Let $F$ be a Siegel cusp form of weight $k$ and degree $n>1$ with Fourier‐Jacobi coefficients ${\{{\varphi }_m\}}_m\in \mathbb{N}$. In this article, we investigate the Ramanujan–Petersson conjecture (formulated by Kohnen) for the Petersson norm of ${\varphi }_m$. In particular, we show that this conjecture is true when $F$ is a Hecke eigenform and a Duke–Imamoğlu–Ikeda lift. This generalizes a result of Kohnen and Sengupta. Further, we investigate an omega result and a lower bound for the Petersson norms of ${\varphi }_m$ as $m\to \infty$. Interestingly, these results are different depending on whether $F$ is a Saito–Kurokawa lift or a Duke–Imamoğlu–Ikeda lift of degree $n⩾4$.

中文翻译：

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Siegel尖点形式的Fourier-Jacobi系数的Ramanujan-Petersson猜想

让 $F$ 成为重量的西格尔尖峰形式 $ķ$ 和程度 $ñ>\mathrm{1个}$ 具有傅里叶-雅各比系数 ${\{{\varphi }_{米}\}}_{米}\in \mathbb{ñ}$。在本文中，我们研究了Ramanujan-Petersson猜想（由Kohnen提出）的Petersson范数。${\varphi }_{米}$。特别是，我们证明了这种猜想是正确的$F$是Hecke本征形和Duke–Imamoğlu–Ikeda升降机。这概括了Kohnen和Sengupta的结果。此外，我们调查了欧米茄的结果和彼得森（Petersson）规范的下界${\varphi }_{米}$ 如 $米\to \infty$。有趣的是，这些结果取决于是否$F$ 是Saito–Kurokawa升降机或Duke–Imamoğlu–Ikeda升降机 $ñ⩾4$。