Jaulent's strong cyclotomic conjecture for S-split T-ramified Iwasawa modules

Let $K$ be an arbitrary number field, let

$$K_\infty=\bigcup_{n\in\mathbb N}K_n$$

be its cyclotomic $\mathbb Z_\ell$-extension, and let $\Lambda$ be the Iwasawa algebra of

$$\Gamma=\operatorname{Gal}(K_\infty/K)=\gamma^{\mathbb Z_\ell}.$$

Let

$$Pl_K^\ell=S_K\sqcup T_K$$

be a partition of the places of $K$ above $\ell$. Write $H^{T_K}_{S_K}(K_\infty)$ for the maximal abelian pro-$\ell$-extension of $K_\infty$ that is $S_K$-split and $T_K$-ramified, and set

$$\mathcal C^{T_K}_{S_K}(K_\infty)=\operatorname{Gal}(H^{T_K}_{S_K}(K_\infty)/K_\infty).$$

Strong cyclotomic conjecture. The characteristic polynomial of the $\Lambda$-module $\mathcal C^{T_K}_{S_K}(K_\infty)$ is not divisible by $\omega=\gamma-1$. Equivalently, its fixed-point submodule is finite:

$$\mathcal C^{T_K}_{S_K}(K_\infty)^\Gamma\sim 1.$$

This conjecture contains the Leopoldt and Gross–Kuz'min conjectures as special cases. For totally real fields, and for CM fields with partitions stable under complex conjugation, it follows from the conjunction of Leopoldt and Gross–Kuz'min; the unrestricted assertion for arbitrary partitions remains the issue addressed by the paper.