Independence of l-adic representations of geometric Galois groups
IIndependence of (cid:96) -adic representationsof geometric Galois groups
G. B¨ockle, W. Gajda and S. PetersenOctober 30, 2018
Abstract
Let k be an algebraically closed field of arbitrary characteristic, let K/k be a finitelygenerated field extension and let X be a separated scheme of finite type over K . For eachprime (cid:96) , the absolute Galois group of K acts on the (cid:96) -adic etale cohomology modules of X .We prove that this family of representations varying over (cid:96) is almost independent in thesense of Serre, i.e., that the fixed fields inside an algebraic closure of K of the kernels ofthe representations for all (cid:96) become linearly disjoint over a finite extension of K . In doingthis, we also prove a number of interesting facts on the images and on the ramification ofthis family of representations. Let G be a profinite group and L a set of prime numbers. For every (cid:96) ∈ L let G (cid:96) be a profinitegroup and ρ (cid:96) : G → G (cid:96) a homomorphism. Denote by ρ : G → (cid:89) (cid:96) ∈ L G (cid:96) the homomorphism induced by the ρ (cid:96) . Following the notation in [37] we call the family ( ρ (cid:96) ) (cid:96) ∈ L independent if ρ ( G ) = (cid:81) (cid:96) ∈ L ρ (cid:96) ( G ). The family ( ρ (cid:96) ) (cid:96) ∈ L is said to be almost independent if thereexists an open subgroup H of G such that ρ ( H ) = (cid:81) (cid:96) ∈ L ρ (cid:96) ( H ).The main examples of such families of homomorphisms arise as follows: Let K be a field withalgebraic closure (cid:101) K and absolute Galois group Gal( K ) = Aut( (cid:101) K/K ). Let
X/K be a separatedalgebraic scheme and denote by L the set of all prime numbers. For every q ∈ N and every (cid:96) ∈ L (cid:114) { char( k ) } we shall consider the representations ρ ( q ) (cid:96),X : Gal( K ) → Aut Q (cid:96) ( H q ( X (cid:101) K , Q (cid:96) )) and ρ ( q ) (cid:96),X,c : Gal( K ) → Aut Q (cid:96) ( H qc ( X (cid:101) K , Q (cid:96) ))of Gal( K ) on the ´etale cohomology groups H q ( X (cid:101) K , Q (cid:96) ) and H qc ( X (cid:101) K , Q (cid:96) ). The following inde-pendence result has recently been obtained. Key words:
Galois representation, ´etale cohomology, algebraic scheme, finitely generated field A scheme
X/K is algebraic if the structure morphism X → Spec K is of finite type (cf. [15, Def. 6.4.1]). a r X i v : . [ m a t h . N T ] O c t ndependence of (cid:96) -adic representations Theorem 1.1.
Let K be a finitely generated extension of Q and let X/K be a separated algebraicscheme. Then the families ( ρ ( q ) (cid:96),X ) (cid:96) ∈ L and ( ρ ( q ) (cid:96),X,c ) (cid:96) ∈ L are almost independent. The proof of this statement in the important special case trdeg( K/ Q ) = 0 is due to Serre(cf. [37]). The case trdeg( K/ Q ) > ρ (cid:96) : Gal( K ) → G (cid:96) ) (cid:96) ∈ L means that after a finite field ex-tension E/K , the image of Gal( E ) under the product representation (cid:81) (cid:96) ∈ L ρ (cid:96) is the product P = (cid:81) (cid:96) ∈ L ρ (cid:96) (Gal( E )) of the images. In particular for any finite extension F/E , the image ofGal( F ) is open in P . This has applications if one has precise knowledge of the shape of the im-ages for all (cid:96) . For instance, suppose that there exists a reductive algebraic subgroup G of someGL n over Q such that for all sufficiently large finite extensions F of K the image ρ (cid:96) (Gal( F )) isopen in G ( Q (cid:96) ) ∩ GL n ( Z (cid:96) ) for all (cid:96) and surjective for almost all (cid:96) . Then almost independenceimplies that for some finite extension E of K the image Gal( E ) is adelically open, i.e., it is openin the restricted product (cid:81) (cid:48) (cid:96) ∈ L G ( Q (cid:96) ). For K a number field, the Mumford-Tate conjecture(cf. [34, C.3.3, p. 387]) predicts a group G as above if ρ (cid:96) arises from an abelian variety over K .The present article is concerned with a natural variant of Theorem 1.1 that grew out of thestudy of independence of families over fields of positive characteristic. For K a finitely generatedextension of F p it has long been known, e.g. [20] or [11], that the direct analogue of Theorem 1.1is false: If ε (cid:96) : Gal( F p ) → Z × (cid:96) denotes the (cid:96) -adic cyclotomic character that describes the Galoisaction on (cid:96) -power roots of unity, then it is elementary to see that the family ( ε (cid:96) ) (cid:96) ∈ L(cid:114) { p } is notalmost independent. It follows from this that for every abelian variety A/K , if we denote by σ (cid:96),A : Gal( K ) → Aut Q (cid:96) ( T (cid:96) ( A )) the representation of Gal( K ) on the (cid:96) -adic Tate module of A ,then ( σ (cid:96),A ) (cid:96) ∈ L(cid:114) { p } is not almost independent. One is thus led to study independence over thecompositum (cid:101) F p K obtained from the field K by adjoining all roots of unity. Having gone thatfar, it is then natural to study independence over any field K that is finitely generated over anarbitrary algebraically closed field k . Our main result is the following independence theorem. Theorem 1.2. (cf. Theorem 7.5)
Let k be an algebraically closed field of characteristic p ≥ .Let K/k be a finitely generated extension and let
X/K be a separated algebraic scheme. Thenthe families ( ρ ( q ) (cid:96),X | Gal( K ) ) (cid:96) ∈ L(cid:114) { p } and ( ρ ( q ) (cid:96),X,c | Gal( K ) )) (cid:96) ∈ L(cid:114) { p } are almost independent. It will be clear that many techniques of the present article rely on [37]. Also, some of the keyresults of [14] will be important. The new methods in comparison with the previous results arethe following: (i) The analysis of the target of our Galois representations, reductive algebraicgroups over Q (cid:96) , will be based on a structural result by Larsen and Pink (cf. [27]) and no longeras for instance in [37] on extensions of results by Nori (cf. [31]). In the technically useful case k (cid:54) = (cid:101) k , this facilitates greatly the passage from Gal( K ) to Gal( K (cid:101) k ) when studying their imageunder ρ ( q ) (cid:96),X, ? . (ii) Since we also deal with cases of positive characteristic, ramification propertieswill play a crucial role to obtain necessary finiteness properties of fundamental groups. Theresults on alterations by de Jong (cf. [6]) will obviously be needed. However we were unableto deduce all needed results from there, despite some known semistability results that follow2ndependence of (cid:96) -adic representationsfrom [6]. Instead we carry out a reduction to the case where K is absolutely finitely generatedand where X/K is smooth and projective (this uses again [6]). (iii) In the latter case, we usea result by Kerz-Schmidt-Wiesend (cf. [24]) that allows one to control ramification on X bycontrolling it on all smooth curves on X . Since X is smooth, results of Deligne show that thesemisimplifications of ρ ( q ) (cid:96),X, ? form a pure and strictly compatible system. On curves, we can thenapply the global Langlands correspondence proved by Lafforgue in [26]. This is a deep tool, butit allows us to obtain a very clean conclusion about the ramification properties of ( ρ ( q ) (cid:96),X, ? ) (cid:96) ∈ L(cid:114) { p } .Part (i) is carried out in Section 3. Results on fundamental groups and first results on ramifica-tion are the theme of Section 4; this includes parts of (ii) and we also refine some results from[24]. Section 5 provides the basic independence criterion on which our proof of Theorem 1.2ultimately rests. Section 6 performs the reductions mentioned in (ii). The ideas described in(iii) are concluded in Section 7, where a slightly more precise form of Theorem 1.2 is proved.We would like to point out that an alternative method for the part (ii) of our approach could bebased on a recent unpublished result by Orgogozo which proves a global semistable reductiontheorem (cf. [32, 2.5.8. Prop.]). When our paper was complete we were informed by AnnaCadoret that, together with Akio Tamagawa, she has proven our Theorem 1.2 by a differentmethod cf. [5]. Acknowledgments:
G.B. thanks the Fields Institute for a research stay in the spring of 2012during which part of this work was written. He also thanks Adam Mickiewicz University inPozna´n for making possible a joint visit of the three authors in the fall of 2012. He is supportedby a grant of the DFG within the SPP 1489. W.G. thanks the Interdisciplinary Center forScientific Computing (IWR) at Heidelberg University for hospitality during a research visit inJanuary 2012 shortly after this project had been started. He was partially supported by theAlexander von Humboldt Foundation and by research grant UMO-2012/07/B/ST1/03541 ofthe National Centre of Sciences of Poland. S.P. thanks the Mathematics Department at AdamMickiewicz University for hospitality and support during several research visits. We thank F.Orgogozo and L. Illusie for interesting correspondence concerning this project.
For a field K with algebraic closure (cid:101) K , we denote by K s ⊂ (cid:101) K a separable closure. Then Gal( K )is equivalently defined as Gal( K s /K ) and as Aut( (cid:101) K/K ), since any field automorphism of K s fixing K has a unique extension to (cid:101) K . If E/K is an arbitrary field extension, and if (cid:101) K is choseninside (cid:101) E , then there is a natural isomorphism Aut( (cid:101) K/ (cid:101) K ∩ E ) (cid:39) −→ Aut( (cid:101)
KE/E ). Composing itsinverse with the natural restriction Gal( E ) → Aut( E (cid:101) K/E ) one obtains a canonical map whichwe denote res
E/K : Gal( E ) → Gal( K ). If E/K is algebraic, then res
E/K is injective and weidentify Gal( E ) with the subgroup res E/K (Gal( E )) = Gal( E ∩ (cid:101) K ) of Gal( K ).Let G be a profinite group. A normal series in G is a sequence G (cid:46) N (cid:46) N (cid:46) · · · (cid:46) N s = { e } of closed subgroups such that each N i is normal in G .3ndependence of (cid:96) -adic representationsA K -variety X is a scheme X that is integral separated and algebraic over K . We denote by K ( X ) its function field. Let S be a normal connected scheme with function field K . A separablealgebraic extension E/K is said to be unramified along S if for every finite extension F/K inside E the normalization of S in F is ´etale over S . We usually consider S as a scheme equipped withthe generic geometric base point s : Spec( (cid:101) K ) → S and denote by π ( S ) := π ( S, s ) the ´etalefundamental group of S . If Ω denotes the maximal extension of K in K s which is unramifiedalong S , then π ( S ) can be identified with the Galois group Gal(Ω /K ). A homomorphism ρ : Gal( K ) → H is said to be unramified along S if the fixed field K ker( ρ ) s is unramified along S .If E/K is an arbitrary algebraic extension, then ρ | Gal( E ) stands for ρ ◦ res E/K . In this section, we prove a structural result for compact profinite subgroups of linear algebraicgroups over (cid:101) Q (cid:96) (cf. Theorem 3.4) that will be crucial for the proof of the main theorem of thisarticle. It is a consequence of a variant (cf. Proposition 3.8) of a theorem of Larsen and Pink(cf. [27, Thm. 0.2, p. 1106]). The proof of Proposition 3.8 makes strong use of the results andmethods in [27], and in particular does not depend on the classification of finite simple groups. Definition 3.1.
For c ∈ N we denote by Σ (cid:96) ( c ) the class of profinite groups M which possess anormal series by open subgroups M (cid:46) I (cid:46) P (cid:46) { } such that M/I is a finite product of finite simple groups of Lie type in characteristic (cid:96) , the group
I/P is finite abelian of order prime to (cid:96) and index [ I : P ] ≤ c , and P is a pro- (cid:96) group. Definition 3.2.
For d ∈ N and (cid:96) a prime we denote by Jor (cid:96) ( d ) the class of finite groups H which possess a normal abelian subgroup N of order prime to (cid:96) and of index [ H : N ] ≤ d . Wedefine Jor( d ) as the union of the Jor (cid:96) ( d ) over all primes (cid:96) . Definition 3.3.
A profinite group G is called n -bounded at (cid:96) if there exist closed compactsubgroups G ⊂ G ⊂ GL n ( (cid:101) Q (cid:96) ) such that G is normal in G and G ∼ = G /G . The following is the main result of this section.
Theorem 3.4.
For every n ∈ N there exists a constant J (cid:48) ( n ) (independent of (cid:96) ) such thatthe following holds: For any prime (cid:96) , any group G that is n -bounded at (cid:96) lies in a short exactsequence → M → G → H → such that M is open normal in G and lies in Σ (cid:96) (2 n ) and H lies in Jor (cid:96) ( J (cid:48) ( n )) . We state an immediate corollary:
Corollary 3.5.
Let G be n -bounded at (cid:96) and define G + (cid:96) as the normal hull of all pro- (cid:96) Sylowsubgroups of G . Then for (cid:96) > J (cid:48) ( n ) , the group G + (cid:96) is an open normal subgroup of M of indexat most n . (cid:96) -adic representationsIn the remainder of this section we shall give a proof of Theorem 3.4. Moreover we shall derivesome elementary permanence properties for the properties described by Σ (cid:96) ( d ) and Jor (cid:96) ( d ).The content of the following lemma is presumably well-known. Lemma 3.6.
For every r ∈ N , every algebraically closed field F and every semisimple algebraicgroup G of rank r the center Z of G satisfies | Z ( F ) | ≤ r .Proof. Lacking a precise reference, we include a proof for the reader’s convenience. Observefirst that the center Z is a finite (cf. [30, I.6.20, p. 43]) diagonalizable algebraic group. Let T be a maximal torus of G . Denote by X ( T ) = Hom( T, G m ) the character group of T and byΦ ⊂ X ( T ) the set of roots of G . Then R = ( X ( T ) ⊗ R , Φ) is a root system. Let P = Z Φ be theroot lattice and Q the weight lattice of this root system. Then P ⊂ X ( T ) ⊂ Q . The center Z of G is the kernel of the adjoint representation (cf. [30, I.7.12, p. 49]). Hence Z = (cid:84) χ ∈ Φ ker( χ )and there is an exact sequence 0 → Z → T → (cid:89) χ ∈ Φ G m where the right hand map is induced by the characters χ : T → G m ( χ ∈ Φ). We apply thefunctor Hom( − , G m ) and obtain an exact sequence (cid:89) χ ∈ Φ Z → X ( T ) → Hom( Z, G m ) → X ( T ) /P . Thus | Z ( F ) | ≤ [ X ( T ) : P ] ≤ [ Q : P ].Furthermore, the root system R decomposes into a direct sum R = s (cid:77) i =1 ( E i , Φ i )of indecomposable root systems R i := ( E i , Φ i ). Let r i = dim( E i ) be the rank of R i . Let P i be the root lattice and Q i the weight lattice of R i . Note that by definition P = ⊕ i P i and Q = ⊕ i Q i . It follows from the classification of indecomposable root systems that | Q i /P i | ≤ r i (cf. [30, Table 9.2, p. 72]) for all i . Hence | Z ( F ) | ≤ | Q/P | ≤ r r · · · r s = 2 r as desired. Remark 3.7.
The semisimple algebraic group (SL , C ) r has rank r and its center ( µ ) r hasexactly r C -rational points. Hence the bound of Lemma 3.6 cannot be improved. The following result is an adaption of the main result of [27] by Larsen and Pink.
Proposition 3.8.
For every n ∈ N , there exists a constant J (cid:48) ( n ) such that for every field F ofpositive characteristic (cid:96) and every finite subgroup Γ of GL n ( F ) , there exists a normal series Γ (cid:46) L (cid:46) M (cid:46) I (cid:46) P (cid:46) { } of Γ with the following properties: (cid:96) -adic representations i) [Γ : L ] ≤ J (cid:48) ( n ) .ii) The group L/M is abelian of order prime to (cid:96) .iii) The group
M/I is a finite product of finite simple groups of Lie type in characteristic (cid:96) .iv) The group
I/P is abelian of order prime to (cid:96) and [ I : P ] ≤ n .v) P is an (cid:96) -group.Furthermore the constant J (cid:48) ( n ) is the same as in [27, Thm. 0.2, p. 1106].Proof. We can assume that F is algebraically closed. Let J (cid:48) ( n ) be the constant from [27,Thm. 0.2, p. 1106]. Larsen and Pink construct in the proof of their Theorem [27, Thm. 0.2, p.1155–1156] a smooth algebraic group G over F containing Γ and normal subgroups Γ i of Γ suchthat there is a normal series Γ (cid:46) Γ (cid:46) Γ (cid:46) Γ (cid:46) { } and such that [Γ : Γ ] ≤ J (cid:48) ( n ), Γ / Γ is a product of finite simple groups of Lie type incharacteristic (cid:96) , Γ / Γ is abelian of order prime to (cid:96) and Γ is an (cid:96) -group. Let R be theunipotent radical of the connected component G ◦ of G . The proof of Larsen and Pink showsthat Γ (cid:47) G ◦ ( F ), Γ = Γ ∩ R ( F ) and Γ / Γ is contained in Z ( F ) where Z denotes the center ofthe reductive group G := G ◦ /R . Let D = [ G, G ] be the derived group of G and D = [ G ◦ , G ◦ ] R .Now define L = Γ , M = Γ ∩ D ( F ), I = Γ ∩ D ( F ) and P = Γ . These groups are normal inΓ, because D ( F ) is characteristic in G ◦ ( F ) and because Γ , Γ , Γ are normal in Γ. The group L/M is a subgroup of the abelian group G ◦ ( F ) /D ( F ). The group M/I is a normal subgroupof Γ / Γ , hence it is a product of finite simple groups of Lie type in characteristic (cid:96) . The group I/P is a subgroup of Γ / Γ , hence I/P is abelian of order prime to (cid:96) . Furthermore
I/P = I/ Γ is a subgroup of G ( F ) which lies in D ( F ) and in Z ( F ). Thus I/P lies in the center Z ( F ) ∩ D ( F )of the semisimple group D ( F ). It follows by Lemma 3.6 that [ I : P ] ≤ rk( D ) .It remains to show that rk( D ) ≤ n . Let T be a maximal torus of D and denote by π : G ◦ → G the canonical projection. Then the algebraic group B := π − ( T ) sits in an exact sequence0 → R → B → T → B is connected smooth and solvable, because R and T have these properties. The aboveexact sequence splits (cf. [10, XVII.5.1]); hence B contains a copy of T . This copy is containedin a maximal torus T (cid:48) of GL n,F . Thus n = dim( T (cid:48) ) ≥ dim( T ) = rk( D ) as desired. (cid:50) Proof of Theorem 3.4.
Suppose G is n -bounded at (cid:96) , so that it is a quotient G /G with G i ⊂ GL n ( (cid:101) Q (cid:96) ). By Lemma 3.9(a) below, it will suffice to prove the theorem for G . Thus we assumethat G is a compact profinite subgroup of GL n ( (cid:101) Q (cid:96) ). By compactness of G and a Baire categorytype argument (cf. [12, proof of Cor. 5]) the group G is contained in GL n ( E ) for some finiteextension E of Q (cid:96) . Let O E be the ring of integers of the local field E . Again by compactness of G one can then find an O E -lattice in E n that is stable under G . Hence we may assume that G is a closed subgroup of GL n ( O E ). 6ndependence of (cid:96) -adic representationsLet p be the maximal ideal of the local ring O E and let F = O E / p be its residue field. The kernel K of the canonical map p : GL n ( O E ) → GL n ( F ) is a pro- (cid:96) group. Hence Q = K ∩ G is pro- (cid:96) andopen normal in G . We now apply Proposition 3.8 to the subgroup G/Q of GL n ( F ) ⊂ GL n ( F )with F = F ∼ = F (cid:96) . This yields a normal series G (cid:46) L (cid:46) M (cid:46) I (cid:46) P (cid:46) Q (cid:46) { } such that the group G/M lies in Jor (cid:96) ( J (cid:48) ( n )), and the group M lies in Σ (cid:96) (2 n ) – for the latter usethat Q is pro- (cid:96) and normal in G and P/Q is a finite (cid:96) -group.The following lemma records a useful permanence property of groups in Σ (cid:96) ( c ) and Jor (cid:96) ( d ). Lemma 3.9.
Fix any e ∈ N . Then for any prime number (cid:96) the following holds:(a) If H (cid:48) (cid:2) H is a normal subgroup of some H ∈ Jor (cid:96) ( e ) , then H (cid:48) and H/H (cid:48) lie in
Jor (cid:96) ( e ) .(b) If M (cid:48) (cid:2) M is a closed normal subgroup of some M ∈ Σ (cid:96) ( e ) , then M (cid:48) and M/M (cid:48) lie in Σ (cid:96) ( e ) . If M (cid:48) in part (b) of the lemma was not normal in M, then clearly M (cid:48) need not lie in Σ (cid:96) ( c ) again. Proof.
We only give the proof of (b), the proof of (a) being similar but simpler. Let M bein Σ (cid:96) ( e ) and consider a normal series M (cid:46) I (cid:46) P (cid:46) { } as in Definition 3.1. Then L := M/I is isomorphic to a product L × · · · × L s for certain finite simple groups of Lie type L i incharacteristic (cid:96) . Suppose M (cid:48) is a closed normal subgroup of M and define M (cid:48) = M (cid:48) I/I . ByGoursat’s Lemma the groups M (cid:48) and M (cid:48) /M (cid:48) are products of some of the L i . From this it isstraightforward to see that both M (cid:48) and M/M (cid:48) lie in Σ (cid:96) ( c ).The following corollary is immediate from Lemma 3.9(b): Corollary 3.10.
Fix a constant c ∈ N . Let G be a profinite group, and for each (cid:96) ∈ L let ρ (cid:96) : G → G (cid:96) be a homomorphism of profinite groups such that Im( ρ (cid:96) ) ∈ Σ (cid:96) ( c ) for all (cid:96) ∈ L .Then for any closed normal subgroup N (cid:2) G one has Im( ρ (cid:96) | N ) ∈ Σ (cid:96) ( c ) for all (cid:96) ∈ L .In particular, if H (cid:2) G is an open subgroup, then the above applies to any normal open subgroup N (cid:2) G that is contained in H . The purpose of this section is to recall some finiteness properties of fundamental groups andto provide some basic results on ramification. Regarding the latter we draw from results byKerz-Schmidt and Wiesend (cf. [24]) and from de Jong on alterations (cf. [6]).We begin with a finiteness result of which a key part is from [14].7ndependence of (cid:96) -adic representations
Proposition 4.1.
Suppose that either k is a finite field and S is a smooth proper k -variety orthat k is a number field and S is a smooth k -variety, and denote by K = k ( S ) the functionfield of S . For d ∈ N , let M d be the set of all finite Galois extensions E/K inside (cid:101) K such that Gal(
E/K ) satisfies Jor( d ) and such that E is unramified along S . Then there exists a finiteGalois extension K (cid:48) /K which is unramified along S such that E ⊂ (cid:101) kK (cid:48) for every E ∈ M d .Proof. Let Ω = (cid:81) E ∈M d E be the compositum of all fields in M d . For every E ∈ M d the groupGal( E/K ) satisfies Jor( d ) and hence there is a finite Galois extension E (cid:48) /K inside E such that[ E (cid:48) : K ] ≤ d and such that E/E (cid:48) is abelian. DefineΩ (cid:48) = (cid:89) E ∈M d E (cid:48) . Then Ω / Ω (cid:48) is abelian. Let k (resp. κ (cid:48) , resp. κ ) be the algebraic closure of k in K (resp. in Ω (cid:48) ,resp. in Ω). K Ω (cid:48) Ω k k κ (cid:48) κ It suffices to prove the following
Claim.
The extension Ω /κK is finite.In fact, once this is shown, it follows that the finite separable extension Ω /κK has a primitiveelement ω . Then Ω = κK ( ω ) and K ( ω ) /K is a finite separable extension. Let K (cid:48) be the normalclosure of K ( ω ) /K in Ω. Then ˜ kK (cid:48) ⊃ κK (cid:48) ⊃ κK ( ω ) = Ω as desired.In the case k = Q the claim has been shown in [14, Proposition 2.2]. Assume from now on that k is finite. It remains to prove the claim in that case. The structure morphism S → Spec( k ) ofthe smooth scheme S factors through Spec( k ) and S is a geometrically connected k -variety.The profinite group π ( S × k Spec( (cid:101) k )) is topologically finitely generated (cf. [18, Thm. X.2.9])and Gal( k ) ∼ = ˆ Z . Thus it follows by the exact sequence (cf. [18, Thm. IX.6.1])1 → π ( S × k Spec( (cid:101) k )) → π ( S ) → Gal( k ) → π ( S ) is topologically finitely generated. Thus there are only finitely many extensions of K in ˜ K of degree ≤ d which are unramified along S . It follows that Ω (cid:48) /K is a finite extension.Thus κ (cid:48) is a finite field. If we denote by S (cid:48) the normalization of S in Ω (cid:48) , then S (cid:48) → S is finiteand ´etale, hence S (cid:48) is a smooth proper geometrically connected κ (cid:48) -variety. Furthermore Ω / Ω (cid:48) isabelian and unramified along S (cid:48) . Hence Ω /κ Ω (cid:48) is finite by Katz-Lang (cf. [22, Thm. 2, p. 306]).As Ω (cid:48) /K is finite, it follows that Ω /κK is finite.Our next aim is to introduce several notions of ramification, that are refinements of [24], usefulfor coverings of general schemes. Let E/K be a separable algebraic extension of fields. Let v be a discrete rank 1 valuation of K and w an extension of v to E . Let (cid:96) be a prime numberdifferent from char( k ( v )). The extension E/K is said to be tame (resp. (cid:96) -tame ) at w if the8ndependence of (cid:96) -adic representationsresidue field extension k ( w ) /k ( v ) is separable and for every finite extension F of K inside E/K the ramification index [ w ( F × ) : v ( K × )] is prime to char( k ( v )) (resp. is a power of (cid:96) ). If E/K is Galois and the ramification group I w is of order prime to char( k ( v )) (resp. pro- (cid:96) ), then theresidue field extension k ( w ) /k ( v ) is automatically separable and E/K is tame (resp. (cid:96) -tame) at w . The extension E/K is tame (resp. (cid:96) -tame ) at v if the extension E/K is tame (resp. (cid:96) -tame)at every extension w of v to E .We fix some terminology for curves. Let k be a field of characteristic p ≥
0. A curve C over k isa smooth (but not necessarily projective) k -variety of dimension 1. Denote by P ( C ) the smoothprojective model of the function field k ( C ) (the model is unique up to isomorphism). Then P ( C )contains C , and we set ∂C := P ( C ) (cid:114) C . Let (cid:96) be a prime number different from p . An ´etalecover C (cid:48) → C is called tame (resp. (cid:96) -tame ) if for any point x ∈ ∂C with valuation v x of k ( C )and every connected component Z (cid:48) of C (cid:48) , the extension k ( Z (cid:48) ) /k ( C ) is tame (resp. (cid:96) -tame) at v x . Definition 4.2.
Let S be a regular variety over a field k of characteristic p ≥ . Let f : T → S be an ´etale cover. Let (cid:96) be a prime different from p .(a) The cover f : T → S is curve tame (resp. curve (cid:96) -tame ) if for all k -morphisms ϕ : C → S with C a smooth curve over k , the base changed cover f C : C × S T → C is tame (resp. (cid:96) -tame).(b) Assume that S is an open subscheme of a regular projective scheme S such that D = S (cid:114) S is a normal crossings divisor (NCD). Then f : T → S is tame (resp. (cid:96) -tame ) if for everydiscrete valuation v of K defined by a generic point of D and every connected component Z of T the extension k ( Z ) /k ( S ) is tame (resp. (cid:96) -tame) at v .We extend the above notions to profinite ´etale covers by saying that such a cover is pro- (cid:96) curvetame or pro- (cid:96) tame if these conditions hold for all subcovers of finite degree. Definition 4.3.
Let k be a field of characteristic p ≥ . Let K/k be a finitely generated extensionand
E/K an algebraic extension. Let (cid:96) be a prime different from p .(a) The extension E/K is generically ´etale if there exists a regular k -variety S with functionfield K such that for every finite extension F of K inside E/K the normalization of S in F is ´etale over S .(b) The extension E/K is divisor tame (resp. divisor (cid:96) -tame ) if it is generically ´etale andtame (resp. (cid:96) -tame) at every discrete rank valuation v of K which is trivial on k . Definition 4.4.
Suppose k is a field of characteristic p ≥ and K/k is a finitely generatedextension. Let (cid:96) be a prime number different from p . We call a homomorphism Gal( K ) → G ofprofinite groups (cid:96) -tame over k if the fixed field E := ( K s ) Ker( ρ ) is divisor (cid:96) -tame over K . Remark . Let k be a field of characteristic p ≥ S a regular k -variety. Let f : T → S be a connected ´etale cover. Let K = k ( S ) and E = k ( T ). Let (cid:96) be a prime number differentfrom p . In [24, p. 12] the cover f : T → S is defined to be divisor tame if for any normal9ndependence of (cid:96) -adic representationscompactification S of S and a point s ∈ S (cid:114) S with codim S s = 1, the extension E/K is tameat the rank 1 valuation v s on K . We claim that the cover T /S is divisor tame in the sense of[24, p. 12] if and only if the extension
E/K is divisor tame in the sense of definition 4.3.Clearly our notion of divisor tameness implies that of [24]. For the converse we follow closelythe argument in [24, Thm. 4.4] proof of (ii) ⇒ (iii) though with different references. Let w bea valuation of E that is trivial on k and denote by v its restriction to K . Let S be a normalcompactification of S , which exists by the theorem of Nagata [28]. By [38, Prop. 6.4], thereexists a blow-up S (cid:48) of S with center outside S such that v is the valuation of a codimension 1point s ∈ S (cid:48) . By normalization, we may further assume that S (cid:48) is normal. Both operations,blow-up and normalization, do not affect S , and so we may take for S a normal compactificationof S that contains a codimension 1 point s with valuation v = v s . But then the divisor tamenessof [24] implies that w/v is at most tamely ramified.The following result is a variant of parts of [24, Thm. 4.4]: Proposition 4.6.
Let k, S, T, K, E, f, (cid:96) be as in Remark 4.5. Then the following hold:(a) The cover
T /S is curve-tame if and only if the extension
E/K is divisor tame.(b) Suppose
E/K is Galois. Then the cover
T /S is curve- (cid:96) -tame if and only if the extension
E/K is divisor (cid:96) -tame.(c) If S is an open subscheme of a regular projective scheme S such that D = S (cid:114) S is a NCD,then both conditions from (b) are equivalent to T /S being (cid:96) -tame.The assertions (a)–(c) extend in an obvious manner to profinite covers.Proof.
By Remark 4.5, part (a) of the proposition follows directly from the equivalence (i) ⇔ (ii)in [24, Thm. 4.4].For the proof of (b), suppose first that T /S is curve- (cid:96) -tame and assume that
E/K is not divisor (cid:96) -tame. By (a) we know that
E/K is divisor tame. So let w be a valuation of E at which E/K is not divisor (cid:96) -tame. Denote by K ⊂ K ⊂ E extensions of K such that E/K is totallyramified (and Galois) at w and K /K is of prime degree (cid:96) (cid:48) (cid:54) = (cid:96) . As in Remark 4.5, there existsa normal compactification S of S and a codimension 1 point s of S (cid:114) S that has a preimage t inthe normalization T of S in E/K with v t = w . We define T i , T i , i = 1 ,
2, as the normalizationsof S or S in K i /K , respectively. We claim that T /T is curve- (cid:96) -tame.To see this, observe first, that as with curve-tameness, it is a simple matter of drawing a suitablecommutative diagram to see that curve- (cid:96) -tameness is stable under base change. In particularthe base change T × S T → T is curve- (cid:96) -tame. However, considering the commutative fiberproduct diagram T (cid:15) (cid:15) (cid:39) (cid:39) s (cid:45) (cid:45) T × S T (cid:111) (cid:111) (cid:15) (cid:15) S T , (cid:111) (cid:111) (cid:96) -adic representationswe see that there is a canonical splitting s : T → T × S T over T . Hence T is a connectedcomponent of T × S T and as such the restriction T → T of the morphism T × S T → T inherits curve- (cid:96) -tameness.Having the claim at our disposal, the hypothesis [ K : K ] = (cid:96) (cid:48) yields that for any curve C mapping to T , the induced cover C × T T → C is everywhere unramified along ∂C . Now T is regular in codimension 1, hence the regular locus W contains T as well as the divisorcorresponding to w | K . Let W be its preimage in T . Now by [24, Prop. 4.1], which can beparaphrased as: curve-unramifiedness implies unramifiedness over a regular base , it follows that W → W is ´etale. But then K /K is ´etale along w , a contradiction.For the converse of (b) suppose that E/K is divisor (cid:96) -tame. We assume that there is a k -morphism C → S for C a smooth curve such that π : C × S T → S is not (cid:96) -tame along ∂C . SinceGal( E/K ) acts faithfully on C × S T → C , by passing to a subgroup and thus an intermediateextension of E/K we may assume that C × S T is irreducible. Since then Gal( E/K ) is also theGalois group of the cover π , some further straightforward reductions allow us to assume that[ E : K ] = (cid:96) (cid:48) (cid:54) = (cid:96) for some prime (cid:96) (cid:48) (which by (a) is different from p ), and that Gal( E/K ) ∼ = Z /(cid:96) (cid:48) is the inertia group above some valuation of k ( C ). Following the argument in the proof of [24,Thm. 4.4] (v) ⇒ (i), we can find a discrete rank d = dim S valuation of E that is ramified oforder (cid:96) (cid:48) (via a Parshin chain through the image of Spec k ( C )). But [24, Lem. 3.5] says that E/K is ramified at a discrete rank d valuation if and only if it is ramified at a discrete rank 1valuation. We reach a contradiction because by hypothesis E/K is unramified at all discreterank 1 valuations.Finally, we prove (c). It is clear that divisor (cid:96) -tameness implies (cid:96) -tameness. The proof that (cid:96) -tameness implies curve (cid:96) -tameness follows from the argument given in [24, Prop. 4.2]: thereit is shown that tameness implies curve tameness. Consider a curve C over k and a morphism ϕ : C → S over k . Then ϕ extends to a morphism ϕ : P ( C ) → S . Denote by ϕ ( C ) the closureof ϕ ( C ) in S . The ramification of T × S C → C occurs precisely at those points of P ( C ) thatunder ϕ map to D ∩ ϕ ( C ). To analyze the ramification, the proof of [24, Prop. 4.2] appeals toAbhyankar’s lemma. In the notation of loc. cit. , the ramification is then governed by indices n i , i = 1 , . . . , r , that are prime to p . By the (cid:96) -tameness of T → S , the n i must all be powers of (cid:96) .But then loc. cit. implies that T × S C → C is (cid:96) -tame, and this completes the proof.Our formulation of divisor-tameness easily transfers under rather general field extensions: Lemma 4.7.
Suppose that char( k ) = p > and consider the following inclusions of fields: K (cid:31) (cid:127) (cid:47) (cid:47) K (cid:48) k (cid:31) (cid:63) (cid:79) (cid:79) (cid:31) (cid:127) (cid:47) (cid:47) k (cid:48) (cid:31) (cid:63) (cid:79) (cid:79) If E/K is Galois and divisor (cid:96) -tame over k , then so is EK (cid:48) /K (cid:48) over k (cid:48) .Proof. It clearly suffices to prove the lemma in the case where
E/K is finite Galois. Then E (cid:48) := EK (cid:48) is finite Galois over K (cid:48) . Let w (cid:48) be any discrete rank one valuation of E (cid:48) trivial on11ndependence of (cid:96) -adic representations k (cid:48) and denote by w its restriction to E , by v (cid:48) the restriction to K (cid:48) and by v the restriction to K . We need to show that [ w (cid:48) ( E (cid:48)∗ ) : v (cid:48) ( K (cid:48)∗ )] is a power of (cid:96) and that the residue extensionis separable. The latter can be taken care of at once: The extension E/K is finite separable.Hence so is E (cid:48) /K (cid:48) , because a primitive element of E/K will be such an element for E (cid:48) /K (cid:48) . Forthe same reason, separability is preserved by the extensions of completions E (cid:48) w (cid:48) /K (cid:48) v (cid:48) at v (cid:48) . Nowby the Cohen structure theorem, the extension of residue fields is a subextension of E (cid:48) w (cid:48) /K (cid:48) v (cid:48) ,and as such it must be separable. It remains to consider the index of the value groups.Suppose first that v ( K ∗ ) = 0. Then we must have w ( E ∗ ) = 0, since otherwise, if α ∈ E would satisfy w ( α ) (cid:54) = 0, then the sequence ( α n ) n ∈ Z would be linearly independent over K , acontradiction. This means that under the residue map of E (cid:48) , the subfield E is mapped injectivelyto the residue field of E (cid:48) at w (cid:48) . But then E/K defines purely a residue extension of E (cid:48) /K (cid:48) , andthus w (cid:48) ( E (cid:48)∗ ) = v (cid:48) ( K (cid:48)∗ ).Next assume that w is non-trivial, so that by the above v is non-trivial as well. We pass tothe completions and note that E w /K v and E (cid:48) w (cid:48) /K (cid:48) v (cid:48) remain Galois extensions. By the Cohenstructure theorem, K v now contains the residue field k ( v ), and E v the residue field k ( w ). Inparticular, F = k ( w ) K v is an unramified extension of K v and E w /F is totally ramified. Wemay thus consider these two cases separately. Suppose first that E w /K v is unramified. Then E (cid:48) w (cid:48) = K (cid:48) v (cid:48) k ( w ) where clearly k ( w ) defines a separable extension of the residue field of K (cid:48) v (cid:48) .Hence E (cid:48) w (cid:48) /K (cid:48) v (cid:48) is unramified. We conclude w (cid:48) ( E (cid:48)∗ ) = v (cid:48) ( K (cid:48)∗ ) which completes the argument.Finally let E w /K v be totally ramified. By our hypothesis, the extension E/K is at most (cid:96) -orderramified at w . It follows that E w /K v is a Galois extension with Gal( E w /K v ) an (cid:96) -group. NowGal( E (cid:48) w (cid:48) /K (cid:48) v (cid:48) ) injects into Gal( E w /K v ) because E (cid:48) = KE , and thus [ E (cid:48) w : K (cid:48) v ] is a power of (cid:96) .Since the order of w (cid:48) ( E (cid:48)∗ ) /v (cid:48) ( K (cid:48)∗ ) divides the degree [ E (cid:48) : K (cid:48) ], the proof is complete.Combining ramification properties with finiteness properties of fundamental groups, we obtainthe following criterion for a family of representations of Gal( K ) with images in Jor (cid:96) ( d ), or withabelian images of bounded order, to become trivial over Gal( K (cid:48) (cid:101) k ) for some finite K (cid:48) /K . Proposition 4.8.
Let k be a field and let S/k be a normal k -variety with function field K . Let L be a set of prime numbers which does not contain char( k ) . Suppose ( ρ (cid:96) : π ( S ) → G (cid:96) ) (cid:96) ∈ L is afamily of continuous homomorphisms such that if char( k ) > each ρ (cid:96) is (cid:96) -tame. Under eitherof the following two conditions there exists a finite extension K (cid:48) of K such that for all (cid:96) ∈ L wehave ρ (cid:96) (Gal( K (cid:48) (cid:101) k )) = { } .(a) The field k is finite or a number field and there exists a constant d ∈ N such that for each (cid:96) ∈ L the group Im( ρ (cid:96) ) lies in Jor (cid:96) ( d ) .(b) The field k is algebraically closed and there exists a constant c ∈ N such that for each (cid:96) ∈ L the group Im( ρ (cid:96) ) is of order at most c .Proof. First we replace K by a finite Galois extension and S be the normalization in thisextension, so that we can assume that ρ (cid:96) ( π ( S )) = { } for all (cid:96) ≤ d or (cid:96) ≤ c , respectively. Nextwe apply the result of de Jong on alterations (cf. [6, Thm. 4.1, 4.2]). It provides us with a finite12ndependence of (cid:96) -adic representationsextension k (cid:48) of k , a smooth projective geometrically connected k (cid:48) -variety T (cid:48) , a non-empty opensubvariety S (cid:48) of T (cid:48) and an alteration f : S (cid:48) → S , such that furthermore D (cid:48) := T (cid:48) (cid:114) S (cid:48) is anormal crossing divisor. We define K (cid:48) to be the function field of S (cid:48) , so that K (cid:48) /K is finite. (If k is perfect, we could also assume that K (cid:48) /K is separable.)Next observe, that if char( k ) is positive, then the (divisor) (cid:96) -tameness of ρ (cid:96) implies the (divisor) (cid:96) -tameness of ρ (cid:96) | π ( S (cid:48) ) by Lemma 4.7, and thus, by Proposition 4.6, for each (cid:96) the extension K (cid:48) (cid:96) = ( K (cid:48) s ) Ker( ρ (cid:96) | π S (cid:48) ) ) of K (cid:48) is (cid:96) -tame. Because of the first reduction step in the previousparagraph, this implies that each ρ (cid:96) | π ( S (cid:48) ) is unramified at the generic points of D (cid:48) . Purity ofthe branch locus (cf. [18, X.3.1]) now implies that all ρ (cid:96) | π ( S (cid:48) ) factors via π ( T (cid:48) ).Now, in case (a), the assertion of the proposition follows from Proposition 4.1. In case (b) weuse that, since k is algebraically closed, the fundamental group π ( T (cid:48) ) is topologically finitelygenerated (cf. [18, Thm. X.2.9]), and that furthermore, if char( k ) = 0, the same holds true for π ( S (cid:48) ) (cf. [19, II.2.3.1]). Hence there are only finitely many possibilities for the fields K (cid:48) (cid:96) and (cid:81) (cid:96) K (cid:48) (cid:96) is a finite extension of K (cid:48) . This completes the proof in case (b). From now on, let k be any field, let K/k be a finitely generated field extension and let L be aset of prime numbers not containing p := char( k ). For every (cid:96) ∈ L let G (cid:96) be a profinite groupand let ρ (cid:96) : Gal( K ) → G (cid:96) be a continuous homomorphism.If for all (cid:96) ∈ L the groups Im( ρ (cid:96) ) are n -bounded at (cid:96) , then by Theorem 3.4 we have a shortexact sequence 1 → M (cid:96) → Im( ρ (cid:96) ) → H (cid:96) → H (cid:96) ∈ Jor (cid:96) ( d ) for d = J (cid:48) ( n ) and M (cid:96) ∈ Σ (cid:96) (2 n ).At the end of the previous section we have seen that a combination of tameness of ramificationand results on fundamental groups allow one to control the H (cid:96) in a uniform manner. In thissection we shall show how to control M (cid:96) in a uniform manner, if one has a uniform control onramification. We begin by introducing the necessary concepts and then give the result.To ( ρ (cid:96) ) (cid:96) ∈ L we attach the family ( (cid:101) ρ (cid:96) ) (cid:96) ∈ L by defining each (cid:101) ρ (cid:96) as the composite homomorphism (cid:101) ρ (cid:96) : Gal( K ) ρ (cid:96) −→ Im( ρ (cid:96) ) → Im( ρ (cid:96) ) /Q (cid:96) where Q (cid:96) is the maximal normal pro- (cid:96) subgroup of im( ρ (cid:96) ). Note that if ρ (cid:96) is an (cid:96) -adic represen-tation, then (cid:101) ρ (cid:96) is essentially the semisimplification of the mod (cid:96) reduction of ρ (cid:96) . Definition 5.1. (a) The family ( ρ (cid:96) ) (cid:96) ∈ L is said to satisfy the condition R ( k ) , if there exist afinite extension k (cid:48) of k , a finite extension K (cid:48) /Kk (cid:48) and a smooth k (cid:48) -variety U (cid:48) with functionfield K (cid:48) such that for every (cid:96) ∈ L the homomorphism (cid:101) ρ (cid:96) | Gal( K (cid:48) ) is unramified along U (cid:48) .(b) The family ( ρ (cid:96) ) (cid:96) ∈ L is said to satisfy the condition S ( k ) , if it satisfies R ( k ) and if one canchoose the field K (cid:48) for R ( k ) such that each (cid:101) ρ (cid:96) | Gal( K (cid:48) ) is (cid:96) -tame. The condition R ( k ) says that each member (cid:101) ρ (cid:96) is up to pro- (cid:96) ramification potentially generically´etale in a uniform way. The condition S ( k ) is a kind of semistability condition.13ndependence of (cid:96) -adic representations Example 5.2.
Set L = L (cid:114) { char( k ) } and let A/K be an abelian variety. For every (cid:96) ∈ L denote by σ (cid:96) : Gal( K ) → Aut Z (cid:96) ( T (cid:96) ( A )) the representation of Gal( K ) on the (cid:96) -adic Tate modulelim ←− i ∈ N A [ (cid:96) i ]. There exists a finite extension k (cid:48) of k and a finite separable extension K (cid:48) /k (cid:48) K suchthat K (cid:48) is the function field of some smooth k (cid:48) -variety V (cid:48) . By the spreading-out principles of[17] there exists a non-empty open subscheme U (cid:48) of V (cid:48) and an abelian scheme A over U (cid:48) withgeneric fibre A . This implies (cf. [19, IX.2.2.9]) that σ (cid:96) is unramified along U (cid:48) for every (cid:96) ∈ L .Hence the family ( σ (cid:96) ) (cid:96) ∈ L satisfies condition R ( k ).In order to obtain also S ( k ), we choose an odd prime (cid:96) ∈ L , and we require the field K (cid:48) aboveto be finite separable over k (cid:48) K ( A [ (cid:96) ]). Now let v (cid:48) be any discrete valuation of K (cid:48) which is trivialon k (cid:48) , and let R v (cid:48) be the discrete valuation ring of v (cid:48) . Let N v (cid:48) / Spec( R v (cid:48) ) be the N´eron modelof A over R v (cid:48) . The condition K (cid:48) ⊃ K ( A [ (cid:96) ]) forces N v (cid:48) to be semistable (cf. [19, IX.4.7]). Thisin turn implies that σ (cid:96) | I ( v (cid:48) ) is unipotent (and hence σ (cid:96) ( I ( v (cid:48) )) is pro- (cid:96) ) for every (cid:96) ∈ L (cf. [19,IX.3.5]). It follows that the family ( σ (cid:96) ) (cid:96) ∈ L satisfies condition S ( k ).The following is the main independence criterion of this section: Proposition 5.3.
Let k be an algebraically closed field and let K/k be a finitely generated ex-tension. Suppose that ( ρ (cid:96) ) (cid:96) ∈ L is a family of representations of Gal( K ) that satisfies the followingconditions:(a) The family ( ρ (cid:96) ) (cid:96) ∈ L satisfies R ( k ) if char( k ) = 0 and S ( k ) if char( k ) > .(b) There exists a constant c ∈ N such that for all (cid:96) ∈ L one has ρ (cid:96) (Gal( K )) ∈ Σ (cid:96) ( c ) .Then there exists a finite abelian Galois extension E/K with the following properties.(i) For every (cid:96) ∈ L the group ρ (cid:96) (Gal( E )) lies in Σ (cid:96) ( c ) and is generated by its (cid:96) -Sylowsubgroups; if (cid:96) > c then the group ρ (cid:96) (Gal( E )) is generated by the (cid:96) -Sylow subgroups of ρ (cid:96) (Gal( K )) .(ii) The group Gal( E ) is a characteristic subgroup of Gal( K ) .(iii) The restricted family ( ρ (cid:96) | Gal( E ) ) (cid:96) ∈ L (cid:114) { , } is independent and ( ρ (cid:96) ) (cid:96) ∈ L is almost independent.Proof. We can assume that ρ (cid:96) is surjective for all (cid:96) ∈ L . Denote by G + (cid:96) the normal subgroup of G (cid:96) which is generated by the pro- (cid:96) Sylow subgroups of G (cid:96) . Then G (cid:96) := G (cid:96) /G + (cid:96) is a finite groupof order prime to (cid:96) . Denote by π (cid:96) : G (cid:96) → G (cid:96) the natural projection. As G (cid:96) lies in Σ (cid:96) ( c ), sodoes its quotient G (cid:96) by Lemma 3.9(b). Now any group in Σ (cid:96) ( c ) of order prime to (cid:96) is abelianof order at most c , and thus the latter holds for G (cid:96) .Let K + (cid:96) be the fixed field in K s of the kernel of the map π (cid:96) ◦ ρ (cid:96) , so that Gal( K + (cid:96) /K ) ∼ = G (cid:96) . Thenthe compositum E = (cid:81) (cid:96) ∈ L K + (cid:96) is an abelian extension of K such that Gal( E/K ) is annihilatedby c !. From Proposition 4.8(b), which uses hypothesis (a), we see that E/K is finite. Assertion(ii) is now straightforward: By definition of the K + (cid:96) the subgroups Gal( K + (cid:96) ) of Gal( K ) arecharacteristic and hence so is their intersection Gal( E ).14ndependence of (cid:96) -adic representationsWe turn to the proof of (i): For every (cid:96) ∈ L , from (ii) the group ρ (cid:96) (Gal( E )) is normal in G (cid:96) ,and hence it lies in Σ (cid:96) ( c ) by Lemma 3.9. By construction ρ (cid:96) (Gal( E )) ⊂ ρ (cid:96) (Gal( K + (cid:96) )) = G + (cid:96) and N (cid:96) := G + (cid:96) /ρ (cid:96) (Gal( E )) is abelian and annihilated by c !. We claim that (1) N (cid:96) is an (cid:96) -group, sothat N (cid:96) is trivial if (cid:96) > c , and that (2) ρ (cid:96) (Gal( E )) is generated by its pro- (cid:96) Sylow subgroups.We argue by contradiction and assume that (1) or (2) fails.If (2) fails, then ρ (cid:96) (Gal( E )) has a finite simple quotient of order prime to (cid:96) . Because ρ (cid:96) (Gal( E ))lies in Σ (cid:96) ( c ), this simple quotient has to be abelian of prime order (cid:96) (cid:48) different from (cid:96) . Againby (b), the Galois closure over K of the fixed field of this (cid:96) (cid:48) -extension is a solvable extension.Denote by F either this solvable extension if (2) fails, or the extension of K + (cid:96) whose Galoisgroup is canonically isomorphic to N (cid:96) if (1) fails. In either case F/K is Galois and solvable,and we have a canonical surjection π (cid:48) (cid:96) : G (cid:96) −→→ Gal(
F/K ). Arguing as in the first paragraph,it follows that I (cid:96) surjects onto Gal( F/K ). By construction Gal(
F/K + (cid:96) ) is not an (cid:96) -group. Itfollows from the definition of K + (cid:96) that the normal subgroup π (cid:48) (cid:96) ( P (cid:96) ) ⊂ Gal(
F/K ) is a propersubgroup of Gal(
F/K + (cid:96) ). But then its fixed field is a proper extension of K + (cid:96) which is at onceGalois and of a degree over K that is prime to (cid:96) . This contradicts the definition of K + (cid:96) , andthus (1) and (2) hold. This in turn completes the proof of (i).We now prove (iii). Denote by Ξ (cid:96) the class of those finite groups which are either a finite simplegroup of Lie type in characteristic (cid:96) or isomorphic to Z / ( (cid:96) ). The conditions in (i) imply thatevery simple quotient of ρ (cid:96) (Gal( E )) lies Ξ (cid:96) . But now for any (cid:96), (cid:96) (cid:48) ≥ (cid:96) (cid:54) = (cid:96) (cid:48) one hasΞ (cid:96) ∩ Ξ (cid:96) (cid:48) = ∅ (cf. [37, Thm. 5], [1], [25]). The first part of (iii) now follows from [37, Lemme2]. The second part follows from the first, the definition of almost independence and from [37,Lemme 3]. Let k , K , L , p and ( ρ (cid:96) ) (cid:96) ∈ L be as at the beginning of Section 5. In the previous two sections wehave described ramification properties of ( ρ (cid:96) ) (cid:96) ∈ L and properties of ( ρ (cid:96) (Gal( K ))) (cid:96) ∈ L that wereessential to control in a uniform way the groups H (cid:96) and M (cid:96) that occur in ρ (cid:96) (Gal( K )) as inTheorem 3.4. The aim of this section is to explain how these properties for a general pair ( K, k )in our target Theorem 1.2, can be reduced to a pair where k is the prime field and K is finitelygenerated over it. Moreover we shall explain how one can reduce the proof of our target theoremto the case where X is a smooth and projective variety over K . Lemma 6.1.
Suppose we have a commutative diagram of fields K (cid:31) (cid:127) (cid:47) (cid:47) K (cid:48) k (cid:31) (cid:127) (cid:47) (cid:47) (cid:31) (cid:63) (cid:79) (cid:79) k (cid:48) (cid:31) (cid:63) (cid:79) (cid:79) such that K (cid:48) is finite over Kk (cid:48) . Then the following properties hold true:(i) If ( ρ (cid:96) ) (cid:96) ∈ L satisfies R ( k ) , then ( ρ (cid:96) | Gal( K (cid:48) ) ) (cid:96) ∈ L satisfies R ( k (cid:48) ) . (cid:96) -adic representations (ii) If char( k ) > and ( ρ (cid:96) ) (cid:96) ∈ L satisfies S ( k ) , then ( ρ (cid:96) | Gal( K (cid:48) ) ) (cid:96) ∈ L satisfies S ( k (cid:48) ) .(iii) If there exists a constant c ∈ N such that for all (cid:96) ∈ L , the group ρ (cid:96) (Gal( K (cid:101) k )) lies in Σ (cid:96) ( c ) , then there exists a finite Galois extension E (cid:48) /K (cid:48) such that for all (cid:96) ∈ L , the group ρ (cid:96) (Gal( E (cid:48) (cid:101) k (cid:48) )) lies in Σ (cid:96) ( c ) .Proof. Considering the diagram
K K (cid:31) (cid:127) (cid:47) (cid:47) k (cid:48) K (cid:31) (cid:127) (cid:47) (cid:47) K (cid:48) k (cid:31) (cid:127) (cid:47) (cid:47) (cid:31) (cid:63) (cid:79) (cid:79) k (cid:48) ∩ K (cid:31) (cid:63) (cid:79) (cid:79) (cid:31) (cid:127) (cid:47) (cid:47) k (cid:48) (cid:31) (cid:63) (cid:79) (cid:79) k (cid:48) , (cid:31) (cid:63) (cid:79) (cid:79) it suffices to prove the lemma in the following three particular cases: (a) K (cid:48) = K , (b) Kk (cid:48) = K (cid:48) and K ∩ k (cid:48) = k (base change), (c) k = k (cid:48) and K (cid:48) is finite over K .For the proof of (iii) note that in case (a) the group Gal( K (cid:48) (cid:101) k (cid:48) ) is a closed normal subgroupof Gal( K (cid:101) k ) and in case (b) the canonical homomorphism Gal( K (cid:48) (cid:101) k (cid:48) ) → Gal( K (cid:101) k ) is surjective.In either case we take E = K (cid:48) . Finally in case (c) we define E as the Galois closure of K (cid:48) over K . Then in cases (a) and (c) assertion (iii) follows from Corollary 3.10 with H = Gal( E )and G = Gal( K ). Case (b) is obvious.In all cases, the proof of (ii) is immediate from Lemma 4.7, once (i) is proved. Therefore itremains to prove (i). We first consider case (a). By replacing k , k (cid:48) and K several times byfinite extensions, we can successively achieve the following, where in each step the previousproperty is preserved: First, using de Jong’s result on alterations (cf. [6, Thm. 4.1, 4.2]), thereexists a smooth projective scheme X/k (cid:48) whose function field is K . Second, by the spreadingout principle, there exists an affine scheme U (cid:48) over k whose function field is k (cid:48) and a smoothprojective U (cid:48) -scheme X whose function field is K . Third, by hypothesis R ( k ) there exists asmooth k -scheme U whose function field is K such that all (cid:101) ρ (cid:96) factor via π ( U ). By shrinking U we may assume it to be affine. Also we choose an affine open subscheme V of X . Thecorresponding coordinate rings we denote by R and R , respectively. Both of these rings arefinitely generated over k . Since the fraction field of both is K , by inverting a suitable element g (cid:54) = 0 of R we have R [ g − ] ⊃ R , and similarly we can find 0 (cid:54) = f ∈ R . such that R [ f − ] ⊃ R .Inverting both elements shows that we can find an affine open subscheme V of both U and V .In particular, the function field of V is K , the scheme V is smooth over k and over U (cid:48) and therepresentations ρ (cid:96) all factor via π ( V ). The following diagram displays the situation: V (cid:77)(cid:109) (cid:124) (cid:124) (cid:17) (cid:113) (cid:35) (cid:35) (cid:25) (cid:25) V (cid:48) (cid:26) (cid:26) (cid:111) (cid:111) U (cid:25) (cid:25) X (cid:15) (cid:15) X (cid:111) (cid:111) (cid:15) (cid:15) Spec K (cid:111) (cid:111) (cid:121) (cid:121) U (cid:48) (cid:124) (cid:124) Spec k (cid:48) (cid:111) (cid:111) (cid:117) (cid:117) Spec k (cid:96) -adic representationsDefine V (cid:48) as the base change V × U (cid:48) Spec k (cid:48) , so that V (cid:48) → Spec k (cid:48) is smooth and affine. Now if W → U is any ´etale Galois cover, then the base change W × U V (cid:48) is an ´etale Galois cover of V (cid:48) .We deduce that (cid:101) ρ (cid:96) factors via π ( V (cid:48) ) for all (cid:96) , and thus we have verified R ( k (cid:48) ) for ( (cid:101) ρ (cid:96) ) (cid:96) ∈ L .Case (b): This is a base change. Therefore if we replace k by a finite extension, then U/k becomesa smooth variety with function field a finite separable extension of Kk (cid:48) . But then U × Spec k Spec k (cid:48) is smooth over k (cid:48) and its function field is a finite separable extension of K (cid:48) = Kk (cid:48) . From this(i) is immediate.Case (c): To see R ( k ) over K (cid:48) , let k (cid:48) ⊃ k and K (cid:48)(cid:48) ⊃ K (cid:48) k (cid:48) be finite extensions such that thereexists a smooth k (cid:48) -scheme U with function field K (cid:48)(cid:48) such that all (cid:101) ρ (cid:96) factor via π ( U ). Let U (cid:48) bethe normalization of U in K (cid:48) K (cid:48)(cid:48) . Now choose k (cid:48)(cid:48) ⊃ k (cid:48) and K (cid:48)(cid:48)(cid:48) ⊃ K (cid:48) K (cid:48)(cid:48) finite such that thereis a smooth k (cid:48)(cid:48) -scheme U (cid:48)(cid:48) with function field K (cid:48)(cid:48) and a finite morphism to U (cid:48) . Then R ( k (cid:48) ) isverified by U (cid:48)(cid:48) .The following is a standard lemma from algebraic geometry about models of schemes over finitelygenerated fields. Lemma 6.2.
Let k be a field, K/k be finitely generated and X be an separated algebraic schemeover K . Then there exists an absolutely finitely generated field K ⊂ K and a separated algebraicscheme X over K such that kK = K and X ⊗ K K = X . If in addition X/K is smoothand/or projective, then one can choose X and K in a way so that X /K is smooth and/orprojective.Proof. Let K be the set of all finitely generated subfields of K . Then K = (cid:83) K (cid:48) ∈ K K (cid:48) andSpec( K ) = lim ←− K (cid:48) ∈ K Spec( K (cid:48) ). There exists K (cid:48) ∈ K and a separated algebraic K (cid:48) -scheme X (cid:48) suchthat X = X (cid:48) K (cid:48) (cf. [17, 8.8.2] and [17, 8.10.5(v)]). If X/K is projective, then one can choose K (cid:48) and X (cid:48) in such a way that X (cid:48) /K (cid:48) is projective (cf. [17, 8.10.5(xiii)]). If X/K is smooth, then X (cid:48) /K (cid:48) is smooth. Furthermore there exist x , · · · , x t ∈ K such that K = k ( x , · · · x t ). Define K := K (cid:48) ( x , · · · , x t ) and X := X (cid:48) K . Then kK = K , the field K is finitely generated and X has the desired properties.For a separated algebraic scheme X over K and any (cid:96) ∈ L (cid:114) { char( k ) } we denote by ρ (cid:96),X therepresentation of Gal( K ) on (cid:76) q ≥ (cid:0) H qc ( X (cid:101) K , Q (cid:96) ) ⊕ H q ( X (cid:101) K , Q (cid:96) ) (cid:1) . Corollary 6.3.
Let p be a prime number or p = 0 . Let L = L (cid:114) { p } . Suppose that for allabsolutely finitely generated fields K with prime field k of characteristic p and for all schemes X that are separated algebraic over K , the following conditions are true:(a) The family ( ρ (cid:96),X ) (cid:96) ∈ L satisfies R ( k ) if p = 0 and S ( k ) if p > .(b) There exists a constant c ∈ N and a finite extension E of K such that for all (cid:96) ∈ L onehas ρ (cid:96),X (Gal( E (cid:101) k )) ∈ Σ (cid:96) ( c ) .Let k, K, X be as in Theorem 1.2 and ρ (cid:96) = ρ (cid:96),X for (cid:96) ∈ L . Then there exists a finite Galoisextension K (cid:48) /K and a finite Galois extension E/K with K (cid:48) ⊂ E such that the assertions (a) (cid:96) -adic representations and (b) and conclusions (i)-(iii) of Proposition 5.3 about ( ρ (cid:96) ) (cid:96) ∈ L hold true if one replaces K by K (cid:48) in them. In particular Theorem 1.2 holds.Proof. Let k be an algebraically closed field, let K over k be a finitely generated extensionfield and let X be a separated algebraic scheme over K . By Lemma 6.2, we can find K ⊂ K absolutely finitely generated and X a separated algebraic scheme over K such that X = X ⊗ K K , and moreover if X is smooth and/or projective over K , then the same can be assumedfor X over K . Next let c be the constant and E the field as guaranteed by our hypotheses.Now Lemma 6.1 yields a finite Galois extension K (cid:48) /E K such that ( ρ (cid:96),X | Gal( K (cid:48) ) ) (cid:96) ∈ L satisfies R ( k ) if p = 0 and S ( k ) if p > (cid:96) ∈ L , the image ρ (cid:96) (Gal( K (cid:48) )) lies inΣ (cid:96) ( c ). We can assume K (cid:48) /K Galois after replacing K (cid:48) by a finite extension. By Proposition 5.3,there exists a finite abelian Galois extension E of K (cid:48) such that the assertions and conclusions ofProposition 5.3 hold true if one replaces K by K (cid:48) in them. Furthermore E/K is Galois becauseGal( E ) is a characteristic subgroup of the normal subgroup Gal( K (cid:48) ) of Gal( K ).We now come to the second reduction step. Definition 6.4.
For a representation ρ (cid:96) : Gal( K ) → GL n ( Q (cid:96) ) we denote by ρ sss (cid:96) its strictsemisimplification , i.e., the direct sum over the irreducible subquotients of ρ (cid:96) where each iso-morphism type occurs with multiplicity one. Note that Im( ρ sss (cid:96) ) = Im( ρ ss (cid:96) ) where ρ ss (cid:96) denotes the usual semisimplification of ρ (cid:96) . Lemma 6.5.
For every (cid:96) ∈ L let ρ (cid:96) and ρ (cid:48) (cid:96) be representations Gal( K ) → GL n ( Q (cid:96) ) . Supposethat one of the following two assertions is true:(a) ρ sss (cid:96) = ( ρ (cid:48) (cid:96) ) sss for all (cid:96) ∈ L , or(b) ρ (cid:96) is a direct summand of ρ (cid:48) (cid:96) for all (cid:96) ∈ L .Then the following hold:(i) if the family ( ρ (cid:48) (cid:96) ) (cid:96) ∈ L satisfies R ( k ) then so does ( ρ (cid:96) ) (cid:96) ∈ L ;(ii) if the family ( ρ (cid:48) (cid:96) ) (cid:96) ∈ L satisfies S ( k ) then so does ( ρ (cid:96) ) (cid:96) ∈ L ;(iii) for any (cid:96) ∈ L , if ρ (cid:48) (cid:96) (Gal( K (cid:101) k )) lies in Σ (cid:96) ( c ) then so does ρ (cid:96) (Gal( K (cid:101) k )) . Note that condition (a) is symmetric, so that under (a) each of (i)–(iii) is an equivalence.
Proof.
The proof of (i)–(iii) under hypothesis (a) is an immediate consequence of the simplefact that the kernel of Im( ρ (cid:96) ) → Im( ρ ss (cid:96) ) = Im( ρ sss (cid:96) ) is a pro- (cid:96) -group. Assertions (i) and (ii)under hypothesis (b) are trivial. For (iii) note that hypothesis (b) implies that ρ (cid:48) (cid:96) (Gal( K (cid:101) k )) isa closed normal subgroup of ρ (cid:96) (Gal( K (cid:101) k )), and so we can apply Lemma 3.9.The following important result is taken from the Seminaire Bourbaki talk of Berthelot on deJong’s alteration technique (cf. [4, Thm. 6.3.2])18ndependence of (cid:96) -adic representations Theorem 6.6.
Let k be a field and K be a finitely generated extension field. Let X be aseparated algebraic scheme over K . Then there exists a finite extension k (cid:48) /k , a finite separableextension K (cid:48) /Kk (cid:48) and a finite set of smooth projective varieties { Y i } i =1 ,...,k over K (cid:48) such thatfor all (cid:96) ∈ L (cid:114) { char( k ) } the representation ( ρ (cid:96),X | Gal( K (cid:48) ) ) sss is a direct summand of (cid:0) (cid:76) i ρ (cid:96),Y i (cid:1) sss . Before giving the proof of Theorem 6.6, let us state the following immediate consequence ofTheorem 6.6, Lemma 6.5 and Corollary 6.3:
Corollary 6.7.
Let p be a prime number or p = 0 . Let L = L (cid:114) { p } . Suppose that for allabsolutely finitely generated fields K with prime field k of characteristic p and for all smoothprojective K -varieties X /K , the following conditions are true:(a) The family ( ρ (cid:96),X ) (cid:96) ∈ L satisfies R ( k ) if p = 0 and S ( k ) if p > .(b) There exists a constant c ∈ N and a finite extension E of K such that for all (cid:96) ∈ L (cid:114) { p } one has ρ (cid:96),X (Gal( E (cid:101) k )) ∈ Σ (cid:96) ( c ) .Let k be an algebraically closed field of characteristic p . Let K/k be a finitely generated extensionand
X/K a separated algebraic scheme. Then there exists a finite Galois extension
E/K suchthat the following holds true:(i) The family ( ρ (cid:96),X ) (cid:96) ∈ L satisfies R ( k ) if p = 0 and S ( k ) if p > .(ii) For every (cid:96) ∈ L the group ρ (cid:96),X (Gal( E )) lies in Σ (cid:96) ( c ) and is generated by its (cid:96) -Sylowsubgroups.(iv) The restricted family ( ρ (cid:96),X | Gal( E ) ) (cid:96) ∈ L (cid:114) { , } is independent and ( ρ (cid:96),X ) (cid:96) ∈ L is almost indepen-dent.In particular Theorem 1.2 holds. Note that the auxiliary field K (cid:48) from corollary 6.3 disappears in corollary 6.7 since we onlyrecord those assertions and conclusions from Proposition 5.3, that are important later, andthese do not involve the field K (cid:48) . Proof of Theorem 6.6.
For completeness we provide details of the proof in [4]. For ? ∈ { c, ∅ } we denote by ρ (cid:96),X, ? the representation of Gal( K ) on (cid:76) q ≥ (cid:0) H q ? ( X (cid:101) K , Q (cid:96) ) (cid:1) . It suffices to prove thetheorem separately for the families ( ρ (cid:96),X, ? ) (cid:96) ∈ L . We also note that whenever it is convenient, weare allowed (by passing from K to a finite extension) to assume that X is geometrically reducedover K . This is so because H q ? ( X (cid:101) K , Q (cid:96) ) ∼ = H q ? ( X (cid:101) K, red , Q (cid:96) ) for any q ∈ Z and ? ∈ { ∅ , c } . Wefirst consider the case of cohomology with compact supports. The proof proceeds by inductionon dim X .We may assume, by passing from K to a finite extension, that X is geometrically reduced.Then X (cid:101) K is generically smooth. After passing from K to a finite extension, we can find a dense19ndependence of (cid:96) -adic representationsopen subscheme U ⊂ X that is smooth over K . By the long exact cohomology sequence withsupports (cf. [29, Rem. III.1.30]) we have for any (cid:96) an exact sequence . . . −→ H ic ( U (cid:101) K , Q (cid:96) ) −→ H ic ( X (cid:101) K , Q (cid:96) ) −→ H ic (( X (cid:114) U ) (cid:101) K , Q (cid:96) ) −→ . . . , so that for all (cid:96) the representation ( ρ (cid:96),X,c ) sss is a direct summand of ( ρ (cid:96),U,c ) sss ⊕ ( ρ (cid:96),X (cid:114) U,c ) sss .By induction hypothesis, it thus suffices to treat the case that U is smooth over K . By theinduction hypothesis it is also sufficient to replace U by any smaller dense open subscheme, andit is clearly also sufficient to treat the case where U is in addition geometrically irreducible.By de Jong’s theorem on alterations (cf. [6, Thms. 4.1, 4.2]), after passing from K to a finiteextension, we can find a smooth projective scheme Y, an open subscheme U (cid:48) of Y and analteration π : U (cid:48) → U . By replacing K yet another time by a finite extension, we can assumethat U (cid:48) → U is generically finite ´etale. And now we pass to an open subscheme V of U and to V (cid:48) := π − ( V ) ⊂ U (cid:48) such that V (cid:48) → V is finite ´etale. By the induction hypothesis applied to Y (cid:114) V (cid:48) and again the long exact cohomology sequence for cohomology with support, we find thatthe assertion of the theorem holds true for ( ρ (cid:96),V (cid:48) ,c ) sss (cid:96) ∈ L . From now on π denotes the restrictionto V (cid:48) and Q (cid:96),X will be the constant sheaf Q (cid:96) on any scheme X . Since π is finite ´etale, say ofdegree d , there exists a trace morphism Trace π : π ∗ Q (cid:96),V (cid:48) → Q (cid:96),V whose composition with thecanonical morphism Q (cid:96),V → π ∗ Q (cid:96),V (cid:48) is multiplication by d (cf. [29, Lem. V.1.12]). In particular,the constant sheaf Q (cid:96),V is a direct summand of π ∗ Q (cid:96),V (cid:48) . Since H ic ( V (cid:48) (cid:101) K , Q (cid:96) ) ∼ = H ic ( V (cid:101) K , π ∗ Q (cid:96) ), wededuce that ( ρ (cid:96),V,c ) sss is a direct summand of ( ρ (cid:96),V (cid:48) ,c ) sss , and this completes the induction step.Now we turn to the case ? = ∅ . The case when X is smooth over K but not necessarilyprojective is reduced, by Poincar´e duality, to the case of compact supports: If X is connectedthen one has H q ( X (cid:101) K , Q (cid:96) ) ∼ = H d − qc ( X (cid:101) K , Q (cid:96) ( d )) ∨ for d = dim X (cf. [29, Cor. VI.11.12]), and onecan reduce to the connected case by considering the connected components of X seperately.Suppose now that X is an arbitrary separated algebraic scheme over K . By what we said above,we may assume that X is geometrically reduced. Again we perform an induction on dim X .The first step is a reduction to the case where X is irreducible, which maybe thought of as aninduction by itself. Suppose X = X ∪ X where X is an irreducible component of X and X is the closure of X (cid:114) X . Consider the canonical morphism f : X (cid:116) X → X . It yields a shortexact sequence of sheaves 0 −→ Q (cid:96),X −→ f ∗ Q (cid:96),X (cid:116) X −→ F −→ F is a sheaf on X . Consider the inclusion i : X (cid:44) → X for X := X ∩ X . We claim that F ∼ = i ∗ Q (cid:96),X . To see this observe first that if we compute the pullback of the sequence along theopen immersion j : X (cid:114) X (cid:44) → X , then F vanishes and the morphism on the left becomes anisomorphism. In particular, F is supported on X . To compute the pullback along the closedimmersion i we may apply proper base change, since f is proper. But now the restriction of f to X is simply the trivial double cover X (cid:116) X −→→ X , so that i ∗ F ∼ = Q (cid:96),X . This proves theclaim because F ∼ = i ∗ i ∗ F , as F is supported on X .By an inductive application of the long exact cohomology sequences to sequences like (1),it suffices to prove the theorem for schemes X that are geometrically integral and separated20ndependence of (cid:96) -adic representationsalgebraic over K . In this case, the proof follows by resolving X by a smooth hypercovering X • see [6, p. 51], [8, 6.2.5] and the proof of [4, Thm. 6.3.2]. Since the hypercovering yieldsa spectral sequence that computes the cohomology of X in terms of the cohomologies of thesmooth X i , and this for all (cid:96) , and since only those X i with i ≤ X , contribute to X , theinduction step is complete since we have reduced the case of arbitrary X to lower dimensionsand to smooth X i . In this final section, we study the family ( ρ (cid:96),X ) (cid:96) ∈ L in the particular case where K is absolutelyfinitely generated with prime field k . We shall establish properties R ( k ) and if char( k ) > S ( k ). This will use Deligne’s purity results from Weil I (cf. [7, Thm. 1.6]) as well asthe global Langlands correspondence for function field proved by Lafforgue (cf. [26]).Let k be a finite field and U a smooth k -variety. For every closed point u ∈ U let k ( u ) be the (fi-nite) residue field of u , and let D ( u ) ⊂ π ( U ) be the corresponding decomposition group (definedonly up to conjugation). Denote by Fr u ∈ D ( u ) the preimage under the canonical isomorphism D ( u ) ∼ = Gal( k ( u )) of the map σ u : x (cid:55)→ x | k ( u ) | , so that within π ( U ), the automorphism Fr u isalso defined only up to conjugation. Proposition 7.1.
Suppose K is a finitely generated field of characteristic p ≥ . Let k be theprime field of K . Let X/K be a smooth projective scheme.a) There exists a finite separable extension K (cid:48) /K and smooth k -variety U (cid:48) /k with functionfield K (cid:48) such that for each q ≥ and every (cid:96) ∈ L (cid:114) { p } , the representation ρ ( q ) (cid:96),X | Gal( K (cid:48) ) of Gal( K (cid:48) ) is unramified along U (cid:48) .b) If p > , then the family of representations ( ρ ( q ) (cid:96),X | Gal( K (cid:48) ) ) (cid:96) ∈ L(cid:114) { p } is strictly compatible andpure of weight q , that is: For every closed point u (cid:48) ∈ U (cid:48) the characteristic polynomial p u (cid:48) ( T ) of ρ ( q ) (cid:96),X (Fr u (cid:48) ) has integral coefficients, is independent of (cid:96) ∈ L(cid:114) { p } , and the reciprocal rootsof p u (cid:48) ( T ) all have absolute value | k ( u (cid:48) ) | q/ .Proof. Note that k is perfect. There exists a finite separable extension E/K such that X E splitsup into a disjoint union of geometrically connected smooth projective E -varieties. Thus, afterreplacing K by a finite separable extension, we can assume that X/K is geometrically connected.Let ( S , · · · , S r ) be a separating transcendence basis of K/k . Identify k ( S , · · · , S r ) with thefunction field of A r and let S be the normalization of A r in K . Then S is a normal k -varietywith function field K . There exists an alteration S (cid:48) → S such that S (cid:48) /k is a smooth variety andthe function field K (cid:48) of S (cid:48) is a finite separable extension of K (cf. [6, Thm. 4.1, Remark 4.2]).Let A be the set of all affine open subschemes of S (cid:48) . Then Spec( K (cid:48) ) = lim ←− U (cid:48) ∈ A U (cid:48) . There exists U (cid:48) ∈ A and a projective U (cid:48) -scheme f : X (cid:48) → U (cid:48) such that X (cid:48) × U (cid:48) Spec( K (cid:48) ) = X K (cid:48) . (cf. [17,8.8.2] and [17, 8.10.5(v) and (xiii)]). By the theorem of generic smoothness, after shrinking U (cid:48) (cid:96) -adic representationsand X (cid:48) , we can assume that f is smooth. Furthermore, after removing the support of f ∗ ( O X (cid:48) )from U (cid:48) and shrinking X (cid:48) accordingly, we can assume that f has geometrically connected fibres.Let q ≥ u (cid:48) ∈ U (cid:48) . Define k (cid:48) := k ( u (cid:48) ) and X u (cid:48) := X (cid:48) × U (cid:48) Spec( k (cid:48) ). Then for every (cid:96) ∈ L (cid:114) { p } the ´etale sheaf R q f ∗ Z (cid:96) is lisse and compatible with any base change (cf. [29, VI.2.,VI.4]). Thus ρ ( q ) (cid:96),X | Gal( K (cid:48) ) factors through π ( U (cid:48) ) and a) holds true. Furthermore it follows that H q ( X (cid:101) K , Q (cid:96) ) can be identified with H q ( X u (cid:48) , (cid:101) k (cid:48) , Q (cid:96) ) in a way compatible with the Galois actions.Assume p >
0. Part b) then follows applying Deligne’s theorem on the Weil conjectures (cf. [7,Thm. 1.6]) to X u (cid:48) . Lemma 7.2.
Let the notation be as in the previous proposition and suppose we are in thesituation of part (b), so that k is a finite field. Fix q ∈ Z and denote by n the dimension of H q ( X (cid:101) K , Q (cid:96) ) which is independent of (cid:96) (cid:54) = p . Then the following hold:(a) For any smooth curve C over k and morphism ϕ : C → U (cid:48) , there exist irreducible cuspidalautomorphic representations ( π C,i ) i =1 ,...,m ϕ of GL n i ( A k ( C ) ) such that (cid:80) i n i = n and for all (cid:96) the representation ( ρ ( q ) (cid:96),X ◦ ϕ ∗ ) ss , where ϕ ∗ : π ( C ) → π ( U (cid:48) ) , agrees with the (cid:96) -adic repre-sentation ⊕ i ρ (cid:96),π C,i attached to π C,i via the global Langlands correspondence in [26, p. 2].(b) If for some prime (cid:96) ≥ there is a Z (cid:96) -lattice Λ of the Q (cid:96) representation space underlying ρ ( q ) (cid:96) ,X that is stabilized by Gal( K (cid:48) ) and such that Gal( K (cid:48) ) acts trivially on Λ /(cid:96) Λ , then in(a) for all ϕ : C → U (cid:48) the representations π C,i are semistable at all places of P ( C ) (cid:114) C .In particular, for all (cid:96) (cid:54) = p , the representation ρ ( q ) (cid:96),X satisfies S ( k ) .Proof. Let C be a smooth curve over k and ϕ : C → U (cid:48) a morphism over k . By Proposition 7.1,the family of representations ( ρ ( q ) (cid:96),X ◦ ϕ ∗ ) ss , where (cid:96) (cid:54) = p, is pure of weight q , semisimple and strictlycompatible as a representation of π ( C ). By the main theorem of [26, p. 2], each ( ρ ( q ) (cid:96),X ◦ ϕ ∗ ) ss being pure and semisimple gives rise to a list of irreducible cuspidal automorphic representationsvia the global Langlands correspondence. By the strict compatibility and the bijectivity of thecorrespondence on simple objects, this list is the same for all (cid:96) (up to permutation). Thisproves (a).Let now (cid:96) be as in (b). For each ϕ : C → U (cid:48) as in (a) consider the representation ( ρ ( q ) (cid:96) ,X ◦ ϕ ∗ ) ss as an action on the lattice Λ, that is trivial modulo (cid:96) Λ. Any filtration of Λ ⊗ Z (cid:96) Q (cid:96) preservedby the action of Gal( k ( C )) induces a filtration of Λ. Denote by Λ C the induced lattice for( ρ ( q ) (cid:96) ,X ◦ ϕ ∗ ) ss . Then it follows that the induced action of Gal( k ( C )) on Λ C /(cid:96) Λ C is trivial.Since (cid:96) >
2, this implies that series of the logarithm converges on the image of ( ρ ( q ) (cid:96) ,X ◦ ϕ ∗ ) ss . We call an automorphic representation π of GL n semistable at a place v if under the bijective local Langlandscorrespondence between local representations and Frobenius semi-simplified Weil-Deligne parameters, the Weil-Deligne parameter of π v is unramified when restricted to the Weil group. This definition is for instance used in[33, § loc. cit. on page 8 it is also pointed out that by [3], this notion of semistability is equivalent to thatof the automorphic representation having an Iwahori fixed vector. Lafforgue in [26] describes only a correspondence between Q (cid:96) -sheaves of weight 0, i.e., Galois representationsto GL n whose determinant has finite image, and automorphic representations with finite order central character.A general statement is given in [9, § (cid:96) -adic representationsThis image being pro- (cid:96) , a standard argument (cf. [39, Cor. 4.2.2]) shows that the Weil-Delignerepresentation at any place of P ( C ) attached to ( ρ ( q ) (cid:96) ,X ◦ ϕ ∗ ) ss is unramified when restricted to theWeil group. But then by the compatibility of the global and local Langlands correspondence(cf. [26, Thme. VII.3, Cor. VII.5]), this means that all π C,i are semistable at the places in ∂C (they are unramified at all other places). We now apply the same argument to all (cid:96) (cid:54) = p in reverse, to deduce that all ramification of ( ρ ( q ) (cid:96),X ◦ ϕ ∗ ) ss is (cid:96) -tame for all (cid:96) (cid:54) = p . The pointsimply is that in a family of Galois representations arising from a set of automorphic forms, theWeil-Deligne representation at a place of P ( C ) is independent of (cid:96) (cid:54) = p .Finally from Proposition 4.6(b), which is a variant of a result of Kerz-Schmidt-Wiesend, wededuce that for all (cid:96) (cid:54) = p , the representation ρ ( q ) (cid:96),X is divisor (cid:96) -tame. Combined with Proposi-tion 7.1(b), this establishes S ( k ). Remark . We would like to point out the parallel between the proof of S ( k ) in part (b) ofthe previous lemma and in Example 5.2. In the proof of (b) we selected a prime (cid:96) and enlarged K so that its image would act trivially on Λ /(cid:96) Λ via ρ (cid:96) for a Gal( K )-stable lattice Λ. Thenwe could use the uniformity provided by automorphic representations (after restricting the ρ (cid:96) to any curve) to deduce from this that all ramification was semistable in a sense.In Example 5.2 we selected a prime (cid:96) and enlarged K so that it contains K ( A [ (cid:96) ]). Thisrequirement is equivalent to asking that Gal( K ) acts trivially on the quotient T (cid:96) ( A ) /(cid:96) T (cid:96) ( A )for the lattice T (cid:96) ( A ) from the Tate-module. Then we used the uniformity provided by theN´eron model N of the abelian variety A over any discrete valuation ring in K to deduce S ( k ).The semistability of N follows from the condition at the single prime (cid:96) – in the analogous waythat the semistability of automorphic representations was deduced from a single prime (cid:96) . Corollary 7.4.
Suppose K is absolutely finitely generated and of characteristic p > . Suppose X/K is smooth projective. Then ( ρ (cid:96),X ) (cid:96) ∈ L(cid:114) { p } satisfies S ( k ) .Proof. Fix a prime (cid:96) ≥
3. Choose a lattice Λ underlying the representation ρ (cid:96) ,X . Replace K by a finite extension such that Gal( K ) acts trivially on Λ /(cid:96) Λ via ρ (cid:96) ,X . Apply now (b) of theprevious lemma to deduce the corollary. Theorem 7.5.
Let k (cid:48) be a field of characteristic p ≥ and let K (cid:48) /k (cid:48) be a finitely generatedextension. Let L = L (cid:114) { p } . Let X (cid:48) /K (cid:48) be a separated algebraic scheme. Then there exists afinite extension E (cid:48) /K (cid:48) and a constant c (cid:48) ∈ N with the following properties:(i) For every (cid:96) ∈ L the group ρ (cid:96),X (cid:48) (Gal( (cid:101) k (cid:48) E (cid:48) )) lies in Σ (cid:96) ( c (cid:48) ) and is generated by its (cid:96) -Sylowsubgroups.(ii) The family ( ρ (cid:96),X (cid:48) | Gal( (cid:101) k (cid:48) E (cid:48) ) ) (cid:96) ∈ L (cid:114) { , } is independent and ( ρ (cid:96),X (cid:48) | Gal( (cid:101) k (cid:48) K ) ) (cid:96) ∈ L is almost indepen-dent.Proof. It suffices to establish conditions (a) and (b) of Corollary 6.7. For this, let K be absolutelyfinitely generated, k its prime field and X/K a smooth projective variety. In this case we have23ndependence of (cid:96) -adic representationsproven that R ( k ) holds in Proposition 7.1 and that S ( k ) holds for k of positive characteristicin Corollary 7.4. This verifies condition (a) of Corollary 6.7.For condition (b) define n to be the dimensions of ρ (cid:96),X . Since X/K is smooth projective, thisdimension is independent of the chosen (cid:96) . Thus all images ρ (cid:96),X (Gal( K )) are n -bounded at (cid:96) .From Theorem 3.4 we deduce for each (cid:96) the existence of a short exact sequence1 → M (cid:96) → ρ (cid:96),X (Gal( K )) → H (cid:96) → M (cid:96) in Σ (cid:96) (2 n ) and H (cid:96) in Jor (cid:96) ( J (cid:48) ( n )), for the constant J (cid:48) ( n ) from Theorem 3.4. Considerthe induced representations τ (cid:96) defined as the composite Gal( K ) ρ (cid:96),X → ρ (cid:96),X (Gal( K )) → H (cid:96) . Since ρ (cid:96),X satisfies R ( k ) if char( k ) = 0 and S ( k ) if char( k ) >
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Gebhard B¨ockleComputational Arithmetic GeometryIWR (Interdisciplinary Center for Scientific Computing)University of HeidelbergIm Neuenheimer Feld 36869120 Heidelberg, Germany
E-mail address: [email protected]
Wojciech GajdaFaculty of Mathematics and Computer ScienceAdam Mickiewicz UniversityUmultowska 8761614 Pozna´n, Poland
E-mail adress: [email protected] (cid:96) -adic representations
Sebastian PetersenInstitut INF1Universit¨at der BundeswehrWerner-Heisenberg-Weg 3985577 Neubiberg, Germany
E-mail adress: [email protected]@unibw.de