On a theorem of Vignéras
Let \(f(z)=\sum_{n\geq 0}a(n) e^{2\pi inz}\) be a non-zero modular form of half-integral weight \(k+{1\over 2}\) on \(\Gamma_1(N)\). The author proves several criteria for the nonvanishing of the coefficients \(a(n)\). One of them is: If \(k\geq 2\), or if \(k=1\) and \(f\) is a cusp form in the orthogonal complement of the space spanned by theta series in one variable, then for any prime \(p\nmid N\) and any \(\varepsilon\in\{1,-1\}\) there is some \(n\) such that \(({n\over p})=\varepsilon\) and \(a(n)\neq 0\). The results imply an earlier theorem of \textit{M. F. Vignéras} on the nonvanishing of \(a(n)\) [Lect. Notes Math. 627, 79-103 (1977; Zbl 0373.10027)]. The proof uses well-known operators on modular forms and the theorem of \textit{J.-P. Serre} and \textit{H. M. Stark} on the generation of modular forms of weight \({1\over 2}\) by theta series [Lect. Notes Math. 627, 27-67 (1977; Zbl 0371.10019)].
scientific article; zbMATH DE number 1416673