(MOD l) REPRESENTATIONS

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1 -INDEPENDENCE FOR COMPATIBLE SYSTEMS OF (MOD ) REPRESENTATIONS CHUN YIN HUI Abstract. Let K be a number fied. For any system of semisimpe mod Gaois representations {φ : Ga( Q/K) GL N (F )} arising from étae cohomoogy (Definition 1), there exists a finite norma extension L of K such that if we denote φ (Ga( Q/K)) and φ (Ga( Q/L)) by respectivey Γ and γ for a, and et S be the F -semisimpe subgroup of GL N,F associated to γ (or Γ ) by Nori [No87] for a sufficienty arge, then the foowing statements hod for a sufficienty arge : A(i) The forma character of S GL N,F (Definition 3) is independent of and is equa to the forma character of (G )der GL N,Q, where (G )der is the derived group of the identity component of G, the monodromy group of the corresponding semi-simpified -adic Gaois representation Φ ss. A(ii) The non-cycic composition factors of γ and S (F ) are identica. Therefore, the composition factors of γ are finite simpe groups of Lie type of characteristic and cycic groups. B(i) The tota -rank rk Γ of Γ (Definition 14) is equa to the rank of S and is therefore independent of. B(ii) The A n -type -rank rk An N\{1, 2, 3, 4, 5, 7, 8} and the parity of (rk A4 Γ of Γ (Definition 14) for n Γ )/4 are independent of. Contents 1. Introduction Agebraic enveope Ḡ Nori s theory Characters of tame inertia group Exponents of characters arising from étae cohomoogy Tame inertia tori and rigidity Mathematics Subject Cassification. 11F80, 14F20, 20D05. Key words and phrases. Gaois representations, -independence, big Gaois image, étae cohomoogy. Major revisions of the paper were done when I was a postdoctora feow at The Hebrew University of Jerusaem supported by Aner Shaev s ERC Advanced Grant no

2 2 CHUN YIN HUI 2.5. Construction of Ḡ independence of Γ Forma character of Ḡ GL N,F Forma character of S GL N,F Proofs of Theorem A and Coroary B Acknowedgments References Introduction Let K be a number fied, P N the set of prime numbers, and X a compete non-singuar variety defined over K. For any integer i beonging to [0, 2dimX], the absoute Gaois group Ga K := Ga( Q/K) acts on the ith -adic étae cohomoogy group H í et (X K, Q ) for each prime number P. The dimension of H í et (X K, Q ) as a Q -vector space is independent of and we denote it by N. We therefore obtain a system of continuous, -adic Gaois representations indexed by P: {Φ : Ga K GL N (Q )} P which satisfies strict compatibiity (Deigne [De74]) in the sense of Serre [Se98, Chapter 1]. There is a conjectura -independence [Se94] on the images of {Φ } which has been studied by many peope. When X is an eiptic curve without compex mutipication, Serre has proved that the Gaois action on the -adic Tate modue T (X) is the whoe GL(T (X)) when is sufficienty arge by showing that the Gaois action φ on -torsion points X[] = T (X)/T (X): φ : Ga K GL(X[]) = GL 2 (F ) is surjective for 1 [Se72]. This paper is motivated by the idea that the argeness of the -adic Gaois image Γ := Φ (Ga K ) can be studied via taking mod reduction. More precisey, given any continuous, -adic representation Φ : Ga K GL N (Q ), one can find a Gaois stabe Z - attice of Q N so that up to some change of coordinates, we may assume Φ (Ga K ) GL N (Z ) since Ga K is compact. Then by taking mod reduction GL N (Z ) GL N (F ) and semi-simpification, we obtain a continuous, semisimpe, mod Gaois representation φ : Ga K GL N (F ) which is independent of the choice of the Z -attice by Brauer-Nesbitt [CR88, Theorem 30.16]. Denote the mod Gaois image φ (Ga K ) by Γ.

3 -INDEPENDENCE FOR COMPATIBLE SYSTEMS OF (MOD ) REPRESENTATIONS3 Definition 1. A system of mod Gaois representations {φ : Ga K GL N (F )} P is said to be arising from étae cohomoogy if it is the semi-simpification of a mod reduction of the -adic system or its dua system: {Φ : Ga K GL(H í et(x K, Q ))} P, {Φ : Ga K GL(H í et(x K, Q ) )} P for a compete non-singuar variety X defined over K and some i, where H í et (X K, Q ) := Hom Q (H í et (X K, Q ), Q ). Let ρ ss denote the semi-simpification for any finite dimensiona representation ρ over a perfect fied (we defined by Brauer-Nesbitt [CR88, Theorem 30.16]). Let {Φ } be a compatibe system of -adic representations of Ga K in Definition 1, the agebraic monodromy group at of the semi-simpified system {Φ ss }, denoted by G, is the Zariski cosure of Φ ss (Ga K) in GL N,Q. Then G is reductive. Denote the set of non-archimedean vauations of K and K by respectivey Σ K and Σ K. The strict compatibiity of {Φ } impies {φ } is stricty compatibe in the foowing sense. Definition 2. A system of mod Gaois representations {φ : Ga K GL N (F )} P is said to be stricty compatibe if {φ } is continuous, semisimpe, and satisfies the foowing conditions: (i) There is a finite subset S Σ K such that φ is unramified outside S := S {v Σ K : v } for a, (ii) For any 1, 2 P and any v Σ K extending some v Σ K \(S 1 S 2 ), the characteristic poynomias of φ 1 (Frob v ) and φ 2 (Frob v ) are the reductions mod 1 and mod 2 of some poynomia P v (x) Q[x] depending ony on v. Let ρ : G GL N,F be a faithfu representation of a rank r reductive agebraic group G defined over fied F. We define in the beginning of 2 the forma character of ρ as an eement of quotient set GL r (Z)\Z[Z r ]. Here Z[Z r ] is the free abeian group generated by Z r and GL r (Z) acts naturay on Z[Z r ]. This aows us to define what is meant by two representations having the same forma character (see Definition 3 ) and the forma character is bounded by a constant C > 0 (see Definition 4,4 ). Let {φ } be a stricty compatibe system of mod representations

4 4 CHUN YIN HUI arising from étae cohomoogy (Definition 1,2), this paper studies - independence of mod Gaois images Γ for a sufficienty arge. Let g be a Lie type. We define tota -rank rk Γ and g-type -rank rk g Γ of a finite group Γ in 3.3 Definition 14. The main resuts are as foows. Theorem A. (Main theorem) Let K be a number fied and {φ : Ga K GL N (F )} P a stricty compatibe system of mod Gaois representations arising from étae cohomoogy (Definition 1,2). There exists a finite norma extension L of K such that if we denote φ (Ga K ) and φ (Ga L ) by respectivey Γ and γ for a, and et S GL N,F be the connected F -semisimpe subgroup associated to γ (or Γ ) by Nori s theory for 1, then the foowing hod for 1 : (i) The forma character of S GL N,F is independent of (Definition 3 ) and is equa to the forma character of (G )der GL N,Q, where (G )der is the derived group of the identity component of G, the agebraic monodromy group of the semi-simpified representation Φ ss. (ii) The composition factors of γ and S (F ) are identica moduo cycic groups. Therefore, the composition factors of γ are finite simpe groups of Lie type of characteristic and cycic groups. Coroary B. Let S be defined as above, then the foowing hod for 1 : (i) The tota -rank rk Γ of Γ (Definition 14) is equa to the rank of S and is therefore independent of. (ii) The A n -type -rank rk An Γ of Γ (Definition 14) for n N\{1, 2, 3, 4, 5, 7, 8} and the parity of (rk A 4 Γ )/4 are independent of. Remark 1.1. As an appication of the main resuts, we prove in [HL14] that Φ (Ga K ), the -adic Gaois image arising from étae cohomoogy has certain maximaity inside the agebraic monodromy group G if is sufficienty arge and G is of type A. This generaizes Serre s open image theorem on non-cm eiptic curves [Se72]. Remark 1.2. For any fied F, define ι to be the invoution of GL N,F that sends A to (A t ) 1. If Γ is a subgroup of GL N (F ), then Γ is semisimpe on F N if and ony if ι(γ) is semisimpe on F N. If φ is the mod Gaois representation arising from the dua representation Het í (X K, Q ) (Definition 1), then the mod representation arising from Het í (X K, Q ) is ι φ under suitabe basis by Brauer-Nesbitt [CR88, Theorem 30.16]. Since ι is an automorphism of GL N, it suffices to prove Theorem A by considering ony the dua mod system {φ } and Gaois images { Γ }. Let φ v be the restriction of φ to inertia subgroup I v such

5 -INDEPENDENCE FOR COMPATIBLE SYSTEMS OF (MOD ) REPRESENTATIONS5 that v Σ K divides. The reason for choosing the dua system is that the characters of φ ss v have bounded exponents in the sense of Definition 8 for 1 by Serre s tame inertia conjecture proved by Caruso [Ca08] (see Theorem 2.3.1). Such boundedness makes our arguments simper. This paper can be considered as mod version of [Hu13] in which we studied -independence of monodromy groups of any compatibe system of -adic representations by the theory of abeian -adic representation [Se98] and the representation theory of compex semisimpe Lie agebra. The main difference between [Hu13] and this paper is that you get nothing new for considering monodromy groups of mod Gaois images because they are just finite groups. The strategy in this paper is to first construct for each 1 a connected F -reductive subgroup Ḡ GL N,F with bounded forma characters (Definition 4,4 ) such that [ Γ : Γ Ḡ(F )] and [Ḡ(F ) : Γ Ḡ(F )] are both uniformy bounded (Theorem 2.0.5). The idea to construct such Ḡ was due to Serre [Se86] where he considered the Gaois action on the -torsion points of abeian varieties A without compex mutipication. In Serre s case, the semisimpe part S of Ḡ is constructed by Nori s theory [No87] and the center C of Ḡ is the mod reduction of some Q-diagonaizabe group C Q which is a subgroup of the centraizer of monodromy G in GL N,Q. Hence, {Ḡ GL N,F } has bounded forma characters. The construction of C Q uses the abeian theory of -adic representations [Se98] and the Tate conjecture for abeian variety (proved by Fatings [Fa83]) which reates the endomorphism ring of A and the commutant of Gaois image Γ in End N (Q ). Athough we don t have the uxury of the Tate conjecture for étae cohomoogy in genera, it is sti possibe to construct reductive Ḡ GL N,F with the above conditions for 1 by Nori s theory, tame inertia tori [Se86], and Serre s tame inertia conjecture (proved by Caruso [Ca08]). The constructions of these agebraic enveopes Ḡ of Γ (see Definition 5) are accompished in 2. Once these nice enveopes are ready, we can use the techniques in [Hu13, 3] to prove that the forma character (Definition 3) of the semisimpe part S GL N,F is independent of 1 (Theorem A). The number of A n factors of S for arge n are then independent of for a 1 by [Hu13, Theorem 2.19]. Since the group of F -rationa points of Ḡ is commensurate to the Gaois image Γ, one deduces -independence resuts on the number of Lie type composition factors of characteristic of Γ for 1 (Coroary B). Section 3 is devoted to the proofs of Theorem A and Coroary B. The foowing summarizes the symbos that are frequenty used within

6 6 CHUN YIN HUI the text. Groups inside GL N,F with charf > 0 have their symbos over-ined and shoud not be confused with base change to an agebraic cosure. Ga F the absoute Gaois group of fied F K, L number fieds v a vauation of K that divides prime I v the inertia subgroup of Ga K at vauation v U, V, W (Ū, V, W ),... vector spaces defined over F (over F ) Γ, γ, Ω, Ω v,... finite subgroups of GL N (F ) G, T,... Ḡ, S, N, Ī, Ī v,... agebraic subgroups of GL N,Q agebraic subgroups of GL N,F Φ, Ψ, Θ,... representations over Q φ, ψ, µ, t, ρ v, f v, w v,... representations over F ρ ss the semi-simpification of representation ρ ρ the dua representation of representation ρ 2. Agebraic enveope Ḡ We define forma character and prove some reated propositions before stating the main resut (Theorem 2.0.5) of this section. Let ρ : G GL N,F be a faithfu representation of a rank r reductive agebraic group G defined over fied F. Choose a maxima torus T of G and denote the character group of T by X. Let {w 1, w 2,..., w N } X be the mutiset of weights of ρ T over F and choose an isomorphism X = Z r. Then the image of w 1 + w w N Z[X] = Z[Z r ] in the quotient set GL(X)\Z[X] = GL r (Z)\Z[Z r ] is independent of the choices of maxima torus T and isomorphism X = Z r. Definition 3. Let ρ be as above. The forma character of ρ is defined to be the image of w 1 + w w N Z[Z r ] in GL r (Z)\Z[Z r ]. This definition of forma character is more genera than the one in [Hu13, 2.1]. It aows us to compare forma characters of two N- dimensiona faithfu representations ρ 1 : G 1 GL N,F1 and ρ 2 : G 2 GL N,F2 over different fieds whenever G 1 and G 2 have the same rank. Let G N m be the diagona subgroup of GL N. Every character χ of G N m can be expressed uniquey as x m 1 1 x m 2 2 x m N N, a product of powers of standard characters {x 1, x 2,..., x N }, where x i maps (a 1,..., a N ) G N m to a i for a i. The foowing proposition (definition) is particuary usefu. Proposition (Definition 3 ) Let ρ 1 and ρ 2 be as above. If T 1 G 1 and T 2 G 2 are maxima tori. The foowing conditions are equivaent:

7 -INDEPENDENCE FOR COMPATIBLE SYSTEMS OF (MOD ) REPRESENTATIONS7 (i) Representations ρ 1 and ρ 2 have the same forma character. (ii) Tori ρ 1 (T 1 ) and ρ 2 (T 2 ) are respectivey conjugate (in GL N, F1 and GL N, F2 ) to some subtori D 1 and D 2 of the diagona subgroup G N m GL N such that D 1 and D 2 are annihiated by the same set of characters of G N m. Hence, forma characters of N-dimensiona faithfu representations are in bijective correspondence with subtori in G N m up to natura action of permutation group Perm(N) of N etters on G N m. Proof. Assume T j = G r m, F j and ρ j (T j ) G N m, F j GL N, Fj from now on by base change to agebraic cosure of F j and diagonaizations for j = 1, 2. Suppose (i) hods, then by taking an automorphism of the character group of T 1 and a permutation of coordinates of G N m we obtain x i ρ 1 = x i ρ 2 for a standard character x i of G N m if we identify the character groups of G r m, F 1 and G r m, F 2 naturay. This impies the set of characters of G N m that annihiate D j := ρ j (T j ) for j = 1, 2 are equa which is (ii). Suppose (ii) hods, it suffices to consider the case that ρ 1 and ρ 2 are standard representations (incusions) since ρ : G GL N,F and ρ(g) GL N,F aways have the same forma character. Condition (ii) impies that there exists an automorphism of G N m such that D j = {(a 1,..., a N ) G N m : a 1 = a 2 = = a N r = 1} for j = 1, 2 because D 1 and D 2 are connected. Then (i) foows easiy. Let ρ : T GL N, F be a representation of a torus T. Since the set of weights of ρ is obtained by diagonaizing ρ(t) and is independent of diagonaizations, subtori of G N m that are conjugate to ρ(t) ony differ by a permutation of N coordinates. Therefore, the map from forma characters of N-dimensiona faithfu representations to subtori of G N m moduo action of Perm(N) is we defined. Since the equivaence of (i) and (ii) impies injectivity and any subtorus of G N m is the forma character of the standard representation of the subtorus, the map is a bijective correspondence. Exampes: Denote standard representation and faithfu representation by respectivey Std and ρ. Beow are some pair of representations that have the same forma character: (i) (GL 2,Q, Std) and (GL 2,F, Std); (ii) (G, ρ) and (H, ρ H ) if H is a reductive subgroup of G of same rank; (iii) (G, ρ) and (G, ρ );

8 8 CHUN YIN HUI (iv) (G, ρ) and (ρ(g), Std). Definition 4. The forma character of ρ is said to be bounded by a constant C > 0 if there exists an isomorphism X = Z r such that the coefficients of the images of weights w 1, w 2,..., w N X in Z r have absoute vaues bounded by C. Let N be a fixed integer and {ρ i : G i GL Ni,F i } i I a famiy of faithfu representations of reductive groups such that N i N for a i I. The famiy is said to have bounded forma characters if the forma character of ρ i is bounded by some constant C > 0 for a i I. Remark Let {ρ i } i I be a famiy of representations in Definition 4 having bounded forma characters. Then the number of distinct forma characters arising from the famiy is finite. Let χ = x m 1 1 x m 2 2 x m N N be a character of GN m expressed as products of standard characters. We ca mutiset {m 1,..., m N } the exponents of χ and say the exponents are bounded by C > 0 if m i < C for a 1 i N. The foowing characterization of Definition 4 is very usefu in this paper. Proposition (Definition 4 ) Let {ρ i } i I be a famiy of faithfu representations of reductive G i such that ρ i is N i -dimensiona and N i N for a i I. Choose a maxima torus T i of G i for each i I. The foowing conditions are equivaent: (i) The famiy has bounded forma characters. (ii) For any i I and any subtorus D i of the diagona subgroup G N i m GL Ni that is conjugate (in GL Ni, F i ) to ρ i (T i ), one can choose a set R i of characters of G N i m such that the common kerne of R i is D i and the exponents of characters in R i are bounded by a constant independent of i I. Proof. It foows easiy from Definition 4, the bijective correspondence in Proposition 2.0.1, and Remark Proposition Let {ρ i } i I and {φ i } i I be two famiies of faithfu representations of reductive G i and H i over fied F i with bounded forma characters such that the target of ρ i and φ i are both equa to GL Ni,F i and ρ i (G i ) commutes with φ i (H i ) for a i I. Then the famiy of standard representations {ρ i (G i ) φ i (H i ) GL Ni,F i } i I aso has bounded forma characters.

9 -INDEPENDENCE FOR COMPATIBLE SYSTEMS OF (MOD ) REPRESENTATIONS9 Proof. It foows easiy from Remark 2.0.2, Proposition 2.0.3, and the fact (by the commutativity hypothesis) that any maxima torus of ρ i (G i ) φ i (H i ) is generated by some maxima torus of ρ i (G i ) and some maxima torus of φ i (H i ). Let {φ } be the stricty compatibe system of mod Gaois representations arising from (Definition 1,2) the dua system of -adic representations {Φ }. Denote the image of φ by Γ and the ambient space of the representation by V = F N for each. Each Γ := φ (Ga K ) is a subgroup of GL N (F ) for a fixed N. Suppose K is a finite norma extension of K. Since [φ (Ga K ) : φ (Ga K )] [K : K] for a and the restriction of {φ } to Ga K is semisimpe [CR88, Theorem 49.2] and satisfies the compatibiity conditions (Definition 2), we are free to repace K by K in the course of proving the main theorem. The main resut of this section states that for 1, Γ can be approximated by some connected, reductive subgroup Ḡ GL N,F with bounded forma characters (Definition 4 ). Theorem Let {φ } P be a system of mod Gaois representations as above. There exist a finite norma extension L of K and a connected, F -reductive subgroup Ḡ of GL N,F for each 1 such that (i) γ := φ (Ga L ) is a subgroup of Ḡ (F ) of uniformy bounded index, (ii) the action of Ḡ on V := V F is semisimpe, (iii) the representations {Ḡ GL N,F } 1 have bounded forma characters in the sense of Definition 4. Definition 5. A system of connected reductive groups {Ḡ} 1 satisfying the conditions in the above theorem is caed a system of agebraic enveopes of { Γ } 1. We say Ḡ is the agebraic enveope of Γ when a system of agebraic enveopes is given. We first estabish in essentia ingredients of the proof of Theorem Then the proof is presented in Nori s theory. The materia in this subsection is due to Nori [No87]. Suppose > N 1. Given a subgroup Γ of GL N (F ), Nori s theory gives us a connected agebraic group S that captures a the order eements of Γ if is bigger than a constant that ony depends on N. Let Γ[] = {x Γ x = 1}. The norma subgroup of Γ generated by Γ[] is denoted by Γ +. Define exp(x) and og(x) by exp(x) = 1 i=0 x i i! 1 (1 x) i and og(x) =. i i=1

10 10 CHUN YIN HUI Denote by S the (connected) agebraic subgroup of GL N,F, defined over F, generated by the one-parameter subgroups t x t := exp(t og(x)) for a x Γ[]. Agebraic subgroups with the above property are said to be exponentiay generated. The theorem we need is stated beow. Theorem [No87, Theorem B(1), 3.6(v)] There is a constant c 0 = c 0 (N) such that if > c 0 and Γ is a subgroup of GL N (F ), then (i) Γ + = S(F ) +, (ii) S(F )/ S(F ) + is a commutative group of order 2 N 1. Proposition Let S be the agebraic group associated to Γ by Nori s theory for a > N 1. There is a constant c 1 = c 1 (N) > c 0 (N) that depends ony on N such that if > c 1, then the foowing hod: (i) S is a connected, exponentiay generated, semisimpe F -subgroup of GL N,F. (ii) S acts semi-simpy on the ambient space V = FN. (iii) [ S (F ) : S (F ) Γ ] 2 N 1. Proof. Since Γ acts semi-simpy on V, so does Γ + [CR88, Theorem 49.2]. Part (ii) then foows from [EHK12, Theorem 24] for some sufficienty arge constant c 1 (N) (> c 0 (N)) depending ony on N, see aso [Se86]. Since > c 0 (N), S (F ) + = Γ + (Theorem 2.1.1) aso acts semisimpy on V. This impies S (F ) + cannot have norma -subgroup. If S has a non-trivia unipotent radica Ū, then Ū is defined over F [Sp08, Proposition (v)] and Ū(F ) is then a non-trivia norma - group of S (F ) + which is a contradiction. Therefore S is reductive. S is actuay semisimpe since it is generated by unipotent eements Γ +. This proves (i). Since > c 0 (N), (iii) is proved by Theorem Definition 6. Define the semisimpe enveope of Γ for a sufficienty arge as the connected, semisimpe F -agebraic group S in Proposition Remark If K is a finite extension of K, then the semisimpe enveopes of φ (Ga K ) and φ (Ga K ) are identica for 1 because the order eements of the two finite groups are the same when is arge Characters of tame inertia group. Let ρ : Ga K GL N (F ) be a continuous representation and I v the inertia subgroup of Ga K at v Σ K that divides. Let I w v be the wid inertia (norma) subgroup of I v and ρ ss v the semi-simpification of the restriction of ρ to I v. Since ρ ss (Iw v ) is an -group and semisimpe on F N, ρss v (I w v ) = {1} and ρ ss v

11 -INDEPENDENCE FOR COMPATIBLE SYSTEMS OF (MOD ) REPRESENTATIONS 11 factors through a representation of the tame inertia group I t v := I v /I w v (sti denoted by ρ ss v ): ρ ss v : I t v GL N (F ). The tame inertia group I t v is a projective imit of cycic groups of order prime to [Se72, Proposition 2] θ v : I t v = im k where the projective system is given by norm maps of finite fieds of characteristic. The isomorphism is unique up to action of Ga F on the target. Definition 7. The fundamenta characters of I t v of eve d [Se72, 1.7] are defined as θd j, j = 0, 1,..., d 1 where θ d : I t v θ v im F F F k k d. Any continuous character χ : I t v F of ρss v factors through a power of some θ d. Character theory says that Hom(F, F d ) = Hom(F, C ) d is cycic generated by θ d of order d 1. Therefore, χ can aways be expressed as a product of fundamenta characters of eve d F k χ = (θ d ) m0 (θd) m1 (θd d 1 ) m d 1 Definition 8. Let χ : I t v F be a character of ρss v and express χ as a product of fundamenta characters of eve d as above. (i) The product is said to be -restricted if 0 m i 1 for a i and not a m i equa to 1. It is easy to see that -restricted expression of χ is unique. (ii) The exponents of χ are defined to be the mutiset of powers {m 0, m 1,..., m d 1 } in the -restricted product. Note that the mutiset is independent of the action of Ga F on the target. Lemma Let V = F n be a continuous, irreducibe subrepresentation of ρ v, then the characters of the representation can be written as a product of fundamenta characters of eve n. Proof. For simpicity, assume ρ v is irreducibe. The image of I t v in GL(V ) is a cycic group of order prime to, therefore V is a F [x]/(f(x))- modue where x corresponds to a generator of the cycic image and the minima poynomia f(x) is separabe. Irreducibiity of V impies f(x) is irreducibe over F. Thus ρ v (I t v) is contained in a maxima subfied F of End(V ) and ρ v : I t v F GL(V ) can be written as a product

12 12 CHUN YIN HUI of fundamenta characters of eve n as above. On the other hand, V has a structure of F -vector space of dimension 1 such that the action of ρ v (I t v) F is through fied mutipication. By tensoring F with F (on the right) over F, we obtain an F -isomorphism F F F F F x y (xy, x y,..., x n 1 y) where x, x,..., x n 1 are just conjugate of x over F. If x ρ v (I t v) F, then we see the action of I t v on V F F is a direct sum of products of fundamenta characters of eve n Exponents of characters arising from étae cohomoogy. Every character χ of ρ ss v : I t v GL N (F ) can be written as χ = (θ n ) m0 (θ n) m1 (θ n 1 n ) m n 1, a product of fundamenta characters of eve n N by Lemma One woud ike to study the exponents m 0,..., m n 1 (Definition 8) and in the case of étae cohomoogy we have the foowing theorem proved by Caruso [Ca08]. Theorem (Serre s tame inertia conjecture) Let X be a proper and smooth variety over a oca fied K (a finite extension of Q ) with semi-stabe reduction over O K, the ring of integers of K and i an integer. The Gaois group Ga K acts on H í et (X K, Z/Z), the F -dua of the ith cohomoogy group with Z/Z coefficients. If we restrict the representation to the inertia group of Ga K, then the exponents of the characters of the tame inertia group on any Jordan-Hoder quotient of H í et (X K, Z/Z) are between 0 and ei where e is the ramification index of K/Q. We now reate our mod Gaois representation φ to representation H í et (X K, Z/Z) in Theorem Cohomoogy group H í et (X K, Z ) is a finitey generated, free Z -modue [Ga83] for 1: Reduction mod gives H í et(x K, Z ) = Z Z. H í et(x K, Z ) F = Z/Z Z/Z and the semi-simpification of H í et (X K, Z ) F is then isomorphic to the semi-simpification of a mod reduction of -adic representation H í et (X K, Q ) by Brauer-Nesbitt [CR88, Theorem 30.16]. Since the sequence H í et(x K, Z ) H í et(x K, Z ) H í et(x K, Z/Z)

13 -INDEPENDENCE FOR COMPATIBLE SYSTEMS OF (MOD ) REPRESENTATIONS 13 is exact [Mi13, Theorem 19.2], H í et (X K, Z ) F is isomorphic to H í et (X K, Z/Z). Reca V is the semi-simpification of a mod reduction of H í et (X K, Q ). Thus, we concude that Proposition For a sufficienty arge, Het í (X K, Z ) F is isomorphic to Het í (X K, Z/Z) and the semi-simpification of Het í (X K, Z/Z) is V. The foowing theorem is the main resut of this subsection. Theorem Let K be a number fied. Let φ : Ga K GL(V ) = GL N (F ) be the mod Gaois representation arising from étae cohomoogy group H í et (X K, Q ) for sufficienty arge. If we restrict φ to the inertia group I v of a vauation v of K and semi-simpify the representation, then every character χ of the representation can be written as χ = (θ N! ) m0 (θ N!) m1 (θ N! 1 N! ) m N! 1 a product of fundamenta characters of eve N! with exponents (Definition 8) m 0,..., m N! 1 (depending on ) beonging to [0, ei] where e is the ramification index of K v /Q, v = v K, and K v is the competion of K with respect to v. Proof. Proposition impies that if is sufficienty arge, then Gaois representations V = (V ) and (Het í (X K, Z/Z) ) ss are isomorphic. Let χ be a character of I t v given by the semi-simpification of the restriction of V to inertia subgroup I v. By Theorem 2.3.1, χ can be written as χ = (θ d ) m0 (θ d) m1 (θ d 1 d ) m d 1, a product of fundamenta characters of eve d ( N by Lemma 2.2.1) with exponents m 0,..., m d 1 beonging to [0, ei] where e is the ramification index of K v /Q. Since d divides N!, θ N! factors through χ. Consider the norm map Nm : F N! F d x x x d x 2d x (N! d). Then we obtain a product of fundamenta characters of eve N! χ = (Nm θ N! ) m 0+m 1 + +m d 1 d 1 = (θ N! ) s0 (θ N!) s1 (θ N! 1 N! ) s N! 1 with exponents s 0,..., s N! 1 beonging to [0, ei].

14 14 CHUN YIN HUI 2.4. Tame inertia tori and rigidity. Tame inertia tori were considered by Serre when he studied Gaois action on -torsion points of abeian varieties without compex mutipication [Se86]. He observed that these tori have certain rigidity which wi be expained in this subsection. Assume > N 1 as in 2.1. Since every non-trivia eement of every -Syow subgroup of Γ is of order and Γ + is contained in S (F ) by Theorem 2.1.1(i), index [ Γ : Γ S (F )] is prime to. Let N be the normaizer of S in GL N,F ; ceary Γ N. Theorem [Se86, 1 Theorem] There are constants c 2 = c 2 (N) and c 3 = c 3 (N) such that if > c 2, S GL N,F is an exponentiay generated semisimpe agebraic group defined over F, and the action on V = FN is semisimpe. If W is the F -subspace of U := c 3 i=1 ( i V ) fixed by S, then t : N / S GL W is an F -embedding. Moreover, if x / S, then there is an eement of W that is not fixed by x. By Theorem 2.4.1, Γ /( Γ S (F )) embeds in GL(W ) with dim(w ) c 4 = c 4 (N) uniformy for some integer c 4. Theorem beow is the main resut of this subsection. Definition 9. Define µ : Ga K GL(W ) to be the composition t φ for each and Ω to be the image µ, where t is defined in Theorem Theorem Let Ī be the agebraic group generated by a set of tame inertia tori Ī v (Definition 10) for 1. There exist constant c 8 = c 8 (N) and a finite norma fied extension L/K such that if 1, then Ī is a torus, caed the inertia torus at, and µ (Ga L ) Ω is a subgroup of Ī(F ) such that (i) {Ī GL W } 1 have bounded forma characters (Definition 4 ), (ii) [Ī(F ) : µ (Ga L )] is bounded by c 8. Theorem [Jo78, Jordan s theorem on finite inear groups] For every n there exists a constant J(n) such that any finite subgroup of GL n over a fied of characteristic zero possesses an abeian norma subgroup of index J(n). The order of Ω is prime to. Ω can thus be ifted to a subgroup of GL N (C) such that N ony depends on N. Theorem (Jordan)

15 -INDEPENDENCE FOR COMPATIBLE SYSTEMS OF (MOD ) REPRESENTATIONS 15 then says that Ω has a abeian norma subgroup J of index ess than a constant c 5 = c 5 (N) := J(N ) depends on N. Since N depends on N, we have [ Ω : J ] c 5. If v divides, then the action of the inertia group I v on W is semisimpe because Ω is prime to. Since dim(w ) c 4! We obtain µ : I t v θ c4! F c 4! GL(W ). By Theorem and W in Theorem 2.4.1, there exist c 6 = c 6 (N) 0 such that if χ is a character, then χ can be written as a product of fundamenta characters of eve c 4! χ = (θ c4!) m0 (θ c 4!) m1 (θ c 4! 1 c 4! ) m c 4! 1 with exponents m 0,..., m c4! 1 beonging to [0, c 6 ] for a 1. Therefore, we make the foowing definition. Definition 10. Denote fied F c 4! by E for a. This gives a homomorphism f v : E GL(W ) if > c 6 (N) + 1. Let Ē denote Res E /F (G m ) (Wei restriction of scaars) for a. We have Ē(F ) = E. Then f v extends uniquey [Ha11, 3] to an -restricted F -morphism beow: w v : Ē := Res E /F (G m ) GL W. Denote the image of w v by Ī v for v 1. It is caed the tame inertia torus at v Σ K. Lemma There exists a constant c 7 = c 7 (N) such that for any v > c 6 (N) + 1, we have (i) {Ī v GL W } v have bounded forma characters (Definition 4 ); (ii) [Ī v(f ) : f v (E )] c 7. Proof. Since dim(w ) and dim(ē) are bounded by a constant independent of and the exponents of the characters of w v in terms of the fundamenta characters [Ha11, 3] beong to [0, c 6 ], we find by Proposition a set of characters R v of uniformy bounded exponents of the diagona subgroup of GL W by diagonaizing Ī v and then obtain assertion (i). For assertion (ii), uniform boundedness of exponents of characters and dim(ē) = c 4! (for a ) impy the number of connected components of Ker(w v ) is uniformy bounded by c 7. On the other hand, the number of F -rationa points of any F -torus of dimension k is between ( 1) k and ( + 1) k by [No87, Lemma 3.5]. Therefore,

16 16 CHUN YIN HUI µ (I t v) = f v (E ) has at east E c 7 ( + 1) = c4! 1 dim(ker(w v)) c 7 ( + 1) dim(ker(w v)) points and [Ī v(f ) : µ (I t v)] is bounded by c 7 ( + 1) dim(ker(w v))+dim(im(w v)) c 4! 1 when is big. This proves (ii). = c 7( + 1) c 4! c 4! 1 c 7 Lemma (Rigidity) [Ha11, 3],[Se86, 3] Let s GL(W ) be a semisimpe eement and f v : E GL(W ) a representation such that the exponents of characters of f v beong to [0, c] for some c > 0. If H E is a subgroup such that f v(h) commutes with s in GL(W ) and c [E : H] 1, then Ī v commutes with s, and hence so does f v (E ). Reca from Definition 2 that there is a finite subset S Σ K such that φ is unramified outside S := S {v Σ K : v } for a. Proof of Theorem The foowing arguments are infuenced by the arguments Serre gave for [Se86, Theorem 1]. Proof. Denote the image of µ (I t v) under the map Γ /( Γ S (F )) GL(W ) by Ω v whenever v. Let J be a maxima abeian norma subgroup of Ω := µ (Ga K ). We first prove that Ω v commutes with J if is arge. Since Ω v and J are abeian and [ Ω v : Ω v J ] c 5 by Theorem (Jordan), the tame inertia torus Ī v at v (Definition 10) and hence f v (E ) = Ω v commute with J if > c 5 c by rigidity (Lemma 2.4.5). For any v 1, v 2, since Ω v1 J commutes with Ω v2 J which are of bounded index in Ω v1 and Ω v2 respectivey, we obtain Ī v 1 commutes with Ī v 2 if 1 by rigidity (Lemma 2.4.5). The subgroup H of Ω generated by the inertia subgroups Ω v for a v is abeian and norma for 1. As J is maxima norma abeian in Ω, H J for a 1. Therefore, Ω / J corresponds to a fied extension of K of degree bounded by c 5 that ony ramifies in S (Definition 2) for 1. By Hermite s Theorem [La94, p.122], the composite of these fieds is sti a finite fied extension K of K. Therefore, µ (Ga K ) J for 1. Since the representations {φ } come from étae cohomoogy and I v Ga K is the inertia subgroup of Ga K at v [Ne99, Proposition 9.5], they

17 -INDEPENDENCE FOR COMPATIBLE SYSTEMS OF (MOD ) REPRESENTATIONS 17 are potentiay semi-stabe which means there exists a finite extension K of K such that φ (I v Ga K ) is unipotent for any v not dividing [dej96, 1]. Therefore, for each 1 we have a finite abeian extension of K with Gaois group µ (Ga K ) contained in J that ony ramifies at v Σ K dividing. Since µ (Ga K ) is an abeian Gaois group over K, each ramified prime v Σ K dividing arge corresponds to an inertia subgroup Ī v µ (Ga K ) and there are at most [K : Q] of them. For each inertia subgroup Ī v, choose a tame inertia torus Ī v such that Ī v Ī v(f ). Since these tame inertia tori commute with each other, the agebraic group Ī generated by them is an F -torus, caed the inertia torus at. Since {Ī v GL W } v 1 have bounded forma characters (Lemma 2.4.4(i)) and each Ī is generated by at most [K : Q] tame inertia tori, {Ī GL W } 1 have bounded forma characters by Proposition This proves (i). Let Ī be the subgroup of µ (Ga K ) generated by Ī v Then, for 1 we have µ (Ga K )/Ī for a v. is the Gaois group of a finite abeian extension of K that is unramified at every non-archimedean vauation. By abeian cass fied theory, these fieds generate a finite extension K of K. Choose L norma over K such that K L. Then, we obtain ( ) : µ (Ga L ) Ī Ī(F ). It remains to prove (ii). Suppose Ī is generated by tame inertia tori for 1 i k for some fixed k [K : Q]. We have Ī vi [Ī(F ) : µ (Ga L )] = [Ī(F ) : Ī(F ) Ω ] [Ī(F ) Ω : µ (Ga L )] [Ī(F ) : f v1 (E ) f vk (E )] [L : K]. It suffices to show [Ī(F ) : f v1 (E ) f v k (E )] is bounded independent of. The proof is identica to Lemma 2.4.4(ii) since f v1 (E ) f v k (E ) is the image of Ī is the image of f v1 f vk : (E ) k GL(W ), w v1 w vk : (Ē) k GL W, k (depending on ) is aways ess than [K : Q], and the exponents of characters (-restricted 10) of w v1 w vk are uniformy bounded. Therefore, there exists c 8 = c 8 (N) such that [Ī(F ) : µ (Ga L )] c 8 for 1.

18 18 CHUN YIN HUI 2.5. Construction of Ḡ. An F -torus Ī GL W is constructed in 2.4 for 1 and we have the foowing map defined in Theorem t : N N / S GL W. One has to show that Ī t ( N ) so that t 1 (Ī) is connected. It suffices to consider tame inertia tori Ī v. Reca vector space U from Theorem Lemma Let H be an agebraic subgroup of GL V. Then H acts on Ū. If H is invariant on the subspace W Ū fixed by S, then H is contained in N. Proof. Let x H \ N. Then there exists s S such that xsx 1 / S. There exists w W such that xsx 1 w w by the ast statement of Theorem Therefore, sx 1 w x 1 w impies x 1 w / W, a contradiction. Hence, H is contained in N. Proposition The F -torus Ī in GL W is a subgroup of the image of t : N N / S GL W defined in Theorem Proof. Let v be a vauation of K and I v the inertia subgroup of Ga K at v. The restriction φ : I v GL(V ) factors through a finite quotient π v : I v J v such that J v = k ( c 4! 1). Reca vector spaces W U from Theorem and f v : E GL(W ) from Definition 10. Consider the foowing diagram so that r φ i v = f v and the actions of E on W via f v and f v are the same. Here r is the obvious map and i v is a spitting of π v. This is possibe because E defined in 2.4 is cycic of order ( c4! 1) prime to. J v i v π v E φ f v GL V r GLU

19 -INDEPENDENCE FOR COMPATIBLE SYSTEMS OF (MOD ) REPRESENTATIONS 19 If is sufficienty arge, then the exponents of the characters (restricted) of representations φ i v and r φ i v beong to [0, i] and [0, ic 3 ] respectivey by Theorem and the construction of U. Reca Ē from definition 10. By Wei restriction of scaars, we obtain two F -morphisms α : Ē GL V β : Ē GL U. Since r α and β are both -restricted [Ha11, 3] and equa to r φ i v when restricting to E, by uniqueness [Ha11, 3] we have r α = β. The image r φ i v (E ) = f v(e ) maps W and hence W to itsef, so β (Ē) aso maps W to itsef. Since r α (Ē) = β (Ē), we concude that α (Ē) N by Lemma One aso observes that the foowing morphism t : N N / S GL W maps α (Ē) to Ī v := w v (Ē). Therefore, tame inertia torus Ī v and thus Ī is a subgroup of t ( N ). Definition 11. Let L be the norma extension of K in Theorem Denote φ (Ga L ) by γ for a. Then [ Γ : γ ] [L : K] for a. Proof of Theorem 2.0.5(i),(ii). Proof. Since S is a connected norma subgroup of N, Ī is a torus, and t is an F -morphism, Proposition impies t 1 (Ī), the preimage of the F -torus Ī is a connected F -reductive group Ḡ. Moreover, γ Ḡ(F ) by construction of Ḡ for 1. We obtain an exact sequences of F agebraic groups for 1 1 S Ḡ Ī 1. and hence 1 S (F ) Ḡ(F ) Ī(F ). Reca µ (Ga L ) = t ( γ ) from Theorem Since the semisimpe enveopes (Definition 6) of Γ and γ are identica for 1 by Remark 2.1.3, the above exact sequence impies [Ḡ(F ) : γ ] [ S (F ) : γ S (F )][Ī(F ) : µ (Ga L )] 2 N 1 c 8 by Proposition 2.1.2(iii) and Theorem for 1. Since the derived group of Ḡ is S, the action of Ḡ on the ambient space is semisimpe if 1 by Proposition 2.1.2(ii). Therefore, we have proved Theorem (i) and (ii).

20 20 CHUN YIN HUI Proof of Theorem 2.0.5(iii). S sc Proof. Let S be the simpy connected cover of S. The representation ( S sc S GL N,F ) F is semisimpe and has a Z-form which beongs to a finite set of Z-representations of simpy-connected Chevaey schemes [EHK12, Theorem 24] if 1. Thus, { S GL N,F } 1 have bounded forma characters (Definition 4 ). Let C be the center of Ḡ. Since S acts semi-simpy on V by Proposition 2.1.2(ii) for 1, we decompose the representation S GL( V ) into a sum of absoutey irreducibe representations M i m i 1 m 1 m 2 m k V = ( M 1 ) ( M 2 ) ( M k ) 1 1 such that Mi M j if i j. If c C, then M i and c( M i ) are isomorphic representations of S for a i. Hence, c is invariant on M i and m i 1 M i is a subrepresentation of Ḡ on V for a i. Let n i be the dimension of Mi. Denote the representation of S on M i under some coordinates by u i : S GL ni ( F ). Then, the representation of Ḡ on m i 1 q i : Ḡ GL ni m i ( F ) 1 M i is given by: so that when restricting to S, the action is diagona q i : S u i GLni ( F ) m i x u i (x) (u i (x),..., u i (x)). 1 GL ni ( F ) GL ni m i ( F ) Since u i is a irreducibe representation and q i (c) commutes with q i ( S ), q i (c) is contained in the subgroup D 11 D12... D1mi D H i = 21 D22... D2mi......, D mi 1 D mi 2... Dmi m i where D jk is the subgroup of scaars of GL ni ( F ) for a 1 j m i, 1 k m i. We see that H i is isomorphic to GL mi ( F ). Since q i ( C ) is a diagonaizabe group which commutes with q i ( S ) and q i S is diagona, we may assume q i ( C ) is contained in the foowing torus

21 -INDEPENDENCE FOR COMPATIBLE SYSTEMS OF (MOD ) REPRESENTATIONS 21 D i for a i D D D i = Dmi m i after a change of coordinates by some eement in H i = GLmi ( F ). Therefore, we may assume that C is a subgroup of B := D 1 D 2 D k GL N ( F ). in suitabe coordinates. Torus B centraizes S impies B N. Denote the restriction t B by s. Since N acts on W, we have s : B GL W. We obtain (s 1 (Ī)) = C because Ker(s ) is discrete. Consider the construction of U from Theorem This impies the exponents of F characters of s on D i mi = 1 are between 0 and c 3 for a i. By Theorem 2.4.2(i) and above, the diagonaizabe groups {s 1 (Ī)} 1 satisfies the bounded exponents condition in Definition 4. Hence, { C = (s 1 (Ī)) B GL V } 1 have bounded forma characters. Since { C GL N,F } 1 and { S GL N,F } 1 both have bounded forma characters and C commutes with S for 1, {Ḡ = C S GL N,F } 1 have bounded forma characters by Proposition This prove Theorem 2.0.5(iii). 3. -independence of Γ 3.1. Forma character of Ḡ GL N,F. A system of agebraic enveopes {Ḡ} 1 of { Γ } 1 (Definition 5) are constructed in 2.5. Let G be the agebraic monodromy group of Φ ss for a. The compatibiity (Definition 2) of the system {φ } impies that the forma characters of {Ḡ GL N,F } 1 {G GL N,Q } 1 are the same in the sense of Definition 3 : Theorem Let {Ḡ} 1 be a system of agebraic enveopes of { Γ } 1 (Definition 5). (i) The forma characters of Ḡ GL N,F and G GL N,Q are the same for 1. (ii) The forma characters of {Ḡ GL N,F } 1 are the same. Proof. The mod system {φ : Ga K GL N (F )} comes from the -adic system {Φ ss : Ga K GL N (Q )} (Definition 1). The agebraic monodromy group G is reductive for a. By taking a finite extension K conn of K [Se81], we may assume G is connected for a. This does

22 22 CHUN YIN HUI not change the forma character of G GL N,Q. It is we known that these agebraic monodromy groups have same reductive rank r. Define Char : GL N G N 1 a G m that maps a matrix to the coefficients of its characteristic poynomia. We know that Char(G ) is a Q-variety of dimension r that is independent of (by the compatibiity conditions) and can be defined over Z[ 1 ] for some positive integer N that is sufficienty divisibe. Let N P Z[ 1 N ] be the Zariski cosure of Char(G ) in the projective P N. Since Z[ 1 N ] φ is continuous, every eement of Γ is the image of a Frobenius eement. Therefore, Char( Γ ) is a subset of the F -rationa points of P F := P Z[ 1 N ] Z F for 1. Generic fatness [DG65, Theorem 6.9.1] impies P Z[ 1 N ] is fat over Z[ 1 ] for sufficienty divisibe N, so the dimension of every irreducibe N component of P Z[ 1 N ] is r + 1 [Ha77, Chapter 3 Proposition 9.5] and hence the dimension of every irreducibe component of P F is r [Ha77, Chapter 3 Coroary 9.6] for 1. Aso, the Hibert poynomia of P F and in particuar the degree (et it be d) of P F P N F is independent of for 1 [Ha77, Chapter 3 Theorem 9.9]. Since d is a positive integer, we concude that the number and degrees of irreducibe components of P F are bounded by d [Ha77, Chapter 1 Proposition 7.6(a),(b)]. By [LW54, Theorem 1] and above, we have P F (F ) 3d r for 1. Let T be a F -maxima torus of Ḡ. [No87, Lemma 3.5] impies T has at east ( 1) dim( T ) F -rationa points. By Theorem (i), there is an integer n > 0 such that the nth power of T (F ) is contained in γ for 1. One sees by diagonaizing T in GL N, F that the order of the kerne of this nth power homomorphism is ess than or equa to n N. Hence, we obtain T (F ) γ ( 1)dim( T ) n N. Aso, morphism Char restricted to any maxima torus of GL N is finite morphism of degree N!. Therefore, there is a constant c > 0 such that c dim( T ) Char( T (F ) γ ) Char( γ ) P F (F ) 3d r for 1. This impies dim( T ) r for 1. On the other hand, we find for each 1 a set R of characters of G N m of exponents bounded by C > 0 such that T is conjugate in GL N, F to the kerne of R by Theorem 2.0.5(iii) and Definition 4. Let L be an infinite subset of prime numbers P such that for a, L,

23 -INDEPENDENCE FOR COMPATIBLE SYSTEMS OF (MOD ) REPRESENTATIONS 23 we have equaity R = R. Denote this common set of characters by R and define Y C = {y G N m,c : χ(y) = 1 χ R} so that dim CY C = dim F T for a L. If v divides v Σ K \S (S in Definition 2), then the characteristic poynomia of φ (Frob v ) is just the mod reduction of the characteristic poynomia of Φ ss (Frob v) = P v (x) Q[x] which depends ony on v (Definition 2). Therefore, for each v / S (Definition 2), we can put the roots of P v (x) in some order α 1, α 2,..., α N such that the foowing congruence equation hods: for any character x m 1 1 x m 2 2 x m N N α m 1 1 α m 2 2 α m N N 1 (mod ) R and L v := L \{ P : v S s.t. v } if v. Since α m 1 1 α m 2 2 α m N N is an agebraic number and L v consists of infinitey many primes, we obtain equaity α m 1 1 α m 2 2 α m N N = 1 for any character x m 1 1 x m 2 2 x m N N R. Therefore, (Char G N m ) 1 ({P v (x) : v Σ K \S}) g Perm(N) g(y C ), where Perm(N) is the group of permutations of N etters permuting the coordinates. Since {P v (x) : v Σ K \S} is Zariski dense in Char(G ) of dimension r and Char G N m is a finite morphism of degree N!, the Zariski cosure of (Char G N m ) 1 ({P v (x) : v Σ K \S}) in G N m,c denoted by D C is aso of dimension r. Since we have obtained dim( T ) r at the end of the second paragraph and any maxima torus of the agebraic monodromy group G is conjugate in GL N,C to an irreducibe component of D C [Se81], the incusion D C g(y C ) g Perm(N) impies the forma characters of Ḡ GL N,F and G GL N,Q are the same in the sense of Definition 3 for a L. There are ony finitey many possibiities for R by Remark and Proposition By excuding the primes such that R appears finitey many times, we concude that the forma characters of Ḡ GL N,F and G GL N,Q are the same for 1. This proves (i) and hence (ii) since forma character of G GL N,Q is independent of [Se81].

24 24 CHUN YIN HUI 3.2. Forma character of S GL N,F. We make the foowing assumptions for this subsection. Assumptions: By taking a fied extension of K, we may assume (i) G, the agebraic monodromy group of Φ ss is connected for a (see [Se81]), (ii) Ω := µ ( Γ ) corresponds to an abeian extension of K that is unramified at a primes not dividing for a (see the first paragraph of the proof of Theorem 2.4.2). Theorem beow is the main resut in this subsection. Denote a finite extension of K by K. Since S is independent of K over K for 1 by Remark 2.1.3, the assumptions above remain vaid for K, and {Ḡ} 1 constructed in 2.5 are sti agebraic enveopes of {φ (Ga K )} 1, we are free to repace K by K in this subsection. Theorem Let S GL N,F be the semisimpe enveope of Γ (Definition 6) for a 1. (i) The forma character of S GL N,F is equa to the forma character of G der GL N,Q for 1, where G der is the derived group of the agebraic monodromy group G of Φ ss. (ii) The forma character of S GL N,F is independent of if 1. In [Hu13, 3], we used mainy abeian -adic representations to prove that the forma character of G der GL N,Q is independent of. To prove Theorem 3.2.1, we adopt this strategy in a mod fashion. The key point is to prove that the inertia characters of µ (Definition 9) for 1 are in some sense the mod reduction of inertia characters of some Serre group S m [Se98, Chapter 2] (Proposition 3.2.4). Definition 12. For each prime P, choose a vauation v of Q that extends the -adic vauation of Q. This vauation on Q is equa to the restriction of the unique non-archimedean vauation on Q (extending the -adic vauation on Q ) to Q with respect to some embedding Q Q. Denote aso this vauation on Q by v. Define the foowing notation. Ga ab K : the Gaois group of the maxima abeian extension of K, I K : the group of idées of K, (x v ) v ΣK : a representation of a finite idée, K v : the competion of K with respect to v Σ K, U v : the unit group of Kv, k v : the residue fied of K v, m 0 : the moduus of empty support,

25 -INDEPENDENCE FOR COMPATIBLE SYSTEMS OF (MOD ) REPRESENTATIONS 25 U m0 := v U v, K := v K v = K Q, Z : the vauation ring of v, p : the maxima idea of v, k : the residue fied of v, x := (x v ) v. Let σ : K Q be an embedding of K in Q. The composition of σ with Q Q extends to a Q -agebra homomorphism σ : K Q. Remark The fied k is an agebraic cosure of F and homomorphism σ is trivia on the components K v of K when v is not equivaent to v σ. Reca representation µ : Ga K GL(W ) (abeian by Assumption (ii)) from Definition 9. Thus, µ induces ρ beow for each by composing with I K Ga ab K : ρ : I K GL(W ). Proposition If χ : I K F is a character of ρ for 1, then for a finite idée x U m0 we have the congruence χ (x) σ (x 1 ) m(σ,) (mod p ) σ Hom(K, Q) such that 0 m(σ, ) c 6. Proof. Since Ω is prime to, the foowing homomorphism U v K v I K ρ GL(W ) factors through α v : k v GL(W ) for a v. On the other hand, et v Σ K divide. Since Ω is abeian and of order prime to, the restriction of µ : Ga K GL(W ) to I v factors through I v I t v = im F k k v and induces β v : k v GL(W ) that depends on v = v K. By [Se72, Proposition 3], α v and β v are inverse of each other. Since f v (Definition 10) factors through β v and the exponents of any character of f v when expressed as a -restricted (Definition 8) product of fundamenta characters of eve c 4! are bounded by c 6 for 1 ( 2.4), the exponents of χ when expressed as a -restricted product of fundamenta characters of eve [k v : F ] are aso bounded by c 6 for 1. Since ρ is unramified at a v not dividing, ρ is trivia on subgroup v U v of U m0 := v U v. Therefore, we concude the congruence for 1.