Iwasawa theory of totally real fields for certain non-commutative \(p\)-extensions. (English) Zbl 1196.11148
The author proves the Iwasawa main conjecture (IMC) for a certain family of non-commutative \(1\)-dimensional \(p\)-adic Lie extensions of totally real fields. Before we state the main theorem of the paper, we briefly recall the main ingredients of the IMC.
Let \(F\) be a totally real number field and \(F^{\infty} / F\) a Galois extension with Galois group \(G\) satisfying the following conditions: \(G\) is a compact \(p\)-adic Lie group; only finitely many primes of \(F\) ramify in \(F^{\infty}\); \(F^{\infty}\) is totally real and contains the cyclotomic \(\mathbb{Z}_p\)-extension of \(F\). Let \(\Lambda(G)\) be the Iwasawa algebra of \(G\) and let \(S\) be the canonical Ore set for \(G\). Then there is a localization sequence \[ K_1(\Lambda(G)) \rightarrow K_1(\Lambda(G)_S) \overset{\partial}{\rightarrow} K_0(\Lambda(G), \Lambda(G)_S) \rightarrow 0 \] and the canonical complex \[ C = C(F^{\infty} / F) = R\mathrm{Hom}(R \Gamma_{et}( \mathrm{Spec} (\mathfrak{o}_{F^{\infty}}[1 / \Sigma], \mathbb{Q}_p / \mathbb{Z}_p),\mathbb{Q}_p / \mathbb{Z}_p) \] defines a class \([C(F^{\infty} / F)] \in K_0(\Lambda(G), \Lambda(G)_S)\). Here, \(\Sigma\) is a finite set of primes of \(F\) containing all the infinite primes and all primes which ramify in \(F^{\infty}\). Let us denote the cyclotomic character by \(\kappa\) and the (complex) \(\Sigma\)-truncated Artin \(L\)-function of an Artin representation \(\rho\) of \(G\) by \(L_{\Sigma}(s, \rho)\). Then the IMC asserts that there is a (unique) element \(\xi_{F^{\infty}/F} \in K_1(\Lambda(G)_S)\) such that \(\partial(\xi_{F^{\infty}/F}) = - [C(F^{\infty} / F)]\) and “evaluation” of \(\xi_{F^{\infty}/F}\) at \(\rho \kappa^r\) yields an equality \(\xi_{F^{\infty}/F}(\rho \kappa^r) = L_{\Sigma}(1-r, \rho)\) for all natural numbers \(r\) divisible by \(p-1\).
The author considers \(1\)-dimensional \(p\)-adic Lie groups \(G = G^f \times \Gamma\), where \(\Gamma \simeq \mathbb{Z}_p\) and \[ G^f = \begin{pmatrix} 1 & \mathbb{F}_p & \mathbb{F}_p & \mathbb{F}_p \\ 0 & 1 & \mathbb{F}_p & \mathbb{F}_p \\ 0 & 0 & 1 & \mathbb{F}_p \\ 0 & 0 & 0 & 1\end{pmatrix} \] and proves the IMC (up to its uniqueness statement) for \(F^{\infty} / F\) provided that \(p \not=2,3\) and the Iwasawa \(\mu\)-invariant vanishes. The main tool of the proof is a theorem of David Burns which reduces the IMC to the commutative case under certain hypotheses. One of them is that each Artin representation of \(G\) is a \(\mathbb{Z}\)-linear combination of induced representations \(\mathrm{ind} (\chi)\), where the \(\chi\)’s are abelian representations. The main part of the paper deals with the verification of these hypotheses in the above mentioned special cases. This yields to some hard computations and in particular to the verification of certain congruences between \(p\)-adic pseudomeasures in the spirit of the congruences of J. Ritter and A. Weiss [Math. Res. Lett. 15, No. 4, 715–725 (2008; Zbl 1158.11047)] and of K. Kato [“Iwasawa theory of totally real fields for Galois extensions of Heisenberg type.” preprint].
The reviewer would like to point out that recent unpublished work of J. Ritter and A. Weiss has led to a full proof of the IMC (up to its uniqueness statement) in the \(1\)-dimensional case provided that \(\mu\) vanishes.
Let \(F\) be a totally real number field and \(F^{\infty} / F\) a Galois extension with Galois group \(G\) satisfying the following conditions: \(G\) is a compact \(p\)-adic Lie group; only finitely many primes of \(F\) ramify in \(F^{\infty}\); \(F^{\infty}\) is totally real and contains the cyclotomic \(\mathbb{Z}_p\)-extension of \(F\). Let \(\Lambda(G)\) be the Iwasawa algebra of \(G\) and let \(S\) be the canonical Ore set for \(G\). Then there is a localization sequence \[ K_1(\Lambda(G)) \rightarrow K_1(\Lambda(G)_S) \overset{\partial}{\rightarrow} K_0(\Lambda(G), \Lambda(G)_S) \rightarrow 0 \] and the canonical complex \[ C = C(F^{\infty} / F) = R\mathrm{Hom}(R \Gamma_{et}( \mathrm{Spec} (\mathfrak{o}_{F^{\infty}}[1 / \Sigma], \mathbb{Q}_p / \mathbb{Z}_p),\mathbb{Q}_p / \mathbb{Z}_p) \] defines a class \([C(F^{\infty} / F)] \in K_0(\Lambda(G), \Lambda(G)_S)\). Here, \(\Sigma\) is a finite set of primes of \(F\) containing all the infinite primes and all primes which ramify in \(F^{\infty}\). Let us denote the cyclotomic character by \(\kappa\) and the (complex) \(\Sigma\)-truncated Artin \(L\)-function of an Artin representation \(\rho\) of \(G\) by \(L_{\Sigma}(s, \rho)\). Then the IMC asserts that there is a (unique) element \(\xi_{F^{\infty}/F} \in K_1(\Lambda(G)_S)\) such that \(\partial(\xi_{F^{\infty}/F}) = - [C(F^{\infty} / F)]\) and “evaluation” of \(\xi_{F^{\infty}/F}\) at \(\rho \kappa^r\) yields an equality \(\xi_{F^{\infty}/F}(\rho \kappa^r) = L_{\Sigma}(1-r, \rho)\) for all natural numbers \(r\) divisible by \(p-1\).
The author considers \(1\)-dimensional \(p\)-adic Lie groups \(G = G^f \times \Gamma\), where \(\Gamma \simeq \mathbb{Z}_p\) and \[ G^f = \begin{pmatrix} 1 & \mathbb{F}_p & \mathbb{F}_p & \mathbb{F}_p \\ 0 & 1 & \mathbb{F}_p & \mathbb{F}_p \\ 0 & 0 & 1 & \mathbb{F}_p \\ 0 & 0 & 0 & 1\end{pmatrix} \] and proves the IMC (up to its uniqueness statement) for \(F^{\infty} / F\) provided that \(p \not=2,3\) and the Iwasawa \(\mu\)-invariant vanishes. The main tool of the proof is a theorem of David Burns which reduces the IMC to the commutative case under certain hypotheses. One of them is that each Artin representation of \(G\) is a \(\mathbb{Z}\)-linear combination of induced representations \(\mathrm{ind} (\chi)\), where the \(\chi\)’s are abelian representations. The main part of the paper deals with the verification of these hypotheses in the above mentioned special cases. This yields to some hard computations and in particular to the verification of certain congruences between \(p\)-adic pseudomeasures in the spirit of the congruences of J. Ritter and A. Weiss [Math. Res. Lett. 15, No. 4, 715–725 (2008; Zbl 1158.11047)] and of K. Kato [“Iwasawa theory of totally real fields for Galois extensions of Heisenberg type.” preprint].
The reviewer would like to point out that recent unpublished work of J. Ritter and A. Weiss has led to a full proof of the IMC (up to its uniqueness statement) in the \(1\)-dimensional case provided that \(\mu\) vanishes.
Reviewer: Andreas Nickel (Regensburg)
MSC:
| 11R23 | Iwasawa theory |
| 11R80 | Totally real fields |
| 11R42 | Zeta functions and \(L\)-functions of number fields |
Citations:
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