Harmonic weak Maass-modular grids in higher level cases (Q2845645)

Let \(p\) be a prime or \(p=1\) and \(\Gamma=\Gamma_0(p)\) or \(\Gamma_0^+(p)=\langle \Gamma_0(p),W_p\rangle\), where \(W_p\) is the Fricke involution. For an integer \(k\), let \(M_k^!(\Gamma)\) and \(H_k(\Gamma)\) be the space of weakly holomorphic modular forms and that of harmonic weak Maass forms of weight \(k\) on \(\Gamma\) respectively. In [\textit{P. Guerzhoy}, Math. Res. Lett. 16, No. 1, 59--65 (2009; Zbl 1243.11058)], two collections \(f_n\in M_k^!(\Gamma)\) and \(g_m\in H_{2-k}(\Gamma),~m>0,n\geq 0\) with \(q\)-expansions \(f_n=q^{-n}+\sum_{m>0}c_{f_n}(m)q^m,~g_m=g_m^-+q^{-m}+\sum_{n\geq 0}c_{g_m}(n)q^n\) is called a harmonic weak Maass-modular grid (or simply a grid) of weight \(k\) on \(\Gamma\), if the identity of Fourier coefficients \(c_{f_n}(m)=-c_{g_m}(n)\) holds. For the group \(\mathrm{SL}_2(\mathbb Z)\), Guerzhoy showed that for a non-negative even integer \(k\) there exists a unique grid of weight \(k\) with indices \(n\geq 0,m>0\) and moreover that \(f_n=-n^{1-k}D^{k-1}g_n^+\), where \(D=q(d/dq)\) and \(g_n^+=g_n-g_n^-\). In this article, the authors obtain similar results for the group \(\Gamma\). Let \(H_k^\infty(\Gamma)\) be the subspace of those \(g\in H_k(\Gamma)\) whose principal parts at the cusps other than \(\infty\) are constant. For even \(k\geq 0\), they showed that if \(k>2\), then there exist unique \(f_n\in M_k^!(\Gamma)\cap H_k^\infty(\Gamma), n\geq 0\) and \(g_m\in H_{2-k}^\infty(\Gamma), m>0\) such that \((f_n,g_m)\) is a grid of weight \(k\) on \(\Gamma\), if \(k=2\) then there exist such a grid for all \(m,n\geq 1\) and \(f_n\) is unique and \(g_m\) is unique up to constants. Furthermore, let \(T_n\) be the Hecke operator. Then they prove that \(f_n=n^{1-k}f_1\mid T_n\) for any positive integer \(n\) relatively prime to the level of \(\Gamma\).