奇异模的Eichler-Selberg关系
Eichler-Selberg relations for singular moduli
AI总结：
本研究将Eichler-Selberg迹公式推广到模函数Hecke系统的奇异模迹，得到一族新的迹关系，包含由对称化移位卷积L-函数值构成的新项，涵盖了Kaneko的经典奇异模公式。
摘要：
Eichler-Selberg迹公式将Hecke算子在尖点形式空间上的迹表示为Hurwitz-Kronecker类数的加权和。我们将该公式推广为一类关于奇异模迹的自然关系，其中类数被视为常函数$j_0(τ)=1$的迹。更一般地，我们考虑模函数Hecke系统的奇异模：\n\\[ j_m(τ) := mT_m \left(j(τ)-744\right). \\]\n对每个$ν\geq 0$和$m\geq 1$，我们得到一个Eichler-Selberg关系。当$ν=0$且$m\in \{1, 2\}$时，这些关系就是Kaneko著名的关于$j(τ)$系数的奇异模公式。对每个$ν\geq 1$和$m\geq 1$，我们得到一个新的关于权$2ν+2$尖点形式空间上Hecke作用的Eichler-Selberg迹公式，其中$j_m(τ)$奇异模的迹替代了Hurwitz-Kronecker类数。这些公式包含一个由对称化移位卷积$L$-函数值构成的新项。
英文摘要：
The Eichler-Selberg trace formula expresses the trace of Hecke operators on spaces of cusp forms as weighted sums of Hurwitz-Kronecker class numbers. We extend this formula to a natural class of relations for traces of singular moduli, where one views class numbers as traces of the constant function $j_0(τ)=1$. More generally, we consider the singular moduli for the Hecke system of modular functions \[ j_m(τ) := mT_m \left(j(τ)-744\right). \] For each $ν\geq 0$ and $m\geq 1$, we obtain an Eichler-Selberg relation. For $ν=0$ and $m\in \{1, 2\},$ these relations are Kaneko's celebrated singular moduli formulas for the coefficients of $j(τ).$ For each $ν\geq 1$ and $m\geq 1,$ we obtain a new Eichler-Selberg trace formula for the Hecke action on the space of weight $2ν+2$ cusp forms, where the traces of $j_m(τ)$ singular moduli replace Hurwitz-Kronecker class numbers. These formulas involve a new term that is assembled from values of symmetrized shifted convolution $L$-functions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
