On the rationality of algebraic monodromy groups of compatible systems

Let $E$ be a number field and $X$ a smooth geometrically connected variety defined over a characteristic $p$ finite field. Given an $n$-dimensional pure $E$-compatible system of semisimple $\lambda$-adic representations of the \'etale fundamental group of $X$ with connected algebraic monodromy groups $G_\lambda$, we construct a common $E$-form $G$ of all the groups $G_\lambda$ and in the absolutely irreducible case, a common $E$-form $G\hookrightarrow\text{GL}_{n,E}$ of all the tautological representations $G_\lambda\hookrightarrow\text{GL}_{n,E_\lambda}$ (Theorem 1.1). Analogous rationality results in characteristic $p$ assuming the existence of crystalline companions in $\text{F-Isoc}^{\dagger}(X)\otimes E_{v}$ for all $v|p$ (Theorem 1.5) and in characteristic zero assuming ordinariness (Theorem 1.6) are also obtained. Applications include a construction of $G$-compatible system from some $\text{GL}_n$-compatible system and some results predicted by the Mumford-Tate conjecture.

1. Introduction 1.1.The Mumford-Tate conjecture.Let A be an abelian variety defined over a number field K ⊂ C, V ℓ := H 1 (A K , Q ℓ ) the étale cohomology groups for all primes ℓ, and V ∞ = H 1 (A(C), Q) the singular cohomology group.The famous Mumford-Tate conjecture [Mu66, §4] asserts that the ℓ-adic Galois representations ρ ℓ : Gal(K/K) → GL(V ℓ ) are independent of ℓ, in the sense that if G ℓ denotes the algebraic monodromy group of ρ ℓ (the Zariski closure of the image of ρ ℓ in GL V ℓ ) and G MT denotes the Mumford-Tate group of the pure Hodge structure of V ∞ , then via the comparison isomorphisms In particular, the representations ρ ℓ are semisimple and the identity components G • ℓ are reductive with the same absolute root datum.This conjectural ℓ-independence of different algebraic monodromy representations can be formulated almost identically for projective smooth varieties Y defined over K, and more generally, for pure motives over K by the universal cohomology theory envisaged by Grothendieck and some deep conjectures in algebraic and arithmetic geometry (see [Se94,§3]).
The same Mumford-Tate type question can also be asked for projective smooth varieties Y defined over a global field K of characteristic p > 0. Since V ℓ = H w (Y K , Q ℓ ) are Weil cohomology theories for Y K only when ℓ = p 1 , one may ask if the algebraic monodromy representations G • ℓ ֒→ GL V ℓ of the Galois representations V ℓ are independent of ℓ for all ℓ = p.This expectation is supported by the philosophy of motives (see [Dr18,§E]).On the other hand, one can always exploit the fact that the system of ℓ-adic Galois representations {V ℓ = H w (Y K , Q ℓ )} ℓ is a Q-compatible system (in the sense of Serre [Se98, Chap.I-11 Definition]) that is pure of weight w (proven by Deligne [De74,De80]) to directly argue ℓ-independence of the algebraic monodromy representations G • ℓ ֒→ GL V ℓ .This approach holds, regardless of the characteristic of the global field K.By utilizing the compatibility and weight conditions of the compatible system, Serre developed the method of Frobenius tori [Se81] to prove the ℓ-independence result below (Theorem A).
Let us define some notation first.If L is a subfield of Q, then denote by P L the set of places of L. Denote by P L,f (resp.P L,∞ ) the set of finite (resp.infinite) places of L. Then P L = P L,f ∪ P L,∞ .Denote by P (p) L,f the set of elements of P L,f not extending p.The residue characteristic of the finite place v ∈ P L,f is denoted by p v .Let V and W be free modules of finite rank over a ring R. Let G m ⊂ • • • ⊂ G 1 ⊂ GL V and H m ⊂ • • • ⊂ H 1 ⊂ GL W be two chain of closed algebraic subgroups over R. We say that the two chain representations (or simply representations if it is clear that they are chains of subgroups of some GL n ) are isomorphic if there is an R-modules isomorphism V ∼ = W such that the induced isomorphism GL V ∼ = GL W maps G i isomorphically onto H i for 1 ≤ i ≤ m.
Theorem A. (Serre) [Se81] (see also [LP97]) (i) (The component groups) The quotient groups G ℓ /G • ℓ for all ℓ are isomorphic.(ii) (Common Q-form of formal characters) For all v in a positive Dirichlet density subset of P K,f , there exist a subtorus T := T v of GL n,Q such that for all ℓ = p v , the representation (T ֒→ GL n,Q ) × Q Q ℓ is isomorphic to the representation T ℓ ֒→ GL V ℓ for some maximal torus T ℓ of G ℓ .
It follows immediately that the connectedness and the absolute rank of G ℓ are both independent of ℓ.Later, Larsen-Pink obtained some ℓ-independence results for abstract semisimple compatible system on a Dirichlet density one set of primes ℓ [LP92] and for the geometric monodromy of {V ℓ } ℓ if Char(K) > 0 [LP95].
When Char(K) = 0, the author proved that the formal bi-character (Definition 2.2(ii)) of G • ℓ ֒→ GL V ℓ is independent of ℓ and obtained ℓ-independence of G • ℓ under some type A hypothesis [Hu13,Hu18].The next result is by far the best result in positive characteristic, in a setting more general than the above étale cohomology case.
Let X be a smooth geometrically connected variety defined over a finite field F q of characteristic p.Let E be a number field.For any λ ∈ P E , denote by E λ the λ-adic completion of E. Let (2) 1 When ℓ = p, one has to consider crystalline cohomology group of Y .
be an E-compatible system of n-dimensional semisimple λ-adic representations of the étale fundamental group π ét 1 (X, x) of X (with base point x) that is pure of integral weight w.Denote by G λ ⊂ GL V λ the algebraic monodromy group of the representation V λ .For simplicity, set π 1 (X) = π ét 1 (X, x) and for all λ ∈ P (p) E,f , choose coordinates for V λ so that G λ is identified as a subgroup of GL n,E λ .The following theorem was obtained by Chin when X is a curve [Ch04] 2 and is true in general by reducing to the curve case by finding a suitable curve S in some covering X ′ of X [BGP19, §3.3], see also [D'Ad20, §4.3].
Theorem B. Let ρ • be an E-compatible system of n-dimensional λ-adic semisimple representations of π 1 (X) that is pure of integer weight w.The following assertions hold in some coordinates of V λ .
(i) (Common E-form of formal characters): There exists a subtorus T of GL n,E such that for all λ ∈ P (p) E,f , T λ := T × E E λ is a maximal torus of G λ .(ii) (λ-independence over an extension): There exist a finite extension F of E and a chain of subgroups T sp ⊂ G sp ⊂ GL n,F such that G sp is connected split reductive, T sp is a split maximal torus of G sp , and for all λ ∈ P (p) E,f , if F λ is a completion of F extending λ on E, then there exists an isomorphism of chain representations: The isomorphisms f F λ in (ii) can be chosen such that the restriction isomorphisms f F λ : E,f and F λ .1.2.The results of the paper.

Theorem B(ii) asserts that the algebraic monodromy representations G •
λ ֒→ GL n,E λ have a common (split) F -model after finite extensions F λ of E λ .The main theme of this article is to remove these extensions.Base on Theorem B(i)-(iii) and some ideas seeded in [Hu18], we prove the following E-rationality result (Theorem 1.1).In case the representations V λ are absolutely irreducible 3 , it answers the Mumford-Tate type question in positive characteristic.
(i) There exists a connected reductive group G defined over E such that G × E E λ ∼ = G • λ for all λ ∈ P (p) E,f .(ii) If moreover G • λ ֒→ GL V λ is absolutely irreducible for some λ, then there exists a connected reductive subgroup G of GL n,E such that for all λ ∈ P (p) E,f , the representations are isomorphic: λ ֒→ GL V λ ).1.2.1.2.Let O λ be the ring of integers of E λ , O E be the ring of integers of E, O E,S be the localization for some finite subset S ⊂ P E,f , and A (p) E be the adele ring of E without factors above p.We construct an adelic representation ρ G A in Corollary 1.2 and in the absolutely irreducible case, a common model G ⊂ GL n,OE,S of the group schemes G λ ֒→ GL n,O λ (with respect to some O λ -lattice in V λ ) for all but finitely many λ in Corollary 1.3.
Corollary 1.2.Let ρ • be a λ-adic compatible system of π 1 (X) as above.Suppose G λ is connected for all λ.Then the following assertions hold.
(i) There exist a connected reductive group G defined over E and an isomorphism G × E E λ φ λ → G λ for each λ ∈ P (p) E,f such that the direct product representation 2 [Ch04] used pivotally Serre's Frobenius tori and Lafforgue's work [La02] on the Langlands' conjectures.In case X is a curve, Theorem B(i),(ii),(iii) follow, respectively, from Lemma 6.4, Thm.1.4, Thm.6.8 and Cor.6.9 of the paper. 3In general, we expect a common E-form of the faithful representations G λ ֒→ GL V λ for all λ ∈ P (p) E,f exists.
factors through a G-valued adelic representation via φ λ : (ii) If the representations V λ are absolutely irreducible, then there exist a connected reductive subgroup G of GL n,E and an isomorphism of representations factors through a G-valued adelic representation via φ λ : Corollary 1.3.Let ρ • be a λ-adic compatible system of π 1 (X) as above.Suppose V λ is absolutely irreducible and G λ is connected for all λ.Then there exists a smooth reductive group scheme G ⊂ GL n,OE,S defined over O E,S (for some finite S) whose generic fiber is G ⊂ GL n,E such that for all λ ∈ P (p) Next corollary is about the G-valued compatibility of the system, motivated by the papers [BHKT19], [Dr18] on Langlands conjectures.As obtained in [BHKT19, §6], the results in [D'Ad20, §4] ([Ch04, §6] when X is a curve) imply that the E-compatible system ρ • (assume connectedness of G λ ), after some finite extension F/E, factors through an F -compatible system ρ G sp • of G sp -representations for some connected split reductive group G sp defined over F .In some situation, we prove that the extension F/E can be omitted.This shows evidence to the motivic hope in [Dr18, §E] that the Tannakian categories T λ (X) of semisimple (weight 0) E λ -representations of π 1 (X), at least for all λ not extending p, should come from a canonical category T (X) over E in a compatible way (see [Dr18,Thm. 1.4.1]).The definition of an E-compatible system of G-representations will be recalled in §3.2.Let π λ : E → E λ be the natural surjection to the λ-component.
Corollary 1.4.Let ρ • be a λ-adic compatible system of π 1 (X) as above.Suppose V λ is absolutely irreducible and G λ is connected for all λ.Let G ֒→ GL n,E be the E-form and ρ G λ be the adelic representation in Corollary is an E-compatible system of G-representations when one of the following holds.
(i) The group G λ is split for all λ.
(ii) The outer automorphism group of the derived group G der × E E is trivial (β λ = id).
Hence, for any E-representation α : G → GL m,E , the system of m-dimensional λ-adic semisimple represen- is also E-compatible.[Ke22a,§2] for definition).Any t ∈ X(F q k ) induces a fiber functor to the category of vector spaces over E v,q k given by the composition where the first one is via the pull-back of i : t → X and the second one is the forgetful functor.The image V t,v := w t (M v ) is an n-dimensional vector space.The Tannakian group of the subcategory generated by M v with respect to w t can be identified as a reductive subgroup G t,v ⊂ GL Vt,v ∼ = GL n,E v,q k and is called the algebraic monodromy group of (M v , w t ).For different closed points t and t ′ in X(F q k ), G t,v and G t ′ ,v differ by an inner twist [DS82, Theorem 3.2].Let λ be a finite place of E not extending p.The absolute root data of G • t,v and G • λ (resp.the component groups of G t,v and G λ ) are proven to be isomorphic independently by Pal [Pa15] and D'Addezio [D'Ad20] (relying on [La02] and [Ab18]).Moreover, given the closed point t one can define the Frobenius tori T t,v in G t,v (see [D'Ad20, §4.2]) and Tt ,λ in G λ (up to conjugation, see §3.3).Assume the crystalline companions of ρ • exist for all v ∈ P E,p and certain conditions, we prove an E-rationality result (existence + uniqueness) for the above algebraic monodromy groups at all finite places of E.
be an E-compatible system of n-dimensional λ-adic semisimple representations of π 1 (X) that is pure of integer weight w and t ∈ X(F q k ) a closed point of X. Suppose the semisimple crystalline companion object for each v ∈ P E,p and the following conditions hold.
(a) The Frobenius torus Tt ,λ is a maximal torus of G λ for some λ.
(b) For all v ∈ P E,p , the field (c) The number field E has at least one real place5 .Then the following assertions hold.(i) There exists a chain (of a connected reductive group together with a maximal torus) T ⊂ G defined over E that is the unique common E-form of the chains Tt ,λ ⊂ G • λ for all λ ∈ P (p) E,f and the chains is absolutely irreducible for some λ, then there exist an inner form GL m,D (for some division algebra D over E) of GL n,E over E containing a chain of subgroups T ⊂ G such that T ⊂ G ֒→ GL m,D is the unique common E-form of the chain representations Tt ,λ ⊂ G λ ֒→ GL V λ for all λ ∈ P (p) E,f and the chain representations T t,v ⊂ G t,v ֒→ GL Vt,v for all v ∈ P E,p .When E has exactly one real place, we have GL m,D ∼ = GL n,E .

Characteristic zero.
1.2.2.1.It turns out that the strategy for proving Theorem 1.1 retains in characteristic zero if ordinary representations enter the picture.This part is influenced by the work of Pink [Pi98].To keep things simple, we only consider the Q-compatible system (with exceptional set S) of n-dimensional ℓ-adic Galois representations arising from a smooth projective variety Y defined over a number field K.The set S consists of the finite places of K such that Y does not have good reduction.Let G ℓ be the algebraic monodromy group at ℓ.The Grothendieck-Serre semisimplicity conjecture asserts that the representation ρ ℓ is semisimple (see [Tat65]), which is equivalent to the algebraic group G • ℓ being reductive.Choose coordinates for V ℓ and identify G ℓ as a subgroup of GL n,Q ℓ for all ℓ.Embed Q ℓ into C for all ℓ.
Let v ∈ P K,f \S with p := p v .Let K v be the completion of K at v, O v the ring of integers, and Y v the special fiber of a smooth model of Y over O v .The local representation is crystalline and corresponds, via a mysterious functor of Fontaine [Fo79,Fo82,Fo83], to the crystalline cohomology [Fa89].The local representation V p is said to be ordinary if the Newton and Hodge polygons of M v coincide [Ma72].This notion originates from ordinary abelian varieties defined over finite fields.It is conjectured by Serre that if K is large enough, then the set of places v in P K,f for which the local representations V p are ordinary is of Dirichlet density one, for abelian varieties of low dimensions, see Serre [Se98], Ogus [O82], Noot [No95,No00], Tankeev [Ta99]; for abelian varieties in general, see Pink [Pi98]; and for K3 surfaces, see Bogomolov-Zarhin [BZ09].
Theorem 1.6.Let ρ • be the Q-compatible system (3) arising from the ℓ-adic cohomology (of degree w) of a smooth projective variety Y defined over a number field K. Suppose G ℓ is connected for all ℓ and the following conditions hold.(a) (Ordinariness): The set of places v in P K,f for which the local representations (i) There exists a connected reductive group G defined over Remark 1.7.The conditions 1.6(a),(b),(c) are to be compared with Theorem B(i),(ii),(iii).Since Theorem A(ii) only gives a common Q-form of formal characters for all but one ℓ, the condition (a) is needed if one aims at a Q-common form for all ℓ.Given 1.6(a) and B(i), then 1.6(b) and B(ii) are easily seen to be equivalent (E = Q).The rigidity assertion B(iii) is not known to hold in characteristic zero, and is now replaced with the invariance of roots condition 1.6(c), which holds if G der C is of certain root system [Hu20, Thm.A1, A2].
Remark 1.8.If ρ ℓ is abelian at one ℓ, then the rationality of G ℓ ֒→ GL n,Q ℓ for all ℓ is obtained by Serre via Serre group S m [Se98].
This hypothesis follows from a Galois maximality conjecture of Larsen [Lar95] (see Theorem 3.9), which has been established for type A representations [HL16], abelian varieties and hyper-Kahler varieties (degree w = 2) [HL20].Further assuming the hypothesis, we obtain the following corollaries which are analogous to Corollaries 1.2 and 1.3.Corollary 1.9.Let ρ • be an ℓ-adic compatible system of Gal(K/K) as above.Suppose G ℓ is connected for all ℓ and Hypothesis H holds. Then the following assertions hold.
(i) There exist a connected reductive group G defined over Q and an isomorphism factors through a G-valued adelic representation via φ ℓ : (ii) If the representations V ℓ are absolutely irreducible, then there exist a connected reductive subgroup factors through a G-valued adelic representation via φ ℓ : Corollary 1.10.Let ρ • be an ℓ-adic compatible system of Gal(K/K) as above.Suppose V ℓ is absolutely irreducible, G ℓ is connected for all ℓ, and Hypothesis H holds. Then there exists a smooth reductive group scheme G ⊂ GL n,ZS defined over Z S (for some finite S ⊂ P Q,f ) whose generic fiber is G ⊂ GL n,Q such that for all ℓ ∈ P Q,f \S, the representations (G ֒→ GL n,ZS ) × Z ℓ and G ℓ ֒→ GL n,Z ℓ are isomorphic, where G ℓ is the Zariski closure of ρ ℓ (Gal(K/K)) in GL n,Z ℓ after some choice of Z ℓ -lattice in V ℓ .
1.2.2.3.Suppose Y = A is an abelian variety defined over K of dimension g and w = 1.We say that A has ordinary reduction at v if the local representation V p of Gal(K v /K v ) is ordinary.The following results are due to Pink.
Theorem C. [Pi98, Thm.5.13(a),(c),(d), Thm.7.1] Let A be an abelian variety defined over a number field K with End(A K ) = Z and suppose G ℓ is connected for all ℓ.There exists a connected reductive subgroup G of GL 2g,Q such that the following assertions hold. (i (iii) If the root system of G is determined uniquely by its formal character, i.e., if G does not have an ambiguous factor (in Theorem E), then we can take L in (i) to contain all but finitely many primes.(iv) If G × Q Q does not have any type C r simple factors with r ≥ 3, then the abelian variety A has ordinary reduction at a Dirichlet density one set of places v of K.
By the Tate conjecture of abelian varieties proven by Faltings [Fa83] and End(A K ) = Z, the representations Consider the representation Gal(K/K) → GL(H 1 (A K , Z ℓ )) and let G ℓ be the Zariski closure of the image in GL H 1 (A K ,Z ℓ ) .Combining the previous results, we obtain Theorem 1.11 below which extends Theorem C(iii) to all ℓ assuming ordinariness.
Theorem 1.11.Let A be an abelian variety defined over a number field K with End(A K ) = Z and suppose G ℓ is connected for all ℓ and the following conditions hold.(a) The set of places v in P K,f for which the local representations V p of Gal(K v /K v ) are ordinary is of positive Dirichlet density.(b) The root system of G ℓ is determined uniquely by its formal character.Then there exists a smooth group subscheme G ⊂ GL 2g,ZS over Z S (for some finite S ⊂ P Q,f ) with generic fiber G ⊂ GL 2g,Q and an isomorphism of representations factors through a G-valued adelic representation via φ ℓ : Remark 1.12.By Theorem C(iv), Theorem E, and the fact that for every ℓ, every simple factor of G ℓ × Q ℓ Q ℓ is of type A, B, C, or D [Pi98, Cor.5.11], the conditions 1.11(a),(b) hold if for some prime ℓ ′ , every simple factor of 1.3.The structure of the paper.The paper is structured on the purely algebraic main theorems I and II in next section.Roughly speaking, it states that if a family of connected reductive algebraic subgroups G λ ֒→ GL n,E λ indexed by λ ∈ P (p) E,f (resp.P E,f ) satisfies some conditions, then there exist a common E-form of the family of the subgroups (resp.the representations).The results in §1.2 are established in two big steps.Firstly, we state and prove the main theorems in §2 which require different techniques from representation theory and Galois cohomology.The notation and diagrams we developed in §2 are very much influenced by the work [Hu18].A crucial step to the existence of a common E-form in the main theorem is based on the local-global aspects of Galois cohomology §2.5.Secondly, we prove Theorems 1.1, 1.5, and 1.6 in §3 by checking that the conditions of the main theorems are satisfied for the corresponding family of algebraic monodromy groups of the E-compatible systems and applying the main theorems.For the characteristic p case, to prove Theorem 1.1 (resp.Theorem 1.5) by main theorem I (resp.II), the required conditions are ensured by Theorem B (resp.recent work [D'Ad20], see Theorem B').The characteristic zero case is more involved.It requires the results of formal bi-character ( §2.2c'-bi) and invariance of roots to compensate for the lack of the rigidity condition B(iii).The information at the real place (Proposition 3.3) and a finite place (ordinary representation V p ) are also needed.The other results in §1.2 will also be established in §3.The statements that we quote are named using alphabets (e.g., Theorem A) and the statements that we prove are named using numbers (e.g., Theorem 1.1).

Main theorems
E,f such that the following conditions hold.
(a) (Common E-form of formal characters): There exists a subtorus T of GL n,E such that for all λ ∈ P (p) (b) (λ-independence absolutely): There exists a chain of subgroups T sp ⊂ G sp ⊂ GL n,E such that G sp is connected split reductive, T sp is a split maximal torus of G sp , and for all λ ∈ P (p) E,f , if E λ is a completion of E extending λ on E, then there exists an isomorphism of chain representations: (c) (Rigidity): The isomorphisms f E λ in (b) can be chosen such that the restriction isomorphisms f E λ : The groups G λ are quasi-split for all but finitely many λ ∈ P Then the following assertions hold.
(i) There exists a connected reductive group G defined over In particular, G λ is unramified for all but finitely many λ.
For any E-algebra B, define GL m,B to be the affine algebraic group over E such that for any Main theorem II.Suppose a connected reductive subgroup G λ ⊂ GL n,E λ is given for each λ ∈ P E,f such that the following conditions hold.(a) (Common E-form of formal characters): There exists a subtorus T of GL n,E such that for all λ ∈ P E,f , (b) (λ-independence absolutely): There exists a chain of subgroups T sp ⊂ G sp ⊂ GL n,E such that G sp is connected split reductive, T sp is a split maximal torus of G sp , and for all λ ∈ P E,f , if E λ is a completion of E extending λ on E, then there exists an isomorphism of chain representations: (c) (Rigidity): The isomorphisms f E λ in (b) can be chosen such that the restriction isomorphisms f E λ : The twisted E-torus µ (T sp /C) is anisotropic at some place of E and all real places of E, where C is the center of G sp and µ ∈ Z 1 (E, Aut E T sp ) the cocycle defined by f E in (c).
Then the following assertions hold.
(i) There exists a unique connected reductive group G defined over for all λ ∈ P E,f .In particular, G λ is unramified for all but finitely many λ.(ii) If moreover G sp ֒→ GL n,E is irreducible, then there exist an inner form GL m,D (for some division algebra . Such a chain of E-groups is unique.
Remark 2.1.There are similarities and differences between the two main theorems.
(1) The index set for main theorem I is E,f and for main theorem II is P E,f .(2) Conditions (a), (b), (c) of the two main theorems are identical except for the index sets.
(3) If we embed E λ into C for all λ, then condition (b) is equivalent to asking that the C-representation T is said to be anisotropic at a place λ of F if it is anisotropic over F λ .The twisted E-torus µ (T sp /C) in main theorem II(d) will be defined in §2.6.1.(6) The conclusion of main theorem II is stronger than that of main theorem I as the E-torus T in condition (a) can be found in the common E-form G in main theorem II.Moreover, if E has only one real place, then the inner form GL m,D in main theorem II is equal to GL n,E by class field theory.
2.2.The rigidity condition.The rigidity condition (c) is important for the construction of the E-form G in the main theorems.It does not come for free.In this section, we would like to prove that the rigidity condition follows from conditions (a),(b) and (c') below.(c') Both the following hold.(c'-bi)=(Common E-form of formal bi-characters): There exists a subtorus T ss of T such that The normalizer N GLn,E (T ssp ) is invariant on the roots of the derived group (G sp ) der of G sp with respect to the maximal torus T ssp := T sp ∩ (G sp ) der .

Formal character and bi
It is clear that (c'-bi) together with (a) in the main theorems mean that there exist a chain of subtori, denoted T ss ⊂ T ⊂ GL n,E , such that Proposition 2.4.If conditions (a) and (b) in the main theorems hold and G sp is irreducible on E n , then (c'-bi) holds.
Proof.Let T ⊂ GL n,E be in (a) and let T ss be the identity component of the kernel of the determinant map Since G λ is connected and the representation G λ ⊂ GL n,E λ is absolutely irreducible for all λ by the assumptions, G λ is either G der λ or G der λ • G m by Schur's lemma.Hence by counting dimension, T ss × E E λ is a maximal torus of G der λ for all λ.2.2.2.Invariance of roots.Let F be a field of characteristic zero and G a connected split semisimple subgroup of GL n,F .Fix a split maximal torus T of G and denote by X the character group of T. Let R ⊂ X be the set of roots of G with respect to T. Let N := N GL n,F (T) be the normalizer of T in GL n,F .Since N acts on T, it also acts on X.We would like to know when R is invariant under N.It is easy to see that this invariance of roots condition (i.e., N • R = R) is independent of the choice of the maximal torus T and is invariant under field extension.So, we take F = C for simplicity.If H is an almost simple factor of G, then by the Cartan-Killing classification the root system of H is one of the following: 1 , and Suppose G is irreducible on the ambient space C n .If G 1 is a connected normal subgroup of G, then there exists an unique complementary connected normal subgroup G 2 of G such that the natural map G 1 × G 2 → G is an isogeny of semisimple groups.Moreover, there exist unique irreducible representations V 1 and V 2 of respectively G 1 and G 2 such that the composition representation [FH91]).We say that the representation are two connected semisimple subgroups with the same formal character T ⊂ GL n,C and are both irreducible on the ambient space C n .Then the roots R and R ′ of respectively G and G ′ (with respect to T) are identical in X and the two representations are isomorphic unless one of the following conditions holds.(a) For r 1 , ..., r m , r ∈ N such that r 1 + • • • + r m = r, the spin representation of B r is a factor of (G, C n ) and the tensor product of the spin representations of Then one is a factor of (G, C n ) and the other one is a factor of (G ′ , C n ).
The following corollary follows directly by taking G ′ = gGg −1 , where g ∈ N.
Corollary 2.5.If G ⊂ GL n,C is a connected semisimple subgroup that is irreducible on the ambient space C n , then the invariance of roots condition holds if the following conditions are satisfied.Then the invariance of roots condition holds.
Proof.Let G 1 , ..., G k be the almost simple factors of G. Then T i = G i ∩ T is a maximal torus of G i for all i.Let X i be the character group of T i and R i the roots of G i with respect to T i .Let Φ ⊂ X (resp.Φ i ⊂ X i ) be the subgroup (root lattice) generated by R (resp.R i ).One can impose a metric on the real vector space X R := X ⊗ Z R such that (R, X R ) is a root system, the normalizer N is isometric on X R , and the decomposition The lemma below is needed.
Lemma 2.7.Suppose G ⊂ GL n,C is a connected semisimple subgroup that that satisfies the assumptions of Theorem 2.6.The following assertions are equivalent.
The set of non-zero elements of Φ i with the shortest length is equal to the set of short roots These facts and assumption (a) imply that R • i remains irreducible for all i.Then the orthogonality of the decomposition (4) and the fact that g is isometric on X R imply that g permutes the set {R Since g is isometric on X R , the Lie type assumptions (a)-(e) and the above facts about short roots imply that R i and By the orthogonality of the decomposition (4), the fact that g is isometric on X R , and induction, we conclude that g permutes R.
Back to the theorem, we have Φ = X because G is adjoint.Since X is invariant under N by definition, Φ is invariant under N. Therefore, R is invariant under N by the lemma.

Conditions for rigidity.
Proposition 2.8.If conditions (a), (b) in the main theorem(s) and (c') hold, then condition (c) in the main theorem(s) also holds.
Proof.By (a) and (c'-bi), we have a chain of subtori T ss ⊂ T ⊂ GL n,E such that for all λ, and a chain T sp ⊂ G sp (over E) such that for all λ, there exists an E λ -isomorphism of representations f E λ taking T sp ⊂ G sp to T λ ⊂ G λ (omitting the extension field for simplicity).This implies that f Eλ maps T ssp := T sp ∩ (G sp ) der to T ss λ = T λ ∩ G der λ for all λ.Hence, we conclude that for all λ, the two chains (5) To finish the proof, it suffices to find for all λ, a matrix B λ ∈ GL n (E λ ), such that the conjugation map by B λ takes M G λ M −1 to G sp and is identity on T sp = M TM −1 .Such B λ exists.Indeed, there exists because the chains in (5) are conjugate in GL n (E λ ).Then (7) and (8) imply that A λ ∈ N GL n (T ssp ) and conjugation by A λ takes the roots of M G der λ M −1 to the roots of (G sp ) der .By (c'-inv), the roots of the two semisimple (derived) groups are identical (in the character group of T ssp ).Hence, [Hu18, Thm.3.8] implies that the absolute root data of M G λ M −1 and G sp are identical with respect to the common maximal torus M T λ M −1 = T sp .By [Sp08, Thm.16.3.2],there exists an E λ -isomorphism b λ taking the pair (M G λ M −1 , M T λ M −1 ) to the pair (G sp , T sp ) inducing the identity map between their root data.Let i 1 and i 2 be the tautological representation of M G λ M −1 and G sp into GL n .Then the two representations 2.3.Forms of reductive chains.This section is foundational to the proofs of the main theorems and is developed from [Hu18, §4].

Galois cohomology.
Let F be a field of characteristic zero, G 1 and G ′ 1 be linear algebraic groups defined over F .The Galois group Gal(F /F ) acts (on the left) on the set of F -homomorphisms φ : and the restriction φ| Gi×F F is an isomorphism for all 1 ≤ i ≤ k.Since the groups are defined over F , the F -homomorphism σ φ is also a F -isomorphism between the two chains.In particular, the automorphism group Aut F (G 1 , G 2 , ..., G k ) of the chain (i.e, the subgroup of the automorphism group Aut for all σ ∈ Gal(F /F ) satisfies the 1-cocycle condition: producing a bijective correspondence (see [Se97, Ch. 3.1, Prop. 5 and its proof]) between the set of isomorphism classes of F -forms of the chain and the outer automorphism group of the chain by Then we obtain a short exact sequence of Gal(F /F )-groups and an exact sequence of pointed set [Se97, Ch. 1.5.5, Prop.38] The exactness means that the preimage π −1 ([id]) is equal to the image Im(i). An is called an inner F -form (or inner form) if there exists an F -isomorphism φ such that in (9), the element a σ belongs to Inn F (G 1 , G 2 , ..., G k ) for all σ.In general, the isomorphism classes of inner F -forms do not form a subset of the isomorphism classes of F -forms since the map i in (11) is not injective.However, the sequence (11) is a short exact sequence of pointed sets (and thus i is injective) if (10) splits.We will see in later sections that the splitting of (10) holds for some chains (e.g., T sp ⊂ G sp ).The following simple lemma is useful to study the conjugacy class of a subgroup in GL n,F .Lemma 2.9.Let D be a central division algebra over Proof.Identify U × F F with GL n,F .The condition implies that there exists an F -inner automorphism which proves (i).If the cocycle is neutral, then there exists γ ∈ Inn F (GL n,F , G, T) ⊂ PGL n (F ) such that a σ = γ −1 • σ γ for all σ ∈ Gal(F /F ).This is equivalent to Hence, ψ • γ −1 ∈ PGL n (F ) and GL n,F and GL m,D are F -isomorphic.Therefore, D = F , U = GL n,F , and ψ • γ −1 is an F -inner automorphism of GL n,F taking G to G ′ as well as T to T ′ , which prove (ii).
2.3.2.Some diagrams.In this section, some diagrams of groups and Galois cohomology will be presented.
Let F be a field.Denote by • G sp a connected split reductive group defined over F , the cohomology if M is a linear algebraic group defined over F .
2.3.2.1.Consider the following diagram of Gal(F /F )-groups: where the top (resp.bottom) row is (10) for T sp ⊂ G sp by [Hu18, Prop.4.3] (resp.G sp ) and the vertical arrows are all natural inclusions induced by restricting automorphisms to G sp : Since G sp is split, the Galois group Gal(F /F ) acts trivially on the outer automorphism group Θ F .The proposition below is well-known.
Proposition F. (see e.g.[Hu18, Prop.4.1]) The automorphism group Aut F G sp contains a Gal(F /F )invariant subgroup that preserves T sp and B and is mapped isomorphically onto Out F G sp .Hence, the top (resp.bottom) row in (12) is a split short exact sequence of Gal(F /F )-groups:

Denote by
• Ω F := Im(Res T sp ), where Res T sp restricts automorphisms to T sp : Then the first row in (12) also fits into the following diagram of Gal(F /F )-groups with exact rows and columns by [Hu18,Prop. 4.3] and j denotes a splitting induced by ( 14). ( . Suppose given a faithful (absolutely) irreducible representation G sp ֒→ GL n,F .Then we have the chain T sp ⊂ G sp ⊂ GL n,F .The irreducibility condition implies that C is contained in the subgroup of scalars in GL n,F and the following inclusions hold . (17) In diagram (16), denote by By diagrams (12), ( 16), ( 17) and the fact that the squares in (17) are Cartesian, we obtain the following two diagrams with exact rows and columns.Moreover, (18) injects naturally into (12), ( 19) injects naturally into (16), and j denotes the splitting induced by ( 14). ( . By taking Galois cohomology on diagrams (12),( 16),( 18),( 19), the splitting j, and Hilbert's Theorem 90: , we obtain the following diagrams of pointed sets such that the rows and columns are all exact.Moreover, there are natural maps from ( 22) to ( 20), ( 23) to (21), and j denotes again the splitting. ( Let G be a profinite group and A be a G-group (a discrete group on which G acts continuously).The Galois cohomology H 1 (G, A) is a pointed set with neutral element given by the trivial class → 1 be a short exact sequence of G-groups.Then one obtains an exact sequence of pointed sets be a cohomology class.To study the image of π as well as the fiber of π([β]), that is, the set π −1 (π([β])), one uses the method of twisting in [Se97, Ch. 1.5.3-1.5.7].This technique will be applied to some short exact sequences in §2.4.2.
2.4.1.Definition.Let G be a group, M a (left) G-group, and A (resp.B) be a M -group on which G acts compatibly on the left, i.e., g(m(a)) = g(m)(g(a)) for g ∈ G, m ∈ M , and a ∈ A. Suppose µ := (m g ) ∈ Z 1 (G, M ) is a 1-cocycle.Then one can define a G-group µ A twisted by µ, which can be viewed as A with a new G-action: as a group µ A = A and the G-action is defined by ).
As M acts on itself by inner automorphism (conjugation): (−) → m(−)m −1 , denote by µ M the twisted G-group.Then µ A is a µ M -group under the identification (25) on which G acts compatibly on the left.If µ, µ ′ ∈ Z 1 (G, M ) are cohomologous, then µ A and µ ′A are isomorphic.The association ).This induces the following commutative diagrams such that the vertical arrows are bijective correspondence taking neutral cocycles (resp.classes) to α, β (resp.[α], [β]). (26) )) is a bijective correspondence between the fibers of classes.
2.4.2.Fibers of π.Given a split short exact sequence of G-groups: Then we obtain a split short exact sequence of pointed sets: Since C acts on itself by inner automorphism, it also acts on B and A by the splitting j.Let χ ∈ Z 1 (G, C) be a cocycle.It can also be seen as a cocycle in B via j.Hence, we let be the split short exact sequence of G-groups constructed by twisting (27) by χ.We obtain the corollary below by Proposition G.
Corollary 2.10.In the diagram below, the rows are split short exact sequence of pointed sets and the vertical arrows are bijective with 1. Let G sp be a connected split reductive group defined over F .By Proposition F, there is a split short exact sequence of Gal(F /F )-groups inducing a split short exact sequence of pointed sets A reductive group G/F is said to be quasi-split if G has a Borel subgroup defined over F .The group Θ F via j is a group of F -automorphisms of G sp /F .The image of j in (31) can be characterized.
Theorem I. (see e.g.[Hu18, Thm.4.2] and its proof ) The set j(H 1 (F, Θ F )) in ( 31) is equal to the set of isomorphism classes of quasi-split F -forms of G sp .Moreover, if χ ∈ Z 1 (F, Θ F ), then the Gal(F /F )group χ G sp (F ) is the F -points of a quasi-split connected reductive group G ′ over F corresponding to the Since the twisted automorphism group χ Aut F G sp acts on χ G sp (F ) = G ′ (F ) by Theorem I, the twisted group χ Aut F G sp is naturally isomorphic to Aut F G ′ .Denote by G ′ ad the adjoint quotient of G ′ .By Corollary 2.10, the following diagram has split short exact rows of pointed sets and the vertical arrows are bijective with τ j(χ) Remark 2.11.
(1) The middle vertical correspondence τ j(χ) in (32) is the identity map if we identify the set of isomorphism classes of F -forms of G ′ with that of G sp in a natural way.
(2) The twisted group Θ ′ F is naturally isomorphic to Out F G ′ and corresponds via j ′ to the set of isomorphism classes of quasi-split F -forms of G ′ .
(3) Let G 1 and G 2 be two F -forms of G sp .The form G 1 is said to be an inner form of . By Theorem I, any F -form G 1 is an inner form of a unique quasi-split F -form G ′ .
2.4.2.2.Similarly, let χ ∈ Z 1 (F, θ F ) and twist the second row of (18) by χ.Then we obtain an F -form , where G ′ is a quasi-split F -form of G sp and GL m ′ ,D ′ is an inner form of GL n,F (for some central division algebra D ′ over F ). Since G ′ is quasi-split and the tautological representation is absolutely irreducible, it follows that GL m ′ ,D ′ = GL n,F [Ti71, Thm.3.3] and the F -form is such that the following diagram has split short exact rows of pointed sets and the vertical arrows are bijective with Corollary 2.12.The fiber π −1 ([χ]) in (32) (resp.(34)) can be identified with H 1 (F, G ′ ad ).
2.4.3.Image of π.Given a short exact sequence of G-groups with A abelian: Then C acts on A naturally and there is the twisted group χ A for every χ ∈ Z 1 (G, C).One associates to χ a cohomology class ∆(χ) ∈ H 2 (G, χ A) as follows.Lift χ to a continuous map g → b g of G into B and define Proposition J. [Se97, Ch. 1.5.6 Prop.41] The cohomology class [χ] belongs to the image of π : Since the middle columns of ( 16) and (19) are short exact sequence of Gal(F /F )-groups with T sp /C abelian, we obtain the following.

Local-global aspects.
2.5.1.The localization map.Let E be a number field and P E be the set of places of E. Let G be a linear algebraic group (or more generally an automorphism group of a reductive chain in §2.3.1)defined over E .For any λ ∈ P E , denote by E λ the completion of E with respect to λ and by i λ : E → E λ the embedding.Let i λ : E → E λ be an embedding extending i λ .Then it induces homomorphisms Gal(E λ /E λ ) → Gal(E/E) and G(E) → G(E λ ) for which the Gal(E λ /E λ )-module G(E λ ) and Gal(E/E)-module G(E) are compatible.We obtain a map of cocycles (k = 0, 1 if G non-abelian) The associated map of Galois cohomology is called the localization map at λ.It is functorial and does not depend on i λ [Se97, Ch. 2.1.1].
2.5.2.Some results.We would like to present some results for the map (39) when G is connected reductive and k = 1 and when G is a torus and k = 2.For simplicity, we use the notation and formulation of [Bo98] For each λ ∈ P E , one has a map [Bo98,5.15](40) where H 1 ab (E λ , G) is the first abelian Galois cohomology group of G [Bo98, Definition 2.2] and (M Gal(E/E) ) tor denotes the the torsion subgroup of the Galois coinvariants of M .The surjectivity of abelianization map ab 1 is by [Bo98,Thm. 5.4 ) tor [Bo98, Propositions 2.8 and 4.1(i)] and cor −1 λ is the natural map [Bo98,4.7].
We have the following result for torus G = T by class field theory and [Bo98, Lemma 5.6.2].
Proposition L. Suppose T is a direct product of a split torus T sp and a torus T ′ such that T ′ is anisotropic over E λ for some place λ of E.
2.6.1.The 1-cocycles µ and χ.According to conditions (a),(b),(c) of the main theorem(s), we have a chain T sp ⊂ G sp ⊂ GL n,E , a chain T ⊂ GL n,E , and an E-isomorphism of representations This produces a 1-cocycle (as well as a Galois representation since Gal(E/E) acts trivially on Aut E T sp ): As Ω E (resp.ω E ) is a subgroup of Aut E T sp ( §2.3.2),we first show the following.
Proposition 2.15.The image of the Galois representation µ : 37), condition (b), and diagram (16) (resp.diagrams ( 17) and ( 19)) for F = E λ .Hence, all the local representations land on Ω E (resp.ω E ).Since Aut E T sp is discrete, the image of µ is finite.We are done by the Chebotarev density theorem.

Proof of main theorem I(i). By condition (b) and diagram (21) for
. By Theorem I for F = E, we obtain a quasi-split connected reductive group G ′ over E such that [G ′ ] = j[χ] in (31).On the one hand, for all λ ∈ P (p) belong to same fiber of π in (31) for F = E λ .On the other hand, for almost all λ ∈ P by Theorem I for F = E λ and condition (d).Hence, by Corollary 2.12 for F = E λ for all λ ∈ P (p) E,f by Theorem K and Proposition 2.14.Here G is an inner form of G ′ ad (Remark 2.11(3)).Therefore, we conclude that G × E E λ ∼ = G λ for all λ ∈ P (p) E,f and G λ is unramified for all but finitely many λ.
E,f , we can impose conditions at other places of E except λ ′ .For example, we can require that loc λ

Proof of main theorem I(ii). By condition (b) and diagram (23) for
belong to same fiber of π in (34) for F = E λ .On the other hand, for almost all λ ∈ P by Theorem I for F = E λ , condition (d), and the proposition below.
Proposition 2.17.[Ti71, Lemma 3.2, Thm.3.3] Let F be a field of characteristic zero and D i (i = 1, 2) be central simple algebras over F .Let H be a connected reductive group over F and ρ i : H → GL mi,Di (i = 1, 2) be two F -representations that are absolutely irreducible.If Hence, by Corollary 2.12 for F = E λ for all λ ∈ P (p) by Theorem K and Proposition 2.14.Here G (resp.GL m,D ) is an inner form of G ′ ad (resp.GL n,E ) and GL m,D = GL n,E by ( 46) and class field theory.By Lemma 2.9, we conclude that (G ֒→ GL n,E ) × E E λ ∼ = (G λ ֒→ GL n,E λ ) as representations for all λ ∈ P (p) E,f and G λ is unramified for all but finitely many λ.

Proof of main theorem II. Consider the cocycle
) for all λ ∈ P E,f .It suffices to show that [µ] belongs to the image of the injection Res T sp (ensuring uniqueness) in diagram (21) (resp.( 23)) for F = E.By Corollary 2.13, this is equivalent to ∆(µ) = 0 in H 2 (E, µ (T sp /C)).By condition (d) and Proposition L, it remains to prove that loc λ (∆(µ)) = 0 for all places λ of E. For a finite place λ, this is true by the fact that the image of Res T sp in (21) (resp.( 23)) contains loc λ [µ] and Corollary 2.13 for F = E λ .For a real place, this is true by (d) and H 2 (R, S R ) = 0 if S R is an R-anisotropic torus (see [Ko86,Lemma 10.4]).Therefore, we obtain a common E-form T ⊂ G (resp.T ⊂ G ֒→ GL m,D by Lemma 2.9) of the chain T λ ⊂ G λ (resp.the chain representation T λ ⊂ G λ ֒→ GL n,E λ ) for all finite places λ of E.

Rationality of algebraic monodromy groups
This section is devoted to the proofs of the statements in §1.2.Fix a number field E and denote by p λ the residue characteristic of the finite place λ ∈ P E,f .3.1.Profinite group Π and Frobenius elements Fr.Consider two cases.

(Characteristic zero).
In this case, Π denotes the absolute Galois group Gal(K/K) of a number field K and P := P E,f .Equip Π with a subset Fr ⊂ Π of Frobenius elements as follows.
For all v ∈ P K,f , let q v be the size of the residue field F qv of K v and consider the natural surjection For any Galois extension L/K that is unramified except finitely many v ∈ P K,f and any finite subset S ⊂ P K,f , the image of v∈P K,f \S Fr v in Gal(L/K) is dense [Se98, Chap.I, §2.2 Cor.2].Assign the number q v to the elements in Fr v .

(Characteristic p).
In this case, Π denotes the étale fundamental group π ét 1 (X, x) (with some base point x) of a smooth geometrically connected variety X/F q in characteristic p and P := P (p) E,f .Equip Π with a subset Fr ⊂ Π of Frobenius elements as follows.
Let X cl be the set of closed points of X.For any geometric point x ′ over x ′ ∈ X cl , let Fr x ′ be the image of the geometric Frobenius Fr −1 where q x ′ is the size of the residue field of x ′ .Note that the change of base point isomorphism σ xx ′ is unique up to an inner automorphism of π 1 (X, x).Since the conjugacy class [Fr x ′ ] depends only on x ′ , write Fr x ′ := [Fr x ′ ] and define Fr := The subset Fr is dense in Π [Se65].Assign the number q x ′ to the elements in Fr x ′ .
3.2.E-compatible systems.Let (Π, Fr, P) be one of the two cases in §3.1.In the characteristic zero case, denote by S a finite subset of P K,f .Otherwise, S is the empty set.
3.2.1.GL n -valued compatible systems.A system of n-dimensional λ-adic (continuous) representations of Π is said to be semisimple (resp.irreducible, absolutely irreducible) if for all λ ∈ P, ρ λ is semisimple (resp.irreducible, absolutely irreducible).The system ρ • is said to be E-compatible (with exceptional set S) if • in the characteristic zero case, ρ λ is unramified outside S ∪ {t ∈ P K,f : p λ |q t } for each λ ∈ P; • for each Frobenius element Fr t ∈ Fr satisfying t / ∈ S and for each λ satisfying p λ ∤ q t , the characteristic polynomial (47) has coefficients in E and depends only on t (independent of λ ∈ P).
The compatible system ρ • is said to be pure of weight w ∈ R (resp.mixed of weights) if for each Fr t ∈ Fr with t / ∈ S and each root α ∈ E of P t (T ), the absolute value |i(α)| is equal to q w/2 t for all complex embedding i : E → C (resp. is independent of the complex embedding i : E → C).

3.2.2.
Coefficient extension and the Weil restriction.Let ρ • be an n-dimensional (semisimple) E-compatible system of Π that is pure of weight w (resp.mixed of weights).For a number field E ′ , denote by P ′ = P E ′ ,f in characteristic zero case and by , where λ is the restriction of λ ′ to E. The system is E ′ -compatible (with exceptional set S), pure of weight w (resp.mixed of weights), and called the coefficient extension of ρ If E ′ is a subfield of E, then we obtain by the Weil restriction of scalars a (semisimple The system is E ′ -compatible (with exceptional set S), pure of weight w (resp.mixed of weights), and called the Weil restriction of ρ • (see [BGP19, Definition 3.4]).

G-valued compatible systems.
Let G be a linear algebraic group defined over E with affine coordinate ring R. Since G acts on itself by conjugation, G acts on R. The subring of invariant functions is denoted by R G .For all g ∈ G, let g s be the semisimple part of g.If g is defined over a field extension F/E, then g s is also defined over F .A system of λ-adic G-representations {ρ λ : Π → G(E λ )} λ∈P of Π is said to be E-compatible (with exceptional set S) if • in the characteristic zero case, ρ λ is unramified outside S ∪ {t ∈ P K,f : p λ |q t } for each λ ∈ P; • for each Frobenius element Fr t ∈ Fr satisfying t / ∈ S, each λ satisfying p λ ∤ q t , and each f ∈ R G the number belongs to E and depends only on t and f [Se98, Chap.I, §2.4] (independent of λ ∈ P)6 .It follows that an n-dimensional E-compatible system is the same as an E-compatible system of GL n,Erepresentations.
3.2.4.Algebraic monodromy groups and connectedness.For all λ ∈ P, the algebraic monodromy group of ρ λ , i.e., the Zariski closure of the image of ρ λ in GL n,E λ , is denoted by G λ .It is an closed subgroup of GL n,E λ .The image ρ λ (Π) is a compact subgroup of the λ-adic Lie group G λ (E λ ).The following result is well-known by using the compatibility condition, see [LP92, Prop.6.14].
Proposition M. The component groups G λ /G • λ are isomorphic for all λ ∈ P. In particular, the connectedness of G λ is independent of λ.
3.2.5.Group schemes.Suppose the algebraic monodromy group G λ is connected reductive for all λ.Let O λ be the ring of integers of E λ with residue field k λ of characteristic p λ .Let Λ λ be an O λ -lattice of E n λ that is invariant under the image ρ λ (Π).Let G λ be the Zariski closure of ρ λ (Π) in GL Λ λ ∼ = GL n,O λ , endowed with the unique structure of reduce closed subscheme.The generic fiber of G λ is G λ .The special fiber, denoted by Proposition N. [LP95, Prop.1.3],[BGP19, Prop.5.51, Thm.5.52] For all but finitely many λ ∈ P, the following assertions hold.
(i) The group scheme G λ is smooth with constant absolute rank over O λ .
(ii) The identity component of the special fiber G k λ ⊂ GL n,k λ is saturated.

3.3.1.
Frobenius torus and maximal torus.For all λ ∈ P, let G λ be the algebraic monodromy group of ρ λ .The identity component of G λ is reductive since ρ λ is semisimple.Let ρ λ be a member of the system and Fr t ∈ Fr be a Frobenius element with t / ∈ S. If p λ ∤ q t , then the Frobenius torus T t,λ of Fr t is defined to be the identity component of the smallest (diagonalizable) algebraic subgroup S t,λ in GL n,E λ containing the semisimple part of ρ λ (Fr t ).It follows that T t,λ ⊂ S t,λ ⊂ G λ .The following theorem is due to Serre.
Theorem O. (see [LP97, Thm.1.2 and its proof], [Ch04, Thm.5.7], [Hu18, Thm.2.6]) Suppose the algebraic monodromy group G λ ′ is connected for some λ ′ ∈ P. Suppose there exists a finite subset Q ⊂ Q such that for all Fr t ∈ Fr with t / ∈ S, the following conditions are satisfied for every root α of the characteristic polynomial P t (T ) in (47): (a) the absolute values of α in all complex embeddings are equal; (b) α is a unit at any finite place not extending p t ; (c) for any finite place w of Q such that w(p t ) > 0, the ratio w(α)/w(q t ) belongs to Q. Then there exists a proper closed subvariety (1) If G λ is connected and the Frobenius torus T t,λ is maximal, then T t,λ = S t,λ .(5) In the characteristic p case, the subset of elements Fr t of Fr whose Frobenius tori T t,λ ′ are maximal in G λ ′ is dense in Π. (6) In the characteristic zero case, the subset of places v ∈ P K,f such that T v,λ ′ is a maximal torus of G λ ′ is of Dirichlet density one (see [Hu18, Cor.2.7]).
Let Fr t be a Frobenius element.There is a semisimple matrix M t of GL n (E) with P t (T ) (47) as characteristic polynomial.For all λ ∈ P with p λ ∤ q t , M t is conjugate to the semisimple part ρ λ (Fr t ) s in GL n (E λ ) by E-compatibility.Hence, if we let S t be the smallest algebraic subgroup of GL n,E containing M t and T t be the identity component of S t , then the chain representations (T t ⊂ S t ֒→ GL n,E ) × E E λ and T t,λ ⊂ S t,λ ֒→ GL n,E λ are isomorphic for all λ ∈ P with p λ ∤ q t .Corollary 3.2.Assume the conditions of Theorem O. Then the following assertions hold.
(i) (Common E-form of formal characters) If the Frobenius torus T t,λ ′ is a maximal torus of G λ ′ , then the Frobenius torus T t,λ is also a maximal torus of G λ for all λ ∈ P with p λ ∤ q t .Moreover, the representation (T t ֒→ GL n,E ) × E E λ is isomorphic to T t,λ ֒→ GL n,E λ for all λ ∈ P with p λ ∤ q t .(ii) (Absolute rank) The absolute rank of G λ is independent of λ.
Proof.Assertion (i) is straight forward by Theorem O and the above construction of T t .Assertion (ii) is obvious by (i) in the characteristic p case and follows from (i) and Remark 3.1(6) in the characteristic zero case.
3.3.2.Anisotropic subtorus.In this subsection, G λ is connected for all λ ∈ P. The subtorus T t ⊂ GL n,E in Corollary 3.2(i) is studied under the following hypothesis.Let k be the order of S t /T t .Then the Zariski closure of M kZ t in GL n,E is T t .
Hypothesis R: Assume for each real embedding E → R, the set of powers det(M t ) Z ⊂ R contains some non-zero integral power of the absolute value |i(α)| for every root α of P t (T ) and every complex embedding i : E → C extending E → R. Proposition 3.3.If Hypothesis R holds, then the subtorus (T t ∩ SL n,E ) • of T t is anisotropic at all real places of E.
is the product of some integral powers of the roots i(α) of the polynomial i(P t (T )) ∈ R[T ].Hence, there exist integers h = 0 and m such that by Hypothesis R.This implies χ 2hk = det 2m on T t,R since (M k t ) Z is Zariski dense in T t,R .Hence, χ 2hk is trivial on the subtorus (T t,R ∩ SL n,R ) • for some 2hk = 0. We conclude that the torus (T t,R ∩ SL n,R ) • is anisotropic.
Corollary 3.4.If Hypothesis R holds and E has a real place, then the subtorus (T t ∩ SL n,E ) • of T t is anisotropic at a positive Dirichlet density subset P ′ of P E,f .Proof.Let r be the absolute rank of the E-torus (T t ∩ SL n,E ) • .Then it is an E-form of the split torus G r m,E with automorphism group GL r (Z).The isomorphism class of (T t ∩ SL n,E ) • is represented by an element of H 1 (E, GL r (Z)), which is a continuous group homomorphism φ : Gal(E/E) → GL r (Z) up to conjugation.Let c ∈ Gal(E/E) be a complex conjugation corresponding to a real place of E. Since (T t ∩ SL n,E ) • is anisotropic over R by Proposition 3.3 and c is of order two, it follows that φ(c) = −I r .Since the image of φ is finite, there is a positive Dirichlet density set P ′ of finite places λ of E such that φ(Fr λ ) = −I r by the Chebotarev density theorem.Therefore, (T t ∩ SL n,E ) • is anisotropic over E λ for all λ ∈ P ′ .Remark 3.5.
(1) Hypothesis R holds for every P t (T ) if the E-compatible system is pure.
(2) If λ ∈ P ′ in Corollary 3.4, then the E λ -subtorus λ is also a maximal torus.(3) Corollary 3.4 is not true for general E since (T t ∩ SL n,E ) • can be a non-trivial split torus over E. This is done by taking a finite extension E ′ /E such that P t [T ] splits and replacing the E-compatible system ρ • with its coefficient extension ρ Let G be a connected reductive group defined over a field F .A torus T ⊂ G is said to be fundamental if it is a maximal torus with minimal F -rank.In the characteristic zero case, let S be the subset of elements v ∈ P K,f such that for some λ ∈ P E,f , the Frobenius torus A Frobenius torus T v,λ ⊂ G λ being fundamental is equivalent to T v,λ is a maximal torus and T v,λ ∩ G der λ is anisotropic [Bo98, Prop.5.3.2].When Hypothesis R holds and E has a real place, Remark 3.1(6), Corollary 3.4, and Remark 3.5(2) imply that S is of Dirichlet density one.
Question Q: Suppose Hypothesis R holds, what is the Dirichlet density of S in P K,f when E is totally imaginary?
We do not know the answer; we even do not know if S is non-empty.If we want to apply main theorem II to the algebraic monodromy representations {G λ ֒→ GL n,E λ } λ∈P E,f when E is totally imaginary, then a positive Dirichlet density of S is necessary.[BGP19,Cor. 7.9], or by Proposition 3.6 below, for almost all λ, the existence of a hyperspecial maximal compact subgroup of G λ (E λ ) implies that G λ is unramified [Mi92,§1].We are done by main theorem I.
Proposition 3.6.If G λ is connected for all λ ∈ P, then the image of ρ λ is contained in a hyperspecial maximal compact subgroup H λ of G λ (E λ ) for almost all λ.
Proof.Since π 1 (X) is compact, we may assume ρ λ (π 1 (X)) ⊂ GL n (O λ ) after some change of coordinates V λ ∼ = E n λ for all λ.The geometric étale fundamental group π geo 1 (X) of X satisfies the short exact sequence implies that the special fiber G k λ has trivial unipotent radical for almost all λ.Therefore, the smooth group scheme G λ is reductive over O λ for almost all λ.A .This proves assertion (i).The proof of (ii) is exactly the same except we want to adjust the isomorphism of representation for some finite S ⊂ P E,f .Then by enlarging S we obtain that the group scheme GL n,OE,S × O λ (resp.G × O λ ) is the group scheme associated to the hyperspecial maximal compact subgroup GL gives the construction G(O λ ) ⊂ GL n,E (O λ ) in (51).Since the λ-component Remark 3.7.The proofs of Corollaries 1.2 and 1.3 are standard in the sense that they only require the common E-forms G and G ⊂ GL n,E in Theorem 1.1, Proposition 3.6, and Bruhat-Tits theory [Ti79].
3.4.4.Proof of Corollary 1.4.By Corollary 1.2(ii), there is a common E-form ι : G ֒→ GL n,E .For each λ ∈ P, choose an embedding E → E λ .We claim that the conjugacy class of the semisimple part ρ G λ (Fr t ) s ∈ G(E λ ) is defined over E for all Frobenius element Fr t and all λ ∈ P. Indeed, by field extension, we obtain It suffices to show that for any irreducible representation ψ of G × E, the trace of ψ(ρ G λ (Fr t ) s ) ∈ E. This is true because the roots α of the characteristic polynomial P t (T ) of ρ G λ (Fr t ) ∈ GL n,E (E λ ) belong to E by E-compatibility and ψ is a subrepresentation of ⊗ r ι E ⊗ s ι * E for some r, s ∈ Z ≥0 .The next step is to show that for a fixed Frobenius element Fr t , the conjugacy class of ρ G λ (Fr t ) s in G is independent of λ.By [D'Ad20, Thm.4.3.2]([Ch04, Thm.6.8, Cor.6.9] when X is a curve), there is a finite extension F of E and a connected reductive subgroup G sp ⊂ GL n,F such that for all λ ∈ P, if F λ is a completion of F extending λ on E, then there exists an isomorphism of representations: Since because the absolute rank of G ℓ is independent of ℓ by Corollary 3.2(ii) and T v,ℓ is a maximal Frobenius torus.Since the characteristic polynomials of Φ v and ρ ℓ (Fr v ) s for all ℓ = p are equal to P v (T ), the tori representations T v,ℓ ֒→ GL V ℓ for all ℓ admit a common Q-form T v ֒→ GL n,Q .
Condition II(b): This is just condition 1.6(b).
Condition II(c): By Proposition 2.8 and condition 1.6(c), it suffices to check condition (c'-bi) in §2.2.Identify GL V ℓ as GL n,Q ℓ for all ℓ.We employ the technique in [Hu13, Prop.3.18, Thm.3.19].Let {ψ ℓ } ℓ∈P be an r-dimensional semisimple Q-compatible system of abelian ℓ-adic representations of Gal(K/K).Let S ℓ ⊂ GL r,Q ℓ be the algebraic monodromy group of ψ ℓ and assume S ℓ is torus and with the largest possible dimension d K ([Hu13, Thm.3.8]) for all ℓ.Consider the semisimple Q Fix a prime ℓ ′′ , there exist a finite extension F of K and an abelian variety A over F that is a direct product of CM abelian varieties with the following properties.Let {ǫ ℓ : Gal(F /F ) → GL(W ℓ )} ℓ∈P be the semisimple compatible system of Galois representations with W ℓ := H 1 (A F , Q ℓ ).Let M ℓ and G ′′ ℓ be respectively the algebraic monodromy group of the Galois representation ǫ ℓ and ρ ℓ ⊕ ǫ ℓ of Gal(F /F ).Then the following assertions hold.
(i) For all ℓ, G ′′ ℓ is connected and M ℓ is a torus with dimension independent of ℓ. (ii) The restriction map Since G ′ ℓ is connected for all ℓ, it is again the algebraic monodromy group of the restriction of Finally, we follow the strategy in condition II(a).If v is a finite place of F such that Y × K F and A have good reduction, then write p := p v and the Frobenius element Fr v have characteristic polynomials v is also a maximal torus as the absolute rank of G ′′ ℓ is independent of ℓ.By using the polynomials for all ℓ such that (65) ker(p 2 : T ′′ v → GL 2 dim A,Q ) • ⊂ p 1 (T ′′ v ) ֒→ GL n,Q is a common Q-form of formal bi-characters of G ℓ ⊂ GL n,Q ℓ for all ℓ, where p 1 , p 2 are the obvious projections.We may replace T v ֒→ GL n,Q constructed in condition II(a) with p 1 (T ′′ v ) ֒→ GL n,Q in (65).
Condition II(d): Let T v ⊂ GL n,Q be the Q-form we found in condition II(a).This part is exactly the same as the verification of condition II(d) for Theorem 1.5 once we replace the field E by Q and the E-torus T t by the Q-torus T v .
3.5.2.Proofs of Corollaries 1.9 and 1.10.Since Corollaries 1.9 and 1.10 (of Theorem 1.6) assume Hypothesis H, they follow along the same lines in the proofs of Corollaries 1.2 and 1.3 by Remark 3.7.3.5.3.Galois maximality and Hypothesis H. Let K be a number field and {ρ ℓ : Gal(K/K) → GL n (Q ℓ )} ℓ∈P be a Q-compatible system of ℓ-adic representations.Let Γ ℓ be the image of ρ ℓ and G ℓ be the algebraic monodromy group of ρ ℓ .Then Γ ℓ is a compact subgroup of G ℓ (Q ℓ ).Suppose for simplicity that G ℓ is connected for all ℓ.Denote by G ss ℓ be the quotient of G ℓ by its radical and by G sc ℓ the simply-connected covering of G ss ℓ .Denote by Γ ss ℓ the image of Γ ℓ in G ss ℓ (Q ℓ ) and by Γ sc ℓ the inverse image of Γ ss ℓ in G sc ℓ (Q ℓ ).When ℓ ≫ 0 compared to the absolute rank of G sc ℓ , a compact subgroup H ℓ of G sc ℓ (Q ℓ ) is hyperspecial maximal compact if the "mod ℓ reduction" of H ℓ is "of the same Lie type" as the semisimple group G sc ℓ (see [HL16]).In [Lar95], Larsen proved that the set of primes ℓ for which Γ sc ℓ ⊂ G sc ℓ (Q ℓ ) is hyperspecial maximal compact is of Dirichlet density one and conjectured the following.
Conjecture S. For all ℓ ≫ 0, Γ sc ℓ is a hyperspecial maximal compact subgroup of G sc ℓ (Q ℓ ).This conjecture is also related to the conjectures of Serre on maximal motives [Se94,11.4,11.8].Suppose the ℓ-adic compatible system is {H w (Y K , Q ℓ )} ℓ∈P , where Y is a smooth projective variety defined over a number field K.When Y is an elliptic curve without complex multiplication and w = 1, a well-known theorem of Serre states that for ℓ ≫ 0, Γ ℓ ∼ = GL 2 (Z ℓ ) is maximal compact in GL(V ℓ ) [Se72].In general, by studying the mod ℓ compatible system {H w (Y K , F ℓ )} ℓ≫0 , we proved that Γ ℓ ⊂ G ℓ (Q ℓ ) is large in the sense that its mod ℓ reduction has "the same semisimple rank" as the algebraic group G ℓ for ℓ ≫ 0 [Hu15, Thm.A].This result is crucial to the following.
Theorem T. [HL16, HL20] Let ρ • be the Q-compatible system (3) arising from a smooth projective variety Y defined over K. Conjecture S holds in the following cases.
Let G red F ℓ be the quotient of G • F ℓ by its unipotent radical.For ℓ ≫ 0, the special fiber G der F ℓ (of G der ℓ ) is a normal connected semsimple subgroup of G • F ℓ (Proposition 3.10), which injects into G red Therefore, (72) is an equality and the special fiber G F ℓ is reductive for ℓ ≫ 0.
Remark 3.11.Let F be a finitely generated field of characteristic p and Y be a smooth projective variety defined over F .Conjecture S holds for the Q-compatible system {H w (Y F , Q ℓ )} ℓ =p [CHT17, Thm.1.2].
One observes that each simple Lie algebra has at most one possible representation.

Final remarks.
(1) We construct a common E-form G ֒→ GL n,E of the algebraic monodromy representations G λ ֒→ GL n,E λ of the system (2) in case it is absolutely irreducible and G λ is connected (for all λ) in Theorem 1.1(ii).The non-absolutely irreducible case and the non-connected case remain open.(2) Let ρ • be the system in Theorem 1.6 and assume Conjecture S. Then Corollary 1.9(i) produces an adelic representation ρ G A : Gal(K/K) → G(A Q ).Let ρ G F ℓ be the mod ℓ reduction of the ℓ-component ρ G ℓ of ρ G A for ℓ ≫ 0. One can deduce by [Hu15, Thm.A, Cor.B] that there is a constant C > 0 such that the index satisfies [G(F ℓ ) : ρ G F ℓ (Gal(K/K))] ≤ C, ∀ℓ ≫ 0. Thus, the composition factors of Lie type in characteristic ℓ of ρ G F ℓ (Gal(K/K)) can be described when ℓ ≫ 0, see a similar result [Hu18, Cor.1.5] for certain type A compatible system.
(3) The smooth subgroup scheme G λ ⊂ GL n,O λ in Corollary 1.3 depends on the choice of an O λ -lattice of V λ .It is shown in [Ca17] that for almost all λ, the subscheme G λ ⊂ GL n,O λ is unique up to isomorphism.(4) The E-forms G and G ⊂ GL n,E we constructed in Theorem 1.1 are not unique for the simple reason that X 1 (E, G ad ) in Theorem K may not be trivial, where G ad denotes the adjoint quotient of G. (5) Let S ′ be a non-empty finite subset of P E,f .Actually, by examining the proof, main theorem I holds if we replace P (p) E,f with P E,f \S ′ .(6) In the characteristic zero case, Question Q in §3.3.2 should be addressed if one wants to apply main theorem II to an E-compatible system when E is totally imaginary.However, one can always use main theorem I by omitting a finite place of E if one knows that G λ is quasi-split for almost all λ, or, one can take the Weil restriction Res E/Q ( §3.2.2) to obtain a Q-compatible system and see if main theorem II can be applied.
2.1.Here are some examples for the invariance of roots condition.Theorem D. [Hu18, Thm.3.10],[Hu20, Thm.A2] The following C-connected semisimple groups G satisfy the invariance of roots condition for all representations G ⊂ GL n,C .(a) (Hypothesis A): G has at most one A 4 almost simple factor and if Pick two out of the three unique dimension 4096 = 2 12 irreducible representations of C 4 , D 4 , and F 4 .
(a) For r 1 , ..., r m , r ∈ N such that r 1 + • • • + r m = r, the spin representation of B r and the tensor product of the spin representations of B rj for all 1 ≤ j ≤ m are not both factors of (G, C n ).(b) For 1 ≤ k ≤ r − 1 and r ≥ 2, the representations of C r and D r with highest weight (k, k − 1, .., 2, 1, ...0) are not both factors of (G, C n ).(c) The unique dimension 27 irreducible representations of A 2 and G 2 are not both factors of (G, C n ).
(d) Any two of the unique dimension 4096 irreducible representations of C 4 , D 4 , and F 4 are not both factors of (G, C n ).2.2.2.2.Inspired by Theorem E, we give more examples for the invariance of roots condition.Theorem 2.6.Suppose G ⊂ GL n,C is a connected adjoint semisimple subgroup that satisfies the following Lie type assumptions:(a) G does not have a factor of type B r (r ≥ 2).(b) If G has a factor of type C 3 , then it cannot have a factor of type A 3 .(c) If G has a factor of type C r , then it cannot have a factor of type D r (r ≥ 4).(d) If G has a factor of type F 4 , then it cannot have a factor of type D 4 .(e) If G has a factor of type G 2 , then it cannot have a factor of type A 2 .

3. 4 .
Proofs of characteristic p results.Let P be P (p) E,f .3.4.1.Proof of Theorem 1.1.By Proposition M and taking a finite Galois covering of X, we assume that G λ is connected for all λ ∈ P. It suffices to check conditions (a),(b),(c),(d) of main theorem I for the system of algebraic monodromy representations {G λ ֒→ GL n,E λ } λ∈P .Conditions (a),(b),(c) follow directly from assertions (i),(ii),(iii) of Theorem B. Condition (d) holds by E λ G λ (E λ ) is transitive on the set of hyperspecial maximal compact subgroups of G λ (E λ ) [Ti79, §2.5].Hence, by Proposition 3.6 and adjusting φ λ for almost all λ, we assume φ λ (G(O λ )) = H λ ⊂ G λ (E λ ) for almost all λ.Then the image of the map λ∈P φ −1 λ • ρ λ is contained in the adelic points G(A (p) E ), which defines the desired G-valued adelic representation ρ G 1.2.1.3.Denote by P E,p the set of finite places of E extending p.Let Q p k be a degree k unramified extension of Q p , v ∈ P E,p , and E v,p k the composed fields E v •Q p k .Let ρ • be in Theorem 1.1.The semisimple crystalline companion object of ρ • at v (whose existence 4 is conjectured by Deligne [De80, Conjecture 1.2.10]) is an object are ordinary is of positive Dirichlet density.(b) (ℓ-independence absolutely): There exists a connected reductive subgroup G C of GL n,C such that the representations G C ֒→ GL n,C and (G ℓ ֒→ GL n,Q ℓ ) × Q ℓ C are isomorphic for all ℓ.
be the Zariski closure of G λ (resp.the identity component of G geo λ ) in GL n,O λ with special fiber G k λ (resp.G geo k λ ).It suffices to prove that for almost all λ, H λ := G λ (O λ ) is a hyperspecial maximal compact subgroup of G λ (E λ ).By Bruhat-Tits theory, this condition follows if we show that the O λ -group scheme G λ is reductive [Ti79, §3.8.1].By [BGP19, Thm.7.3], the O λ -group scheme G geo λ is semisimple for almost all λ.Let k λ be the residue field of E λ .Since the O λ -group scheme G λ is smooth with constant absolute rank for almost all λ by Proposition N(i) and contains G geo 3.4.2.Proof of Corollary 1.2.By Theorem 1.1(i), there is a connected reductive group G defined over E and an isomorphism φ λ : G × E E λ → G λ for each λ ∈ P. For almost all λ, the O λ -points G(O λ ) is well-defined (by finding some integral model G of G) and is a hyperspecial maximal compact subgroup of the E λ -points G(E λ ) [Ti79, §3.8.1].Let G ad λ be the adjoint group of G λ .The subgroup G ad λ (E λ ) of Aut the local representation V p is ordinary, H Vp is solvable [Pi98, Prop.2.9] and its image H red Vp in GL V ss p is a torus.Since the conjugacy class of Φ Vp in H Vp is defined over Q p [Pi98, Prop.2.2] and H red Vp is abelian, the image of Φ Vp in H red Vp , denoted by Φ red Vp , belongs to H red Vp (Q p ).By the splitting of the surjection H Vp ։ H red Vp , there is a semisimple element connected by (61).Hence, G ′ ℓ is connected for all ℓ by Proposition M. Since the dimension of the center of G ′ ℓ is d K = dim S ℓ for all ℓ [Hu13, Prop.3.8, Thm.3.19], it follows that for all ℓ the projection to the ith factor, i = 1, 2. Since there exists a surjective map G ′′ ℓ ′′ → G ′ ℓ ′′ by Proposition P(ii), it follows from (62) and the connectedness ofG ′′ ′′ ) • = G der ℓ ′′ = (G ′′ ℓ ′′ ) der is the semisimple part of G ′′ ℓ ′′ .Since {ρ ℓ ⊕ ǫ ℓ } ℓ∈Pis a compatible system of representations of Gal(F /F ), the semisimple rank and the dimension of the center of G ′′ By condition 1.6(a), there exists v ∈ P F,f such that the Frobenius torus T ′′ v,ℓ ⊂ G ′′ ℓ is maximal for all ℓ = p and the local representation V p of Gal(F v /F v ) is ordinary.Then we let H Vp⊕Wp ⊂ GL Vp × GL Wp be the algebraic monodromy group of the local crystalline representationρ p ⊕ ǫ p : Gal(F v /F v ) → GL(V p ) × GL(W p )and H red Vp⊕Wp its image (semisimplification) in the (abelian) diagonalizable subgroupH red Vp × M p ⊂ GL Vp × GL Wp ,where H redVp is defined in condition II(a).Since the local representation V p ⊕ W p is crystalline, we conclude by repeating the arguments in the second and third paragraphs of condition II(a) that there exists an element in H red,• Vp⊕Wp