Invent. math. 130, 99±152 (1997)
Heights of Heegner cycles and derivatives of L-series
Shouwu Zhang
Department of Mathematics, Columbia University, New York City, NY 10027, USA
(e-mail: szhang@math.columbia.edu)
Oblatum 21-III-1997 & 2-IV-1997
Contents
0. Introduction ................................................................................................................. 99
1. Intersections and heights ............................................................................................. 103
2. Kuga-Sato varieties and CM-cycles ............................................................................ 117
3. Heights of CM-cycles .................................................................................................. 124
4. Heights of Heegner cycles............................................................................................. 137
5. Proof of the main identity and the consequences ....................................................... 147
References ........................................................................................................................ 151
0. Introduction
In [18], Gross and Zagier proved an identity on modular curves between the
height pairings of certain Heegner points and coecients of certain cusp
forms of weight 2. As a consequence, they showed that any modular elliptic
curve over an imaginary quadratic ®eld whose L-function has a simple zero
at s ˆ 1 contains a Heegner point of in®nite order. This result plays a crucial
rule in the solution of the Gauss class number problem by Goldfeld-GrossZagier [15, 18], and in the solution of the Birch and Swinnerton-Dyer
conjecture [24] by Kolyvagin when the L-series of the modular elliptic curve
over Q has order 1.
In this paper, we will extend Gross and Zagier's result to higher weights
by using the arithmetic intersection theory. More precisely, we will de®ne
the (global) height pairing between CM-cycles in certain Kuga-Sato varieties, and show an identity between the height pairings of Heenger cycles
and coecients of certain cusp forms of higher weights.
Research support by NSF under the grant numbers DMS-9303475 and DMS 9796021
100
S. Zhang
In the following, we will ®rst describe our main result and applications,
then summarize the main contents of the remaining sections.
0.1 De®nitions of p
CM-cycles
and height pairings. For an elliptic curve E

with a CM by D0 , let Z…E† denote the divisor
pclass on E E of
CÿE f0g ÿ D0 f0g E; where C is the graph of D0 . For k a positive
integer, then Z…E†kÿ1 is a cycle of codimension k ÿ 1 in E2kÿ2 . Let Sk …E†
denote the cycle
X
sgn g …Z…E†kÿ1 †;
c
g2G2kÿ2
where G2kÿ2 denotes the symmetric group of 2k ÿ 2 letters which acts on
E2kÿ2 by permuting the factors, and c is a real number such that the selfintersection of Sk …E† on each ®ber is …ÿ1†kÿ1 .
For N a product of two relatively prime integers 3, one can show that
the universal elliptic curve over the non-cuspidal locus of X …N †Z can be
extended uniquely to a regular semistable elliptic curve E…N † over whole
X …N †. The Kuga-Sato variety Y ˆ Yk …N † will be de®ned to be a canonical
resolution of the 2k ÿ 2-tuple ®ber product of E…N † over X …N †. If y is a CMpoint on X …N †, the CM-cycle Sk …y† over x will be de®ned to be Sk …Ey † in Y .
If x a CM-divisor
p on X0 …N †Z , the CM-cycle Sk …x† over x will be de®ned to
P
be Sk …xi †=P deg p, where p denotes the canonical morphism from X …N † to
xi ˆ p x. One can show that Sk …x† has zero intersection with
X0 …N †, and
any cycle of Y supported in the special ®ber of YZ , and that the class of Sk …x†
in H 2k …Y …C†; C† is zero. So there is a green's current gk …x† on Y …C† such that
@ @
b
pi gk …x† ˆ dSk …x† . The arithmetic CM-cycle Sk …x† over x, in the sense of Gillet
and Soule [13], is de®ned to be …Sk …x†; gk …x††.
If x and y are two CM-points on X0 …N †, then the height pairing of the
CM-cycles Sk …x† and Sk …y† will be de®ned to be
hSk …x†; Sk …y†i :ˆ …ÿ1†k b
Sk …y†:
Sk …x† b
0.2. Main identity. Let K be an imaginary quadratic ®eld with the discriminant D, such that every prime factor of N is split in K. Let H denote the
Hilbert class ®eld of K. Let r be a ®xed element Gal…H =K†,
P and A the ideal
class in OK corresponding to r via the Artin map. Let f ˆ n1 a…n†e2pinz be a
new form of weight 2k on C0 …N †. De®ne the L-series associated to f and A by
X D
X
LA … f ; s† ˆ
nÿ2s‡2kÿ1
a…m†rA …m†mÿs ;
n
n1
m1
n;ND†ˆ1
where rA …m† is the number of integral ideals in A with the norm m. Then
Gross and Zagier proved that the function LA … f ; s† has analytical continuation to the entire complex plane, and satis®es a functional equation when
s is replaced by 2k ÿ s, and vanishes at the point s ˆ k. They have
Heights of Heegner cycles and derivatives of L-series
101
P
new
constructed explicitly an element U ˆ m1 aA …m†qm 2 S2k
C0 …N †† to
represent the linear functional
p
2k ÿ 2†! jDj 0
LA … f ; k†
f ÿ!
24kÿ1 p2k
new
on the hermitian space S2k
C0 …N †† with Petersson product. The main identity in this paper is as follows:
Theorem 0.2.1 Let x be a Heegner point on X0 …N † with discriminant D, and m
an integer prime to N . Then
hsk …x†; Tm sk …xr †i ˆ u2 aA …m†
where u ˆ jOK j=2.
Notice that when k ˆ 1 Gross and Zagier showed that
hx ÿ 1; Tm …xr ÿ 1†i ˆ u2 aA …m†;
where h; i is the NeÂron - Tate pairing on the Jacobian of X0 …N †.
0.3 Consequences. Let V be the subspace of Heegner cycles generated by
Tm xr r 2 Gal …H =K†; …m; N † ˆ 1 :
Let V 0 be the quotient of V modulo the null subspace with respect to the
height pairing. Then we will show the following
Theorem. 0.3.1. The Hecke module V 0 is isomorphic to a sub-quotient module
of S2k …C0 …N ††h .
P
Let v a character of G, set s0v ˆ r2G vÿ1 …r†s0k …xr † where s0k …xr † is the image
new
C0 …N †† be a normalized eigenform. Extend ff g
of sk …xr † in V 0 . Let f 2 S2k
to an orthogonal basis f1 ˆ f ; , of S2k …C0 …N ††, then the cycle s0v can be
written
P as a sum of fi -isotropic components transform like that of fj :
s0v ˆ djˆ1 s0v fj with Tm s0v;fj ˆ am … fj †s0v;fj : By the same reasoning as GrossZagier, we obtain the following corollaries:
Corollary 0.3.2.
L0 … f ; v; k† ˆ
24kÿ1 p2k … f ; f †
p hs0v;f ; s0v;f i:
2k ÿ 2†!u2 h jDj
Corollary 0.3.3. If L0 … f ; v; k† 6ˆ 0 then s0v;f 6ˆ 0.
Corollary 0.3.4. Let f 2 S2k …C0 …N †† be any newform and v any character of
have a
Gal…H =K†. Then either all conjugates L… f a ; va ; † (a 2 Gal…Q=Q††
simple zero at s ˆ k or else all have a zero of order 3.
102
S. Zhang
Corollary 0.3.5. Let f be any new form of weight 2k and f a (a 2 Gal…Q=Q†)
any conjugate of f . Then
ordsˆk L… f ; s† ˆ 0()ordsˆk L… f a ; s† ˆ 0
ordsˆk L… f ; s† ˆ 1()ordsˆk L… f a ; s† ˆ 1
ordsˆk L… f ; s† 2()ordsˆk L… f a ; s† 2
ordsˆk L… f ; s† 3()ordsˆk L… f a ; s† 3
Finally we give the following two consequences of the following index
conjecture of Gillet-Soule and Beilinson-Bloch:
Corollary 0.3.6. Assume the height pairing on sk …X † is positively de®nite. Then
(a) V is ®nitely dimensional.
new
C0 …N †† and any character v of Gal…H =K†,
(b) for any eigenform f 2 S2k
L0 … f ; v; k† 0:
Notice that the inequality here is already predicted by the general Riemann
hypothesis.
We will also obtain some conditional results about the algebraicity
conjecture of Gross-Zagier, and the generalized Birch and Swinnerton-Dyer
conjecture of Beilinson and Bloch. Since the statements of these results need
some extra de®nitions, we postpone them until 5.
0.4. Plan of the proof. As in weight 2 case treated by Gross and Zagier, for m
prime to N , we need to de®ne the global pairing hSk …x†; Tm Sk …xr †i, and
compute it as a sum of local pairings hSk …x†; Tm Sk …xr †iv , even when Sk …x† and
Tm Sk …xr † are not disjoint.
In Sect. 1, we will de®ne the global and local height pairings for general
cycles Z1 ; Z2 on an arithmetic varieties over a number ®eld, provided that
they have good models over the ring of integers. Here we will use the
arithmetic intersection theory introduced by Gillet and Soule [13].
In Sect. 2, we will study Kuga-Sato varieties and CM-cycles. Here we will
follow closely the work of Deligne-Rapoport [9], Katz-Mazur [23], Deligne
[7], and Scholl [33].
In Sect. 3, we will study global and local height of CM-cycles. First we
will de®ne CM-cycles Sk …x† and a height pairing on the group Sk …X † of CMcycles, and show that both Beilinson's index conjecture and Gillet-SouleÂ's
index conjecture imply the positivity of the height pairing. Then we will give
formulas for local heights. At Archimedean place, we will show that the
local height pairings are given by certain standard green's functions. At the
nonarchimedean place, the local heights are related to deformations of elliptic curves. The ®nal formulas are similar to those given by Gross-Zagier
[18] and Brylinski [5].
Heights of Heegner cycles and derivatives of L-series
103
In Sect. 4, we will compute hSk …x†; Tm Sk …xr †i for a Heegner point x on X .
Here we will use our local formulas and some computations of Gross and
Zagier.
In Sect. 5, we will ®rst prove the main identity and its corollaries. Then
we will give the three applications.
Acknowledgments. It should be mentioned that Perrin-Riou [32] has proved
a p-adic version of Gross-Zagier's formula, and Nekovar [31] has extended
Perrin-Riou's work to high weights.
It should be also mentioned that in [5], Brylinski worked some de®nitions
of local heights suggested by Deligne. However, since the lack of the global
theory of pairing, as well as the theory of self-pairing (e.g. adjunction formula), the results of Gross and Zagier seems dicult to be extended to
higher weights by just using these de®nitions. At this point, our work should
be considered a continuation of his work, even though our proof doesn't
depend on any of his result.
I would like to thank J.-L. Brylinski, B. H. Gross, G. Pappas, A. Wiles,
and D. Zagier for useful discussions. The work was partially prepared
during my visits to the Universite de Paris-Sud, and the Insititut des Hautes
EÂtudes Scienti®que. I would like to express my gratitude to these institutions
for their hospitalities.
1 Intersections and heights
1.1. Intersections. Let us ®rst review the arithmetic intersection theory of
Gillet and Soule [13]. Let Y be a regular arithmetic scheme of dimension d
over Spec OF . This means that the morphism Y ! Spec OF is projective and
¯at and that Y is regular. For any integer p 0, let Ap;p …Y † (resp. Dp;p …Y ††
denote the real vector space of real di€erential forms (resp. currents) a
a ˆ …ÿ1†p a, where
which are of type … p; p† on Y …C† and such that F1
F1 : Y …C† ! Y …C† denotes the complex conjugation.
AP
cycle of codimension p on Y with real coecients is a ®nite formal sum
Z ˆ a ra Za , where ra 2 R, and Za are closed irreducible subvarieties of
codimension p in Y . Such a cycle de®nes a current of integration
dZ 2 D p;p …YR †, whose value on a form g of complementary degree is
X Z
dZ …g† ˆ
ra
pa …g†:
a
Za …C†
A green current for Z is any current g 2 Dpÿ1;pÿ1 …YR † such that the
curvature
@ @
g
h Z ˆ dZ ÿ
pi
is a smooth form in Ap;p …Y † Dp;p …Y †.
104
S. Zhang
The (real) arithmetic group of codimension p is the real vector space
c p …Y † generated by pairs …Z; g†, where Z is a real cycle of codimension p
Ch
R
on Y and g is a green current for Z, the addition being de®ned compo is
nentwise, with the following relation over R. First any pair …0; @u ‡ @v†
p
0
c
trivial in Ch …Y †R . Second, if Y Y is a irreducible subscheme of codimension p ÿ 1 on Y , f 2 k…Y 0 † a nonzero rational function on Y , then the
c p …Y † .
pair …div… f †; ÿ log jf jdY …C† † is zero in Ch
R
b
b
Let Z1 ˆ …Z1 ; g1 † and Z 2 ˆ …Z2 ; g2 † be two arithmetic cycles of Y of codimensions p and d ÿ p. Assume both Z1 and Z2 are irreducible and intersect
properly. Then we de®ne the intersection of …Z1 ; g1 † and …Z2 ; g2 † as follows:
b2 ˆ log jC…Z1 Z2 ; O†j ‡
b1 Z
Z
Z
Z
Z2 …C†
g1 ‡
Y …C†
g2 hZ :
b1 :
For later use, we also de®ne the intersection of Z2 and Z
Z
b
g1 :
Z2 Z1 ˆ log jC…Z1 Z2 ; O†j ‡
Z2 …C†
1:1:1†
c …Y † , and
b2 depends only on the classes of Z
bi in Ch
b1 Z
One can show that Z
R
c
b
b
Z1 Z2 depends only on Z2 and the classes of Z1 in Ch …Y †R . Since Y is
cp
regular, for
P any0 class y in Ch …Y †R and any cycle Z2 , one can ®nd a cycle
ri …Zi ; gi † (with real coecients) representing y such that Zi0 is
Z1 ; g† ˆ
irreducible and intersects every irreducible component of Z2 properly. In this
b2 y by linearity.
way we may de®ne Z2 y and Z
More generally, as showing in [13], there is an (associative and commutative) intersection product
c q …Y † ! Ch
c p‡q …Y †
c p …Y † Ch
Ch
R
R
R
such that if …Z1 ; g1 † and …Z2 ; g2 † are two cycle such that cod…Z1 \ Z2 † ˆ p ‡ q
then
Z1 ; g1 † …Z2 ; g2 † ˆ …Z1 Z2 ; g2 dZ1 …C† ‡ h2 g1 †:
c d …Y † with R by taking intersection with Y as (1.1.1), then
If we identify Ch
R
the intersection product of cycles with complementary degrees gives the
intersection pairing of these cycles.
We now state the following index conjecture of Gillet and SouleÂ. Given a
line bundle L on Y , equipped with a smooth hermitian metric invariant
c 1 …Y † ; de®ned as the class
2 Ch
under F1 , one gets a ®rst Chern class bc1 …L†
R
div…s†; ÿ log ksk†, for any nonzero rational section s of L on Y . Denote by
c p‡1 …Y †
c p …Y † ! Ch
L : Ch
R
R
i.e., L…Y † ˆ Y bc1 …L†.
the product b
c…L†,
Heights of Heegner cycles and derivatives of L-series
105
is positive if the following three conditions are satis®ed:
We say that L
is ample on Y ,
(a) L
is a positive 1±1 form on Y …C†,
(b) the curvature c1 …L†
(c) for any subvariety Y 0 of Y of dimension n and ¯at over Spec Z,
n Y 0 > 0.
c1 …L†
be a positive hermitian line
Conjecture 1.1.1. (Gillet and Soule [14]). Let L
bundle on Y and x an arithmetic Chow cycle of codimension p. If 2p d,
x 6ˆ 0, and Ldÿ2p‡1 …x† ˆ 0, then
ÿ1†p deg…xLdÿ2p x† > 0
Remarks. (a) The case that Y is an arithmetic surface, the index conjecture is
a theorem of Faltings [10] and Hriljac [21].
(b) KuÈnnermann [25, 26] and Moriwaki [28] have proved the conjecture
in several cases.
(c) In their original conjecture, instead of conditions above, Gillet and
Soule actually stated that for an ample line bundle L on Y , there is a metric
on L such that Conjecture 1.1.1 is true.
b2 into the local
b1 Z
1.2. Local decompositions. We would like to decompose Z
b2 † for places v of F :
b1 Z
intersections …Z
v
b2 ˆ
b1 Z
Z
X
v
b1 Z
b2 † v :
Z
v
1:2:1†
If Z1 and Z2 are disjoint at the generic ®ber then the intersection Z1 Z2 with
support de®nes an element in ChdjZ1 j\jZ2 j …Y †, (see 4.1.1 in [13]). Since jZ1 j\jZ2 j
is supported in special ®bers, one has well de®ned xv 2 ChdjY k…v†j …Y † for each
®nite place v such that
Z1 Z2 ˆ
X
v
xv :
We de®ne
b2 † ˆ deg xv
b1 Z
Z
v
if v is ®nite, and
b2 † ˆ
b1 Z
Z
v
Z
Z
g‡
Z2v …C†
g Z2 h 1
Yv …C†
if v is in®nite, where Yv denotes Y OF ;r C for an embedding r : F ! C
inducing v and Z2v is the pullback of Z2 on Yv . Notice that Yv can be
considered as a component of
106
S. Zhang
Y Spec Z Spec C ˆ
a
X OF ;r C;
r:F !C
and that the integrals do not depend on the choice of r.
When jZ1F j \ jZ2F j 6ˆ ;, we will try to de®ne the local intersection in the
following situation: there is a morphism p : Y ! X from Y to a regular
arithmetic surface X such that both Z1 and Z2 are contained in a ®ber YD of
Y over an integral divisor D of X , and that the morphism pD : YD ! D is
~ ! D be the normalization of D, i : D ! X the inclusion,
smooth. Let u : D
and v ˆ iu. We will use letters uY , iY and vY for their pullbacks under the
base change Y ! X .
uY
Y
?D
?
?pD
y
ƒƒƒƒ!
u
D
ƒƒƒƒ!
YD~
?
?
?p ~
y D
ƒƒƒƒ!
~
D
ƒƒƒƒ!
iY
?Y
?
?p
y
i
X
Fix a local coordinate t for DF in XF by which we mean an element t in
the algebraic closure of the function ®eld F …X † such that
(a) Some positive power te is in F …X †;
(b) The divisor 1e div…te † ÿ D is disjoint with D on XF .
Let div…dD t† denote …1e divte ÿ D†jD . De®ne ordv …dD t† to be a rational number
such that the pushforward of the 0-cycle div…dD t† to Spec OF is
P
v ordv ord…dD t†‰vŠ. Notice that when D is a section of X over Spec OF ,
e div…dD t† is a section of …X1X =OF †
e .
By de®nitions, one has
Z
b2 ˆ Z2 Z
b1 ‡
b1 Z
g2 h1 ;
Z
Y …C†
and
b1 ˆ u Z2 v~ Z
b
Z2 Z
Y
Y 1:
Now in Ch …YD~ †, one has
v …Z1 † ˆ Z1 vY p c1 …O…D†† ˆ ÿZ1 uY pD div dD t:
It follows that there is a current g0 for cycle 0 such that
vY …Z1 ; g1 † ˆ…ÿZ1 uY pD div dD t; g0 †
X
0
ordv dD t†Z1 k…v†; g ;
ˆ ÿ
vj1
1:2:2†
Heights of Heegner cycles and derivatives of L-series
107
where Z1 k…v† denotes the pullback of Z1 to the ®ber YD k…v† of YD on the
closed point Spec k…v† corresponding to v. Since Z1 and Z2 are ¯at over D,
one has
X
v …ordv dD t†…Z1F Z2F †YD;F
uY Z2 vY …Z1 ; g1 † ˆ ÿ
v-1
‡
X
Z
v
vj1
g0 ;
Z2v …C†
where …Z1F Z2F †YD;F is the intersection number being taking
R in YD;F .
To de®ne the local intersections we have to describe Z2v …C† g0 explicitly
for each ®x each in®nite place v. Let g be a di€erential form such that
ˆ 0 on each ®ber Y …C† when q is near jDj…C†, and that its
@g ˆ @g
q
Then
restriction on Y …C†RD…C† is dZ2 …C† modulo the images of @ and @.
the function q ! Yv …C† gg is well de®ned
R ÿ f pg. We hope
R on Xv …C†
q
this function gives some information for Z2v …C† g0 ˆ YD;v …C† gg0 . We would
like to give the following
Conjecture 1.2.1. With notations and assumptions as above, for any p in
jDv j…C†, one has
Z
Z
Z
gg ‡ log jtj…q†
gˆ
g0 g ‡ o…1†
Yv …C†q
Zv …C†
Yv …C†p
as q ! p.
Suppose that the conjecture is true. We want to de®ne the local intersection at a place v as follows. If v is ®nite, then de®ne
b2 † ˆ ÿ…Z1F Z2F † ordv dD t;
b1 Z
Z
v
YD;F
If v is in®nite, let G be a function on jDj…C† de®ned by
Z
G… p† ˆ lim
q!p
gg ‡ log jt…q†j…Z1F Z2F †YD;F ;
Y …C†q
then we de®ne
b2 † ˆ G…Dr …C†† ‡
b1 Z
Z
v
Z
g2 h1 :
Yv …C†
Remarks. (a) When Z1 moves in a family, the local intersection at
archimedean place has been de®ned by B. Harris and B. Wang [19]. It is not
dicult to show that the above conjecture is true.
108
S. Zhang
(b) In this paper we need to compute the intersection pairing of CMcycles. In this case, the ®ber of Y over D is an abelian scheme and Zi are
sums of subabelian schemes. However Zi do not move in families. In 1.4,
we will prove that above conjecture is true in this case. Moreover, we will
prove
Theorem 1.2.2. The conjecture is true in the following case: each irreducible
component Z 0 of Z1 is regular and cnÿ1 …NZF0 …YD;F †† is trivial as a Chow
cycle.
We now try to de®ne the intersections over valuation ®elds. In nonarchimedean case, we will consider a regular, projective, and ¯at scheme V
over a discrete ring R. Let Z1 and Z2 be two irreducible cycles on V with
complementary dimensions. If they are disjoint at the generic ®ber, we can
de®ne the intersection Z1 Z2 as usual. Otherwise, we assume that V has a
®beration to a regular surface S over R, and that Z1 and Z2 are supported in
the ®ber VD over an integral divisor D of S which is ¯at over R, and that
every components of Z1 and Z2 are ¯at over D. Let t be a coordinate for Dg ,
where g is the generic point of Spec R. Then we can de®ne the local intersection with respect to t by
Z1 Z2 † ˆ ÿ…Z1g Z2g †YDg ordD t:
1:2:3†
If v is a ®nite place of F , let Wv denote the completion of maximal
b1 and Z
b2 on Y
unrami®ed extension of Ov . Then any two arithmetic cycles Z
will induce two cycles Z1v and Z2v on Y Wv . One has
b 2 † ˆ D1v D2v :
b1 D
D
v
In archimedean case, we consider a regular and projective variety V over
b ˆ …Z; g† of a
C. By an arithmetic cycle of codimension p, we mean a pair Z
cycles Z of codimension p on V and a Green's current g for Z as before. Let
b2 ˆ …Z2 ; g2 † be two arithmetic cycles on V with
b1 ˆ …Z1 ; g1 † and Z
Z
dim Z1 ‡ dim Z2 ˆ dim V ÿ 1. If Z1 and Z2 have disjoint support then we can
de®ne intersection as usual. Otherwise, we will assume that V has a ®beration to a regular curve C over C and Z1 and Z2 are contained in a ®ber Vp
over a point p. Let t be a local coordinate for p, let g be an di€erential form
on Y …C† which is @ and @ closed over ®bers Y …C†q for q 2 C…C† near p, and
whose restriction on Y …C†p represents the cohomology class of Z2 …C†.
Assume that the limit
Z
g1 g ÿ …Z1p Z2p † log jtj
G… p† :ˆ lim
q!p
Y …C†q
exists. Then we de®ne
Heights of Heegner cycles and derivatives of L-series
b2 ˆ G… p† ‡
b1 Z
Z
109
Z
g2 h1 :
Y …C†
b1 and
If v is an archimedean place of F , then any two arithmetic cycles Z
b2 in Y induce two arithmetic cycles Z
b1v and Z
b2v on Yv . One has
Z
b2 † ˆ Z1v Z2v :
b1 Z
Z
v
For a morphisms of arithmetic varieties, one can de®ne the pushforward
and pullback maps on the groups of arithmetic cycles. One has projection
formulas for intersections and local intersections. The formulas for cycles Z1
and Z2 with disjoint supports at the generic ®ber are obvious. Otherwise, we
will consider the case that cycles are contained in a ®ber of a ®beration. For
example we have the following projection formula for archimedean local
intersections:
Proposition 1.2.3. Consider the following diagram of morphisms of regular and
projective varieties over C:
Y?0
?
f 0?
y
ƒƒƒƒ!
pY
Y?
?
f?
y
X0
ƒƒƒƒ!
pX
X;
where X and X 0 are curves. Let D and D0 be irreducible divisors on X and X 0
respectively. Assume that the following induced diagrams are cartesian:
YD0?0
?
fD0 ?
y
D0
ƒƒƒƒ!
ƒƒƒƒ!
Y?
pD0
?
?
y
pD;
Yp0?ÿ1 D
?
?
y
ƒƒƒƒ!
YD
?
?
?fD
y
pÿ1 D
ƒƒƒƒ!
D;
b ˆ …Z; g† be an arithmetic
and that the morphism fD0 and fD are smooth. Let Z
b0 ˆ …Z 0 ; g0 † an
cycle of Y with Z supported in YD and ¯at over D, and Z
0
0
0
¯at over D0 . If
arithmetic cycle of Y with Z supported in the ®ber YD0 and
1
0
0
e
pD ˆ D, let t be a local coordinate for D and t ˆ …p t† be the local coordinate for D0 , where e is the rami®cation index of pX along D0 . One has
b p Z
b0 ˆ p Z
bZ
b0 :
Z
Proof. By de®nitions.
(
1.3. Heights. Now assume that the dimension of Y is d ˆ 2n. Let L be an
ample line bundle on Y . Let Z1F and Z2F be two cycles of YF of codimension
n. Assume the following conditions:
110
S. Zhang
(a) Z1F has an integral model Z in Y which has zero intersection number
with any cycle of Y of dimension n supported in special ®bers; and
(b) the class of Z1 vanishes in H 2n …Y …C†; C†. So there is a green's current
g1 on Y …C† such that @pi@ g ˆ dZ1 .
We de®ne the global height pairing
hZ1F ; Z2F i ˆ …ÿ1†n …Z1 ; g1 † …Z2 ; g2 †
where g2 is any current for Z2 . It is not dicult to show that hZ1F ; Z2F i
depends only on the rationally equivalent classes of Z1F and Z2F . The commutativity of the height pairing follows from that of intersection pairing.
Concerning the positivity of height pairing, one has the Gillet-SouleÂ's
Conjecture 1.1.1 and the following
Conjecture 1.3.1 (Beilinson [2]). Let Chn …X †00 denote the subgroup of Chow
cycles satisfying the above two conditions and having 0 intersections with
c1 …L†. Then the height pairing is positively de®nite on Chn …X †00 .
Remark. The conjecture implies in particular that the pairing is nondegenerate. This nondegeneracy is already conjectured by Bloch. Notice that
Beilinson and Bloch independently de®ned their pairings by cohomological
method. It is believed that their pairings are coincide with our pairing in
Arakelov theory. See [2, 3, 34] for discussions.
One can also de®ne the local height pairing by
hZ1F ; Z2F iv ˆ …ÿ1†n ……Z1 ; g1 † …Z2 ; g2 ††v ;
when the right hand side is de®ned.
Let Y ! X be a morphism over Spec OF from an arithmetic variety Y to
an arithmetic surface X . Assume that Z1 and Z2 satisfy the condition (a) and
(b) in Sect. 1.3 and are contained in ®bers over two irreducible divisors D1
and D2 of X over Spec OF . From the de®nition one can show
Proposition 1.3.2. For Z1 and Z2 and v as above, the pairing hZ1F ; Z2F iv de®ned
as above depends only on the classes of Z1 and Z2 in Ch …YD1 ;F † and
Ch …YD2 ;F †.
Similarly, we can de®ne height pairing for cycles over valuation ®elds using
Sect. 1.2.
1.4. Proof of Theorem 1.2.2. The morphism vY : YD~ ! Y is a composition of
~ ! Y and an embedding ~i : Y ~ ! Y~ . One has a decomposiu~ : Y~ ˆ Y D D
D
tion u~ …Z; g† ˆ …Za ; ga † ‡ …Zb ; gb † such that Za is supported in YD~ and that
Zb;F is disjoint with YD;F
~ . So we may assume that D is a section of X over
Spec OF . By linearity, we may also assume that Z :ˆ Z1 is integral.
In general, a direct computation for iT …Z; g† is dicult to ®nd. We now
®rst blow-up Y along Z to obtain the following diagrams:
Heights of Heegner cycles and derivatives of L-series
E
iE
fE #
Z
,!
iZ
,!
111
iZ
Y0
#f
Z0
fZ 0 #
,!
Y
Z
,!
iZ
T0
# fT 0
T
where E is the exceptional divisor, T 0 is the proper transformation of T , and
Z 0 is the intersection of E and T 0 . We then show that f …Z† has more space to
move, more precisely, whose restriction on the generic ®ber is a multiple of a
divisor:
Lemma 1.4.1. In the Chow group of Y 0 , one has
f …Z† ˆ T 0 iE ffE c…NZ …T ††…1 ‡ iE E†ÿ1 gkÿ1 ‡ iE fE ckÿ1 …NZ …T ††
where k is the codimension of Z in Y and fgkÿ1 denotes the part of degree
k ÿ 1.
Proof. As Chow cycle, one has
f ‰ZŠ ˆ iE …ckÿ1 … fE NZ …Y †=NE …Y 0 †††:
See [12] for a proof. From the exact sequence
0 ! NZ …T † ! NZ …Y † ! NT …Y †jZ ! 0;
we have
f …Z† ˆ iE ffE c…NZ …T ††…1 ‡ c… fE …NT …Y †jZ †††…1 ‡ c1 …NE …Y 0 †††ÿ1 gkÿ1 :
Since
fE …NT …Y †jZ † ˆ fE …O…T †jZ † ˆ O…E†jE O…T 0 †jE
and NE …Y 0 † ˆ O…E†, we have
f …Z† ˆ iE ffE c…NZ …T ††…1 ‡ iE E ‡ iE T 0 †…1 ‡ iE E†ÿ1 gkÿ1
ˆ T 0 iE ffE c…NZ …T ††…1 ‡ iE E†ÿ1 gkÿ1 ‡ iE fE ckÿ1 …NZ …T ††
Let U be a cycle in Y representing
(
ffE c…NZ …T ††…1 ‡ iE E†ÿ1 gkÿ1 ;
and V be a cycle representing of ckÿ1 …NZ …T †† with support in the special
®bers. Choose currents gU , gE , gT 0 for U ; E; T 0 such that gE ‡ gT 0 ˆ log jtj for
points near T . Then there is a smooth form a such that
112
S. Zhang
f …Z; gZ † ˆ …T 0 ; gT † …U ; gU † ‡ …iE fE V ; a†:
1:4:1†
To get a formula for gZ , we may assume that U is supported in E and
properly intersect with Z 0 in E. Then f …Z† is represented by
T 0 iE …U † ‡ iE fE V . It is not dicult to show that as cycles,
f …T 0 iE U ‡ iE fE V † ˆ Z:
It follows that
gZ ˆ f …gU dT 0 ‡ hU gT 0 ‡ a†:
1:4:2†
To compute iT …Z; gZ †, we ®rst pullback to T 0 and then pushforward to Y .
Let U 0 be a cycle in Y representing
iE ffE c…NZ …T ††…1 ‡ c1 …E††ÿ1 gkÿ1
such that U 0 properly intersects with Z 0 : Write div…t† ˆ T 0 ‡ E ‡ R: Then by
(1.4.1), one has
iT 0 fE …Z; gZ † ˆ iT 0 …ÿE ÿ R; gT 0 ‡ log jtj† iT 0 …U 0 ; gU 0 † ‡ …iZ 0 fZ0 V ; ajT 0 †
or simply
iT 0 fE …Z; gZ † ˆ …ÿZ 0 U 0 ‡ iZ 0 fZ0 V ÿ R U 0 T 0 ; ÿgE gU 0 ‡ ajT 0 †
1:4:3†
We claim that ÿZ 0 U 0 ‡ iZ 0 fZ0 V can be moved away, more precisely, we
have
Lemma 1.4.2. The cycle iZ 0 …ÿU 0 ‡ fE V † vanishes in the Chow group of Z.
Proof. Using O…E†jZ 0 ˆ NZ 0 …T 0 †, one knows that iZ 0 …ÿU 0 † ‡ fZ0 V represents
ffZ0 c…NZ …T ††…1 ‡ c1 …NZ 0 …T 0 †††ÿ1 …ÿc1 …NZ 0 …T 0 †† ‡ fZ0 c…NZ …T †††gkÿ1
ˆ ffZ0 c…NZ …T ††…1 ‡ c1 …NZ 0 …T 0 †††ÿ1
‰ÿc1 …NZ 0 …T 0 †† ‡ 1 ‡ c1 …NZ 0 …T 0 ††Šgkÿ1
ˆ ffZ0 c…NZ …T ††…1 ‡ c1 …NZ 0 …T 0 †††ÿ1 gkÿ1
fZ 0 NZ …T †
ˆ 0;
ˆ ckÿ1
NZ 0 …T 0 †
where in the last equation, we use the fact that the bundle
k ÿ 2.
fZ0 NZ …T †
NZ 0 …T 0 †
has rank
(
This implies that there are subvarieties Wi (1 i l) in Z 0 and functions
fi on Wi such that as cycles
Heights of Heegner cycles and derivatives of L-series
ÿU 0 Z 0 ‡ iZ 0 fZ0 V ˆ
l
X
113
divWi … fi †:
1:4:4†
iˆ1
Now by (1.4.2) we have
iT 0 fE …Z; gZ †
ˆ
0
0
ÿ R U T ; ÿg g jT 0 ‡ ajT ‡
Z0
X
U0
1:4:5†
log jfi jdWi
We pushforward this cycle to Y and get
iT …Z; gZ † ˆ…ÿf …R U 0 T †; ÿfT 0 …gE gU jT 0 †
X
0
log jfi jdWi :
‡ fT …ajT 0 † ‡ fT …1:4:6†
We need to compute the current
fT X
log jfi jdWi
ˆ f
X
log jfi jdWi
in terms of the given currents gU , gT and a.
c …Z†, one has
Lemma 1.4.3 As cycles in Ch
fE …gT 0 gU 0 jE ‡ ajE † ˆ f
X
log jfi jdWi :
1:4:7†
Proof. We will prove this by the equation fE iE …Z; gZ † ˆ 0. From (1.4.1),
one has
iE …Z; gZ † ˆ …T 0 ; gT 0 †jE …U 0 ; gU 0 †jE ‡ … fE V c1 …NE …Y 0 ††; ajE †
ˆ …Z 0 U 0 ‡ fE V c1 …NE …Y ††; gT 0 gU 0 jE ‡ ajE †:
By (1.4.4), this equals to
or
fE V
0
Z ÿ
fE V
l
X
iˆ1
divWi … fi † ‡
fE V
c1 …NE …Y ††; gT 0 gU 0 jE ‡ ajE
0
Z ‡ c1 …NE …Y †††; gT gU 0 jE ‡ ajE ÿ
X
log jfi jdWi :
(
fE iE …Z; gZ †
Since
Z, we have
c
0
ˆ 0 in Ch …Z†, and fE …Z ‡ c1 …NE …Y ††† ˆ 0 as cycles in
114
S. Zhang
X
fE gT 0 gU 0 jE ‡ ajE ÿ
log jfi jdWi ˆ 0:
From (1.4.6) and (1.4.7), it follows that
iT …Z; gZ † ˆ…ÿZ R; ÿfT 0 …gE gU 0 jT 0 †
‡ fT 0 …ajT 0 † ‡ fE …gT 0 gU 0 jE ‡ ajE ††
or simply
iT …Z; gZ † ˆ …ÿZ div…dt††; ÿhU gE jT 0 ‡ ajT 0 ‡ fE …hU gT 0 ‡ a†jE dZ :
We are ready to give an asymptotic formula for
R
Y …C†q
1:4:8†
gg. De®ne
YE;q ˆ fy 2 Yq …C†; gu …y† > gT 0 …y†g
YT 0 ;q ˆ fy 2 Yq …C†; gT …y† > gu …y†g
for any q 2 X …C† near p. By (1.4.2),
Z
Z
Z
gZ g† ˆ
hU gT 0 ‡ a†g ˆ
Y …C†q
(since
R
Y …C†q
ÿgE hU ‡ a†g;
Y …C†q
hU g ˆ 0)
Z
Z
ÿgE hU ‡ a†g ‡
ÿgE hU ‡ a†g
ˆ
Yq
YE;q
YT 0 ;q
Z
Z
ÿhU g† ‡
ˆ ÿ log jtj
YE;q
Z
gT 0 hU ‡ a†g ‡
YE;q
ÿgE hU ‡ a†g
YT 0 ;q
ˆ ÿ log jtjc…q† ‡ d…q†
where
Z
c…q† ˆ
ÿhU g;
YE;q
Z
Z
gT hU ‡ a†g ‡
d…q† ˆ
YE;q
ÿgE hU ‡ a†g:
YT 0 ;q
We need a asymptotic formula for c and d as q ! p.
Lemma 1.4.4. Let g0 be the green current in (1.4.8), then
Z
g ‡ O…jtj log jtj†
c…q† ˆ
Z…C†
Heights of Heegner cycles and derivatives of L-series
Z
115
g0 g ‡ O…jtj…log jtj†2 †
d… p† ˆ
T 0 …C†
If we assume this lemma, then we have
Z
Z
Z
gZ g† ˆ ÿ log jtj g ‡ g0 g ‡ O…jtj…log jtj†2 †:
Yq
Z
1:4:9†
T
The theorem follows.
It remains to prove the lemma. We compute the leading terms ®rst.
Z
ÿhU g ˆ
c… p† ˆ
E…C†
Z
Z
ÿ
hU
E…C†=Z…C†
g
Z…C†
ˆ ÿfE c1 …O…E†jE †kÿ1 …ÿ1†kÿ2
ˆ fE c1 …O…1††kÿ1
g
Z…C†
Z
g
Z
Z
ˆ
Z
g;
Z…C†
Z
d… p† ˆ
Z
gT 0 hU ‡ a†g ‡
E…C†
Z
ÿgZ hU ‡ a†g
T 0 …C†
Z
fE …gT 0 hU ‡ a†g ‡
ˆ
Z…C†
ÿgE hU ‡ a†g:
T 0 …C†
Now we need to estimate the error term as q ! p. By a partition of the
unit, we reduce the question to the same estimates for smooth g supported in
any chosen neighborhood U of a given point p on Y . Here we drop the
assumption on closeness of g on ®bers. If p 62 Z 0 , we may choose U disjoint
with U 0 . It is easy to show that both c and d are smooth. If p 2 Z 0 , then
locally in a neighborhood of p, Y 0 has the equation z1 z2 ˆ t in Cd‡1 with
coordinates …t; z1 ; z2 ; zd †, E has the equation z1 ˆ 0, T 0 has the equation
z2 ˆ 0, and Z 0 has the equations z1 ˆ z2 ˆ 0. We therefore reduce the
question to the following lemma.
Lemma 1.4.5. On Cd‡1 …d 2†, let / be a smooth real function and let g be a
smooth form of degree 2…d ÿ 1† with compact support. For each t 2 C, de®ne a
subset Ut of Cd by
Ut ˆ f…z1 ; z2 ; ; zd † : jz1 je/ > jz2 jeÿ/ ; z1 z2 ˆ tg;
116
S. Zhang
and numbers
Z
log jz1 jg;
G…t† ˆ
Ut
Z
H …t† ˆ
g:
Ut
Then as t ! 0, one has
G…t† ˆ G…0† ‡ O…jtj…log jtj†2 †;
H …t† ˆ H …0† ‡ O…jtj log jtj†:
Proof. We ®rst reduce the problem to case d ˆ 2 and / ˆ 0. Write
g ˆ g1 ‡ g2 such that the support of the restriction of g1 on Ut is relatively
compact, and that the points of the support of the restriction of g2 on Ut
have small coordinates z1 and z2 . The functions G and H for g ˆ g1 are
smooth. So we may assume g ˆ g2 . We can change coordinates with z1
replacing z1 e/ and z2 replacing z2 eÿ/ and with zi unchanged if i > 2. In these
new coordinates we have / ˆ 0. Let Ut0 be in C2 de®ned by
Ut0 ˆ f…z1 ; z2 † : jz1 j > jz2 j; z1 z2 ˆ tg
and g0 a 2-form on C3 de®ned by
Z
0
g …z1 ; z2 † ˆ
g;
C
dÿ2
where the integral is over the variables z3 ; ; zd . Then g0 is smooth and has
compact support and
Z
log jz1 jg0 ;
G…t† ˆ
Ut0
Z
H …t† ˆ
g0 :
Ut0
This reduces to the case d ˆ 2 and / ˆ 0.
Now we assume d ˆ 2 and / ˆ 0. The only nontrivial contribution to the
integral is the (1, 1) part of g. We may assume that g is of type (1,1) of the
form
g ˆ a11 dz1 ^ dz1 ‡ a12 dz1 ^ dz2 ‡ a21 dz2 ^ dz1 ‡ a22 dz2 ^ dz2 ;
where aij 's are smooth functions of …t; z1 ; z2 † with compact supports. We
want to change the integrals to one variable integrals by substituting z1 ˆ z
and z2 ˆ t=z. Then
Heights of Heegner cycles and derivatives of L-series
Ut ˆ f…z; t=z† : jzj >
and
117
p
jtjg
!
t
t
jtj2
g ˆ a11 ÿ a12 2 ÿ a21 2 ‡ a22 4 dzdz
z
z
jzj
t
ˆ h z; 2 dz ^ dz
z
where h is a smooth function of two variables with compact support, for
example we assume h ˆ 0 if the norm of the ®rst variable is > A.
The error term of G can be written as
Z
t log jzj h z; 2 ÿ h…z; 0† dz ^ dz
G…t† ÿ G…0† ˆ
z
p
A>jzj> jtj
Z
log jzjh…z; 0†dz ^ dz:
ÿ
p
jzj<
jtj
Since h…z; zt2 † ÿ h…z; 0† is dominated by j zt2 j and h…z; 0† by 1, it follows that the
®rst integral is dominated by
Z
ÿ
A>jzj>
p
jtj
log jzj† 2 dz ^ dz ˆ ÿ2pijtj‰…log A†2 ÿ …log jtj†2 Š;
jzj
p
jtj
and the second integral is dominated by
p 1
log jzjdz ^ dz ˆ ÿ2pijtj log jtj ÿ
:
2
p
Z
jzj<
jtj
We therefore have the following estimate:
G…t† ÿ G…0† ˆ O…jtj…log jtj†2 †:
Similarly we have the estimate:
H …t† ÿ H …0† ˆ O…jtj log jtj†:
(
2. Kuga-Sato varieties and CM-cycles
2.1. Universal semistable elliptic curves. Let N 3 be a positive integer and
fN a primitive N -th root of the unity. Let X …N †Q‰fN Š be the compacti®cation
118
S. Zhang
of the moduli of elliptic curves E over a Q‰fN Š-scheme S, with a canonical
full level N structure, i.e., an isomorphism of groups
/ : …Z=N †2 ! E‰N Š
over S, such that the Weil pairing of /…1; 0† and /…0; 1† is fN . Let E…N †0Q‰fN Š
be the universal elliptic curve over the noncuspidal part X …N †0Q‰fN Š of
X …N †Q‰fN Š , and E…N †Q‰fN Š be the Kodaira-NeÂron minimal compacti®cation of
E0 …N †Q‰fN Š , which is a semistable elliptic curve over X …N †Q‰fN Š with N -polygons at cusps. Let pQ‰fN Š denote the structure morphism E…N †Q‰fN Š !
X …N †Q‰fN Š . We want to construct certain model over Z‰fN Š for E…N †Q‰fN Š
following [9] and [23].
Theorem 2.1.1. Assume that N is a product of two relatively prime integers
N1 3. Then the morphism pQ‰fN Š can be uniquely extended to a morphism
p ˆ pZ‰fN Š : E…N † ! X …N † of regular, ¯at, and projective Z‰fN Š-schemes, such
that p makes E…N † a semistable elliptic curves over X …N †.
Proof. The uniqueness of p is clear. We need only show the existence. Write
N0 ˆ N ˆ N1 N2 with N1 and N2 relatively prime and 3. Over
Spec Z fNi ; N1i …i ˆ 0; 1; 2†, Deligne and Rapoport have shown that
E…Ni †Q‰fN Š ! X …Ni †Q‰fN Š have integral models
i
i
E…Ni †Z‰fN ; 1 Š ! X …Ni †Z‰fN ; 1 Š ;
i Ni
i Ni
satisfying corresponding properties as stated in the theorem with the base
Z‰fNi ; N1i Š. One has the following commutative diagram:
E…N1 †Z‰fN ; 1 Š
1 N1
#
X …N1 †Z‰fN ; 1 Š
1 N1
#
Spec Z‰fN1 ; N11 Š
E…N †Z‰fN ; 1 Š
!
N
#
X …N1 †Z‰fN ; 1 Š !
N
#
Spec Z‰fN ; N1 Š !
E…N2 †Z‰fN ; 1 Š
2 N2
#
X …N1 †Z‰fN ; 1 Š
2 N2
#
Spec Z‰fN2 ; N12 Š:
Let X …N †Z‰fN ;N1 Š be the normalization of X …Ni †Z‰fN ;N1 Š in the function
i i
i
®eld of X …N †Q‰fN Š , and let E…N †0Z‰fN ; 1 Š be the pullback of E…Ni †Z‰fN ; 1 Š on
Ni
i Ni
X …N †Z‰fN ; 1 Š , then we obtain the following diagram:
Ni
E…N †0Z‰fN ; 1 Š
N1
#
X …N1 †Z‰fN ; 1 Š
N1
E…N †Z‰fN ; 1 Š
N
#
X …N †Z‰fN ; 1 Š
N
!
!
E…N †0Z‰fN ; 1 Š
N2
#
X …N2 †Z‰fN ; 1 Š
N2
It is not dicult to see that the two lower arrows are open embeddings.
So we can de®ne scheme X …N † over Spec Z‰fN Š by gluing two schemes at two
lower ends by the scheme at the lower middle.
Heights of Heegner cycles and derivatives of L-series
119
The two upper arrows are open embedding over non cuspidal points of
X 's but not over cusps. At a cusp of X …N †, the ®bers of E…N †0 ‰fN ;N1i Š have
Ni sides. Locally near the intersection of two sides over a cusp of X …N †,
where X …N † is de®ned by an equation t ˆ 0, the scheme E…N †0‰fN ; 1 Š has an
Ni
equation xy ˆ tN =Ni . We blow-up E…N †0 ‰fN ;N1i Š along intersections several
times, we will obtain a scheme E…N †‰fN ;N1i Š which has an equation xy ˆ t near
the intersection of two sides near a ®ber. It is obvious that over
Spec Z‰fN ; N1 Š, these blow-ups are just E…N †Z‰fN ; 1 Š so we obtain two open
N
embeddings
E…N †Z‰fN ; 1 Š ! E…N †Z‰fN ; 1 Š :
E…N †Z‰fN ; 1 Š
N1
N
N2
Gluing two schemes at sides by the middle one, we obtain a scheme E…N †
over Spec Z‰fN Š. The regularity of X …N † can be found in [23]. The regularity
of E…N † follows from the fact that the morphism E…N † ! X …N † is smooth at
non cuspidal point, and that E…N † locally in eÂtale topology has an equation
xy ˆ t near a cuspidal point.
(
Remark. Let E…N † be the smooth locus of the morphism E…N † ! X …N †.
The full level structure N on the generic ®ber de®nes a homomorphism of
group schemes:
/ : …Z=N †2 ! E…N † :
P
One can show that
a;b†2…Z=N †2 ‰/…a; b†Š equals to E…N † ‰N Š as Cartier
divisors. It is an interesting question to show that the morphism
E…N † ! X …N † represents the moduli stack which assigns to a scheme S the
category of semistable elliptic curves E over S with a morphism of group
schemes over S
/ : …Z=N †2 ! E
such that the following properties are veri®ed:
P
(a) as a Cartier divisor, …a;b†2…Z=N †2 ‰/…a; b†Š equals to E ‰N Š and that
(b) im …/† meets every irreducible component of each geometric ®ber of
E ! S.
Following [9] and [23], we can also describe the ®bers of X …N † and E…N †
over a closed point Spec k ! Spec Z‰fN Š. Let p be the characteristic of k. If
p-N , then Xk is smooth and the cuspidal divisor is reduced, so E…N †k is
smooth. If pjN , say N ˆ pn M with …M; p† ˆ 1, then X …N † is the disjoint
union, with crossing at the supersingular points, of smooth curves X …N †k;A
over k indexed by the set of cyclic subgroups A of …Z=pn †2 of order pn . For
each cyclic subgroup A of …Z=pn †2 , the non-cuspidal and non-supersingular
points of X …N †k;A parameterized elliptic curves E with a canonical full level
M structure and a morphism w : …Z=pn †2 ! E of group scheme, such that
the following conditions are veri®ed:
120
S. Zhang
(a) as Cartier divisors
X
‰w…a; b†Š ˆ E‰pn Š;
a;b†2…Z=pn †2
(b) the Weil pairing of w…1; 0† and w…0; 1† is 1.
(c) the image of A is connected.
Proposition 2.1.2. At each closed point Spec k ! Spec Z‰fN Š, each irreducible
component C of Xk …N † is a regular and projective curve over k and the ®ber EC
of E…N † over C is a regular and semistable elliptic curve over C.
2.2. Kuga-Sato varieties. Let X be a regular scheme and E be a regular and
semistable elliptic curve over X . Assume that the cuspidal divisor of X (over
which the morphism E ! X is not smooth) is smooth in X . Let w be an
positive integer, then w-tuple ®ber product scheme Ew over X is not regular,
it has a regular resolution of singularities as follows [7].
Let e be a closed point in E over which the morphism / : E ! X is not
smooth. Then x ˆ /…e† is in the cuspidal divisor of X . Let t be a parameter
b e;E , A ˆ O
b x;X be the
on Ox;X de®ning the cuspidal divisor. Let B ˆ O
completions of local rings, then one has
B ˆ A‰‰u; vŠŠ=…uv ÿ t†:
Now in eÂtale topology, at each closed point Ew has the singularity like that of
V ˆ Spec
A‰s1 ; ; sq ; u1 ; v1 ; ; ur ; vr Š
u1 v1 ÿ t; u2 v2 ÿ t; ; ur vr ÿ t†
where q; r are nonnegative integers.
Q
Let I be the ideal of OV generated by monomials riˆ1 ruiiÿ1 where r is a
permutation of coordinates which preserves the set of pairs fui ; vi g
1 i r†. Then the variety Ve induced from V by blowing-up I isQregular.
To see this let U be the open ane subscheme of Ve over which riˆ1 uiiÿ1
de®nes an invertible section of O…1†. The structure algebra O…U † over O…V †
is generated by elements
Q iÿ1
rui
Q iÿ1
ui
where r are permutations of the set fu1 ; v1 ; ; ur ; vr g preserving the set of
pairs f…u1 ; v1 †; …ur ; vr †g. The regularity will follows from the following
lemma
Lemma 2.2.1 (Deligne [7]).
O…U † ˆ A‰s1 ; ; sw ; v1 =u2 ; u1 =u2 ; ; urÿ1 =ur ; ur Š:
Heights of Heegner cycles and derivatives of L-series
121
Proof. We prove ®rst that RHS LHS. For 0 j r ÿ 1 we de®ne
permutations rj as follows. When j ˆ 0,
8
< v2
r0 …ui † ˆ v1
:
ui
8
if i ˆ 1,
< u2
if i ˆ 2, r0 …vi † ˆ u1
:
otherwise.
vi
if i ˆ 1,
if i ˆ 2,
otherwise.
When j > 0,
8
< ui‡1
rj …ui † ˆ uiÿ1
:
ui
8
if j ˆ i,
< vi‡1
if j ˆ i ‡ 1, rj …vi † ˆ viÿ1
:
otherwise.
vi
if j ˆ i,
if j ˆ i ‡ 1,
otherwise.
It is easy to see that
Q
rj …ui †iÿ1
Q iÿ1 ˆ
ui
v1 =u2
uj =uj‡1
if j ˆ 0,
if j > 1.
This proves that RHS LHS.
Now we want to prove that LHS RHS. It is obvious that si 's and ui 's
are in RHS.QFor a permutation
r preserving the set of pairs, we need to
Q
is
in RHS. If for some j 1 and k > 1,
show that
r…ui †iÿ1 = uiÿ1
i
r…ui † ˆ vk then we may replace vk by uk , since vk ˆ …v1 =uk †…u1 =uk †uk and
v1 =uk †…u1 =uk † is in RHS. Finally we may assume that r takes u's to u's, or to
v1 . We have a permutation a of f1; ; rg such that r…ua…i† † ˆ ui if i > 1 and
r…ua…1† † ˆ u1 or v1 and we have
Q iÿ1 aÿ1 …1†ÿ1 aÿ1 …1†‡aÿ1 …2†ÿ1ÿ2
rui
u1 or v1
u2
Q iÿ1 ˆ
u2
u3
ui
r…1†‡r…rÿ1†ÿ1ÿÿ…rÿ1†
urÿ1
:
ur
Since the exponents are all nonnegative, this is in RHS. This proves that
LHS RHS.
(
Now on Ew , we may blow up the ideal J ˆ Ox;Ew \ I and get a regular
scheme Y :
Notation. For k a positive integer and N a product of two coprime integers
3, we let Yk …N † denote the canonical resolution of the 2k ÿ 2-tuple product
of E…N † over X …N †.
Let us now de®ne the Hecke correspondences. Write X :ˆ X …N † and
E :ˆ E…N †. For m a positive integer prime to N , Let Xm0 be the moduli
scheme classifying elliptic curves E with a C…N † structure and an isogeny
0
E ! E0 of degree m. Write E0m for the universal curve over Xm0 and E0m ! Em0
122
S. Zhang
0
for the universal isogeny. Then Em0 has a level C…N † structure coming from
that on E0m . Consider the diagram
E0 †2kÿ2
#
X0
/1
E0m †2kÿ2
#
Xm0
w
!
ˆ
Em0 †2kÿ2
#
Xm0
0
/2
!
!
E0 †2kÿ2
#
X0
On Y …M; N † we de®ne Hecke correspondence Tm as the Zariski closure in
E2kÿ2 E2kÿ2 of the correspondence /1 w /2 on …E0 †2kÿ2 .
2.3. -component of cohomology. In this subsection we will review some
constructions and results of Scholl [33]. Let N be a product of two relatively
prime integers 3. Let X be an irreducible component of a geometric ®ber
of X …N † ! Spec Z‰fN Š. Let E denote the ®ber of E…N † on X and Y the ®ber
of the Kuga-Sato variety Y …N † on X . By Proposition 2.1.2, X and Y are
smooth over the base ®eld k with characteristic p. As in [33], the full level N
structure on E de®nes a homomorphism of group schemes / : …Z=N †2 ! E ,
where E denote the smooth locus of the morphism E ! X . Therefore,
Z=N †2 acts on E by translations. Combining with multiplication by 1, this
also gives an action on E by the semiproduct …Z=N †2 l2 on E. The
symmetric group Gw of w-letters acts on Ew by permuting the factors. Hence
the semiproduct
Dw :ˆ ……Z=N †2 l2 †w Gw
acts on Ew . As the resolution introduced in Sect. 2.2 is canonical, this
semiproduct also acts on Y . Let : Dw ! f1g be the homomorphism
which is trivial on …Z=N †w , is the product map on lw2 , and is the sign
character on Gw . For any Q‰Dw Š-module V , write V …† for - isotropic
component of V .
Let p denote the projection Y ! X . Let j : X 0 ! X denote the inclusion
of the complement X 0 of cusps in X . Let H be either l-adic cohomology
theory (l 6ˆ p) or Betti cohomology …when k ˆ C†. So the coecient F of H is either Ql or Q. One main result proved in [33] is as follows.
Theorem 2.3.1 (Scholl [33]). There are canonical isomorphisms:
H 1 …X ; j Symw Rw p F † ! H …Y ; F †…†:
Moreover when characteristic p - N , the actions of the Hecke correspondences
on the right hand side are compatible with actions of Hecke correspondences
on the left hand side de®ned in [7].
Strictly speaking, Scholl only stated his theorem when k has characteristic 0, but his proof is valid for our general case without any change, as in
the case that k has positive characteristic, Het is the cohomological part of a
twisted Poincare duality [4].
Heights of Heegner cycles and derivatives of L-series
123
2.4. CM-cycles. Let E be a CM elliptic curve over an integral ring R whose
generic point g has characteristic 0. The ring EndS …E† Q which depends
only on the isogeny class of E, is isomorphic to an imaginary quadratic
extension. Fix an embedding sE : EndS …E† Q ! C. As the NeÂron-Sevri
group NS…Eg Eg † of Eg Eg has rank 4, there is a divisor of E S E whose
E g † is perpendicular to the diagonal Dg , f0g Eg , and
image in NS…E
pg
0) be
Eg f0g. Let ÿD0 (D0 >
 an element in EndS …E†. Let C be the graph
p
of the multiplication by ÿD0 then we can choose this divisor as
Z…E† ˆ C ÿ E f0g ÿ D0 f0g E:
Let k be a positive integer, then Z…E†kÿ1 de®nes a cycle of codimension
k ÿ 1 in E2kÿ2 . Notice that the symmetric group G2kÿ2 has an action on E2kÿ2
by permuting the factors. We de®ne
X
Wk …E† ˆ
sgn…g†g …Z…E†kÿ1 †
g2G2kÿ2
and
Sk …E† ˆ cWk …E†
where c is a positive constant such that the self intersection of Sk …E†g in
Eg2kÿ2 is …ÿ1†kÿ1 . The existence of c follows from the fact that the cohomology class of Sk …E†g is nonzero and primitive with respect to the product
polarization. By the Hodge index theorem the self intersection of Sk …E†g is
non-zero and has signature …ÿ1†kÿ1 . Actually, one can compute c by representing Z…E† by di€erential forms on E E as in Sect. 3.4. The following
proposition follows from the de®nitions.
k
2kÿ1
† does not depend
Proposition 2.4.1.p(a)
The
 class ‰Sk …E†Š of Sk …E† in Ch …E
0
on the choice of ÿD when s is ®xed.
(b) If s changed to its complex conjugate then in the Chow group, ‰Sk …E†Š is
changed to …ÿ1†kÿ1 ‰Sk …E†Š.
(c) Let / : E1 ! E2 be an isogeny of CM-elliptic curves over two CMdivisors x1 and x2 of X . For a ®xed embedding
s : End…E1 † Q ˆ End…E2 † Q ! C;
one has / ‰Sk …E2 †Š ˆ …deg /†kÿ1 ‰Sk …E1 †Š in the Chow group of E12kÿ1 .
Let N be a product of two relatively prime integers 3. Write E :ˆ
E…N † R and Y :ˆ Yk …N † R. So one has the following morphisms of
schemes:
X
E2kÿ2
Y:
Let R be an integral and ¯at algebra over Z which is unrami®ed over all
primes dividing N . We want to de®ne the following objects:
124
S. Zhang
(a) the space sk …X † of CM-cycles over Q and
(b) for each CM-divisor x on X , the pair of elements sk …x† in sk …X †R .
Let x be an irreducible CM divisor in X , by this we mean that x is ¯at
over Spec R and the corresponding elliptic curve E :ˆ Ex has a complex
multiplication. Write Wk …x† for Wk …E† and Sk …x† for Sk …E†. We call the class
sk …x† of Sk …x† in Chk …Y † the CM-Chow-cycles over x.
We de®ne the space of CM-cycles sk …X † to be a subspace of Chk …Y †
generated by the classes of Wk …x† over all CM-divisors x of X . So
sk …x† 2 sk …X † R. By the above proposition, sk …x† depends only x.
Let N be a product of two relatively prime integers 3.
Proposition 2.4.2. Let x be an irreducible CM-divisor of point X . One has
Tm …‰Sk …x†Š† ˆ
if Tm x ˆ
P
X
mkÿ1 ‰Sk …xi †Š
i
i xi
Proof. The actions of Hecke operators are compatible with map p and p .
By functoriality, we may assume that N is a product of two integers 3.
Then the proposition follows from Proposition 2.4.1 (c).
(
Concerning action of D2kÿ2 one has
Lemma 2.4.3 The action of D2kÿ2 on sk …X † has character .
Proof. It is easy to see that sk …X † has signature character under G2kÿ2 .
For ……Z=N 0 †2 l2 †2kÿ2 part, we need only work on Z…E† in Ch1 …E E†
for an elliptic curve de®ned over an integral domain. But here the fact is
obvious.
(
3. Heights of CM-cycles
3.1. Global heights. Let N be a product of two relatively prime integers 3.
Let F be a number ®eld in which all prime factors of N are unrami®ed. Write
X :ˆ X …N † OF . We want to de®ne the height pairing on sk …XF †. Let x and y
be two CM-divisors on XF . Let x and y be their Zariski closures on X . To
de®ne the global height pairing hSk …x†; Sk …y†i of CM-cycles in YF , we have to
check two conditions in Sect. 1.3.1. For the ®rst condition, we consider the
integral CM-cycles Sk …x† and show that the restriction of this cycle on any
component W of a geometric ®ber of Y over Spec OF is numerically
equivalent to 0. For this we need only consider the case that F ˆ Q. Let p be
the characteristic of the ground ®eld of W . Then for l 6ˆ p, the restriction of
Sk …x† on W de®nes an element in Hetk …W ; Ql …k††. This class is -isotropical by
Lemma 2.4.3, so is 0 by Theorem 2.3.1. By the similar way, the second
condition is also veri®ed. So we can de®ne global height pairing in Sk …XF †.
Heights of Heegner cycles and derivatives of L-series
125
More precisely, let gk …x† be the Greens current for the cycle Sk …x† with the
following properties:
(a) @pi@ gk …x† ˆ dSk …x† ,Rand
(b) the integration gk …x†g is 0 for any @pi@ closed form g on Y …C†. Such
gk …x† is determined up to images of @ and @.
We de®ne the arithmetic CM-cycles over x as
b
Sk …x† ˆ …Sk …x†; gk …x††:
Now the height pairing of CM-cycles Sk …x† and Sk …y† is
Sk …y†:
hSk …x†; Sk …y†i ˆ …ÿ1†k b
Sk …x† b
3:1:1†
We are going to prove the positive de®niteness of the height pairing on
the groups of CM-cycles under Conjecture 1.2.1 of Gillet and SouleÂ, or
under Conjecture 1.4.1 of Beilinson-Bloch. We need to show that all these
arithmetic cycles are primitive for some ®xed positive hermitian line bundle
ˆ …L; kk†.
L
as a sum of the following line bundles:
We may choose L
the pull back of a line bundle on X with divisor supported at
(a) M:
cusps, and
i …1 i 2k ÿ 2†: the pull-backs of a line bundle on E with a
(b) L
divisor supported on the unit section O, and
a line bundle which has a divisor supported in the exceptional
(c) N:
2kÿ2
.
divisor of the blow-up
P Y !E
we may assume that L
is invariant under
by
r
L
Replacing L
r2D2kÿ2
x†
is
-isotropic.
D2kÿ2 . Similarly, we may assume that b
Sk
Sk …x† in
Proposition 3.1.1. For any CM-cycle Sk …x†, the intersection b
c1 …L†
c
Ch…Y † is zero.
Lemma 3.1.2. For any CM-cycle Sk …x†, the intersection c1 …L† Sk …x† in Ch…Y †
is 0.
Let us ®rst show that this lemma implies the proposition. Actually, this
b
c …Y † is represented by
Sk …x† in Ch
statement implies that the element b
c1 …L†
@ @
and b
Sk …x†, so it must
0; g†. Since pi g is the product of the curvatures of L
be 0. It follows that g de®nes an element in H k …Y ; C†…†. By Theorem 2.3.1,
Therefore …0; g† is 0
this group is 0. So g is in the sum of images of @ and @.
c …Y †R .
in Ch
To show the lemma, it suce to show that Sk …x† has 0 intersection in the
Chow group of Y with all c1 …Li †, c1 …M†, and c1 …N†. It is easy for c1 …M†
and c1 …N†, as M and N are supported in the ®bers of Y over the cusps of X
which are disjoint with Sk …x†. To show that c1 …Li † Sk …x† ˆ 0; it suce to
show for each r 2 G2kÿ2 that
126
S. Zhang
Z…E†2kÿ2 r c1 …Li † ˆ 0;
where E is the elliptic curve corresponding to x. Up to a permutation of
factors, it is easy to see that the left hand side of this equation is
Z…E† f0g E†p34
Z…E†† p2kÿ3;2kÿ2
Z…E††
p12
or
Z…E† E f0g†p34
Z…E†† p2kÿ3;2kÿ2
Z…E††
p12
Now our lemma follows from the following lemma:
Lemma 3.1.3. LetpE
! Spec OF be a smooth elliptic curve with a complex

multiplication by ÿD0 . Then in Ch…E Spec OF E†Q ,
Z…E† f0g E ˆ 0;
Z…E† E f0g ˆ 0
0 ÿ E f0g ÿ D0 f0g E.
where Z…E† ˆ CpÿD
Proof. Indeed,
Z…E† f0g E ˆ ÿD0 …f0g f0g† E ˆ 0;
as f0g f0g in Ch…E†Q is 0. Similarly
Z…E† E f0g ˆ ÿE …f0g f0g† ˆ 0:
(
3.2. Local decompositions. The notations and assumptions are as above. We
want to decompose global heights into local heights
hSk …x†; Sk …y†i ˆ
X
hSk …x†; Sk …y†iv v :
3:2:1†
v
Let x and y be two irreducible CM-divisors of XF . Then we can de®ne as
in Sect. 1.3.3,
Sk …x† b
Sk …y††v ;
hSk …x†; Sk …y†iv ˆ …ÿ1†k …b
3:2:2†
when the right hand side is de®ned. The right hand side is always de®ned
and (3.2.1) is true when x 6ˆ y. When x ˆ y, we need to check the assumption
of Theorem 1.2.2. The cycle Sk …x† ˆ cWk …E† is supported in the ®ber E2kÿ2 of
Y over x, where E is the elliptic curve corresponding to x. Notice that E2kÿ2 is
an abelian variety and Wk …E† is a sum of abelian subvarieties Ai . Since all Ai
have trivial normal bundle in E2kÿ2 over F , it follows from Theorem 1.2.2
Heights of Heegner cycles and derivatives of L-series
127
that the Conjecture 1.3.1 is true for Ai (with any current). It follows that
Conjecture 1.3.1 is true for Sk …x†. Now we can de®ne local intersection as in
Sect. 1.2.
To choose a local coordinate t for x we introduce the di€erential
4
Y
dq
n
:
1 ÿ q †
xˆ q
q
n1
1
24
We choose a local coordinate t at x such that
1
x ˆ …1 ‡ O…t††t1ÿu…x† dt;
3:2:3†
in a neighborhood of x, where u…x† ˆ ordx x. The pairing hSk …x†; Sk … y†iv does
not depend on the choice of t satisfying the above condition.
We may also formally de®ne the heights for CM-cycles de®ned over
valuation ®eld, as local intersections can be de®ned in this case as in
Sect. 1.2.
3.3. Nonarchimedean formulas. Let W be a complete discrete valuation
ring. Assume that the ®eld Q of fractions of W has characteristic 0, and
the residue ®eld W0 of W has characteristic not dividing N and is algebraically closed. Write X for X …N † W . We want to compute the local
height hSk …x†; Sk … y†i for two irreducible representable CM-divisors x and y
on X .
Let Wx , Wy be the normalizations of the the structure rings of x and y. Let
p be an uniformizer of Wx . For any integer n 0 and any Wx scheme or
algebra Z, write Zn for Z Wx =pn‡1 . We de®ne Homn …x; y†deg m to be the set
of pair … f ; g† of an embedding f : Spec Wxn ! Spec Wy over W , and a homomorphisms g of group schemes over Wxn which makes the following
diagram commutative
2
Z=N
? †
?
id?
y
Z=N 2 †
axn
ƒƒƒƒ!
ayn
ƒƒƒƒ!
E?
xn
?
?
gy
Eyn
0
Where Eyn ˆ Ey f Wxn , Eyn
ˆ Ey0 f Wxn , and axn , ayn are full level structure
representing x and y. We write Isomn …x; y† for Homn …x; y†deg 1 , Autn …x; y† for
Isomn …x; x†. If we drop subscript ``n'' in these de®nitions, we will obtain (old)
sets Hom…x; y†deg m , Isom…x; y†, and Aut…x; y†. These sets can be considered as
subsets of previous ones with subscript ``n''. We will use supscript ``new'' to
denote the complements of old subsets:
Homnew
n …x; y†deg m ˆ Homn …x; y†deg m nHom…x; y†deg m :
128
S. Zhang
Proposition 3.3.1. (a) If x 6ˆ y then
hSk …x†; Sk …y†i ˆ
ÿ1†k
2
X
n0
w2Isomn …x;y†
Sk …y†0 w0 Sk …x†0 :
Here for a morphism w : x ! y over Wxn which induces particular a morphism
2kÿ2
2kÿ2
! Ey0
over Wx0 ˆ W0 , the cycle w0 Sk …x†0 is the pullback of Sk …x†0 .
w0 : Ex0
2kÿ2
.
The intersection Sk … y†0 w0 Sk …x†0 is taking in Ex0
(b) If x ˆ y then
hSk …x†; Sk …x†i ˆ ord dx t:
The case x ˆ y follows from (3.2.2), (1.3.10), and the fact that the self
intersection of Sk …x†Q is …ÿ1†kÿ1 . So we may assume x 6ˆ y. We claim that the
intersection cycle Sk …x† Sk … y† has a ¯at presentation over z ˆ x \ y. Since
Sk …x† and Sk … y† has presentations by abelian subvarieties we need only prove
the following
Lemma 3.3.2. Let R be a complete discrete valuation ring with algebraic closed
residue ®eld k ˆ R=m. Let A ! Spec R be an abelian scheme, and A0 ! Spec R
be an abelian subscheme of A. Let n be a positive integer and A00 ! Spec R=mn
be an abelian subscheme of A Spec R Spec R=mn . Assume that dimR A0 ‡
dimR=mn A00 ˆ dimR A:
(a) If A0 R=m ‡ A00 R=m ˆ A R=m, then the schematic intersection of
0
A R=mn and A00 is ¯at over R=mn .
(b) If A0 R=m ‡ A00 R=m 6ˆ A R=m, then for any prime l not dividing
char …R=m† and any element d in A‰lŠ whose restriction dk on the special ®ber
of A is not in A0 ‰lŠ…R=m† ‡ A00 ‰lŠ…R=m†, the scheme A0 ‡ d is disjoint with A00 .
Proof. (a) The intersection of A0 R=mn and A00 is the kernel of the projection
A00 R=mn ! …A=A0 † R=mn :
So it is ¯at over Spec R=mn .
(b) We need to check the set theoretical intersection of schemes in A k.
If there are a0 2 A0 …k†, a00 2 A00 …k† such that dk ‡ a0 ˆ a00 then dk ˆ
a00 ÿ a0 2 A0 ‡ A00 . Since d 2 A‰lŠ and Ak ‰lŠ \ …A0k ‡ A00k † ˆ A0k ‰lŠ ‡ A00k ‰lŠ so
(
dk 2 A0k ‰lŠ ‡ A00k ‰lŠ. This contradicts the property of d.
By this lemma, we may represent U and V for Sk …x† and Sk … y† in the
®bers Yx and Yy respectively such that U and V intersect properly and the
intersection is ¯at over the z. So we have
Sk …x† Sk … y† ˆ …x y†…Sk …x†0 Sk … y†0 †:
3:3:1†
Heights of Heegner cycles and derivatives of L-series
129
If x and y don't intersect then x y ˆ 0. If x and y intersect and let h
denote …x y†, then Isomn …x; y† is empty if n > h and has one element / if
n h. The assertion of the proposition follows.
For an isogeny p
/ 
de®ned
over W0 between two elliptic curves E1 and E2

having CM by
ÿD0 over W , we want to give a formula for
Sk …E1 †0 / Sk …E2 †0 . For this, let l 6ˆ p be a prime. We want to compute the
intersection number through the pairing of l-adic cohomology:
H 2kÿ2 …E12kÿ2 ; Ql …k ÿ 1†† H 2kÿ2 …E12kÿ2 ; Ql …k ÿ 1†† ! Ql :
p
p
0
For convenience we choose
p an l such that 1 D is in Ql but i ˆ ÿ1 is
not in Ql . Write F ˆ Ql … ÿ1†.pand
 Hj ˆ Het …Ej ; Ql † F . Let ‰iŠ be the
endomorphism
on Hj such that D0 ‰iŠ is induced from the endomorphism
p
ÿD0 on E.
Let H ˆ F X F Y be a vector space of dimension two over F with an
alternate pairing
; † : H H ! F
such that …X ; Y † ˆ i and an endomorphism ‰iŠ such that ‰iŠX ˆ ÿiX ;
‰iŠY ˆ Y . Then we can ®x isomorphisms from Hj to H which is compatible
with the pairings and the actions by ‰iŠ.
Let / is an isogeny between E1 and E2 over W0 , then the induced
endomorphism / on H is given by a matrix
Mˆ
a b
:
c d
Proposition 3.3.3. Let Pkÿ1 …t† denote a constant multiple of dtd kÿ1 …t2 ÿ 1†kÿ1
such that Pkÿ1 …1† ˆ 1 then
bc ‡ ad
:
Sk …E1 †0 / Sk …E2 †0 ˆ …ÿ det M†kÿ1 Pkÿ1
det M
kÿ1
Proof. We ®rst claim that the class Sk …Ej † in Het2kÿ2 …Ej2kÿ2 ; F † is a constant
multiple of X kÿ1 Y kÿ1 in Sym2kÿ2 H . By construction of Sk …Ej † we need only
Also
check that the class of Z…Ej † is a multiple of XY . p
by construction
Z…Ej † is the projection of the class of the graph of ÿD0 on H H . We
need only check that the endomorphism by ‰iŠ on H is represented by a
multiple of XY . Indeed, the endomorphism ‰iŠ is given by X Y ‡ Y X in
H H.
Since the class / Sk …E2 † is a constant multiple of …aX ‡ bY †kÿ1
cX ‡ dY †kÿ1 , the intersection Sk …xr † / Sk …x† is a constant multiple of
the coecient of X kÿ1 Y kÿ1 in …aX ‡ bY †kÿ1 …cX ‡ dY †kÿ1 . This coecient
is equal to
130
S. Zhang
d kÿ1 …ax ‡ b†kÿ1 …cx ‡ d†kÿ1 :
kÿ1
dx
xˆ0
b
d
‡ 2c
. Then the last expression equals
Write u ˆ x ‡ 2a
ac†kÿ1
d kÿ1 b
d kÿ1 d
b kÿ1
ÿ
ÿ
u
‡
u
‡
dukÿ1 uˆ b ‡ d
2a 2c
2c 2a
2a
or
ac†kÿ1
2c
d kÿ1 det M kÿ1 det M kÿ1
:
u
‡
u
ÿ
dukÿ1 uˆ b ‡ d
2ac
2ac
2a
2c
M
Write u ˆ det
2ac t. Then the last expression equals
!
kÿ1
d kÿ1 2
t ÿ1
dtkÿ1 tˆbc‡ad
det M
bc ‡ ad
kÿ1
:
ˆ const…det M† Pkÿ1
det M
det M
Sk …x † / Sk …x†† ˆ const
2
r
kÿ1
Taking / as the identity map, we obtain that the constant in the last line
should be …ÿ1†kÿ1 as Pkÿ1 …1† ˆ 1 and the self-intersection of Sk …x† is
(
ÿ1†kÿ1 . The lemma follows.
3.4. Archimedean formulas. Let x and y be two points on X …C† where
X :ˆ X …N † Z‰fN Š C where fN ˆ exp…2pi=N †. We want to identify hx; yi with
a certain Green's function Gk constructed in page 238±239 in [18]. We recall
the de®nition as follows. Write X 0 ˆ H=C…N †. Denote by Q…t† the Legendre
function of the second kind de®ned for t > 1 by
Z1
p
Q…t† ˆ …t ‡ t2 ÿ 1 cosh u†ÿk du;
0
and de®ne
gk …z; z † ˆ ÿ2Q 1 ‡
0
0 2
jz ÿ z0 j2
;
2 im z im z0
jzÿz j
where 1 ‡ 2 im
z im z0 is the hyperbolic cosine of the distance between the
0
points z, z of H. Then we de®ne a function on H Hn``diagonal''
by
X
gk …z; cz0 †:
Gk …z; z0 † ˆ
c2C
Heights of Heegner cycles and derivatives of L-series
131
Proposition 3.4.1 For two CM-points x and y on X , one has
1
hSk …x†; Sk … y†i ˆ Gk …x; y†
2
3:4:1†
1
hSk …x†; Sk …x†i ˆ lim…Gk …x; y† ÿ log jtj2 … y††:
2 y!x
3:4:2†
if x 6ˆ y, and
We want to extend the function hSk …x†; Sk … y†i to a continuous function of y
on X nfxg when x is a ®xed CM-point. Let E be the elliptic
curve correR
sponding to y. Let b be a holomorphic form on E such that bb ˆ 1. If y is a
CM-point, then the class of ZE in H 1;1 …E E; C† is represented by a constant
multiple of b1 b2 ‡ b1 b2 . It follows that the class of ZE is actually included in
Sym H 1 …E; C† H 2 …E E; C†, and that the class of Wk …E† is included in
Sym kÿ1 H 1 …E; C† H 2kÿ2 …E2kÿ2 ; C†.
For any I ‰1; 2k ÿ 2Š with jIj ˆ k ÿ 1,
Let us write b1 ˆ b and bÿ1 ˆ b.
let I : ‰1; 2kŠ ! f1; ÿ1g be a function such that I …i† ˆ 1 i€ i 2 I. Then the
class of Sk … y† in H 2kÿ2 …E2kÿ2 ; C† is represented by
gy ˆ c k
X
I
1†
b1I
2kÿ2†
I
b2kÿ2
;
where the constant ck is a constant independent of y subject to the condition
that the self intersection of Sk … y† in E2kÿ2 is …ÿ1†kÿ1 .
Since gy is de®ned for all noncuspidal y in X , we de®ne a function of
noncuspidal points in X 0 ÿ fxg by
Hk …x; y† ˆ …ÿ1†k
Z
gy gk …x†:
3:4:3†
Yy
Since gk …x† is
@ @
pi
closed, if y is a CM-point then
Z
Z
gy gk …x† ˆ
Yy
gk …x†
Sk … y†
or
hSk …x†; Sk … y†i ˆ Hk …x; y†:
3:4:4†
We claim that gy for y near x is the restriction of a smooth form g on Y
independent of y. Locally, on a neighborhood V of x, the universal elliptic
curve E can be written as a quotient
EV ˆ C V =Z2 ;
132
S. Zhang
where …m; n† 2 Z2 acts as …u; z† ! …u ‡ m ‡ nz; z†. Then the form du ÿ yt dz
p
on C V descends to a di€erential form a on E, where u ˆ s ‡ ÿ1t. The
di€erential form we need can be de®ned as
g ˆ ck y 1ÿk
X
I
1†
a1I
2kÿ2†
I
a2kÿ2
:
Since the self-intersection of Sk …x† in D ˆ Yx is …ÿ1†kÿ1 , we have
hSk …x†; Sk …x†i ˆ lim…Hk …x; y† ÿ log jtj… y††:
y!x
24†
By (3.4.4) and (3.4.5), to show Proposition 3.4.1, it suces to show the
equality
1
Hk …x; y† ˆ Gk …x; y†:
2
It has been shown in [20], Chapter Six, that Gk …z; z0 † on X has the
following properties:
(a) Gk …z; z0 † is killed by D ÿ k…k ÿ 1† operating on the ®rst variable
@2
@2
when D ˆ y 2 …@x
2 ‡ @y 2 †, where z ˆ x ‡ yi.
(b) Gk …z; z0 † ˆ log jz ÿ z0 j2 ‡ O…1† as z ! z0 .
(c) In a neighborhood of a cusp cÿ1 1, the function im…cz†kÿ1 Gk …z; z0 †
extends to a continuous function.
We claim that these properties characterize Gk …z; z0 † uniquely. Indeed, if
0
Gk …z; z0 † is another function satisfying same properties, then for each z0 the
di€erence f …z† :ˆ Gk …z; z0 † ÿ G0k …z; z0 † is a continuous function on X killed by
D ÿ k…k ÿ 1† for all noncuspidal point z 6ˆ z0 . Let g be a continuous function
on a neighborhood U of z0 such that Dg ˆ k…k ÿ 1†f as a distribution acting
on the smooth forms with compact supports in U , then we have
D…g ÿ f † ˆ 0 away from z0 . It follows that g ÿ f is continuous at z0 and
harmonic away from z0 . By a standard fact of harmonic functions, g ÿ f is
harmonic at z0 . So f is killed by D ÿ k…k ÿ 1† at all points of H. Since D is a
negative operator on L2 …X † it follows f ˆ 0. Our claim follows.
To show the equality Hk …z; z0 † ˆ 12 Gk …z; z0 † it suce to check that Hk …z; z0 †
has corresponding properties. Notice that Property (b) is already shown in
the formulas in Sect. 1.4.
3.5. Continuity at cusps. In this subsection, we want to show Property (c) in
Sect. 3.4.4 of Hk …x; y†. Now ®x a cusp of X . Choose a parameter q on X for
the cusp such that the smooth locus of the universal elliptic curve near the
1 dw
cusp can be written as E ˆ C =qN Z . The form du ˆ 2pi
w gives a section of
2piu
2 C , and xE=X ˆ XE=X …log 1† is the
C…xE=X †, where u 2 C and w ˆ e
relative dualizing sheaf. Now for a CM point x, when q 6ˆ 0 we have
Z X
1ÿk
I …i†
^2kÿ2
gk …q†:
Hk …x; q† ˆ const…ÿ log jqj†
iˆ1 du
Yq
I
Heights of Heegner cycles and derivatives of L-series
133
By partition of the unit, to prove the asymptotic formula of Hk …x; q† as
q ! 0, we need only prove for each point p 2 Y …C†, that the integral
Z
X
G…q† ˆ
I
Y …C†q
i†
I
^2kÿ2
g
iˆ1 dui
de®nes a continuous function in a neighborhood of 0, where g is any smooth
form of Y supported in any neighborhood U of p.
Let p0 be the image of p in E2kÿ2 . Then locally near p0 , E2kÿ2 has equations w1 v1 ˆ w2 v2 ˆ ˆ wr vr ˆ q in C2kÿ1‡r with coordinates
q; w1 ; ; w2kÿ2 ; v1 ; ; vr †
i
and dui ˆ dw
wi . By Lemma 2.2.1, after a possible permutation of indices, U is
smooth with coordinates z0 ; ; z2kÿ1 such that z0 ˆ v1 =w2 , zi ˆ wi =wi‡1 if
1 i < 2k ÿ 2, and zi ˆ wi otherwise. The ®ber of U over q is de®ned by
(
Uq ˆ
z1 ; ; z2kÿ1 † : q ˆ z0 z1
r
Y
iˆ2
)
z2i
:
Therefore,
r
X
dz0 dz1
dzi
‡
‡2
ˆ 0:
z0
z1
z
iˆ0 i
For 0 i r, let Ui be the subset of U de®ned by
Ui ˆ f…z1 ; ; z2kÿ2 † 2 U : jzi j jzj j for any i 6ˆ j rg:
Then over Uiq ˆ Ui \ Uq , one has
dul ˆ
X dzj
dwi
ˆ
tlj
;
2piwi
zj
j6ˆi
where tij are integers. In the following expansion over Uiq :
X
I
1†
a1I
2kÿ2†
I
a2kÿ2
ˆ
X
aI; J ^l2I
I; J
dzj
dzl
^j2J
;
zl
zj
one must have aI; J ˆ 0 if I \ J 6ˆ ;, where I and J are sets of integers l such
that 1 l 2k ÿ 2 and l 6ˆ i. It follows that
Z X
Uiq
I
1†
a1I
2k†
akI g
Z
ˆ
X
Uiq I\ Jˆ ;
gI; J ^i2I
dzj
dzi
^j 2 J
;
zi
zj
134
S. Zhang
where gI; J are smooth forms supported in U . The continuity of this function
z
z
and dz^d
are integrable in a
follows from the fact that the form dz^d
z
z
neighborhood of 0 in C.
3.6. A di€erential equation. In this paragraph we will show that Hk …z; z0 †
satis®es the di€erential equation
Dz Hk …z; z0 † ˆ k…k ÿ 1†Hk …z; z0 †:
As before, locally on a neighborhood V of z in H, we may write
EV ˆ C U =Z2 :
p
Let u ˆ t ‡ ÿ1s be the coordinate of C. Then du ÿ yt dz descends to a
, fa1 ; ; a2kÿ2 ; dzg form a
di€erential form a on EV . Now on YV ˆ E2kÿ2
V
basis for …1; 0†-form, where ai is the pullback of a via the i-th projection
from E2kÿ2 ! E. Now write
X
X
X
X
gk …z0 † ˆ
aI aI ‡ dz
bJ aJ ‡ dz
cK aK ‡ dzdz
d L aL
K
where aI …resp. aJ ; aK ; aL † runs through the set of … p; p†-type [resp.
p ÿ 1; p†-type, …p; p ÿ 1†-type, …p ÿ 1; p ÿ 1†-type ] monomials of 1-forms
a1 ; ; a2kÿ2 ; a1 ; ; a2kÿ2 :
P
Since @pi@ gk …z0 † ˆ 0, on ®bers E2kÿ2
, aI aI jE2kÿ2
is @pi@ -closed. Since aI jE2kÿ2
are
z
z
z
we
may
monomials from fdu1 ; ; du2kÿ2 g, replacing gz by gz ‡ @a ‡ @b,
assume that aI are constants on ®bers. In other words, aI are functions of
z only. Since r gk …z0 † ˆ sgn…r†gk …z0 †, we may write
gk … p† ˆf …z†
X
I
‡ dz
1†
a1I
X
2kÿ2†
I
a2kÿ2
bJ aJ ‡ dz
X
‡
X
aI 0 a I 0
cK aK ‡ dzdz
X
3:6:1†
g L aL
where aI 0 is a multiple of ai ^ ai for some i. Now @pi@ gz ˆ 0 implies that the
zero in @pi@ gz and in particular,
Rcoecients M of dzdzdu1 d u1 du2kÿ2 d u2kÿ2 is
Mdu ˆ 0, where du is a volume form on E2kÿ2
.
z
Lemma 3.6.1. The only non-trivial contribution to
the above expression (3.6.1).
R
M is from the ®rst term of
Proof. (a) Contribution of @pi@ aI 0 aI 0 to M: Write aI 0 ˆ ^i1 2I1 ai1 ^ ^i2 2I2 ai2 , then
X
sj
si
aI 0 ÿ dz
duIÿfjg aI 0
y
y
i2 I1
j2I2
X
si sj
aI :
‡ dzdz
duIÿfi; jg
y y
i2 I1 ; j2 I2
aI 0 aI 0 ˆ duI 0 ÿ dz
X
duIÿfig
Heights of Heegner cycles and derivatives of L-series
135
Since aI 0 is constant on ®bers, the contribution to M is the sum of terms like
@ aI @…si †
; where i; i0 2 I1 ; i 6ˆ i0 ;
@z y @ui0
@ aI @…sj †
; where j; j0 2 I2 ; j ˆ
6 j0 ;
@z y @uj0
and
@2
si sj † aI 0 ; where …i0 ; j0 † 6ˆ …i; j†:
@ui @uj
They are all 0.
(b) Contribution of
then
@ @
pi dzbJ aJ
to
R
M: Write aJ ˆ ^j1 2J1 aj1 ^ ^j2 2J2 aj2 ,
dzbJ aJ ˆ dzbJ duJ ÿ dzdz
X
j2J2
The contribution to
R
duJ ÿfjg
sj
bJ :
y
M is the sum of the terms like
Z
@2
bJ ;
@z@u
Z
@ 2 bJ s j
; where i 6ˆ k:
@ ui @ uk y
Since @bJ is a function on E2kÿ2 , the integral
@bJ sj =y @z
2kÿ2
,
@ uk is a function of the i-th factor of E
Z
@ 2 bJ s j
ˆ
@ui @ uk y
Z
@
@ui
vanishes. Now if i 6ˆ j, then
so
@ bJ s j
@ uk y
ˆ 0:
R
Same for k 6ˆ j. So the contribution of @pi@ dzbz ak to M is 0.
Similarly the contribution of @pi@ dzdcK aK is 0.
@ @
(c) Contribution of pi dzdzaL . Write aL ˆ ^l1 2L1 ^ ^l2 2L2 al2 , then
gL dzdzaL ˆ gL dzdzduL :
So the contribution is
R
@2
@ui @uj gL .
This is 0.
Lemma 3.6.2.
@ 2 f …z† k ÿ 1 @ f …z† k ÿ 1 @ f …z† k ÿ 1 f …z†
‡
ÿ
ÿ
ˆ 0:
@z@z y kÿ1
2iy @z y kÿ1
2i @z y k
4 y k‡1
(
136
S. Zhang
Proof. By Lemma 3.6.1 and the fact that
@ @
@ @
pi g
ˆ 0 we have
f …z† X I …1†
I …2kÿ2†
a
a2kÿ2
ˆ 0:
y kÿ1 I 1
3:6:2†
Notice that for a ˆ du ÿ ys dz,
1
adz;
2iy
1
adz;
@ a ˆ ÿ
2iy
@a ˆ ÿ
ˆ 1 adz;
@a
2iy
1
@a ˆ
adz:
2iy
Using these formulas to compute @ part of (3.6.2), we obtain
@ f …z† X I …1†
I …2kÿ2†
a
a2kÿ2
dz
@z y kÿ1 I 1
ÿ1
f …z† X X I …1†
I …iÿ1† ÿ1 I …i‡1†
a1 aiÿ1
a ai‡1 dzI …i†
‡ p @ k
2 ÿ1 y
i
I
@
3:6:3†
3:6:4†
Computing @ part of (3.6.3), we obtain
@ 2 f …z† k ÿ 1 @ f …z† X I …1†
I …2kÿ2†
‡
a1 a2kÿ2
dzdz
@z@z y kÿ1
2iy @z y kÿ1
I
3:6:5†
Similarly, computing @ part of (3.6.4) we obtain
ÿ
k ÿ 1 @ f …z† k ÿ 1 f …z† X I …1†
I …2kÿ2†
ÿ
a1 a2kÿ2
dzdz
2i @z y k
4 y k‡1
I
The lemma follows from (3.6.5) and (3.6.6).
(
By lemma one has
y 2 fzz ˆ
Since
D ˆ y2
k…k ÿ 1†
f:
4
@2
@2
‡ 2
2
@x
@y
i
ˆ 4y 2
it follows that
Df ˆ k…k ÿ 1†f :
3:6:6†
@2
@z@z
Heights of Heegner cycles and derivatives of L-series
137
4. Heights of Heegner cycles
4.1. Heights of CM-cycles over X0 …N†: Let N be a positive integer. In this
subsection we want to de®ne the space sk …X0 …N †† cycles and the height
pairing on this space. For this let N 0 be any multiple of N such that N 0 is the
product of two relatively prime integers 3. Let p denotePthe natural
xi is a sum
morphism from X …N 0 † to X0 …N †. If x is a CM-point, then p x ˆ
ÿ1
of CM-points
x
.
Then
the
elliptic
scheme
E
over
p
x
has
an
endomorphism
i
p
0
by
P endomorphism to de®ne Wk …xi †, Wk … p x† ˆ
P some ÿD . We use this
Sk …xi †. Let sk …X † be the subspace of sk …X 0 †
Wk …xi †, and Sk … p x† ˆ
generated by Wk … p x† for all CM-divisors x on X . This space does not
Finally
we de®ne
depend on the choice of N 0 up to canonical isomorphisms.
p

Sk …x† to be the image in sk …X † of Sk …x† ˆ Sk … p x†= deg p. Then the restriction of the height papring on CM-cycles over X …N 0 † gives a height
papring on sk …X †. In the following we want to ®nd formulas for these local
intersections.
In the nonarchimedean case, let W be a complete discrete valuation ring,
such that the ®eld Q of fractions of W has characteristic 0, and the residue
®eld W0 of W has characteristic not dividing N and is algebraically closed.
Write X for X …N † W . We say an irreducible CM-divisor x on X is representable, if x represents an a pair …Ex ; bx † of an elliptic curve Ex , and a
cyclic isogeny bx : Ex ! Ex0 of degree N . We want to compute the local
height hSk …x†; Sk … y†i for two irreducible representable CM-divisors x and y
on X .
Let Wx , Wy be the normalizations of the the structure rings of x and y. Let
p be an uniformizer of Wx . For any integer n 0 and any Wx scheme or
algebra Z, write Zn for Z Wx =pn‡1 . We de®ne Homn …x; y†deg m to be the set
of triples … f ; g; h† of an embedding f : Spec Wxn ! Spec Wy over W , and
homomorphisms g; h of group schemes over Wxn which makes the following
diagram commutative
Exn
?
?
g?
y
Eyn
bxn
ƒƒƒƒ!
byn
ƒƒƒƒ!
0
Exn
?
?
h?
y
0
Eyn
0
Where Eyn ˆ Ey f Wxn and Eyn
ˆ Ey0 f Wxn . We write Isomn …x; y† for
Homn …x; y†deg 1 , Autn …x; y† for Isomn …x; x†. If we drop subscript ``n'' in these
de®nitions, we will obtain (old) sets Hom…x; y†deg m , Isom…x; y†, and Aut…x; y†.
These sets can be considered as subsets of previous ones with subscript ``n''.
We will use supscript ``new'' to denote the complements of old subsets such
as
Homnew
n …x; y†deg m ˆ Homn …x; y†deg m nHom…x; y†deg m :
138
S. Zhang
Proposition 4.1.1. (a) If x 6ˆ y then
hSk …x†; Sk … y†i ˆ
ÿ1†k
2
X
n0
Sk … y†0 w0 Sk …x†0 :
w2Isomn …x;y†
Here for a morphism w : x ! y over Wxn which induces particular a morphism
2kÿ2
2kÿ2
! Ey0
over Wx0 ˆ W0 , the cycle w0 Sk …x†0 is the pullback of Sk …x†0 .
w0 : Ex0
2kÿ2
.
The intersection Sk … y†0 w0 Sk …x†0 is taking in Ex0
(b) If x ˆ y then
hSk …x†; Sk …x†i ˆ ord dx t:
Proof. Choose N 0 as above such that N 0 is prime to the characteristic of W0 .
Write X 0 for X …N 0 † W and let p : X 0 ! X be the canonical unrami®eld
morphism.
P
P
Then p x ˆ u…x† i xi and p y ˆ u… y† j yj , where u…x† ˆ j Aut…x†= 1j,
u… y† ˆ jAut… y†= 1j, and xi and yj are distinct components of p x and p y
respectively. They are de®ned over Wx and Wy respectively, and represent all
di€erent C…N 0 † structures on Ex and Ey with ®xed C0 …N † structures respectively. By de®nition we have
*
u…x† X
u… y† X
Sk …xi †; p
Sk … yj †
hSk …x†; Sk … y†i ˆ p
deg p i
deg p j
X
hSk …xi †; Sk … y1 †i;
ˆ u…x†
+
i
as C…N 0 † acts transitively on xi 's.
If x 6ˆ y, the proposition in the case of X 0 implies
hSk …x1 †; Sk … y†i ˆ
u…x†
2
X
Sk …xi †0 /0 Sk … y1 †0 :
n0
/2Isomn …xi ;y1 †
For any / 2 Isomn …x; y†, the C…N 0 † structure of Ey corresponding to y1 gives
a C…N 0 † structure on x. This induces a map
Isomn …x; y†=Aut…x† !
a
i
Isomn …xi ; y1 †:
It is easy to see that this map is bijective. The ®rst part of the proposition
follows.
1
If x ˆ y, we can choose
a local coordinate t1 ˆ tu…x† for x1 . Write
P
div t ˆ x ‡ x0 . Then div t1 ˆ i xi ‡ p x0 =u…x†: It follows that
Heights of Heegner cycles and derivatives of L-series
139
h
hSk …x†; Sk …x†i ˆ …ÿ1†k u…x† …Sk …x1 † Sk …x1 ††
i
X
xi x1 †…Sk …xi †0 Sk …x1 †0 †
‡
i>1
ˆ u…x† orddx1 t1 ÿ
X
!
xi x1
i>1
ˆ u…x†x1 div t1 ÿ x1 ÿ
X
!
xi
i>1
ˆ x1 p x0 ˆ x x 0
ˆ ord …dx t†:
The second part of the proposition follows.
(
For archimedean place, we can also de®ne Gk …x; y† as in X …N † case. We
will also have
Proposition 4.1.2. For two CM-points x and y on X, one has
1
hSk …x†; Sk … y†i ˆ Gk …x; y†
2
4:1:1†
1
hSk …x†; Sk …x†i ˆ lim…Gk …x; y† ÿ log jtj2 … y††:
2 y!x
4:1:2†
if x 6ˆ y, and
Proof. Let N 0 3 be a multiple of N . Let X 0 denote the modular curve
0
X …N 0 † Z‰fN Š C. Consider
P the map p : X ! XP. For two CM-points x and y
on X , write p x ˆ u…x† i xi and p y ˆ u… y† j yj . Where u…x† and u… y† are
the cardinalities of the stabilizers of C at x and y, and xi and yj are distinct
components of p x and p y respectively. By de®nition, we have
*
+
u…x† X
u… y† X
Sk …xi †; p
Sk … y j †
hSk …x†; Sk … y†i ˆ p
deg p i
deg p j
X
hSk …xi †; Sk … y1 †i:
ˆ u…x†
i
0
0
G0k …x; y†
We write C for C…N † and
for green's function de®ned in X 0 , and
0
choose points z and z in H projecting to x1 and y1 . If x 6ˆ y one has
X
1
hSk …x†; Sk … y†i ˆ u…x†
G0k …xi ; y1 †
2
i
1 X X
ˆ
gk …c0 c00 z; z0 †
2 c00 2C0 nC c0 2C0
140
S. Zhang
ˆ
1X
gk …cz; z0 †
2 c2C
1
ˆ Gk …x; y†:
2
1
If x ˆ y, then t1 ˆ tu…x† gives a local coordinate for x1 , and one has
"*
+#
X
1
hSk …x†; Sk …x†i ˆ u…x† Sk …x1 †; Sk …x1 †i ‡
hSk …xi †; Sk …x1 †
2
i2
1
ˆ u…x† lim G0k …x1 ; y† ÿ log jt1 j2 … y†
y!x1
2
X
1
G0k …xi ; x1 †
‡ u…x†
2
i2
"
#
X
1
2u…x†
0
ˆ lim u…x†
Gk …xi ; y† ÿ log jt1 j
2 y!x1
i
h
i
1
ˆ lim Gk …x; y† ÿ log jtj2 … y†
2 y!x
The proposition follows.
(
4.2. Heights of Heegner cycles. Let N be a positive integer and K an imaginary quadratic ®eld with discriminant D, such that every prime factor of N
of discriminant D. This
splits in K. Let x be a Heegner point on X0 …N † Q
means that in the corresponding isogeny E ! E0 , both E and E0 have
complex multiplications by the full ring of integers of K. Then x is rational
over the Hilbert class ®eld H of K. Let m be a integer prime to N and r be an
element in Gal…H =K†.
Proposition 4.2.1. Over X :ˆ X0 …N † H , with the notations are adopted from
[18], one has
kÿ1
log
hSk …x†; Tm Sk …xr †i ˆ mkÿ1 cm
N;k …A† ‡ hurA …m†m
ÿm
kÿ1 2
u
X
0<n<
mjDj
N
r0A …n†rA …mjDj
N
m
ÿ nN †Pkÿ1
2nN
1ÿ
:
mjDj
As in weight 2 case [18], we will deduce this equality from the total local
heights hSk …x†; Tm Sk …xr †ip over each place p of Q, where
hSk …x†; Tm Sk …xr †ip ˆ
X
hSk …x†; Tm Sk …xr †iv dv ;
vjp
where v runs through the set of places of H , and dv ˆ 2 if v is complex, and
dv ˆ ‰k…v† : Z=pŠ if v is ®nite.
Heights of Heegner cycles and derivatives of L-series
141
Proposition 4.2.2.
hx; Tm xr i1 ˆ mkÿ1 cm
N ;k …A†:
Proposition 4.2.3. Let p be a prime.
(a) If p splits in K then
hx; Tm xr ip ˆ ÿurA …m†h ordp …m=N †mkÿ1 :
(b) If p is inert in K then
hx; Tm xr ip ˆ ÿ rA …m†hu ordp …m†mkÿ1
X
ÿ u2 mkÿ1
ordp … pn†rA …mjDj ÿ nN †d…n†
mjDj
N
n0 mod p
0<n<
2nN
;
RfAqng …n=p†Pkÿ1 1 ÿ
mjDj
where q is an ideal of OK whose norm is a prime q with property that …ql† ˆ …ÿp
l †
for all ljD.
(c) If p is rami®ed in K, then
hx; Tm xr ip ˆ ÿ rA …m†hu ordp …m†mkÿ1
X
ÿ u2 mkÿ1
ordp … pn†rA …mjDj ÿ nN †d…n†
mjDj
N
n0 mod p
0<n<
RfAqpng …n=p†Pkÿ1
2nN
1ÿ
;
mjDj
where p2 ˆ p and q is an ideal whose norm
is a prime q such that
all prime factor l 6ˆ p of D, and that ÿq
ˆ ÿ1.
p
Let
hSk …x†; Tm Sk …xr †ifinite :ˆ
X
p:prime
ÿq
l
ˆ
ÿÿ1
l
for
hSk …x†; Tm Sk …xr †ip log p:
As in [18], from the second part of Proposition (4.6) in p. 285 in [18], one can
show that Proposition 4.1.3 implies
hSk …x†; Tm Sk …xr †ifinite ˆ hurA …m†mkÿ1 log
ÿm
kÿ1 2
u
X
0<n<
mjDj
N
N
m
r0A …n†rA …mjDj
2nN
ÿ nN †Pkÿ1 1 ÿ
:
mjDj
Now Proposition 4.1.1 follows from this equality, Proposition 4.2.2, and the
fact
142
S. Zhang
hSk …x†; Tm Sk …xr †i ˆ hSk …x†; Tm Sk …xr †i1 ‡ hSk …x†; Tm Sk …xr †ifinite :
4.3. Proof of Proposition 4.2.2. We identify the noncuspidal complex points
of X0 …N † with C0 …N †nH. As in (1.2) in [18] p. 235, the Hecke correspondence Tm (m 2 N, …m; N † ˆ 1) acts on X0 …N † by
X
cz
Tm …z† ˆ
c2C0 …N †nRN
det cˆm
Z Z
. By Proposition 2.4.2 and Proposition 4.1.2, for
NZ Z
0
any two point z and z in H, one has
mkÿ1
0
0 Gk …z; z † Tm
hSk …z†; Tm Sk …z †i ˆ
2
z0
mkÿ1 X
ˆ
Gk …z; cz0 †
2 C …N†nR
where RN ˆ
0
N
det cˆm
ˆ
mkÿ1 X
gk …z; cz0 †:
2 c2R =1
N
det rˆm
By (2.24) in [18] p. 242, we have
hSk …z†; Tm Sk …z0 †i ˆ
mkÿ1 m
GN ;k :
2
The set of Heegner points in X …C† is parameterized by the set of pairs
A; n† with A an ideal class of OK and n an integral ideal of OK of norm N
in the way that each pair …A; n† corresponds to the point sA;n which represents the point …C=a; anÿ1 =a†, where a is an ideal in A. If we identify
Gal…H =K† with the group of ideal class via Artin map, then Gal…H =K† acts
on the set of Heegner points by multiplication on A and trivially on n.
Assume rA …m† ˆ 0 and x ˆ sA;n for one embedding H C. Then
X
hSk …x†; Tm Sk …xr †iv
hSk …x†; Tm Sk …xr †i1 ˆ 2
vj1
ˆ2
X
hSk …xa †; Tm Sk …xar †i
a2Gal…H =K†
ˆ2
X
hSk …sA0 C;n †; Tm Sk …sA0 CA;n †i
C2Pic…OK †
X
ˆ2
A1 ;A2 2Pic…OK †
A1 Aÿ1 ˆA
2
By (3.3) in [18] p. 242, we have
hSk …sA1 ;n †; Tm Sk …sA2 ;n †i:
Heights of Heegner cycles and derivatives of L-series
143
hSk …x†; Tm Sk …xr †i1 ˆ mkÿ1 cm
N ;k :
Proposition 4.2.1 therefore follows from (3.17) in [18] p. 247.
Now we assume that rA …m† 6ˆ 0. Here we observe that
log jt… y†jv ÿ u log j2pig4 …z†…w ÿ z†jv ! 0
as y ! x, where z and w are points in the upper half-plane which map to x
and y on X0 …N †…C†.
By Proposition 4.4.2, we ®nd
8
>
>
>
>
kÿ1 < X
m
0
gk …z; cz0 †
hSk …z†; Tm Sk …z †i ˆ
2 >
>
c2RN =1
>
>
: det cˆm
cz0 6ˆz
9
>
>
>
=
h
i>
2
‡urA …m† lim gk …z; w† ÿ log j2pig4 …z†…w ÿ z†j
w!z
>
>
>
>
;
kÿ1
0
By (5.7) in [18] p. 251, this is again m 2 Gm
N;k …z; z †. Also by Proposition 5.8 in
[18] p. 252, the same computation gives
hSk …x†; Tm Sk …xr †i1 ˆ mkÿ1 cm
N ;k :
4.4. Intersections and homomorphisms. Let v be a ®nite place of H , W the
completion of the maximal unrami®ed extension of Hv , and p a prime
of W . In this subsection, we want to show the following formula for
hSk …x†; Tm Sk …xr †iv .
Proposition 4.4.1. Let p denote the characteristic of the residue ®eld of v.
(a) If p is inert in K then
hSk …x†; Tm Sk …xr †iv ˆ
ÿ1†k
2
X
n0
Sk …xr †0 /0 Sk …x†0
/2Homnew …xr; x†deg m
Wn
1
ÿ urA …m†ordp …m†mkÿ1 :
2
(b) If p is rami®ed in K then
hSk …x†; Tm Sk …xr †iv ˆ
ÿ1†k
2
X
n0
Sk …xr †0 /0 Sk …x†0
/2Homnew n …xr; x†deg m
W =p
ÿ urA …m†ordp …m†mkÿ1 :
144
S. Zhang
(c) If p ˆ p
p is split in K and vjp then
hSk …x†; Tm Sk …xr †iv ˆ ÿurA …m† ordp …m†jp mkÿ1 :
where jp are integers depending on p such that jp ‡ jP ˆ ordp …m=N †.
Proof. We ®rst assume that rA …m† ˆ 0. If m is prime to p, then components
of Tm …xr † are de®ned over W , and then for any n 0, the canonical map
a
Isomn … y; x† ! Homn …xr ; x†deg m
y2Tm …xr †
is bijective. The proposition therefore follows from Proposition 4.1.1.
Now we write m ˆ pt r with r prime to p. The points z in thePdivisor Tr xr
are rational over W , but the points y in the divisor Tm xr ˆ z Tpt …z† are
rational over rami®ed extensions Wy of W . These y are the quasi-canonical
liftings of their reductions. Let y…s† be the divisor over W obtaining by
taking the sum of a point of level s with all of its conjugates over W . When p
is split in K the proposition is true since x and Tm …xr † has no intersection and
Homn …xr ; x† is empty for every n, as explained in [18].
Now we assume that p is inert in K. The number hSk …x†; Tm Sk …xr †iv is the
sum of rkÿ1 hSk …x†; Tpt Sk …z†iv , and therefor the sum of mkÿ1 hSk …x†; Sk … y…s††iv ,
for all z in Tm …xr † and all irreducible components y…s† of Tpt z. If s > 0 and
n > 0, then IsomWn …x; y…s†† is empty. Since all y…s† are congruent to a ®xed y0
of level 0, by Proposition 4.1.1, it follows that
hSk …x†; Sk … y…s††iv ˆ
ÿm†k
2
X
Sk … y0 †0 w0 Sk …x†0
n1
w2IsomW =pn … y0 ; x†
if s ˆ 0, and
hSk …x†; Sk … y…s††iv ˆ
X
ÿm†k
2 w2Isom … y
0
Sk … y0 † w Sk …x†
0 ;x†
k
if s > 0. Therefore hSk …x†; Tm Sk …xr †iv is the sum of …ÿm†
2 …Sk … y0 †0 w0 Sk …x†0 †
with w runs through a disjoint union of sets Isomn … y0 ; x† with certain
multiplicities. The arguments in [18] pp. 260±261 yields a bijective map from
this disjoint union to Hom…xr ; x†deg m , by composing each w in the disjoint
union with a certain homomorphism w0 from xr to y0 de®ned over W . Since
0
mkÿ1 …Sk … y0 †0 w0 Sk …x†† ˆ Sk …xr † w w Sk …x†;
the proposition therefore follows.
Now we drop the assumption rA …m† ˆ 0. The proof of the case v - mN is
similar to the case rA …m† ˆ 0. When v - N but vjm, the formula is changed
Heights of Heegner cycles and derivatives of L-series
145
slightly, because urA …m† elements in HomW …xr ; x†deg m =…1† contribute to
the isomorphism over W =p between quasi-canonical liftings in Tm xr to x.
For any homomorphism / from xr to x of degree m, / Sk …x† ˆ mkÿ1 Sk …xr †.
The proposition follows in this case.
(
4.5. Proof of Proposition 4.2.3. If p has a unique prime factor p in K, then x
and xr have supersingular reductions (mod p) and EndW =p …x† ˆ R is an
order in the quaternian algebra B over Q which is rami®ed at 1 and p. Then
the embedding O ˆ EndW …x† ! R ˆ EndW =p …x†, given by the reduction of
the endomorphisms, extends to a linear map K ! B. This in turn yields a
decomposition
B ˆ B‡ ‡ Bÿ ˆ K ‡ Kj;
where j is an element in the nontrivial coset of NB …K †=K : The decomposition is represented by the reduced norm
N…b† ˆ N…b‡ † ‡ N…bÿ †:
Proposition 4.5.1. With notation as above, one has
(a) If p is inert in K then
X
Nb‡ ÿ Nbÿ 1
mkÿ1 Pkÿ1
ordp … pNbÿ †
hSk …x†; Tm Sk …xr †iv ˆ ÿ
Nb
2
b2Ra=
NbˆmNa
bÿ 6ˆ0
1
ÿ urA …m† ordp …m†mkÿ1
2
(b) If p is rami®ed in K then
X
Nb‡ ÿ Nbÿ
ordp …DNbÿ †
mkÿ1 Pkÿ1
hSk …x†; Tm Sk …xr †iv ˆ ÿ
Nb
b2Ra=
NbˆmNa
bÿ 6ˆ0
ÿ urA …m† ordp …m†mkÿ1 :
(c) If p ˆ p
p is split in K as vjp, then
hSk …x†; Tm Sk …xr †iv ˆ ÿurA …m†jp mkÿ1 :
where jp ‡ jp ˆ ordp …m=N †.
To prove the proposition we need the explicit descriptions of
r
r
Homnew
n …x ; x†deg m and …Sk …x †0 /0 Sk …x†0 †:
The ®rst one is already given in [18]:
146
S. Zhang
Proposition 4.5.2. (Gross-Zagier [18]). (a)
n
o
EndW =pn …x† ˆ b 2 R; DNb ˆ 0 mod p…Np†nÿ1
(b)
HomW =pn …xr ; x† ' EndW =pn …x†a
in B, where a is any ideal in the class A. If an isogeny / : xr ! x corresponds
to b 2 B, then deg / ˆ Nb=Na.
Now we want to give a formula for …Sk …xr †0 /0 Sk …x†0 †:
Proposition 4.5.3. If b ˆ b‡ ‡ bÿ then
Sk …x†0 b Sk …x†0 † ˆ
ÿNb
Na
kÿ1
Pkÿ1
Nb‡ ÿ Nbÿ
:
Nb
Proof. We want to apply Proposition 3.3.3. First of all we make identi®cations from HEx and HExr to H . Then there are u and v in F such that for
any a 2 a as an morphism from Exr to Ex , one has
a …X † ˆ auX ; a …Y † ˆ avY
Taking care of the pairings one has
a …X †; a …Y †† ˆ deg a…X ; Y †;
so deg a ˆ N…a†uv: It follows that the isogeny b ˆ b‡ ‡ bÿ has matrix
p
b‡ = pNa

ÿbÿ = Na
p bÿ =pNa
 :
b‡ = Na
The proposition follows from Proposition 3.3.3.
(
When p is split in K, Proposition 4.5.1 follows from Proposition 4.4.1.(c).
Assume p is inert or rami®ed in K. The ®rst term of the expression is easy to
obtain. We compute the sum of the second term in Proposition 4.5.1 over all
places v with weights dv ˆ ‰k…v† : Z=pŠ as follows.
Fix a place v of H over a prime p. In the proof of proposition (9.2) in [18],
p. 265±266, Gross and Zagier gave the following description for the set
Sv ˆ fb 2 Ra= 1; Nb ˆ mNa; bÿ 6ˆ 0g:
First of all, Sv is a disjoint union of subsets
Sv;n ˆ
b 2 Ra= 1; Nb‡ ˆ
mjDj ÿ nN
nN
a; Nbÿ ˆ
Na
jDj
jDj
Heights of Heegner cycles and derivatives of L-series
indexed by the set
Iˆ
147
mjDj
; n 0… p† :
n 2 Z; 0 < n <
N
It follows that the sum we want to compute is
X
X
2nN 1
ordp … pn†
mkÿ1 Pkÿ1 1 ÿ
jSv;n jdv
jDj 2
n2I
when p is inert, and
X
kÿ1
m
Pkÿ1
n2I
2nN
1ÿ
jDj
X
1
ordp …n†
jSv;n jdv
2
4:5:1†
4:5:2†
when p is rami®ed.
For a choosing q as in the proposition, Gross and Zagier have shown
X
jSv;n jdv ˆ 2u2 d…n†rA …mjDj ÿ nN †RAqpn …n=p†:
4:5:3†
Proposition 4.5.1 follows from (4.5.1), (4.5.2), and (4.5.3).
5. Proof of the main identity and the consequences
5.1. Main identity. Let N be a positive integer, K an imaginary quadratic
®eld, such that every prime divisor of N splits in K. Let D be the discriminant of K and H the Hilbert class ®eld of K. Let x 2 X0 …N †…H † be one of the
Heegner points associated to K, A an ideal class of K, and r the corresponding element of G ˆ Gal…H
P =K†.
As in weight 2, let f ˆ n1 an e2pinz be an element in the space of new
form of weight 2k on C0 …N †. De®ne the L-series associated to f and A by
LA … f ; s† ˆ
X
n1
where …n† :ˆ
ÿD
n
n;ND†ˆ1
n†nÿ2s‡2kÿ1
X
a…m†rA …m†mÿs
m1
the associated quadratic character of K.
Theorem 5.1.1 (Gross-Zagier [18]). (a) The function LA … f ; s† has an analytical continuation to the entire complex plane, satis®es the functional equation
LA :ˆ …2p†ÿ2s N s jDjs C…s†2 LA … f ; s† ˆ ÿLA … f ; 2k ÿ s†:
In particular, LA … f ; k† ˆ 0.
P
(b) There is a holomorphic cusp form U ˆ m1 am;A qm 2 S2k …C0 …N ††
satisfying
148
S. Zhang
p
2k ÿ 2†! jDj 0
f ; U† ˆ
LA … f ; k†
24kÿ1 p2k
new
for all new form f 2 S2k
C0 …N †† and with am;A (m prime to N ) given by
am;A ˆ
mkÿ1 m
h
N
r …A† ‡ rA …m†mkÿ1 log
u2 N ;k
u
m
X
2nN
kÿ1
:
rA …mjDj ÿ nN †Pkÿ1 1 ÿ
ÿm
mjDj
mjDj
0<n<
N
By Proposition 4.2.1, we therefore prove Theorem 0.2.1 in Introduction.
Now we want to prove Theorem 0.3.1. By Theorem 0.2.1, it suces to show
the following.
Proposition 5.1.2. Let v1 ; ; vh be CM-cycles of weight 2k for C0 …N †. Assume
for each i; j there is an element gi;j 2 S2k …C0 …N †† such that for any positive
integer m prime to N the number hvi ; Tm …vj †i is the m-th coecient of gi;j . Let
V be the subspace of Heegner cycles generated by
Tm vj j1 j h; …m; N † ˆ 1g:
Let V 0 be the quotient of V modulo the null subspace with respect to the height
pairing on V V . Then the Hecke module V 0 is isomorphic to a sub-quotient
module of S2k …C0 …N ††h .
Proof. De®ne the action
of Hecke algebra T (generated by correspondences
Tm , …m; N † ˆ 1) on C ‰qŠ by the usual formula:
Tm
X
!
a…n†qn
n
2
3
mn
X X
4
d kÿ1 a 2 5qn :
ˆ
d
n
dj…m;n†
Let S 0 be the image of S2k …C0 …N †† of the map
X
X
a…n†e2pnz !
a…n†qn :
n
n;N †ˆ1
De®ne h map /i …1 i h† from V to C‰‰qŠŠ by
v ! /i …v† ˆ
X
hv; Tm …vi ††iqm :
m1
m;N†ˆ1
It is easy to see that these maps induce an embedding of V 0 into C‰‰qŠŠh
as C-vector space. To prove the theorem it suces to show that /i is a
morphism of Hecke modules and the image of /i is in S 0 .
Heights of Heegner cycles and derivatives of L-series
149
Now for any …m; N † ˆ 1 and any v 2 V , since Tm is self-adjoint on V with
respect to the height pairing, it follows that
/i …Tm …v†† ˆ
X
X
hTm …v†; Tn …vi †iqn ˆ
n;N†ˆ1
hv; Tm Tn …vi †iqn :
n;N †ˆ1
To show that /i is a morphism of Hecke modules it suces to show that the
action of Hecke algebra on V satis®es the property
X
d kÿ1 Tmn2 :
Tm Tn ˆ
dj…m;n†
d
But this follows from the corresponding property of the Hecke correspondences on X0 …N †, and from Proposition 2.4.2, where we have shown that for
any CM-point on X0 …N †,
Tm sk …x† ˆ mkÿ1 sk …Tm …x††:
This ®nishes the proof of that / is a morphism of Hecke modules.
By the assumption of the theorem /i …vj † (1 j h) is in S0 . It follows
that /i …V † is in S 0 as /i …V † is generated by /i …xj † : j ˆ 1; ; h as a Hecke
module. This completes the proof of the theorem.
(
The proof of corrolaries in Introduction uses the same argument as
Gross-Zagier. We omit all details.
new
C0 …N †† is
5.2. Algebraicity conjecture. Recall from [18] that a relation of S2k
a sequence of integers k ˆ …km †m1 satisfying the following conditions:
(a) P
km 2 Z; km ˆ 0 for all but ®nitely many
P m.
am qm 2 S2k …C0 …N ††.
(b)
m1 km am ˆ 0 for any cusp form
(c) km ˆ 0 for m not prime to N or rA …m† 6ˆ 0.
We write
GN ;k;k …z; w† ˆ
1
X
mˆ1
km mkÿ1 Gm
N ;k …z; w†:
Conjecture 5.2.1 (Gross and Zagier) Let k ˆ …km †m1 be a relation for
new
C0 …N ††. Fix a Heegner point x and an embedding H C. Then there
S2k
exists an element a 2 H such that
GN ;k;k …xs ; xrs † ˆ u2 D1ÿk log jas j
for all s 2 G ˆ Gal…H =K†.
150
S. Zhang
Theorem 5.2.2. Assume that the height pairing on CM-cycles is non-degenerate. Fix a Heegner point x and an embedding H C. Then there exist a
rational number r and an element a 2 H such that
GN ;k;k …xs ; xrs † ˆ r log jas j
for all s 2 G ˆ Gal…H =K†.
P
Proof. The condition on k implies that m km s0k …x† ˆ 0 in Ch …Y †. It follows
that there are subvarieties Wi of Y and functions fi on Wi such that
X
X
div… fi jWi†
km Tm Sk …x† ˆ c
i
where c is a rational number. Now for any embedding H ,!C inducing an
in®nite place v of H , and any r 2 Gal…H =K†, the local height pairing is given
by
*
+
X
X
r
km Tm sk …x†; sk …x † ˆ c
log j fi …Sk …xr † Wi †jv
m1
v
i
Y
r
ˆ c log fi …Sk …x † Wi † :
i
v
(
5.3. Beilinson-Bloch conjecture. We will combine our result with a result of
NekovaÂrÏ to give some application to Beilinson-Bloch conjecture. We start
with a review
NekovaÂrÏ .
P of a result of
new
C0 …N †† be a normalized eigenform of weigh 2k
Let f ˆ n1 an qn 2 S2k
on C0 …N † with coecients in Q. Let M ˆ M… f † be the Grothendieck motive
over Q constructed by U. Jannsen and A.J. Scholl. The l-adic realization Ml
is a two dimensional representation of Gal…Q=Q†
corresponding to f and
Ql †, where
Y Q;
appearing as a factor in the cohomology group He2kÿ1
t
Y ˆ Y …N 0 † Q for multiple N 0 of N such that N 0 is a product of two relatively prime integers 3. For any number ®eld F , let Chk …YF †0 be the group
of homologically trivial cycles of codimension k in YF modulo the rational
equivalence. The l-adic Abel-Jacobi map
1
Ql †…k††
UF : Chk …YF †0 Ql ! Hcont
F ; Het2kÿ1 …Y Q;
induces a map
1
UF ;f : Chk …YF †0 Ql ! Hcont
F ; Ml …k††:
Conjecture 5.3.1. (Beilinson and Bloch) The dimension of im UF ;f is equal to
the order of L… f F ; s† at s ˆ k.
If F ˆ K, x a Heegner point on X as before then
Heights of Heegner cycles and derivatives of L-series
sK ˆ
X
151
sk …xr †
r2Gal…H =K†
de®nes a Heegner cycle in Chk …YK †0 Ql .
Theorem 5.3.2. (NekovaÂrÏ [30] [31]). Suppose that l does not divide 2N . If
y0 ˆ Uf …sK † is nonzero then im Uf ˆ Ql y0 ; and an analogue of the l-primary
part of the Tate-Shafarevich group is ®nite.
By a theorem of Scholl, we know that sk …XK † is included on
Chk …YK †0 R.
Theorem 5.3.3. Assume that k > 1. If UK is injective on the subgroup sk …XK † of
CM-cycles for every imaginary quadratic ®eld K, then the equality
rankQl im…Uf ;Q † ˆ ordsˆk L… f ; s†
holds if ordsˆk L… f ; s† 1 and l does not divide 2N .
Proof. If ordsˆk L… f ; s† 1 then this number is 0 or 1 depending totally on
the sign of the functional equation of L… f ; s†. By the theorems of Waldspurger ‰36Š, Murty-Murty ‰29Š, and Bump-Friedberg-Ho€stein ‰6Š, one has
in®nitely many imaginary quadratic ®elds K in which every prime factor of
N is split, such that
ordsˆk L … f ; s† ‡ ordsˆk L… f ; s† ˆ 1;
P
where Lÿ …f ; s† ˆ
n†an nÿs is the L-function of f twisted by the character
D
n† ˆ n associated to K. It follows that ordsˆk L… f ; Id; s† ˆ 1 as
L… f ; Id; s† ˆ L… f K; s† ˆ L … f ; s†L… f ; s†:
By Corollary 5.1.6, the Heegner point sK;f is not zero, so is y0 ˆ UK;f …sK; f †.
Then by NekovaÂrÏ 's theorem, im Uf ; K ˆ Ql y0 . Let s be the nontrivial
element in Gal…K=Q†, then sy0 ˆ y0 with opposite sign as the functional
(
equation for L… f ; s†. It follows that dim im Uf ;Q ˆ ordsˆk L… f ; s†.
References
1. S. J. Arakelov, Intersection theory of divisors on an arithmetic surfaces, Math. U.S.S.R.
Izvestija 8, pp. 1167±1180 (1974)
2. A. A. Beilinson, Heights pairing between algebraic cycles, Comtemp. Math. 67, pp 1±24
(1987)
3. S. Bloch, Height paring for algebraic cycles, J. Pure Appl. Algebra 34, pp. 119±145 (1984)
4. S. A. Bloch and O. Ogus, Gerstein's conjecture and homology of schemes, Ann. Sci. EÂc.
Norm. Super., IV. Ser. 7, pp. 181±202 (1974)
5. J.-L. Brylinski, Heights for local system on curves, Duke Math. J. 59, pp. 1±26 (1989)
6. D. Bump, S. Friedberg, and J. Ho€stein, Non-vanishing theorems for L-functions of
modular forms and their derivatives, Inventiones Math. 102, 543±618 (1990)
152
S. Zhang
7. P. Deligne, Formes modulaire et repreÂsentation l-adiques, SeÂminaire Bourbaki 1968/69 exp.
355. (Lect. Notes Math., vol 179, pp. 139±172) Berlin-Heidelberg-New York: Springer 1971
8. P. Deligne, Le deÂterminant de la cohomologie, Comtemporary Mathematics, vol 67, pp.
93±178
9. P. Deligne and M. Rapoport, Les scheÂmas de modules de courbes elliptiques, in: Modular
function of one variable II (ed. P. Deligne, W. Kuyk, Lect. Notes Math. vol 349, pp. 143±
316), Berlin-Heidelberg-New York: Springer 1973
10. G. Faltings, Calculus on arithmetic surfaces, Ann. of Math. 119, pp. 387±424 (1984)
11 G. Faltings, Lectures on arithmetic Riemann-Roch theorem Ann. Math. studies 127 (1992)
12 Fulton, Intersection theory Ergebn. Math. Grenzgeb. (3), Band 2, Springer-Verlag, Berlin,
Heidelberg, and New York (1984)
13 H. Gillet and C. SouleÂ, Arithmetic intersection theory Publ. Math. I.H.E.S 72, pp. 94±174
(1990)
14. H. Gillet and C. SouleÂ, Arithmetic analog of the standard conjectures Proceedings of
Symposia in mathematics, vol 55, part 1, (1994)
15. D. Goldfeld, The class numbers of quadratic ®elds and the conjectures of Birch and
Swinnerton-Dyer, Ann. Sc. Norm. Super. Pisa 3, pp. 623±663 (1976)
16. B. H. Gross, On canonical and quasi-canonical liftings, Invent. Math. 84, pp. 321±326
(1986)
17. B. H. Gross, Local heights on curves in: Arithmetic Geometry (ed. Cornell and Silverman)
Springer-Verlag
18. B. H. Gross and D. B. Zagier, Heegner points and derivatives of L-series, Invent. Math. 84,
pp. 225±320 (1986)
19. B. Harris and B. Wang, Archimedean height pairing of intersection cycles, Internat. Math.
Res. Notices 4, pp. 107±111 (1993)
20. D. A. Hejhal,The Selberg trace formula for P SL…2; R†, Lect. Notes Math. 1001. BerlinHeidelberg-New York-Tokyo: Springer 1983
21. P. Hriljac, Heights and Arakelov's intersection theory, Amer. J. math. 107, pp. 23±38
22. U. Jannsen, Mixed Motives and algebraic K-theory. Lect. Notes Math, vol. 1400 BerlinHeidelberg-New York: Springer 1990
23. N. Katz and B. Mazur, Arithmetic moduli of elliptic curves, Ann. Math. Study. 108.
Princeton University Press 1985
24. V. A. Kolyvagin, Euler system, in : The Grothendieck Festschrift, vol. II. (Prog. Math), vol.
87, pp. 435-483) Boston Basel Berlin: BirkhaÈuser 1990
25. K. KuÈnnermann, Some remarks on the arithmetic Hodge index theorem, Preprint MuÈnster
(1994)
26. K. KuÈnnermann, Higher picard varieties and the height pairing, Preprint (1995)
27. S. Lang, Introduction to Arakelov theory, Springer-Verlag (1988)
28. A. Moriwaki, Remarks on arithmetic analogues of Grothendieck's standard conjectures,
Preprint UCLA (1994)
29. M. R. Murty and V. K. Murty, Mean values of derivatives of modular L-series. Ann. Math.
133, 447±475 (1991)
30. J. Nekovar, Kolyvagin's method for Chow groups of Kuga-Sato varieties, Invent. Math.
107, pp. 99±125 (1992)
31. J. Nekovar, On the p-adic heights of Heegner cycles, Math. Ann. 302, 609±686 (1995)
32 B. Perrin-Riou, Points de Heegner et deÂriveÂes de fonction L p-adiques, Invent. Math. 89,
455±510 (1987)
33. A. J. Scholl, Motives for modular forms, Invent. Math. 100, pp. 419±430 (1990)
34. A. J. Scholl, Height pairings and special values of L-functions, Proc. Sym. AMS. 55, part 1,
571±598 (1994)
35. L. Szpiro, DegreÂs, intersections, hauteurs, in SeÂminaire sur les pinceaux arithmeÂtiques: la
conjecture de Mordell, AsteÂrisque 127, pp. 11±28 (1985)
36. J.-L. Waldspurger, Sur les values de certaines fonctions L automorphes en leur centre de
symmetre, Comp. math. 54, pp. 173±242 (1985)