#### Vol. 15, No. 10, 2021

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Remarks on generating series for special cycles on orthogonal Shimura varieties

### Stephen S. Kudla

Vol. 15 (2021), No. 10, 2403–2447
DOI: 10.2140/ant.2021.15.2403
##### Abstract

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

$\left({\left(m,2\right)}^{{d}_{+}},{\left(m+2,0\right)}^{d-{d}_{+}}\right),\phantom{\rule{1em}{0ex}}1\le {d}_{+}

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 ${O}_{\phantom{\rule{-0.17em}{0ex}}F}$. 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 $1\le n\le m$.

##### Keywords
orthogonal Shimura varieties, special cycles, Hilbert–Siegel modular forms
##### Mathematical Subject Classification 2010
Primary: 14C25
Secondary: 11F27, 11F46, 14G35