Inductive construction of the \(p\)-adic zeta functions for noncommutative \(p\)-extensions of exponent \(p\) of totally real fields. (English) Zbl 1238.11100

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 \(\Gamma \simeq \mathbb{Z}_p\) is the Galois group of the cyclotomic \(\mathbb Z_p\)-extension and \(G^f\) is a finite \(p\)-group of exponent \(p\) and proves the IMC (up to its uniqueness statement) for \(F^{\infty} / F\) provided that \(p \not=2\) and the Iwasawa \(\mu\)-invariant vanishes. The main tool of the proof is a technique first proposed by David Burns. This reduces the IMC to certain congruences between abelian \(p\)-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)].
As an application, the author proves the \(p\)-part of the equivariant Tamagawa number conjecture for the Tate motive \(h^0(\text{Spec}(F'))(1-r)\) with coefficients in \(\mathbb Z[\text{Gal}(F'/F)]\) for any finite Galois extension \(F'\) of \(F\) inside \(F^{\infty}\) and any positive integer \(r\) divisible by \(p-1\) (under the same hypotheses).
The reviewer would like to point out that recent work of J. Ritter and A. Weiss [J. Am. Math. Soc. 24, No. 4, 1015–1050 (2011; Zbl 1228.11165)] has led to a full proof of the IMC (up to its uniqueness statement) in the \(1\)-dimensional case provided that \(\mu\) vanishes. Independently, M. Kakde [The main conjecture of Iwasawa theory for totally real fields, preprint; see J. Algebr. Geom. 20, No. 4, 631–683 (2011; Zbl 1242.11084)] gave a proof which covers the case of higher dimensional \(p\)-adic Lie extensions as well. Finally, D. Burns [“On main conjectures in non-commutative Iwasawa theory and related conjectures”, preprint] has also shown that the higher dimensional case reduces to the \(1\)-dimensional case, and that the \(p\)-part of the equivariant Tamagawa number conjecture for the Tate motive \(h^0(\text{Spec}(F'))(1-r)\) with coefficients in \(\mathbb Z[Gal(F'/F)]\) holds (if \(\mu=0\)) for any finite extension of totally real number fields \(F'/F\) and any positive even integer \(r\).


11R23 Iwasawa theory
11R42 Zeta functions and \(L\)-functions of number fields
11R80 Totally real fields
19B28 \(K_1\) of group rings and orders
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