Basis for the space of weakly holomorphic modular forms in higher level cases
Let \(p\) be either \(1\) or a prime number. Let \(\Gamma_0(p)^+\) be the group generated by the group \(\Gamma_0(p)\) and the Fricke involution \(W_p\) and \(M_k^!(\Gamma_0(p)^+)\) be the space of weakly holomorphic modular forms (that is, meromorphic with poles only at the cusps) of even integral weight \(k\) with respect to \(\Gamma_0(p)^+\). In this article, the results of \textit{W. Duke} and \textit{P. Jenkins} [Pure Appl. Math. Q. 4, No. 4, 1327--1340 (2008; Zbl 1200.11027)] on a canonical basis of \(M_k^!(\Gamma_0(1)^+)\)~(\(\Gamma_0(1)^+= \mathrm{SL}_2(\mathbb Z))\) are extended to those of \(M_k^!(\Gamma_0(p)^+)\), where \(\Gamma_0(p)^+\) is of genus \(0\). These groups have only one cusp \(i\infty\) up to equivalence and have the Hauptmodul with a pole at the cusp. Let \(m_k\) be the maximal value of order of Fourier expansions at the cusp of non-zero forms belong to \(M_k^!(\Gamma_0(p))\). The authors give explicitly the form \(F_k\in M_k^!(\Gamma_0(p)^+)\) of order \(m_k\) with the leading coefficient \(1\). By using \(F_k\) and the Hauptmodul, a canonical basis \(\{f_{k,n}\} (n\in \mathbb Z,-n\leq m_k)\) of the space \(M_k^!(\Gamma_0(p))\) is constructed such that \(f_{k,n}=q^{-n}+\sum_{m>m_k}a_k(n,m)q^m,~a_k(n,m)\in\mathbb Z,~q=e^{2\pi i z}\) and \(\{f_{k,n}\}\) have a generating function and the duality \(a_k(n,m)=-a_{2-k}(m,n)\) of Fourier coefficients.
scientific article; zbMATH DE number 6138271
scientific article