Kummer Theory of Drinfeld Modules

Transcription

1 Kummer Theory of Drinfeld Modules Master Thesis Simon Häberli Department of Mathematics ETH Zürich Advisor: Prof. Richard Pink May, 2011

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3 Contents 1 Introduction 5 2 Ingredients from the theory of Drinfeld modules 9 3 An ingredient from group cohomology 12 4 Setting 15 5 An A p -linear independence 24 6 A p -adic result of Kummer-type 28 7 An adelic result of Kummer-type 35 Bibliography 39 3

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5 1 Introduction In [5] Ribet deals with the Kummer theory on extensions of abelian varieties by tori over a field k of characteristic 0. For any such extension V and any prime number p he considers the extension k(v p ) of k obtained by adjoining to k the p-division points V p on V. To this extension he adjoins p-th roots of finitely many points P 1,..., P d V (k), which are linearly independent over the ring of k-rational endomorphisms of V. There exists a Kummer type embedding of the Galois group of the resulting Galois extension ( k V p ; 1 p P 1,..., 1 ) p P d /k (V p ) into V d p. Using what is known as Bashmakov s method, Ribet proves that, under certain restrictions on V, this embedding is surjective for almost all prime numbers p (see [5, Theorem 1.2]). Our goal is to obtain a Kummer style result for Drinfeld modules using an analogous method. Let A := F q [X] be the polynomial ring over a finite field F q with q elements and denote by M A the set of its maximal ideals. For p M A let A p be the p -adic completion of A and let π p denote a generator of p. Moreover let F be the quotient field of A. Denote by  := p M A A p the profinite completion of A. Let K be a finite extension of F. Let K s K be a separable and an algebraic closure of K respectively. For any field extension K L K s denote by G L := Gal(K s /L) the corresponding absolute Galois group. Consider a Drinfeld module ϕ : A K{τ} over K of degree r 1 with generic characteristic and consider K s as an A-module by means of ϕ. Moreover, we assume that End K (ϕ) = A. Given this assumption Pink and Rütsche proved (see [7, Theorem 3.3.1]) the openness of the image of the adelic representation ρ ad : G K GL r (Â), 5

6 which corresponds to the action of G K on torsion points of ϕ. Let K ad K s be the subfield with G Kad = Ker(ρ ad ). For any maximal ideal p M A let T p [ϕ] denote the p -adic Tate module of ϕ, which is a free A p -module of rank r. Furthermore, let d 1 be a natural number and let Λ denote an A-submodule of K generated by A-linearly independent elements x 1,..., x d. For any 1 i d consider a system {x i,n } n 0 K s with ϕ πp (x i,n+1 ) = x i,n and x i,0 = x i. Let us introduce the map Ψ p : G K (T p [ϕ]) d = Matr d (A p ) ) σ ((σ(x 1,n ) x 1,n ) n 0,..., (σ(x d,n ) x d,n ) n 0. Taking the product over all p M A yields a map Ψ ad : G K Mat r d (Â). Let Ψ ad and Ψ p be the group homomorphism obtained by restricting Ψ ad and Ψ p, respectively, to G Kad. Theorem 1.1. (see Theorem 4.4) The image of Ψ ad is open. In particular, the image of Ψ p is equal to Mat r d (A p ) for almost all p M A and is open for all p M A. By means of Ψ ad and ρ ad we will define a group homomorphism Φ ad : G K Mat r d (Â) GLr(Â), where GL r (A p ) acts on Mat r d (A p ) by matrix multiplication. By the result of Pink and Rütsche, Theorem 1.1 is equivalent to the following theorem, which claims that the image of Φ ad, too, is as large as possible. Theorem 1.2. (see Corollary 4.14) The image of Φ ad is open. We briefly illustrate our method in a less extensive setting. Let p M A be any maximal ideal and let W p K s denote the kernel of ϕ πp which is a vector space of dimension r over the finite residue field κ p := A/p. Denote by K(W p ) the extension of K obtained by adjoining to K the elements of W p. Let G p be the image of the representation G K Aut(W p ). As the actions of G K and A on K s commute, κ p is contained in the commutant of G p in End(W p ). To obtain a result of Kumme type we need to verify the following crucial axioms: 6

7 B 1. For almost all prime ideals p, the residue field κ p is equal to the commutant of G p in End κp (W p ). B 2. For almost all p, the cohomology group H 1 (G p, W p ) vanishes. B 3. For the division hull of Λ in K Div K (Λ) := {y K ϕ a (y) L, for some 0 a A} there exists an element c A which annihilates the A-module Div K (Λ)/Λ. As W p = κ r p we may see G p as a subgroup of GL r (κ p ). As a consequence of Pink and Rütsche s result we have G p = GL r (κ p ) for all but finitely many p s. Thus B1 follows immediately and B2 turns into a well-known result on cohomology groups which we state in Corollary 3.3. Axiom B3 will be the content of Lemma 6.9 which is a direct consequence of a result of Poonen (see [3, Lemma 5]). Our axioms thus hold. It is not difficult to see that, moreover, (1.3) ϕ 1 π p (Λ) K = Λ for any p M A which does not divide c. For x Λ let w x denote a choice of an element in ϕ 1 π p (x) K s. Considering Hom(G K(Wp), W p ) as an A-module by means of the action of A on W p we have an A-module homomorphism ε : Λ Hom(G K(Wp), W p ), [σ σw x w x ]. Set ψ i := ε(x i ). This yields the homomorphism of groups ψ p : G K(Wp) W d p, σ (ψ i (σ)) 1 i d. Proposition 1.4. The homomorphism ψ p is surjective for almost all p M A. We sketch its proof and restrict ourselves to maximal ideals which satisfy the properties described in B1 and B2 as well as (1.3). Step I. Let us first show that the ψ i are κ p -linearly independent. For this purpose we show (1.5) Ker(ε) = ϕ πp (Λ). 7

8 To see (1.5), let x Ker(ε). By definition of ε we get ϕ 1 π p (x) K(W p ). As in the proof of Lemma 6.13 the cohomology axiom B2 can be used to show ϕ 1 π p (x) K, trivializing an appropriate cocycle in H 1 (G p, W p ). We conclude by (1.3) that there exists an element w ϕ 1 π p (x) Λ. Hence x = ϕ πp (w) ϕ πp (Λ). The other inclusion of (1.5) follows straight from the definition of ε. Let ε be the factorization of ε through Λ/ϕ πp (Λ) which is injective by (1.5). We may set x i := x i mod ϕ πp (Λ) and note that ψ i = ε(x i ). The linear independence now follows immediately from the injectivity of ε together with the κ p -linear independence of the x i. Step II. We show that Im(ψ p ) is a Mat r r (κ p )-submodule of Wp d = Mat r d (κ p ). An elementary calculation yields σ ψ p (τ) = ψ p (στσ 1 ) for all σ G K(Wp), τ G K. Thus Im(ψ p ) is a G p -submodule of Mat r d (κ p ), where G p acts on the latter by matrix multiplication. By B1 we have G p = GL r (κ p ). Moreover, as we can consider Im(ψ p ) as a subgroup of Mat r d (κ p ) it thus has an F p [GL r (κ p )]- module structure. The natural map F p [GL r (κ p )] Mat r d (κ p ), which is a surjection, finally turns Im(ψ p ) into a Mat r d (κ p )-module. Steps I and II provide exactly what we need to deduce Im(ψ p ) = Mat r d (κ p ) from a general result stated in Lemma 6.3. Finally, one might ask about possible generalizations of our results. We expect Theorems 1.1 and 1.2 to hold also if we let F be any finite extension of the quotient field of F q [X] and A the ring of elements of F which are integral at all places except possibly at a fixed nontrivial place. We even expect the statement of Theorem 1.1 to be true for such a ring A and for arbitrary End K (ϕ). Moreover, in [1] Devic and Pink recently proved a result for Drinfeld modules with special characteristic analogous to the one of Pink and Rütsche. Thus, analogues are likely to be found in special characteristic. Acknowledgments. I would like to thank Professor Pink very much for his support. I also thank Hadi Hedayatzadeh for our discussions. 8

9 2 Ingredients from the theory of Drinfeld modules Let A := F q [X] be the polynomial ring over a finite field F q with q elements and denote by M A the set of its maximal ideals. Let F be the quotient field of A. For any maximal ideal p M A we denote by A p and F p the p -adic completion of A and F, respectively, and by κ p := A/p the residue field of A p. Let π p denote a generator of p. Let K be a finite field extention of F. Let K s K be a separable and an algebraic closure of K, respectively. For any field extension K L K s denote by G L := Gal(K s /L) the corresponding absolute Galois group. Consider a Drinfeld module ϕ : A K{τ} over K of degree r 1 with generic characteristic. Moreover, we assume that ϕ has smallest possible endomorphism ring, i.e., End K (ϕ) = A. In the sequel, we consider K s with the A-module structure given by the Drinfeld module. We call a subgroup of K s an A-module if it is an A-submodule of K s. For any nonzero ideal I A denote by ϕ I the unique monique generator of the left ideal generated by {ϕ a a I} K{τ}. Consider ϕ I as an endomorphism of the additive group of K s, and let Ker(ϕ I ) K s be its kernel, which carries a natural A-module structure. Lemma 2.1. We have Ker(ϕ I ) = (A/I) r as A-modules for any nonzero ideal I A. Proof. See for example [6, Corollary to Theorem 13.1]. The absolute Galois group G K acts A-linearly on Ker(ϕ I ). This yields a group homomorphism ρ I : G K Aut A (Ker (ϕ I )) = GL r (A/I). Similarly, fixing any maximal ideal p, we can define a continuous representation of G K on the p -adic Tate module. The p -adic Tate module is defined as the inverse limit T p [ϕ] := lim Ker ( ) ϕ π n p n 0 9

10 of the inverse system {Ker ( ) ϕ π n p }n 1 of A-modules with homomorphisms ( ) ϕ πp : Ker Ker ( ) ϕ π n p. ϕ π n+1 p The Tate module T p [ϕ] is a free A p -module of rank r. The action of G K on Ker ( ϕ π n p ) respects the structure of the inverse system, i.e., we have σϕ πp (y) = ϕ πp σ(y) for all σ G K andy Ker ( ϕ π n p ) and n 1, and commutes with the action of A p on Ker ( ϕ π n p ) given by the projection of A p onto A/p n. Hence, any element of G K induces an A p -linear automorphism of T p [ϕ]. This assignment yields a group homomorphism ρ p : G K Aut Ap (T p [ϕ]) = GL r (A p ), which is continuous with respect to the profinite topology on GL r (A p ). Let ν p denote the projection of GL r (A p ) onto GL r (κ p ) and note that ρ p = ν p ρ p. The p -adic representation leads to the adelic representation of G K ρ ad : G K p M A GL r (A p ) σ (ρ p (σ)) p, which is continuous with respect to the product topology on p M A GL r (A p ). Set Γ ad := Im(ρ ad ). In [7, Theorem 3.3.1], using End K (ϕ) = A, Pink and Rütsche proved Theorem 2.2. The image Γ ad is open. Throughout the thesis we only make use of the assumption on the endomorphism ring at this point. By definition of the product topology this means that Γ ad contains a subgroup of the form p M A Γ p for open subgroups Γ p GL r (A p ), such that Γ p = GL r (A p ) for all p outside some finite subset of M A. As Γ p is an open subgroup in the compact group GL r (A p ), we have Remark 2.3. The subgroup Γ p GL r (A p ) has finite index for any maximal ideal p M A, and is equal to GL r (A p ) for all but finitely many p M A. Remark 2.4. Often, the symbol Γ p is used to denote the image of ρ p. However, in our case, we only have Γ p Im(ρ p ). 10

11 We will also need a few results about the structure of K as an A-module. Let M be an A-module. Recall that the rank of M is the F -vector space dimension of M A F. An A-module M is tame if any submodule N M of finite rank is finitely generated as an A-module. Furthermore let Div M (N) := {m M am N, for some 0 a A} denote the division hull of an A-submodule N M in M. Remark 2.5. An A-module M is tame if and only if Div M (N) is finitely generated for any finitely generated A-submodule N M. The following theorem is due to Poonen (see [3, Lemma 5]). Theorem 2.6. The field K considered as an A-module is tame. With the aid of this theorem Poonen concluded the following Theorem 2.7. (see [3, Theorem 1]) The A-module K is the direct sum of a finite torsion submodule with a free A-module of countably infinite rank. As a direct consequence we have Corollary 2.8. For any natural number d there exist A-linearly independent elements x 1,..., x d K. 11

12 3 An ingredient from group cohomology The proofs of Proposition 4.6 and Lemma 6.13 will depend on the vanishing of first group cohomology groups. With this we deal in this section. Let us first recall the definition of the first group cohomology of a group G and a G-module M. Consider the abelian group Map(G, M) of maps f : G M. The subgroup Z 1 (G, M) of 1-cocycles and the subgroup B 1 (G, M) of 1- coboundaries are defined by and Z 1 (G, M) := {f f(στ) = σ f(τ) + f(σ), for all σ, τ G} B 1 (G, M) := {f m M with f(σ) = m σ m, for all σ G}. Note that B 1 (G, M) Z 1 (G, M). The first group cohomology of G and M is defined by H 1 (G, M) := Z 1 (G, M)/ B 1 (G, M). In the sequel we will only be concerned with cases in which M is a vector space. Lemma 3.1. Let k be a field and n 1 a natural number. Let H be a group with a group homomorphism H α GL n (k) and let H act on k n by means of α. Let λ GL n (k) be a nontrivial scalar. If there exists an element h in the center of H with α(h) = λ, then H 1 (H, k n ) = {0}. Proof. Let f Z 1 (H, k n ) and set a := f(h). We claim that 1 λ (3.2) f(g) = a α(g) a for all g H. Indeed, we have f(h 1 ) = f(h) λ since λf(h 1 ) + f(h) = f(hh 1 ) = f(1) = 0. 12

13 This together with the cocycle property of f and the fact that h is in the center of H implies for all g H: f(g) = f(ghh 1 ) = f(hgh 1 ) = λα(g) f(h 1 ) + f(hg) = λα(g) ( 1 f(h)) + λ f(g) + f(h) = (1 α(g)) f(h) + λ f(g) λ = (1 λ)(1 α(g)) a + λ f(g). From this, we conclude (3.2) as desired. For any maximal ideal p M A the natural action of Γ p on κ r p is given via the projection GL r (A p ) νp GL r (κ p ). Moreover, the action of Γ ad on F r p and κ r p is given by and α 1 : Γ ad α 2 : Γ ad p M A GL r (A p ) GL r (A p ) respectively. We obtain the following p M A GL r (A p ) GL r (κ p ) Corollary 3.3. We have the following vanishing of the first group cohomologies: (i) H 1 (GL r (κ p ), κ r p) = {0} for all p M A, such that κ p 3, (ii) H 1 (Γ p, F r p ) = {0} for all p M A, (iii) H 1 (Γ p, κ r p) = {0} for all p M A, such that ρ p is surjective and κ p 3, (iv) H 1 (Γ ad, F r p ) = {0} for all p M A, and (v) H 1 (Γ ad, κ r p) = {0} for all p M A, such that ρ p is surjective and κ p 3. Proof. In the first part we may apply the lemma to k = κ p and H = GL r (κ p ), in the second to k = F p and H = Γ p GL r (A p ). Indeed, GL r (κ p ) contains a non-trivial scalar since κ p 3. Note that also Γ p contains non-trivial scalars for any p, since [ A p : A p Γ p ] [GLr (A p ) : Γ p ] < and A p =, where we have identified A p with the scalars in GL r (A p ). 13

14 To see (iii) consider GL r (A p ) νp GL r (κ p ) and let α := ν p Γp. Since ρ p = π p ρ p, we conclude by our assumption that α is surjective. We may take h α 1 (λ) A p of any nontrivial scalar λ in GL r (κ p ). The latter exists since κ p 3. As h is a scalar, it is contained in the center of Γ p. Setting H = Γ p, the lemma applies. For parts (iv) and (v) we set H := Γ ad and α := α 1 and α := α 2 respectively. In both cases we can chose a nontrivial scalar h p Γ p which, by the proof of (iii), can be chosen in the latter case such that ν p (h p ) is a nontrivial scalar. By Corollary??, the element h := (h p ) p M A p M A GL r (A p ) with h p = 1 for p p is contained in Γ ad. As h is a product of scalars it is moreover contained in the center of Γ ad. These choices of H, α and h are thus appropriate to apply the lemma once more. 14

15 4 Setting Let d 1 be a natural number and let x 1,..., x d K be A-linearly independent elements in K, the existence of which is guaranteed by Corollary 2.8. Consider the A-submodule Λ := x 1,..., x d K generated by the x i. Consider a nonzero ideal I A. Note that ϕ I x i is a separable polynomial when considered as an element in K[X] (its derivative is a non-zero constant). We can therefore choose elements y i ϕ 1 I ({x i }) K s for 1 i d. We get σy i y i Ker (ϕ I ) for all σ G K, as σ commutes with ϕ I. Thus, we can define the map η : G K (Ker(ϕ I )) d = Matr d (A p ) σ (σy 1 y 1,..., σy d y d ). Next, consider the action of GL r (A/I) on Mat r d (A/I) given by the matrix multiplication, which induces a group homomorphism from GL r (A/I) to Aut(Mat r d (A/I)). We denote by Mat r d (A/I) GL r (A/I) the semidirect product of Mat r d (A/I) and GL r (A/I) with respect to this homomorphism. We may define the map Φ I : G K Mat r d (A/I) GL r (A/I) σ (η(σ), ρ I (σ)). which is in fact a group homomorphism. We do not prove this property at this point. It will follow as a consequence of Lemma 4.3, (ii). Let p M A be a maximal ideal. Replacing A/I by A p we can define a map analogous to Φ I. For this purpose let us first introduce some notation. Let ϕ p 0 := id K s and define the A-submodule of K s Ker(ϕ p ) := n 0 Ker ( ϕ π n p ) K s. Set K p := K(Ker(ϕ p )) and let K ad denote the composite of all K p for p M A (A). Let M K s be an A-submodule of K s (we will in fact only be concerned with the cases M = {0} and M = Λ). For n 0 consider the A-module ϕ 1 π n p (M) Ks, 15

16 where we interpret ϕ π n p as an A-modul endomorphism of K s. The set of A-modules {ϕ 1 π (M)} p n n 0 together with the homomorphisms ϕ πp : ϕ 1 πp n+1 (M) ϕ 1 π n p (M) form an inverse system, the inverse limit of which we shall denote by T p [M] := lim ϕ 1 π (M). p n n 0 Note that T p [ϕ] = T p [0]. Moreover, let K L K s be a field extension of K with M L. Note that the action of ϕ πp and the action of σ G L on elements in K s commute. Hence we get an action of G L on T p [M] given by σ (w n ) n 0 = (σ(w n )) n 0, (w n ) n 0 T p [M]. Consider the projection homomorphism π M : T p [M] M, (w n ) n 0 w 0 which yields a G L -equivariant short exact sequence of A-modules 0 T p [ϕ] T p [M] π M M 0, where G L acts trivially on M. Observe further that for any σ G L and (w n ) n 0 T p [M] we have (4.1) σw n w n Ker ( ϕ π n p ) and (4.2) ϕ πp (σw n+1 w n+1 ) = σϕ πp (w n+1 ) ϕ πp (w n+1 ) = σw n w n. Hence we get a map ξ L,M : T p [M] Map(G L, T p [ϕ]) (w n ) n 0 [σ (σ(w n ) w n ) n 0 ]. Note that Map(G L, T p [ϕ]) carries a natural A-module structure via T p [ϕ]. Lemma 4.3. Let M and L be as above. Assume M K p L. Then: 16

17 (i) The map ξ L,M is constant on the residue classes mod T p [ϕ]. (ii) Im(ξ L,M ) Hom(G L, T p [ϕ]). (iii) The map ξ L,M is an A-module homomomorphism. In particular, we have an A-module homomorphism ξ L,M : M Hom(G L, T p [ϕ]), which is the factorization of ξ L,M through M. Proof. For (w n ) n 0 T p [ϕ] we have w n K p L for all n 0. Thus the action of elements of G L on (w n ) n 0 is trivial. This shows (i). Next, fix (w n ) n 0 T p [M] and note that σ (w n ) n 0 and (w n ) n 0 belong to the same conjugacy class mod T p [ϕ] for all σ G L. Hence, using (i), we get τσ(w n ) w n = (τσ(w n ) σ(w n )) + (σ(w n ) w n ) = (τ(w n ) (w n )) + (σ(w n ) w n ) for all τ, σ G L, which implies (ii). It is easy to see that ξ L,M is a homomorphism of groups. As the action of A and the action of G L on T p [M] commute, ξ L,M is also A-linear. This proves the last part. We first apply this construction to M = Λ and L = K. For all 1 i d choose elements (w i,n ) n 0 π 1 Λ (x i) and consider the map η p : G K ( T p [ ϕπp ]) d = Matr d (A p ) σ (ξ L,Λ ((w 1,n ) n 0 ),..., ξ L,Λ ((w d,n ) n 0 )). Note that we cannot apply Lemma 4.3 to this setting. The map η p is not a group homomorphism and even depends on the choices of the (w i,n ) n 0. We may, however, apply Lemma 4.3 to M = Λ and L = K ad to obtain the A-module homomorphism ξ Kad,Λ : Λ Hom(G Kad, T p [ϕ]) w [ σ (σ(w n ) w n ) n 0 ]. 17

18 Consider the images ψ i := ξ Kad,Λ(x i ) Hom(G Kad, T p [ϕ]) of the fixed generators x 1,..., x d of Λ and let us consider the group homomorphism Ψ p : G Kad (T p [ϕ]) d = Matr d (A p ) σ (ψ 1 (σ),..., ψ d (σ)). We remark that Ψ p = η p GKad. In particular, η p restricted to G Kad is a group homomorphism, whereas η p itself is not. More generally, for any set T of prime ideals of A we have the homomorphism Ψ T : G Kad p T Mat r d (A p ), For Ψ ad := Ψ MA we will show σ (Ψ p (σ)) p T. Theorem 4.4. The image of Ψ ad is open. We make the general Remark 4.5. All the group homomorphisms from G K (or subgroups of G K ) to profinite groups with which we are concerned in this thesis are continous maps from compact spaces to Hausdorff spaces. Their images are thus closed. Moreover, by a well-known result, closed subgroups of a profinite group are open if and only if they have finite index (See for instance [4, Lemma and Theorem 2.1.3]). On our way to prove Theorem 4.4 we will first determine in Section 6 the images of Ψ p for p M A and show Proposition 4.6. The image of Ψ p is equal to Mat r d (A p ) for almost all p M A and is open for all p M A. The following Lemmas on the module structures of Im(Ψ T ) for T M A will be crucial ingredients to both the proof of Proposition 4.6 as well as to deduce Theorem 4.4 from this Proposition in Section 7. For any subset T M A we set Γ T := p T Γ p. 18

19 Lemma 4.7. For any subset T M A the image Im(Ψ T ) p T Mat r d (A p ) is a Γ T -submodule, where the action of the Γ p on Mat r d (A p ) is given by matrix multiplication. Proof. Let (X p ) p T Γ T. We may thus write X p = ρ p (σ) for some σ G K and for any p T. For any τ G Kad we claim that (X p ) p T Ψ T (τ) Im ( Ψ T ), which is sufficient for the proof of the lemma. For p T and (w i,n ) n 0 π 1 Λ (x i) we have ρ p (σ) n Ψ p (τ) n = σ(τw 1,n w 1,n,..., τw d,n w d,n ) = σ(τσ 1 w 1,n σ 1 w 1,n,..., τσ 1 w d,n σ 1 w d,n ) = (στσ 1 w 1,n w 1,n,..., στσ 1 w d,n w d,n ) = Ψ p (στσ 1 ) n for any n 0. In the second equation we used the first part of Lemma 4.3. As στσ 1 G Kad, the claim follows. The module structure of Im(Ψ T ) stated in the lemma turns Im(Ψ T ) naturally into an F p [Γ T ]-submodule of Mat r d (A p ), where p is the characteristic of A. Let G T := p T GL r (A p ). Let Mat T := p T Mat r r(a p ). Moreover, set Mat p κ T the natural ring homomorphism := Mat {p }. Denote by F p [G T ] Mat T. Consider F p [Γ T ] as a subring of F p [G T ] and denote by B T its image under κ T. Set B p := B {p }. Let B T denote the completion of B T in Mat T. Lemma 4.8. Let T M A. Then Im(Ψ T ) carries a B T -module structure by means of κ T. In particular, this turns Im(Ψ T ) into a B T -module. 19

20 Proof. The first assertion follows from the observation that the kernel of κ T acts trivially on Im(Ψ T ), which is not difficult to see. By continuity of matrix multiplication and because, by Remark 4.5, the image Im(Ψ T ) is closed, we conclude that Im(Ψ T ) also carries a module structure with respect to the closure of B T which extends the one of B T. In the following Lemma we determine B T and B T for any finite set T. Lemma 4.9. For any finite subset T M A we have (4.10) [Mat T : B T ] <. In particular, B T is open. Proof. For the proof of (4.10) note that it is enough to show (4.11) [Mat T : Im(κ T )] < since we already have [Im(κ T ) : B T ] <. The latter is true because F p is finite and because the index of Γ T in G T is finite by Theorem 2.2. To see (4.11) we first consider the case where T = {p } for some p M A. We first show that any matrix X Mat p with only one non-zero component which lies outside the diagonal is contained in the image. Let a be the value of this specified component and let E denote the identity matrix. Then (4.12) X = (E + X) E Im(κ T ). Similarly, (4.12) holds for any matrix with only one non-zero component, which lies in the diagonal, but whose value a is contained in A p \ A p. Here we use that for any such a we have 1 + a A p. Now let k := κ p and (a i ) 1 i k A p be a set of representatives for A p mod A p. Consider the set S of diagonal matrices whose entries in the diagonal are contained in (a i ) 1 i k. From the above reasonning we see that S contains a set of representatives 20

21 for Mat p mod Im(κ T ). This set, consequently, has less than k r elements, which shows (4.11) for T = 1. For a general T = {p 1,..., p n } observe that for any 1 i n the image of the subring F p [ {1} {1} GLr (A p i ) {1} {1} ] under κ T is contained in the subgroup H i := F p F p Mat p i F p F p and has finite index in H i by the case T = 1 and because F p and T are both finite. Let (X i,j ) 1 j ki H i be a set of representatives for H i mod(h i Im(κ T )) for some natural number k i. Since Mat T = H i, it is not difficult to see that { 1 i n 1 i n Y i Y i (X i,j ) 1 j ki } contains a set of representatives for Mat T mod Im(κ T ). This shows (4.11) for general T. The openess of B T now follows from Remark 4.5. Before we proceed, we point out the following: It is due to the applicability of Lemmas 4.7 and 4.8 that we have defined Ψ p on the Galois group G Kad rather than on the larger group G Kp which seems to be a more natural choice at first sight. In fact, Proposition 4.6 would still be valid for the latter choice and its proof could be simplified for instance in Section 3. As in the case of A/I, the action of GL r (A p ) on Aut(Mat r d (A p )) by matrix multiplication gives a group homomorphism from GL r (A p ) to the automorphisms Aut(Mat r d (A p )). We denote the semidirect product of 21

22 Mat r d (A p ) and GL r (A p ) with respect to this homomorphism by Mat r d (A p ) GL r (A p ). This yields the map Φ p : G K Mat r d (A p ) GL r (A p ) σ (η p (σ), ρ p (σ)), which is a group homomorphism by the next lemma. Lemma The map Φ p is a group homomorphism. Proof. Let σ, τ G K. We need to show (ρ p (σ)η p (τ) + η p (σ), ρ p (στ)) = (η p (σ), ρ p (σ)) (η p (τ), ρ p (τ)) = (η p (στ), ρ p (στ)). But by the definition of the maps ρ p and η p this follows from στ(w i,n ) w i,n = στ(w i,n ) σ(w i,n ) + σ(w i,n ) w i,n = σ(τ(w i,n ) w i,n ) + σ(w i,n ) w i,n for 1 i d and (w i,n ) n 0 π 1 Λ (x i). For any subset T M A define the product Φ T : G K p T Mat r d (A p ) GL r (A p ) σ (Φ p (σ)) p T. For Φ ad := Φ MA we get as a consequence of Theorem 2.2 and Theorem 4.4 Corollary The image of Φ ad is open. Proof. By Remark 4.5 it is enough to show that the image of Φ ad has finite index. By Theorem 2.2, Theorem 4.4 and Remark 4.5 we know that Im(Ψ ad ) and Im(ρ ad ) have both finite index. Let n and m be the respective indexes. Write  := p M A A p. 22

23 and let (X i,p ) p Mat r d (Â), 1 i n, be a set of representatives for Mat r d (Â) mod Im(Ψ ad). Moreover, let (Y j,p ) p GL r (Â), 1 j m, be a set of representatives for GL r (Â) mod Im(ρ ad). It is not difficult to see that ((X i,p, Y j,p )) p Mat r d (Â) GLr(Â), 1 i n, 1 j m, then contains a set of representatives for Mat r d (Â) GL r(â) mod Im(Φ ad), which, consequently, is finite. In terms of ideals I A and corresponding homomorphisms Φ I this reads as follows: Corollary There exists a constant C such that [Mat r d (A/I) GL r (A/I) : Im(Φ I )] < C for all ideals I A. Moreover, there exists a finite set T of prime ideals of A, such that for any ideal I A for which none of the prime ideals in T appears in its prime ideal factorization, we have Im(Φ I ) = Mat r d (A/I) GL r (A/I). Proof. By Remark 4.5 and Corollary 4.14, the index of Φ ad is finite. Let C denote this index. Moreover, we can conclude from the openess of Im(Φ ad ) that there exists a set S M A which contains almost all maximal ideals and such that Φ S is surjective. Let T be the complement of S in M A. We show that for these choices of C and T the corollary holds. Thus let I A be an ideal and denote by T (I) the set of maximal ideals which divide I. Moreover, let π(i) denote the projection Mat r d (A p ) GL r (A p ) Mat r d (A/I) GL r (A/I). p T (I) We have π(i) Φ T (I) = Φ I. Corollary 4.14 particularly implies that Im(Φ T (I) ) has finite index. Thus also Im(Φ I ) has finite index. Moreover, if T (I) S, then Φ T (I) is surjective and consequently also Φ I is surjective. 23

24 5 An A p -linear independence An important technical step toward Proposition 4.6 is to show the A p - linear independence of the ψ i, which will be a first confirmation that, indeed, the image of Ψ p is large. Proposition 5.1. The ψ i are A p -linearly independent. We want to give an indirect proof of this Proposition and thus make first the following Assumption 5.2. There exist r i A p for 1 i d which satisfy equation (5.3) d r i ψ i (σ) = 0 i=1 for all σ G Kad. Multiplying the r i by an appropriate element in F p if necessary, we may further assume that r 1 A p. For all n 0 and 1 i d choose r i,n A such that r i,n = r i mod p n. Furthermore, choose elements (x i,n ) n 0 ϕ 1 p (Λ) with x i,0 = x i for 1 i d. Define d y n := ϕ ri,n (x i,n ) for all n 0. Lemma 5.4. For any n 0 we have: (i) y n K ad, and i=1 (ii) ϕ πp (σ(y n+1 ) y n+1 ) = σ(y n ) y n for all σ Gal(K ad /K). Proof. Let σ G Kad and let r i and ψ i (σ) denote the residue classes of r i and ψ i (σ) mod p n respectively. We get σ(y n ) y n = d ϕ ri,n (σ(x i,n ) x i,n ). i=1 Under the isomorphism Ker(ϕ π n p ) = (A/p ) r this element is mapped to d i=1 r i ψ i (σ), where r i ψi (σ) = r i ψ i (σ) mod p. But the latter element is 0 by 24

25 our assumption. Since σ G Kad was arbitrary, the first statement of the lemma follows. To see the second part let σ Gal(K ad /K) and note that r i,n+1 r i,n p n for all n 0, and 1 i d. It follows c n := ϕ πp (y n+1 ) y n = d ϕ (ri,n+1 r i,n )(x i,n ) Λ K. i=1 Hence the c n are fixed by σ. We get ϕ πp (σ(y n+1 ) y n+1 ) = σ(ϕ πp (y n+1 )) ϕ πp (y n+1 ) = σ(y n + c n ) (y n + c n ) = σ(y n ) y n. This proves the second part. To proceed further let us fix an A p -basis of T p [ϕ]. We thus fix elements (C j,n ) n 0 T p [ϕ] which are A p -linearly independent and generate T p [ϕ] as an A p -module. For A r p we may take the standard basis. Consider the isomorphisms T p [ϕ] = A r p and Aut Ap (T p [ϕ]) = GL r (A p ) with respect to these fixed bases. ρ p Note that G K GLr (A p ) factors through Gal(K ad /K). For σ Gal(K ad /K) let (B i,j (σ)) = (B i,j,n (σ)) n 0 denote the corresponding element in GL r (A p ). By the second part of Lemma 5.4 we have (σ(y n ) y n ) n 0 T p [ϕ], for any σ Gal(K ad /K). By means of the above isomorphism we obtain an element (a 1 (σ),..., a r (σ)) A r p such that σ(y n ) y n = r ϕ aj,n (σ)(c j,n ). j=1 The corresponding map f : Gal(K ad /K) A r p Fp r σ (a 1 (σ),..., a r (σ)) is a cocycle by the next lemma. 25

26 Lemma 5.5. The map f is a cocycle, i.e. f Z 1 (Gal(K ad /K), F r p ). Proof. Let σ, γ Gal(K ad /K). Then r στ(y n ) = σ(y n + ϕ aj,n (τ)(c j,n )) = y n + j=1 r ϕ aj,n (σ)(c j,n ) + r j=1 j=1 ϕ aj,n (τ) r ϕ Bi,j,n (σ)c i,n, i=1 and hence r ϕ ai,n (στ)c i,n = i=1 r ϕ ai,n (σ)c i,n + r i=1 j=1 ϕ aj,n (τ) r ϕ Bi,j,n (σ)c i,n. i=1 In F r p this reads as follows: (a 1 (στ),..., a r (στ)) = (a 1 (σ),..., a r (σ)) + (B i,j (σ)) (a 1 (τ),..., a r (τ)). This is exactly f(στ) = f(σ) + (B i,j (σ)) f(τ). Recall from Corollary 3.3 that we have H 1 (Gal(K ad /K), F r p ) = {0}. Thus there exists an element e = (e 1,..., e r ) F r p with the property that for all σ Gal(K ad /K) we have (5.6) f(σ) = e (B i,j (σ)) e. We assume without loss of generality that e Fp r \A r p. Let m be the minimal natural number with p m e A r p and choose s p m \ p m+1. Define the elements z n := ϕ s (y n ) + ϕ s ei,n (C i,n ) K p for all n 0. 1 i r Lemma 5.7. For all n 0 we have z n K. 26

27 Proof. Let σ Gal(K ad /K) be arbitrary. Equation (5.6) yields σ(z n ) = ϕ s σ(y n ) + ϕ sej,n σ(c j,n ) Thus z n K. = ϕ s ( 1 j r y n + = ϕ s y n + 1 j r 1 j r (5.6) = ϕ s y n + 1 j r ϕ aj,n (σ)c j,n ) ϕ saj,n (σ)c j,n + ϕ sej,n C j,n = z n. + 1 j r 1 j r ϕ sej,n 1 i r ϕ s((bi,j (σ)) e) j C j,n ϕ Bi,j,n (σ)c i,n We can now finish the proof of the linear independence of the ψ i. Proof of Proposition 5.1. Assumption 5.2 allows to consider the A-module M := {z n } n 0 generated by the z n, which is a submodule of K by Lemma 5.7. It is even an a submodule of the division hull Div K (Λ) of Λ in K since ϕ π n p (z n ) Λ for all n 0. By remark 2.5 we know that Div K (Λ) is a finitely generated A-module. Consequently, as A is noetherian, also M is finitely generated. Thus there exists a natural number k > 0 such that z k+1 = ϕ ai (z i ) for some a 1,..., a k A. We get From this we conclude that u := 1 i k ϕ π k p (z k+1 ) = 1 i r 1 i k ϕ ai (ϕ π k p (z i )). ϕ s ei,k+1 (C i,1 ) ϕ m+k π p (Λ). By the A-linear independence of the x i and since u Ker(ϕ πp ) this is only possible if u = 0. By the κ p -linear independence of the C i,1 and because e Fp r \ A r p, this implies s e i,k+1 p for all i. But this contradicts the definition of s. 27

28 6 A p -adic result of Kummer-type Fix a maximal ideal p M A and let M p := Im(Ψ p ). In this section we show Proposition 4.6. We split its proof into two parts and show first that M p has finite index in Mat r d (A p ) for all p M A. Then we show that for almost all p M A we even have M p = Mat r d (A p ). Part I. By the use of the linear independence of the ψ i we will now show that [Mat r d (A p ) : M p ] <. Recall from Section 4 the subring B p Mat r r (A p ) for which M p is a B p - module by Lemma 4.8. By Lemma 4.9 we further know that the index [Mat r r (A p ) : B p ] is finite. Let C p := B p A p, where we identify A p with the scalars in Mat r r (A p ). Lemma 4.9 thus particularly implies (6.1) [A p : C p ] [Mat r r (A p ) : B p ] <. We have Lemma 6.2. We have F p B p = Mat r r (F p ). Proof. Let 0 X Mat r r (F p ) be arbitrary and choose an element t A p with Y := tx Mat r r (A p ). As the set A p Y of A p -multiples of Y is infinite and the index of B p in Mat r r (A p ) is finite we get A p Y B p {0}. Thus there exists a nonzero s A p with sy B p. This yields X = 1 st (sy ) F pb p. As the zero matrix is also contained in F p B p, this finishes the proof. Consider the F p B p -submodule F p M p Mat r d (F p ) which is a Mat r r (F p )- module by Lemma 6.2. We will obtain F p M p = Mat r d (F p ) as a consequence of the general Lemma 6.3. Let k be a field and let r and d be positive natural numbers. Let U be an r-dimensional k-vectorspace and let V := U U 28

29 be the product of d copies of U which is equipped with a natural Mat r r (k)- module structure. Let W be a proper Mat r r (k)-submodule of V. Then there exist elements c 1,..., c d k, not all zero, such that all (w 1,..., w d ) W satisfy d c i w i = 0. i=1 Proof. First note that Mat r r (k) is a noetherian ring and V is a finitely generated Mat r r (k)-module. Thus V is noetherian and hence also V/W. We therefore obtain a Mat r r (k)-submodule W W V such that V/W is a simple module. As k r is the unique simple Mat r r (k)-module this yields a surjective Mat r r (k)-module homomorphism f : V k r, whose restriction to W vanishes. Let V i V denote the i-th copy of U considered as a Mat r r (k)-submodule of V. Thus V = V i. 1 i d Let f i denote the restiction of f to V i. We have V i = k r as Mat r r (k)- modules and hence Hom Matr r (k)(v i, k r ) = k. In other words the Mat r r (k)- module homomorphisms f i correspond to the multiplication by elements c i k. Note that, because f 0, not all of the c i s vanish. For all v = (v 1,..., v d ) V we get f(v) = f i (v i ) = c i v i, 1 i d 1 i d where for the last equation we have used the identification V i = k r. As f restricted to W vanishes, the c i s have the desired properties. Corollary 6.4. We have F p M p = Mat r d (F p ). Proof. Assume by contradiction that F p M p Mat r d (F p ). Since F p M p is a Mat r r (F p )-submodule, the lemma applies and yields elements c 1,..., c d F p, not all zero, such that (6.5) d c i w i = 0, i=1 29

30 for all (w 1,..., w d ) F p M p. Multiplying, if necessary, the c i by an appropriate element of A p, we can assume that the c i are elements in A p. Equation (6.5) particularly holds for all (ψ 1 (σ),..., ψ d (σ)), σ G Kp and thus contradicts Proposition 5.1. From Corollary 6.4 we can deduce Proposition 6.6. The index of M p in Mat r d (A p ) is finite. Proof. We proceed in two steps and show first (6.7) [Mat r d (A p ) : A p M p ] <, and then (6.8) [A p M p : M p ] <, which together yield the statement. To see (6.7) let us fix a finite set of generators (X i ) 1 i l for some l > 0 of the A p -module Mat r d (A p ). By Corollary 6.4 we obtain elements s i A p \ {0} with s i X i A p M p. Set We thus get s := 1 i l s i 0. sx A p M p for any X Mat r d (A p ). Let n > 0 be such that s is a unit mod p n. We claim that p n Mat r d (A p ) A p M p. Indeed, if X p n Mat r d (A p ) we may write X = ty for some Y Mat r d (A p ) and t p n. By the choice of n, we have u := t s A p and thus Therefore we get X = usy ua p M p A p M p. [Mat r d (A p ) : A p M p ] [Mat r d (A p ) : p n Mat r d (A p )] <. Let us turn to (6.8). We use [2, Theorem 3], which in particular claims that, whenever S is a subring of a commutative noetherian ring R, such that 30

31 R is a finitely generated S-module, then S as well is a noetherian ring. In our case note that A p is noetherian. As A p is finitely generated over the subring C p, we conclude that also C p is Noetherian. Therefore, the finitely generated C p -module Mat r d (A p ) is noetherian and hence, M p is a finitely generated C p -module. Let us choose generators Y 1,..., Y m of the C p -module M p. By (6.1) we may also choose a finite set of representatives {λ 1,..., λ n } of A p mod C p. It is not difficult to see that { } a ij λ i Y j a ij {0, 1} i,j is a set of representatives for A p M p mod M p. Since it is finite, we are finished. Part II. Next, we will show that M p = Mat r d (A p ) for all but finitely many maximal ideals p M A. The method will be similar to the one we used to conclude Proposition 6.6 with only slight adaptions to the new setting. To specify the finite set of primes which has to be excluded from our considerations, we need Lemma 6.9. There exists an element 0 c A which annihilates Div K (Λ)/Λ. Moreover, we have (6.10) ϕ 1 π p (Λ) K = Λ. for any maximal ideal p M A which does not divide c. Proof. By Remark 2.5 we know that Div K (Λ) is finitely generated. Let {µ 1,..., µ k } denote a set of generators. For all 1 i k let 0 c i A be such that c i µ i Λ. The element c := c i A \ {0} 1 i k therefore annihilates Div K (Λ)/Λ. Moreover, for a maximal ideal p M A which does not divide c let us choose a 1, a 2 A with a 1 π p + a 2 c = 1. 31

32 If we take x ϕ 1 π p (Λ) K, we get x = ϕ a1 π p (x) + ϕ a2 c(x). By the choice of c and the definition of x both summands are contained in Λ. This shows ϕ 1 π p (Λ) K Λ. The reverse inclusion of (6.10) is easy to see. Denote by S the set of prime ideals which satisfy both of the following two properties: Γ p = GL r (A p ), ϕ 1 π p (Λ) K = Λ. By Corollary 2.3 and Lemma 6.10 it contains all but finitely many prime ideals. Proposition We have M p = Mat r d (A p ) for all p S. To give an indirect proof, we make the following Assumption There exists a prime ideal p S for which M p Mat r d (A p ). We first claim that Mat r d (A p )/M p is a noetherian Mat r r (A p )-module. Indeed, as A p is noetherian, so is Mat r r (A p ). The Mat r r (A p )-module Mat r d (A p ) is finitely generated and thus noetherian. Hence also the quotient Mat r d (A p )/M p is noetherian. Note further, that κ r p is the unique simple Mat r r (A p )-module. Thus, assuming M p Mat r d (A p ), there exists a surjective Mat r r (A p )-module homomorphism g : Mat r d (A p )/M p κ r p, which factors through Mat r d (κ p )/M p, where M p denotes the image of M p Mat r d (A p ) Mat r d (κ p ). In particular M p Mat r d (κ p ). Note that the factorization of g, g : Mat r d (κ p )/M p κ r p, 32

33 is also a Mat r r (κ p )-module homomorphism. Writing ψ i := ψ i mod p and we get Ψ := ( ψ 1,..., ψ d ), M p = Im(Ψ). Recall further the A-module homomorphism ξ Kad,Λ : Λ Hom(G Kad, T p [ϕ]). The projection of T p [ϕ] onto Ker(ϕ πp ) induces an A-module homomorphism π : Hom(G Kad, T p [ϕ]) Hom(G Kad, Ker(ϕ πp )). Set ε := π ξ Kad,Λ and note that ψ i = ε(x i ) for all 1 i d. Once more we need the vanishing of a first group cohomology group. By a method similar to the one we used to show the linear independence of the ψ i, but applied to an easier setting, we can prove Lemma Let x K such that ϕ 1 p (x) K ad. Then ϕ 1 p (x) K. Proof. Choose an element w ϕ 1 p (x) and consider the map f : Gal(K ad /K) Ker(ϕ πp ), σ σ(w) w. We have f Z 1 (Gal(K ad /K), Ker(ϕ πp )), since f(στ) = στw σw + σw w = σ(τ(w) w) + σw w = σf(τ) + τ. for all σ, τ Gal(K ad /K). By Corollary 3.3, (iii) and since Γ p = GL r (A p ), we have H 1 (Gal(K ad /K), Ker(ϕ πp )) = {0}. We thus obtain an element z Ker(ϕ πp ) satisfying This yields f(σ) = z σz for all σ Gal(K ad /K). σ(w + z) = σw + σz = w + f(σ) + σz = w + z σz + σz = w + z, for all σ Gal(K ad /K), and hence w + z ϕ 1 p (x) K. 33

34 As a consequence of Lemma 6.13 we get Corollary We have Ker(ε) = ϕ πp Λ. Moreover, the ψ i are κ p -linearly independent. Proof. Let x Ker(ε). Thus ϕ 1 p (x) K ad. By Lemma 6.13 and since we get an element ϕ 1 π p (Λ) K = Λ w ϕ 1 p (x) Λ. Hence x = ϕ πp w ϕ πp Λ. The reverse inclusion is easy to see. To see the second assertion, set x i := x i mod ϕ πp Λ and denote by ε the factorization of ε through Λ/ϕ πp Λ which is an injective κ p -module homomorphism. As the x i are A-linear independent, the x i are κ p -linear independent. The κ p -linear independence of the ψ i thus follows immediately from ψ i = ε(x i ). By the same reasoning as in the proof of Corollary 6.4 the κ p -linear independence of the ψ i together with M p Mat r d (κ p ) yields a contradiction to Lemma 6.3. Consequently, Assumption 6.12 is false. This finishes the proof of Proposition

35 7 An adelic result of Kummer-type Let us finally tackle the proof of Theorem 1.2. Consider again M p := Im Ψ p for p M A. Lemma 7.1. Let p be a prime ideal with κ p > 3 and Γ p = GL r (A p ). Consider M p as a Γ p -module. Then every element in M p is of the form gx X for some g Γ p and X M p. In particular, M p has no coinvariants. Proof. Let X be an element of M p. We need to find elements g Γ p and X M p with (7.2) X = gx X. We again identify A p with the scalars in GL r (A p ). By our assumption on p we may take an element λ A p Γ p such that λ ±1 mod p. Thus g := λ + 1 is an element in A p. Setting additionally X := 1 λ X M p yields elements which satisfy (7.2). Let T be any set of maximal ideals of A. In the sequel we set M T := p T Mat r d (A p ), and M p := M {p }. Recall the group homomorphism Ψ T : G Kad M T from Section 4. σ (Ψ p (σ)) p T. Recall from Section 6, Part II the set S of prime ideals p for each of which we have shown that Ψ p is surjective. Let S denote the subset of S obtained by removing all the prime ideals p S with κ p 3 and let T be its complement in M A. Note that T is finite and write T := {p 1,..., p l } for some natural number l. Recall from Lemma 4.8 that Im(Ψ T ) has a module structure over the open subring B T p T Mat r r (A p ). 35

36 Let us again identify A p with the scalars in Mat r r (A p ) for any p M A. Consider 1 i l A p i as a subgroup of p T Mat r r (A p ) and let C T := ( B T 1 i l A p i ) 1 i l A p i. Hence C T is an open subgroup of 1 i l A p i with respect to the subspace topology. As the subspace topologies of a profinite group coincides with the profinite topology of its subgroups, there exist elements 0 λ p i for 1 i l with (7.3) A p i λ p i C T B T. 1 i l A p i This we need to show Lemma 7.4. The index of Im(Ψ T ) in M T is finite. Proof. We fix a p i T and set p := p i. By Proposition 6.6 we now that (7.5) [M p : Im(Ψ p )] <. Consider M p as a subgroup of M T. As T is a finite set and since p T is arbitrary, it is enough to show Hence, by (7.5), it is enough to prove However, we know by (7.3) that [M p : (M p Im(Ψ T ))] <. [Im(Ψ p ) : (M p Im(Ψ T ))] <. A p λ p Im(Ψ p ) (M p Im(Ψ T )). We can therefore make another reduction and do only have to show (7.6) [Im(Ψ p ) : A p λ p Im(Ψ p )] <. To see (7.6), recall the subring C p := B p A p A p from Section 6 and recall that Im(Ψ p ) is a C p -submodule of M p. By (6.1) we know that C p has finite 36

37 index. Moreover, in the proof of Proposition 6.6 we argued that C p is a noetherian ring. From (6.1) we know that M p is a finitely generated C p -module. As C p is noetherian thus also Im(Ψ p )/A p λ p Im(Ψ p ) is a finitely generated C p -module. Consequently, it is even a finitely generated module over the ring R := C p /C p λ p. Note that R is well-defined since λ p is contained in C p. To conclude (7.6), it is therefore enough to show that R is a finite ring. Note that since λ p 0. Moreover, we have [A p : A p λ p ] < [A p λ p : C p λ p ] [A p : C p ] <, by (6.1). Combining the two preceding statements, we get indeed which finishes the proof. R [A p : C p λ p ] <, We are finally prepared to prove Theorem 4.4, i.e. to show that Ψ ad has an open image. Proof of Theorem 4.4. As the image of Ψ ad is closed it only remains to be shown by Remark 4.5 that the image has finite index in M MA. Set For 1 i N let N := [M T : Im(Ψ T )]. ˆX i = (X i,p ) p MA M MA be elements such that X i,p = 0 for all p S and such that their projection on M T form a set of representative for M T mod Im(Ψ T ). We claim that for any finite set T M A which contains T the projections of the ˆX i on M T form as well a set of representatives for M T mod Im(Ψ T ). We prove this claim by induction on t := T T. For T = T this follows straight from the definitions. To see the induction step, let t > T and write T = {p 1,..., p t } T, 37

38 with p t / T. Consider M p i as a subgroup of M T for any 1 i t. Note that it is enough to show (7.7) M p t Im(Ψ T ). By Lemma 4.7, the image Im(Ψ T ) M T is a Γ p i -submodule. 1 i t Since p t / T we have Γ p t = GL r (A p t ). In particular, Im(Ψ T ) is a GL r (A p t )- submodule, where we identify GL r (A p t ) with {1} {1} GL r (A p t ) Γ p i. 1 i t Thus the action of GL r (A p t ) on M p i is the trivial action for 1 i t 1 and the usual matrix multiplication for i = t. To see (7.7) let X p t be an element in M p t. By Proposition 6.11 we have Im(Ψ p t ) = M p t. Thus there exists a σ G Kad with Ψ p t (σ) = X p t. For any element g GL r (A p t ) we get ( ) ( 0,..., 0, gxp t X p t = Ψp 1 (σ),..., Ψ p (σ), gψ t 1 p t (σ) ) ( Ψ p 1 (σ),..., Ψ p t 1 (σ), Ψ p t (σ) ) = gψ T (σ) Ψ T (σ) Im(Ψ T ). By Lemma 7.1 every element in M p t is of the form gx p t X p t for some g GL r (A p t ) and X p t M p t. Thus, indeed, (7.7) holds, which proves our claim. Using the closedness of Im(Ψ ad ) M MA, we conclude from the claim that the ( ˆX i ) 1 i N form a set of representatives for M MA mod Im(Ψ ad ). More precisely, given any Ŷ i = (Y i,p ) p MA M MA, there exists an ˆX i for some 1 i N such that (Y i,p ) p T (X i,p ) p T Im(Ψ T ) for any finite set T T M A. The closedness implies Ŷi ˆX i Im(Ψ ad ). We have thus shown that the image of Ψ ad has finite index. 38

39 References [1] Devic, A., Pink, R., Adelic Openess for Drinfeld Modules in Special Characteristic, Diss. ETH No (2010). [2] Formanek, E., Jategaonkar, A., Subrings of Noetherian Rings, Proceedings of the American Mathematical Society, Vol. 46, No. 2 (1974), [3] Poonen, B., Local height functions and the Mordell-Weil theorem for Drinfeld modules, Compositio Mathematica, tome 97 (1995), [4] Ribes, L., Zalesskii, P., Profinite Groups, Springer (2010). [5] Ribet, K., Kummer Theory on Extensions of Abelian Varieties by Tori, Duke Mathematical Journal, Vol. 46, No. 4 (1979), [6] Rosen, M., Number Theory in Function Fields, Springer (2002), [7] Pink, R., Rütsche, E., Absolute Irreducibility of the Residual Representation and Adelic Openness in generic characteristic for Drinfeld modules, Diss. ETH No ,