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An equivariant main conjecture in Iwasawa theory and applications. (English) Zbl 1330.11070
Inspired by the Deligne-Tate proof of the Brumer-Stark conjecture in function fields via \(p\)-adic realizations of Picard 1-motives, the authors introduce “abstract 1-motives” and study their “\(p\)-adic realizations”, in [Int. Math. Res. Not. 2012, No. 5, 986–1036 (2012; Zbl 1254.11063)] for smooth projective curves defined over arbitrary algebraically closed fields, and in the present article for Iwasawa modules over \(\mathbb Z_p\)- fields \(\mathcal K\), i.e., cyclotomic \(\mathbb Z_p\)- extensions of number fields. We do not recall the general definitions and properties of such 1-motives, the main interest lying rather in the construction of their \(p\)-adic realizations \(T_p (\mathcal M^{\mathcal K}_{\mathcal S,\mathcal T} )\). Here \(\mathcal K\) is a \(\mathbb Z_p\)- field \((p \neq 2)\) such that \(\mu_{\mathcal K} = 0, \mathcal S\) and \(\mathcal T\) are two finite sets of finite primes in such that \(\mathcal T\cap (\mathcal S \cup \mathcal S_p )= \emptyset\), \(\mathcal S_p\) being the set of primes of \(\mathcal K\) above \(p ; \mathcal M^{\mathcal K}_{\mathcal S,\mathcal T}\) denotes the 1-motive, for which one can define a Tate module \(T_p (\mathcal M^{\mathcal K}_{\mathcal S,\mathcal T})\), which has an arithmetic meaning. More specifically, suppose that \(K/k\) is a Galois extension of number fields, where \(K\) is CM and \(k\) is totally real, and \(\mathcal K\) is the cyclotomic \(\mathbb Z_p\)-extension of \(K (p \neq 2)\); let \(G = \text{Gal}(K/k),\mathcal G = \text{Gal}(\mathcal K/k), \Gamma = \text{Gal}(\mathcal K/K), \Lambda = \mathbb Z_p [[ \Gamma]]\). Assuming that \(\mathcal S\) and \(\mathcal T\) are \(\mathcal G\)-equivariant, \(T_p (\mathcal M^{\mathcal K}_{\mathcal S,\mathcal T} )\) is naturally by construction a \(\mathbb Z_p [[\mathcal G]]\)-module, and one of the authors’ main results when \(\mathcal T \neq \emptyset\) is that \(T_p (\mathcal M^{\mathcal K}_{\mathcal S,\mathcal T} )^{-}\) has projective dimension 1 over \(\mathbb Z_p [[\mathcal G]]^{-}\) (unless it is trivial of course). The \(p\)-adic realization can be related to more classical Iwasawa modules.
Denote by \(\mathcal X_\mathcal S\) the Galois group of the maximal abelian pro-\(p\)-extension of \(K\) which is unramified outside \(\mathcal S \cup \mathcal S_p\), endowed with the usual canonical \(\mathbb Z_p [[\mathcal G]]\)-module structure, and put \(\mathcal X^{\ast}_\mathcal S = \operatorname{Hom}(\mathcal X_\mathcal S , \mathbb Z_p )\). If moreover \(K\) contains \(\mu_p\) , there is an exact sequence of \(\mathbb Z_p [[\mathcal G]]\)-modules \[ 1 \rightarrow \mathbb Z_p (1) T_p (\Delta_{\mathcal K,\mathcal T} )^{-} \rightarrow T_p (\mathcal M^{\mathcal K}_{\mathcal S,\mathcal T} )^{-} \rightarrow T_p (\mathcal M^{\mathcal K}_{\mathcal S,\emptyset} )^{-} = \mathcal X^{ +}_{\mathcal S} (-1)^{\ast} \rightarrow 1 , \tag{*} \] with \(\Delta_{\mathcal K,\mathcal T} := \oplus_{\mathcal T} \kappa (v)^\times \), \(\kappa( v) =\) the residue field at \(v\). Note that \(\mathcal X^{+}_{\mathcal S} (-1)^\ast\) is very rarely of finite projective dimension over \(\mathbb Z_p [[\mathcal G]]^{-}\). When \(\mathcal G\) is abelian, it follows from the projective dimension 1 property that the initial Fitting ideal \(\text{Fit}_{\mathbb Z_ p} [[\mathcal G]]^{-} (\mathcal T_p (\mathcal M^{\mathcal K}_ {\mathcal S,\mathcal T} )^{-} )\) is principal when \(\mathcal T \neq \emptyset\). The authors propose and prove the following Equivariant Main Conjecture (EMC for short) which gives a distinguished generator of this ideal. Let \(\mathcal S\) and \(T\) in \(k\) sit under \(\mathcal S\) and \(\mathcal T\). Recall that for a finite set \(\mathcal S\) of primes in \(k\), containing the Archimedean primes and the primes which ramify in \(K/k\), one can define the so called \(\mathcal S\)-incomplete \(G\)-equivariant complex \(L\)-function \(\Theta_{\mathcal S,K/k} (s) := \sum_{\chi}L{}_{\mathcal S} (\chi^{-1} , s)\cdot e_\chi\), where \(\chi\) runs through all the irreducible \(C\)-valued characters of \(\mathcal G\) and \(e_\chi\) is the usual idempotent of \(\mathcal G\) attached to \(\chi\). For \(T \neq \emptyset\) and \(T \cap (\mathcal S \cup \mathcal S_p ) = \emptyset\), let \(\delta_{\mathcal T,K/k} (s) := \prod_{\mathcal T} (1- \sigma ^{-1}_v \cdot (Nv)^{1-s })\), where \(\sigma_v\) denotes the Frobenius automorphism at \(v\), and define the \(\mathcal S\)-incomplete \(T\)-modified \(G\)-equivariant \(L\)-function \(\Theta_{\mathcal S,\mathcal T,K/k} (s) := \delta_{\mathcal T,K/k} (s) \cdot\Theta_{\mathcal S,K/k} (s)\). Taking projective limits along the tower \(\mathcal K/K\), one gets, for any integer \(m \geq 1\), elements \(\Theta^\infty_\mathcal S (1 - m )\) and \(\Theta^\infty_{\mathcal S,\mathcal T} (1 - m )\) which live a priori in the total ring of fractions of \(\mathcal O[[\mathcal G]]\) because of Siegel’s theorem (here \(\mathcal O\) is a finite extension of \(\mathbb Z_p\) containing the values of all the characters of \(G\)). But the additional set of primes \(\mathcal T\) allows to eliminate the denominators, so that actually \(\Theta^\infty_{\mathcal S,\mathcal T} (1 - m) \in \mathbb Z_p [[ \mathcal G]]\). Starting from Wiles’ theorem (= the classical non equivariant Main Conjecture), the authors show that \(\text{Fit} \mathbb Z_{p [[\mathcal G]]}- (T_p (\mathcal M^{\mathcal K}_{\mathcal S,\mathcal T} ) ^{-}) = (\Theta^\infty_{\mathcal S, \mathcal T} (0))\). For the relationship with other existing EMC’s, see the remark below.
As applications of their EMC, the authors use (co)descent to prove refinements of the Brumer-Stark conjecture and the Coates-Sinnott conjecture away from the prime 2:
(i) Keeping the current hypotheses (in particular \(K\) is CM and \(k\) is totally real,), assume further that \(\mathcal S_p \subseteq \mathcal S,\mathcal T \cap \mathcal S = \emptyset\), and either \(T\) contains at least two primes of distinct residual characteristics or \(K\) contains no non-trivial root of unity which is congruent to 1 modulo all primes in \(T\). The authors show by descent that the following property \(\overline{\mathbf {BrSt}}( K/k, \mathcal S, \mathcal T, p)\) holds: \(\Theta_{\mathcal S,\mathcal TK/k} (0) \in \text{Fit}_{\mathbb Z_p [\mathcal G]} (A^{-}_{\mathcal K,\mathcal T })^\vee )\), where \(A_{\mathcal K,\mathcal T}\) is the \(p\)-part of the generalized \(\mathcal T\)-ideal class group (in Galois terms, unramified outside T and at most tamely ramified inside) and its dual is endowed with the covariant \(G\)-action.
(ii) Let \(K/k\) be an abelian extension and \(\mathcal S\) be as before, and suppose moreover that \(\mathcal S\) contains \(\mathcal S_p\). If \(k\) is totally real, assume that the \(\mu\)-invariant of the maximal CM-subfield of \(K(\zeta_p )\) vanishes. Then the following property \(\overline{\mathbf{CS}} ( K/k, \mathcal S, p, n)\) holds: \[ \text{Ann}_ {\mathbb Z_p [\mathcal G]} (H^1_{\text{et}} (\mathcal O_{K,\mathcal S} [1/p], \mathbb Z_p ( n))_{\text{tors}} )\cdot \Theta_{\mathcal S,K/k} (1 n) = e_n (K/k)\cdot (H^2_{\text{et}} (\mathcal O_{K,\mathcal S} [1/p], \mathbb Z_p ( n)) \]
for any \(n \geq 2\). Here \(\mathcal O_{\mathcal K,\mathcal S}\) is the ring of \(\mathcal S\)-integers of \(K\), and \(e_n (K/k)=\prod\frac12 (1 + (-1)^n \sigma_v)\) if \(k\) is totally real (where \(\sigma_v\) is the generator of the decomposition subgroup \(G_v\) for all Archimedean primes \(v\) of \(k\)), \(0\) otherwise. Note that because of the order of vanishing of the \(\mathcal S\)-incomplete \(L\)-function at \(s = 1 - n\), it suffices to prove the statement for \(k\) totally real, and the authors proceed by successive reduction steps to bring the problem back to the case where \(K\) is imaginary, which is the usual formulation of the (étale) Coates-Sinnott conjecture.
Remark. The authors point out that A. Nickel [Proc. Lond. Math. Soc. (3) 106, No. 6, 1223–1247 (2013; Zbl 1273.11155)], has given a precise relationship between their EMC and the work of J. Ritter and A. Weiss [Manuscr. Math. 109, No. 2, 131–146 (2002; Zbl 1014.11066)]. Actually Nickel has obtained non abelian generalizations of the results reviewed here, using his “non-commutative Fitting invariants” in the framework of the non-commutative EMC proved by Ritter and Weiss, and independently by Kakde (let us call it EMC1). On the way, he has shown that this EMC 1 is equivalent to a non-commutative version of the authors’ EMC. It seems worthwhile to explain the main point of Nickel’s argument [loc. cit., §5]. On one side, for \(\mathcal K/k\) totally real, Ritter and Weiss construct in [loc. cit.] a complex \(\mathcal C_{\mathcal S} (\mathcal K/k)\) of \(\mathbb Z_p [[ \mathcal G]]\)-modules of projective dimension \(\leq 1\), concentrated in degrees \(-1, 0\), such \(H^{ -1} (\mathcal C_{ \mathcal S }(K/k))=\mathcal X_{\mathcal S}\) and \(H^0 (\mathcal C_{\mathcal S} (\mathcal K/k))= \mathbb Z_p\). On the other side, in the setting of the exact sequence (*) above, one can apply to (*) the functor \(\alpha(-)(1)\), where \(\alpha(-)\) is the Iwasawa adjoint, to produce a complex \(\tilde{C} (\mathcal K^{+} /k) : \alpha(\mathcal T_p (\mathcal M^{\mathcal K}_ {\mathcal S,\mathcal T})^{-})(1) \to \alpha \mathcal T_p (\Delta_{\mathcal K,\mathcal T} )^{-1} (1)\), concentrated in degrees \(-1, 0\), such that \(H^{ -1} (\tilde{C} (\mathcal K^{ +} /k)) = \mathcal X^{ +}_{\mathcal S}\) and \(H^0 (\tilde{C} (\mathcal K^{ +} /k))= \mathbb Z_p\). The vanishing of the \(\mu\)-invariant then implies that the Fitting invariants over \(\mathbb Z_p [[\mathcal G^{ +} ]]\) of the complexes \(C_{\mathcal S} (\mathcal K^{ +} /k)\) and \(\tilde{C}( \mathcal K + /k)\) coincide. According to the formalism of Nickel’s Fitting invariants, this means that \(\text{ Fit}_{\mathbb Z_p [[ G]]^{-}} (\mathcal T_p (\mathcal M^{\mathcal K}_{\mathcal S,\mathcal T} )^{-} )\) has the desired generator. The argument can be reversed to show that this generation property holds if and only if EMC1 holds [loc. cit., remark 3.5]. In the commutative case, Nickel’s formalism equally shows that EMC1 is equivalent to the formulation (also in terms of Fitting ideals) given by the reviewer in [J. Théor. Nombres Bordx. 17, No. 2, 643–668 (2005; Zbl 1098.11054)], and used by him to show by descent the classical (étale) Coates-Sinnott conjecture .

11R23 Iwasawa theory
11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
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