Large Values of Arithmetic Hyperbolic Three-Manifold Eigenfunctions

Definitions and mathematical statement

Let $M$ be a compact arithmetic hyperbolic $3$-manifold, and let $d\mu$ denote the volume measure intended in the source, whose normalization is unspecified. A Hecke--Laplace eigenfunction is a nonzero smooth function $f:M\to{\mathbb{C}}$ that is an eigenfunction of the Laplace operator and a simultaneous eigenfunction of the relevant Hecke operators. Writing the Laplace operator with the nonnegative sign convention, this means \[ \Delta_M f=\lambda_f f,\qquad T f=a_T f\quad(T\in\mathcal H), \] where $\mathcal H$ denotes the intended family of Hecke operators. The arithmetic data and the family $\mathcal H$ are not specified in the supplied record. Assume the stated normalization \[ \|f\|_{L^2(M,\mu)}^2=\int_M |f(x)|^2\,d\mu(x)=1. \] Define the maximum magnitude and root-mean-square magnitude by \[ \|f\|_\infty=\max_{x\in M}|f(x)|, \qquad \operatorname{rms}_\mu(f)=\left(\frac{1}{\mu(M)}\int_M|f(x)|^2\,d\mu(x)\right)^{1/2} =\mu(M)^{-1/2}. \] Thus the root-mean-square magnitude equals $1$ if $\mu(M)=1$. The stated integral normalization alone does not identify the average magnitude $\mu(M)^{-1}\int_M|f|\,d\mu$ with $1$.

Source question, with unresolved qualifications. Must $|f|$ attain values much larger than its root-mean-square magnitude? Equivalently, the qualitative question concerns the size of the dimensionless ratio \[ \frac{\|f\|_\infty}{\operatorname{rms}_\mu(f)} =\mu(M)^{1/2}\|f\|_\infty. \] The supplied wording does not define "much larger": it specifies neither a numerical threshold nor an asymptotic lower bound, a limiting parameter, or quantifiers over eigenfunctions. It also does not specify whether $M$ is fixed, whether a positive-eigenvalue restriction is intended, or whether the assertion should concern every eigenfunction, a subsequence, or some other family. Consequently, the record supplies a qualitative question rather than a uniquely determined quantitative conjecture. No particular growth assertion is imposed here.

Short English statement

The source asks whether an $L^2$-normalized Hecke--Laplace eigenfunction on a compact arithmetic hyperbolic three-manifold must have unusually large peak values. The comparison size is $1$ only when the volume measure has total mass $1$. The meaning of "much larger" and the intended family and quantifiers remain unspecified.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json, permanent ID $30004496$, title "Large Values of Arithmetic Hyperbolic Three-Manifold Eigenfunctions," source problem code catalogue entry. This draft uses the supplied complete source record.
• Supplied report citation: Automorphic Forms and Arithmetic (2021), Oberwolfach Reports, DOI 10.4171/owr/2020/26, pp. 1317--1383. Supplied source and literature link, labeled "Oberwolfach Report 26/2020": https://doi.org/10.4171/owr/2020/26. The record attributes the question to the Oberwolfach Reports workshop collection; no individual proposer is supplied. The proposed year is recorded as $2020$. The citation's $2021$ title date and its $2020$ report designation are retained as supplied.
• Literature assessment dated 2026-08-22: the supplied curation labels the question open, describes it as a qualitative large-values question, and states that the extracted wording supplies no quantitative threshold that has been proved. This is supplied dated curation, not an independently verified literature assessment; the record requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and describes its editing as preserving mathematical content while repairing presentation or omitted definitions. This is a supplied editorial label, not independent mathematical verification. The original wording compares with an "average value 1," whereas the cleaned wording compares with a "root-mean-square size 1." Their measure-normalization qualification remains unresolved.