Subconvex Pointwise Equidistribution of Hecke–Laplace Eigenfunctions

Definitions and mathematical statement

Let $\mathbb A_{\mathbb Q}$ denote the ring of adeles of $\mathbb Q$. The supplied statement uses a symbol $D$, its group of units $D^\times$, and the quotient \[ X=D^\times(\mathbb Q)\backslash D^\times(\mathbb A_{\mathbb Q}). \] The record does not define $D$ or specify hypotheses on it. If $f$ is a function on $X$, interpret $f(y)$, for $y\in D^\times(\mathbb A_{\mathbb Q})$, as its value at the coset represented by $y$. The symbols $\mu$, the Laplace operator, and the Hecke operators are not specified. Formally, a Hecke--Laplace eigenfunction is a simultaneous eigenfunction of the intended Hecke operators and Laplace operator, with $\lambda$ its Laplace eigenvalue; the operators and their common function space remain to be supplied.

The source asks whether there exists $\delta>0$ such that, for $y\in D^\times(\mathbb A_{\mathbb Q})$ and every relevant Hecke--Laplace eigenfunction $f$ of eigenvalue $\lambda$, \[ f(y)=\int_X f(x)\,\mathrm d\mu(x) +O\!\left(\lambda^{1/4-\delta}\right). \] The single existential quantifier for $\delta$ precedes the quantifiers over $y$ and $f$. The record does not specify whether the implied constant is uniform in $y$, which eigenfunction family is intended, or which other parameters the constant may depend on. The apparent spectral asymptotic $\lambda\to+\infty$ is not explicitly stated in the source.

This formula is retained as a proposed assertion with unresolved hypotheses. No normalization of $f$ is supplied. Since scalar multiples of an eigenfunction remain eigenfunctions, an additive error bound uniform over $f$ cannot be interpreted without a normalization or another explicit restriction on scaling. The record also does not specify the measure $\mu$, its normalization, a treatment of the center, or assumptions ensuring convergence of the integral. In particular, the integral cannot be identified with a probability average from the supplied information. These choices must be resolved before this becomes a fully specified mathematical conjecture.

Short English statement

The supplied question asks for a fixed positive power saving in the difference between a Hecke--Laplace eigenfunction's value at an adelic point and its integral over the displayed quotient. The algebra $D$, eigenfunction normalization, operators, measure, convergence conditions, and error-term uniformity are unspecified, so the intended conjecture requires clarification. The supplied open label is a dated curation claim, not an independently verified mathematical conclusion.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $30004497$, exact source problem code catalogue entry, titled "Subconvex Pointwise Equidistribution of Hecke--Laplace Eigenfunctions," in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2020.
• Supplied report citation: Automorphic Forms and Arithmetic (2021), Oberwolfach Reports, pp. 1317--1383, DOI 10.4171/owr/2020/26. Supplied source URL and literature-source link, labeled "Oberwolfach Report 26/2020": https://doi.org/10.4171/owr/2020/26. No specific page or passage containing this question is identified. The background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature assessment, checked on 2026-08-22: the record labels the problem open, describes the proposed power-saving estimate for point values and the period integral of all relevant eigenfunctions as a question in the report, and reports that no unconditional resolution of this exact assertion was found. This is supplied dated curation derived from the OWR material, not independently verified here; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and states that its mathematical content was preserved while notation, typography, omitted definitions, or duplicate clauses were repaired for self-contained publication. This supplied editorial designation does not resolve the missing hypotheses in the supplied record or constitute mathematical review of this draft.