On the Congruence Kernel for Simple Algebraic Groups

Prasad, Gopal; Rapinchuk, Andrei S.

doi:10.1134/s0371968516010143

Cited by 4 publications

(2 citation statements)

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“…Some more scattered results are available in the literature. Recently, it was established that anisotropic inner forms of type A n have trivial S-congruence kernel C(k, G, S) for certain infinite sets S of places, for example those S which contain all primes in an arithmetic progression [24,26].…”

Section: Appendix a The Congruence Subgroup Problemmentioning

confidence: 99%

On the profinite rigidity of lattices in higher rank Lie groups

Kammeyer¹,

Kionke²

2020

Preprint

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Section: Appendix a The Congruence Subgroup Problemmentioning

confidence: 99%

On the profinite rigidity of lattices in higher rank Lie groups

Kammeyer¹,

Kionke²

2020

Preprint

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“…The congruence kernel C(Γ) had been completely computed (or proved to be infinite) in many cases, notably, for G = SL n (n ≥ 3) and G = Sp 2n (n ≥ 2) in [BLS, BMS], G = SL 2 in [S70], for all k-split simply connected simple groups G in [Ma], and for all k-isotropic simply connected simple groups G in [Ra76,Ra86,PRa83,PR96]. For several classes of k-anisotropic groups the problem is still open; we refer to [PR10,PR16] for a detailed exposition of available results.…”

Section: Introductionmentioning

confidence: 99%

On the congruence kernel of isotropic groups over rings

Stavrova

2020

Trans. Amer. Math. Soc.

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Let R be a connected noetherian commutative ring, and let G be a simply connected reductive group over R of isotropic rank ≥ 2. The elementary subgroup E(R) of G(R) is the subgroup generated by U P + (R) and U P − (R), where U P ± are the unipotent radicals of two opposite parabolic subgroups PWe prove that the congruence kernel of E(R), defined as the kernel of the natural homomorphism E(R) → E(R) between the profinite completion of E(R) and the congruence completion of E(R) with respect to congruence subgroups of finite index, is central in E(R). In the course of the proof, we construct Steinberg groups associated to isotropic reducive groups and show that they are central extensions of E(R) if R is a local ring. , and the research program 6.38.74.2011 "Structure theory and geometry of algebraic groups and their applications in representation theory and algebraic K-theory" at St. Petersburg State University. Proof. See [St14, Theorem 2.1]. Definition 2.5. Under the assumptions of Theorem 2.4, we call the subgroup E(A) = E P (A), where P is a strictly proper parabolic subgroup of G, the elementary subgroup of G(A).2.2. Abstract relative roots. In order to better understand systems of relative roots Φ(S, G) of Definition 2.2, we consider the abstract systems of relative roots derived from abstract root systems in the sense of [Bou]. The corresponding notion was first introduced in [PeSt09].Let Φ be a reduced root system in the sense of [Bou] with an inner product (−, −). Let Π = {a 1 , . . . , a l } be a fixed system of simple roots of Φ; if Φ is irreducible, we assume that the numbering follows Bourbaki [Bou]. Let D be the Dynkin diagram of Φ. We identify nodes of D with the corresponding simple roots in Π.Definition 2.6. Let Γ ≤ Aut (D) be a subgroup, and let J ⊆ Π be a Γ-invariant subset. Consider the projectionProof. The proof of (i) and (ii) for the case of maximal roots is literally the same as in [KaSt, Lemma 1], which deals with the case where α is a simple relative root. The case of minimal roots is symmetric. To establish (iii), we observe that Ψ = Φ Π,Γ is naturally a root system, and Ψ π Π,Γ (J),{id} = Φ J,Γ . Applying (i) to Ψ π Π,Γ (J),{id} , we conclude that the images under π Π,Γ of all Π-maximal (respectively, Π-minimal) roots in π −1 J,Γ (α) coincide. By [PeSt09, Lemma 3] this implies that Γ acts transitively on such roots. The claim (iv) follows similarly from (ii).Lemma 2.8. Assume that rank(Φ J,Γ ) ≥ 2 and it is irreducible. Let α, β ∈ Φ J,Γ be two relative roots. Let a max ∈ π −1 J,Γ (α) be a Π-maximal root, and let b min ∈ π −1 J,Γ (β) be a Π-minimal root. Let a 1 , . . . , a n ∈ Φ + be a sequence of roots such that b min = a max − a 1 − . . . − a n ,

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