論文の公開元へBirational geometry and compactifications of modular varieties and arithmetic of modular forms
概要
G(Af ), the Shimura datum (G,D) gives a orthogonal Shimura variety MKf over C, whose
C-valued points are given as follows:
MKf (C) = G(Q)\(D × G(Af ))/Kf .
Here Af is the ring of finite ad`eles of Q. We remark that MKf has a canonical model over
a number field called the reflex field. Hence MKf is canonically defined over Q. In this
subsection, Q is an algebraic closure of Q embedded in C. By abuse of notation, in this
chapter, the canonical model of MKf over Q is also denoted by the same symbol MKf .
Then the Shimura variety MKf is a projective variety over Q since 1 ≤ e < d. It is a
smooth variety over Q if Kf is sufficiently small.
For i = 1, . . . , e, let Li ∈ Pic(Di ) be the line bundle which is the restriction of
OP(Vσi ,C ) (−1) to Di . By pulling back to D, we get p∗i Li ∈ Pic(D), where pi : D → Di
are the projection maps. These line bundles descend to LKf ,i ∈ Pic(MKf ) ⊗Z Q and thus
we obtain L := LKf ,1 ⊗ · · · ⊗ LKf ,e on MKf .
We shall define special cycles following Kudla [92], [95] and Rosu-Yott [131]. Let
W ⊂ V be a totally positive subspace over E0 . We denote GW := ResF/Q GSpin(W ⊥ ). Let
DW := DW,1 × · · · × DW,e be the Hermitian symmetric domain associated with GW , where
DW,i := {w ∈ Di | ∀v ∈ Wσi , hv, wi = 0} (1 ≤ i ≤ e).
Then we have an embedding of Shimura data (GW , DW ) ,→ (G, D). For any open compact
subgroup Kf ⊂ G(Af ) and g ∈ G(Af ), we have an associated Shimura variety MgKf g−1 ,W
over C:
MgKf g−1 ,W (C) = GW (Q)\(DW × GW (Af ))/(gKf g −1 ∩ GW (Af )).
Assume that Kf is neat so that the following morphism
MgKf g−1 ,W (C) → MKf (C)
[τ, h] 7→ [τ, hg]
is a closed embedding [95, Lemma 4.3]. Let Z(W, g)Kf be the image of this morphism. We
consider Z(W, g)Kf as an algebraic cycle of codimension e dimE0 W on MKf defined over
Q.
For any positive integer r and x = (x1 , . . . , xr ) ∈ V r , let U (x) be the E0 -subspace of V
spanned by x1 , . . . , xr . We define the special cycle in the Chow group
Z(x, g)Kf ∈ CHer (MKf )C := CHer (MKf ) ⊗Z C
by
Z(x, g)Kf := Z(U (x), g)Kf (c1 (LK∨f ,1 ) · · · c1 (LK∨f ,e ))r−dim U (x)
if U (x) is totally positive. Otherwise, we put Z(x, g)Kf := 0.
For a Bruhat-Schwartz function φf ∈ S(V (Af )r )Kf , Kudla’s generating function is
defined to be the following formal power series with coefficients in CH er (MKf )C in the
variable τ = (τ1 , . . . , τd ) ∈ (Hr )d :
X
X
φf (g −1 x)Z(x, g)Kf q T (x) . ...
