Local Maxima of Special Eigenfunctions at Spherical Points

Definitions and mathematical statement

Let ${\mathbb{A}}_{{\mathbb{Q}}}$ denote the adele ring of ${\mathbb{Q}}$, and let $Z(G)$ denote the center of a group $G$. Using the symbols in the supplied record, put \[ G=D^{\times}({\mathbb{A}}_{{\mathbb{Q}}}),\qquad X=D^{\times}({\mathbb{Q}})\backslash G/\left((K_\infty\cap G)Z(G)\right). \] The record does not specify $D$, the ambient group containing $K_\infty$, or the action relevant to the representation condition below. These symbols therefore retain their unresolved source meanings.

The supplied hypothesis is that there are no nonzero functions on $X$ transforming according to the stated $3$-dimensional spherical representation of $\operatorname{SO}(3)$. Denote this hypothesis by $H$, and denote the unspecified collection of special eigenfunctions by $\mathcal E$. The cleaned statement asks whether \[ H\ \Longrightarrow\ \exists\,\mathcal E_0\subseteq\mathcal E,\quad |\mathcal E_0|<\infty,\quad \forall f\in\mathcal E\setminus\mathcal E_0\quad \forall x\in G,\quad \text{$x$ is a local maximum point of $f$}. \] This is only a transcription of the cleaned formulation: the original statement instead quantifies over "every such point $x$", whose antecedent is absent from the supplied continuation fragment. Consequently, the intended set of points may be a previously defined subset rather than all of $G$.

For a real-valued function $f$ on a topological space $M$, a point $x\in M$ is a local maximum point if there exists a neighborhood $U$ of $x$ such that $f(y)\leq f(x)$ for every $y\in U$. The record does not identify the domain or codomain of the special eigenfunctions, the topology in which maxima are taken, or whether representatives $x\in G$ are to be evaluated through a quotient map. If the eigenfunctions are complex-valued, it also does not specify an ordered real-valued quantity, such as their absolute value, to maximize. No such interpretation is imposed here. The missing context prevents a fully specified mathematical statement.

Short English statement

Under a stated vanishing condition for functions transforming according to a $3$-dimensional spherical representation of $\operatorname{SO}(3)$, the question asks whether every relevant point is a local maximum for all but finitely many special eigenfunctions. The eigenfunctions, representation action, and relevant points require missing context; the original wording does not justify replacing "every such point" by every adelic group element.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004499$, title Local Maxima of Special Eigenfunctions at Spherical Points, source code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year $2020$ and names no proposer.
• Supplied source citation: Automorphic Forms and Arithmetic (2021), Oberwolfach Reports, pp. 1317--1383, DOI 10.4171/owr/2020/26. Source URL and supplied literature reference, labeled Oberwolfach Report 26/2020: https://doi.org/10.4171/owr/2020/26. The citation's year $2021$, report designation $26/2020$, and proposed year $2020$ are retained as supplied.
• The registry labels the statement corrected_verified and reports that the official report was checked against Question 3 to restore the adelic quotient, spherical representation, finite exceptional set, and local-maximum question. This is supplied verification metadata, not an independent verification performed for this draft.
• Literature assessment dated exactly 2026-08-22: supplied dated curation labels the problem open and describes the extracted text as a continuation fragment concerning conditional methods for period integrals and a local-maximum question, with no proof of the underlying general claim recorded. The background identifies Oberwolfach Reports as workshop reports containing open-problem discussions. The supplied assessment expressly requires independent verification of time-sensitive status and cited claims; it is not independently verified here.