Structure of the space of polyharmonic Maass forms with an application to \(L\)-values (Q6558334)

The authors describe bases of some spaces of polyharmonic Maaß forms in terms of parabolic Poincaré series. Polyharmonic Maaß forms generalize harmonic (weak) Maaß forms, vanishing under some power of the Laplace operator. Since the Laplace operator can be factored in terms of the \(\xi\)-operator, one finds a filtration on the space of polyharmonic Maaß forms by the exponent of the Laplace operator that annihilates them. This leads to the notion of depth which takes values in \(\frac{1}{2} \mathbb{Z}\). The filtration quotients associated with this a priori inject into spaces of weakly holomorphic forms.\N\NA classification of polyharmonic Maaß forms in level one was given in [\textit{C. Alfes} et al., J. Algebra 661, 713--756 (2025; Zbl 07938146)], where also a construction via parabolic Poincaré series was given. As in the case of harmonic Maaß forms, the only arithmetic obstruction that arises originates in Serre duality [\textit{J. H. Bruinier} and \textit{J. Funke}, Duke Math. J. 125, No. 1, 45--90 (2004; Zbl 1088.11030)]. The present work implicitly describes these obstructions via possible orders of poles of weakly holomorphic modular forms and then applies the Poincaré series construction to obtain bases for spaces of polyharmonic Maaß forms of square-free level.