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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 .

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