Simultaneous nonvanishing of quadratic twists via Rankin-Cohen brackets

Abstract

Let $D$ be an odd fundamental discriminant, with $D=1$ permitted, and let $r\geq 1$ be fixed. We prove that, for every sufficiently large integer $\ell$ satisfying $(-1)^\ell D>0$, the first $r$ traced diagonal Rankin--Cohen brackets \[ \mathrm{Tr}_1^{|D|}[G_{\ell-2e,D},G_{\ell-2e,D}]_{2e}, \qquad 1\leq e\leq r, \] are linearly independent in $S_{2\ell}(SL_2(\mathbb Z))$. Here $G_{k,D}$ is the Eisenstein series of weight $k$, level $|D|$, and nebentypus $\chi_D$. The Petersson formula of Kayath, Lane, Neifeld, Ni, and Xue then implies that at least $r$ normalized Hecke eigenforms $f\in S_{2\ell}(SL_2(\mathbb Z))$ satisfy $L(f\otimes\chi_D,\ell)\neq 0$. For $D=1$, this gives, for every fixed $r$ and every sufficiently large $K\equiv 0\pmod 4$, at least $r$ level-one eigenforms of weight $K$ with nonzero central value.