## Equivariant Bloch-Kato conjecture and non-abelian Iwasawa main conjecture

Venue: | in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 149–162, Higher Ed |

Citations: | 9 - 0 self |

### BibTeX

@INPROCEEDINGS{Huber_equivariantbloch-kato,

author = {A. Huber and G. Kings},

title = {Equivariant Bloch-Kato conjecture and non-abelian Iwasawa main conjecture},

booktitle = {in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 149–162, Higher Ed},

year = {},

publisher = {Press}

}

### OpenURL

### Abstract

In this talk we explain the relation between the (equivariant) Bloch-Kato conjecture for special values of L-functions and the Main Conjecture of (nonabelian) Iwasawa theory. On the way we will discuss briefly the case of Dirichlet characters in the abelian case. We will also discuss how “twisting ” in the non-abelian case would allow to reduce the general conjecture to the case of number fields. This is one the main motivations for a non-abelian Main

### Citations

101 |
Higher regulators and values of L-functions
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- 1984
(Show Context)
Citation Context ...Let M ∨ be the dual motive. In the p-adic realization it corresponds to the dual Galois module. We denote by H1 M (Z, M(k)) the “integral” motivic cohomology of the motive M in the sense of Beilinson =-=[1]-=-. For any finite Galois extension K/Q with Galois group G, let Q[G] be the group ring of G. It is a non-commutative ring with center denoted Z(Q[G]). We consider the deformation Q[G] ⊗ M := h0 (K) ⊗ M... |

97 |
A.: Class fields of abelian extensions
- Mazur, Wiles
- 1984
(Show Context)
Citation Context ...The zeta distribution LS(G∞, V (χ) ∨ , 1 − k) generates det −1 Λ H1 (Z[1/S], Λ ⊗ Tp(χ)(k)) ⊗ detΛ H 2 (Z[1/S], Λ ⊗ Tp(χ)(k)). Remark: This is a reformulation of the main theorem of Mazur and Wiles in =-=[29]-=-. There is an extension to the case of totally real fields by Wiles [37] and an equivariant version by Burns and Greither [6]. Non-critical case: χ(−1) = (−1) k−1 . Here H 1 M (Z, E[Gn] ⊗ V (χ)(k)) ha... |

81 |
Methods of Representation Theory, vol
- Curtis, Reiner
- 1987
(Show Context)
Citation Context ...n element of the abelian group K1(A) for all rings A. Let Bn = Im K1(Zp[Gn]) → K1(Qp[Gn]). By assumption δ(n) ∈ Bn. There is a system of short exact sequences 0 → SK1(Zp[Gn]) → K1(Zp[Gn]) → Bn → 0 By =-=[11]-=- 45.22 the groups SK1(Zp[Gn]) are finite. The system of these groups is automatically Mittag-Leffler. Hence we get a surjective map lim ←− K1(Zp[Gn]) → lim Bn ←− The system (δ(n))n has a preimage ( ˜ ... |

81 |
p-adic Hodge theory and values of zeta functions of modular forms. Cohomologies p-adiques et applications arithmetiques
- Kato
(Show Context)
Citation Context ...onjecture [34]. 2) In [25] the (absolute) Bloch-Kato conjecture for V (ψ) is deduced from this under the condition that H 2 (Z[1/S], Tp(ψ)(k)) is finite (fulfilled for almost all k for fixed p). Kato =-=[24]-=- has investigated the case of elliptic curves over Q and the cyclotomic tower. His approach to the Birch-Swinnerton-Dyer conjecture uses the ideaEquiv. Bloch-Kato conjecture and non-abelian Iwasawa M... |

76 | A First Course in Noncommutative Rings - Lam - 2001 |

55 |
The “main conjectures” of Iwasawa theory for imaginary quadratic
- Rubin
- 1991
(Show Context)
Citation Context ...is a canonical isomorphism of determinants ( 1 detΛ H (Z[1/S], Λ ⊗ Tp(ψ)(k))/ek(G∞, tp(ψ)) ) ∼ = detΛ H 2 (Z[1/S], Λ ⊗ Tp(ψ)(k)). Remark: 1) This is a reformulation of Rubin’s Iwasawa Main Conjecture =-=[34]-=-. 2) In [25] the (absolute) Bloch-Kato conjecture for V (ψ) is deduced from this under the condition that H 2 (Z[1/S], Tp(ψ)(k)) is finite (fulfilled for almost all k for fixed p). Kato [24] has inves... |

55 |
The Iwasawa conjecture for totally real fields
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- 1990
(Show Context)
Citation Context .../S], Λ ⊗ Tp(χ)(k)) ⊗ detΛ H 2 (Z[1/S], Λ ⊗ Tp(χ)(k)). Remark: This is a reformulation of the main theorem of Mazur and Wiles in [29]. There is an extension to the case of totally real fields by Wiles =-=[37]-=- and an equivariant version by Burns and Greither [6]. Non-critical case: χ(−1) = (−1) k−1 . Here H 1 M (Z, E[Gn] ⊗ V (χ)(k)) has E[Gn]-rank 1. It is a theorem of Borel (resp. Soulé) that rD ⊗ R (resp... |

37 | Tamagawa numbers for motives with (noncommutative) coefficients
- Burns, Flach
(Show Context)
Citation Context ...e the element with ρ-component the leading coefficient at s = k of the E ⊗Q Cvalued L-functions LS(V (ρ)⊗M, s) without the Euler factors at S. Then LS(G, M, k) ∗ has actually values in Z(R[G]) ∗ (see =-=[4]-=- Lemma 7) and is independent of the choice of E. We will always consider LS(G, M, k) ∗ as an element in Z(R[G]) ⊂ R[G]. Remark: In [22] Kato uses a different description of this equivariant L-function... |

37 |
Le déterminant de la cohomologie” in Current Trends in Arithmetical Algebraic Geometry
- DELIGNE
- 1985
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Citation Context ...function. 2.3. Non-commutative determinants We follow the point of view of Burns and Flach. Let A be a (possibly noncommutative) ring and V (A) the category of virtual objects in the sense of Deligne =-=[12]-=-. V (A) is a monoidal tensor category and has a unit object 1A. Moreover it is a groupoid, i.e., all morphisms are isomorphisms. There is a functor detA : {perfect complexes of A-modules and isomorphi... |

36 |
Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions via BdR
- Kato
- 1993
(Show Context)
Citation Context ...sense correspond to A-generators of L. 2.4. Formulation of the conjecture The original conjecture dates back to Beilinson [1] and Bloch-Kato [3]. The idea of an equivariant formulation is due to Kato =-=[23]-=- and [22]. Fontaine and Perrin-Riou4 A. Huber and G. Kings gave a uniform formulation for mixed motives and all values of L-functions at all integer values [14], [15]. The generalization to non-abeli... |

33 |
Autour des conjectures de Bloch et Kato: cohomologie galoisienne et valeurs de fonctions L
- Fontaine, Perrin-Riou
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Citation Context ...riant formulation is due to Kato [23] and [22]. Fontaine and Perrin-Riou4 A. Huber and G. Kings gave a uniform formulation for mixed motives and all values of L-functions at all integer values [14], =-=[15]-=-. The generalization to non-abelian coefficients is due to Burns and Flach [4]. For simplicity of exposition, we restrict to values at very negative integers. In the absolute case this coincides with ... |

31 | Euler Systems
- Rubin
- 2000
(Show Context)
Citation Context ...of Λ-determinants ( 1 detΛ H (Z[1/S], Λ ⊗ Tp(χ)(k))/ck(G∞, tp(χ)) ) ∼ = detΛ H 2 (Z[1/S], Λ ⊗ Tp(χ)(k)). Remark: For p ∤ ord(χ) this is a consequence of theorem 5.1..1 and was shown directly by Rubin =-=[33]-=- with Euler system methods. The restriction at the order of χ is removed in Burns-Greither [5] and Huber-Kings [20] by different methods. The Tamagawa number conjecture for V (χ)(r) (and hence for h 0... |

25 |
Higher regulators and Hecke L-series of imaginary quadratic fields
- Deninger
- 1989
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Citation Context ... ψ, which has a zero of order 1 at 2 − k, where k ≥ 2. Let S = Np, where N is the conductor of ψ and let Kn := K(E[p n ]). It is not known if H 1 M (OK, K[Gn] ⊗V (ψ)(k)) has K[Gn]-rank 1 but Deninger =-=[13]-=- shows that rD ⊗ R is surjective and that the Beilinson conjecture holds. It is a result of Kings [25] that the image in étale cohomology of the zeta element δp(Gn, V (ψ), 2 − k) given by Beilinson’s ... |

23 | Classical motivic polylogarithm according to Beilinson and
- Huber, Wildeshaus
- 1998
(Show Context)
Citation Context ...case: χ(−1) = (−1) k−1 . Here H 1 M (Z, E[Gn] ⊗ V (χ)(k)) has E[Gn]-rank 1. It is a theorem of Borel (resp. Soulé) that rD ⊗ R (resp. rp ⊗ Qp) is an isomorphism. By a theorem of BeilinsonDeligne (see =-=[21]-=- or [19]), the image of δp(Gn, V (χ), k) under rp is given by where ck(Gn, tp(χ)) −1 ⊗ tp(χ)(k − 1), ck(Gn, tp(χ)) ∈ H 1 (Z[1/S], Op[Gn] ⊗ Tp(χ)(k)) is a twist of a cyclotomic unit and tp(χ)(k − 1) is... |

16 |
Valeurs spéciales des fonctions L des motifs, Astérisque 206
- Fontaine
- 1992
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Citation Context ...equivariant formulation is due to Kato [23] and [22]. Fontaine and Perrin-Riou4 A. Huber and G. Kings gave a uniform formulation for mixed motives and all values of L-functions at all integer values =-=[14]-=-, [15]. The generalization to non-abelian coefficients is due to Burns and Flach [4]. For simplicity of exposition, we restrict to values at very negative integers. In the absolute case this coincides... |

16 | On the structure theory of the Iwasawa algebra of a p-adic Lie group
- Venjakob
(Show Context)
Citation Context ...is the classical Iwasawa algebra. For the tower of p n -torsion points of E, the Iwasawa algebra was studied by Coates and Howson [8], [9]. Modules over such algebras are studied recently by Venjakob =-=[36]-=- and by Coates-Schneider-Sujatha [10]. We are concerned with the complex of Λ-modules RΓ(Z[1/S], Λ⊗Zp Tp(k)) and (Λ ⊗Z TB(k − 1)) + . They are perfect complexes. Note that RΓ(Z[1/S], Λ ⊗Zp Tp(k)) = li... |

15 |
Euler characteristics and elliptic curves
- Coates, Howson
(Show Context)
Citation Context ...ular and local ring. For the cyclotomic tower, Λ ∼ = Zp[G1][[t]] is the classical Iwasawa algebra. For the tower of p n -torsion points of E, the Iwasawa algebra was studied by Coates and Howson [8], =-=[9]-=-. Modules over such algebras are studied recently by Venjakob [36] and by Coates-Schneider-Sujatha [10]. We are concerned with the complex of Λ-modules RΓ(Z[1/S], Λ⊗Zp Tp(k)) and (Λ ⊗Z TB(k − 1)) + . ... |

15 |
Correction to: “Twisted Sunits, p-adic class number formulas, and the Lichtenbaum conjectures
- Kolster, Do, et al.
- 1996
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Citation Context ... from theorems 5.1..1 and 5.1..2, see Burns-Greither [5] or Huber-Kings [20]. Previous partial results were proved in Mazur-Wiles [29], Wiles [37], Kato [22], [23], Kolster-Nguyen Quang Do-Fleckinger =-=[26]-=- and Benois-Nguyen Quang Do[2]. We would like to stress that the strategy 3.3. is used in Huber-Kings [20] to prove theorems 5.1..1, 5.1..2 and the Tamagawa number conjecture from the class number for... |

14 | Modules over Iwasawa algebras
- Coates, Schneider, et al.
(Show Context)
Citation Context ... the tower of p n -torsion points of E, the Iwasawa algebra was studied by Coates and Howson [8], [9]. Modules over such algebras are studied recently by Venjakob [36] and by Coates-Schneider-Sujatha =-=[10]-=-. We are concerned with the complex of Λ-modules RΓ(Z[1/S], Λ⊗Zp Tp(k)) and (Λ ⊗Z TB(k − 1)) + . They are perfect complexes. Note that RΓ(Z[1/S], Λ ⊗Zp Tp(k)) = lim ←− RΓ(OKn[1/S], Tp(k)) where OKn is... |

14 | Bloch-Kato conjecture and Main Conjecture of Iwasawa theory for Dirichlet characters
- Huber, Kings
(Show Context)
Citation Context ...r p ∤ ord(χ) this is a consequence of theorem 5.1..1 and was shown directly by Rubin [33] with Euler system methods. The restriction at the order of χ is removed in Burns-Greither [5] and Huber-Kings =-=[20]-=- by different methods. The Tamagawa number conjecture for V (χ)(r) (and hence for h 0 (F)(r) with F an abelian number field) can be deduced from theorems 5.1..1 and 5.1..2, see Burns-Greither [5] or H... |

13 | Degeneration of l-adic Eisenstein classes and of the elliptic polylog
- Huber, Kings
- 1999
(Show Context)
Citation Context ...−1) = (−1) k−1 . Here H 1 M (Z, E[Gn] ⊗ V (χ)(k)) has E[Gn]-rank 1. It is a theorem of Borel (resp. Soulé) that rD ⊗ R (resp. rp ⊗ Qp) is an isomorphism. By a theorem of BeilinsonDeligne (see [21] or =-=[19]-=-), the image of δp(Gn, V (χ), k) under rp is given by where ck(Gn, tp(χ)) −1 ⊗ tp(χ)(k − 1), ck(Gn, tp(χ)) ∈ H 1 (Z[1/S], Op[Gn] ⊗ Tp(χ)(k)) is a twist of a cyclotomic unit and tp(χ)(k − 1) is a gener... |

12 |
Iwasawa theory and p-adic Hodge theory
- Kato
- 1993
(Show Context)
Citation Context ...rs at S. Then LS(G, M, k) ∗ has actually values in Z(R[G]) ∗ (see [4] Lemma 7) and is independent of the choice of E. We will always consider LS(G, M, k) ∗ as an element in Z(R[G]) ⊂ R[G]. Remark: In =-=[22]-=- Kato uses a different description of this equivariant L-function. 2.3. Non-commutative determinants We follow the point of view of Burns and Flach. Let A be a (possibly noncommutative) ring and V (A)... |

11 |
le residue de la fonction zeta p-adique d’un corps de nombers, C.R.Aca.Sci.Paris287 series A(1978),183–188
- Serre, Sur
(Show Context)
Citation Context ...t is not clear to us if the class of f/g ∈ K1(D) = (D∗) ab is uniquely determined by the sequence fng−1 n . In the abelian case this is true and f/g is a generalization of Serre’s pseudo measure (cf. =-=[35]-=-). In this case the complexes RΓ(OKn[1/S], Tp(k)) are torsion. Hence the complex RΓ(Z[1/S], Λ ⊗ Tp(k)) = ←− lim nRΓ(OKn[1/S], Tp(k)) is bounded and its cohomology is Λ-torsion (see [18]). The main con... |

9 | Fragments of the GL2 Iwasawa theory of elliptic curves without complex multiplication - Coates - 1997 |

9 | The Tamagawa Number Conjecture for CM elliptic curves
- Kings
(Show Context)
Citation Context ...et Kn := K(E[p n ]). It is not known if H 1 M (OK, K[Gn] ⊗V (ψ)(k)) has K[Gn]-rank 1 but Deninger [13] shows that rD ⊗ R is surjective and that the Beilinson conjecture holds. It is a result of Kings =-=[25]-=- that the image in étale cohomology of the zeta element δp(Gn, V (ψ), 2 − k) given by Beilinson’s Eisenstein symbol is given by ek(Gn, tp(ψ)) −1 ⊗ tp(ψ) , where ek(Gn, tp(ψ)) ∈ H 1 (Z[1/S], Op[Gn]⊗Tp(... |

8 |
Iwasawa theory for motives, in L-Functions and Arithmetic
- Greenberg
- 1991
(Show Context)
Citation Context ...gy over K∞ to a module of p-adic analytic nature. It would be interesting to compare her approach with the above. c) A Main Conjecture for motives and the cyclotomic tower was formulated by Greenberg =-=[16]-=-, [17]. Ritter and Weiss consider the case of the cyclotomic tower over a finite non-abelian extension [32]. Proposition 3.2..2 (see section 6.) The equivariant Bloch-Kato conjecture for M, k and all ... |

7 | Equivariant Weierstrass preparation and Values of L-functionsat negativeintegers
- Burns, Greither
(Show Context)
Citation Context ...Remark: This is a reformulation of the main theorem of Mazur and Wiles in [29]. There is an extension to the case of totally real fields by Wiles [37] and an equivariant version by Burns and Greither =-=[6]-=-. Non-critical case: χ(−1) = (−1) k−1 . Here H 1 M (Z, E[Gn] ⊗ V (χ)(k)) has E[Gn]-rank 1. It is a theorem of Borel (resp. Soulé) that rD ⊗ R (resp. rp ⊗ Qp) is an isomorphism. By a theorem of Beilins... |

6 |
p-adic representations arising from descent on Abelian varieties
- Harris
- 1979
(Show Context)
Citation Context ...o measure (cf. [35]). In this case the complexes RΓ(OKn[1/S], Tp(k)) are torsion. Hence the complex RΓ(Z[1/S], Λ ⊗ Tp(k)) = ←− lim nRΓ(OKn[1/S], Tp(k)) is bounded and its cohomology is Λ-torsion (see =-=[18]-=-). The main conjecture 3.2..1 takes the following form: Conjecture 4.2..1 Let M be an Artin motive, k > 1, S, G∞ as before (in particular G∞ pro-p and without p-torsion) and Q[Gn] ⊗ M(k) critical for ... |

5 |
L-functions and Tamagawa numbers of motives,The Grothendieck Festschrift
- Bloch, Kato
- 1990
(Show Context)
Citation Context ...mmutative case. Generators of detA X = (L, 0) in the above sense correspond to A-generators of L. 2.4. Formulation of the conjecture The original conjecture dates back to Beilinson [1] and Bloch-Kato =-=[3]-=-. The idea of an equivariant formulation is due to Kato [23] and [22]. Fontaine and Perrin-Riou4 A. Huber and G. Kings gave a uniform formulation for mixed motives and all values of L-functions at al... |

4 |
Iwasawa theory and p-adic deformations of motives” in Motives
- Greenberg
- 1991
(Show Context)
Citation Context ...r K∞ to a module of p-adic analytic nature. It would be interesting to compare her approach with the above. c) A Main Conjecture for motives and the cyclotomic tower was formulated by Greenberg [16], =-=[17]-=-. Ritter and Weiss consider the case of the cyclotomic tower over a finite non-abelian extension [32]. Proposition 3.2..2 (see section 6.) The equivariant Bloch-Kato conjecture for M, k and all Gn is ... |

4 |
p-adic L-functions and p-adic representations
- Perrin-Riou
- 2000
(Show Context)
Citation Context ...See section 5. for more details. Remark: a) The conjecture is independent of the choice of lattice TB. The correction factor (Λ ⊗ TB(k − 1)) + compensates different choices of lattice. b) Perrin-Riou =-=[31]-=- has defined a p-adic L-function and stated a Main Conjecture for motives in the abelian case. She starts at the other side of the functional equation, where the exponential map of Bloch-Kato comes in... |

3 | Quang Do, Universal distribution lattices for abelian number fields - Kolster, Nguyen - 2001 |

2 |
Thong Nguyen Quang Do, La conjecture de Bloch et Kato pour les motifs Q(m) sur un corps abélien
- Benois
- 2000
(Show Context)
Citation Context ...2, see Burns-Greither [5] or Huber-Kings [20]. Previous partial results were proved in Mazur-Wiles [29], Wiles [37], Kato [22], [23], Kolster-Nguyen Quang Do-Fleckinger [26] and Benois-Nguyen Quang Do=-=[2]-=-. We would like to stress that the strategy 3.3. is used in Huber-Kings [20] to prove theorems 5.1..1, 5.1..2 and the Tamagawa number conjecture from the class number formula. 5.2. Elliptic curves Let... |

2 |
On the equivariant Tamagawa conjecture for Tate motives
- Burns, Greither
- 2001
(Show Context)
Citation Context ...p(χ)(k)). Remark: For p ∤ ord(χ) this is a consequence of theorem 5.1..1 and was shown directly by Rubin [33] with Euler system methods. The restriction at the order of χ is removed in Burns-Greither =-=[5]-=- and Huber-Kings [20] by different methods. The Tamagawa number conjecture for V (χ)(r) (and hence for h 0 (F)(r) with F an abelian number field) can be deduced from theorems 5.1..1 and 5.1..2, see Bu... |

1 |
Euler characteristics and elliptic curves, Elliptic curves and modular forms
- Coates, Howson
- 1996
(Show Context)
Citation Context ...a regular and local ring. For the cyclotomic tower, Λ ∼ = Zp[G1][[t]] is the classical Iwasawa algebra. For the tower of p n -torsion points of E, the Iwasawa algebra was studied by Coates and Howson =-=[8]-=-, [9]. Modules over such algebras are studied recently by Venjakob [36] and by Coates-Schneider-Sujatha [10]. We are concerned with the complex of Λ-modules RΓ(Z[1/S], Λ⊗Zp Tp(k)) and (Λ ⊗Z TB(k − 1))... |

1 | On the structure of Selmer groups of p-adic Lie extensions, to appear - Ochi, Venjakob |

1 |
Toward equivariant Iwasawa theory, to appear: Manuscripta Mathematica
- Ritter, Weiss
(Show Context)
Citation Context ...above. c) A Main Conjecture for motives and the cyclotomic tower was formulated by Greenberg [16], [17]. Ritter and Weiss consider the case of the cyclotomic tower over a finite non-abelian extension =-=[32]-=-. Proposition 3.2..2 (see section 6.) The equivariant Bloch-Kato conjecture for M, k and all Gn is equivalent to the Main Conjecture for M, k and G∞. 3.3. Twisting Assume that Tp becomes trivial over ... |