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

Abstract

Let F be a Siegel cusp form of weight k and degree n > 1 with Fourier-Jacobi coefficients {φm}m ∈ N. In this article, we investigate the Ramanujan–Petersson conjecture (formulated by Kohnen) for the Petersson norm of φm. In particular, we show that this conjecture is true when F is a Hecke eigenform and a Duke–Imamo˘glu–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 φm as m → ∞. Interestingly, these results are different depending on whether F is a Saito–Kurokawa lift or a Duke–Imamo˘glu–Ikeda lift of degree n 4.