Remarks on generating series for special cycles on orthogonal Shimura varieties

Kudla, Stephen S.

doi:10.2140/ant.2021.15.2403

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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

({(m, 2)}^{d_{+}}, {(m + 2, 0)}^{d - d_{+}}), 1 \leq d_{+} < d .

For each $n$ , $1 \leq n \leq m$ , there are special cycles $Z (T)$ 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 $H^{2 n d_{+}} (S)$ 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 ${CH}^{n d_{+}} (S)$ . 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 $1 \leq n \leq m$ .

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Keywords

orthogonal Shimura varieties, special cycles, Hilbert–Siegel modular forms

Mathematical Subject Classification 2010

Primary: 14C25

Secondary: 11F27, 11F46, 14G35

Milestones

Received: 10 September 2019

Revised: 25 February 2021

Accepted: 1 April 2021

Published: 8 February 2022

Authors

Stephen S. Kudla
Department of Mathematics University of Toronto Toronto ON Canada

Vol. 15, No. 10, 2021