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OP-11911Status unassessedNumber theory

A short Fourier-coefficient bound for Petersson norm

A short Fourier-coefficient bound for Petersson norm. Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today.

MSC 11 · Number theory

Editorial MSC2020 mapping of A short Fourier-coefficient bound for Petersson norm. https://ems.press/content/serial-article-files/52435#page=56 Not an author-supplied code.

Source question · editorial summaryAs of 2025; current status unassessedSource & evidence ↓
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Problem statement

Write a weight-kk, level-NN cusp form as f(z)=∑n≥1af(n)n(k−1)/2e2πinzf(z)=\sum_{n\ge1}a_f(n)n^{(k-1)/2}e^{2\pi inz}. For every ε>0\varepsilon>0, does some C=C(k,ε)C=C(k,\varepsilon) give ∥f∥Pet2≪k,ε∑n≤CN1+ε∣af(n)∣2\|f\|_{\mathrm{Pet}}^2\ll_{k,\varepsilon}\sum_{n\le CN^{1+\varepsilon}}|a_f(n)|^2 uniformly for all such ff?

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Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today.

References
  1. [1]Problem-session contributors; compiled by Thomas F. Bloom. Oberwolfach 2025 — Analytic Number Theory Problem Session (2025).; A short Fourier-coefficient bound for Petersson norm linkCC BY-SA 4.0; source contributors, compiled by Thomas F. Bloom. These attributed adaptations are available under the same license.

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