On the integrality of characteristic elements in nonabelian Iwasawa theory
For a compact \(p\)-adic Lie group \( G \) having a closed normal subgroup \(H\) such that \(G / H\) is isomorphic to \(\mathbb Z_p\), and a canonical Ore set \( S^{*} \) in the Iwasawa algebra \(\Lambda(G)\), in this paper the author gives several conjectures on the integrality of elements in \(K_1(\Lambda(G)_{S^{*}})\), and prove certain part of them. These conjectures are almost equivalent to the previous one formulated by \textit{J.~Coates} et al. in [Publ. Math., Inst. Hautes Étud. Sci. 101, 163--208 (2005; Zbl 1108.11081)]. Let \(K\) be a finite extension of \(\mathbb Q_p\), \(\mathbb O\) the ring of integers of \(K\). For a \(p\)-adic representation \(\rho: G \to \text{GL}_n(\mathbb O)\) of \(G\), define the induced homomorphisms \(\Phi _{\rho}: \Lambda (G) \to M_n(\Lambda _{\mathbb O}(\Gamma))\) and \(\Phi' _{\rho}: K_1(\Lambda(G)_{S^{*}}) \to \mathbb Q_{\mathbb O}(\Gamma)^{\times}\) \((\Gamma = G/H \simeq \mathbb Z_p)\). We also denote \(A_1 = \Lambda (G)_S\), \(A_2 = \Lambda (G) [\frac{1}{p}]\), \(A_3 = \Lambda (G)_{S^{*}}\), \(R_1 = \Lambda (\Gamma)_{(p)}\), \(R_2 = \Lambda (\Gamma) [\frac{1}{p}]\), \(R_3 = \mathbb Q(\Gamma)\), \(R_{i, \mathbb O} = R_{i} \otimes _{\mathbb Z_p} \mathbb O\) \((i= 0, 2, 3)\), \(R_{1, \mathbb O} = \) the localization of \(\Lambda _{\mathbb O} (\Gamma)\) at the prime \((\pi_K)\). Conjecture 1.1 states that if \(G\) has no elements of order \(p\), then in each of the following new cases, the assertions (a), (b) and (c) are equivalent for any element \(\xi \in K_1 (A_3)\): Case V: (a) \(\xi \in \alpha (A_1 \cap A_3^{\times})\), (b) \(\Phi' _{\rho}(\xi) \in R_{1, \mathbb O}\) for all \(\rho\), (c) \(\Phi' _{\rho}(\xi) \in R_{1, \mathbb O}\) for all Artin representations \(\rho\). Case VI: (a) \(\xi\in \alpha (A_{1}^{\times})\), (b) \( \Phi _{\rho }^{'} (\xi) \in R_{1, \mathbb O}^{\times}\) for all \(\rho\), (c) \(\Phi '_{\rho} (\xi ) \in R_{1, \mathbb O}^{\times}\) for all Artin representations \(\rho\) (where \(\alpha : A_{3}^{\times} \to K_{1} ( A_{3})\) is the canonical homomorphism). The other four cases I--IV in conjecture 1.1 not being stated here are known and copied from Conjecture 4.8 in the above mentioned paper of Coates et al. To study conjecture 1.1, the author reduces the assumption `\(G\) has no elements of order \(p\)' to the condition that `\(H\) is finite', as claimed in his conjecture 2.2, and proves that all the known results for conjecture 1.1 are also true for conjecture 2.2 (Theorem 3.3). He also gives a reduction conjecture (i.e., conjecture 2.1) and proves it in Cases V and VI under the condition that \(\dim G> 2\). Finally he obtains a result on the relation between the original conjecture (conjecture 1.1) and the two new conjectures 2.1 and 2.2 (Theorem 3.5).
scientific article; zbMATH DE number 5286517