Computation of pro-locally analytic vectors in perfect relative period rings

Determine how to compute the pro-locally analytic vectors in perfect relative period rings in general.

Background

The paper compares its decompletion techniques with the theory of pro-locally analytic vectors, which provides another possible framework for decompletion. A stated obstacle is that the relevant analytic vectors are not generally computable in the perfect relative period rings.

Resolving this question would clarify the relationship between the authors’ decompletion formalism and the pro-locally analytic-vector approach, and could potentially provide an alternative method for constructing relative pp-adic differential equations.

References

One central difficulty of the latter is that we do not know in general how to compute the (pro)-locally analytic vectors in perfect relative period rings.

— Relative $(\varphi, Γ)$-modules and $p$-adic differential equations  (2609.08179 - Diao et al., 8 Sep 2026) in Remark in Section 1, subsection “Relative \(p\)-adic differential equations”