The purpose of this paper is to prove the main conjecture of noncommutative Iwasawa theory for p-adic Lie extensions for totally real fields, for an odd prime p, assuming Iwasawa's conjecture on vanishing of the cyclotomic µ-invariant for certain CM extension of the base field.
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We begin a study of mth Chern classes and mth characteristic symbols for Iwasawa modules which are supported in codimension at least m. This extends the classical theory of characteristic ideals and their generators for Iwasawa modules which are torsion, i.e., supported in codimension at least 1. We apply this to an Iwasawa module constructed from an inverse limit of p-parts of ideal class groups of abelian extensions of an imaginary quadratic field. When this module is pseudo-null, which is conjecturally always the case, we determine its second Chern class and show that it has a characteristic symbol given by the Steinberg symbol of two Katz p-adic L-functions.In the language of the classical main conjectures, an element f ∈ Q × such that ν 1 (f ) = c 1 (M ) is a characteristic power series for M when R is a formal power series ring. A main conjecture for M posits that there is such an f which can be constructed analytically, e.g. via p-adic L-functions.The key to generalizing this is to observe that Q × is the first Quillen K-group K 1 (Q) and ν 1 is a tame symbol map. To try to relate c m (M ) to analytic invariants for arbitrary m, one can consider elements of K m (Q) which can be described by symbols involving m-tuples of elements of Q associated to L-functions. The homomorphism ν 1 is replaced by a homomorphism ν m involving compositions of tame symbol maps. We now describe one way to do this.Suppose that η = (η 0 , . . . , η m ) is a sequence of points of Y with codim(η i ) = i and such that η i+1 lies in the closure η i of η i for all i < m. Denote by P m (Y ) the set of
In the setting of Brumer-Stark, ρ 1 is the trivial character and ρ 2 is a totally odd character χ. When p is odd, χ(c) = −1 ≡ 1 (mod p), where c denotes complex conjugation. Ribet's method therefore carries through. However when p = 2, we may have χ ≡ 1 (mod 2), and a new construction is necessary. This issue was handled in our paper [12], whose main theorem is a general version of Ribet's Lemma that applies in the residually indistinguishable case.The next issue at p = 2 concerns the Ritter-Weiss module ∇ Σ ′ Σ (H) appearing in our previous work. It turns out that this module is not what appears in the Galois representations that we study locally at 2. We must alter the module ∇ Σ ′ Σ (H) by modifying the local conditions at the archimedean places. On the analytic side this modification is reflected in the power of 2 in the denominator of the constant terms of Eisenstein series for F .These modifications have an important effect on our constructions with group-ring valued Hilbert modular forms. In the case that the extension H/F is not ramified at any primes above 2, the set Σ that one would naturally like to take is empty; however it is an important assumption in our study of Ritter-Weiss modules that Σ is non-empty. One is therefore led to introduce an auxiliary prime l that does not divide the conductor of the extension H/F into the set Σ. The inclusion of this prime leads to the presence of oldforms in our constructions with Hilbert modular forms, which in turn yields a Hecke algebra T that is not reduced. Working with a non-reduced Hecke algebra involves new ideas to show that the Galois representations we construct have the desired properties. Therefore, the main technical contribution of this paper is the introduction of our new generalized Ritter-Weiss module ∇ Σ ′ Σ (H) in which the depletion set Σ need not contain the archimedean primes, as well as a delicate handling of the constructions with modular forms and Galois representations that ensue; the connection between our Ritter-Weiss modules and Galois cohomology is handled in a more conceptual way than in [11], and requires new results on class formations.We conclude the introduction by stating precisely our main arithmetic results. First we recall the statement of the Brumer-Stark conjecture, and next we describe our stronger results involving Ritter-Weiss modules.
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