Toric modular forms and nonvanishing of \(L\)-functions (Q2744712)

Let \(l>1\) be an integer, let \({\mathcal M}(l)\) be the space of weight two modular forms on the congruence subgroup \(\Gamma_1(l)\subset \text{SL}_2({\mathbb Z})\) and let \({\mathcal S}(l)\) be the subspace of cusp forms. For \(a=1,\ldots,l-1\) let NEWLINE\[NEWLINEs_{a/l}(\tau)=\frac{1}{2\pi i}\partial_z(\log \theta)(\frac{a}{l},\tau),NEWLINE\]NEWLINE where \(\theta(z,\tau)\) is the standard theta function (\(z\in {\mathbb C}\), Im \(\tau>0\)). The main result of the paper is that the space spanned by pairwise products of \(s_{a/l}\) is the direct sum of the span of all Hecke eigenforms \(f\in {\mathcal S}(l)\) whose \(L\)-function does not vanish at \(s=1\) and some space of Eisenstein series.