Construct A_K^+ in the ramified case

Construct a ramified analog of the Wach-module base ring A_K^+ (which equals W(k)[[mu]] when K is absolutely unramified) for a general p-adic field K with ramification, suitable for defining Wach modules that classify crystalline Z_p-representations of G_K.

Background

Berger’s theory of Wach modules over A_K+ = W(k)[[mu]] gives a classification of crystalline representations when K is absolutely unramified. Extending this to ramified K would require an appropriate construction of A_K+ with analogous Frobenius and Gamma_K-structures.

The paper circumvents this difficulty by working over :=W(mathcal{O}{K}\flat) and introducing crystalline (phi,Gamma)-modules there, but explicitly notes that constructing A_K+ in the ramified case remains an obstacle.

References

The difficulty is the construction of $\mathbb{A}_K+$. There are some conditions which $\mathbb{A}_K+$ should satisfy, but the author does not know how to construct $\mathbb{A}_K+$ in the ramified case.

— On the $(\varphi,Γ)$-modules corresponding to crystalline representations  (2405.19829 - Watanabe, 2024) in Introduction

One would want to have a prism $(\A_K+, \varphin([p]_q))$ which satisfies \A_K+[1/\varphi{n}([p]_q)]_p{\wedge} \simeq \A_K. However it seems such a prism cannot exist in general without dropping some desirable properties. See Appendix 2, Proposition 23 for more details. The argument there showing nonexistence uses that the mod $p$ reduction should be $\E_K+$, the ring of integers of the field of norms. It is still possible that $\A_K+$ exists by dropping this condition, but this seems unnatural to do.

— A prismatic Riemann-Hilbert correspondence  (2610.20466 - Pentland, 14 Oct 2026) in Remark immediately following Corollary \ref{cor:other_bases}, in the section discussing the main equivalence