Toward equivariant Iwasawa theory.

*(English)*Zbl 1014.11066Let \(K/k\) 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 \(X_\infty\).

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 \(X_\infty\).

Reviewer: T.Nguyen Quang Do (Besançon)

##### MSC:

11R23 | Iwasawa theory |

11R33 | Integral representations related to algebraic numbers; Galois module structure of rings of integers |