The four-case integrality conjecture for characteristic elements

Let $G$ be a compact $p$-adic Lie group with no element of order $p$, and define

$$R_1=\Lambda(G),\qquad R_2=\Lambda(G)[1/p],\qquad R_3=\Lambda(G)_{S^*},$$

as well as

$$A_1=\Lambda_O(\Gamma),\qquad A_2=\Lambda_O(\Gamma)[1/p].$$

Let $\alpha:R_3^{\times}\to K_1(R_3)$ be the canonical surjection, and let ${\frak R}(G)$ be the continuous representations of $G$ of the prescribed form, with ${\frak A}(G)$ its subset of Artin representations. For $\xi\in K_1(R_3)$, write $\xi(\rho)$ for its evaluation and $\Phi'_{\rho}(\xi)$ for the associated element of the quotient field of $A_1$.

The four-case integrality conjecture. In each of the following four cases, the assertions (a), (b), (c), and (d) are equivalent for every $\xi\in K_1(R_3):

1. (a) $\xi\in\alpha(R_2\cap R_3^{\times})$; (b) $\xi(\rho)\ne\infty$ for every $\rho$; (c) $\Phi'_{\rho}(\xi)\in A_2$ for every $\rho$; (d) $\Phi'_{\rho}(\xi)\in A_2$ for every Artin $\rho$.

2. (a) $\xi\in\alpha(R_1\cap R_3^{\times})$; (b) $\xi(\rho)$ is finite and belongs to $O$ for every $\rho$; (c) $\Phi'_{\rho}(\xi)\in A_1$ for every $\rho$; (d) $\Phi'_{\rho}(\xi)\in A_1$ for every Artin $\rho$.

3. (a) $\xi\in\alpha(R_2^{\times})$; (b) $\xi(\rho)\ne0,\infty$ for every $\rho$; (c) $\Phi'_{\rho}(\xi)\in A_2^{\times}$ for every $\rho$; (d) $\Phi'_{\rho}(\xi)\in A_2^{\times}$ for every Artin $\rho$.

4. (a) $\xi\in\alpha(R_1^{\times})$; (b) $\xi(\rho)$ is finite and belongs to $O^{\times}$ for every $\rho$; (c) $\Phi'_{\rho}(\xi)\in A_1^{\times}$ for every $\rho$; (d) $\Phi'_{\rho}(\xi)\in A_1^{\times}$ for every Artin $\rho$.

The conjecture is supported by the case $G\cong\mathbb Z_p^d\times\Delta$, where $d\geq1$ and $\Delta$ is finite of order prime to $p; the supplied text does not establish the general equivalences.