Pointwise Asymptotics of Special Quaternionic Eigenfunctions

Definitions and mathematical statement

Let $\mathbb A_{{\mathbb{Q}}}$ denote the ring of adeles of ${\mathbb{Q}}$. The source uses the notation $D^{\times}$ and the quotient \[ X_D=D^{\times}({\mathbb{Q}})\backslash D^{\times}(\mathbb A_{{\mathbb{Q}}}), \] but does not specify $D$. The title suggests a quaternionic setting, without supplying the algebra or its hypotheses. The terms "special eigenfunction" and $\mu$ are also undefined in the supplied record. For the displayed integral to be meaningful, the function must descend to $X_D$ and be integrable with respect to a specified measure $\mu$.

The supplied question asks whether, for $y\in D^{\times}(\mathbb A_{{\mathbb{Q}}})$ and every special eigenfunction $f$, \[ f(y)=(1+o(1))\int_{X_D}f(x)\,\mathrm d\mu(x). \] Here evaluation at $y$ is understood through its quotient class if $f$ is a function on $X_D$. This reproduces the requested relation but does not specify an asymptotic proposition: the limiting parameter, the varying family, the normalization of $f$ and $\mu$, and the intended dependence on $y$ are absent. The multiplicative error also requires clarification when the integral vanishes. No limiting regime, replacement error term, or additional hypotheses are imposed in this draft.

Short English statement

The question asks whether each special eigenfunction on the indicated adelic quotient is pointwise asymptotic to its integral. The supplied record does not define the algebra, the special eigenfunctions, the measure, or the limiting process, so a precise asymptotic statement requires further source clarification.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $30004498$, title "Pointwise Asymptotics of Special Quaternionic Eigenfunctions," exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is named; the supplied proposed year is 2020.
• Supplied report citation: Automorphic Forms and Arithmetic (2021), Oberwolfach Reports, pp. 1317--1383, DOI 10.4171/owr/2020/26. Supplied source and literature link, labeled "Oberwolfach Report 26/2020": https://doi.org/10.4171/owr/2020/26. The background describes these reports as documenting mathematical workshops and their open-problem discussions. The report contents have not been independently inspected for this draft.
• Supplied literature assessment dated 2026-08-22: the record labels the problem open and describes the special-eigenfunction asymptotic and required period-integral estimates as expectations and conditional reduction targets, with the exact question unresolved. This is supplied dated curation derived from the OWR record, not an independently verified assessment; the supplied background explicitly requests independent verification of time-sensitive status and cited claims.