It is known that among Siegel modular forms of degree 2 and level 1 the only functions that violate the Ramanujan conjecture are Saito–Kurokawa lifts of modular forms of level 1. These are precisely the functions whose Fourier coefficients satisfy Maass relations. More generally, the Ramanujan conjecture for \(\mathrm {GSp}_4\) is predicted to fail only in case of CAP representations. It is not known though whether the associated Siegel modular forms (of various levels) still satisfy a version of Maass relations. We show that this is indeed the case for the ones related to P-CAP representations. Our method generalises an approach of Pitale, Saha and Schmidt who employed representation–theoretic techniques to (re)prove this statement in case of level 1. In particular, we compute and express certain values of a global Bessel period in terms of Fourier coefficients of the associated Siegel modular form. Moreover, we derive a local–global relation satisfied by Bessel periods, which allows us to combine those computations with a characterization of local components of CAP representations.

众所周知，在2级和1级的Siegel模块化形式中，唯一违反Ramanujan猜想的函数是1级模块化形式的Saito-Kurokawa提升。这些正是傅里叶系数满足Maass关系的函数。更一般而言，\（\ mathrm {GSp} _4 \）的Ramanujan猜想预计仅在CAP表示的情况下失败。但是，尚不知道相关的Siegel模块化形式（各种级别）是否仍满足Maass关系的版本。我们表明，与P-CAP表示相关的确实如此。我们的方法概括了Pitale，Saha和Schmidt的方法，他们使用表示理论技术在1级的情况下（重新）证明了这一说法。特别是，我们根据贝叶斯的傅立叶系数来计算和表达全球贝塞尔周期的某些值相关的Siegel模块化形式。此外，我们得出贝塞尔周期满足的局部-全局关系，这使我们能够将这些计算与CAP表示的局部分量的特征结合起来。