## Toward equivariant Iwasawa theory.(English)Zbl 1014.11066

Let $$K/k$$ be a Galois extension of totally real number fields, $$K_\infty$$ the cyclotomic $$\mathbb{Z}_\ell$$-extension of $$K$$ ($$\ell$$ being an odd prime), $$\Gamma= \text{Gal} (K_\infty/K)$$, $$G_\infty= \text{Gal} (K_\infty/k)$$, $$X_\infty$$ the Galois group over $$K_\infty$$ of the maximal Abelian $$\ell$$-extension of $$K_\infty$$ which is unramified outside a finite set of primes containing the Archimedean primes, the primes above $$\ell$$ and those which ramify in $$K/k$$. One could say that the main goal of Iwasawa theory is the description of the Galois module structure of $$X_\infty$$ in algebraic-analytical terms. The Usual Main Conjecture (UMC, or Wiles’ theorem) does this over the Iwasawa algebra $$\mathbb{Z}_\ell [[\Gamma]]$$. An Equivariant Main Conjecture (EMC) should do the same thing over the complete algebra $$\mathbb{Z}_p [[G_\infty]]$$ and should provide a powerful tool towards important conjectures in Galois module theory such as the “lifted root number conjecture” or more generally, the “equivariant Tamagawa number conjecture”.
In their “tripod” paper [Mem. Am. Math. Soc. 748 (2002; Zbl 1002.11082)], the authors had proposed a formulation of such an EMC in terms of the localization exact sequence relating $$K_0$$ and $$K_1$$. More precisely, let $$\partial$$ be the connecting map $$K_1 (Q\mathbb{Z}_\ell [[G_\infty]])\to K_0T (\mathbb{Z}_\ell [[G_\infty]])$$, where $$Q(.)$$ denotes the total quotient ring and $$K_0T(.)$$ the Grothendieck group of the category of finitely generated torsion modules of finite projective dimension. In their “tripod” paper, the authors constructed:
i) A Galois module invariant $$\mho_S\in K_0T (\mathbb{Z}_\ell [[G_\infty]])$$ attached to $$X_\infty$$. The fancy notation is meant to recall that this construction is mimicking, at the infinite level, the “lifted $$\Omega$$-construction” at the finite level.
ii) Assuming that $$G_\infty$$ is Abelian, an analytical element $$\Theta_S\in K_1(Q \mathbb{Z}_\ell [[G_\infty]])$$, built from $$\ell$$-adic $$L$$-functions.
The analytical object is related to the algebraic object by means of the conjectural equation $$\partial(\Theta_S)= \mho_S$$. In the present paper, the authors prove this EMC assuming the vanishing of the $$\mu$$-invariant attached to $$X_\infty$$. The essential tools are localization at height one prime ideals and, of course, the UMC (Wiles’ theorem) which reads, in this setting: $$\partial (\varepsilon \Theta_S)= \mho_S$$, where $$\varepsilon$$ is a certain unit of the integral closure of $$\mathbb{Z}_\ell [[G_\infty]]$$ in its total quotient ring.
Note that K. Kato had proposed, in terms of perfect complexes, a (probably equivalent) EMC, which has been proved recently by D. Burns and C. Greither for cyclotomic fields [“On the equivariant Tamagawa number conjecture for Tate motives”, preprint, 10/2000]. Note also that the above equation $$\partial (\Theta_S)= \mho_S$$ can be translated in terms of Fitting ideals and used to compute the Fitting ideals of some Iwasawa modules related to $$X_\infty$$.

### MSC:

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

Zbl 1002.11082
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