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Some relations between the analytical modular forms and Maass waveforms for \(\text{PSL}(2,{\mathbb{Z}})\). (English) Zbl 0814.11028

Holomorphic modular forms for \(\Gamma = \text{SL}_ 2(\mathbb{Z})\) can be expressed as polynomials in the holomorphic Eisenstein series \(E_ 4\) and \(E_ 6\) of weights \(4\) and \(6\). In the present paper a similar approach is discussed for real analytic modular forms of weight 0.
Let \(\mathcal H\) be the Hilbert space of square integrable \(\Gamma\)-invariant functions on the upper half plane that satisfy the condition \(f(- \overline{z}) = f(z)\). Let \(M\) be the space of linear combinations of \[ y^ k \widetilde{E}_ 2(z)^ l E_ 4(z)^ m E_ 6(z)^ n \overline{\widetilde{E}_ 2(z)^ p E_ 4(z)^ q E_ 6(z)^ r} \] with non-negative integral exponents satisfying \(k = 2l + 4m + 6n = 2p + 4q + 6r \geq 2\). (\(\widetilde{E}_ 2\) is an Eisenstein series of weight 2 that is not completely holomorphic.) The elements of \(M\) are \(\Gamma\)- invariant, but not necessarily square integrable. The main result is that the closure \(M^{\mathcal H}\) of \(M \cap {\mathcal H}\) in \(\mathcal H\) has finite codimension and contains the subspace corresponding to the continuous spectrum of the Laplacian.
The author indicates a proof in which he uses the spectral theory in \(\mathcal H\) of the operator \(-y^ 2 \partial_ x^ 2 - y^ 2\partial_ y^ 2 + q\) with \(q \in M^{\mathcal H}\), and its perturbation under changes of the potential \(q\). I look forward to future work of the author with the complete proof.

MSC:

11F11 Holomorphic modular forms of integral weight
11F37 Forms of half-integer weight; nonholomorphic modular forms
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
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