Ramanujan–Petersson conjecture for Fourier–Jacobi coefficients of Siegel cusp forms

Let F be a Siegel cusp form of weight k and degree n > 1 with Fourier‐Jacobi coefficients{ ϕ m } m ∈ N . In this article, we investigate the Ramanujan–Petersson conjecture (formulated by Kohnen) for the Petersson norm of ϕ 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 ϕ m as m → ∞ . Interestingly, these results are different depending on whether F is a Saito–Kurokawa lift or a Duke–Imamoğlu–Ikeda lift of degree n ⩾ 4 .