##
**Bloch-Kato conjecture and Main Conjecture of Iwasawa theory for Dirichlet characters.**
*(English)*
Zbl 1044.11095

Suppose given a smooth projective variety \(X \to \text{ Spec}(\mathbb Q)\) and integers \(i,j \in \mathbb Z.\) To the “motive” \(M = h^{i}(X) (j)\) are attached a complex function \(L (M, s)\) and all the data conjecturally describing the leading coefficient \(L^{\ast}(M)\) of this function at \(s = 0.\) Here we need not appeal to any more elaborate notion of a motive than that which identifies \(M\) with a collection of “realizations” by cohomological spaces: \(M_{B} = H^{i} (X(\mathbb C), \mathbb Q) (j)\) (Betti), \(M_{dR} = H^{i}_{dR} (X/{\mathbb Q}) (j)\) (de Rham), \(H^{0}_{f}(M)\) and \(H^{1}_{f} (M)\) (“motivic”). Very roughly speaking, Beilinson’s conjecture describes \(L^{\ast}(M) \in {\mathbb R}^{\ast}\) in terms of regulators up to a rational factor, and the Bloch-Kato conjecture gives the rational factor in terms of so-called Tamagawa numbers (commutative algebraic groups being replaced by motivic cohomology groups). The reformulation by Fontaine and Perrin-Riou of this last conjecture in terms of complexes could be viewed (roughly speaking again) as some kind of very elaborate global-local principle:

- taking the “alternate probelowduct” of determinants of some (six) cohomological realizations of \(M,\) one constructs a one-dimensional \({\mathbb Q}\)-vector space \(\Delta_{f}(M)\) called the “fundamental line”.

- in the archimedean world, tensorization of these cohomological spaces by \({\mathbb R}\) over \({\mathbb Q}\) should give an exact sequence via period and cycle maps, height pairing, and the Beilinson regulator, hence an isomorphism \(\iota_{\infty}\;:\;{\mathbb R} \simeq \Delta_{f}(M) \otimes {\mathbb R}.\) Beilinson’s conjecture then amounts to \(r_{\infty}(L^{\ast}(M)^{-1}) \in \Delta_{f} (M) \otimes 1.\)

- in the \(p\)-adic world, there should be natural isomorphisms \(H^0_f (M) \otimes \mathbb Q_p \simeq H^0_f (\mathbb Q, M_p)\) (cycle class map) and \(H^1_f (M) \otimes \mathbb Q_p \simeq H^1_f (\mathbb Q, M_p)\) (Chern class map). Let \(S\) be a finite set of primes containing \(p, \infty\) and primes of bad reduction. Introducing cohomology with compact support, one gets an isomorphism \(\iota_p: \Delta_f (M) \otimes \mathbb Q_p \simeq \text{ Det}_{\mathbb Q_p} R \Gamma_c (\mathbb Z [{1 \over S}], M_p).\) Let \(T_p \subset M_p\) any Galois stable \(\mathbb Z_p\)-lattice. The \(p\)-part of the Bloch-Kato conjecture then amounts to \(\mathbb Z_p \iota_p \iota^{-1}_{\infty} (L^{\ast} (M)^{-1}) = \text{ Det}_{{\mathbb Z}_{p}}\;R\;\Gamma_c (\mathbb Z [{1 \over S}], T_p).\) This statement is independent of the choice of \(S\) and \(T_p.\) Its validity for all \(p\) determines \(L^{\ast}(M)\) up to sign.

Number theorists are mainly interested in motives arising from \(X = \text{ Spec }F\) (\(F\) a number field), equipped or not with additional structures. For example, if \(M = h^0 (\text{Spec }F) (0),\) then \(L(M, s)\) is the Dedekind zeta function. In this case, the Bloch-Kato conjecture for \(M\) and \(M(1)\) is nothing but the class number formula (modulo the functional equation); for negative twists \(j,\) it is equivalent to the (cohomological version of the) Lichtenbaum conjecture, which was proved, for abelian \(F\) and up to powers of 2, by M. Kolster, T. Nguyen Quang Do and V. Fleckinger [Duke Math. J. 84, 679–717 (1996; Zbl 0863.19003)] (but with some errors; see remarks below); for positive twists, and still in the abelian case, the conjecture was proved by P. D. Benois and T. Nguyen Quang Do [Ann. Sci. Éc. Norm. Supér., IV. Sér. 35, 641–672 (2002; Zbl 1125.11351)], by showing compatibility with the functional equation.

In the present paper, the authors prove the Bloch-Kato conjecture, up to powers of 2, for the so-called Dirichlet motives \(h(\chi).\) With our previous notations, \(h(\chi) = h^0 (\text{Spec}(\mathbb Q(\mu_N))(j),\) \(N\) being a multiple of the conductor of the Dirichlet character \(\chi,\) the motive coming equipped with a “projector”, the usual idempotent associated with \(\chi^{-1}.\) Actually, \(h^{0} (\text{spec}(\mathbb Q(\mu_N))(j) = \oplus_{\chi} h(\chi),\) for all characters \(\chi\) of conductor dividing \(N.\) The starting point is the same as in all previously proved cases of the conjecture: use special (= Beilinson’s) elements and their archimedean realizations (related to polylogarithm values) and \(p\)-adic realizations (related to cyclotomic elements in Iwasawa theory). Knowing the \(p\)-part of the Bloch-Kato conjecture for all fields in the cyclotomic tower is equivalent to the Iwasawa Main Conjecture formulated appropriately. But the accomplishment of this program is fraught with great technical difficulties, mainly because one needs to show the Main Conjecture in the non semi-simple case (i.e. \(p\) can divide the order of \(\chi),\) and perform “descent” even in the presence of “trivial zeros” of \(p\)-adic \(L\)-functions. An overview of how the authors manage to overcome these difficulties is given on p. 412:

(1) Prove directly the Bloch-Kato conjecture for \(h^{0} (\text{spec }F)(j),\;j=0, 1,\) for any number field \(F\) (this amounts to the analytical class number formula).

(2) Use Euler system methods to establish a divisibility statement for Iwasawa modules in the case \(\chi (-1) = (-1)^j.\) Prove the Iwasawa Main Conjecture from it by applying the usual “class number trick”, but replacing the class number formula by the validity of the Bloch-Kato conjecture for \(F = \mathbb Q (\mu_N)\) and \(j = 1.\)

(3) Using Kato’s explicit reciprocity law, prove the Bloch-Kato conjecture for \(j \geq 1\) and \(\chi (-1) = (-1)^j\) from the Main Conjecture.

(4) Using the precise understanding of the regulators of cyclotomic elements, prove the Bloch-Kato conjecture for \(j \leq 0\) and \(\chi (-1) = (-1)^j\) form the Main Conjecture. Note that the argument does not work for \(j= 0\) and \(\chi (p) = 1,\) the case of “trivial zeros” (see (7)).

(5) By the compatibility with the functional equation (shown in appendix B), deduce the Bloch-Kato conjecture for the remaining cases \(j \neq 0, 1.\)

(6) From an equivariant version of (5), deduce a second version of the Main Conjecture, which in turn allows to show the Bloch-Kato conjecture for \(j = 0, 1,\) except in the case of trivial zeroes.

(7) The last exceptional case, namely \(j = 0\) and \(\chi (p) = 1,\) follows again by the functional equation.

This is a clever piece of work. To put the results in perspective, we must compare them with previous, or concomittant, independent ones. The equivariant Kato-Burns-Flach conjecture provides a convenient framework, in which the motive comes equipped with “extra symmetries”, more precisely with the action of a semi-simple, finite dimensional \(\mathbb Q\)-algebra \(A,\) and a distinguished \(\mathbb Z\)-order \(\mathcal A\) of \(A.\) Then:

(i) For \(A = \mathbb Q\) and \(\mathcal A = \mathbb Z,\) the equivariant conjecture is nothing else but the Bloch-Kato conjecture for \(h^0 (\text{Spec }F)(j).\) Reviewer’s remark: The comments on the status of the Lichtenbaum conjecture (p. 396) are misleading. The erroneous Euler factors in Kolster-Nguyen-Fleckinger were due to a wrong statement on Galois generators of cyclotomic units, and were corrected in Benois-Nguyen, appendix A3. The reference to théorème A-2-3 is irrelevant (and the comments unfounded).

(ii) For \(G = \text{ Gal }(\mathbb Q(\mu_N)/{\mathbb Q}),\;A = \mathbb Z[G]\) and \(\mathcal A =\) the maximal order of \(A\) inside \(\mathbb Q[G],\) the equivariant conjecture is equivalent to the Bloch-Kato conjecture for all Dirichlet motives \(h(\chi)\) and all characters \(\chi\) of conductor dividing \(N.\)

(iii) For \(A = \mathcal A = \mathbb Z[G],\) the equivariant conjecture is now a theorem of Burns and Greither (to appear in Invent. Math.).

Note that by general functoriality, (iii) \(\Longrightarrow\) (ii) \(\Longrightarrow\) (i).

- taking the “alternate probelowduct” of determinants of some (six) cohomological realizations of \(M,\) one constructs a one-dimensional \({\mathbb Q}\)-vector space \(\Delta_{f}(M)\) called the “fundamental line”.

- in the archimedean world, tensorization of these cohomological spaces by \({\mathbb R}\) over \({\mathbb Q}\) should give an exact sequence via period and cycle maps, height pairing, and the Beilinson regulator, hence an isomorphism \(\iota_{\infty}\;:\;{\mathbb R} \simeq \Delta_{f}(M) \otimes {\mathbb R}.\) Beilinson’s conjecture then amounts to \(r_{\infty}(L^{\ast}(M)^{-1}) \in \Delta_{f} (M) \otimes 1.\)

- in the \(p\)-adic world, there should be natural isomorphisms \(H^0_f (M) \otimes \mathbb Q_p \simeq H^0_f (\mathbb Q, M_p)\) (cycle class map) and \(H^1_f (M) \otimes \mathbb Q_p \simeq H^1_f (\mathbb Q, M_p)\) (Chern class map). Let \(S\) be a finite set of primes containing \(p, \infty\) and primes of bad reduction. Introducing cohomology with compact support, one gets an isomorphism \(\iota_p: \Delta_f (M) \otimes \mathbb Q_p \simeq \text{ Det}_{\mathbb Q_p} R \Gamma_c (\mathbb Z [{1 \over S}], M_p).\) Let \(T_p \subset M_p\) any Galois stable \(\mathbb Z_p\)-lattice. The \(p\)-part of the Bloch-Kato conjecture then amounts to \(\mathbb Z_p \iota_p \iota^{-1}_{\infty} (L^{\ast} (M)^{-1}) = \text{ Det}_{{\mathbb Z}_{p}}\;R\;\Gamma_c (\mathbb Z [{1 \over S}], T_p).\) This statement is independent of the choice of \(S\) and \(T_p.\) Its validity for all \(p\) determines \(L^{\ast}(M)\) up to sign.

Number theorists are mainly interested in motives arising from \(X = \text{ Spec }F\) (\(F\) a number field), equipped or not with additional structures. For example, if \(M = h^0 (\text{Spec }F) (0),\) then \(L(M, s)\) is the Dedekind zeta function. In this case, the Bloch-Kato conjecture for \(M\) and \(M(1)\) is nothing but the class number formula (modulo the functional equation); for negative twists \(j,\) it is equivalent to the (cohomological version of the) Lichtenbaum conjecture, which was proved, for abelian \(F\) and up to powers of 2, by M. Kolster, T. Nguyen Quang Do and V. Fleckinger [Duke Math. J. 84, 679–717 (1996; Zbl 0863.19003)] (but with some errors; see remarks below); for positive twists, and still in the abelian case, the conjecture was proved by P. D. Benois and T. Nguyen Quang Do [Ann. Sci. Éc. Norm. Supér., IV. Sér. 35, 641–672 (2002; Zbl 1125.11351)], by showing compatibility with the functional equation.

In the present paper, the authors prove the Bloch-Kato conjecture, up to powers of 2, for the so-called Dirichlet motives \(h(\chi).\) With our previous notations, \(h(\chi) = h^0 (\text{Spec}(\mathbb Q(\mu_N))(j),\) \(N\) being a multiple of the conductor of the Dirichlet character \(\chi,\) the motive coming equipped with a “projector”, the usual idempotent associated with \(\chi^{-1}.\) Actually, \(h^{0} (\text{spec}(\mathbb Q(\mu_N))(j) = \oplus_{\chi} h(\chi),\) for all characters \(\chi\) of conductor dividing \(N.\) The starting point is the same as in all previously proved cases of the conjecture: use special (= Beilinson’s) elements and their archimedean realizations (related to polylogarithm values) and \(p\)-adic realizations (related to cyclotomic elements in Iwasawa theory). Knowing the \(p\)-part of the Bloch-Kato conjecture for all fields in the cyclotomic tower is equivalent to the Iwasawa Main Conjecture formulated appropriately. But the accomplishment of this program is fraught with great technical difficulties, mainly because one needs to show the Main Conjecture in the non semi-simple case (i.e. \(p\) can divide the order of \(\chi),\) and perform “descent” even in the presence of “trivial zeros” of \(p\)-adic \(L\)-functions. An overview of how the authors manage to overcome these difficulties is given on p. 412:

(1) Prove directly the Bloch-Kato conjecture for \(h^{0} (\text{spec }F)(j),\;j=0, 1,\) for any number field \(F\) (this amounts to the analytical class number formula).

(2) Use Euler system methods to establish a divisibility statement for Iwasawa modules in the case \(\chi (-1) = (-1)^j.\) Prove the Iwasawa Main Conjecture from it by applying the usual “class number trick”, but replacing the class number formula by the validity of the Bloch-Kato conjecture for \(F = \mathbb Q (\mu_N)\) and \(j = 1.\)

(3) Using Kato’s explicit reciprocity law, prove the Bloch-Kato conjecture for \(j \geq 1\) and \(\chi (-1) = (-1)^j\) from the Main Conjecture.

(4) Using the precise understanding of the regulators of cyclotomic elements, prove the Bloch-Kato conjecture for \(j \leq 0\) and \(\chi (-1) = (-1)^j\) form the Main Conjecture. Note that the argument does not work for \(j= 0\) and \(\chi (p) = 1,\) the case of “trivial zeros” (see (7)).

(5) By the compatibility with the functional equation (shown in appendix B), deduce the Bloch-Kato conjecture for the remaining cases \(j \neq 0, 1.\)

(6) From an equivariant version of (5), deduce a second version of the Main Conjecture, which in turn allows to show the Bloch-Kato conjecture for \(j = 0, 1,\) except in the case of trivial zeroes.

(7) The last exceptional case, namely \(j = 0\) and \(\chi (p) = 1,\) follows again by the functional equation.

This is a clever piece of work. To put the results in perspective, we must compare them with previous, or concomittant, independent ones. The equivariant Kato-Burns-Flach conjecture provides a convenient framework, in which the motive comes equipped with “extra symmetries”, more precisely with the action of a semi-simple, finite dimensional \(\mathbb Q\)-algebra \(A,\) and a distinguished \(\mathbb Z\)-order \(\mathcal A\) of \(A.\) Then:

(i) For \(A = \mathbb Q\) and \(\mathcal A = \mathbb Z,\) the equivariant conjecture is nothing else but the Bloch-Kato conjecture for \(h^0 (\text{Spec }F)(j).\) Reviewer’s remark: The comments on the status of the Lichtenbaum conjecture (p. 396) are misleading. The erroneous Euler factors in Kolster-Nguyen-Fleckinger were due to a wrong statement on Galois generators of cyclotomic units, and were corrected in Benois-Nguyen, appendix A3. The reference to théorème A-2-3 is irrelevant (and the comments unfounded).

(ii) For \(G = \text{ Gal }(\mathbb Q(\mu_N)/{\mathbb Q}),\;A = \mathbb Z[G]\) and \(\mathcal A =\) the maximal order of \(A\) inside \(\mathbb Q[G],\) the equivariant conjecture is equivalent to the Bloch-Kato conjecture for all Dirichlet motives \(h(\chi)\) and all characters \(\chi\) of conductor dividing \(N.\)

(iii) For \(A = \mathcal A = \mathbb Z[G],\) the equivariant conjecture is now a theorem of Burns and Greither (to appear in Invent. Math.).

Note that by general functoriality, (iii) \(\Longrightarrow\) (ii) \(\Longrightarrow\) (i).

Reviewer: Thong Nguyen Quang Do (Besançon)

### MSC:

11R23 | Iwasawa theory |

19F27 | Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects) |

11G55 | Polylogarithms and relations with \(K\)-theory |

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\textit{A. Huber} and \textit{G. Kings}, Duke Math. J. 119, No. 3, 393--464 (2003; Zbl 1044.11095)

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