Vol. 15, No. 10, 2021

Download this article
Download this article For screen
For printing
Recent Issues

Volume 18, 1 issue

Volume 17, 12 issues

Volume 16, 10 issues

Volume 15, 10 issues

Volume 14, 10 issues

Volume 13, 10 issues

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Author Index
To Appear
Other MSP Journals
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

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.

orthogonal Shimura varieties, special cycles, Hilbert–Siegel modular forms
Mathematical Subject Classification 2010
Primary: 14C25
Secondary: 11F27, 11F46, 14G35
Received: 10 September 2019
Revised: 25 February 2021
Accepted: 1 April 2021
Published: 8 February 2022
Stephen S. Kudla
Department of Mathematics
University of Toronto
Toronto ON