Heights of CM points I
Gross–Zagier formula
Xinyi Yuan, Shou-wu Zhang, Wei Zhang
September 24, 2008
Contents
1 Introduction and state of main
1.1 L-function and root numbers .
1.2 Waldspurger’s formula . . . .
1.3 Gross–Zagier formula . . . . .
1.4 Applications . . . . . . . . . .
1.5 Plan of proof . . . . . . . . .
1.6 Notation and terminology . .
results
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2
2
3
4
6
6
7
2 Weil representation and kernel functions
2.1 Weil representations . . . . . . . . . . .
2.2 Theta series . . . . . . . . . . . . . . . .
2.3 Shimizu lifting . . . . . . . . . . . . . . .
2.4 Integral representations of L-series . . . .
2.5 Proof of Waldspurger formula . . . . . .
2.6 Vanishing of the kernel function . . . . .
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10
10
11
12
14
17
18
3 Derivatives of kernel functions
3.1 Discrete series of weight two at infinite Places . . . . . . .
3.2 Derivatives of Whittaker Functions . . . . . . . . . . . . .
3.3 Degenerate Schwartz functions at non-archimedean places .
3.4 Non-archimedean components . . . . . . . . . . . . . . . .
3.5 Archimedean Places . . . . . . . . . . . . . . . . . . . . . .
3.6 Holomorphic projection . . . . . . . . . . . . . . . . . . . .
3.7 Holomorphic kernel function . . . . . . . . . . . . . . . . .
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21
21
24
28
30
33
34
39
4 Shimura curves, Hecke operators, CM-points
4.1 Shimura curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Hecke correspondences and generating series . . . . . . . . . . . . . . . . . .
44
44
46
1
4.3
4.4
4.5
4.6
CM-points and height series . . . . . . .
Arithmetic intersection pairing . . . . . .
Decomposition of the height pairing . . .
Hecke action on arithmetic Hodge classes
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48
49
50
53
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55
55
58
65
67
70
6 Pseudo-theta series and Gross–Zagier formula
6.1 Pseudo-theta series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Proof of Gross–Zagier formula . . . . . . . . . . . . . . . . . . . . . . . . . .
73
73
77
5 Local heights of CM points
5.1 Archimedean places . . . .
5.2 Supersingular case . . . .
5.3 Superspecial case . . . . .
5.4 Ordinary case . . . . . . .
5.5 The j -part . . . . . . . . .
1
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Introduction and state of main results
In 1984, Gross and Zagier [GZ] proved a formula that relates the Neron–Tate heights of
Heegner points to the central derivatives of some Rankin L-series under certain ramification
conditions. Since then some generalizations are given in various papers [Zh1, Zh2, Zh3].
The methods of proofs of the Gross–Zagier theorem and all its extensions depend on some
newform theories. There are essential difficulties to remove all ramification assumptions in
this method. The aim of this paper is a proof of a general formula in which all ramification
condition are removed. Such a formula is an analogue of a central value formula of Waldspurger [Wa] and has been more or less formulated by Gross in 2002 in term of representation
theory. In the following, we want to describe statements of the main results and main idea
of proof.
1.1
L-function and root numbers
Let F be a number field with adele ring A = AF . Let π = ⊗v πv be a cuspidal automorphic
×
representation of GL2 (A). Let E be a quadratic extension of F , and χ : E × \A×
E → C be a
character of finite order. We assume that
χ|A× · ωπ = 1
where ωπ is the central character of π.
Denote by L(s, π, χ) the Rankin-Selberg L-function. Denote by ( 21 , πv , χv ) = ±1 the
local root number at each place v of F , and denote
Σ = {v : (1/2, πv , χv ) 6= χv ηv (−1)},
2
where ηv is the quadratic character associated to the extension Ev /Fv . Then Σ is a finite set
and the global root number is given by
Y
(1/2, π, χ) =
(1/2, πv , χv ) = (−1)#Σ .
v
The following is a description of root numbers in terms of linear functionals:
Proposition 1.1.1 (Saito–Tunnell [Tu, Sa]). Let v be place of F and Bv a quaternion
division algebra over Fv . Let πv0 be the Jacquet–Langlands correspondence of πv on Bv× if πv
is square integrable. Fix embeddings Ev× ⊂ GL2 (Fv ) and Ev× ⊂ Bv× as algebraic subgroups.
Then v ∈ Σ if and only if
HomEv× (πv ⊗ χv , C) = 0.
Moreover
dim HomEv× (πv ⊗ χv , C) + HomEv× (πv0 ⊗ χv , C) = 1.
Here the second space is treated as 0 if πv0 is not square integrable.
Arithmetic properties of the L-function depends heavily on the parity of #Σ. When #Σ
is even, the central value L( 21 , π, χ) is related to certain period integral. Explicit formulae
have been given by Gross, Waldspurger and S. Zhang. We will recall the treatment of
Waldspurger [Wa] in next section.
When #Σ is odd then L( 12 , π, χ) = 0. Under the assumption that E/F is a CM-extension,
that πv is discrete of weight 2 for all infinite place v, and that χ is of finite order, then
the central derivative L0 ( 12 , π, χ) is related to the height pairings of some CM divisors on
certain Shimura curves. Explicit formulae have been obtained by Gross-Zagier and one of
the authors under some unramified assumptions. The goal of this paper is to get a general
explicit formula in this odd case without any unramified assumption.
1.2
Waldspurger’s formula
Assume that the order of Σ is even in this subsection. We introduce the following notations:
1. B is the unique quaternion algebra over F with ramification set Σ;
2. B × is viewed as an algebraic group over F ;
3. T = E × is a torus of G for a fixed embedding E ⊂ B;
4. π 0 = ⊗v πv0 is the Jacquet–Langlands correspondence of π on B × (A).
Define a period integral `(·, χ) : π 0 → C by
Z
`(f, χ) =
f (t)χ(t)dt,
Z(A)T (F )\T (A)
Here the integral uses the Tamagawa measure.
3
f ∈ π0.
Assume that ωπ is unitary. Then π 0 is unitary with Petersson inner product h·, ·i using
Tamagawa measure which has volume 2 on A× B × \B × (A). Fix any non-trivial Hermitian
form h·, ·iv on πv0 so that their product gives h·, ·i. Waldspurger proved the following formula
when ωπ is trivial:
Theorem 1.2.1 (Walspurger when ωπ = 1). Assume that f = ⊗v fv ∈ π 0 is decomposable
and nonzero. Then
ζF (2)L( 21 , π, χ) Y
|`(f, χ)|2 =
α(fv , χv ),
2 L(1, π, ad) v≤∞
where
L(1, ηv )L(1, πv , ad)
α(fv , χv ) =
ζv (2)L( 12 , πv , χv )
Z
Fv× \Ev×
hπv0 (t)fv , fv iv χv (t)dt.
Moreover, α(fv , χv ) is nonzero and equal to 1 for all but finitely many places v.
We interpret the formula as a result on bilinear functionals. As π 0 is unitary, the contragredient π
e0 is equal to π̄ 0 . Thus we have two bilinear functionals on π 0 ⊗ π
e0 . The first one
is
Z
−1
f1 (t)f2 (t)χ(t1 )χ̄(t2 )dt1 dt2 , f1 ∈ π 0 , f2 ∈ π
e0 .
`(f1 , f2 ) = `(f1 , χ)`(f2 , χ ) =
(Z(A)T (F )\T (A))2
And the second one is the product of local linear functionals:
Y
α(f1 , f2 ) =
α(f1v , f2v )
v
L(1, ηv )L(1, πv , ad)
α(f1v , f2v ) =
ζv (2)L( 12 , πv , χv )
Z
Fv× \Ev×
hπv0 (t)f1,v , f2,v iv χv (t)dt.
It is easy to see that both pairings are bilinear and (χ−1 , χ)-equivariant under the action of T (A) × T (A). But we know such functionals are unique up to scalar multiples by
the uniqueness theorem of the local linear functionals of Saito–Tunnell (Proposition 1.1.1).
Therefore, these two functionals must be proportional. Theorem 1.1 says that their ratio is
recognized as a combination of special values of L-functions.
1.3
Gross–Zagier formula
Now assume that Σ is odd. We further assume that
1. F is totally real and E is totally imaginary.
2. πv is discrete of weight 2 at all infinite places v of F .
3. χ is a character of finite order.
4
In this case, the set Σ must contain all infinite places. We have a totally definite quaternion
algebra B over A with ramification set Σ which does not have a model over F . For each open
×
compact subgroup U of B×
f := (B ⊗A Af ) , we have a Shimura curve XU whose complex
points at a real place τ of F can be described as
XU,τ (C) = B(τ )× \H ± × B×
f /U∞ U
where B(τ ) is a quaternion algebra over F with ramification set Σ \ {τ }, Bf is identified with
B(τ )Af as an Af -algebra, and B(τ )× acts on H ± through an isomorphism B(τ )τ ' M2 (R).
Fix an embedding EA −→B, then we have a set CU ⊂ XU (E ab ) of CM-points by E which
has a (non-canonical) identification:
CU ' E × \E × (Af )/U ∩ E × (Af )
(1.3.1)
with Galois action of Gal(E ab /E) given by left multiplication of E × (Af ) and the class field
theory. More precisely, at a complex place τE of E over τ of F , CτE (C) in XU,τ (C) is
represented by (z0 , t) with t ∈ E × (Af ) and z0 ∈ H ± is the unique point fixed by E × such
that the action of E × on the tangent space TH ± ,z0 is given by inclusion τE .
Assume that U ∩ E × (Af ) is included into ker χ. Define a divisor of degree zero
X
χ(t)([t] − ξt )
YU =
t∈E × \E × (Af )/U ∩E × (Af )
where [t] denote the CM-point in CU via identification (1.3.1) and ξt is the Hodge class of
degree 1 in the fiber of XU containing [z0 , t]. The divisor YU up to a multiple by a root of
unity does not depend on the choice of identification (1.3.1).
0 e
Let πf0 denote the Jacquet–Langlands correspondence of πf on B×
ef0 .
f . Let f ∈ πf , f ∈ π
e
We then have a Hecke operator Tf ⊗fe attached to a φ ∈ C0∞ (B×
f ) which has image f ⊗ f in
0
0
πf ⊗ π
ef via the projection
0
C0∞ (B×
ef0
f )−→πf ⊗ π
and image 0 in the projections σf0 ⊗e
σf0 for σf0 the Jacquet–Langlands correspondences of finite
parts of all other cuspidal representations σ of GL2 (A) with discrete components of weight
2 at all archimedean places. It is easy to shown that the Neron–Tate height hY, Tf ⊗feY iN T
does not depend on the choice of the lifting φ.
Theorem 1.3.1. Assume that f = ⊗fv ∈ πf0 and fe = ⊗fev ∈ π
ef0 are decomposable. Then
hY, Tf ⊗feY iN T =
ζF (2)L0 (1/2, π, χ) Y
α(fv , fev ).
4L(1, π, ad)
v<∞
Remark. For new form f , some partial results have been proved in [Zh1, Zh2, Zh3] with
more precise formula under some unramified assumptions.
5
1.4
Applications
Let π and χ satisfy the same condition as in §1.3. Then we have an abelian variety A defined
over E such that
Y
1
L(s, A) =
L(s − , π σ , χσ )
2
σ
where (π σ , χσ ) are conjugates of (π, χ) for automorphisms σ of C in the sense that the Heck
eigenvalues of π σ and χσ are σ-conjugates of those of π and χ. Let Q[π, χ] denote the subfield
generated by Heck eigenvalues of π and values of χ. Then A has a multiplication by an order
in Q[π, χ]. Replace A by an isogenous one, we may assume that A has multiplication by
Z[π, χ].
Theorem 1.4.1 (Tian–Zhang). Under the assumption above, we have:
1. If ords=1/2 L(s, π, χ) = 1, then the Mordell-Weil group A(E) as a Z[π, χ]-module has
rank 1 and the Shafarevich-Tate group X(A) is finite.
2. If ords=1/2 L(s, π, χ) = 0 and A does not have CM-type, then the Mordell-Weil group
A(E) is finite. Furthermore, if χ is trivial, and A is geometrically simple, then the
Shafarevich-Tate group X(A) is also finite.
Remark. For π as above, there is an abelian variety Aπ defined over F with L-series given
by
Y
L(s, Aπ ) =
L(s − 1/2, π σ ).
σ
We may assume that Aπ has multiplications by the ring Z[π] of the Hecke eigenvalues in π.
The variety A up to an isogeny can be obtained by the algebraic tensor product:
A := Aπ ⊗Z[π] Z[π, χ].
In terms of algebraic points,
A(Ē) = Aπ (Ē) ⊗Z[π] Z[π, χ]
with usual Galois action by Gal(Ē/E) on A(Ē) and by character χ on Z[π, χ].
Example. The theorem applies to all modular F -elliptic curves: the elliptic curve A over a
Galois extension of F such that Aσ is isogenous to A for all σ ∈ Gal(F̄ /F ).
1.5
Plan of proof
The ideal of proof of Theorem 1.3.1 which is still based on Gross–Zagier’s paper [GZ] is to
show that the kernel functions representing the the central derivative of L-series is equal to
the generating series of height pairings of CM-points. The new idea here is to construct the
kernel function and generating series systematically using Weil representations. This is a
variation of an idea used by Waldspurger in the central value formula [Wa].
6
First of all, for given Σ, we will have a quaternion algebra B over A with ramification
set Σ. The reduced norm on B defines an orthogonal space V with group GO of orthogonal
similitudes. Then we have a Weil representation of GL2 (A)×GO(A) on the space S (V×A× ).
For each φ ∈ S (V × A× ) we will construct an automorphic form I(s, g, φ) as a mixed
Eisenstein and theta series to represent the L-series L(s, π, χ).
If Σ is even, then B = BA for some quaternion algebra B over F . Then by Siegel–Weil
formulae, I(0, g, χ, φ) is equal to a period integral θ(g, φ, χ) of a theta series associate to φ.
This is essentially the Waldspurger formula.
If Σ is odd, I(0, g, χ, φ) = 0 and I 0 (0, g, χ, φ) will represent L0 (1/2, π, χ). In the case
where E/F is a CM-quadratic extension, where π has weight 2 at each archimedean place,
and where χ is a finite character, we will write I 0 (0, g, χ) as a sum I 0 (0, g, χ, φ)(v) indexed
over places v of F and a finite sum of Eisenstein series and their derivations defined in §4.4.4
in our previous paper [Zh1].
The role of theta series in Waldspurger’s work is played by the following generating series
of heights of CM-points:
Zφ (g, χ) = hZφ (g)YU , YU i
where Zφ (g) is aPgenerating series of Hecke operators on XU defined by Kudla extending the
classic formula n Tn q n . The modularity of this generating series is proved in our previous
paper [YZZ]. Using Arakelov theory, we may decompose Zφ (g, χ) into a sum of pairings
involving the Hodge class ξ’s and pairings only involving CM-points. The pairings involves
ξ’s give a linear combination of Eisenstein series and their derivations. The pairing over
CM-points can be further decomposed into a sum Zφ (g, χ)(v) indexed over places of F .
For good v, one can show that I 0 (0, g, χ)(v) is essentially Zφ (g, χ)(v). For bad v’s, we
can show that they are essential equal to a pseudo-theta series associate to quadratic space
over F . Since both I 0 (0, g, χ, φ) and Zφ (g, χ) are both automorphic, some density argument
can be used to show that their difference is essentially a linear combination of theta series
and Eisenstein series. Now the main theorem follows from the result of Saito and Tunnel on
linear forms, Proposition 1.1.1.
Acknowledgement
This research has been supported by some grants from the National Science Foundation and
Chinese Academy of Sciences.
1.6
Notation and terminology
Local fields and global fields
We normalize the absolute values, additive characters, and measures following Tate’s thesis.
Let k be a local field of characteristic zero.
• Normalize the absolute value | · | on k as follows:
It is the usual one if k = R.
7
It is the square of the usual one if k = C.
It takes the uniformizer to the inverse of the cardinality of the residue field if k is
non-archimedean.
• Normalize the additive character ψ : k → C× as follows:
If k = R, then ψ(x) = e2πix .
If k = C, then ψ(x) = e4πiRe(x) .
If k is non-archimedean, then it is a finite extension of Qp for some prime p. Take
ψ = ψQp ◦trk/Qp . Here the additive character ψQp of Qp is defined by ψQp (x) = e−2πiι(x) ,
where ι : Qp /Zp ,→ Q/Z is the natural embedding.
• We take the Haar measure dx on k to be self-dual with respect to ψ. More precisely,
If k = R, then dx is the usual Lebesgue measure.
If k = C, then dx is twice of the usual Lebesgue measure.
1
If k is non-archimedean, then vol(Ok ) = |dk | 2 . Here Ok is the ring of integers and
dk ∈ k is the different of k over Qp .
• We take the Haar measure d× x on k × as follows:
d× x = ζk (1)|x|−1
v dx.
Recall that ζk (s) = (1 − Nv−s )−1 if v is non-archimedean with residue field with Nv elements, and ζR (s) = π −s/2 Γ(s/2), ζC (s) = 2(2π)−s Γ(s). With this normalization, if
k is non-archimedean, then vol(Ok× ) = vol(Ok ).
Now go back to the totally real field F . For each place v, we choose | · |v , ψv , dxv , d× xv
as above. By tensor products, they induce global | · |, ψ, dx, d× x.
Quadratic extensions
As for the CM field E, we view V1 = (E, q) as a two-dimensional vector space over F , which
uniquely determines self-dual measures dx on AE and Ev for each place v of F . We define
the measures on the corresponding multiplicative groups by the same setting as above.
Let E 1 = {y ∈ E × : q(y) = 1} act on E by multiplication. It induces an isomorphism
SO(V1 ) ' E 1 of algebraic groups over F . We also have SO(V1 ) = E × /F × given by
E × /F × → E 1 ,
t 7→ t/t̄.
It is an isomorphism by Hilbert Theorem 90.
We have the following exact sequence
q
1 → E 1 → E × → q(E × ) → 1.
8
For all places v, we endow Ev1 the measure such that the quotient measure over q(Ev× ) is the
subgroup measure of Fv× . On the other hand, we endow Ev× /Fv× with the quotient measure.
It turns out that these two measures induce the same one on SO(V1 ).
Let v be a non-archimedean place of F , denote by dv ∈ OFv the local different of Fv ,
1
and by Dv ∈ OFv the local discriminant of Ev /Fv . Then vol(OFv ) = vol(OF×v ) = |dv |v2 and
1
vol(OEv ) = vol(OE×v ) = |Dv | 2 |dv |v . We set dv and Dv to be 1 when v is archimedean.
If v is a non-archimedean place inert in E, then
vol(Ev1 ) =
vol(OE×v )
vol(OE×v )
1
1
1
=
= |Dv | 2 |dv | 2 = |dv | 2 .
×
×
vol(q(OEv ))
vol(OFv )
If v is a non-archimedean place ramified in E, then
vol(Ev1 ) =
vol(OE×v )
vol(OE×v )
1
1
=
2
= 2|Dv | 2 |dv | 2 .
×
×
vol(q(OEv ))
vol(OFv )
If v is archimedean, then vol(Ev1 ) = 2.
Notation on GL2
We introduce the matrix notation:
a
1
a
∗
m(a) =
, d(a) =
, d (a) =
a−1
a
1
1 b
cos θ sin θ
1
n(b) =
, kθ =
, w=
.
1
− sin θ cos θ
−1
We denote by P ⊂ GL2 and B ⊂ SL2 the subgroups of upper triangular matrices. And N
be the standard unipotent subgroup.
For any place v, the character δv : P (Fv ) → R× defined by
a1
a b
2
δv :
7→ d
d
which extends to a function δv : GL2 (Fv ) → R× by Iwasawa decomposition.
If v is a real place, we define a function λv : GL2 (Fv ) → C by λv (g) = eiθ if
c
a b
cos θ sin θ
h=
c
1
− sin θ cos θ
in the form of Iwasawa decomposition.Q
Q
For g ∈ GL2 (A), we define δ(g) = v δv (gv ) and λ∞ (g) = v|∞ λv (gv ).
9
2
Weil representation and kernel functions
In this section, we will review the theory of Weil representation and its applications to integral representations of Rankin–Selberg L-series L(s, π, χ) and to a proof of Waldspurger’s
the central value formula. We will mostly follow Waldspurger’s treatment with some modifications including the construction of incoherent Eisenstein series from Weil representation.
We will start with the classical theory of Weil representation of O(F ) × SL2 (F ) on S (V )
for an orthogonal space V over a local field F and its extension to GO(F ) × GL2 (F ) on
S (V, F ) by Waldspurger. We then define theta function, state Siegel–Weil formulae, and
define normalized local Shimizu lifting. The main result of this section is an integral formula
for L-series L(s, π, χ) using a kernel function I(s, g, φ, χ). This kernel function is a mixed
Eisenstein and theta series attached to each φ ∈ S (V × A× ) for V an orthogonal space
obtained form a quaternion algebra over A. The Waldspurger formula is a direct consequence
of the Siegel–Weil formula. We conclude the section by proving the vanishing of the kernel
function at s = 0 using information on Whittaker function of incoherent Eisenstein series.
2.1
Weil representations
Let us start with some basic set up on Weil representation. We follow closely from Waldspurger’s paper [Wa] but we will work on a slightly bigger space S (V, F × ).
Non-archimedean case
Let F be a non-archimedean local field and (V, q) a quadratic space over F . Let O = O(V, q)
f 2 the double cover of SL2 . Then for any nondenote the orthogonal group of (V, q) and SL
f 2 (F ) × O(F ) has an action r on the space S (V ) of
trivial character ψ of F , the group SL
locally constant with compact support as follows:
• r(h)φ(x) = φ(h−1 x),
h ∈ O(F );
a ∈ F ×;
• r(m(a))φ(x) = χV (a)φ(ax)|a|dim V /2 ,
• r(n(b))φ(x) = φ(x)ψ(bq(x)),
b
• r(w)φ = γ φ(x),
b ∈ F;
γ 8 = 1.
Archimedean case
When F ' R, we may define an analogous representation of the pair (G , K ) of the Lie
f 2 (R) × K 0 of SL
f 2 (R) × O(R). More
algebra G and a maximal compact subgroup K = SO
0
precisely, the maximal compact subgroup K stabilizes a unique orthogonal decomposition
V = V + + V − such that the restrictions of q on V ± are positive and negative definite
respectively. Then we take S (V ) as to be the space of functions of the form
P (x)e−2π(q(x
+ )−q(x− ))
x = x+ + x− , x ± ∈ V ±
,
10
where P is a polynomial function on V . The action of (G , K ) on S (V ) can be deduced
formally from the same formulae as above. Notice that the space S (V ) depends on the
choice of the maximal compact group K 0 of O(V ) and ψ.
Extension to GL2
Assume that dim V is even, we want to extend this action to an action r of GL2 (F ) × GO(F )
or more precisely their (G , K ) analogue in real case. If F is non-archimedean, let S (V, F × )
be the space of smooth functions φ(x, u) on V × F × which is in S (V ) for any fixed u. Then
the Weil representation is defined by the following formulae:
• r(h)φ(x, u) = φ(h−1 x, ν(h)u),
• r(g)φ(x, u) = ru (g)φ(x, u),
h ∈ GO(F );
g ∈ SL2 (F );
• r(d(a))φ(x, u) = φ(x, a−1 u)|a|− dim V /4 ,
a ∈ F×
where in the second formula φ(x, u) is a function of x, and ru is the Weil representation on
V with new norm uq.
Remark. In Waldspurger’s paper [Wa], he only considers the Weil representation on the the
space S (V × F × ) consisting of locally constant functions with compact support if F is
non-archimedean, and linear combinations of functions of the form
h(u)P (x)e−2π|u|(q(x
+ )−q(x− ))
with h(u) ∈ C0∞ (R) if F is archimedean.
2.2
Theta series
Now we assume that F is a totally real number field and (V, q) is an orthogonal space over
F . Then we can define a Weil representation r on S (VA ) (which actually depends on ψ)
f 2 (A) × O(VA ). When dim V is even, we can define an action r of GL2 (A) × GO(VA )
of SL
on S (VA , A× ) which is the restricted tensor product of S (Vv , Fv ) with base element as
characteristic function of VOFv × OF×v .
Notice that the representation r depends only on the quadratic space (VA , q) over A.
Thus we may define representations directly for a pair (V, q) of a free A-module V with
non-degenerate quadratic form q.
If (V, q) is a base change of an orthogonal space over F , then we call this Weil representation is coherent; otherwise it is called incoherent.
Let F be a number field, and (V, q) a quadratic space over F . Fix a nontrivial additive
character ψ of F \A. Then for any φ ∈ S (VA ), we can form a theta series as a function on
f 2 (A) × O(F )\O(A):
SL2 (F )\SL
X
f 2 (A) × O(A).
θφ (g, h) =
r(g, h)φ(x),
(g, h) ∈ SL
x∈V
11
Similarly, when V has even dimension we can define theta series for φ ∈ S (VA × A× ) as an
automorphic form on GL2 (F )\GL2 (A) × GO(F )\GO(A):
X
θφ (g, h) =
r(g, h)φ(x, u).
(x,u)∈V ×F ×
Siegel–Weil formulae
Still in the global case. Let V be an orthogonal space over F . For φ ∈ S (VA ), s ∈ C, we
have a section
g 7→ δ(g)s r(g)φ(0)
dim V /2+s
2
in IndSL
) where δ is the modulo function which assigns g to |a|s in the Iwasawa
B (χV |·|
decomposition g = m(a)n(b)k. Thus we can form an Eisenstein series
X
E(s, g, φ) =
δ(γg)s r(γg)φ(0).
γ∈P (F )\SL2 (F )
This Then we have Siegel–Weil formula in the following cases:
A. (V, q) = (E, NE/F ) with E a quadratic extension of F , and
B. V is the space of the trace free elements in a quaternion algebra over F and q is the
the restriction of the reduced norm.
Theorem 2.2.1 (Siegel–Weil formula). Let φ ∈ S (VA ). Then
Z
θφ (g, h)dh = mE(0, g, φ)
SO(F )\SO(A)
where dh is the Tamagawa measure on SO(A) which has values 2L(1, χ) and 2 on SO(F )\SO(A)
for dim V = 2 and 3 respectively, and m = L(1, η) and 2 respectively.
Notice that the Eisenstein series can be defined for any quadratic space (V, q) with
rational det V . But its value vanishes on s = 0. Kudla has proposed a connection between
the derivative of Eisenstein series and arithmetic intersection of certain arithmetic cycles on
Shimura varieties.
2.3
Shimizu lifting
Let F be a local field and B a quaternion algebra over F . Write V = B as an orthogonal
space with quadratic form q defined by the reduced norm on B. Let B × × B × act on V by
x 7→ h1 xh−1
2 ,
x ∈ B,
hi ∈ B × .
Then we have an exact sequence:
1−→F × −→(B × × B × ) o {1, ι}−→GO(F )−→1.
12
ι−1
Here ι acts on V = B by the canonical involution, and on B × ×B × by (h1 , h2 ) 7→ (hι−1
2 , h1 ),
and here F × is embedded into the middle group by x 7→ (x, x) o 1. The theta lifting of any
representation π of GL2 (F ) on GO is induced by representation JL(e
π ) ⊗ JL(π) on B × × B × ,
JL(π) is the Jacquet–Langlands lifting of π. Recall that JL(π) 6= 0 only if B = M2 (F ) or π
is discrete, and that if JL(π) 6= 0,
dim HomGL2 (F )×B × ×B × (S (V × F × ) ⊗ π, JL(π) ⊗ JL(e
π )) = 1.
This space has a normalized form θ as follows: for any ϕ ∈ π realized as W−1 (g) in a
Whittaker model for the additive character ψ −1 , φ ∈ S (V × F × ), and for F the canonical
form on JL(π) ⊗ JL(e
π ),
Z
ζ(2)
W−1 (g)r(g)φ(1, 1)dg
(2.3.1)
F θ(φ ⊗ ϕ) =
L(1, π, ad) N (F )\GL2 (F )
The constant before the integral is used to normalize the form so that F (θφϕ ) = 1 when every
thing is unramified.
Let F be a number field and B a quaternion algebra over F . Then the Shimuzu lifting
can be realized as a global theta lifting:
Z
ζ(2)
ϕ(g)θφ (g, h)dg,
h ∈ BA× × BA× .
(2.3.2)
θ(φ ⊗ ϕ)(h) =
2L(1, π, ad) GL2 (F )\GL2 (A)
Here is the relation between global theta lifting and normalized local theta lifting:
Proposition 2.3.1. We have a decomposition θ = ⊗θv in
π ) ⊗ JL(π)).
HomGL2 (A)×B × ×B × (S (VA × A× ) ⊗ π, JL(e
A
A
Proof. It suffices to show that for decomposable φ = ⊗φv and ϕ = ⊗ϕv ,
Y
F θ(φ ⊗ ϕ) =
F θv (φv ⊗ ϕv ).
The inner product between JL(π) and JL(e
π ) is given by integration on the diagonal (A× B × \BA× )2 .
Let V = V0 ⊕ V1 correspond to the decomposition B = B0 ⊕ F with B0 the subspace of trace
free elements. Then the diagonal can be identify with SO0 = SO(V0 ) thus we have globally
Z
Z
F θ(φ ⊗ ϕ) =
dh
θφ (g, h)ϕ(g)dg.
SO0 (F )\SO0 (A)
GL2 (F )\GL2 (A)
To compute this integral, we interchange the order of integrals and apply the Siegel–Weil
formula:


Z
X
X
θφ (g, h)dh = 2 
δ(γg)s
r(γg)φ(x, u) .
SO0 (F )\SO0 (A)
γ∈P (F )\GL2 (F )
13
(x,u)∈V1 ×F
s=0
Thus we have
F θ(φ ⊗ ϕ) =2
Z
X
ϕ(g)
GL2 (F )\GL2 (A)
r(γg)φ(x, u)dg
γ∈P (F )\GL2 (F ) (x,u)∈V1 ×F
Z
=2
X
X
ϕ(g)
P (F )\GL2 (A)
(x,u)∈V1
r(g)φ(x, u)dg
×F ×
The sum here can be written as a sum of two parts I1 + I2 invariant under P (F ) where I1 is
the sum over x 6= 0 and I2 over x = 0. It is easy to see that I2 is invariant under N (A) which
contributes 0 to the integral as ϕ is cuspidal. The sum I1 is a single orbit over diagonal
group. Thus we have
Z
X
ϕ(g)
r(g)φ(1, 1)dg
2
N (F )\GL2 (A)
(x,u)∈V1 ×F
Z
=2
W−1 (g)r(g)φ(1, 1)dg.
N (A)\GL2 (A)
2.4
Integral representations of L-series
In the following we want to describe an integral representation of the L-series L(s, π, χ). Let
F be a number field with ring of adeles A. Let B be a quaternion algebra with ramification
set Σ. Fix an embedding EA −→B. We have an orthogonal decomposition
B = EA + EA j,
j2 ∈ A× .
Write V for the orthogonal space B with reduced norm q, and V1 = EA and V2 = EA j as
subspaces of VA . Then V1 is coherent and is the base change of F -space V1 := E, and V2 is
coherent if and only if Σ is even.
For φ ∈ S (V × A× ), we can form a mixed Eisenstein–Theta series:
X
X
I(s, g, φ) =
δ(γg)s
r(γg)φ(x1 , u)
(x1 ,u)∈V1 ×F ×
γ∈P (F )\GL2 (F )
Z
I(s, g, φ, χ) =
χ(t)I(s, g, r(t, 1)φ)dt.
T (F )\T (A)
For ϕ ∈ π, we want to compute the integral
Z
P (s, ϕ, φ, χ) =
ϕ(g)I(s, g, φ, χ)dg.
Z(A)GL2 (F )\GL2 (A)
14
Proposition 2.4.1 (Waldspurger). If φ = ⊗φv and ϕ = ⊗ϕv are decomposable, then
Y
P (s, ϕ, φ, χ) =
Pv (s, ϕv , φv , χv )
v
where
Z
Z
Pv (s, ϕv , φv , χv ) =
δ(g)s W−1,v (g)r(g)φv (t−1 , q(t))dg.
χ(t)dt
Z(Fv )\T (Fv )
N (Fv )\GL2 (Fv )
Proof. Bring the definition formula of I(s, g, φ, χ) to obtain an expression for P (s, ϕ, φ, χ):
Z
Z
X
s
ϕ(g)δ(g)
χ(t)
r(g, (t, 1))φ(x, u)dtdg.
Z(A)P (F )\GL2 (A)
T (F )\T (A)
(x,u)∈V1 ×F ×
We decompose the first integral as a double integral:
Z
Z
dg =
Z(A)P (F )\GL2 (A)
dndg
Z(A)N (A)P (F )\GL2 (A)
and perform the integral on N (F )\N (A) to obtain:
Z
Z
X
s
δ(g) dg
χ(t)
Z(A)N (A)P (F )\GL2 (A)
Z
T (F )\T (A)
N (F )\N (A)
Wq(x)−1 u (g)r(g, (t, 1))φ(x, u)dt.
(x,u)∈V1 ×F ×
Here as ϕ is cuspidal, the term x = 0 has no contribution to the integral. In this way, we
may change variable (x, u) 7→ (x, q(x−1 )u) to obtain the following expression of the sum:
X
W−u (g)r(g, (t, 1))φ(x, q(x−1 )u)
(x,u)∈E × ×F ×
=
X
W−u (g)r(g, (tx, 1))φ(1, u)).
(x,u)∈E × ×F ×
The sum over x ∈ E × collapses with quotient T (F ) = E × . Thus the integral becomes
Z
Z
X
s
δ(g) dg
χ(t)
W−u (g)r(g, (t, 1))φ(1, u)dt.
Z(A)N (A)P (F )\GL2 (A)
T (A)
u∈F ×
The expression does not change if we make the substitution (g, au) 7→ (gd(a)−1 , u). Thus we
have
Z
Z
X
s
δ(g) dg
χ(t)
W−u (d(u−1 )g)r(d(u−1 )g, (t, 1))φ(1, u)dt.
Z(A)N (A)P (F )\GL2 (A)
T (A)
u∈F ×
The sum over u ∈ F × collapses with quotient P (F ), thus we obtain the following expression:
Z
Z
s
P (s, ϕ, φ, χ) =
δ(g) dg
χ(t)W−1 (g)r(g)φ(t−1 , q(t))dt.
Z(A)N (A)\GL2 (A)
T (A)
15
We may decompose inside integral as
Z
Z
Z(A)\T (A)
and move the integral
obtain
R
Z(A)\T (A)
Z(A)
to the outside. Then we use the fact that ωπ · χ|A× = 1 to
Z
Z
P (s, ϕ, φ, χ) =
δ(g)s W−1 (g)r(g)φ(t−1 , q(t))dg.
χ(t)dt
Z(A)\T (A)
N (A)\GL2 (A)
When everything is unramified, Waldspurger has computed these integrals:
Pv (s, ϕv , φv , χv ) =
L((s + 1)/2, πv , χv )
.
L(s + 1, ηv )
Thus we may define a normalized integral Pv0 by
Pv (s, ϕv , φv , χv ) =
L((s + 1)/2, πv , χv ) 0
Pv (s, ϕv , φv , χv ).
L(s + 1, ηv )
This normalized local integral Pv0 will be regular at s = 0 and equal to
Z
L(1, ηv )L(1, πv , ad)
χv (t)F (JL(π)(t)θ(φv ⊗ ϕv )) dt.
ζv (2)L(1/2, πv , χv ) Z(Fv )\T (Fv )
We may write this as αv (θ(φv ⊗ ϕv )) with
αv ∈ Hom(JL(π) ⊗ JL(e
π ), C)
given by the integration of matrix coefficients:
L(1, ηv )L(1, πv , ad)
αv (ϕ ⊗ ϕ)
e =
ζv (2)L(1/2, πv , χv )
Z
Z(Fv )\T (Fv )
χv (t)(π(t)ϕ, ϕ)dt.
e
And we define an element α := ⊗v αv in Hom(JL(π) ⊗ JL(e
π ), C).
We now take value or derivative at s = 0 to obtain
Proposition 2.4.2.
P (0, ϕ, φ, χ) =
L(1/2, π, χ) Y
αv (θ(φv ⊗ ϕv )).
L(1, χE ) v
If Σ is odd, then L(1/2, π, χ) = 0, and
P 0 (0, ϕ, φ, χ) =
L0 (1/2, π, χ)
α(θ(φ ⊗ ϕ)).
2L(1, χE )
16
(2.4.1)
Remark. Let AΣ (GL2 , χ) denote the direct sum of cusp forms π on GL2 (A) such that
Σ(π, χ) = Σ. If Σ is even, let I (g, φ, χ) be the projection of I(0, g, φ, χ) on AΣ (GL2 , χ−1 ).
If Σ is odd, let I 0 (g, φ, χ) denote the projection of I 0 (0, g, φ, χ) on AΣ (GL2 , χ−1 ). Then have
shown that I (g, φ, χ) and I 0 (g, φ, χ) represent the functionals
ϕ 7→
L(1/2, π, χ) e
α(θ(φ ⊗ ϕ))
L(1, χE )
if Σ is even
ϕ 7→
L0 (1/2, π, χ) e
α(θ(φ ⊗ ϕ))
2L(1, χE )
if Σ is odd
on π respectively.
2.5
Proof of Waldspurger formula
Assume that Σ is even and we recall Waldspurger’s proof of his central value formula. Now
the space V = V (A) is coming from a global V = B over F . For φ ∈ S (V × A× ) and ϕ ∈ π,
we want to compute the double period integral of θ(φ ⊗ ϕ):
Z
θ(φ ⊗ ϕ)(t1 , t2 )χ(t1 )χ−1 (t2 )dt1 dt2 .
(T (F )Z(A)\T (A))2
Using definition, this equals
ζ(2)
2L(1, π, ad)
where
Z
ϕ(g)θ(g, φ, χ)dg
Z(A)GL2 (F )\GL2 (A)
Z
θφ (g, (t1 , t2 ))χ(t1 t−1
2 )dt1 dt2 ,
θ(g, φ, χ) =
Z ∆ (A)T (F )2 \T (A)2
where Z ∆ is the image of the diagonal embedding F × −→(B × )2 . We change variable t1 = tt2
to get a double integral
Z
Z
θ(g, φ, χ) =
χ(t)dt
θφ (g, (tt2 , t2 ))dt2 ,
T (F )\T (A)
Z(F)T (F )\T (A)
Notice that the diagonal the diagonal embedding Z\T can be realized as SO(Ej) in the
decomposition V = B = E + Ej . Thus we can apply Siegel Weil formula 2.2.1 to obtain
θ(g, φ, χ) = L(1, χ)I(0, g, φ, χ).
Here recall
I(s, g, φ) =
X
γ∈P (F )\GL2 (F )
X
δ(γg)s
(x,ψ)∈E×F ×
17
r(γg, (t, 1)φ)(x, u).
Combining with Proposition 2.4.2, we have
Z
ζ(2)L(1/2, π, χ)
θ(φ ⊗ ϕ)(t1 , t2 )χ(t1 )χ−1 (t2 )dt1 dt2 =
α(θ(φ ⊗ ϕ)).
2L(1, π, ad)
(T (F )Z(A)\T (A))2
This is certainly an identity as functionals on JL(π) ⊗ JL(e
π ):
`χ ⊗ `χ−1 =
2.6
ζ(2)L(1/2, π, χ)
· α.
2L(1, π, ad)
(2.5.1)
Vanishing of the kernel function
We want to compute the Fourier expansion of I(s, g, φ, χ) and I(s, g, φ) in the case that Σ
is odd, and prove that they vanish at s = 0. We will mainly work on φ ∈ S (V × A× ).
By writing φ as a linear combination, we may reduce the computation to the decomposable case: φ(x1 + x2 , u) = φ1 (x1 , u)φ2 (x2 , u) for xi ∈ Vi . Then we have
X
θ(g, u, φ1 )E(s, g, u, φ2 )
I(s, g, φ) =
u∈F ×
and
Z
I(s, g, φ, χ) =
χ(t)I(s, g, r(t, 1)φ)dt
T (F )\T (A)
where
θ(g, u, φ1 ) =
X
r(g)φ1 (x, u),
x∈E
E(s, g, u, φ2 ) =
X
δ(γg)s r(γg)φ2 (0, u).
γ∈P (F )\GL2 (F )
It suffices to study the behavior of Eisenstein series at s = 0. The work of Kudla-Rallis
in more general setting shows that the incoherent Eisenstein series E(s, g, u, φ2 ) vanishes at
s = 0 by local reasons. For readers’ convenience, we will show this fact by detailed analysis
of the local Whittaker functions. We will omit φ1 and φ2 in the notation.
We start with the Fourier expansion of Eisenstein series:
X
E(s, g, u) = E0 (s, g, u) +
Ea (s, g, u)
a∈F ×
where the a-th Fourier coefficient is
Z
1 b
Ea (s, g, u) =
E s,
g, u ψ(−ab)db, a ∈ F.
1
F \A
An easy calculation gives the following result.
18
(2) Assume a ∈ Fv× .
(a) If au−1 is not represented by (V2,v , q2,v ), then Wa,v (0, g, u) = 0.
(b) Assume that there exists ξ ∈ V2,v satisfying q(ξ) = au−1 . Then
Z
1
Wa,v (0, g, u) =
r2 (g)φ2,v (ξτ, u)dτ.
L(1, ηv ) Ev1
Proof. It is easy to reduce to the case u = 1. By Siegel-Weil theorem, we know that all the
equalities hold up to constants independent of g and φv . Hence we only need to check the
constants here. Many cases are in the literature (see [KRY] for example).
Proposition 2.6.3. E(0, g, u) = 0, and thus I(0, g, φ) = 0.
Proof. It is a direct corollary of Proposition 2.6.2. The Eisenstein series has Fourier expansion
X
E(0, g, u) =
Ea (0, g, u).
a∈F
The vanishing of the constant term easily follows Proposition 2.6.2. For a ∈ F × ,
Y
Wa,v (0, g, u)
Ea (0, g, u) = −
v
is nonzero only if au−1 is represented by (Ev , q2,v ) at each v.
In fact, Bv is split if and only if there exists a non-zero element ξ ∈ V2,v (equivalently,
for all nonzero ξ ∈ V2,v ) such that −q(ξ) ∈ q(Ev× ). In another word, the following identities
hold:
ε(Bv ) = ηv (−q(ξ)),
where ε(Bv ) is the Hasse invariant of Bv and ηv is the quadratic character induced by Ev .
Now the existence of ξ at v such that q2,v (ξ) = au−1 is equivalent to
ε(Bv ) = ηv (−au−1 ).
If this is true for all v, then
Y
v
ε(Bv ) =
Y
ηv (−au−1 ) = 1.
v
It contradicts to the incoherence condition that the order of Σ is odd.
20
3
Derivatives of kernel functions
In this section, we want to study the derivative of the kernel function for L-series when Σ is
odd and π has discrete components of weight 2 at archimedean places. The main content of
this section is various local formulae.
In §3.1, we will extend the result in the last section to functions in φ ∈ S (V×A× )GO(F∞ ) ,
the maximal quotient of S (V × A× ) with trivial action by GO(F∞ ). In §3.2, we decompose
the derivative I 0 (0, g) of the kernel function into a sum of finitely many usual theta series,
their derivations, and infinite many local terms I 0 (0, g)(v) indexed by places of F . Each
local term is a period integral of some kernel function K (v) (g, (t1 , t2 )). In §3.3, to simplify
the kernel function, we introduce a class of “degenerate” Schwartz functions and proves
a non-vanishing result concerning them. We will give some precise formulae for the kernel
functions in §3.4 and 3.5. In the last two section we will define and compute the holomorphic
projection of I 0 (0, g).
3.1
Discrete series of weight two at infinite Places
Discrete series of weight two at infinity
If V is positive over an F ' R of even dimension 2d, the subspace S (V, F )GO(F ) of functions
invariant under GO(V ) will be an representation of GL2 (F ) isomorphic to the discrete series
of weight d. In fact this space consists of functions on V × F × of the form:
(P1 (|u|q(x)) + sgn(u)P2 (|u|q(x)))e−2π|u|q(x)
with polynomials Pi on R. The space S (V, F )GO(F ) can be a quotient of S (V × R× ) (resp.
S (V × R× )O(F ) ) via integration over GO(F ):
Z
Z
r(h)φdh).
r(h)φdh,
(resp.
φ 7→
Z(F )
GO(F )
Thus we have an equality
O(F )
S (V, F × )GO(F ) = S (V × F × )GO(F ) = S (V × F × )Z(F ) .
where the second (resp. third) the space denote the maximal quotient of S (V × F × ) (resp.
S (V × F × )O(F ) with trivial action of GO(F ) (resp. Z(F )).
The the standard Schwartz function φ ∈ S (V, R× ) is the Gaussian
1
φ0 (x, u) = (1 + sgn(u))e−2|u|q(x)
2
Then one verifies that
(d)
r(g)φ(x, u) = Wuq(x) (g)
21
(d)
where Wa (g) is the standard Whittaker function of weight d for character e2πiax :

d
diθ

if a = 0
|y0 | 2 e
d
(d)
2πia(x
+iy
)
diθ
0
0
Wa (g) = |y0 | 2 e
e
if ay0 > 0


0
if ay0 < 0
for any a ∈ R and
g=
z0
z0
y0 x0
1
cos θ sin θ
− sin θ cos θ
∈ GL2 (R)
in the form of Iwasawa decomposition.
We may extend the result in the last section to functions in
× GO(F∞ )
S (V × A× )GO(F∞ ) := S (V∞ , F∞
)
⊗ S (Vf × A×
f ).
Theta series
First let us consider the definition of theta series. Let V be a positive definite quadratic space
over a totally real field F . Let φ ∈ S (V (A)×A× )GO(F∞ ) . There is an open compact subgroup
K ⊂ GO(V )(Af ) such that φf is invariant under the action of K by Weil representation.
Denote µK = F × ∩ K. Then µK is a subgroup of the unit group OF× , and thus is a finitely
generated abelian group. Our theta series is of the following form:
X
θφ (g, h) =
r(g, h)φ(x, u),
(x,u)∈µK \(V ×F × )
The definition here depends on the choice of K. We may normalize this by adding a factor
[OF× : µK ]−1 in front of sum. But we will not use this normalization.
By choosing a different fundamental domain, we can write
X X
θφ (g, h) = ε−1
r(g, h)φ(x, u).
K
u∈µ2K \F × x∈V
Here εK = |{1, −1}∩K| ∈ {1, 2}. In particular, εK = 1 for K small enough. The summation
over u is well-defined since φ(x, u) = r(α)φ(x, u) = φ(αx, α−2 u) for any α ∈ µK . It is an
automorphic form on GL2 (A) × GO(V )(A), provided the absolute convergence.
To show the convergence, we claim that the summation over u is actually a finite
sum depending on (g, h). For fixed (g, h), there is a compact subset K 0 ⊂ A×
f such that
r(g, h)φf (x, u) 6= 0 only if u ∈ K 0 . Thus the summation is taken over u ∈ µ2K \(F × ∩ K 0 ),
which is a finite set. In fact, it is a result of finiteness of µK \(F × ∩ K 0 ) and µ2K \µK . The
0
0
former follows from the injection µK \(F × ∩ K 0 ) ,→ (A×
f ∩ K)\K and compactness of K .
×
2
And µK \µK is finite because µK ⊂ OF is a finitely generated abelian group.
22
Alternatively, we may construct theta series for above φ ∈ S (V (A), A× ) by some function
φe = φ∞ ⊗ φf ∈ S (V × A× )O(F∞ ) such that
Z
r(h)φdh = φ0 .
Z(F∞ )
In this case,
Z
θφ (g, h) =
Z(F∞ )/µK
θφe(g, zh)dz.
Kernel functions
For φ ∈ S (V × A× )GO(F∞ ) , we may define the mixed Eisenstein-theta series as
I(s, g, φ) =
X
X
δ(γg)s
r(γg)φ(x1 , u).
(x1 ,u)∈µF \V1 ×F ×
γ∈P (F )\GL2 (F )
If χ has trivial component at infinity, then we may define
Z
I(s, g, φ, χ) = vol(KZ )
χ(t)I(s, g, r(t, 1)φ)dt.
T (F )\T (A)/Z(F∞ )KZ
It is clear that I(s, g, φ) is a finite linear combination of I(s, g, φ, χ). The definitions here do
not depend on the choice of K. Moreover if
φe = φe∞ ⊗ φf ∈ S (V × A× )O(F∞ )
such that
Z
φ∞ =
r(z)φe∞ dz
Z(F∞ )
then
Z
I(s, g, φ) =
×
Z(F∞
)/µF
e
I(s, g, r(z)φ)dz
e χ).
I(s, g, φ, χ) = I(s, g, φ,
Indeed, in the definition of I(s, g, φ, χ) in section 2.4, we may decompose the integral over T (F )\T (A) into double integrals over T (F )\T (A)/Z(F∞ )KZ and integrals over
Z(F∞ )KZ T (F )/T (F ) to obtain
Z
Z
e
e
I(s, g, φ, χ) =
χ(t)dt
I(s, g, r(tz, 1)φ)dz.
T (F )\T (A)/Z(F∞ )KZ
T (F )\T (F )Z(F∞ )KZ
The inside integral domain can be identified with
T (F )\T (F )Z(F∞ )KZ ' µZ \Z(F∞ )KZ
23
with has a fundamental domain Z(F∞ )/µK × KZ . Thus the second integral can be written
as
Z
e
vol(KZ )
I(s, g, r(tz, 1)φ)dz.
Z(F∞ )/µK
e as a double
For this last integral, we write the sum over V1 × F × in the definition of I(s, g, φ)
×
sum over µZ \V1 ×F and a sum of µZ . The first sum commutes with integral over Z(F∞ )/µK
while the second the sum collapse with quotient Z(F∞ )/µF to get an simple integral over
Z(F∞ ) which changes φe to φ.
3.2
Derivatives of Whittaker Functions
We now compute the derivative of the kernel function I(s, g, φ) for φ ∈ S (V × A× )GO(F∞ ) .
It suffices to do this for φ of the form φ = φ1 ⊗ φ2 with φi ∈ S (Vi × A× )GO(Vi∞ ) . In this
case,
X
I(s, g) =
θ(g, u, φ1 )E(s, g, u, φ2 )
u∈µ2K \F ×
where KZ is an open subset of Z(Af ) not including such that φf is invariant under KZ . It
amounts to compute the derivative of the Eisenstein series E(s, g, u, φ2 ). We may further
assume that both φi have standard components φi∞ defined in the last subsection. As before
we will suppress the dependence on φ.
Let us start with the Fourier expansion:
X
Wa (s, g, u).
E(s, g, u) = δ(g)s r2 (g)φ2 (0, u) −
a∈F
Denote by F (v) the set of a ∈ F × that is represented by (E(Av ), uq2v ) but not by (Ev , uq2,v ).
Then F (v) 6= ∅ only if E is non-split at v. By Proposition 2.6.2, Wa,v (0, g, u) = 0 for any
a ∈ F (v). Then taking the derivative yields
0
Wa0 (0, g, u) = Wa,v
(0, g, u)Wav (0, g, u).
It follows that
E 0 (0, g, u) = log δ(g)r2 (g)φ2 (0, u) − W00 (0, g, u) −
X
X
0
Wa,v
(0, g, u)Wav (0, g, u).
v nonsplit a∈F (v)
Notation. For any non-split place v, denote the v-part by
X
0
E 0 (0, g, u)(v) :=
Wa,v
(0, g, u)Wav (0, g, u).
a∈F (v)
I 0 (0, g)(v) :=
X
θ(g, u)E 0 (0, g, u)(v).
u∈µ2K \F ×
24
Then we have a decomposition
X
I 0 (0, g) = −
I 0 (0, g)(v) −
X
W00 (0, g, u)θ(g, u)+
u∈µ2K \F ×
v nonsplit
X
+ log δ(g)
r2 (g)φ2 (0, u)θ(g, u).
u∈µ2K \F ×
We first take care of I 0 (0, g)(v) for any fixed non-split v. Denote by B = B(v) the nearby
quaternion algebra. Then we have a splitting B = E + Ej. Let V = (B, q) be the corresponding quadratic space with the reduced norm q, and V = V1 + V2 be the corresponding
orthogonal decomposition. We identify the quadratic spaces V2,w = V2,w unless w = v. It
follows that for a ∈ F × , a ∈ F (v) if and only if a is represented by (V2 , uq).
Proposition 3.2.1. For non-split v,
Z
I (0, g)(v) = 2
(v)
Kφ (g, (t, t))dt,
0
Z(A)T (F )\T (A)
Z
where the integral
(v)
employs the Haar measure of total volume one, and
X
(v)
Kφ (g, (t1 , t2 )) = Kr(t1 ,t2 )φ (g) =
X
kr(t1 ,t2 )φv (g, y, u)r(g, (t1 , t2 ))φv (y, u).
u∈µ2K \F × y∈V −V1
Here for any y = y1 + y2 ∈ Vv − V1v ,
kφv (g, y, u) =
Proof. By Proposition 2.6.2,
X
E 0 (0, g, u)(v) =
L(1, ηv )
0
r(g)φ1,v (y1 , u)Wuq(y
(0, g, u).
2 ),v
vol(Ev1 )
0
v
Wuq(y
(0, g, u)Wuq(y
(0, g, u)
2 ),v
2)
y2 ∈E 1 \(V2 −{0})
1
=
v
L (1, η)
=
X
0
Wuq(y
(0, g, u)
2 ),v
y2 ∈E 1 \(V2 −{0})
1
1
vol(Ev )Lv (1, η)
1
=
1
vol(Ev )Lv (1, η)
Z
X
y2 ∈E 1 \(V2 −{0})
Z
E 1 (A)
X
E 1 \E 1 (A) y ∈V −{0}
2
2
Z
E 1 (Av )
r(g)φv2 (y2 τ, u)dτ
0
Wuq(y
(0, g, u)r(g)φv2 (y2 τ, u)dτ
2 τ ),v
0
Wuq(y
(0, g, u)r(g)φv2 (y2 τ, u)dτ.
2 τ ),v
Therefore, we have the following expression for I 0 (0, g)(v):
X X
1
I 0 (0, g)(v) =
r(g)φ1 (y1 , u)·
vol(Ev1 )Lv (1, η)
2
×
y
∈V
1
1
u∈µK \F
Z
X
0
·
Wuq(y
(0, g, u)r(g)φv2 (y2 τ, u)dτ
2 τ ),v
E 1 \E 1 (A) y ∈V −{0}
2
2
25
Move two sums inside integral to obtain:
Z
X
X
0
v −1
r(g)φ1,v (y1 , u)Wuq(y
yt, u)dt
−1 t̄),v (0, g, u)r(g)φ (t
2t
A× E × \A×
E
u∈µ2K \F × y=y1 +y2 ∈V
y2 6=0
(v)
By definition of kφv and Kφ , we have
Z
X
X
1
0
I (0, g)(v) = v
kφv (g, t−1 yt, u)r(g)φv (t−1 yt, u)dt
L (1, η) Z(A)T (F )\T (A)
u∈µ2K \F × x∈V −V1
Z
1
(v)
=
Kφ (g, (t, t))dt.
L(1, η) Z(A)T (F )\T (A)
Since vol(Z(A)T (F )\T (A)) = 2L(1, η), we get the result. Here we have used the relation
kφv (g, t−1 yt, u) = kr(t,t)φv (g, y, u) in the lemma below.
Lemma 3.2.2. The function kφv (g, y, u) behaves like Weil representation under the action
of P (Fv ) and Ev× × Ev× . Namely,
kφv (m(a)g, y, u)
kφv (n(b)g, y, u)
kφv (d(c)g, y, u)
kr(t1 ,t2 )φv (g, y, u)
=
=
=
=
|a|2 kφv (g, ay, u), a ∈ Fv×
ψ(buq(y))kφv (g, y, u), b ∈ Fv
|c|−1 kφv (g, y, c−1 u), c ∈ Fv×
−1
kφv (g, t−1
(t1 , t2 ) ∈ Ev× × Ev×
1 yt2 , q(t1 t2 )u),
Proof. These identities follow from the definition of Weil representation and some transformation of integrals. We will only verify the first identity. By definition, we can compute the
transformation of m(a) on Whittaker function directly:
Z
Wa0 ,v (s, m(a)g, u) =
δ(wn(b)m(a)g)s r2 (wn(b)m(a)g)φ2,v (0, u)ψv (−a0 b)db
ZFv
=
δ(m(a−1 )wn(ba−2 )g)s r2 (m(a−1 )wn(ba−2 )g)φ2,v (0, u)ψv (−a0 b)db
Fv
Z
−s
= |a|
δ(wn(ba−2 )g)s r2 (wn(ba−2 )g)φ2,v (0, u)|a|−1 ηv (a)ψv (−a0 b)db
Fv
Z
−s−1
= |a|
ηv (a)
δ(wn(b)g)s r2 (wn(b)g)φ2,v (0, u)ψv (−a0 a2 b)|a|2 db
Fv
= |a|−s+1 ηv (a)Wa2 a0 ,v (s, g, u).
It follows that
Wa0 0 ,v (0, m(a)g, u) = |a|ηv (a)Wa0 2 a0 ,v (0, g, u).
This implies the result by combining
r(m(a)g)φ1,v (y1 , u) = |a|ηv (a)φ1,v (ay1 , u).
26
Now we rewrite the second and the third summations, which is less complicated than the
first one.
Proposition 3.2.3.
X
I 0 (0, g) = −
v nonsplit
− c0
X
X
u∈µ2K \F ×
y∈V1
I 0 (0, g)(v) + 2 log δ(g)
X
X
r(g)φ(y, u) −
X
v
u∈µ2K \F × y∈V1
where
X
r(g)φ(y, u)
X
cφv (g, y, u)r(g v )φv (y, u),
u∈µ2K \F × y∈V1
L(s, η)
d
c0 = |s=0 log
ds
L(s + 1, η)
is a constant, and
cφv (g, y, u) = r1 (g)φ1,v (y, u)
◦ 0
W0,v
(0, g, u)
1
1
|Dv | 2 |dv | 2
+ log δ(gv )r(g)φv (y, u).
Moreover, cφv = 0 identically for all archimedean places v and almost all non-archimedean
places v.
Proof. Take derivative on
W0 (s, g, u) =
L(s, η)
L(s, η) Y ◦
W0◦ (s, g, u) =
W0,v (s, g, u).
L(s + 1, η)
L(s + 1, η) v
By Proposition 2.6.2,
Y
d
L(s, η)
L(0, η) X ◦ 0
◦
0
W0,v
W0 (0, g, u) = |s=0
W0,v (0, g, u)
W0◦ (0, g, u) +
0 (0, g, u)
ds
L(s + 1, η)
L(1, η) v
0
v 6=v
◦ 0
X
W0,v (0, g, u) v v
d
L(s, η)
r(g )φ (0, u).
= |s=0 log
r(g)φ(0, u) +
1
1
ds
L(s + 1, η)
2 |d | 2
|D
|
v
v
v
It gives the desired equality. The map
δ(g)s r(g)φ2 (0, u) 7→ W0 (s, g, u, φ2 ),
I(s, η) → I(−s, η)
is an intertwining operator. For archimedean places v and almost all non-archimedean places
v, one has the local component
◦
W0,v
(s, g, u, φ2,v ) = δ(g)−s r(g)φ2 (0, u).
And thus we have cφv = 0. See [KRY].
27
3.3
Degenerate Schwartz functions at non-archimedean places
In this subsection we introduce a class of “degenerate” Schwartz functions at a non-archimedean
place. It is generally very difficult to obtain an explicit formula of I 0 (0, g)(v) for a ramified
finite prime v. When we choose a degenerate Schwartz function at v, the function I 0 (0, g)(v)
turns out to be easier to control as we will see in the next subsection.
Define
(
{φv ∈ S (Bv × Fv× ) : φv |Bv sing ∪Ev ×Fv× ) = 0}, if Ev /Fv is nonsplit;
S 0 (Bv × Fv× ) =
{φv ∈ S (Bv × Fv× ) : φv |Bv sing ×Fv× = 0},
if Ev /Fv is split.
Here Bv sing = {x ∈ Bv : q(x) = 0}.
By compactness, for any φv ∈ S 0 (Bv × Fv× ), there exists a constant C > 0 such that
φv (x, Fv× ) = 0 if |v(q(x))| > C. The following result will be useful when we extend our final
results to general φv .
Proposition 3.3.1. Let v be a non-archimedean place. Then for any nonzero GL2 (Fv )equivariant homomorphism S (Bv × Fv× ) → πv , the image of S 0 (Bv × Fv× ) in πv is nonzero.
We will prove a more general result. For simplicity, let F be a non-archimedean local
field and let (V, q) be a non-degenerate quadratic space over F of even dimension. Then
we have the Weil representation of GL2 (F ) on S (V × F × ), the space of Bruhat-Schwartz
functions on V × F × . Let α : S (V × F × )−→σ be a surjective morphism to an irreducible
and admissible representation of GL2 (F ). We will prove the following result which obviously
implies Proposition 3.3.1.
Proposition 3.3.2. Let W be a proper subspace of V of even dimension. Assume that σ
is not one dimensional, and that in case W 6= 0, W is non-degenerate and its orthogonal
complement W 0 is anisotropic. Then there is a function φ ∈ S (V × F × ) such that α(φ) 6= 0
and that the support supp(φ) of φ contains only elements (x, u) such that q(x) 6= 0 and
W (x) := W + F x
is non-degenerate of dimension dim W + 1.
Let us start with the following Proposition which allows us to modify any test function
to a function with support at points (x, u) ∈ V × F × with components x of nonzero norm
q(x) 6= 0.
Proposition 3.3.3. Let φ ∈ S (V × F × ) be an element with nonzero image in σ. Then
there is a function φe ∈ S (V × F × ) supported on
e ⊂ supp(φ) ∩ Vq6=0 × F × .
supp(φ)
The key to prove this proposition is the following lemma. It is well-known but we give a
proof for readers’ convenience.
28
Lemma 3.3.4. Let π be an irreducible and admissible
representation of GL2 (F ) with dim >
1 t
1. Then the group N (F ) of matrixes n(t) :=
does not have any non-zero invariant
0 1
on π.
Proof. If there exists such an invariant vector v 6= 0, then by the smoothness, v is also fixed by
some compact open subgroup. In particular, v is fixed by some element γ ∈ SL2 (F ) − B(F ).
Then v is in fact SL2 -invariant since SL2 (F ) is generated by N and any one element γ in
SL2 (F ) − B(F ). Then such π must be one dimensional. Contradiction!
Proof of Proposition 3.3.3. Applying the lemma above, we obtain elements t ∈ F such that
σ(n(t))α(φ) − α(φ) 6= 0.
e with
The left hand side is equal to α(φ)
φe = n(t)φ − φ.
By definition, we have
e u) = (ψ(tuq(x)) − 1)φ(x).
φ(x,
Thus such a φe has support
e ⊂ supp(φ) ∩ Vq6=0 × F × .
supp(φ)
Proof of Proposition 3.3.2. Choose any φ such that α(φ) 6= 0. If W = 0, then we apply
Proposition 3.3.3 to obtain a function φe satisfying the proposition. Thus we assume that
W 6= 0. Since W 0 is anisotropic, we have an orthogonal decomposition V = W ⊕ W 0 , and
an identification S (V × F × ) = S (W × F × ) ⊗S (F × ) S (W 0 × F × ). The action of GL2 (F ) is
given by actions on S (W × F × ) and S (W 0 × F × ) respectively. We may assume that φ is
a pure tensor:
φ = f ⊗ f 0,
f ∈ S (W × F × ),
f 0 ∈ S (W 0 × F × ).
Since W 0 is anisotropic, S (Wq60=0 × F × ) is a subspace in S (W 0 × F × ) with quotient
S (F × ). The quotient map is given by evaluation at (0, u). Thus
S (W 0 × F × ) = S (Wq60=0 × F × ) + wS (Wq60=0 × F × )
as w acts as the Fourier transform up to a scale multiple. In this way, we may write
f 0 = f10 + wf20 ,
fi0 ∈ S (Wq60=0 × F × ).
Then we have decomposition
φ = φ1 + wφ2 .
φ1 := f ⊗ f10 , φ2 = w−1 f ⊗ f20 .
One of α(φi ) 6= 0. Thus we may replace φ by this φi to conclude that the support of φ
consists of points x = (w, w0 ) with W (x) = W ⊕ F w0 which is non-degenerate. Applying
Proposition 3.3.3 again, we may further assume that φ has support on Vq6=0 × F × .
29
3.4
Non-archimedean components
Assume that v is a non-archimedean place non-split in E. Resume the notations in the last
section. We now compute the local kernel function:
kφv (g, y, u) =
L(1, ηv )
0
r(g)φ1,v (y1 , u)Wuq(y
(0, g, u),
2 ),v
1
vol(Ev )
1
y = y1 + y2 ∈ Vv − V1v .
1
Recall that vol(OFv ) = |dv |v2 and vol(OEv ) = |Dv | 2 |dv |v under the self-dual measures.
Here dv ∈ OFv is the local different of Fv , and Dv ∈ OFv is the local discriminant of Ev /Fv .
Proposition 3.4.1. Assume a ∈ F (v) for a finite prime v.
(1) If v - 2 or E is unramified at v, then for g = 1,
v(adv )
0
Wa,v
(0, 1, u)
1
2
= |dv | log Nv
X
n=0
Nvn
Z
φ2,v (x2 , u)du x2 ,
Dn
and thus
1
L(1, ηv )
kφv (1, y, u) =
|dv | 2 log Nv φ1,v (y1 , u)
1
vol(Ev )
v(uq(y2 )dv )
X
Nvn
n=0
Z
φ2,v (x2 , u)du x2 .
Dn
Here the domain
Dn = {x2 ∈ V2,v : uq(x2 ) ∈ pnv d−1
v }
is an OEv -lattice of V2,v , and the measure du x2 is the self-dual measure of (V2,v , uq)
1
which gives vol(OEv x2 ) = |Dv | 2 |dv uq(x2 )| for any x2 ∈ V2,v .
(2) In all cases including v | 2, if φv ∈ S 0 (Vv ×Fv× ), then kφv (1, y, u) extends to a Schwartz
function of (y, u) ∈ Vv × Fv× .
Proof. It suffices to verify the formulae for Whittaker functions under the condition that
u = 1. The general case is obtained by replacing q by uq. We will drop the index u to
simplify the notation. Recall
Z
Z
s
Wa,v (s, 1) =
δ(wn(b))
φ2,v (x2 )ψv (b(q(x2 ) − a))dx2 db.
Fv
V2,v
By
δ(wn(b)) =
1
|b|−1
if b ∈ OFv ,
otherwise,
we will split the integral over Fv into the sum of an integral over OFv and an integral over
Fv − OFv . Then
Z
Z
Wa,v (s, 1) =
φ2,v (x2 )ψv (b(q(x2 ) − a))dx2 db
OFv V2,v
Z
Z
−s
+
|b|
φ2,v (x2 )ψv (b(q(x2 ) − a))dx2 db
Fv −OFv
V2,v
30
The second integral can be decomposed as
∞ Z
X
n=1
=
−n+1
p−n
v −pv
∞ Z
X
n=1
−
Nv−ns
p−n
v
Nv−ns
φ2,v (x2 )ψv (b(q(x2 ) − a))dx2 db
V2,v
Z
φ2,v (x2 )ψv (b(q(x2 ) − a))dx2 db
V2,v
∞ Z
X
−(n−1)
Nv−ns
Z
pv
n=1
Z
φ2,v (x2 )ψv (b(q(x2 ) − a))dx2 db.
V2,v
Combine with the first integral to obtain
Wa,v (s, 1) =
∞ Z
X
n=0
−
p−n
v
Nv−ns
=(1 −
φ2,v (x2 )ψv (b(q(x2 ) − a))dx2 db
V2,v
∞ Z
X
n=0
Z
p−n
v
Nv−s )
Nv−(n+1)s
Z
φ2,v (x2 )ψv (b(q(x2 ) − a))dx2 db
V2,v
∞
X
Nv−ns
n=0
Z
Z
φ2,v (x2 )ψv (b(q(x2 ) − a))dx2 db.
p−n
v
V2,v
As for the last double integral, change the order of the integration. The integral on b is
nonzero if and only if q(x2 ) − a ∈ pnv d−1
v . Here dv is the local different of F over Q, and also
the conductor of ψv . Denote
Dn (a) = {x2 ∈ V2,v : q(x2 ) − a ∈ pnv d−1
v }.
Then we have
Wa,v (s, 1) = (1 −
Nv−s )
∞
X
Nv−ns vol(p−n
v )
Z
φ2,v (x2 )dx2
Dn (a)
n=0
1
2
= |dv | (1 −
Nv−s )
∞
X
Nv−ns+n
Z
φ2,v (x2 )dx2 .
Dn (a)
n=0
We will see that Dn (a) is empty for n large enough by the condition that a ∈ F (v). Then
the sum above have only finitely many terms, and thus Wa,v (0, 1) = 0 just by the vanishing
of the factor 1 − Nv−s . Hence,
0
(0, 1)
Wa,v
1
2
= |dv | log Nv
∞
X
n=0
Nvn
Z
φ2,v (x2 )dx2 .
Dn (a)
We first show (1). We claim that v(q(x2 ) − a) = min{v(a), v(q(x2 ))} for any x2 ∈ V2,v .
Actually, if this is not true, we will have v(a−1 q(x2 )) = 0 and v(1 − a−1 q(x2 )) > 0. Then
31
a−1 q(x2 ) ∈ 1 + pv ⊂ q(Ev× ), and thus a is represented by (V2,v , q), which contradicts to the
assumption that a ∈ F (v). This simple fact implies that
n −1
Dn (a) = {x2 ∈ V2,v : q(x2 ) ∈ pnv d−1
v , a ∈ pv dv }.
We see that Dn (a) is empty if v(a) < n − v(dv ). Otherwise, Dn (a) = Dn . It proves (1).
Now we consider (2). If v satisfies the assumption of (1), then by the results we know
0
(0, 1) = 0 if v(a) is too small. On ther other hand, we have φ2,v (Dn , u) = 0 if n is
that Wa,v
0
(0, 1)
large enough by the vanishing property of the Schwartz function. It follows that Wa,v
is constant when v(a) is large enough. By both facts, we can obtain the result.
If v divides 2 and ramifies in E, then Dn (a) has no simple expression as above since we
don’t have v(q(x2 ) − a) = min{v(a), v(q(x2 ))} any more. However, we have
v(q(x2 ) − a) ≤ min{v(a), v(q(x2 ))} + v(8)
by a similar argument: if this were not true, then v(a−1 q(x2 )) = 0 and v(1−a−1 q(x2 )) > v(8).
It follows that a−1 q(x2 ) ∈ 1 + 8pv ⊂ q(Ev× ), and thus a is represented by (Ev , q), which
contradicts the assumption that a ∈ F (v).
Therefore, Dn (a) is non-empty only if v(a) > −v(8), and
Dn (a) ⊂ {x2 ∈ V2,v : 8q(x2 ) ∈ pnv d−1
v }.
It is easy to obtain (2) by a similar argument.
Corollary 3.4.2. Assume that v is a finite prime nonsplit in E such that:
• Bv is split at v;
• v - 2 if Ev /Fv is ramified;
• the local different dv is trivial (which makes the additive character ψv unramified);
• φv is the characteristic function of OBv × OF×v .
Then kφv (1, y, u) 6= 0 only if (y, u) ∈ OBv × OF×v . Here Bv the quaternion over Fv that is
non-isomorphic to Bv . Under this condition, we have
kφv (1, y, u) =
v(q(y2 )) + 1
log Nv .
2
Proof. Note that φ1,v (y1 , u) is a factor of kφv (1, y, u). To make it nonzero, we need to assume
(y1 , u) ∈ OEv × OF×v . Then
Z
v(q(y2 ))
v(q(y2 ))
X
X
L(1, ηv )
L(1, ηv )
n
φ
(x
,
u)d
x
=
kφv (1, y, u) =
log
N
N
log
N
Nvn vol(Dn )
v
2,v
2
q 2
v
v
1)
vol(Ev1 )
vol(E
Dn
v
n=0
n=0
32
with
Dn = {x2 ∈ V2,v : q(x2 ) ∈ pnv }.
It is nonzero only if v(q(y2 )) ≥ 0, which we now assume.
Recall V2,v = Ev jv . Since Bv is a matrix algebra, we can assume that q(jv ) ∈ OF×v . Then
Dn = {x2 jv : x2 ∈ Ev , q(x2 ) ∈ pnv }.
−2[ n+1 ]
[ n+1 ]
If E is unramified at v, we have Dn = pv 2 OEv jv and vol(Dn ) = Nv 2 . Then
n
Nv vol(Dn ) = 1 for even n, and Nvn vol(Dn ) = Nv−1 for odd n. Note that v(q(y2 )) is odd since
Bv is nonsplit. Hence,
kφv (1, y, u) = L(1, ηv ) log Nv
v(q(y2 )) + 1
v(q(y2 )) + 1
(1 + Nv−1 ) =
log Nv .
2
2
n
1
1
If E is ramified at v, then Dn = pv2 OEv jv and vol(Dn ) = Nv−n |Dv | 2 . By vol(Ev1 ) = 2|Dv | 2 ,
we still get the result.
3.5
Archimedean Places
For an archimedean place v, the quaternion algebra Bv is isomorphic to the Hamiltonian
quaternion. We will compute kφv (g, y, u) for standard φv of weight 2 introduced in Section
3.1. The computation here is done by [KRY].
The result involves the exponential integral Ei defined by
Z z t
e
Ei(z) =
dt, z ∈ C.
−∞ t
Another expression is
z
et − 1
dt,
t
0
where γ is the Euler constant. It follows that it has a logarithmic singularity near 0. This
fact is useful when we compare the result here with the archimedean local height, since we
know that Green’s functions have a logarithmic singularity.
Z
Ei(z) = γ + log(−z) +
Proposition 3.5.1.

− 1 Ei(4πuq(y )y ) |y |e2πiuq(y)(x0 +iy0 ) e2iθ
2 0
0
kφv (g, y, u) =
2
0
if uy0 > 0
if uy0 < 0
for any
g=
z0
z0
y0 x0
1
in the form of the Iwasawa decomposition.
33
cos θ sin θ
− sin θ cos θ
∈ GL2 (Fv )
Proof. It suffices to show the formula in the case g = 1. The general case is obtained by
Proposition 3.2.2 and the fact that r(kθ )φv = e2iθ φv .
Now we show that
(
− 12 Ei(4πuq(y2 ))e−2πuq(y) if u > 0;
kφv (1, y, u) =
0
if u < 0.
It amounts to show that, for any a ∈ F (v),
(
−πe−2πa Ei(4πa)
0
(0, 1, u) =
Wa,v
0
if u > 0;
if u < 0.
Assume u > 0 since the case u < 0 is trivial. Then a < 0 by the condition a ∈ F (v).
We compute explicitly as follows. First of all, it is easy to check δ(wn(b)) = 1/(1 + b2 ). It
follows that
Z
Z
s
δ(wn(b))
Wa,v (s, 1, u) =
φ2,v (x2 , u)ψv (b(uq(x2 ) − a))du x2 db
Fv
Z =
R
V2v
1
1 + b2
2s Z
e−2πuq(x2 )+2πib(uq(x2 )−a) du x2 db
C
Take an isometry of quadratic space (V2v , uq) ' (C, | · |2 ) to obtain
Z R
Z =
ZR
=
1
1 + b2
2s Z
1
1 + b2
2s Z
2 −a)
dx2 db
C
− 2s
(1 + ib)
2
e−2π|x2 | e2πib(|x2 |
−2π(1−ib)|x2 |2 −2πiab
e
e
Z dx2 db =
C
R
− 2s −1
(1 − ib)
1
1 + b2
2s
1
e−2πiab db
1 − ib
e−2πiab db.
R
By the computation in [KRY], page 19,
Z
s
s
d
|s=0 (1 + ib)− 2 (1 − ib)− 2 −1 e−2πiab db = −πe−2πa Ei(4πa).
ds
R
Thus
0
Wa,v
(0, 1, u) = −πe−2πa Ei(4πa).
3.6
Holomorphic projection
In this subsection we consider the general theory of holomorphic projection which we will
apply to the form I 0 (0, g, χ) in the next subsection. Denote by A (GL2 (A), ω) the space of
34
automorphic forms of central character ω, by A0 (GL2 (A), ω) the subspace of cusp forms,
(2)
and by A0 (GL2 (A), ω) the subspace of holomorphic cusp forms of parallel weight two.
The usual Petersson inner product is just
Z
(f1 , f2 ) =
f1 (g)f2 (g)dg, f1 , f2 ∈ A (GL2 (A), ω).
Z(A)GL2 (F )\GL2 (A)
(2)
Denote by Pr0 : A (GL2 (A), ω) → A0 (GL2 (A), ω) the orthogonal projection. Namely, for
(2)
any f ∈ A (GL2 (A), ω), the image Pr0 (f ) is the unique form in A0 (GL2 (A), ω) such that
(Pr0 (f ), ϕ) = (f, ϕ),
(2)
∀ϕ ∈ A0 (GL2 (A), ω).
We simply call Pr0 (f ) the holomorphic projection of f . Apparently Pr0 (f ) = 0 if f is an
Eisenstein series.
For any automorphic form f for GL2 (A) we define a Whittaker function
Z
deg F
(2)
fψ,s (g) = (4π)
W (g∞ )
δ(g)s fψ (gf h)W (2) (h)dh.
Z(F∞ )N (F∞ )\GL2 (F∞ )
Here W (2) is the standard holomorphic Whittaker function of weight two at infinity, and fψ
denotes the Whittaker function of f . As s → 0, the limit of the integral is holomorphic at
s = 0.
Proposition 3.6.1. Let f ∈ A (GL2 (A), ω) be a form with asymptotic behavior
a 0
f
g = O(|a|1− )
0 1
as |a| → ∞ for some > 0. Then the holomorphic projection Pr0 (f ) has Whittaker function
Pr0 (f )ψ (gf g∞ ) = lim fψ,s (g).
s→0
Proof. For any Whittaker function W of GL2 (A) with decomposition W (g) = W (2) (g∞ )Wf (gf )
such that W (2) (g∞ ) is standard holomorphic of weight 2 and that Wf (gf ) compactly supported modulo Z(Af )N (Af ), the Poincaré series is defined as follows:
X
ϕW (g) := lim
W (γg)δ(γg)s ,
s→0+
γ∈Z(F )N (F )\G(F )
where
1
2
δ(g) = |a∞ /d∞ | ,
g=
a b
k,
0 d
k∈U
where U is the standard maximal compact subgroup of GL2 (A). Assume that W and f have
the same central character. Since f has asymptotic behavior as in the proposition, their
35
inner product can be computed as follows:
Z
(f, ϕW ) =
f (g)ϕ̄W (g)dg
Z(A)GL2 (F )\GL2 (A)
Z
= lim
f (g)W̄ (g)δ(g)s dg
s−→0 Z(A)N (F )\GL (A)
2
Z
= lim
fψ (g)W̄ (g)δ(g)s dg.
s−→0
(3.6.1)
Z(A)N (A)\GL2 (A)
We may apply this formula to Pr0 (f ) which has the same inner product with ϕW as f .
Write
Pr0 (f )ψ (g) = W (2) (g∞ )Pr0 (f )ψ (gf ).
Then the above integral is a product of integrals over finite places and integrals at infinite
places:
Z
Z ∞
(2)
2
|W (g)| dg =
y 2 e−4πy dy/y 2 = (4π)−1 .
Z(R)N (R)\GL2 (R)
0
In other words, we have
−g
Z
Pr0 (f )ψ (gf )W̄ (gf )dgf .
(f, ϕW ) = (4π)
(3.6.2)
Z(Af )N (Af )\GL2 (Af )
As W̄ can be any Whittaker function with compact support modulo Z(Af )N (Af ), the combination of above formulae give the proposition.
We introduce an operator Pr formally defined on the function space of N (F )\GL2 (A).
For any function f : N (F )\GL2 (A) → C, denote as above
Z
deg F
(2)
δ(g)s fψ (gf h)W (2) (h)dh
fψ,s (g) = (4π)
W (g∞ )
Z(F∞ )N (F∞ )\GL2 (F∞ )
if it has meromorphic continuation around s = 0. Here fψ denotes the first Fourier coefficient
of f . Denote
f s→0 fψ,s (g),
Pr(f )ψ (gf g∞ ) = lim
f s→0 denotes the constant term of the Laurent expansion at s = 0. Finally, we write
where lim
X
Pr(f )(g) =
Pr(f )ψ (d∗ (a)g).
a∈F ×
The the above result is just Pr(f ) = Pr0 (f ). In general, Pr(f ) is not automorphic when
f is automorphic but fails the growth condition of Proposition 3.6.1.
36
Now we want to consider the holomorphic projection of I 0 (0, g, φ) for φ with standard
infinite component at infinity. First, let us compute the asymptotic behavior of I(s, g, χ)
near a cusp for small s. This function is a finite linear combination of I(s, g, χ)’s:
Z
I(s, g, χ) =
I(s, g, r(t, 1)φ)χ(t)dt
T (F )Z(F∞ )\T (A)
with central character χ|A× . Thus we can define and compute the holomorphic projection
F
using Proposition 3.6.1.
×
Proposition 3.6.2. If χ does not factor through norm NE/F : A×
E −→A , then I(s, g, χ) has
an asymptotic O(|a|1− ). Otherwise, let χ be induced from two different characters µi on
F × \A× :
χ = µ1 ◦ NE/F = µ2 ◦ NE/F .
Then there is an Eisenstein series J(s, g, χ) for the characters
a ∗
7→ |a/b|1/2±s µ−1
i (ab)
0 b
such that I(s, g, χ) − J(s, g, χ) has asymptotic behavior O(|a|1− ).
Proof. Recall that I(s, g, χ) can be rewritten as an integration of I(s, g):
Z
I(s, g, χ) =
I(s, g, r(t, 1)φ)χ(t)dt
T (F )Z(F∞ )\T (A)
and that
I(s, g) =
X
θ(g, u)E(s, g, u).
u∈µ2K \F ×
It suffices to get an asymptotic
for the theta series and the Eisenstein series for left
behavior
a 0
action by diagonal elements
. In their explicit Fourier expansions, the non-constant
0 b
term exponentially decays in as |a/b| → ∞. The constant terms have asymptotical behavior:
a 0
θ0
g, u = |a/b|1/2 θ0 (g, a−1 b−1 u) = r(g)φ(0, a−1 b−1 u)|a/b|1/2
0 b
a 0
E0 s,
g, u = |a/b|1/2+s f (g, a−1 b−1 u) + |a/b|1/2−s fe(g, a−1 b−1 u)
0 b
where
s
f (g, u) = δ(g) r2 (g)φ2 (0, u),
Z
fe(g, u) = −
N (A)
37
δ(wmg)s r2 (wng)φ2 (0, u).
It follows that I(s, g) has the following asymptotic behavior modulo a function with exponential decay,
X
X
a 0
I0 s,
g = |a/b|1+s
θ0 f (g, a−1 b−1 u) + |a/b|1−s
θ0 fe(g, a−1 b−1 u).
0 b
2
2
×
×
µK \F
µK \F
Similarly I(s, g, χ) has an asymptotic behavior modulo a function with exponential decay,
a 0
I0 s,
g, χ = |a/b|1+s I0+ (s, g, ab, χ) + |a/b|1−s I0− (s, g, ab, χ)
0 b
where
I0+ (s, g, α, χ)
Z
X
=
T (F )\T (A)
I0− (s, g, α, χ)
Z
X
=
T (F )\T (A)
θ0 f (g, q(t)uα)χ(t)dt
u∈µ2K \F ×
θ0 fe(g, q(t)uα)χ(t)dt.
u∈µ2K \F ×
It is clear that these two function vanish if χ does not factor through the the norm NE/F .
If χ factors through the norm, then χ is induced from exactly two characters µi on F × \A×
F:
χ(t) = µ1 (NE/F t) = µ2 (NE/F t). Then the above integral can be written as integrals over
F × \A× :
I0+ (s, g, α, χ) = µ1 (α)−1 I0+ (s, g, µ1 ) + µ2 (α)−1 I0+ (s, g, µ2 )
I0− (s, g, α, χ) = µ1 (α)−1 I0− (s, g, µ1 ) + µ2 (α)−1 I0− (s, g, µ2 )
where
1
I (s, g, µi ) =
2
Z
1
I (s, g, µi ) =
2
Z
+
−
X
F × \A×
X
F × \A×
θ0 f (g, zu)µi (z)dz
u∈µ2K \F ×
θ0 fe(g, zu)µi (z)dz.
u∈µ2K \F ×
In this way, we see that I0 (s, g, χ) is a linear combination of principal series induced from
characters on the upper triangular groups:
a ∗
7→ |a/b|1/2±s · µ−1
i (ab).
0 b
Let J(s, g, χ) be the Eisenstein series formed by I0 (s, g, χ):
X
J(s, g, χ) =
I0 (s, γg, χ).
γ∈P (F )\GL2 (F )
The constant term of J(s, g, χ) has two terms I0 (s, g, χ) and Ie0 (s, g, χ) which is a linear
combination of series in the characters:
a ∗
7→ |a/b|−1/2±s · µ−1
i (ab).
0 b
In this way, the proof of the proposition is complete.
38
We now are ready to apply Proposition 3.6.1 to the form I 0 (0, g, χ) − J 0 (0, g, χ). More
precisely,
Pr0 (I 0 (0, g, χ)) = Pr0 (I 0 (0, g, χ) − J 0 (0, g, χ)) = Pr(I 0 (0, g, χ)) − Pr(J 0 (0, g, χ))
where in the right-hand side the operator Pr is defined after Proposition 3.6.1.
In the following we want to study Pr(J 0 (0, g, χ)), which is computed from the limit
0
0
of Jψ,s
Writing J(s, g, χ) as a linear combination of Eisenstein series
0 (0, g, χ) as s → 0.
J ± (s, g, µi ) defined by functions in I ± (s, g, µi ), we need only treat the Fourier coefficients
±
Jψ,s
0 (s, g, µi ). From the proof of Proposition 3.6.2, we see that their Whittaker functions
have a decomposition
Jψ± (s, g, µi ) = W (2) (±s, g∞ )Jψ± (s, gf , µi )
where W (2) (s,
at each archimedean place has weight 2, trivial central character and value
g∞ ) y
0
y 1+s at g =
. In this way
0 1
±
(2)
Jψ,s
(g∞ )Jψ± (s, gf , µi )
0 (s, g, µi ) = W
Γ(1 + s + s0 )
.
(4π)s+s0
Take the limit as s0 → 0, and the derivative at s = 0 to obtain
0
(2)
Pr(J 0 (0, g, χ))ψ := 0lim Jψ,s
(g∞ )
0 (0, g, χ) = W
s −→0
Γ(1 + s)
d
|s=0 Jψ± (s, gf , µi )
.
ds
(4π)s
It is clear that it is a sum of Fourier coefficients of Eisenstein series and their derivations
(§4.4.4 in [Zh1]). In summary, we have shown:
Proposition 3.6.3. The difference Pr0 (I 0 (0, g, χ))−Pr(I 0 (0, g, χ)) is a finite sum of Eisenstein series and their derivations.
3.7
Holomorphic kernel function
We now apply holomorphic projection formula to I 0 (0, g) which has the following expression
by Proposition 3.2.3,
X
X
I 0 (0, g) = −
I 0 (0, g)(v) + 2 log δ(g)
r(g)φ(y, u)
u∈µ2K \F × ,y∈E
v nonsplit
− c0
X
r(g)φ(y, u) −
X
X
v
u∈µ2K \F × ,y∈E
u∈µ2K \F × ,y∈E
cφv (g, y, u)r(g v )φv (y, u).
For a function on f (g) on GL2 (A) which is invariant under N (F ) on the left, we let f ∗ (g)
denote the sum of non-constant coefficients in the Fourier coefficients under left translation
by N (F )\N (A).
39
Proposition 3.7.1. Modulo a finite sum of Eisenstein series and their derivation,
X
X
Pr(I 0 (0, g)) = −
Ie0 (0, g)(v) −
I 0 (0, g)(v)
v-∞ nonsplit
v|∞
X
+2
(y,u)∈µK \E × ×F ×
−
1
log |uq(y)|f )r(g)φ(y, u)
2
X
cφv (g, y, u)r(g v )φv (y, u) − c1
(log δf (gf ) +
X
X
v
u∈µ2K \F × ,y∈E ×
r(g)φ(y, u).
u∈µ2K \F × ,y∈E ×
Here for an archimedean v,
Z
I (0, g)(v) = 2
(v)
Kf
φ (g, (t, t))dt,
e0
Z(A)T (F )\T (A)
(v)
Kf
φ (g, (t1 , t2 ))
=
X
X
f s→0
lim
a∈F ×
r(g, (t1 , t2 ))φ(y)a kv,s (y),
×
y∈µK \(B(v)×
+ −E )
Z
Γ(s + 1) ∞
1
kv,s (y) =
dt,
s
2(4π)
t(1 − ξv (y)t)s+1
1
L(s, η)
1
d
c1 = |s=0 log
+ (γ + log 4π)[F : Q].
ds
L(s + 1, η)
2
Other terms are the same as in Proposition 3.2.3.
Proof. We will consider the image under Pr of each term on the right-hand side. We will see
that just a few terms are changed since the operator Pr only changes the non-holomorphic
terms.
First let’s look at I 0 (0, g)(v), which is always the main term. By Proposition 3.2.1,
Z
(v)
0
I (0, g)(v) = 2
Kφ (g, (t, t))dt.
Z(A)T (F )\T (A)
with
(v)
(v)
Kφ (g, (t1 , t2 )) = Kr(t1 ,t2 )φ (g) =
X
X
kr(t1 ,t2 )φv (g, y, u)r(g, (t1 , t2 ))φv (y, u).
u∈µ2K \F × y∈B(v)−E
Note that the integral above is just a finite sum.
We have a simple rule
r(n(b)g, (t1 , t2 ))φv (y, u) = ψ(uq(y)b) r(g, (t1 , t2 ))φv (y, u),
and its analogue
kr(t1 ,t2 )φv (n(b)g, y, u) = ψ(uq(y)b)kr(t1 ,t2 )φv (g, y, u)
40
showed in Proposition 3.2.2. By these rules it is easy to see that the first Fourier coefficient
is given by
X
(v)
Kφ (g, (t1 , t2 ))ψ =
kr(t1 ,t2 )φv (gv , yv , uv )r(g, (t1 , t2 ))φv (y, u).
(y,u)∈µK \((B(v)−E)×F × )1
If v is non-archimedean, all the infinite components are already holomorphic of weight
(v)
two. So the operator Pr doesn’t change Kφ (g, (t1 , t2 ))ψ at all. Thus
X
(v)
(v)
Pr(Kφ (g, (t1 , t2 ))) =
Kφ (d∗ (a)g, (t1 , t2 ))ψ
a∈F ×
=
X
X
kr(t1 ,t2 )φv (d∗ (a)gv , yv , uv )r(d∗ (a)g, (t1 , t2 ))φv (y, u)
a∈F × (y,u)∈µK \((B(v)−E)×F × )1
=
X
X
kr(t1 ,t2 )φv (gv , ayv , a−1 uv )r(g, (t1 , t2 ))φv (ay, a−1 u)
a∈F × (y,u)∈µK \((B(v)−E)×F × )1
=
X
X
kr(t1 ,t2 )φv (gv , yv , uv )r(g, (t1 , t2 ))φv (y, u)
a∈F × (y,u)∈µK \((B(v)−E)×F × )a
(v)
=Kφ (g, (t1 , t2 )).
Here the third equality follows from Proposition 3.2.2 that kr(t1 ,t2 )φv transforms according to the Weil representation under upper triangular matrices. We conclude that Pr
(v)
doesn’t change Kφ (g, (t1 , t2 )) if v is non-archimedean, and thus we have Pr(I 0 (0, g)(v)) =
I 0 (0, g)(v).
Now we look at the case that v is archimedean. The only difference is that we need to
f It suffices to
kφv ,s (g, y, u), and then take a “quasi-limit” lim.
replace kφv (g, y, u) by some e
consider the case that uq(y) = 1, it is given by
Z
dy0
(2)
e
y0s e−2πy0 kφv (d∗ (y0 ), y, u)
kφv ,s (g, y, u) = 4πW (gv )
.
y0
Fv,+
Then e
kφv ,s (g, y, u) 6= 0 only if u > 0, since kφv (d∗ (y0 ), y, u) 6= 0 only if u > 0.
Assume that u > 0, which is equivalent to q(y) > 0 since we assume uq(y) = 1 for the
moment. By Proposition 3.5.1,
Z
Z
dy0
1
dy0
s −2πy0
∗
y0 e
kφv (d (y0 ), y, u)
=−
y0s e−2πy0 Ei(4πuq(y2 )y0 ) y0 e−2πy0
y0
2 Fv,+
y0
Fv,+
Z ∞
1
dy0
q(y2 )
=−
y0s+1 e−4πy0 Ei(−4παy0 )
(α = −uq(y2 ) = −
> 0)
2 0
y0
q(y)
Z
Z
Z
Z
1 ∞ s+1 −4πy0 ∞ −1 −4παy0 t dy0
1 ∞ −1 ∞ s+1 −4π(1+αt)y0 dy0
=
y e
t e
dt
=
t
y0 e
dt
2 0 0
y0
2 1
y0
1
0
Z
Γ(s + 1) ∞
1
=
dt.
s+1
2(4π)
t(1 + αt)s+1
1
41
Hence,
Γ(s + 1)
e
kφv ,s (g, y, u) = W (2) (gv )
2(4π)s
Z
∞
1
t(1 −
1
q(y2 ) s+1
t)
q(y)
dt = W (2) (gv )kv,s (y).
This matches the result in the proposition. Since kv,s (y) is invariant under the multiplication
action of F × on y, it is easy to get
(v)
(v)
Pr(Kφ (g, (t1 , t2 ))) = Kf
φ (g, (t1 , t2 )),
Similarly, we have

Pr(I 0 (0, g)(v)) = Ie0 (0, g)(v).

X
Pr 
X
r(g)φ(y, u) =
u∈µ2K \F × ,y∈E
r(g)φ(y, u),
u∈µ2K \F × ,y∈E ×


X
Pr 
X
cφv (g, y, u)r(g v )φv (y, u) =
cφv (g, y, u)r(g v )φv (y, u).
u∈µ2K \F × ,y∈E ×
u∈µ2K \F × ,y∈E
It remains to take care of
X
log δ(g)
r(g)φ(y, u).
u∈µ2K \F × ,y∈E
Its first Fourier coefficient is just
X
log δ(g)
r(g)φ(y, u)
(y,u)∈µK \(E × ×F × )1
X
=
X
log δ(gf )r(g)φ(y, u) +
(y,u)∈µK \(E × ×F × )1
r(g)φf (y, u) · log δ(g∞ )W (2) (g∞ ).
(y,u)∈µK \(E × ×F × )1
Then Pr doesn’t change the first sum of the right-hand side, but changes log δ(g∞ )W (2) (g∞ )
in the second sum to some multiple c2 W (2) (g∞ ) = c2 r(g)φ∞ (y, u), where c2 is some constant.
Hence we see that

∗
X
Pr log δ(g)
r(g)φ(y, u)
u∈µ2K \F × ,y∈E
=
X
X
log δ(d∗ (a)gf )r(d∗ (a)g)φ(y, u)
a∈F × (y,u)∈µK \(E × ×F × )1
+ c2
X
X
r(d∗ (a)g)φ(y, u)
a∈F × (y,u)∈µK \(E × ×F × )1
=
X
1
(log δ(gf ) + log |uq(y)| 2 )r(g)φ(y, u) + c2
X
(y,u)∈µK \E × ×F ×
(y,u)∈µK \E × ×F ×
42
r(g)φ(y, u).
As for the constant, we have
Z
Z ∞
1
1
c2
dy
s −2πy
−2πy
y e
log |y| 2 e
e−4πy log ydy = − (γ + log 4π).
= 4π lim
= 2π
s→0 F
[F : Q]
y
2
0
v,+
43
4
Shimura curves, Hecke operators, CM-points
In this section, we will review the theory of Shimura curves, CM-points, generating series
of Hecke operators, and a preliminary decomposition of height pairing of CM-points using
Arakelov theory.
In §4.1, we will describe a projective system of Shimura curves XU for a totally definite incoherent quaternion algebra B over a totally real field F indexed by compact open
subgroups U of B×
f . In §4.2, we will define a generating function Zφ (g) with coefficients in
Pic(XU × XU ) for each φ ∈ S (V × A× ), and will show that it is automorphic in g ∈ GL2 (A).
This series is an extension of Kudla’s generating series for Shimura varieties of orthogonal
type [Ku1]. In this case, the modularity of its cohomology class is proved by Kudla-Millson
[KM1, KM2, KM3]. The modularity as Chow cycles are proved in our previous work [YZZ].
In §4.3, for E an imaginary quadratic extension of F , we define a set of CM-points by E bijec×
tive to E × \B×
f . For two CM-points represented by βi ∈ Bf , we define a function Zφ (g, β1 , β2 )
for g ∈ GL2 (A) by means of height pairing. It can be viewed as a function in g ∈ GL2 (A) and
(β1 , β2 ) ∈ GO(A) compatible with Weil representation on S (V × A× )GO(F∞ ) . This function
is automorphic for g and left invariant under the diagonal action of A×
E on (β1 , β2 ). In §4.4
and §4.5, using Arakelov theory, we will decompose Zφ (g, t1 , t2 ) into a finite sum of the form
b ηbi where ξb is the arithmetic Hodge class, a theta series, and an infinite sum of local
hZφ (g)ξ,
pairings:
X
jv (Zφ∗ (g)t1 , t2 ) log Nv + i(Zφ∗ t1 , t2 ).
hZφ∗ t1 , t2 i =
v
b ηbi is equal to a finite sum of Eisenstein series and their
In §4.6, we show that hZφ (g)ξ,
derivations in the sense of §4.4.4 in our previous paper [Zh1]. The computation of the local
pairing is the main content in the next section.
4.1
Shimura curves
In the following, we will review the theory of Shimura curves following our previous paper
[Zh2]. Let F be a totally real number field. Let B be a quaternion algebra over A with
odd ramification set Σ including all archimedean places. Then for each open subset U of B×
f
we have a Shimura curve XU . The curve is not connected; its set of connected components
is can be parameterized by F+× \A×
f /q(U ). For each archimedean place τ of F , the set of
complex points at τ is given as follows:
×
XU,τ (C) = B(τ )×
+ \H × Bf /U ∪ {Cusps}
where B(τ )×
+ is the group of totally positive elements in a quaternion algebra B(τ ) over F
with ramification set Σ \ {τ } with an action on H ± by some fixed isomorphisms
B(τ ) ⊗τ R = M2 (R)
B(τ ) ⊗ Af ' Bf ,
44
and where {Cusp} is the set of cusps which is non-empty only when F = Q and Bf = M2 (Af ).
For two open compact subgroups U1 ⊂ U2 of B×
f , one has a canonical morphism πU1 ,U2 :
XU1 −→XU2 which satisfies the composition property. Thus we have a projective system X
of curves XU . For any x ∈ Bf , we also have isomorphism Tx : XU −→Xx−1 U x which induces
an automorphism on the projective system X and compatible with multiplication on B×
f:
Txy = Tx · Ty . All of these morphisms on XU ’s has obvious description on complex manifolds
XU,τ (C). The induced actions are the obvious one on the sets of connected components after
taking norm of Ui and x.
An important tool to study Shimura curves is to use modular interpretation. For a fixed
archimedean place τ , the space H ± parameterizes Hodge structures on V0 := B(τ ) which
has type (−1, 0) + (0, −1) (resp (0, 0)) on V0 ⊗τ R (resp. V0 ⊗σ R for other archimedean
places σ 6= τ ). The non-cuspidal part of XU,τ (C) parameterizes Hodge structure and level
structures on a B(τ )-module V of rank 1.
Due to the appearance of type (0, 0), the curve XU does not parameterize abelian varieties
unless F = Q. To get a modular interpretation, we use an auxiliary imaginary quadratic
extension K over F with complex embeddings σK : K−→C for each archimedean places
σ of F other than τ . These σK ’s induce a Hodge structure on K which has type (0, 0) on
K ⊗τ R and type (−1, 0) + (0, −1) on K ⊗σ R for all σ 6= τ . Now the tensor product of Hodge
structures on VK := V ⊗F K is of type (−1, 0) + (0, −1). In this way, XU parameterizes some
abelian varieties with homology group H1 isomorphic to VK . The construction makes XU a
curve over the reflex field for σK ’s:
!
X
K] = Q
σ(x), x ∈ K .
σ6=τ
√
for
a
construction
following
Carayol
in
the
case
K
=
F
(
d) with d ∈ Q
See our paper
[Zh1]
√
where σK ( d) is chosen independent of σ.
The curve XU has a canonical integral model XU over OF . At each finite place v of
F , the base change XU,v over OFv will parameterizes some p-divisible group with some level
structure. Locally at a geometric point, XU,v is the universal deformation of the represented
p-divisible group.
Hodge class and Neron–Tate height pairing
The curve XU has a Hodge class LU ∈ Pic(XU ) ⊗ Q which is compatible with pull-back
morphism and such that LU ' ωXU when U is sufficiently small.
We also define a class ξU ∈ Pic(XU )⊗Q which has degree 1 on each connected component
and is proportional to LU . One can use ξ to define an projection Div(XU )−→Div0 (XU ) by
sending D to D −deg(D)ξU where deg D is the degree function on the connected components
{Xi } of XU : deg D(Xi ) := deg(D|Xi ). The class deg D · ξU has restriction deg(D|Xi )ξXi on
Xi .
The Jacobian variety JU of XU is defined to be the variety over F to represent line
bundles on XU (or base change ) of degree 0 on every connected components Xi . Over an
45
extension of F , JU is the product of Jacobian varieties Ji of the connected components Xi .
The Neron–Tate height h·, ·i on JU (F̄ ) is defined using the Poincare divisor on Ji × Ji .
4.2
Hecke correspondences and generating series
We want to define some correspondences on XU , i.e., some divisor classes on XU × XU . The
×
projective system of surfaces XU × XU has an action by B×
f × Bf . Let K denote the open
compact subgroup K = U × U .
Hecke operators
For any double coset U xU of U \B×
f /U , we have a Hecke correspondence
Z(x)U ∈ Z 1 (XU × XU )
defined as the image of the morphism
(πU ∩xU x−1 ,U , πU ∩x−1 U x,U ◦ Tx ) :
ZU ∩xU x−1 −→XU2 .
In terms of complex points at a place of F as above, the Hecke correspondence Z(x)U
takes
X
(z, g)−→
(z, gxi )
i
for points on XU,τ (C) represented by (z, g) ∈ H ± × Bf where xi are representatives of
U xU/U .
Hodge class
On MK := XU × XU , one has a Hodge bundle LK ∈ Pic(MK ) ⊗ Q defined as
1
LK = (p∗1 LU + p∗2 LU ).
2
Generating Function
Let V denote the orthogonal space B with quadratic form q. Let S (V × A× )GO(V∞ ) denote
× GO(F∞ )
the space S (V∞ , F∞
)
⊗ S (VAf × A×
f ) which is isomorphic to the maximal quotient
×
of S (V × A ) via integration on GO(F∞ ).
For any (x, u) ∈ V × A× , let us define a cycle Z(x, u)K on XU × XU as follows. This cycle
is non-vanishing only if q(x)u ∈ F × or x = 0. If q(x)u ∈ F × , then we define Z(x, u)K to be
the Hecke operator U xU defined in the last subsection. If x = 0, then we define Z(x, u)K to
be the push-forward of the Hodge class on the subvariety Mα which is union of connected
×
×
components Xα × Xβ with α, β ∈ F × \A× /F∞,+
ν(U ) such that u = αβ mod F+× q(U )F∞,+
.
46
For φ ∈ S (V×A× )GO(F∞ ) which is invariant under K ·GO(F∞ ), we can form a generating
series
X
Zφ (g) =
r(g)φ(x, u)Z(x, u)K .
(x,u)∈(K·GO(F∞ ))\V×A×
It is easy to see that this definition is compatible with pull-back maps in Chow groups in the
projection MK1 −→MK2 with Ki = Ui × Ui and U1 ⊂ U2 . Thus it defines an element in the
2
direct limit Ch1 (M )Q := limK Ch1 (MK ). For any h ∈ (B×
f ) , let ρ(h) denote the pull-back
morphism on Ch1 (M ) by right translation of h. Then it is easy to verify
Zr(h)φ = ρ(h)Zφ .
Proposition 4.2.1. The series Zφ is absolutely convergent and defines an automorphic form
for GL2 (A).
We will reduce the proposition to the modularity proved in [YZZ].
Let MK1 be the subvariety of the union of connected components Xα × Xβ with α, β ∈
×
1
F+× \A×
f /ν(U ) such that α ∈ F+ ν(U ). Then MK is a Shimura variety of orthogonal type
×
×
associated to the subgroup GSpin(V) of B × B of pairs (g1 , g2 ) with same norm. For any
element φ1 ∈ S (V)O(F∞ ) then one can define the generating series
X
Zφ1 (g) =
r(g)φ1 (x)Z(x)K 1
x∈K 1 ·O(F∞ )\V
where K 1 = K ∩GSpin(Vf ) and Z(x)K 1 is non-zero only if q(x) ∈ F × or x = 0. If q(x) ∈ F × ,
then Z(x)K 1 is defined as Heck operator U xU ; if x = 0, then Z(x)K 1 is defined as c1 (LK∨1 ).
In [YZZ], we have shown that Zφ1 (g) is absolutely convergent and defines an automorphic
form on SL2 (A).
For any h ∈ GO(Vb ), let ih denote the composition of the embedding M 1 −→M and the
translation Th on M . Then we can show that MK is covered by i(MK1 h ) for h runs through a
×
×
×
set of elements in GO(Vb ) = B×
f × Bf whose similitudes represents the cosets F+ \Af /ν(K),
and K h = GSpin(Vb ) ∩ hKh−1 . Thus it is clear that
X
Zφ =
ih∗ i∗h Zφ .
h
Let us compute i∗h Zφ . By definition,
X
i∗h Zφ (g) =
r(g)φ(x, u)i∗ Z(x, u)K
(x,u)∈K·GO(F∞ )\V×A×
If x = 0, then i∗h Z(x, u) 6= 0 only if u ∈ q(h)F+× ν(K) in which case it is given as Z(0)K h ;
if q(x)u ∈ F × , then i∗ Z(x, u)K 6= 0 only if ν(h)q(x) ∈ F+× ν(K) in which case it is given by
Z(y)K h , where y ∈ Kx with norm in F+× . In this way the sum above becomes
X
X
X
i∗h Zr(g)φ =
r(g, h)φ(x, u)Z(y)K h =
Zr(g,h)φ(·,u) .
× x∈K h O(F )\V
u∈µ0K \F+
∞
×
u∈µ0K \F+
47
where µ0K = F+× ∩ ν(K). We have thus shown that Zφ is a finite sum of generating series for
M 1 . It follows that Zφ (g) is invariant under left translation by elements in SL2 (F ) and is
absolutely convergent. By definition,
it isclear that Zφ is also invariant under left translation
1 0
by elements of the form d(a) =
. Thus Zφ is invariant under left translation by
0 a
GL2 (F ).
4.3
CM-points and height series
Let E be an imaginary quadratic extension of F with an embedding E(Af ) ⊂ Bf . Then
XU (F̄ ) has a set of CM-points defined over E ab , the maximal abelian extension of E. The
set CMU is stable under action of Gal(E ab /F ) and Hecke operators. More precisely, we have
a projective system of bijections
CMU ' E × \B×
f /U
(4.3.1)
which is compatible with action of Hecke operators and such that the Galois action is given
as follows: the action of Gal(E ab /E) acts by right multiplication of elements of E × (Af ) via
the reciprocity law in class field theory:
E × \E × (Af )−→Gal(E ab /E).
Such a bijection is unique up to left multiplication by elements in E × (Af ).
If τ is a real place of F , then the set CU can be described as a subset of
XU,τ (C) = B(τ )× \H ± × B×
f /U ∪ {cusps}
as the set of points
×
×
CMU = B(τ )× \B(τ )× z0 × B×
f /U ∪ {cusps} ' E \Bf /U
where E is embedded into B(τ ) compatible with isomorphism B(τ )Af = Bf and z0 ∈ H is
unique fixed point of E × .
×U
For any φ ∈ S (BA × A× )UGO(F
, we define the Neron–Tate height pairing
∞)
Zφ (g, β1 , β2 ) ∈ hZφ (g)([β1 ]U − deg([β1 ]U )ξ),
[β2 ]U − deg([β2 ]U )ξiU ,
β1 , β2 ∈ B×
f.
Using the projection formula of height pairing, we see that this definition does not depend
on the choice of U . Also this definition does not depend on the choice of the bijection 4.3.1
since the height pairing is invariant under Galois action. It follows that
Zφ (g, tβ1 , tβ2 ) = Zφ (g, β1 , β2 ).
In this way, we may view this as a function on GO(V) through projection
×
GO(V) = ∆(A)\B× × B× −→GO(Vf ) = ∆(Af )\B×
f × Bf
48
Using the projection formula and the formula Zr(h)φ = ρ(h)Zφ (g) where ρ(h) is the pullback morphism of right translation by h ∈ GO(V), one can show that the resulting function
Zφ (g, h) is equivariant under Weil representation:
Zr(g1 ,h1 )φ (g2 , h2 ) = Zφ (g2 g1 , h2 h1 ),
gi ∈ GL2 (A),
hi ∈ GO(V).
This function is automorphic for the first variable and invariant for the second variable under
left diagonal multiplication by A×
E.
×
Let T denote E as an algebraic group over F . Any β ∈ B×
f gives a CM-point in CMU
which is denoted by [β]U or just β if U is clear. We are particularly interested in the case
that β ∈ T (Af ), i.e. CM-points that are in the image in XU of the zero-dimensional Shimura
variety
CU = T (F )\T (Af )/UT
associated to T = E × . Here UT = U ∩ T (Af ). Notice that the set CU does not depend on
the choice of bijection 4.3.1. For a finite character χ of T (F )\T (A), we can define
Z
Zφ (g, χ) =
Zφ (g, t1 , t2 )χ(t1 t−1
2 )dt1 dt2 .
(T (F )\T (Af ))2
4.4
Arithmetic intersection pairing
The Hodge bundle LU can be extended into a metrized line bundle on XU . More precisely,
at an archimedean place, the metric can defined by using Hodge structure or equivalently
normalized such that its pull-back on Ω1H takes the form:
kf (z)dzk = 4π · Imz · |f (z)|.
At a finite place, we may take an extension LU,v by using the fact that LU is twice of
the cotangent bundle of the divisible groups. The resulting bundle on XU with metrics at
cU . Also we have an arithmetic class ξbU induced from
archimedean places is denoted by L
cU .
L
The Hodge index theorem on the Jacobian Ji of geometric connected components Xi of
X implies a Hodge index theorem on XU : for any two divisors D1 and D2 of degree 0 on
every connected component. Then their Neron–Tate height pairing is given by the following
formula:
b1 · D
b2
hD1 , D2 i = −D
b i are some “flat” arithmetic divisors extending Di . Here, D
b is flat in the sense that
where D
its curvature at archimedean places and its intersection with vertical divisors are all 0.
c U )ξ of arithmetic
For the computation in the next section, we will consider the group Pic(X
c such that b
c) − deg D · b
b is flat, we call such that line bundle ξb
line bundles L
c1 (L
c1 (ξ)
b
b which is ξ-admissible
admissible. Moreover for any divisor D, there is a unique extension D
b Such
such that the vertical component in every fiber of XU has zero intersection with ξ.
a construction is totally local and compatible with pull-back morphism for the projections
49
XU1 −→XU2 and Hecke operators. In particular, we have a well defined Green’s function
gv (x, y) on Xv (F̄v ) such that for any two distinct points x, y ∈ XU (K),
x
b · yb = −
X
1
[K : F ] v
X
gv (σ(x), σ(y)).
σ:K−→F̄v
On XU , one has a decomposition
gv (x, y) = iv (x, y) + jv (x, y)
where iv (x, y) is the intersection of the Zariski closures of x and y and jv (x, y) is a locally
constant function on XU,v (F̄v ) × XU,v (F̄v ).
4.5
Decomposition of the height pairing
Our goal in the geometric side is to compute
Zφ (g, t1 , t2 ) = hZφ (g)(t1 − deg(t1 )ξ), t2 − deg(t2 )ξi,
t1 , t2 ∈ CU .
ˆ we get the decomposition:
Once ξ is extended to an arithmetic class ξ,
hZφ (g)t1 , t2 i − hZφ (g)t1 , deg(t2 )ξi − hZφ (g) deg(t1 )ξ, t2 i + hZφ (g) deg(t1 )ξ, deg(t2 )ξi.
We will take care of the last three terms, but let us first consider hZφ (g)t1 , t2 i which is
the main term. We start with some general discussion on decomposition of intersection of
CM-points.
Decomposition of the height pairing
Now we want to write the height pairing into a sum of local terms in the case that β1 6= β2 ∈
B×
f , and get a similar decomposition even in the case β1 = β2 with an“error term” i0 (β, β).
We can always write
hβ1 , β2 i = i(β1 , β2 ) + j(β1 , β2 )
as in [Zh2].
We first assume that β1 6= β2 . Then we have decompositions
X
X
i(β1 , β2 ) =
iv (β1 , β2 ) log Nv ,
j(β1 , β2 ) =
jv (β1 , β2 ) log Nv
v∈ΣF
v∈ΣF
with
iv (β1 , β2 ) =
X
1
iw (β1 , β2 ),
#ΣEv w∈Σ
jv (β1 , β2 ) =
Ev
X
1
jw (β1 , β2 ).
#ΣEv w∈Σ
Ev
Here ΣEv denotes the set of places of E lying over v.
50
To compute these local pairings, we will use:
Z
X
1
σ
σ
iw (β1 , β2 ) =
iw̄ (β1 , β2 ) =
[L : E]
σ:L,→E w
Z
X
1
σ
σ
jw (β1 , β2 ) =
jw̄ (β1 , β2 ) =
[L : E]
σ:L,→E w
iw̄ (tβ1 , tβ2 )dt,
T (F )\T (Af )
jw̄ (tβ1 , tβ2 )dt.
T (F )\T (Af )
Here L is any Galois extension of E that contains the residue fields of β1 , β2 , and the integral
on T (F )\T (Af ) takes the Haar measure with total volume one.
The pairing jw is zero identically if w is archimedean or XU,E has good reduction at w.
Furthermore, all the decompositions above for j still makes sense even if β1 = β2 .
Now we consider the self-intersection i(β, β) for any β ∈ CMU . In next section, we will
have an extended definition of iw̄ (β, β) for each place w of E. Then formally, we still define
iw (β, β) and iv (β, β) by the above averaging formulas. The difference
X
iv (β, β) log Nv
i0 (β, β) := i(β, β) −
v∈ΣF
is not zero any more. We will see that it is essentially the Faltings height of β by an arithmetic
adjunction formula. But for the Gross-Zagier formula, it is enough to use the property that
i0 (β, β) depends only on the Galois orbit of β.
Decomposition of the kernel Function
Rewrite Zφ (g) according to a = q(x)u ∈ F to obtain
X X
Zφ (g) = Zφ,0 (g) +
r(g)φ(x)a Z(x)U
a∈F × x∈K\B×
f
where
X
Zφ,0 (g) =
r(g)φ(0, u)[Lu−1 ],
×
u∈ν(K)F∞
\A×
and
φ(x)a = φ(x, q(x)−1 a).
The Hecke operator Z(x)U corresponds to the double coset U xU , i.e.,
X
a
Z(x)U t1 =
t1 αj , if U xU =
αj U.
j
j
It follows that
Zφ (g)[t1 ] = Zφ,0 (g)[t1 ] +
X
X
r(g)φ(x)a Z(x)U [t1 ]
a∈F × x∈K\B×
f
= Zφ,0 (g)[t1 ] +
X
X
a∈F ×
x∈B×
f /U
51
r(g)φ(x)a [t1 x].
Since [t1 x] = [t2 ] in CMU if and only if x ∈ t−1
1 t2 T (F )U , the part of self-intersection
comes from
X
X
r(g)φ(x)a [t1 x]
Rφ (g)[t1 ] =
a∈F × x∈t−1
1 t2 T (F )U/U
=
X
X
r(g)φ(t−1
1 yt2 )a [t2 ]
a∈F × y∈E × /(E × ∩U )
=
X
1
[E × ∩ U : µU ]
×
a∈F
X
r(g, (t1 , t2 ))φ(y)a [t2 ].
y∈E × /µ
U
The index [E × ∩ U : µU ] = 1 when U is small enough. In the following, for simplicity we
always assume that E × ∩ U = µU .
Now we consider the decomposition of the global height hZφ (g)t1 , t2 i. We first write
hZφ (g)t1 , t2 i = hZφ,0 (g)t1 , t2 i + hZφ∗ (g)t1 , t2 i
X
jv (Zφ∗ (g)t1 , t2 ) log Nv + i(Zφ∗ (g)t1 , t2 ),
= hZφ,0 (g)t1 , t2 i +
v
where
Zφ∗ (g) = Zφ (g) − Zφ,0 (g) =
Zφ∗ (g)[t1 ] =
X
X
a∈F ×
x∈B×
f /U
X
X
a∈F ×
x∈K\B×
f
r(g)φ(x)a Z(x)U ,
r(g)φ(x)a [t1 x].
Then
i(Zφ∗ (g)t1 , t2 )
X X
r(g)φ(x)a i(t1 x, t2 )
=
a∈F × x∈B× /U
f
=
X
a∈F ×
=
X
X
r(g, (t1 , t2 ))φ(y)a i0 (t2 , t2 ) +
y∈E × /µ
U
iv (Zφ∗ (g)t1 , t2 ) log Nv + i0 (1, 1)
X
X
X
a∈F ×
x∈B×
f /U
X
r(g)φ(x)a
X
iv (t1 x, t2 ) log Nv
v
r(g, (t1 , t2 ))φ(y)a .
a∈F × y∈E × /µU
v
Here we denote
iv (Zφ∗ (g)t1 , t2 ) =
X
X
a∈F × x∈B× /U
f
r(g)φ(x)a
X
iv (t1 x, t2 ).
v
And we have i0 (t2 , t2 ) = i0 (1, 1) since [1] and [t] are Galois conjugate CM points.
52
4.6
Hecke action on arithmetic Hodge classes
In this subsection, we want to discuss the action of Hecke operator Zφ on the arithmetic
classes. More precisely, we want to study the pairing
hZφ ξbf , ηbi
where f is a function on the set of connected components and ηb is an arithmetic divisor
class. Recall using the usual uniformization at an archimedean place, the set of connected
components can be indexed by a group
F+× \A×
f /ν(U ).
The function on this set is generated by characters ω. Thus it suffices to treat the case where
f = ω is a character.
The action of Zφ on the geometric cycle ξω is given by the action on ω, the function on
the set of connected components. By a result of Kudla [Ku1], the ratio of Zφ∗ (ξω ) over ξω is
given by an Eisenstein series E(g, φ, ω) with central character ω −2 :
Zφ∗ (g)ξω = E ∗ (g, φ, ω)ξω .
The Whittaker function of this Eisenstein series is given by
X
Eψ (g, φ, ω) =
r(g)φ(x, q(x)−1 )Z(x, ω)K
x∈KGO(F∞ )\V×
where
Z
Z(x, ω)K =
ω(h)dh.
Z(x)K
It follows that the action on the arithmetic cycle ξb will have formula
Zφ∗ (g)ξbω = E ∗ (g, φ, ω)ξbω + V
where V is a vertical cycle. In the following we want to discuss in more details the Whittaker
coefficient function of Zφ (g).
Assume that φ = ⊗φv . By definition, the Whittaker function of Zφ (g) is given by
Y
Zφ,ψ (g) =
Zφv ,ψv (gv )
v
where
Zv (gv ) =
X
r(gv )φv (xv , q(xv )−1 )Z(xv )Kv .
x∈Kv \V×
v
Here Kv = GO(Fv ) for v | ∞. Notice that for almost all v, Zv (gv ) is a trivial operator.
Indeed, for almost all v, gv ∈ GL2 (Ov ), φv is the characteristic function of OBv × Ov× where
53
×
×
. Thus the sum Zv (gv ) has only one
× OB
OBv is a maximal order of Bv , and Kv = OB
v
v
term Z(1)Kv which is an identity operator. For each v, Zv (gv ) is an operator which is étale
outside v.
Let S be a finite subset of F including all places in Σ and places where Uv is not maximal.
Then the Shimura curve XU has good reduction outside S. Using the same argument as in
our previous paper [Zh1] we can show the following identity:
Zv (gv )ξbω = Zv (g, ω)ξbω + D(gv )
where D(gv ) is a constant multiple of the divisor ωXv supported on the special fiber of X
over v whose connected components is also indexed by F+× \A×
f . The composition of two
operator at places u 6= v outside S is given as
Zu (gu )Zv (gv )ξbω − Zu (gu , ωu )Zv (gv , ωv )ξbω = Zu (gu , ωu )D(gv ) + Zv (gv , ωv )D(gu ).
We have shown that
g−→D(g) := Z S (g)ξbω − Z S (g, ω S )ξbω
behaves like derivative for g ∈ GL2 (AS ): for any two coprime g1 , g2 ∈ GL2 (AS ) in the sense
that for all v ∈
/ S, either g1 or g2 is in GL2 (Ov ) one has
D(g1 g2 ) = Z S (g1 , ω S )D(g2 ) + Z S (g2 , ω S )D(g1 ).
Thus D(g) is a derivation for Z S (g, ω S ) as defined in §4.4.4 in our previous paper [Zh1].
Let us apply this to the function
b ηbi.
g−→hZφ (g)ξ,
This function may not be automorphic but invariant under P (F ). Thus it has a Whittaker
function
b ηbi.
hZφ,ψ (g)ξ,
Let us fix gS ∈ GL2 (AS ). Then we obtain a function on g S ∈ GL2 (AS ):
b ηbi = hZ S (g S )ξ,
b Z t (gS )b
g S −→hZφ,ψ (gS g S )ω ξ,
ηi
S
where ZSt (gS ) is the transpose of ZS (gS ). We have just shown that this is a derivation
function for the Whittaker function of the Eisenstein series of E(g, φ, ω). Applying this to
the decomposition of Zφ (g, t1 , t2 ) we obtain the following:
Proposition 4.6.1. For fixed φ, there is an S such that for any t1 , t2 , the Whittaker function
of the difference
Zφ (g, t1 , t2 ) − hZφ (g)t1 , t2 i
is a finite linear combination of the Whittaker functions of the Eisenstein series E(g, φi , ωi )
and their derivations where φi are pure tensors and ωi are finite idele class characters.
54
5
Local heights of CM points
In this section, we compute the local heights of CM-points
hZφ∗ t1 , t2 i = i(Zφ∗ t1 , t2 ) + j(Zφ∗ t1 , t2 )
(v)
and compare it with the local analytic kernel function Kφ appeared in the decomposition
of I 0 (0, g) representing series L0 (1/2, π, χ). We will follow the work of Gross–Zagier and its
extension in our previous paper [Zh2].
In §5.1, we study the case of archimedean place. We obtain an explicit formula which
agrees with the analytic kernel. In §5.2, we study the supersingular case, namely the finite
place not in Σ and inert in E. In the unramified case, the formula is also explicit and agree
with corresponding analytic kernels. In the ramified case, under the condition that φv is
supported in certain non-degenerate locus of Bv × Fv× , we can show that the i-part of the
local intersection is given by certain pseudo-theta series defined in the next section. The
same result holds at a superspecial place v in §5.3 (namely, a finite place in Σ). In §5.4, we
treat ordinary case, namely finite places split in E. Again the formula in the unramified case
is explicit and agrees with analytic kernels. The i-part of the ramified case will contribute
pseudo-theta series. In §5.5, we will show that the j-part of local intersection contribute
pseudo-theta series for φv with non-generate supports.
5.1
Archimedean places
In this subsection we want to describe local heights of CM points at any archimedean place
v. Fix an identification B(Af ) = Bf . We will use the uniformization
×
XU,v (C) = B+
\H × B × (Af )/U
where B = B(v). We follow the treatment of Gross-Zagier [GZ]. See also [Zh2].
Basic formula
For any two points z1 , z2 ∈ H , the hyperbolic cosine of the hyperbolic distance between
them is given by
|z1 − z2 |2
d(z1 , z2 ) = 1 +
.
2Im(z1 )Im(z2 )
It is invariant under the action of GL2 (R). For any s ∈ C with Re(s) > 0, denote
ms (z1 , z2 ) = Qs (d(z1 , z2 )),
where
Z
Qs (t) =
∞
t+
√
t2
0
55
− 1 cosh u
−1−s
du
is the Legendre function of the second kind. Note that
Q0 (1 + 2λ) =
1
1
log(1 + ), λ > 0.
2
λ
We see that m0 (z1 , z2 ) has the right logarithmic singularity.
For any two distinct points of
×
\H × B × (Af )/U
XU,v (C) = B+
represented by (z1 , β1 ), (z2 , β2 ) ∈ H × B × (Af ), we denote
X
gs ((z1 , β1 ), (z2 , β2 )) =
ms (z1 , γz2 ) 1U (β1−1 γβ2 ).
×
γ∈µU \B+
It is easy to see that the sum is independent of the choice of the representatives (z1 , β1 ), (z2 , β2 ),
and hence defines a pairing on XU,v (C). Then the local height is given by
f s→0 gs ((z1 , β1 ), (z2 , β2 )).
iv̄ ((z1 , β1 ), (z2 , β2 )) = lim
f s→0 denotes the constant term at s = 0 of gs ((z1 , β1 ), (z2 , β2 )), which converges for
Here lim
Re(s) > 0 and has meromorphic continuation to s = 0 with a simple pole.
The definition above uses adelic language, but it is not hard to convert it to the classical
×
language. We first observe that gs ((z1 , β1 ), (z2 , β2 )) 6= 0 only if there is a γ0 ∈ B+
such that
β1 ∈ γ0 β2 U , which just means that (z1 , β1 ), (z2 , β2 ) are in the same connected component.
Assuming this, then (z2 , β2 ) = (z20 , β1 ) where z20 = γ0 z2 . we have
X
X
gs ((z1 , β1 ), (z2 , β2 )) = gs ((z1 , β1 ), (z20 , β1 )) =
ms (z1 , γz20 ) 1U (β1−1 γβ1 ) =
ms (z1 , γz20 ).
×
γ∈µU \B+
γ∈µU \Γ
×
Here we denote Γ = B+
∩ β1 U β1−1 . The connected component of these two points is exactly
×
×
B+
\H × B+
β1 U/U ≈ Γ\H ,
(z, bβ1 U ) 7→ b−1 z.
The stabilizer of H in Γ is exactly Γ ∩ F × = µU . Now we see that the formula is the same
as those in [GZ] and [Zh2].
×
− Ev× , we have
Next we consider the special case of CM points. For any γ ∈ Bv,+
1+
|z0 − γz0 |2
= 1 − 2ξ(γ).
2Im(z0 )Im(γz0 )
Thus it is convenient to denote
ms (γ) = Qs (1 − 2ξ(γ)),
56
γ ∈ Bv× − Ev× .
For any two distinct CM points β1 , β2 ∈ CMU , we obtain
X
gs (β1 , β2 ) =
ms (γ) 1U (β1−1 γβ2 ),
×
γ∈µU \B+
and
f s→0 gs (β1 , β2 ).
iv̄ (β1 , β2 ) = lim
Note that ms (γ) is not defined for γ ∈ E × . The summation above makes sense because
β1 6= β2 implies that 1U (β1−1 γβ2 ) = 0 for all γ ∈ E × . But this is not true for self-intersections.
In general, for any β1 , β2 ∈ CMU , we introduce formally
X
gs (β1 , β2 ) =
ms (γ) 1U (β1−1 γβ2 ),
×
−E × )
γ∈µU \(B+
and
f s→0 gs (β1 , β2 ).
iv̄ (β1 , β2 ) = lim
Then this definition is the same as the before if β1 6= β2 . In the case that β1 = β2 , it will
occur from the arithmetic adjunction formula.
Kernel function
In this section we compute the local height
X
iv̄ (Zφ∗ (g)t1 , t2 ) =
X
r(g)φ(x)a iv̄ (t1 x, t2 ).
a∈F × x∈B× /U
f
for any archimedean place v of F .
(v)
The goal is to show that it is equal to Kf
φ (g, (t1 , t2 )) obtained in Proposition 3.7.1.
Proposition 5.1.1. For any t1 , t2 ∈ CU ,
X
(v)
f s→0
lim
iv̄ (Zφ∗ (g)t1 , t2 ) = Mφ (g, (t1 , t2 )) =
a∈F ×
(v)
X
r(g, (t1 , t2 ))φ(y)a ms (y).
×
y∈µU \(B+
−E × )
(v)
Therefore, Mφ (g, (t1 , t2 )) = Kf
φ (g, (t1 , t2 )).
Proof. By the above formula,
X X
f s→0
iv̄ (Zφ∗ (g)t1 , t2 ) =
r(g)φ(x)a lim
a∈F × x∈B× /U
f
=
X
f s→0
lim
a∈F ×
=
X
a∈F ×
X
ms (γ) 1U (x−1 t−1
1 γt2 )
×
γ∈µU \(B+
−E × )
X
r(g)φ(t−1
1 γt2 )a ms (γ)
×
γ∈µU \(B+
−E × )
f s→0
lim
X
r(g, (t1 , t2 ))φ(γ)a ms (γ).
×
γ∈µU \(B+
−E × )
Here the second equality is obtained by replacing x by t−1
1 γt2 .
57
We want to compare the above result with the holomorphic projection
X
X
(v)
f
Kf
(g,
(t
,
t
))
=
lim
r(g, (t1 , t2 ))φ(y)a kv,s (y)
1 2
s→0
φ
a∈F ×
×
−E × )
y∈µU \(B+
computed in Proposition 3.7.1.
It amounts to compare
ms (y) = Qs (1 − 2ξ(y))
with
Γ(s + 1)
kv,s (y) =
2(4π)s
Z
∞
1
1
dt.
t(1 − ξv (y)t)s+1
By the result of Gross-Zagier,
Z ∞
1
dt = 2Qs (1 − 2ξ) + O(|ξ|−s−2 ),
s+1
t(1
−
ξt)
1
ξ → −∞,
and the error term vanishes at s = 0. We conclude that
(v)
∗
Kf
φ (g, (t1 , t2 )) = iv̄ (Zφ (g)t1 , t2 ).
5.2
Supersingular case
Let v be a finite prime of F non-split in E but split in B. We consider the local height at
the unique place of E lying over v. We will use the local multiplicity functions treated in
Zhang [Zh2]. For more details, we refer to that paper.
Multiplicity function
Assume that U is decomposable at v:
U = Uv · U v .
Let B = B(v) be the nearby quaternion algebra over F . Make an identification B(Av ) = Bv .
Then the set of supersingular points on XK over v is parameterized by
v
SSU = B × \(Fv× / det(Uv )) × (Bv×
f /U ).
Then we have a natural isomorphism
v×
×
×
×
v
v̄ : CMU = E × \B×
f /U → B \(B ×E × Bv /Uv ) × Bf /U
sending β to (1, βv , β v ). The reduction map CMU → SSU is given by taking norm on the
first factor:
×
q : B × ×E × B×
(b, β) 7→ q(b)q(β).
v → Fv ,
58
The intersection pairing is given by a multiplicity function m on
Hv := Bv× ×Ev× B×
v /Uv .
More precisely, the intersection of two points (b1 , β1 ), (b2 , β2 ) ∈ Hv is given by
−1
gv ((b1 , β1 ), (b2 , β2 )) = m(b−1
1 b2 , β1 β2 ).
The multiplicity function m is defined everywhere in Hv except at the image of (1, 1). It
satisfies the property
m(b, β) = m(b−1 , β −1 ).
Lemma 5.2.1. For any two distinct CM-points β1 ∈ CMU and t2 ∈ CU , their local height
is given by
X
−1
)1U v ((β1v )−1 γtv2 ).
iv̄ (β1 , t2 ) =
m(γt2v , β1v
γ∈µU \B ×
Proof. Like the archimedean case, we compute the height by pulling back to Hv × Bv×
f .
v
×
v
The height is the sum over γ ∈ µU \B of the intersection of (1, β1v , β1 ) with γ(1, t2v , t2 ) =
(γ, t2v , γtv2 ) = (γt2v , 1, γtv2 ) on Hv × Bv×
f .
Analogous to the archimedean case, the summation is well-defined for all β1 6= t2 . But
if β1 = t2 , then the summation is not defined at γ ∈ E × such that β1−1 γt2 ∈ U . Hence, we
extend the definition as
=
iv̄ (β1 , t2 )
X
−1
m(γt2v , β1v
)1U v ((β1v )−1 γtv2 )
γ∈µU \(B × −E × ∩β1 U t−1
2 )
=
X
−1
m(γt2v , β1v
)1U v ((β1v )−1 γtv2 ) +
γ∈µU \(B × −E × )
=
X
X
−1
m(γt2v , β1v
)1U v ((β1v )−1 γtv2 )
γ∈µU \(E × −β1 U t−1
2 )
−1
m(γt2v , β1v
)1U v ((β1v )−1 γtv2 ) +
γ∈µU \(B × −E × )
X
−1
m(γt2v , β1v
)1U v ((β1v )−1 γtv2 ).
γ∈µU \(E × −β1 Uv t−1
2 )
The definition is the equal to the previous one if β1 6= t2 .
The kernel function
Now we compute
iv̄ (Zφ∗ (g)t1 , t2 ) =
X
X
a∈F ×
x∈B×
f /U
59
r(g)φ(x)a iv̄ (t1 x, t2 ).
By the above formula,
X
iv̄ (Zφ∗ (g)t1 , t2 ) =
X
a∈F × x∈B× /U
f
X
+
X
r(g)φ(x)a
X
−1 −1
m(γt2 , x−1 t−1
1 )1U v (x t1 γt2 )
γ∈µU \(B × −E × )
X
r(g)φ(x)a
a∈F × x∈B× /U
f
−1 −1
m(γt2 , x−1 t−1
1 )1U v (x t1 γt2 ).
γ∈µU \(E × −t1 xUv t−1
2 )
In the first triple sum, we replace xv by t−1
1 γt2 and thus get
X
X
X
−1
r(g)φv (xv )a m(t−1
r(g)φv (t−1
1 γt2 , x )
1 γt2 )a
a∈F × γ∈µU \(B × −E × )
=
X
X
xv ∈B×
v /Uv
X
r(g, (t1 , t2 ))φv (γ, u)
−1
r(g)φv (xv , uq(γ)/q(xv ))m(t−1
1 γt2 , x ).
xv ∈B×
v /Uv
u∈µ2U \F × γ∈B−E
In the second triple sum, we still replace xv by t−1
/ t1 xUv t−1
is
1 γt2 . The condition γ ∈
2
−1
equivalent to xv ∈
/ t1 γt2 Uv . We get
X X
X
−1
r(g)φv (t−1
γt
)
r(g)φv (xv )a m(t−1
2 a
1
1 γt2 , x )
a∈F × γ∈µU \E ×
=
X
X
−1
xv ∈(B×
v −t1 γt2 Uv )/Uv
X
r(g, (t1 , t2 ))φv (γ, u)
u∈µ2U \F × γ∈E ×
−1
r(g)φv (xv , uq(γ)/q(xv ))m(t−1
1 γt2 , x ).
−1
xv ∈(B×
v −t1 γt2 Uv )/Uv
For convenience, we introduce:
Notation.
Z
mφv (y, u) =
×
ZBv
=
B×
v
m(y, x−1 )φv (x, uq(y)/q(x))dx
m(y −1 , x)φv (x, uq(y)/q(x))dx,
Z
nφv (y, u) =
B×
v −yUv
m(y, x−1 )φv (x, uq(y)/q(x))dx
Z
=
B×
v −Uv
(y, u) ∈ (Bv − Ev ) × Fv×
m(1, x)φv (yx, u/q(x))dx,
(y, u) ∈ Ev× × Fv×
By this notation, we obtain:
Proposition 5.2.2.
(v)
(v)
iv̄ (Zφ∗ (g)t1 , t2 ) = Mφ (g, (t1 , t2 )) + Nφ (g, (t1 , t2 ))
where
(v)
Mφ (g, (t1 , t2 )) =
X
X
r(g, (t1 , t2 ))φv (y, u) mr(g,(t1 ,t2 ))φv (y, u),
u∈µ2U \F × y∈B−E
(v)
Nφ (g, (t1 , t2 )) =
X
X
r(g, (t1 , t2 ))φv (y, u) r(t1 , t2 )nr(g)φv (y, u).
u∈µ2U \F × y∈E ×
60
Here we use the convention
−1
r(t1 , t2 )nr(g)φv (y, u) = nr(g)φv (t−1
1 yt2 , q(t1 t2 )u).
Note that in the above series, we write the dependence on (t1 , t2 ) in different manners for mφv
and nφv . This is because mφv (y, u) translates well under the action of P (Fv )×(Ev× ×Ev× ), but
nφv (y, u) only translates well under the action of P (Fv ). We should compare the following
result with Lemma 3.2.2 for kφv (g, y, u).
Lemma 5.2.3. (1) The function mr(g,(t1 ,t2 ))φv (y, u) behaves like Weil representation under
the action of P (Fv ) × (Ev× × Ev× ) on (y, u). Namely,
mr(g,(t1 ,t2 ))φv (y, u) = r(g, (t1 , t2 ))mφv (y, u),
(g, (t1 , t2 )) ∈ P (Fv ) × (Ev× × Ev× ).
More precisely, it means that
mr(m(a))φv (y, u)
mr(n(b))φv (y, u)
mr(d(c))φv (y, u)
mr(t1 ,t2 )φv (g, y, u)
=
=
=
=
|a|2 mφv (ay, u), a ∈ Fv×
ψ(buq(y))mφv (y, u), b ∈ Fv
|c|−1 mφv (y, c−1 u), c ∈ Fv×
−1
mφv (t−1
(t1 , t2 ) ∈ Ev× × Ev×
1 yt2 , q(t1 t2 )u),
(2) The function nr(g)φv (y, u) behaves like Weil representation under the action of P (Fv )
on (y, u), i.e.,
nr(m(a))φv (y, u) = |a|2 nφv (ay, u), a ∈ Fv×
nr(n(b))φv (y, u) = ψ(buq(y))nφv (y, u), b ∈ Fv
nr(d(c))φv (y, u) = |c|−1 nφv (y, c−1 u), c ∈ Fv× .
Proof. They follow from basic properties of the multiplicity function m(x, y). We only verify
the first identity.
Z
mr(m(a))φv (y, u) =
m(y −1 , x)r(gv )φv (ax, uq(y)/q(x))|a|2 dx
GL2 (Fv )
Z
2
= |a|
m(y −1 , a−1 x)r(gv )φv (x, uq(y)/q(a−1 x))dx
GL (F )
Z 2 v
= |a|2
m((ay)−1 , x)r(gv )φv (x, uq(ay)/q(x))dx
GL2 (Fv )
2
= |a| mφv (ay, u).
61
Unramified Case
Fixing an isomorphism Bv = M2 (Fv ). In this subsection we compute mφv (y, u) and nφv (y, u)
in the following unramified case:
1. φv is the characteristic function of M2 (OFv ) × OF×v ;
2. Uv is the maximal compact subgroup GL2 (OFv ).
By [Zh2], there is a decomposition
GL2 (Fv ) =
∞
a
Ev× hc GL2 (OFv ),
c=0
hc =
1 0
0 $c
(5.2.1)
We may assume that β = hc . For any b ∈ Bv we set
ξ(b) =
q(b)
q(b2 )
=
q(b)
q(b)
for the orthogonal decomposition b = b1 + b2 . The following result is Lemma 5.5.2 in [Zh2].
There is a small mistake in the original statement. Here is the corrected one.
Lemma 5.2.4. The multiplicity function m(b, β) 6= 0 only if q(b)q(β) ∈ OF×v . In this case,
assume that β ∈ Ev× hc GL2 (OFv ). Then

1

 2 (ordv ξ(b) + 1) if c = 0;
m(b, β) = Nv1−c (Nv + 1)−1 if c > 0, Ev /Fv is unramified;

 1 −c
N
if c > 0, Ev /Fv is ramified.
2 v
Proposition 5.2.5.
case,
(1) The function mφv (y, u) 6= 0 only if (y, u) ∈ OBv × OF×v . In this
1
mφv (y, u) = (ordv q(y2 ) + 1).
2
(2) The function nφv (y, u) 6= 0 only if (y, u) ∈ OEv × OF×v . In this case,
1
nφv (y, u) = ordv q(y).
2
Proof. We will use Lemma 5.2.4. Recall that
X
mφv (y, u) =
m(y −1 , x)φv (x, uq(y)/q(x)).
x∈GL2 (Fv )/Uv
Note that m(y −1 , x) 6= 0 only if ord(q(x)/q(y)) = 0. Under this condition, φv (x, uq(y)/q(x)) 6=
0 if and only if u ∈ OF×v and x ∈ M2 (OFv ). It follows that mφv (y, u) 6= 0 only if u ∈ OF×v and
n = ord(q(y)) ≥ 0. Assuming these two conditions, we have
X
m(y −1 , x),
mφv (y, u) =
x∈M2 (OFv )n /Uv
62
where M2 (OFv )n denotes the set of integral matrices whose determinants have valuation n.
Now we use decomposition (2.3), we obtain
mφv (y, u) =
∞
X
m(y −1 , hc )vol(Ev× hc GL2 (OFv ) ∩ M2 (OFv )n ).
c=0
We first consider the case that Ev /Fv is unramified. The set in the right hand side is
non-empty only if n − c is even and non-negative. In this case it is given by
$(n−c)/2 OE×v hc Uv .
The volume of this set is 1 if c = 0 and Nvc−1 (Nv + 1) if c > 0 by the computation of [Zh2,
p. 101]. It follows that, for c > 0 with 2 | (n − c),
m(y −1 , hc )vol(Ev× hc GL2 (OFv ) ∩ M2 (OFv )n ) = 1.
If n is even,
1
n
1
mφv (y, u) = (ordv ξ(y) + 1) + = (ordv q(y2 ) + 1).
2
2
2
1
If n is odd, mφv (y, u) = (n + 1). It is easy to see that ordv q(y1 ) is even and ordv q(y2 ) is
2
odd. Then n = ordv q(y2 ), since n = ordv q(y) = min{ordv q(y1 ), ordv q(y2 )} is odd. We still
get mφv (y, u) = 21 (ordv q(y2 ) + 1) in this case.
Now assume that Ev /Fv is ramified. Then the condition that 2 | (n − c) is unnecessary,
and vol(Ev× hc GL2 (OFv ) ∩ M2 (OFv )n ) = Nvc . Thus
1
1
1
mφv (y, u) = (ordv ξ(y) + 1) + n · = (ordv q(y2 ) + 1).
2
2
2
As for nφv , still write n = ordv q(y), and we have
nφv (y, u) =
∞
X
m(y −1 , hc )vol(Ev× hc GL2 (OFv ) ∩ M2 (OFv )n ).
c=1
n
If Ev /Fv is unramified, n is always even, and we have mφv (y, u) = . The ramified case is
2
similar.
We immediately see that in the unramified case, mφv matches the analytic kernel kφv
computed in Proposition 3.4.2. As for nφv (y, u), its counterpart coming from PrI 0 (0, g) is
1
exactly log δ(gv ) + log |uq(y)|v in Proposition 3.7.1. The good news is that the extra term
2
log δ(gv ) cancels the action of g. In summary, we have the following result:
Proposition 5.2.6.
(1) Let v be unramified as above. Then
nr(g)φv (y, u) log Nv = −(log δ(gv ) +
63
1
log |uq(y)|v ) r(g)φv (y, u).
2
(2) Let v be unramified as above with further conditions:
• v - 2 if Ev /Fv is ramified;
• the local different dv is trivial.
Then
kr(t1 ,t2 )φv (g, y, u) = mr(g,(t1 ,t2 ))φv (y, u) log Nv ,
and thus
(v)
(v)
Kφ (g, (t1 , t2 )) = Mφ (g, (t1 , t2 )) log Nv .
Proof. We first consider (2). The case (g, t1 , t2 ) = (1, 1, 1) follows from the above result
and Corollary 3.4.2. It is also true for g ∈ GL2 (OFv ) since it is easy to see that such g
will not change the local kernel functions for standard φv . For the general case, apply the
action of P (Fv ) and Ev× × Ev× . The equality follows from Proposition 3.2.2 and Lemma 5.2.3.
Similarly, by Lemma 5.2.3 we can show (1).
General case
Now we consider general Uv . By Proposition 5.2.5 for the unramified case, we know that
mφv may have logarithmic singularity around the boundary Ev × Fv× , and nφv may have
logarithmic singularity around the boundary {0} × Fv× . These singularities are caused by
self-intersections in the computation of local multiplicity. However, we will see that there is
no singularity if φv ∈ S 0 (Bv × Fv× ).
Proposition 5.2.7. Assume that φv ∈ S 0 (Bv × Fv× ) is invariant under the right action of
Uv . Then:
(1) The function mφv (y, u) can be extended to a Schwartz function for (y, u) ∈ Bv × Fv× .
(2) The function nφv (y, u) can be extended to a Schwartz function for (y, u) ∈ Ev × Fv× .
Proof. We only consider mφv . The proof for nφv is similar.
By the choice of φv , there is a constant C > 0 such that φv (x, u) 6= 0 only if −C <
v(q(x)) < C and −C < v(u) < C. Recall that
Z
mφv (y, u) =
m(y −1 , x)φv (x, uq(y)/q(x))dx.
B×
v
In order that m(y −1 , x) 6= 0, we have to make q(y)/q(x) ∈ q(Uv ) and thus v(q(y)) =
v(q(x)). It follows that φv (x, uq(y)/q(x)) 6= 0 only if −C < v(u) < C. The same is true for
mφv (y, u) by looking at the integral above. Then it is easy to see that mφv (y, u) is Schwartz
for u ∈ Fv× .
On the other hand, mφv (y, u) 6= 0 only if −C < v(q(y)) < C, since φv (x, uq(y)/q(x)) 6= 0
only if −C < v(q(x)) < C. Extend mφv to Bv × F × by taking zero outside Bv× × F × . We
only need to show that it is locally constant in Bv× × F × .
64
We have φv (Ev Uv , Fv× ) = 0 by the assumption that φv (Ev , Fv× ) = 0 and φv is invariant
under Uv . Thus
Z
mφv (y, u) =
m(y −1 , x)(1 − 1Ev× Uv (x))φv (x, uq(y)/q(x))dx.
B×
v
It is locally constant in Bv× × Fv× , since m(y −1 , x)(1 − 1Ev× Uv (x)) is locally constant as a
function on Bv× × B×
v . This completes the proof.
5.3
Superspecial case
Let v be a finite prime of F non-split in both B and E. In this section we consider the local
height
X X
r(g)φ(x)a iv̄ (t1 x, t2 ).
iv̄ (Zφ∗ (g)t1 , t2 ) =
a∈F × x∈B× /U
f
Since most of the computations are similar to the supersingular case, sometimes we will only
list the results.
Denote by B = B(v) the nearby quaternion algebra. We fix identifications Bv ' M2 (Fv )
and B(Avf ) ' Bvf . The intersection pairing is given by a multiplicity function m on
Hv := Bv× ×Ev× B×
v /Uv .
More precisely, the intersection of two points (b1 , β1 ), (b2 , β2 ) ∈ Hv is given by
−1
gv ((b1 , β1 ), (b2 , β2 )) = m(b−1
1 b2 , β1 β2 ).
The multiplicity function m is defined everywhere on Hv except at the image of (1, 1). It
satisfies the property
m(b, β) = m(b−1 , β −1 ).
For any two distinct CM-points β1 ∈ CMU and t2 ∈ CU , their local height is given by
X
−1
iv̄ (β1 , t2 ) =
m(γt2v , β1v
)1U v ((β1v )−1 γtv2 ).
γ∈µU \B ×
If β1 = t2 , then the summation is not defined at γ ∈ E × such that β1−1 γt2 ∈ U . So we
extend the definition as
=
iv̄ (β1 , t2 )
X
−1
m(γt2v , β1v
)1U v ((β1v )−1 γtv2 )
γ∈µU \(B × −E × ∩β1 U t−1
2 )
=
X
−1
m(γt2v , β1v
)1U v ((β1v )−1 γtv2 ) +
γ∈µU \(B × −E × )
X
γ∈µU \(E × −β1 Uv t−1
2 )
65
−1
m(γt2v , β1v
)1U v ((β1v )−1 γtv2 ).
Analogous to Proposition 5.2.2, we have the following result.
Proposition 5.3.1.
(v)
(v)
iv̄ (Zφ∗ (g)t1 , t2 ) = Mφ (g, (t1 , t2 )) + Nφ (g, (t1 , t2 ))
where
X
(v)
Mφ (g, (t1 , t2 )) =
X
r(g, (t1 , t2 ))φv (y, u) mr(g,(t1 ,t2 ))φv (y, u),
u∈µ2U \F × y∈B−E
X
(v)
Nφ (g, (t1 , t2 )) =
X
r(g, (t1 , t2 ))φv (y, u) r(t1 , t2 )nr(g)φv (y, u).
u∈µ2U \F × y∈E ×
Here we use the same notations:
Z
mφv (y, u) =
m(y −1 , x)φv (x, uq(y)/q(x))dx, (y, u) ∈ (Bv − Ev ) × Fv×
×
ZBv
m(y −1 , x)φv (x, uq(y)/q(x))dx, (y, u) ∈ Ev× × Fv×
nφv (y, u) =
B×
v −yUv
Lemma 5.2.3 is still true. It says that the action of P (Fv )×(Ev× ×Ev× ) on mr(g,(t1 ,t2 ))φv (y, u)
and the action of P (Fv ) on nr(g)φv (y, u) behave like Weil representation. The main result
below is parallel to Proposition 5.2.7.
Proposition 5.3.2. Assume φv ∈ S 0 (Bv × Fv× ) is invariant under the right action of Uv .
Then:
(1) The function mφv (y, u) can be extended to a Schwartz function for (y, u) ∈ Bv × Fv× .
(2) The function nφv (y, u) can be extended to a Schwartz function for (y, u) ∈ Ev × Fv× .
Proof. The proof is very similar to Proposition 5.2.7. We only sketch one for mφv . The proof
for nφv is similar.
By the argument of Proposition 5.2.7, there is a constant C > 0 such that mφv (y, u) 6= 0
only if −C < v(q(y)) < C and −C < v(u) < C. Extend mφv to Bv × F × by taking zero
outside Bv× × F × . The same method shows that it is locally constant on Bv× × F × . But this
is not enough to conclude that it is compactly supported in y, since Bv is the matrix algebra
this time. However, there is a compact subgroup Dv of Bv× such that m(y, x−1 ) 6= 0 only if
y ∈ Ev× Dv . By this we can have our conclusion.
66
5.4
Ordinary case
Assume that v is a finite prime of F such that Ev is split. Then Bv is split because of
the embedding Ev → Bv . Let v1 and v2 be the two primes
of E lying over v. Fix an
F
v
identification Bv ∼
. Assume that v1 corresponds to
= M2 (Fv ) under which Ev =
Fv
Fv
0
the ideal
and v2 corresponds to
of Ev .
0
Fv
We will make use of results of [Zh2]. The reduction map of CM-points to ordinary points
at v1 is given by
v×
×
E × \B×
f /U −→ E \(N (Fv )\GL2 (Fv )) × Bf /U.
The intersection multiplicity is a function on N (Fv )Uv /Uv . By [Zh2, Lemma 6.3.2], the
local pairing of β1 , β2 ∈ GL2 (Fv )/Uv is given by
Z
Z
Z
˜
˜
gv1 (β1 , β2 ) =
1β1 Uv (hy)1β2 Uv (y)dy dh =
1Uv (β1−1 hβ2 )dh.
GL2 (Fv )
GL2 (Fv )
GL2 (Fv )
˜ is defined
Here the measure dy is the usual Haar measure on GL2 (Fv ), and the measure dh
as follows:
Z
Z
Z
1 b
×
˜
f
d× b.
f (n(b))d b =
f (h)dh :=
1
×
×
Fv
Fv
GL2 (Fv )
˜ is invariant under the substitution h 7→ tht−1 for any t ∈ E × .
A nice property is that dh
v
And thus
Z
Z
˜ =
˜
f (ht)dh
f (th)dh.
GL2 (Fv )
GL2 (Fv )
By this pairing, the local height pairing of two distinct CM points β1 , β2 ∈ E × \B×
f /U is
given by
X
iv̄1 (β1 , β2 ) =
gv1 (β1 , γβ2 )1U v (β1−1 γβ2 ).
γ∈µU \E ×
Just like the other cases, the above summation is only well-defined for β1 6= β2 . But we
can extend the definition as
X
X
iv̄1 (β1 , β2 ) =
gv1 (β1 , γβ2 )1U v (β1−1 γβ2 ) =
gv1 (β1 , γβ2 )1U v (β1−1 γβ2 ).
γ∈µU \(E × −β1 U β2−1 )
We want to compute
X
iv̄1 (Zφ∗ (g)t1 , t2 ) =
=
X
a∈F ×
x∈B×
f /U
X
X
a∈F × x∈B× /U
f
γ∈µU \(E × −β1 Uv β2−1 )
r(g)φ(x)a iv̄1 (t1 x, t2 )
X
r(g)φ(x)a
γ∈µU \(E × −t1 xUv t−1
2 )
67
gv1 (t1 x, γt2 )1U v (x−1 t−1
1 γt2 ).
−1
v
v
By 1U v (x−1 t−1
/ t1 xUv t−1
/ t−1
1 γt2 ) = 1, we have x ∈ t1 γt2 U ; by γ ∈
2 , we have xv ∈
1 γt2 Uv .
Thus
X X
X
r(g)φv (xv )a gv1 (t1 xv , γt2 ).
iv̄1 (Zφ∗ (g)t1 , t2 ) =
r(g)φv (t−1
1 γt2 )a
a∈F × γ∈µU \E ×
−1
xv ∈(B×
v −t1 γt2 Uv )/Uv
Recall that
Z
˜
1Uv (b−1
1 hb2 )dh.
gv1 (b1 , b2 ) =
GL2 (Fv )
The last summation is equal to
Z
X
GL2 (Fv )
−1
xv ∈(B×
v −t1 γt2 Uv )/Uv
Z
X
=
GL2 (Fv )
−1
˜
1Uv (x−1
v t1 hγt2 )dh
r(g)φv (xv )a
−1
˜
r(g)φv (xv )a 1Uv (x−1
v t1 γt2 h)dh
−1
xv ∈(B×
v −t1 γt2 Uv )/Uv
Z
˜
r(g)φv (t−1
1 γt2 h)a dh.
=
GL2 (Fv )−Uv
−1 −1
Here we explain the last equality above. We have xv ∈ t−1
1 γt2 hUv by 1Uv (xv ht1 γt2 ) = 1.
Then h ∈
/ Uv by xv ∈
/ t−1
1 γt2 Uv .
In summary, we have obtained:
Proposition 5.4.1.
iv̄1 (Zφ∗ (g)t1 , t2 ) =
X
X
r(g, (t1 , t2 ))φv (y, u) r(t1 , t2 )nr(g)φv ,v1 (y, u).
u∈µ2U \F × y∈E ×
where
Z
˜
φv (yh, u)dh
Z
0 b
1 b
=φ1,v (y, u)
φ2,v y
,u
1 − 1Uv
d× b.
0
1
×
Fv
nφv ,v1 (y, u) =
GL2 (Fv )−Uv
for any (y, u) ∈ Ev× × Fv× .
The above result is only for the place v1 . By symmetry, we have a formula for v2 after
replacing the upper triangular matrices by lower triangular matrices. The average of these
two results yields:
Proposition 5.4.2.
(v)
iv̄ (Zφ∗ (g)t1 , t2 ) = Nφ (g, (t1 , t2 )) =
X
X
u∈µ2U \F × y∈E ×
68
r(g, (t1 , t2 ))φv (y, u) r(t1 , t2 )nr(g)φv (y, u)
where
Z
1
1 b
0 b
d× b
φ2,v y
,u
1 − 1Uv
nφv (y, u) = φ1,v (y, u)
1
0
×
2
F
Z v
1
0
1
+ φ1,v (y, u)
φ2,v y
,u
1 − 1Uv
d× b
b
0
b
1
×
2
Fv
for any (y, u) ∈ Ev× × Fv× .
In the unramified case, we have
Proposition 5.4.3. Assume that Uv is maximal and φv is the standard characteristic function. Then nφv (y, u) 6= 0 only if (y, u) ∈ OEv × OF×v . In this case,
1
nφv (y, u) = ordv q(y).
2
×
Proof.
Assume that (y, u) ∈ OEv × OFv . Otherwise, both sides are zero. Assume that
a1
y=
under the identification.
a2
1 b
Since Uv is maximal,
∈ Uv if and only if b ∈ OFv . Then
1
Z
0 ba1
nφv ,v1 (y, u) =
φ2,v
, u d× b = ordv (a1 ).
0
Fv −OF
v
Similarly, the second integral is equal to ordv (a2 ).
The above result looks exactly the same as the result of nφv (y, u) in Proposition 5.2.5
in the supersingular case. This is the reason that we use the same notation for them. For
example, we have the counterpart of Lemma 5.2.3 (2) as follows:
Lemma 5.4.4. The function nr(g)φv (y, u) behaves like Weil representation under the action
of P (Fv ) on (y, u), i.e.,
nr(m(a))φv (y, u) = |a|2 nφv (ay, u), a ∈ Fv×
nr(n(b))φv (y, u) = ψ(buq(y))nφv (y, u), b ∈ Fv
nr(d(c))φv (y, u) = |c|−1 nφv (y, c−1 u), c ∈ Fv× .
Analogous to Proposition 5.2.6, we have:
Proposition 5.4.5. For the maximal Uv and the standard φv as in Proposition 5.4.3, we
have
1
nr(g)φv (y, u) = −(log δ(gv ) + log |uq(y)|v ) r(g)φv (y, u).
2
69
Proof. We have the identity for g = 1 by Proposition 5.4.5. To extend the result to general
g ∈ GL2 (Fv ), we only need to apply Lemma 5.4.4 to the Iwasawa decomposition of g.
Parallel to Proposition 5.2.7, we have
Proposition 5.4.6. For general Uv , the function nφv (y, u) can be extended to a Schwartz
function for (y, u) ∈ Ev × Fv× if φv ∈ S 0 (Bv × Fv× ).
Proof. It suffices to show the result for
Z
a1 a1 b
1 b
nφv ,v1 (y, u) =
φv
,u
1 − 1Uv
d× b.
a
1
×
2
Fv
a1
Here we still write y =
.
a2
It is easy to see that nφv ,v1 is locally constant at any (y, u) with a1 6= 0. The problem is
nφ
to show that, for fixed (y0 , u0 ) ∈ Fv × Fv× , the function
v ,v1 extends to a locally constant
0
function at (ỹ0 , u0 ). Here ỹ0 denotes the element
∈ Ev .
y0
Consider the behavior when
(y,u) is closed to (ỹ0 , u
0 ). Since v(a
1 ) islarge, we see that
1 b
a1 a1 b
a1 b is closed to zero when
∈ Uv , and thus φv
, u = φv (ỹ0 , u0 ) = 0.
1
a2
It follows that
Z
Z
a1 a1 b
a1 b
×
nφv ,v1 (y, u) =
φv
φv
,u d b =
, u d× b.
a
a
×
×
2
2
Fv
Fv
It is independent of (y, u) in a neighborhood of (ỹ0 , u0 ), and we can extend the function to
(ỹ0 , u0 ).
5.5
The j -part
Now we consider the pairing jv̄ (Zφ∗ (g)t1 , t2 ). It is nonzero only if v is a finite prime such that
Uv is not maximal or Bv is nonsplit. The goal is to write it as a pseudo-theta series and
control the singularities. It turns out that it has no singularity if φv ∈ S 0 (Bv × Fv× ). In
that case, an integration of the associated theta series becomes an Eisenstein series.
For any β1 , β2 ∈ CMU , the pairing j(β1 , β2 ) 6= 0 only if β1 , β2 are in the same geometrically
connected components of XU . Once this is true, the pairing is given by a symmetric locally
2
constant function l(·, ·) on (B×
v /Uv ) . In summary, we have
jv̄ (β1 , β2 ) = l(β1v , β2v )1F+× q(β2 )q(U ) (q(β1 )) = l(β1v , β2v )1(Bf )ad U (β2−1 β1 ).
70
Here we denote
(Bf )ad ={x ∈ Bf : q(x) ∈ F+× }
(Bf )a ={x ∈ Bf : q(x) = a}, a ∈ A×
f.
Now we go back to
jv̄ (Zφ∗ (g)t1 , t2 )
X X
=
r(g)φ(x)a jv̄ (t1 x, t2 )
a∈F × x∈B× /U
f
X
=
X
r(g)φ(x)a l(t1 xv , t2 )1(Bf )ad U (t−1
2 t1 x)
a∈F × x∈B× /U
f
X
=
X
r(g)φ(t−1
1 t2 x)a l(t2 xv , t2 )
a∈F × x∈(Bf )ad U/U
X
X
=
r(g, (t1 t−1
2 , 1))φ(x)a l(t2 xv , t2 ).
a∈F × x∈(Bf )ad /Uad
Apparently µU U 1 ⊂ Uad . The quotient Uad /µU U 1 = (q(U ) ∩ F+× )/µ2U is finite since it is a
quotient of two finitely generated abelian groups of the same rank. Hence we have
[q(U ) ∩ F+× : µ2U ] jv̄ (Zφ∗ (g)t1 , t2 )
X
X
=
r(g, (t1 t−1
2 , 1))φ(x)a l(t2 xv , t2 )
a∈F × x∈(Bf )ad /µU U 1
=
X
X
r(g, (t1 t−1
2 , 1))φ(x, u) l(t2 xv , t2 )
u∈µ2U \F × x∈(Bf )ad /U 1
=
X
X
X
r(g, (t1 t−1
2 , 1))φ(x, u) l(t2 xv , t2 )
× x∈(B ) /U 1
u∈µ2U \F × a∈F+
f a
1
=
vol(U 1 )
X
X Z
×
u∈µ2U \F × a∈F+
r(g, (t1 t−1
2 , 1))φ(x, u) l(t2 xv , t2 )dx.
(Bf )a
The integral above is a local product, which is nice at every place except v.
We first consider the case that Bv is a matrix algebra. We introduce a function
Z
1
lφv (t2 , y, u) =
φv (x, u) l(t2 x, t2 )dx, (y, u) ∈ Bv× × Fv× .
vol(Bv1 ) (Bv )q(y)
Note that Bv is a division algebra and thus Bv1 is compact and has finite volume.
Fix t2 ∈ T (Af ). In general, lφv (t2 , y, u) is not defined at y = 0. But if φv ∈ S 0 (Bv × Fv× ),
then φv (x, Fv× ) = 0 when v(q(x)) is large. It follows that lφv (t2 , y, u) = 0 for y closed to 0.
It is automatically a Schwartz function for (y, u) ∈ Bv × Fv× .
71
Now we go back to the computation. We have:
[q(U ) ∩ F+× : µ2U ]vol(U 1 ) jv̄ (Zφ∗ (g)t1 , t2 )
X X Z
v
(t2 , y, u)dy
=
r(g, (t1 t−1
2 , 1))φ (y, u)lr(g,(t1 t−1
2 ,1))φv
×
u∈µ2U \F × a∈F+
=
X
(Bf )a
X
Z
B 1 (Af )
u∈µ2U \F × y∈B × /B 1
=
X
u∈µ2U \F ×
Z
X
B 1 \B 1 (A
f)
v
(t2 , yh, u)dh
r(g, (t1 t−1
2 , 1))φ (yh, u)lr(g,(t1 t−1
2 ,1))φv
v
(t2 , yh, u)dh.
r(g, (t1 t−1
2 , 1))φ (yh, u)lr(g,(t1 t−1
2 ,1))φv
y∈B ×
The integral on B 1 \B 1 (Af ) essentially reduces to a finite sum. When lφv is a Schwartz
function, we may replace the pseudo-theta series above by the associated theta series. We
will get an integral of this theta series, which gives an Eisenstein series.
Next we consider the case that Bv is a division algebra. In this case, Bv is a matrix
algebra and the function lφv above is never Schwartz in y. To resolve this problem, we take
an open compact subset D of Bv such that Da = D ∩ B(Fv )a is nonempty for all a in the
range
{q(x) : x ∈ Bv , φv (x, Fv× ) 6= 0}.
This is possible for φv ∈ S 0 (Bv × Fv× ) and D depends on φv . Then we define
Z
1
lφv (t2 , y, u) =
φv (x, u) l(t2 x, t2 )dx, (y, u) ∈ Bv× × Fv× .
1D (y)
vol(Dq(y) )
(Bv )q(y)
It is easy to check that it is a Schwartz function for (y, u) ∈ Bv × Fv× . And we also have
Z
Z
φv (x, u) l(t2 x, t2 )dx.
lφv (t2 , y, u)dy =
B(Fv )a
(Bv )a
Then the argument is the same as before. We still get an integral of a non-singular pseudotheta series. The integration of the associated theta series becomes an Eisenstein series.
Remark. The matching between lφv (t2 , y, u) and φv (x, u) l(t2 x, t2 ) so that they have the same
integral looks very similar to the situation of Fundamental Lemma.
72
6
Pseudo-theta series and Gross–Zagier formula
In this section we prove the main result, Theorem 1.3.1. We argue that the computation
and estimation of singularities we have made are enough to make the conclusion. Taking
advantage of the modularity of both sides, we use some approximation to take care the terms
we don’t know how to compute.
Besides the linear independence of quasi-multiplicative Whittaker functions and their
quasi-derivatives used in our previous paper [Zh1], the new ingredient involved in the precise
proof is a theory of pseudo-theta series described in §6.1. Such series arise naturally in the
computation of the derivative and the local height pairing. Roughly speaking, there are
many series we don’t know how to compute explicitly, and their local components are really
messed. Then Lemma 6.1.1 asserts that, if a finite sum of pseudo-theta series is automorphic,
then it can actually be written as a finite sum of usual theta series. The lemma simplifies the
proof significantly. We expect that it also helps with the computation of derivative formula
for triple product L-functions in our another joint work.
6.1
Pseudo-theta series
We use the name “pseudo-theta series” because it looks like theta series and can be compared
with some theta series associated to it. Note that it is usually not automorphic.
Pseudo-theta series
Now we introduce pseudo-theta series. Let V be a positive definite quadratic space over F ,
and V0 ⊂ V1 ⊂ V be two subspaces over F with induced quadratic forms. We allow V0 to
be empty. Let S be a finite set of finite places of F , and φS ∈ S (V (AS ) × AS× )GO(V∞ ) be a
Schwartz function with the standard infinite components.
A pseudo-theta series is a series of the form
X
X
(S)
Aφ0 (g) =
φ0S (g, x, u)rV (g)φS (x, u), g ∈ GL2 (A).
u∈µ2 \F × x∈V1 −V0
We explain the notations as follows:
• The Weil representation rV is not attached to the space V1 but to the space V ;
Q
• φ0S (g, x, u) = v∈S φ0v (gv , xv , uv ) as a product of local terms;
• For each v ∈ S, the function
φ0v : GL2 (Fv ) × (V1 − V0 )(Fv ) × Fv× → C
is locally constant. And it is smooth in the sense that there is an open compact
subgroup Kv of GL2 (Fv ) such that
φ0v (gκ, x, u) = φ0v (g, x, u),
∀(g, x, u) ∈ GL2 (Fv ) × (V1 − V0 )(Fv ) × Fv× , κ ∈ Kv .
73
• µ is a subgroup of OF× with finite index such that φS (x, u) and φ0S (g, x, u) are invariant
under the action α : (x, u) 7→ (αx, α−2 u) for any α ∈ µ. This condition makes the
summation well-defined.
• For any v ∈ S and g ∈ GL2 (Fv ), the support of φ0S (g, ·, ·) in (V1 − V0 )(Fv ) × Fv× is
bounded. This condition makes the sum convergent.
The pseudo-theta series A(S) sitting on the triple V0 ⊂ V1 ⊂ V is called non-degenerate
if V1 = V , and is called non-truncated if V0 is empty. It is called to be of Whittaker type if
φ0S (g, ·, ·) transfer according to Weil representation under the unipotent group, i.e.,
φ0S (n(b)g, x, u) = φ0S (g, x, u) ψ(buq(x)),
∀b ∈ A.
We use this name because, if the above is true, then A(S) is invariant under the left action
of N (F ), and its first Fourier coefficient is exactly
X
φ0S (g, x, u)rV (g)φS (x, u).
(x,u)∈µ\((V1 −V0 )×F × )
uq(x)=1
We call it the Whittaker function of A(S) .
The pseudo-theta series A(S) is called non-singular if for each v ∈ S, the local component
0
φv (1, x, u) can be extended to a Schwartz function on V1 (Fv ) × Fv× .
(S)
Assume that Aφ0 is non-singular. Then there are two usual theta series associated to
A(S) . Extending by zero we may view φ0v (1, ·, ·) as a Schwartz function on V1 (Fv ) × Fv× for
each v ∈ S, and by restriction we may view φw as a Schwartz function on V1 (Fw ) × Fw× for
each w ∈
/ S. Then the theta series
X X
θA,1 (g) =
rV1 (g)φ0S (1, x, u)rV1 (g)φS (x, u)
u∈µ2 \F × x∈V1
(S)
is called the first theta series associated to Aφ0 . Replacing the space V1 by V0 , we get the
theta series
X X
θA,0 (g) =
rV0 (g)φ0S (1, x, u)rV0 (g)φS (x, u).
u∈µ2 \F × x∈V0
(S)
We call it the second theta series associated to Aφ0 . We set θA,0 = 0 if V0 is empty.
We introduce these theta series because the difference between θA,1 and θA,0 somehow
approximates A(S) .
Basic properties
(S)
Now we consider the relation between the non-singular pseudo-theta series Aφ0 and its
associated theta series θA,1 and θA,0 .
74
We first consider the non-truncated case. Then V0 is empty, and
X X
(S)
Aφ0 (g) =
φ0S (g, x, u)rV (g)φS (x, u).
u∈µ2 \F × x∈V1
(S)
Apparently we have Aφ0 (1) = θA,1 (1). But of course we can get more.
A simple computation using Iwasawa decomposition asserts that, if φw is the standard
Schwartz function on V (Fw ) × Fw× , then for any (x, u) ∈ V1 (Fw ) × Fw× ,
(
d−d1
if w - ∞;
δw (g) 2 rV1 (g)φw (x, u)
rV (g)φw (x, u) =
d−d1
d−d1
λw (g) 2 δw (g) 2 rV1 (g)φw (x, u) if w | ∞.
Here we write d = dim V and d1 = dim V1 . Here δw and λw coming from Iwasawa decomposition are explained in Section 1.6.
This result implies that,
(S)
Aφ0 (g) = λ∞ (g)
d−d1
2
δ(g)
d−d1
2
0
∀ g ∈ 1S 0 GL2 (AS ).
θA,1 (g),
Here S 0 is a finite set consisting of finite places v such that v ∈ S or φv is not standard.
Now we consider the general
X
X
(S)
Aφ0 (g) =
φ0S (g, x, u)rV (g)φS (x, u).
u∈µ2 \F × x∈V1 −V0
We have to compare it with the difference between the theta series
X X
θA,1 (g) =
rV1 (g)φ0S (1, x, u)rV1 (g)φS (x, u)
u∈µ2 \F × x∈V1
and the non-truncated pseudo-theta series
X X
(S)
Bφ0 (g) =
rV1 (g)φ0S (1, x, u)rV1 (g)φS (x, u).
u∈µ2 \F × x∈V0
Note that B (S) is just part of θA,1 , where the summation is taken over V0 but the representation is taken over V1 . By the discussion above, we should compare B (S) with the associated
theta series
X X
θB,1 (g) =
rV0 (g)φ0S (1, x, u)rV0 (g)φS (x, u).
u∈µ2 \F × x∈V0
But this is exactly the same as θA,0 . By the same argument, there exists a finite set S 0 of
finite places such that
(S)
d−d1
2
(S)
d1 −d0
2
Aφ0 (g) = λ∞ (g)
Bφ0 (g) = λ∞ (g)
δ(g)
d−d1
2
δ(g)
(S)
(θA,1 (g) − Bφ0 (g)),
d1 −d0
2
θA,0 (g),
75
0
∀ g ∈ 1S 0 GL2 (AS );
0
∀ g ∈ 1S 0 GL2 (AS ).
0
Our conclusion is that for any g ∈ 1S 0 GL2 (AS ),
(S)
Aφ0 (g) = λ∞ (g)
d−d1
2
δ(g)
d−d1
2
θA,1 (g) − λ∞ (g)
d−d0
2
δ(g)
d−d0
2
θA,0 (g).
(6.1.1)
By the smoothness condition of pseudo-theta series, there exists an open compact subgroup
0
KS 0 of GL2 (FS 0 ) such that the above identity is actually true for any g ∈ KS 0 GL2 (AS ).
0
If furthermore A(S) is of Whittaker type, the above remains true for any g ∈ KS 0 GL2 (AS )
if we replace A(S) , θA,1 , θA,0 by their Whittaker functions. It holds by the same argument
since the Whittaker functions are just a part of their corresponding pseudo-theta series or
theta series.
Now we can state our main result for pseudo-theta series.
(S )
Lemma 6.1.1. Let {A` ` }` be a finite set of non-singular pseudo-theta series of Whittaker
type sitting on vector spaces V`,0 ⊂ V`,1 ⊂ V` . Let {(Ei , Di )} be a finite set of pairs of
Eisenstein series for some non-equivalent pairs of characters (µi , νi ) and their derivations.
Assume that the sum
X
X (S )
A` ` (g) +
(Ei + Di )(g)
f :=
i
`
is cuspidal in g ∈ GL2 (A). Then
X (S )
X
(1)
A` ` =
θA` ,1 ,
`
(2)
X
`∈L0,1
θA` ,1 −
`∈Lk,1
X
θA` ,0 = 0,
∀k ∈ Z>0 .
`∈Lk,0
(3) Di = 0.
Here Lk,1 is the set of ` such that dim V` − dim V`,1 = k, and Lk,0 is the set of ` such that
dim V` − dim V`,0 = k. In particular, L0,1 is the set of ` such that V`,1 = V` .
(S )
Proof. In the equation of f , replace each A` ` by its corresponding combinations of theta series on the right-hand side of equation (6.1.1). After recollecting these theta series according
to the powers of λ∞ (g)δ(g), we end up with an equation of the following form:
n
X
λ∞ (g)k δ(g)k fk (g) = 0,
∀g ∈ KS GL2 (AS ).
(6.1.2)
k=0
Here S is some finite set of finite places, KS is an open compact subgroup of GL2 (FS ),
and f0 , f1 , · · · , fn are some automorphic forms on GL2 (A) coming from combinations of f
and theta series. In particular,
X
X
θA` ,1 −
f0 = f −
(Ei + Di ).
`∈L0,1
76
We will show that fk = 0 = Di identically, which is exactly the result of (1), (2), and (3).
As all theta series fk are defined by positive definite quadratic space, fk has a decomposition
X
fk =
fk,m ,
k>0
m
f0 =
X
f0,m +
X
Dm .
m
into a finite sum of forms and their derivations in the irreducible automorphic representations
πm of GL2 (A). The equation of the Whittaker functions corresponding to 6.1.2 is given by
X
X
WDm (g) = 0, ∀g ∈ KS GL2 (AS ).
λ∞ (g)k δ(g)k Wfk,m (g) +
(6.1.3)
m
k,m
This is a sum of quasi-multiplicative functions on GL2 (AS ) as δ(g) is already multiplicative.
Now we can apply Lemma 4.5.1 in our previous paper [Zh1] to functions on Kirillov models
a 0
a 0
k
αk,m (a) := δ Wfk,m
,
hm (a) := WDm
0 1
0 1
to conclude the following:
• Dm = 0 for all m;
• for any pair (k, m), there is another pair (k 0 , m0 ) such that
0
δ(g)k Wfk,m (g) + δ(g)k Wfk0 ,m0 (g) = 0
for g ∈ GL2 (AS ).
The second equation implies an equality between spherical Whittaker functions Wm0 (g) for
almost all unramified places v:
0
0
Wm,v
(g) = cδ(g)k −k Wm0 0 ,v (g)
where c is a nonzero constant. It follows
mthatπm,v and πm0 ,v have the same central character.
π
0
to conclude that πm = πm0 ⊗ | · |(k −k)/2 . Thus
Evaluate the above identity for g =
1
we must have k = k 0 , πm,v = πm0 ,v . By strong multiplicity one, we have π = π 0 and then
fk,m = 0.
6.2
Proof of Gross–Zagier formula
Recall that
Z
I(s, g, r(t, 1)φ)χ(t)dt, ϕ(g) .
P (φ, ϕ, χ, s) = vol(UT )
CU
We are going to show:
77
Theorem 6.2.1.
Z
P (φ, ϕ, χ, 0) = − 2vol(UT )
hZφ (g)t , 1 iχ(t)dt, ϕ(g)
CU
Z Z
vol(UT )
◦
◦
−1
hZφ (g)t1 , t2 iχ(t1 t2 )dt1 dt2 , ϕ(g) .
=−2
|CU |
CU CU
0
◦
◦
Here we write t◦ = t − deg(t)ξ.
Note that for any c ∈ A×
f , we have
X X
X
Zφ∗ (cg)[t] =
r(cg)φ(x)a [tx] =
=
a∈F ×
x∈B×
f /U
X
X
a∈F ×
X
r(g)φ(cx)a [tx]
x∈B×
f /U
r(g)φ(x)a [tc−1 x] = Zφ∗ (g)[c−1 t].
a∈F × x∈B× /U
f
It already implies that Zφ (cg)[t] = Zφ (g)[c−1 t] by modularity. It follows that
Z
hZφ (g)t◦ , 1◦ iχ(t)dt
CU
has the same central character as ϕ, and thus the Petersson inner product in the theorem
makes sense.
The theorem is equivalent to that
Z
(I 0 (0, g, r(t, 1)φ) + 2hZφ (g)t◦ , 1◦ i) χ(t)dt
CU
is perpendicular to ϕ. Now we are going to compare those two kernels.
By Proposition 3.7.1, replacing φ by r(t, 1)φ, we get
Pr(I 0 (0, g, r(t, 1)φ))
X
=−
Ie0 (0, g, r(t, 1)φ)(v) −
X
1
log |uq(y)|f )r(g, (t, 1))φ(y, u)
2
X
cφv (g, y, u)r(g, (t, 1))φv (y, u) − c1
(log δf (gf ) +
(y,u)∈µU \E × ×F ×
−
I 0 (0, g, r(t, 1)φ)(v)
v-∞ non−split
v|∞
+2
X
X
X
v
u∈µ2U \F × ,y∈E ×
r(g, (t, 1))φ(y, u)
u∈µ2U \F × ,y∈E ×
+ Eisenstein series and their derivations.
Here
Z
0
e
I (0, g, r(t, 1)φ)(v) = 2
Z
0
I (0, g, r(t, 1)φ)(v) = 2
(v)
0 0
0
Kf
φ (g, (tt , t ))dt ,
v|∞,
Z(A)T (F )\T (A)
(v)
Kφ (g, (tt0 , t0 ))dt0 ,
Z(A)T (F )\T (A)
78
v - ∞.
Now we list our computational results for the geometric part. Our place-to-place computation and comparison show that we also have
X
X
hZφ (g)t◦ , 1◦ i =
iv (Zφ∗ (g)t, 1) log Nv +
jv (Zφ∗ (g)t, 1) log Nv
v
v
X
+i0 (1, 1)
X
r(g, (t, 1))φ(y)a + hZφ,0 (g)t, 1i
a∈F × y∈E × /µU
+ Eisenstein series
Z and their derivations X
Z
X
(v)
0 0
0
=
Mφ (g, (tt , t ))dt +
log Nv
log Nv
CU
v non−split
+
X
Z
CU
jv̄ (Zφ∗ (g)tt0 , t0 )dt0
log Nv
CU
v
+i0 (1, 1)
v ∈Σ
/
(v)
Nφ (g, (tt0 , t0 ))dt0
X
X
r(g, (t, 1))φ(y)a + hZφ,0 (g)t, 1i
a∈F × y∈E × /µU
+ Eisenstein series and their derivations.
Since the archimedean parts are standard, we see that the average integrals
Z
Z
Z
Z
=
=
=
.
Z(A)T (F )\T (A)
Z(A)T (F )\T (A)/UT
T (F )\T (A)/UT
CU
Furthermore, CU is finite and the integral reduces to a finite sum.
In summary, we have proved that
Z
Z
0
0
◦
◦
Pr
(I (0, g, r(t, 1)φ) + 2hZφ (g)t , 1 i) χ(t)dt =
Dφ (g, (t, 1))χ(t)dt,
CU
CU
where
Dφ (g, (t, 1)) = Pr(I 0 (0, g, r(t, 1)φ)) + 2hZφ (g)t◦ , 1◦ i
is a sum of non-singular pseudo-theta series, Eisenstein series and their derivations.
Here is a checklist of the comparison:
(1) For v|∞, Proposition 5.1.1 shows that
(v)
(v)
Kf
φ (g, (t1 , t2 )) − Mφ (g, (t1 , t2 )) = 0.
(2) For v finite and non-split in E, we have showed that
(v)
(v)
Kφ (g, (t1 , t2 )) − Mφ (g, (t1 , t2 )) log Nv
is a nonsingular pseudo-theta series and vanishes for almost all v. This pseudo-theta
series is non-degenerate. See Proposition 5.2.6 and Proposition 5.2.7(1) for the supersingular case and Proposition 5.3.2 for the superspecial case.
79
(3) For v ∈
/ Σ, we have showed that the difference
(v)
Nφ (g, (t1 , t2 )) +
X
(log δf (gf ) +
(y,u)∈µU \E × ×F ×
1
log |uq(y)|f )r(g, (t1 , t2 ))φ(y, u)
2
is a nonsingular pseudo-theta series and vanishes for almost all v. This pseudo-theta series is degenerate. See Proposition 5.2.6 and Proposition 5.2.7(2) for the supersingular
case and Proposition 5.4.5 and Proposition 5.4.6 for the ordinary case.
(4) For all finite v, we have showed that jv̄ (Zφ∗ (g)tt0 , t0 ) is an integral of a nonsingular
pseudo-theta series on B(v) if φv . This pseudo-theta series is non-degenerate. The
integration of the associated theta series gives a Siegel–Einseisten series. See Section
5.5.
(5) The rest of terms are explicitly expressed as non-singular pseudo-theta series. Some
are non-singular automatically, and some are non-singular if φv ∈ S 0 (Bv × Fv× ).
(6) The Einsenstein series and their derivation comes from Proposition 3.6.3 for holomorphic projection and Proposition 4.6.1 for the height pairing related to the Hodge class
ξ. See §4.4.4 in our previous paper [Zh1] for a general explanation of derivations.
Only finitely many terms are left after the comparison. Note that the integral on CU is
just a finite sum. Then we are in the situation to apply Lemma 6.1.1 to
Z
X
X
Dφ (g, (t, 1))χ(t)dt =
A` +
(Ei + Di ).
CU
i
Since their sum is modular and cuspidal, it is equal to a finite sum of usual theta series
(v)
and Eisenstein series by Lemma 6.1.1. These theta series are coming from Kφ (g, (t1 , t2 )) −
(v)
Mφ (g, (t1 , t2 )) log Nv and jv̄ (Zφ∗ (g)tt0 , t0 ), the only non-degenerate pseudo-theta series in the
list. Note that the quadratic spaces for both of them are the nearby quaternion algebra B(v).
Therefore, we end up with
Z
X
Dφ (g, (t, 1)) =
θφ(v) (g, (tt0 , t0 ))dt0 + Eisenstein Series.
CU
v
We understand that the equality is true after integrating against χ(t) on CU , and that the
theta series is given by
X
X
θφ(v) (g, (t1 , t2 )) =
r(g, (t1 , t2 ))φ0v (yv , uv )r(g, (t1 , t2 ))φv (y, u)
u∈µ2U \F × y∈B(v)
(v)
(v)
for some Schwartz function φ(v) = φv ⊗ φv ∈ S (B(v)A × A× ) with φv ∈ S (B(v)v × Fv× ).
80
Therefore, we get that
Z
ϕ(g)
χ(t)Dφ (g, (t, 1))dt,
X Z Z
=
χ(t)θφ(v) (g, (tt0 , t0 ))dt0 dt,
CU
v
CU
ϕ(g) .
CU
Now the vanishing of the right-hand side following from the criterion of the existence of local
χ-linear vectors by the result of Tunnell [Tu] and Saito [Sa], Proposition 1.1.1.
Note that to control the singularities for some ramified non-archimedean place v, we
have assumed that φv ∈ S 0 (Bv × Fv× ). But we can easily extend the result to all Schwartz
function φv ∈ S (Bv ×Fv× ). We have showed that the main theorem is true for all pairs (ϕ, φ)
with ϕ ∈ π and φ = ⊗v φv such that φv is standard for good v, and φv ∈ S 0 (Bv × Fv× ) for
bad v. By Proposition 3.3.1, all theta liftings θφϕ given by such pairs span the space π 0 ⊗ π̄ 0 .
On the other hand, the result of the main theorem depends only on the theta lifting θφϕ . We
conclude that the main theorem is true for all ϕ, φ.
81
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