Iwasawa theory and \(p\)-adic Hodge theory
The Iwasawa main conjecture for varieties (or motives) over arbitrary number fields is formulated using \(p\)-adic Hodge theory. The classical Iwasawa main conjecture gives a relation between the special values of partial Riemann zeta-functions to the Galois module structures of the ideal class groups of cyclotomic fields over \(\mathbb{Q}\). In this paper, the Iwasawa main conjecture for motives is formulated, where \(\mathbb{Q}\) is replaced by an arbitrary number field \(K\), and a cyclotomic field over \(\mathbb{Q}\) by a finite abelian extension \(L\) of \(K\), and partial Riemann zeta-functions by partial \(L\)-functions \(L_ S (M, \sigma\)-part, \(s)\) of a motive \(M\) over \(K\) for \(\sigma \in \text{Gal} (L/K)\). (Here \(S\) is a finite set of finite places of \(K\) including ``bad primes'', and \(L_ S\) means the \(L\)-function without the \(S\)-part.) The main conjecture of Iwasawa in this setting describes the special values of \(L_ S (M, \sigma\)-part, \(s)\) in terms of the \(\text{Gal} (L/K)\)- module structures of the etale cohomology of \(\text{Spec} ({\mathcal O}_{L,S})\) with coefficients in an etale sheaf coming from \(M\). (Here \({\mathcal O}_{L,S}\) denotes the ring of elements of \(L\) which are integral outside \(S\).) In order to state the main conjecture, some notation ought to be fixed. Let \(G=\text{Gal} (L/K)\), and let \(M\) be a pure motive over \(K\) of weight \(\leq - 1\). Let \(p\) be a prime fixed once and for all. Let \(S\) be a finite set of finite places of \(K\) containing all finite places at which \(M\) has bad reduction and all finite places which ramify in \(L/K\). Let \(L_ S (M,s) = \sum_{\mathfrak A} a({\mathfrak A}) N({\mathfrak A})^{-s}\) be the Hasse-Weil \(L\)-function of \(M\) without Euler factors for places in \(S\). Here the sum runs over all nonzero ideals \({\mathfrak A}\) of \({\mathcal O}_ K\), \(a({\mathfrak A}) \in \mathbb{Q}\), and \(N( {\mathfrak A})\) denotes the norm of \({\mathfrak A}\). For \(\sigma \in G\), define the partial \(L\)-function \[ L_ S (M, \sigma \text{-part},s) = \sum_{\mathfrak A} a({\mathfrak A}) N({\mathfrak A})^{- s} \] where \({\mathfrak A}\) runs over all nonzero ideals of \({\mathcal O}_ K\) which are prime to \(S\) and such that the Artin symbol \(((L/K)/{\mathfrak A}) \in G\) coincides with \(\sigma\). This partial \(L\)-function converges absolutely when \(\text{Re} (s) > wt(M)/2+1\), and is conjectured to extend to the whole complex plane as a meromorphic function. The main conjecture concerns the interpretation of the values of \(L_ S (M, \sigma\)-part,\(s)\) at \(s = 0\), in terms of the Galois module structures. The Galois modules needed for this purpose are constructed: the free \(\mathbb{Q} [G]\)-module \(\Phi^{\text{mot}}\) of rank 1 and an \(\mathbb{R} [G]\)-module isomorphism \[ \Phi^{\text{mot}} \otimes_ \mathbb{Q} \mathbb{R} \to \mathbb{R} [G]. \tag{*} \] Locally, the free \(\mathbb{Q}_ p [G]\)-module \(\Phi_ p^{\text{ar}}\) of rank 1 and a \(\mathbb{Q}_ p [G]\)-module isomorphism \[ \Phi^{\text{mot}} \otimes_ \mathbb{Q} \mathbb{Q}_ p \to \Phi_ p^{\text{ar}} \tag{**} \] are also constructed. With the appropriate Galois modules and isomorphisms at our disposal, an analytic zeta element is then defined as follows: \[ \zeta^{\text{an}}_{L/K,S} (M) = \sum_{\sigma \in G} L_ S(M, \sigma \text{-part}, 0) \cdot \sigma \in \mathbb{R} [G] \quad \text{ if } \text{ wt} (M) \leq-3. \] (Similarly when \(\text{wt} (M) = - 2\) (resp. \(-1)\), an analytic zeta element can be defined.) Conjecture. The image of \(\zeta^{\text{an}}_{L/K,S} (M)\) under isomorphism \((*)\) is contained in \(\Phi^{\text{mot}} \subset \Phi^{\text{mot}} \otimes_ \mathbb{Q} \mathbb{R}\). This conjecture is a generalization of the Beilinson conjecture which corresponds to the case \(L=K\). Assuming the validity of this conjecture, let \(\zeta^{\text{mot}}_{L/K,S} (M)\) denote the element of \(\Phi^{\text{mot}}\) corresponding to \(\zeta^{\text{an}}_{L/K,S} (M)\) via the isomorphism \((*)\). Passing to the local situation, let \(\zeta^{\text{an}}_{L/K,S} (M)_ p\) denote the image of \(\zeta^{\text{mot}}_{L/K,S} (M)\) under the isomorphism \((**)\) in \(\Phi^{\text{ar}}_ p\) and call it the \((p\)-adic) arithmetic zeta element. Now the Iwasawa main conjecture can be formulated. The Iwasawa main conjecture. Assume that \(p \neq 2\), and assume that \(S\) contains all places of \(K\) lying over \(p\), all finite places at which \(M\) has bad reduction and all finite places of \(K\) which ramify in \(L/K\). Then, the arithmetic zeta element \(\zeta^{\text{ar}}_{L/K,S} (M)_ p \in \Phi^{\text{ar}}_ p\) is a \(\mathbb{Z}_ p [G]\)-basis of \(\Phi^{\text{ar}}_{p,S,T}\). (Here \(T\) is a \(\text{Gal} (\overline K/K)\)-stable \(\mathbb{Z}_ p\)-lattice of \(V_ p (M)\) (the \(p\)-adic etale realization of \(M\) which is a \(\mathbb{Q}_ p\)-sheaf on \(\text{Spec} (K)_{\text{et}}\).)) It is proved that the main conjecture coincides with the classical Iwasawa conjecture in the case \(M = \mathbb{Q}(r)\) over \(\mathbb{Q}\) with \(r\) even and positive, and \(L\) real abelian extension of \(\mathbb{Q}\). Also it is shown that the main conjecture is a generalization of the Tamagawa number conjecture of motives which corresponds to the case \(L=K\) [\textit{S. Bloch} and \textit{K. Kato}, Grothendieck Festschrift, Vol. I, 333-400 (1990; Zbl 0768.14001)].
scientific article; zbMATH DE number 409639
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