## Bloch-Kato conjecture and Main Conjecture of Iwasawa theory for Dirichlet characters (2002)

Citations: | 14 - 2 self |

### BibTeX

@MISC{Huber02bloch-katoconjecture,

author = {Annette Huber and Guido Kings},

title = {Bloch-Kato conjecture and Main Conjecture of Iwasawa theory for Dirichlet characters },

year = {2002}

}

### OpenURL

### Abstract

The Tamagawa number conjecture proposed by S. Bloch and K. Kato describes the “special values ” of L-functions in terms of cohomological data. The main conjecture of Iwasawa theory describes a p-adic L-function in terms of the structure of modules for the Iwasawa algebra. We give a complete proof of both conjectures (up to the prime 2) for L-functions attached to Dirichlet characters. We use the insight of Kato and B. Perrin-Riou that these two conjectures can be seen as incarnations of the same mathematical content. In particular, they imply each other. By a bootstrapping process using the theory of Euler systems and explicit reciprocity laws, both conjectures are reduced to the analytic class number formula. Technical problems with primes dividing the order of the character are avoided by using the correct cohomological formulation of the main conjecture.

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(Show Context)
Citation Context ... −1 )). By the main conjecture 5.1.3, it is indeed generated by ̺(Lp(χ, 1)) where ̺ : Ep[[Γ]] → Ep is the projection. The formula ̺(Lp(χ, 1)) = −(1 − χ(p)p −1 ) τ(χ) N ∑ χ −1 (σ)log p(1 − ζ σ N) from =-=[Wa]-=- theorem 5.18 together with τ(χ)τ(χ −1 ) = N implies the claim. 5.4 End of proof of the Bloch-Kato conjecture Theorem 5.4.1. The Bloch-Kato conjecture is true for all abelian Artin motives. Proof. Ext... |

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(Show Context)
Citation Context ...art of the equivariant conjecture for Q(µp n)+ for r ≥ 2 even. (3) The Beilinson conjecture for Dirichlet motives, that is, the first part of Conjecture 1.2.8, has already been proved by Beilinson in =-=[B]-=- (with some corrections in [N] and [E]). (4) Fontaine [F, §10] mentions the case of Dirichlet motives V (χ) but assumes that p 2 (resp., p) does divide N or φ(N) (where N is the conductor of χ) depend... |

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(Show Context)
Citation Context ...e sequence induce an isomorphism ( det� RƔloc(Tp(χ)(k))/�ck(χ)[−1] ) ( ∼ = det� RƔgl(Tp(χ) ∗ (1 − k)) ∗) . Remark. For p ∤ �(N) the result follows from the main conjecture as shown by Mazur and Wiles =-=[MW]-=-. Under this condition it was also proved directly by Rubin [Ru].BLOCH-KATO AND MAIN CONJECTURE FOR DIRICHLET CHARACTERS 433 Let Ɣ = Gal(Q∞/Q). As before, we decompose the cyclotomic character as the... |

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(Show Context)
Citation Context .... . . . . . . . . . . . . . . . . . 459 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 0. Introduction The Tamagawa number conjecture proposed by Bloch and Kato in =-=[BlK]-=- describes the “special values” of L-functions of motives in terms of cohomological data. Special value means here the leading coefficient of the Taylor series of an L-function at an integral point. T... |

62 |
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(Show Context)
Citation Context ...d kind. Then Tw(τ −1εk−1 ∞ )( f (T, θ)) = Lp(χ, 1 − k). (Note that by our normalization of the reciprocity homomorphism, ε∞ operates as −1 on 1-units and ε is the inverse of the Teichmüller ω used in =-=[W]-=-.) Definition 4.2.3 Let χ be a Dirichlet character with χ(−1) = (−1) k . For k > 1 we call the above Lp(χ, 1−k) ∈ Q(�) the p-adic L-function at k −1. For k ≤ 1 we define Lp(χ, 1− k) ∈ Q(�) as Tw(ε k−k... |

39 | Cohomologie de SLn et valeurs de fonctions zêta aux points entiers - Borel - 1977 |

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(Show Context)
Citation Context ...ture There is also an equivariant version of the Tamagawa number conjecture. It was first introduced by Kato for abelian groups in [K1]. The case of nonabelian groups is treated by Burns and Flach in =-=[BuF]-=-. We treat the abelian case. As before, let M be an Artin motive over Q with coefficients in E. Let K be an abelian extension of Q with Galois group G = Gal(K/Q). Let S be a finite set of primes conta... |

36 |
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(Show Context)
Citation Context ...s prime to p in this case. We get the expected index. In the ramified case n ≥ 1, we follow an approach also used by Benois and Nguyen in [BenNg] 2.4.5. We use Kato’s higher explicit reciprocity law, =-=[Ka2]-=- 2.1.7. The map sKn expp is given by ( α ↦→ β ↦→ 1 (r − 1)! p−rnTrKn/Qp (α · (1 − p r−1 φ) −1 D r ) (β(1 + T))(ζpn − 1) where we consider β · (1 + T) ∈ OK0[[T]] and D = (1 + T)d/dT, φ Frobenius on OK0... |

33 |
Autour des conjectures de Bloch et Kato: cohomologie galoisienne et valeurs de fonctions L
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(Show Context)
Citation Context ... Theorem 5.4.1). 2.2. Functional equation We first study the compatibility of the equivariant Bloch-Kato conjecture under the functional equation. The corresponding statements can already be found in =-=[FP]-=- in the absolute case (K = Q) and in [BuF] even for the case of noncommutative coefficients. Let K/Q be abelian and M an Artin motive with coefficients in E. As before, let X (K/Q) be the set of C-val... |

29 | On the equivariant Tamagawa number conjecture for Tate motives
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(Show Context)
Citation Context ...urns and M. Flach in general (see also Conjecture 1.5.2). A proof of the equivariant conjecture for Tate motives Q(r) and r ≤ 0 over abelian number fields and p ̸= 2 is given by Burns and Greither in =-=[BuG]-=-. Our Theorem 1.3.1 is equivalent to the equivariant conjecture with respect to the maximal order ˜Z[G] in Q[G], and hence for r ≤ 0, a consequence of the result of Burns and Greither. Using a key obs... |

27 |
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(Show Context)
Citation Context ... given by ( α ↦→ β ↦→ 1 (r − 1)! p−rm TrKm/Qp (α · (1 − p r−1 φ) −1 D r ) (β(1 + T))(ζpm − 1)tp(r) where we consider β · (1 + T) ∈ OK0[[T]] and D = (1 + T)d/dT, φ Frobenius on OK0[[T]] (see also also =-=[PR2]-=- for facts about Coleman power series.) We evaluate with α = pχ−1(ζN) and β = ˜ ζN ′ and get 1 (r − 1)!p rm Tr Km/Qp (p χ −1(ζN)pχym) tp(r) 54with ym = [(1 − φpr−1 ) −1˜ ζN ′(1 + T)](ζpm − 1). Using ... |

23 | Classical motivic polylogarithm according to Beilinson and
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(Show Context)
Citation Context ...olyvagin, Rubin, Kato, and PerrinRiou; (5) a comparison result on the image of cyclotomic elements in Deligne cohomology and p-adic cohomology known as Bloch-Kato Conjecture 6.2 from [BlK] (proved in =-=[HW]-=- following A. Beilinson and P. Deligne [BD]; a second proof is given in [HK]). Shortly before finishing the first version of this paper, we learned at the Obernai conference that D. Burns and C. Greit... |

22 |
Fonctions L p-adiques des représentations p-adiques, Astérisque 229
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(Show Context)
Citation Context ... that the Main conjecture for a Dirichlet character and a prime p should be used in order to prove the p-part of the Bloch-Kato conjecture (see e.g. the list in Fontaine’s Bourbaki talk [Fo] § 10 and =-=[PR3]-=-). There are two versions of the Main Conjecture depending on parity. Both are needed in order to deduce the Bloch-Kato conjecture in general. The first version involving the p-adic L-function was pro... |

18 |
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Citation Context ...ices to compute the corank of the cokernel. We compute the rank of H 1 (Z[ζN][1/p], Tp(χ) ∨ ) via the Euler Poincaré characteristic ∑ i≥0 (−1) i rkEp H i (Z[ζN][1/p], Vp(χ) ∨ ) which is equal to 0 by =-=[Ja]-=-, lemma 2. We know by [Sch] paragraph 7, Satz 2 that H 2 (Z[ζN][1/p], Ep) + is zero because Leopoldt’s conjecture is true for Q(ζN). This implies that H 2 (Z[ζN][1/p], Vp(χ) ∨ ) is zero, because it is... |

17 |
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Citation Context ...loch-Kato conjecture for Artin motives Our first aim is to present a formulation of the Bloch-Kato conjecture for Artin motives and their L-values at integral points. We follow Fontaine’s approach in =-=[F]-=-. 1.1. Artin motives, realizations, and regulators We work over the base field Q and with coefficients in some number field E. Let GQ be the absolute Galois group of Q and O be the ring of integers of... |

17 |
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(Show Context)
Citation Context ...at the elements cr(ζm)(χ) form an Euler system for for (Tp(χ), p). Euler systems were invented by Kolyvagin. A general theory of Euler systems was developed by Kato [Ka4], Perrin-Riou [PR2] and Rubin =-=[Ru]-=-. We follow Rubin, because his approach is closest to our setting. Let us recall the definition of an Euler system: Definition 3.1.3. An Euler system for (Tp(χ), pN) is a collection of elements for al... |

15 |
Motivic polylogarithms and Zagier’s conjecture, manuscript (version of
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Citation Context ...a comparison result on the image of cyclotomic elements in Deligne cohomology and p-adic cohomology known as Bloch-Kato Conjecture 6.2 from [BlK] (proved in [HW] following A. Beilinson and P. Deligne =-=[BD]-=-; a second proof is given in [HK]). Shortly before finishing the first version of this paper, we learned at the Obernai conference that D. Burns and C. Greither had been working independently on nearl... |

15 |
Correction to: “Twisted Sunits, p-adic class number formulas, and the Lichtenbaum conjectures
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(Show Context)
Citation Context ...tive values to the Lichtenbaum conjecture was shown independently by D. Benois and T. Nguyen Quang Do [BeN]. The Lichtenbaum conjecture is considered by M. Kolster, Nguyen Quang Do, and V. Fleckinger =-=[KNF]-=- and [KN], where the Euler factors are corrected. As far as we can see, these references prove the cohomological Lichtenbaum conjecture for abelian number fields up to an explicit set of bad primes. T... |

15 | Arithmetic Duality Theorems, Perspect - Milne - 1986 |

14 |
Values of zeta-functions, étale cohomology, and algebraic K-theory, Algebraic K-theory, II: “Classical” algebraic K-theory and connections with arithmetic
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Citation Context ...(OF[1/p], Zp(k))tors. Note finally that a finite Zp-module of order h has determinant h −1 Zp. Many cases were known before (see the discussion after Theorem 1.3.1). Lichtenbaum’s original conjecture =-=[L]-=- involves the Borel regulator rather than the Beilinson regulator, and K -groups rather than Galois cohomology. The missing ingredient is the Quillen-Lichtenbaum conjecture for i = 1, 2 and r ≥ 2. K2r... |

13 | Degeneration of l-adic Eisenstein classes and of the elliptic polylog
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- 1999
(Show Context)
Citation Context ...of cyclotomic elements in Deligne cohomology and p-adic cohomology known as Bloch-Kato Conjecture 6.2 from [BlK] (proved in [HW] following A. Beilinson and P. Deligne [BD]; a second proof is given in =-=[HK]-=-). Shortly before finishing the first version of this paper, we learned at the Obernai conference that D. Burns and C. Greither had been working independently on nearly the same problem. They prove th... |

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Citation Context ...e identify H 1 /f (F ⊗ Qp, Qp(1)) ∼ = ⊕ v|p Qp via the valuation maps. Thus the kernel of H 1 (OF[1/p], Zp(1)) → H 1 /f (F ⊗ Qp, Qp(1)) is (O ∗ F )∧ ∼ = O ∗ F ⊗ Zp. On the other hand, it follows from =-=[S]-=- that one has an exact sequence 0 → Cl(OF[1/p]) ⊗ Zp → H 2( OF[1/p], Zp(1) ) → H 0 (F ⊗ Qp, Qp/Zp) ∗ → H 0 (OF[1/p], Qp/Zp) ∗ → 0. Let j : Spec OF[1/p] → Spec OF. The long exact sequence in Zariskicoh... |

12 |
Iwasawa theory and p-adic Hodge theory
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Citation Context ...r the condition that p does not divide Φ(N). What is needed is to overcome the restrictions on the prime p in these results. What we want to advocate strongly is the insight, due to Kato in [Ka2] and =-=[Ka1]-=-, that the Bloch-Kato conjecture and the Main Conjecture are two incarnations of the same mathematical content. Knowing the p-part of the Bloch-Kato conjecture for all fields in the cyclotomic tower i... |

10 | Higher explicit reciprocity laws - Wiles - 1978 |

9 |
The Beĭlinson conjecture for algebraic number fields
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(Show Context)
Citation Context ...ure for Q(µp n)+ for r ≥ 2 even. (3) The Beilinson conjecture for Dirichlet motives, that is, the first part of Conjecture 1.2.8, has already been proved by Beilinson in [B] (with some corrections in =-=[N]-=- and [E]). (4) Fontaine [F, §10] mentions the case of Dirichlet motives V (χ) but assumes that p 2 (resp., p) does divide N or φ(N) (where N is the conductor of χ) depending on parity conditions. (5) ... |

8 |
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Citation Context ...Q(µp n)+ for r ≥ 2 even. (3) The Beilinson conjecture for Dirichlet motives, that is, the first part of Conjecture 1.2.8, has already been proved by Beilinson in [B] (with some corrections in [N] and =-=[E]-=-). (4) Fontaine [F, §10] mentions the case of Dirichlet motives V (χ) but assumes that p 2 (resp., p) does divide N or φ(N) (where N is the conductor of χ) depending on parity conditions. (5) The Bloc... |

8 |
Théorie d’Iwasawa et hauteurs p-adiques
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Citation Context ...y Galois-stable lattice in Vp. Theorem 2.2.2 (Kato). The above conjecture holds for abelian Artin motives and r ≥ 2. 14Proof. This was shown for the motive Q by Bloch and Kato in [BlKa]. Perrin-Riou =-=[PR1]-=- deduced it in the case that p is unramified in V . Benois and Nguyen Quang Do in [BenNg] show it for the full motive h 0 (F) of an abelian number field. Finally, there is unpublished work of Kato [Ka... |

6 |
Systèmes d’Euler p-adiques et théorie d’Iwasawa
- Perrin-Riou
- 1998
(Show Context)
Citation Context ...te duality; 2. the analytic class number formula for Dedekind-ζ-functions; 3. the explicit reciprocity law for Z(r) with r ≥ 2 over unramified cyclotomic fields as proved by Kato [Ka2] or Perrin-Riou =-=[PR2]-=-; 4. the theory of Euler systems due to Kolyvagin, Rubin, Kato and Perrin-Riou; 5. a comparison result on the image of cyclotomic elements in Deligne-cohomology and p-adic cohomology. It was first for... |

5 |
on the approach to Iwasawa theory for Hasse-Weil L-functions via BdR. I, Arithmetic algebraic geometry
- Lectures
- 1991
(Show Context)
Citation Context ... same restriction, it was also proved directly by K. Rubin using Kolyvagin’s Euler systems. The missing cases need to be addressed. What we would like to advocate strongly is the insight, due to Kato =-=[K2]-=-, [K1] and Perrin-Riou [P3], that the Bloch-Kato conjecture and the main conjecture are two incarnations of the same mathematical content. Knowing the p-part of the Bloch-Kato conjecture for all field... |

5 |
Euler systems, Iwasawa theory, and Selmer groups
- Kato
(Show Context)
Citation Context ...tomic elements. Our aim is to show that the elements cr (ζm)(χ) form an Euler system for (Tp(χ), pN). Euler systems were invented by Kolyvagin. A general theory of Euler systems was developed by Kato =-=[K4]-=-, Perrin-Riou [P4], and Rubin [Ru]. We follow Rubin because his approach is closest to our setting. Let us recall the definition of an Euler system. Definition 3.1.4 An Euler system for (Tp(χ), pN) is... |

5 | Higher explicit reciprocity - Wiles |

5 |
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- 1976
(Show Context)
Citation Context ...ic N 1 ( Li1−r(α) + (−1) 2 r Li1−r((α −1 )) ) = holds. ( −2πi N ) −r lim s→r Γ(s)(ζN(s, c) + (−1) r ζN(s, −c)) Proof. This is the well known functional equation of the Hurwitz zeta function, see e.g. =-=[Ap]-=- thm. 12.6. Note that for r ≥ 1 the value of Li1−r(α) is in fact an element in Q(ζN). Let LN(χ, s) be the L-function of χ with the Euler factors at the primes dividing N removed. The theorem immediate... |

4 |
Théorie d’Iwasawa et hauteurs p-adiques Invent
- Perrin-Riou
- 1992
(Show Context)
Citation Context ...n Mp. THEOREM 2.2.2 (Kato) Conjecture 2.2.1 holds for K/Q abelian, M an abelian Artin motive, r > 1, and p ̸= 2. Proof This was shown for the motive Q and K = Q by Bloch and Kato in [BlK]. PerrinRiou =-=[P1]-=- deduced it in the case where p is unramified in M. Benois and Nguyen Quang Do in [BeN] show it for the full motive h0 (F) of an abelian number field. Finally, there is unpublished work of Kato [K3] w... |

4 |
Comparison of the regulators of Beilinson and of Borel
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- 1988
(Show Context)
Citation Context ...fication H 1 ( D R, VR(r) ) ∼ = VDR(r)R/VB(r) + R ∼ = ( )+, VB(r − 1)R where + denotes the invariants under complex conjugation and where the second isomorphism is induced by C = R⊕R(−1). By [Bo] and =-=[R]-=-, r∞ ⊗R is an isomorphism for r > 1. For r = 0, the cycle class map to singular cohomology induces z : H 0 M (Spec Z, V ) → (VB ⊗ R) + = H 1 ( D R, VR(1) ) . Let S be a finite set of rational primes a... |

3 |
Iwasawa theory and p-adic Hodge theory”, Kodai Math
- Kato
- 1993
(Show Context)
Citation Context ...restriction, it was also proved directly by K. Rubin using Kolyvagin’s Euler systems. The missing cases need to be addressed. What we would like to advocate strongly is the insight, due to Kato [K2], =-=[K1]-=- and Perrin-Riou [P3], that the Bloch-Kato conjecture and the main conjecture are two incarnations of the same mathematical content. Knowing the p-part of the Bloch-Kato conjecture for all fields in t... |

3 |
et T. Nguy˜ên-Quang- ¯ D˜ô. Formules de classes pour les corps abéliens réels
- Belliard
(Show Context)
Citation Context ...rove this conjecture for abelian number fields. However, the formula given there has erroneous Euler factors. (There is work in progress to fix these and a mistake in the proof of the Main Conjecture =-=[BelNg]-=- and [KoNg]). It follows as a corollary of our main theorem 1.3.1. 6. In recent work [BenNg], Nguyen and Benois show how to reduce the the BlochKato conjecture for h 0 (F) and r ≥ 1 to the case of r ≤... |

3 |
Quang Do, Universal distribution lattices for abelian number fields
- Kolster, Nguyen
- 2001
(Show Context)
Citation Context ...njecture for abelian number fields. However, the formula given there has erroneous Euler factors. (There is work in progress to fix these and a mistake in the proof of the Main Conjecture [BelNg] and =-=[KoNg]-=-). It follows as a corollary of our main theorem 1.3.1. 6. In recent work [BenNg], Nguyen and Benois show how to reduce the the BlochKato conjecture for h 0 (F) and r ≥ 1 to the case of r ≤ 0 by showi... |

2 |
On higher p-adic regulators” in Algebraic K -Theory
- SOULÉ
- 1980
(Show Context)
Citation Context ...ements er (m) ∈ H 1( Q(ζm), Tp(χ)(r) ) , for all m ≥ 1, such that, for all primes l, ( coresQ(ζml)/Q(ζm) er (ml) ) { er (m) if l | mpN, = (1 − χ −1 (l)l r−1 )er (m) if l ∤ mpN. LEMMA 3.1.5 (see Soulé =-=[So1]-=-) The elements cr (ζm)(χ) form an Euler system for (Tp(χ)(r), pN). Proof This follows from [Ru, Proposition 2.4.2] and the norm compatibility of the cyclotomic units. We also need the following varian... |

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 ...e are the same as ours). Their proof 2is quite different, for example they use what was previously known on the Main conjecture and difficult computations of µ-invariants. Benois and Nguyen Quang Do =-=[BenNg]-=- have recently shown how the Bloch-Kato conjecture for Dedekind-ζ-functions can be reduced to the cohomological Lichtenbaum conjecture. The proof of the latter conjecture is claimed by Kolster, Nguyen... |

1 | Formules de classes pour les corps abéliens réels, Ann - BELLIARD, DO |

1 |
nombres de Tamagawa locaux et la conjecture de Bloch et Kato pour les motifs Q(m) sur un corps abélien, Ann
- BENOIS, DO, et al.
(Show Context)
Citation Context ... cohomological Lichtenbaum conjecture (see Theorem 1.4.1). The reduction of the conjecture at positive values to the Lichtenbaum conjecture was shown independently by D. Benois and T. Nguyen Quang Do =-=[BeN]-=-. The Lichtenbaum conjecture is considered by M. Kolster, Nguyen Quang Do, and V. Fleckinger [KNF] and [KN], where the Euler factors are corrected. As far as we can see, these references prove the coh... |

1 | Algebraic and etale K -theory - DWYER, FRIEDLANDER - 1985 |

1 |
Universal distribution lattices for abelian number fields, preprint
- KOLSTER, DO
- 2000
(Show Context)
Citation Context ...s to the Lichtenbaum conjecture was shown independently by D. Benois and T. Nguyen Quang Do [BeN]. The Lichtenbaum conjecture is considered by M. Kolster, Nguyen Quang Do, and V. Fleckinger [KNF] and =-=[KN]-=-, where the Euler factors are corrected. As far as we can see, these references prove the cohomological Lichtenbaum conjecture for abelian number fields up to an explicit set of bad primes. These are ... |

1 |
L p-adiques des représentations p-adiques, Astérisque 229
- Fonctions
- 1995
(Show Context)
Citation Context ...he main conjecture for a Dirichlet character and a prime p should be used in order to prove the p-part of the Bloch-Kato conjecture (see, e.g., the list in J.-M. Fontaine’s Bourbaki talk [F, §10] and =-=[P3]-=-). There are two versions of the main conjecture depending on parity. Both are needed in order to deduce the Bloch-Kato conjecture in general. The first version involving the p-adic L-function was pro... |

1 |
d’Euler p-adiques et théorie d’Iwasawa, Ann
- Systèmes
- 1998
(Show Context)
Citation Context ... duality; (2) the analytic class number formula for Dedekind ζ -functions; (3) the explicit reciprocity law for Z(r) with r ≥ 2 over unramified cyclotomic fields as proved by Kato [K2] or Perrin-Riou =-=[P4]-=-; (4) the theory of Euler systems due to V. Kolyvagin, Rubin, Kato, and PerrinRiou; (5) a comparison result on the image of cyclotomic elements in Deligne cohomology and p-adic cohomology known as Blo... |

1 | on étale K -theory: Applications” in Algebraic K -Theory (Oberwolfach - “Operations - 1980 |