$L^4$-norms of automorphic forms in the depth aspect

Marius Fischer

$L^4$-norms of automorphic forms in the depth aspect

Number Theory 2026-02-24 v1

Authors: Marius Fischer

Abstract

Let $p$ be an odd prime, and suppose $f$ is an $L^2$ -normalised newform for $\Gamma_0(p^n)$ with bounded spectral parameters and trivial central character. We prove the optimal $L^4$ -norm bound $\lVert f \rVert_4 \ll_{p,\varepsilon}(p^{n})^\varepsilon$ for all $\varepsilon >0$ as $n \rightarrow \infty$ .

Keywords

$L^4$-norm automorphic forms depth aspect newform optimal bound

Cite

@article{arxiv.2602.19646,
  title  = {$L^4$-norms of automorphic forms in the depth aspect},
  author = {Marius Fischer},
  journal= {arXiv preprint arXiv:2602.19646},
  year   = {2026}
}

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51 pages

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