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 : | = d, of signature

((m,2)d+ ,(m + 2,0)dd+ ),1 d+ < d.

For each n, 1 n m, there are special cycles Z(T) in S of codimension nd+, indexed by totally positive semidefinite matrices with coefficients in the ring of integers OF. The generating series for the classes of these cycles in the cohomology group H2nd+(S) are Hilbert–Siegel modular forms of parallel weight m 2 + 1. One can form analogous generating series for the classes of the special cycles in the Chow group CHnd+(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 nd+ on S is modular for all n with 1 n 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