Proposition 6.4 (Prop. 7.2 of [DW11]).Let L/Qbe a finite field extension which contains a cyclic extension M/Qof degree d. Then the set of primes ℓsuch that there is an ideal λin OLof residue degree dhas a positive density. The following is our main result in this part about the application to the inverse Galois problem. Theorem 6.5. Let ρ•= (ρλ)λbe an n-dimensional a. e. symplectic compatible system as in Definition 4.1 with K=Q. Let ψ:GF→L×be a character of finite order. Assume that the multiplier of ρ•is ψχa ℓ. Assume, moreover, that for all but a density zero set of places λof Lthe residual representation ρλhas huge image. Assume that for primes Lof Flying over the residue characteristic of λ, the representation ρλ satisfies Assumption 3.2 (this is automatic if FL=Qℓ). Moreover, assume that there is an integer k, independent of λ, such that the numbers ai,j appearing in Assumption 3.2 are bounded by k. Assume also that there is a place q∈Ssuch that ρ•is maximally induced at qof order p. 22 Then for any d|p−1 nthere exists a set Ldof rational primes ℓof positive density such that for all ℓ∈ Ldthere is a place λof Labove ℓsatisfying that the image of ρproj λis PGSpn(Fℓd)or PSpn(Fℓd). Proof. Let ξp:= Pq−1 i=0 ζqi p. By Lemma 6.2 we know that ξphas degree p−1 nover Qand by Lemma6.3 we also know that ξplies in Kρ•. Consequently, Kρ•contains a cyclic extension M/Qof degree d. The theorem is now just a combination of Proposition 6.4 and Corollary 5.8. References [AdDSW13] Sara Arias-de-Reyna, Luis Dieulefait, Sug Woo Shin, and Gabor Wiese, Compatible systems of symplectic Galois representations and the inverse Galois problem III. Automorphic construction of compatible systems with suitable local properties, Preprint (2013). [AdDW13] Sara Arias-de-Reyna, Luis Dieulefait, and Gabor Wiese, Compatible systems of symplectic Galois representations and the inverse Galois problem II. 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