p-Adic Period Maps and Hodge-Theoretic Variations

Definitions and mathematical statement

For a prime number $p$, write ${\mathbb{Q}}_p$ for the completion of ${\mathbb{Q}}$ with respect to the absolute value normalized by $|p|_p=p^{-1}$. The source does not specify a particular prime or a field over ${\mathbb{Q}}_p$.

To explain the classical terminology, a pure rational variation of Hodge structure of weight $w\in{\mathbb{Z}}$ on a complex manifold $S$ consists of a finite-rank rational local system $\mathbb V$ and a decreasing holomorphic filtration $F^\bullet$ by subbundles of $\mathcal V=\mathbb V\otimes_{{\mathbb{Q}}}\mathcal O_S$. At each $s\in S$, this filtration determines a decomposition \[ \mathbb V_s\otimes_{{\mathbb{Q}}}{\mathbb{C}}=\bigoplus_{a+b=w}H_s^{a,b},\qquad \overline{H_s^{a,b}}=H_s^{b,a},\qquad F_s^r=\bigoplus_{a\ge r}H_s^{a,w-a}. \] The flat connection $\nabla$ induced by $\mathbb V$ satisfies Griffiths transversality: \[ \nabla(F^r)\subseteq F^{r-1}\otimes_{\mathcal O_S}\Omega_S^1. \] After a local flat identification of the fibers, its period map sends $s$ to the filtration $F_s^\bullet$ in the corresponding space of Hodge filtrations. These definitions provide one classical reference setting; the source does not restrict its question to pure rational variations or specify polarization or additional structures.

Source question. What is the $p$-adic analogue of the classical theory of period maps and variations of Hodge structure?

This is a request for a theory, rather than a specified theorem or conjecture. The record does not prescribe a category of base spaces or families, coefficient fields, a notion of $p$-adic variation, a period target, or the classical properties that an analogue must reproduce. Consequently, it does not determine a unique mathematical assertion with explicit hypotheses and conclusion. No particular construction or completeness criterion is imposed here.

Short English statement

The problem asks for a $p$-adic counterpart of classical varying Hodge structures and their period maps. Its intended scope and success criteria are unspecified, so the supplied question remains a research program requiring clarification and human mathematical review.

Sources

• Supplied UnsolvedMath local registry record, permanent ID $30003191$, title "p-Adic Period Maps and Hodge-Theoretic Variations," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2016.
• Supplied report citation: Arithmetic Geometry (2017), Oberwolfach Reports, DOI 10.4171/owr/2016/38, pp. 2171--2224. https://doi.org/10.4171/owr/2016/38. The background describes these reports as documenting mathematical workshops and their open-problem discussions. The citation's year 2017 and the record's proposed year 2016 are retained without reconciliation.
• Supplied literature reference: "Scholze, p-adic period maps." https://doi.org/10.4007/annals.2015.182.3.3. This is the reference label supplied by the record; fuller bibliographic details are not supplied.
• Supplied literature reference: "Oberwolfach Report 38." https://doi.org/10.4171/owr/2016/38.
• Literature assessment dated exactly 2026-08-21: the supplied status is partially_solved. The supplied assessment attributes broad $p$-adic Hodge--Tate period-map constructions to Scholze and subsequent work, while asserting that there is no single complete $p$-adic replacement for every classical variation-of-Hodge-structure feature. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its time-sensitive status and cited claims require independent checking.
• The registry labels the statement corrected_verified and reports that the official report was checked and that this is its organizing open question for $p$-adic period maps. These are supplied verification claims, not verification performed for this draft.