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

TL;DR

The paper resolves the depth-aspect $L^4$-norm problem for $L^2$-normalized newforms $f$ on $\Gamma_0(p^n)$ with bounded spectral parameters by proving the optimal bound $\lVert f \rVert_4 \ll_{p,\\varepsilon}(p^n)^{\\varepsilon}$ for odd $p$ as $n\to\infty$. It develops an adelic framework, introduces a balanced level variant, and derives explicit ($p$-adic) Whittaker newvector formulas using $p$-adic stationary phase and the $p$-adic Airy function, enabling precise control of the key $p$-adic integrals $I_p(\\mathbf m)$. The core technical advance is showing that many $I_p(\\mathbf m)$ vanish away from a diagonal, which together with a fourth-moment Hecke bound yields the desired bound on $\\mathcal N(W)$ and hence on $\\lVert f \rVert_4$. The work extends the eigenvalue-parameter paradigm of KiKi (2023) to the depth aspect, highlighting a deep analogy between large eigenvalues and large depth, and provides explicit $p$-adic Whittaker formulas that may be useful for related problems in automorphic forms and $p$-adic harmonic analysis.

Abstract

Let $p$ be an odd prime, and suppose $f$ is an $L^2$-normalised newform for $Γ_0(p^n)$ 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 $n \rightarrow \infty$.

Theorems & Definitions (56)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Lemma 4.1
• proof
• Proposition 5.1
• proof
• Lemma 5.2
• Lemma 5.3