In this note, we consider special algebraic cycles on the Shimura variety
associated to a
quadratic space
over
a totally real field
,
, of
signature
For each
,
, there are
special cycles
in
of
codimension
,
indexed by totally positive semidefinite matrices with coefficients in the ring of integers
. The
generating series for the classes of these cycles in the cohomology group
are Hilbert–Siegel modular forms of parallel weight
. One can
form analogous generating series for the classes of the special cycles in the Chow group
. For
and
, 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
on
is modular
for all
with
.
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