## Weighted Completion of Galois Groups and Some Conjectures of Deligne (2000)

Citations: | 7 - 1 self |

### BibTeX

@MISC{Hain00weightedcompletion,

author = {Richard Hain and Makoto Matsumoto},

title = { Weighted Completion of Galois Groups and Some Conjectures of Deligne },

year = {2000}

}

### OpenURL

### Abstract

### Citations

224 | Rational homotopy theory - Quillen - 1969 |

205 | On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal(Q/Q), Algebra i Analiz 2 - Drinfeld - 1990 |

149 |
Etale Homotopy
- Artin, Mazur
- 1969
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Citation Context ...ts on and clarification about the origin and history of the conjectures considered in this paper. 2. Preliminaries on Proalgebraic Groups A proalgebraic variety X over a field k is a pro-object (c.f. =-=[1]-=-) in the category of varieties over k: X = (Xα) where {Xα} is a projective system of k-varieties. The morphism set between the two proalgebraic varieties X = (Xα) and Y = (Yβ) is defined by Hom(X, Y )... |

111 |
Cohomologie Galoisienne
- Serre
- 1997
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Citation Context ... K(X) unramified at z. To prove the second vanishing we consider the Hochshild-Serre spectral sequence (cf. [15, p. 105, Thm. 2.20]) H á et (z, Hb ét (Uur, Qℓ(n))) ⇒ H a+b ét (U, Qℓ(n)). According to =-=[19]-=-, U ur has cohomological dimension at most 1, so that H b ét (Uur , Qℓ(n)) = 0 when b > 1. Since the pro-ℓ abelianization of the inertia group π1(U ur ) is Zl(1) as a Galois module, we have H 1 ét(U u... |

105 | Le groupe fondamental de la droite projective moins trois points - Deligne - 1999 |

57 | Linear algebraic groups, Second edition, Graduate Texts in Mathematics 126 - Borel - 1991 |

26 | Local fields. Translated from the French by Marvin Jay Greenberg - Serre - 1979 |

25 |
Completions of mapping class groups and the cycles C
- Hain
- 1993
(Show Context)
Citation Context ... universal for homomorphisms of Γ into such “negatively weighted extensions” of S by a prounipotent group. It is a variant of Deligne’s notion of the relative Malcev completion, which is developed in =-=[8, 9]-=-. Weighted completions arise naturally in Galois theory in many contexts, the simplest of which is the following: (i) ℓ is a prime number, (ii) Γ = Gℓ, the Galois group of the maximal algebraic extens... |

25 |
Lie algebras and Lie groups. 1964 lectures given at Harvard University. Second edition
- Serre
- 1992
(Show Context)
Citation Context ...Iℓ GF,S ∼ = Gr n Iℓ GF . Since [Iℓ m GF,Iℓ n GF] ⊆ Iℓ m+n GF, the associated graded groups Gr >0 Iℓ GF ∼ = Gr >0 Iℓ GF,S are Lie algebras over Zℓ; the bracket is induced by the group commutator. (See =-=[20]-=-, for example.)26 RICHARD HAIN AND MAKOTO MATSUMOTO 10.2. Galois actions on unipotent completions. Set P = π1(X, x) un , the /Qℓ ℓ-adic unipotent completion (see Paragraph A.2) of π1(X, x). Remark 10... |

23 | Revêtement Etales et Groupe Fondamental (SGA - Grothendieck, Raynaud - 1971 |

17 |
On higher p-adic regulators
- Soulé
- 1981
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Citation Context ...hara [11], one can see (cf. Remark 10.7) that there is a GQ-equivariant isomorphism H1(Gr >0 Iℓ GQ, Qℓ) ∼ = ⊕ H 1 cts(GQ, Qℓ(2n + 1)) ∗ ⊗ Qℓ(2n + 1) ∼ = ⊕ Qℓ(2n + 1). n≥1 The value of Soulé’s element =-=[22]-=- of H1 cts(GQ, Qℓ(2n + 1)) on the image of s2n+1 in H1 cts (GQ, Qℓ(2n + 1)) ∗ ⊗ Qℓ(2n + 1) is non-zero. In [11] Ihara constructs explicit elements σ3, σ5, σ7, . . . of Gr >0 Iℓ GQ, which are sometimes... |

16 |
Théorie de Hodge II, Inst
- Deligne
- 1972
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Citation Context ...has a natural weight filtration, and that the weight graded functor is exact on the category of such modules. This exactness property, called strictness, is familiar from Deligne’s mixed Hodge theory =-=[3]-=- as well as Galois theory. It is a key ingredient in the proof of Theorem 1.1. The reason for taking the central cocharacter to be x ↦→ x −2 in the previous paragraph is to make the representation the... |

16 |
Étale cohomology, Princeton mathematical series 33
- Milne
- 1980
(Show Context)
Citation Context ... that H 0 ét (Uur , Qℓ(n)) is Qℓ(n), into the spectral sequence. B.2. Proof of Proposition B.2. The continuous cohomology H • cts is the one defined in [23, Sect. 2]. In étale cohomology we have (cf. =-=[15]-=-) a canonical isomorphism H í et(Spec OF,S, Qℓ(n)) ∼ = [lim H ←− m í et(Spec OF,S, Zℓ(n)/ℓ m )] ⊗Zℓ Qℓ. A similar result is valid for continuous cohomology. Lemma B.5. When i = 1, 2, there is a natura... |

15 |
Profinite braid groups, Galois representations and complex multiplications
- Ihara
- 1986
(Show Context)
Citation Context ...)kℓ and projects to a non-zero element in the −2(2n + 1)th weight graded quotient of kℓ. These together generate Gr W • kℓ. The first step in the proof of Theorem 1.1 is to use the fact, due to Ihara =-=[10]-=-, that the Galois representation (1) factors through Gℓ. This outer action induces one on the unipotent completion P of the geometric fundamental group of P1 − {0, 1, ∞}. One can define the filtration... |

12 |
Some arithmetic aspects of Galois actions on the pro-p fundamental group of P
- Ihara
- 1999
(Show Context)
Citation Context ... Gr >0 Iℓ GQ, which are sometimes called Soulé elements. He studies the non-vanishing of the Lie brackets of these elements and asks [11, p. 300] whether the σj generate (Gr >0 Iℓ GQ) ⊗ Qℓ freely. In =-=[12]-=-, he proves that if (Gr >0 Iℓ GQ) ⊗ Qℓ is free, then it is generated by the Soulé elements. He also shows that the σj generate an open subgroup of the image of the homomorphism Gℓ → Out ( π (ℓ) /L m+1... |

9 |
Galois Rigidity of pro-l Pure Braid Groups of Algebraic Curves
- Nakamura, Takao
- 1998
(Show Context)
Citation Context ...(ℓ) → Aut(Γ un /Q ⊗Q Qℓ) → Out(Γ un /Q ⊗Q Qℓ). Proof. This follows from Theorem A.4 as there is a natural action of Aut Γ (ℓ) on Aut(Γ (ℓ)un /Qℓ ). From this we recover a result of Nakamura and Takao =-=[16]-=-. Corollary A.9. Suppose that k is a subfield of C and that X is a variety over k. Then for each k-rational point x of X, there is a natural homomorphism Gal( ¯ k/k) → Autπ1(X(C), x) un /Q ⊗ Qℓ. Appen... |

8 |
Multiple zeta-values, Galois groups, and geometry of modular varieties
- Goncharov
- 2001
(Show Context)
Citation Context ...mod 691 when ℓ = 691. This, and his conjecture about the stable derivation algebra [12, p. 11, Conjecture 1], imply that Gr >0 Iℓ Gℓ is not generated by σ2m+1. 10.5. Goncharov’s Conjecture. Goncharov =-=[7]-=- considers the varieties XN := P 1 − {0, µN, ∞} where µN denotes the group of Nth roots of unity. Take F to be Q(µN) and S to be the set of primes in OF that lie over Nℓ. As above, let v = −→ 01, the ... |

6 |
On ranks of the stable derivation algebra and Deligne’s problem
- Tsunogai
- 1997
(Show Context)
Citation Context ...m + 1) where the copy of Qℓ(2m + 1) is spanned by σ2m+1. Ihara [11] also uses power series methods to establish nonvanishing results for some brackets of the σj. Improvements can be found in [14] and =-=[24]-=-). Ihara considered the problem of whether the Zℓ-Lie algebra Gr >0 Iℓ Gℓ is generated by such σ2m+1. He found a mysterious congruence m≥1 2[σ3, σ9] − 27[σ5, σ7] ≡ 0 mod 691 when ℓ = 691. This, and hi... |

4 |
Hodge-de Rham theory of relative Malcev completion
- Hain
- 1998
(Show Context)
Citation Context ... universal for homomorphisms of Γ into such “negatively weighted extensions” of S by a prounipotent group. It is a variant of Deligne’s notion of the relative Malcev completion, which is developed in =-=[8, 9]-=-. Weighted completions arise naturally in Galois theory in many contexts, the simplest of which is the following: (i) ℓ is a prime number, (ii) Γ = Gℓ, the Galois group of the maximal algebraic extens... |

4 |
The Galois representations arising from P1 − {0, 1, ∞} and Tate twists of even degree
- Ihara
(Show Context)
Citation Context ...1.1. For all rational primes ℓ, the graded Lie algebra (Gr >0 Iℓ GQ) ⊗ Qℓ is generated by elements s3, s5, s7, . . . where s2n+1 ∈ Gr 2n+1 Iℓ GQ. In fact, combining this result with the work of Ihara =-=[11]-=-, one can see (cf. Remark 10.7) that there is a GQ-equivariant isomorphism H1(Gr >0 Iℓ GQ, Qℓ) ∼ = ⊕ H 1 cts(GQ, Qℓ(2n + 1)) ∗ ⊗ Qℓ(2n + 1) ∼ = ⊕ Qℓ(2n + 1). n≥1 The value of Soulé’s element [22] of H... |

4 | Relations between K2 - Tate - 1976 |

3 |
On the Galois image in derivation of π1 of the projective line minus three points
- Matsumoto
- 1993
(Show Context)
Citation Context ... = ⊕ Qℓ(2m + 1) where the copy of Qℓ(2m + 1) is spanned by σ2m+1. Ihara [11] also uses power series methods to establish nonvanishing results for some brackets of the σj. Improvements can be found in =-=[14]-=- and [24]). Ihara considered the problem of whether the Zℓ-Lie algebra Gr >0 Iℓ Gℓ is generated by such σ2m+1. He found a mysterious congruence m≥1 2[σ3, σ9] − 27[σ5, σ7] ≡ 0 mod 691 when ℓ = 691. Thi... |

3 |
Letter to
- Sharifi
- 2000
(Show Context)
Citation Context .... This theorem is also true when ℓ = 2, something which appears to be known to specialists but for which there seems to be no appropriate reference. An explicit proof has been written down by Sharifi =-=[21]-=-. Theorem B.1 follows directly from this computation when S is the set of primes lying over ℓ. For the general case, it is enough to prove the following statement. r2WEIGHTED COMPLETION OF GALOIS GRO... |

2 |
Nilpotent torsion-free groups, (Russian) Izvestiya
- Malcev
- 1949
(Show Context)
Citation Context ...ipotent. Since unipotent groups are torsion free, and since Γ/Dm k Γ is a subgroup of P(k)/LmP(k), it follows that Dm k Γ ⊇ RmΓ. It follows from Quillen’s version [17, Appendix A] of Malcev’s Theorem =-=[13]-=- that there is a unipotent k-group U that contains Γ/RmΓ as a Zariski dense subgroup. The density implies that the length of U is < m. The homomorphism Γ → U(Qℓ) induces a homomorphism P → U which fac... |