In this note, we consider special algebraic cycles on the Shimura variety $S$ associated to a quadratic space $V$ over a totally real field $F$, $|F:\mathbb{Q}|=d$, of signature

For each $n$, $1\le n\le m$, there are special cycles $Z\left(T\right)$ in $S$ of codimension $n{d}_{+}$, indexed by totally positive semidefinite matrices with coefficients in the ring of integers $O_F$. The generating series for the classes of these cycles in the cohomology group $H^{2n{d}_{+}}\left(S\right)$ are Hilbert–Siegel modular forms of parallel weight $\frac{m}{2}+1$. One can form analogous generating series for the classes of the special cycles in the Chow group ${CH}^{n{d}_{+}}\left(S\right)$. For $d_{+}=1$ and $n=1$, the modularity of these series was proved by Yuan, Zhang and Zhang. In this note we prove the following: Assume the Bloch–Beilinson conjecture on the injectivity of Abel–Jacobi maps. Then the Chow group valued generating series for special cycles of codimension $n{d}_{+}$ on $S$ is modular for all $n$ with $1\le n\le m$.

在本笔记中，我们考虑了 Shimura 变体的特殊代数循环$\mathrm{小号}$与二次空间相关联$五$在一个完全真实的领域$F$,$|F：\mathbb{Q}|=d$, 的签名

对于每个$n$,$1\le n\le 米$, 有特殊的循环$Z\left(吨\right)$在$\mathrm{小号}$维数$n{d}_{+}$，由具有整数环中的系数的完全正半定矩阵索引${○}_F$. 上同调群中这些环类的生成序列$H^{2n{d}_{+}}\left(\mathrm{小号}\right)$是 Hilbert-Siegel 模形式的平行权重$\frac{米}{2}+1$. 可以为 Chow 群中的特殊循环的类形成类似的生成序列${甲烷}^{n{d}_{+}}\left(\mathrm{小号}\right)$. 为了$d_{+}=1$和$n=1$，这些系列的模块化被袁、张和张证明了。在这篇笔记中，我们证明了以下几点：假设关于 Abel-Jacobi 映射的注入性的 Bloch-Beilinson 猜想。然后 Chow 小组重视生成序列的特殊循环的 codimension$n{d}_{+}$在$\mathrm{小号}$对所有人都是模块化的$n$和$1\le n\le 米$.