Title: Eichler-Selberg relations for singular moduli Show Chinese title

Title: Eichler-Selberg 关系式对于奇异模数

Abstract: 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 $j_0(\tau)=1$. More generally, we consider the singular moduli for the Hecke system of modular functions \[ j_m(\tau) := mT_m \left(j(\tau)-744\right). \] For each $\nu\geq 0$ and $m\geq 1$, we obtain an Eichler-Selberg relation. For $\nu=0$ and $m\in \{1, 2\},$ these relations are Kaneko's celebrated singular moduli formulas for the coefficients of $j(\tau).$ For each $\nu\geq 1$ and $m\geq 1,$ we obtain a new Eichler-Selberg trace formula for the Hecke action on the space of weight $2\nu+2$ cusp forms, where the traces of $j_m(\tau)$ singular moduli replace Hurwitz-Kronecker class numbers. These formulas involve a new term that is assembled from values of symmetrized shifted convolution $L$-functions. Show Chinese abstract

Abstract: 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。 我们将这个公式扩展到奇异模量迹的自然类关系中，其中将类数视为常函数$j_0(\tau)=1$的迹。 更一般地，我们考虑模函数\[ j_m(\tau) := mT_m \left(j(\tau)-744\right). \]的Hecke系统中的奇异模量。 对于每个$\nu\geq 0$和$m\geq 1$，我们得到一个Eichler-Selberg关系。 对于$\nu=0$和$m\in \{1, 2\},$这些关系是Kaneko的著名奇异模形式公式，用于$j(\tau).$系数。对于每个$\nu\geq 1$和$m\geq 1,$我们得到一个新的Eichler-Selberg迹公式，用于权为$2\nu+2$的尖点形式空间上的Hecke作用，其中$j_m(\tau)$奇异模形式的迹取代了Hurwitz-Kronecker类数。 这些公式涉及一个新项，该新项由对称化平移卷积$L$-函数的值组成。