On ``good'' half-integral weight modular forms
The authors prove a result on the nonvanishing of Fourier coefficients of modular forms of half-integral weight. Recently, there have been quite a few papers in this field by, among others, \textit{J. Bruinier} [Abh. Math. Sem. Univ. Hamb. 68, 163--168 (1998; Zbl 0954.11016)] and \textit{K. Ono} [C. R. Math. Acad. Sci., Soc. R. Can. 20, 103--107 (1998; Zbl 0929.11017)]. For integers \(k\geq 0\) and \(N> 0\), let \(M_{k+ \frac 12} (N)\) denote the space of modular forms of weight \(k+ \frac 12\) on \(\Gamma_1(N)\), and let \(\Theta(N)\) be the space which is generated by theta series in one variable of weight \(\frac 12\) or \(\frac 32\) and some level dividing \(N\). Functions in \(\Theta(N)\) are superlacunary (K. Ono, loc. cit.). For \(g\) in \(M_{k+ \frac 12} (N_1)\) and \(h\) in \(\Theta(N_2)\), let \(g_h(n)\) be the \(n\)th Fourier coefficient of \(g(z)h(z)\), and put \(G_h(z)= \sum_n g_h(n) e^{2\pi inz}\) where the summation is restricted to \(n\) with \(\gcd (n,N_1N_2)= 1\). Then \(G_h\) is a modular form (of integral weight \(k+1\) or \(k+2\)) on \(\Gamma(N_1^2 N_2^2)\). As in \textit{K. Ono} and \textit{C. Skinner} [Ann. Math. (2) 147, 453--470 (1998; Zbl 0907.11017)], the modular form \(g\) is called good if there is some \(N_2\) and \(h\in \Theta(N_2)\) such that \(G_h\) is a cusp form and not a linear combination of forms of complex multiplication type. It is shown that every \(g\in M_{k+ \frac 12} (N)\) with \(g\not\in \Theta(N)\) is good. This was conjectured by K. Ono and C. Skinner (loc. cit.).
scientific article; zbMATH DE number 1506007