## 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 $$r$$.

### MSC:

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

### Citations:

Zbl 1158.11047; Zbl 1228.11165; Zbl 1242.11084
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