AI 中文总结
本文通过Rankin-Cohen括号构造线性无关的迹，证明二次扭量L函数在中心点的同时非消失，并给出level-one情形下的具体结果。
AI 中文摘要
设$D$为奇数基本判别式，允许$D=1$，并设$r\geq 1$固定。我们证明，对于每个满足$(-1)^\ell D>0$的足够大的整数$\ell$，前$r$个迹对角Rankin-Cohen括号 \\[ \mathrm{Tr}_1^{|D|}[G_{\ell-2e,D},G_{\ell-2e,D}]_{2e}, \qquad 1\leq e\leq r, \\] 在$S_{2\ell}(SL_2(\mathbb Z))$中线性无关。这里$G_{k,D}$是权重$k$、水平$|D|$、nebentypus $\chi_D$的Eisenstein级数。Kayath、Lane、Neifeld、Ni和Xue的Petersson公式随后蕴含至少$r$个归一化Hecke本征形式$f\in S_{2\ell}(SL_2(\mathbb Z))$满足$L(f\otimes\chi_D,\ell)\neq 0$。对于$D=1$，这给出，对于每个固定的$r$和每个足够大的$K\equiv 0\pmod 4$，至少$r$个权重$K$的level-one本征形式具有非零中心值。
英文摘要
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 $χ_D$. 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 $L(f\otimesχ_D,\ell)\neq 0$. 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.
Comments9 pages, first draft
