Abstract
We investigate Poincar\'e series, where we average products of terms of Fourier series of real-analytic Siegel modular forms. There are some (trivial) special cases for which the products of terms of Fourier series of elliptic modular forms and harmonic Maass forms are almost holomorphic, in which case the corresponding Poincar\'e series are almost holomorphic as well. In general this is not the case. The main point of this paper is the study of Siegel-Poincar\'e series of degree $2$ attached to products of terms of Fourier series of harmonic Siegel-Maass forms and holomorphic Siegel modular forms. We establish conditions on the convergence and nonvanishing of such Siegel-Poincar\'e series. We surprisingly discover that these Poincar\'e series are almost holomorphic Siegel modular forms, although the product of terms of Fourier series of harmonic Siegel-Maass forms and holomorphic Siegel modular forms (in contrast to the elliptic case) is not almost holomorphic. Our proof employs tools from representation theory. In particular, we determine some constituents of the tensor product of Harish-Chandra modules with walls.
Highlights
We investigate Poincaré series, where we average products of terms of Fourier series of real-analytic Siegel modular forms
Kaneko and Zagier’s paper was stimulated by Dijkgraaf [7], who conjectured a relation between almost holomorphic modular forms and numbers of topologically inequivalent branched covers of an elliptic curve
It is not difficult to construct almost holomorphic Poincaré series by observing that almost holomorphic modular forms vanish under a power of the Bringmann et al Res Math Sci(2016)3:30 lowering operator L := −2iy2∂τ, where throughout τ = x+iy ∈ H
Summary
We suggest another Poincaré series that is almost holomorphic. In light of what we discuss in Sect. 1.4, it is natural to replace ψk by ψk (n; τ ) := y−k (1 + k, 4π ny) e2πi nτ , k ≥ 0, n > 0. The Harish-Chandra module attached to Pk[d], ;n,m is a (finite) direct sum of (limits of) holomorphic discrete series Graded components of such discrete series correspond to almost holomorphic modular forms, and this proves again what we have already observed in Sect. The inconclusive cases To analyze the Poincaré series in (1.4) and (1.6), we have to consider the tensor product of the C[L, R]-modules generated by φk and φ , and by ψk and φ , respectively. These tensor products are supported on all weights and all weight spaces are infinite dimensional.
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