Iwasawa theory and $p$-adic Hodge theory
Kodai Mathematical Journal
The aim of this paper is to formulate the Iwasawa main conjecture for varieties (or motives) over arbitrary number fields. See (4.9) for the statement of the conjecture, and (4.15) for "philosophical comments" on it. To formulate our conjecture, we need the p-aάic Hodge theory developed by Tate [Tα 2 ], Fontaine and Messing [FM], and Faltings [FαJ. The classical Iwasawa theory relates special values of partial Riemann zeta functions to the Galois module structures of the ideal class groups of
... clotomic extensions of Q. In our conjecture, we replace Q by an arbtirary number field K, a cyclotomic extension of Q by a finite abelian extension L of K, and partial Riemann zeta functions by partial L-functions L S (M, σ-part, s) of a motive M over K for σ^Gai(L/K). (Here S is a finite set of finite places of K including "bad places", and L s means the L-function without the S-part. For the meaning of the (τ-part, see (4.6).) Our conjecture relates special values of L(M, σpart, s) to the Gal(L//0-module structures of the etale cohomology of Spec(O L>< s) with coefficients in an etale sheaf coming from M, where OL.S is the ring of elements of L which are integral outside S. In \_BK~\, Bloch and the author formulated a conjecture on Tamagawa numbers of motives which generalizes the Birch Swinnerton-Dyer conjecture to general Hasse-Weil L-functions. The Iwasawa main conjecture in this paper is a natural generalization of the conjecture of [BK~\ (which is regarded as the case L-K in the above description). In our conjecture, we assume the variety is smooth proper over K but we put no other assumption on our variety. We do not assume the variety is of ordinary reduction at the prime number p in problem. We do not assume our motive is critical in the sense of Deligne [Ztei]. However this does not mean that we can define p-aάic L-functions without these assumptions. Our conjecture treats directly the special values of complex L-functions, but do not treat p-aάic interpolations of special values. In § l- § 3, we review known results and conjectures on p-aάic Hodge theory, /f-theory, and the duality in Galois cohomology of number fields. We formulate the Iwasawa main conjecture in §4. In §5 we consider the case of Tate motives over Q, and in §6 we show that in case of the motive Q(r) over Q with r even and positive, and L real abelian extensions of Q, our Iwasawa main conjecture Received November 4, 1992.