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

TL;DR

This paper establishes an optimal bound for the $L^4$-norm of automorphic newforms in the depth aspect, showing that the norm grows at most polylogarithmically with the level, which advances understanding of automorphic form behavior.

Contribution

The paper proves the first optimal $L^4$-norm bound for automorphic newforms in the depth aspect, extending previous bounds to the sharpest possible growth rate.

Findings

Proves $ orm{f}_4 \,\ll_{p,\varepsilon} (p^n)^{\varepsilon}$ for all $\\varepsilon > 0$.

Establishes bounds that are sharp up to arbitrary small powers of the level.

Advances the understanding of automorphic forms' norms in the depth aspect.

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 $n \rightarrow \infty$.