On coefficients, potentially abelian quotients, and residual irreducibility of compatible systems

A family of n-dimensional λ-adic representations of K indexed by a subset Π E ⊂ Σ E ,

(1)

is called an E-rational compatible system (in the sense of Serre) if there exist a finite S ⊂ Σ K and a polynomial P v (T ) ∈ E[T ] for each v ∈ Σ K \S such that the following conditions hold. (a) For each λ, the representation ρ λ is unramified outside S ∪ S ℓ . (b) For each λ and v ∈ Σ K (S ∪ S ℓ ), the characteristic polynomial of Frobenius at v satisfies

In this case, we also call (1) an E-compatible system or just a compatible system.

Compatible systems of Galois representations can be attached to smooth projective varieties [De74] and they also arise from certain cuspidal automorphic representations of GL n , see [BLGGT14], [HLTT16], [Sc15]. We say that the family (1) is semisimple (resp. abelian, potentially abelian) if ρ λ is semisimple (resp. abelian, potentially abelian) for all λ ∈ Π E . We call (1) a strongly E-rational compatible system (or strongly E-compatible) if it is Ecompatible and up to some change of coordinates, ρ λ (Gal K ) ⊂ GL n (E λ ) 1 for all λ ∈ Π E . This condition on (1) is more restrictive than being E-compatible. The Zariski closure of the image ρ λ (Gal K ) in GL n,E λ , denote by G λ , is called the algebraic monodromy group of ρ λ . It is a subgroup of GL n,E λ . 1.2. Potentially abelian quotients and λ-independence. If (1) is semisimple and strongly E-compatible, then the algebraic monodromy group G λ is naturally a reductive subgroup of GL n,E λ . One can ask, as in [Se94] for motives, if there exists a common E-model G ⊂ GL n,E for the family {G λ ⊂ GL n,E λ } Π E in the sense that G × E E λ and G λ are conjugate in GL n,E λ for all λ ∈ Π E . When ρ λ is abelian for some λ, such a λ-independence problem has an affirmative answer by Serre-Waldschmidt theory [Se98], [Wa81]. In general, it is known that the component group π 0 (G λ ) := G λ /G • λ , the rank and semisimple rank of G λ are independent of λ [Se81], [Hu13]. There have been a lot of studies on the λ-independence of the identity component G • λ (and the construction of G • ⊂ GL n,E ) under specific situations, e.g., see [Mo17] for a survey on the Mumford-Tate conjecture for abelian varieties [Mu66], see [LP90,LP92], [Hu25] when the tautological representation of G • λ is absolutely irreducible for all λ, and see [HL25, §1.2] for recent progress under some type A assumption.

Suppose (1) is semisimple and only E-compatible. In subsection 3.1, we shall provide a canonical construction of the algebraic monodromy group G λ := G ρ λ ,E λ defined over E λ together with a morphism Gal K ρ λ → G λ (E λ ) (Definition 3.5) with natural properties (Proposition 3.4). Our construction builds on the theory of pseudocharacters [Ro96], [Ch14], whereas the only conditions we need are the semisimplicity of ρ λ and that the characteristic polynomial of ρ λ (g) takes values in E λ [T ] for all g ∈ Gal K . This construction is compatible with base change, but the group G λ is in general not a subgroup of GL n,E λ . Define the following E λ -quotients of G λ for all λ ∈ Π E .

, the maximal potentially abelian quotient of G λ . • G ab λ := G λ /[G λ , G λ ], the maximal abelian quotient of G λ . We prove that the systems {G pab λ } Π E and {G ab λ } Π E are independent of λ in a strong sense. Theorem 1.1. Let K be a number field and {ρ λ : Gal K → GL n (E λ )} Π E be a semisimple Erational compatible system of K with {G λ } Π E as the system of algebraic monodromy groups defined over E λ (Definition 3.5).

(i) There exists a family of faithful E λ -representations {f λ :

is a semisimple potentially abelian (strongly) E-rational compatible system of K. (ii) There exists a family of faithful E λ -representations {r λ :

is a semisimple abelian (strongly) E-rational compatible system of K. (iii) There exist reductive groups G pab and G ab defined over E such that

Remark 1.2. Observe that the targets of f λ and r λ are GL k,E λ and not GL k,E λ . But note that we have some freedom in choosing k; see for instance Theorem 5.7.

1.3. Descent of coefficients and residual irreducibility. Suppose (1) is semisimple and Ecompatible. For any finite field extension E ′ /E, one obtains from (1) a semisimple E ′compatible system

(2)

3). We say that (1) can be descended to a strongly E ′ -rational compatible system if there exists E ′ /E such that (2) is strongly E ′ -compatible. This is true if (1) is a regular weakly compatible system, see [BLGGT14, Lemma 5.3.1]. We establish that such descent holds in complete generality, enabling its application to the se