Short Coefficient Detection of Cusp-Form Norms

Definitions and mathematical statement

Let $N\in{\mathbb{Z}}_{\geq 1}$, let $\mathbb H=\{x+iy\in{\mathbb{C}}:y>0\}$, and set \[ \Gamma_0(N)=\left\{\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\operatorname{SL}_2({\mathbb{Z}}):N\mid c\right\}. \] Write $S_k(N)$ for the source's space of cusp forms of weight $k$ and level $N$. The source does not explicitly specify the permitted weights or the character convention, so these remain qualifications requiring review. For $f\in S_k(N)$, use the unnormalized Petersson norm and the normalized Fourier coefficients specified in the source: \[ \|f\|^2=\int_{\Gamma_0(N)\backslash\mathbb H}|f(x+iy)|^2y^{k-2}\,{\,\mathrm{d}} x\,{\,\mathrm{d}} y, \qquad f(z)=\sum_{n=1}^{\infty}a_f(n)n^{(k-1)/2}e(nz), \qquad e(t)=\exp(2\pi i t). \] The requested assertion is that, for each admissible fixed weight $k$ and every $\varepsilon>0$, there exist constants $C=C(k,\varepsilon)>0$ and $B=B(k,\varepsilon)>0$ such that, for every positive integer $N$ and every $f\in S_k(N)$, \[ \|f\|^2\leq B\sum_{\substack{n\in{\mathbb{Z}}_{\geq1}\\n\leq C N^{1+\varepsilon}}}|a_f(n)|^2. \] Both constants must be independent of $N$ and $f$. No square-free restriction on $N$, and no requirement that $f$ be an eigenfunction of the Atkin--Lehner operators, is imposed.

The supplied source attributes the problem to Edgar Assing. It reports the displayed bound when $N$ is square-free and $f$ is an eigenfunction of all Atkin--Lehner operators, and reports a bound for general $f\in S_k(N)$ with the cutoff $C N^{2+\varepsilon}$ in place of $C N^{1+\varepsilon}$, citing an unidentified reference [1]. These are supplied background claims, not independently verified results here. The source also motivates the requested length by its stated estimate $\dim S_k(N)\asymp_k N^{1+o(1)}$, describing the length as essentially minimal; the weight hypotheses for this dimensional assertion are not supplied.

Short English statement

For each fixed admissible weight and every $\varepsilon>0$, ask whether the unnormalized Petersson norm of every level-$N$ cusp form is bounded uniformly by the sum of squares of its normalized Fourier coefficients through $C N^{1+\varepsilon}$. The claim covers all positive integer levels and all forms in the stated space. The supplied record labels it open; weight and character conventions still require clarification.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $30006464$, source problem code catalogue entry, title Short Coefficient Detection of Cusp-Form Norms, in the Oberwolfach Reports Open Problems collection. The record labels the problem open and labels the statement corrected_verified; its verification note describes only repairs to formula delimiters, notation, grammar, and typography.
• Supplied report citation: Analytic Number Theory (2026), Oberwolfach Reports, pp. 2701--2764, DOI 10.4171/owr/2025/51. Supplied source URL and literature-source link, labelled "Oberwolfach report 51/2025": https://doi.org/10.4171/owr/2025/51. The supplied proposed year is 2025; the citation's 2026 title and the 2025 report identifiers are retained without reconciliation.
• Attribution: the original statement explicitly says "Proposed by Edgar Assing", although the structured proposer field is empty.
• Literature assessment checked at 2026-08-22: the supplied curation reports the $N^{1+\varepsilon}$ bound for square-free levels and Atkin--Lehner eigenforms, the $N^{2+\varepsilon}$ bound in general, and an unresolved uniform $N^{1+\varepsilon}$ bound. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment.
• The original statement cites reference [1] for the general $N^{2+\varepsilon}$ bound, but the supplied record contains no bibliographic entry identifying [1].