Heegner Points on Modular Curves

0.28 MB·

by  Li Cai

HEEGNER POINTS ON MODULAR CURVES
LICAI,YIHUACHENANDYULIU
Abstract. In this paper, we study the Heegner points on more general modular curves other than
X0(N),whichgeneralizesGross’work”Heegner points onX0(N)”. TheexplicitGross-Zagierformula
andtheEulersystempropertyarestatedinthiscase. UsingsuchkindofHeegnerpoints,weconstruct
certain families of quadratic twists of X0(36), with the ranks of Mordell-Weil groups being zero and
onerespectively, andshowthatthe2-partoftheirBSDconjectures hold.
6
1
0 Contents
2
1. Introduction 1
n
2. The Modular Curve and Heegner points 3
a
J 2.1. The Modular Curve XK(N) 3

8 2.2. Gross-ZagierFormula 7
1 2.3. Euler System 7
2.4. Waldspurger Formula and Gross Points 8
]
T 3. Quadratic Twists of X0(36) 9
N 3.1. The Waldspurger Formula 9
3.2. Rank Zero Twists 12
.
h 3.3. The Gross-Zagier Formula 13
t
a 3.4. Rank One Twists 14
m References 16
[

1
v 1. Introduction
5
1 Letφ= ∞n=1anqn beanewformofweight2,levelΓ0(N),normalizedsuchthata1 =1. LetK bean
4 imaginaryquadraticfieldofdiscriminantD andχa (primitive)ringclasscharacteroverK ofconductor
4 P
c, i.e. a characterof Pic( ) where is the order Z+c of K. Let L(s,φ,χ) be the Rankin-Selberg
0 Oc Oc OK
convolution of φ and χ. Assume the Heegner condition:
.
1
(1) (c,N)=1,
0
6 (2) Any prime pN is either split in K or ramified in K with ordp(N)=1 and χ([p])=ap, where p
| 6
1 is the unique prime ideal of above p and [p] is its class in Pic( ).
K c
O O
:
v Under this condition, the sign of L(s,φ,χ) is 1 and Gross studies the Heegner points on X0(N)
−
i in [7]. It’s well known that X (N)(C) parameterizes the pairs (E,C), with E an elliptic curve over C
X 0
and C a cyclic subgroup of E of order N. By the Heegner condition, there exists a proper ideal of
r N
a Oc such that Oc/N ∼= Z/NZ. For any proper ideal a of Oc, let Pa ∈ X0(N) be the point representing
(C/a,a 1/a), which is defined over the ring class field H , the abelian extension of K with Galois
− c
N
group Pic( ) by class field theory. Such points are called Heegner points over K of conductor c and
c
O
only depends on the class of a in Pic( ).
c
O
Let J (N) be the Jacobianof X (N) and the cusp [ ] on X (N) defines a morphismfrom X (N) to
0 0 0 0
∞
J (N) over Q: P [P ]. Let P be the point
0 χ
7→ −∞
Pχ = [Pa ] χ([a]) J0(N)(Hc) ZC
−∞ ⊗ ∈ ⊗
[a]∈XPic(Oc)
and Pφ the φ-isotypical component of P . Then under the Heegner condition, Cai-Shu-Tian [3] gives
χ χ
an explicit form of Gross-Zagier formula which relates height of Pφ to L(1,φ,χ). In fact, they give an
χ ′
explicit form of Gross-Zagierformula in general Shimura curve case.
Let the data (φ,K,χ) be as above, and generalize the Heegner condition to the following one ( ):
∗
LiCaiwassupportedbytheSpecialFinancialGrantfromtheChinaPostdoctoral ScienceFoundation 2014T70067.
1
(i) (c,N)=1,
(ii) if prime pN is inert in K, then ord N is even; if pN is ramified in K, then ord N = 1 and
p p
| |
χ([p])=a , where p is the unique prime ideal of above p and [p] is its class in Pic( ).
p K c
6 O O
By this assumption, write N = N N2, with pN if and only if p is inert. Given an embedding
0 1 | 1
K ֒ M (Q) such that K M (Z)=K R (N )= , where
2 2 0 0 c
→ ∩ ∩ O
R0(N0)= A∈M2(Z) A≡ ∗0 ∗ (modN0) .
(cid:26) (cid:12) (cid:18) ∗(cid:19) (cid:27)
Then R= +N R (N ) is an order of M (Z). D(cid:12)efine
c 1 0 0 2 (cid:12)
O
ΓK(N)=R×∩SL2(Z)=A∈SL2(Z)(cid:12) A≡(cid:18)∗0 ∗∗(cid:19) (modN0) .
 (cid:12)(cid:12) AmodN1 ∈R/N1R 
(cid:12)
The modular curve now we have to consider is XK(N)=(cid:12)(cid:12)ΓK(N)\H ∪{cusps}. 
This modularcurve is notthe usualmodular curveofthe formX (M) anylonger,ifN =1. X (N)
0 1 K
6
parameterizes(E,C,α) where E is an elliptic curve over C, C is a cyclic subgroupof E of order N and
0
α is an H-orbit of an isomorphism (Z/N Z)2 E[N ], where
1 1
≃
H =( /N ) GL (Z/N Z).
K 1 K × 2 1
O O ⊂
The readers are referred to [9]. Let h be the fixed point of H under the action of K . Note that
0 ×
Z+Zh−01 is an invertible ideal of Oc, then the triple P = (cid:18)C/Z+Zh0, N10 ,H(cid:18)NNh1011(cid:19)(cid:19) is a Heegner
point on X (N). By CM theory, P X (N)(Kab). The conductor of(cid:10)P is(cid:11)also defined to be c, the
K K
∈
conductor of . For details, see Section 2.
c
O
AssumeφcorrespondstoanellipticcurveE/Q,thenthereisamodularparametrizationf :X (N)
K
→
E, taking to identity in E. It is unique in the sense that given two parametrizations f ,f , there
1 2
∞
exist integers n ,n such that n f = n f [3, Proposition 3.8]. Now we can formulate the following
1 2 1 2 2 2
Gross-Zagierformula:
Theorem 1.1 ([3]). Under the assumption ( )
∗
8π2(φ,φ) h (P (f))
L(1,E,χ)=2 µ(N,D) Γ0(N) K χ (GZ)
′ −
· u2c D degf
K
| | b
where ( , ) is the Petersson inner product, ˆh ispthe N´eron-Tate height over K, µ(N,D) is the
Γ0(N) K
number of prime factors of (N,D), u=[ : 1 ].
Oc× {± }
Inanother way,following the idea ofKolyvagin,these Heegnerpoints forman“Eulersystem”. There
is a norm compatible relation between Heegner points of different conductors. See Theorem 2.13.
As an application of such kind of parametrization, we will construct a family of quadratic twists of
an elliptic curve with Mordell-Weil groups being rank one. The action of complex conjugation on the
CM-points of modular curve is a crucialpoint in the proofofthe nontriviality ofthe Heegner point. For
usualmodularcurveX (N),thecomplexconjugationisessentiallytheAtkin-Lehneroperator. However,
0
it does not keep for the modular curve X (N). We find the correct one, namely, the combination of
K
local Atkin-Lehner operators? and the nontrivial normalizer of K in GL (Q). Denote this operator
× 2
by w. Then f +fw is a constant map with its image a nontrivial 2-torsion point; see Lemma 3.8. This
phenomenon and the norm compatible relation control the divisibility of Heegner cycles. Together with
the Gross-Zagierformula,the divisibility ofHeegner cyclesdeduces the 2-partofthe BSDconjecture for
our family of quadratic twists.
For each square free non-zero integer d = 1, we write E(d) for the twist of an elliptic curve E/Q by
6
the quadratic extension Q(√d)/Q. The results of [26], [2], [5], [14] shows that there are infinitely many
d such that L(E(d),s) is nonvanishing at s=1, and infinitely many d such that L(E(d),s) has a simple
zero at s=1.
The workof[21], [22]for the elliptic curve(X (32),[ ]):y2 =x3 xconstructsexplicitly families of
0
∞ −
d with ord L(E(d),s)= 1. The work of [4] deals with the elliptic curve E = (X (49),[ ]) which has
s=1 0
∞
CM by √ 7, gets the similar results to [21].
−
Here, we construct a family of quadratic twists of E =(X (36),[ ]) such that the ranks of Mordell-
0
∞
Weil groups for this twists are one.
2
Theorem 1.2. Let ℓ be a prime such that 3 is split in Q(√ ℓ) and 2 is unramified in Q(√ ℓ). Let
− −
M =q q be a positive square-free integer with prime factors q all inert in Q(√ 3), q 1(mod4)
1 r i i
··· − ≡
and q inert in Q(√ ℓ). Then
i
−
(1) ord L(s,E( ℓM))=1=rankE( ℓM)(Q);
s=1 − −
(2) #X(E( ℓM)/Q) is odd, and the p-part of the full BSD conjecture of E( ℓM) holds for p∤3ℓM.
− −
The nontriviality of Heegner cycles and Gross-Zagier formula also implies the rank part of BSD
conjecture, for E(M), namely,
ord L(s,E(M))=0=rankE(M)(Q).
s=1
However, the proof of the 2-part of the BSD conjecture for E( ℓM) needs that for E(M).
−
A new feature of this paper is that we give a parallel proof of the BSD conjecture for E(M), that is,
using the Waldspurger formula and the norm property of Gross points. For the induction method using
in [4], [23], there is an embedding problem of imaginary quadratic fields to quaternion algebras which
relating with the problem of representation integers by ternary quadratic forms (See also the argument
before[4,Definition5.5],[23,Section2.1]and[13]). TheuseofthenormpropertyofGrosspointsavoids
this emebdding problem.
If the data (φ,K,χ) satisfies that the root number ǫ(φ,χ)=+1, by [3] we can choose an appropriate
definitequaternionalgebraB overQcontainingK,anorderRofB ofdiscriminantN withR K =
K
∩ O
and a “unique” function f : B B /R C. Assume the conductor c of χ satisfies (c,N) = 1. Let
× × ×
\ →
x K B /R be a Gross point of conductor c, that is x Rx 1 K = . Denote by
c ∈ ×\ × × c −c ∩ Oc
b b
P (f)= f(σ.x )χ(σ).
b b χ bc b b
σ∈GaXl(Hc/K)
ThenwithsimilarnotationsasforGross-Zagierformula,wehavetheWaldspurgerformula(SeeTheorem
2.14)
8π2(φ,φ) P (f)2
L(1,E,χ)=2−µ(N,D) Γ0(N) | χ | ,
· u2 Dc2 · f,f
| | h i
Moreover, the Gross points of different conductors also form an “Euler system”(See Section 2). The
p
following theorem can be viewed as the rank zero version of Theorem 1.2:
Theorem 1.3. Let M = q q be a positive square-free integer with prime factors q all inert in
1 r i
···
Q(√ 3) and q 1(mod4), then
i
− ≡
(1) ord L(s,E(M))=0=rankE(M)(Q);
s=1
(2) #X E(M)/Q is odd, and the p-part of the full BSD conjecture of E( ℓM) holds for p=3.
−
6
Acknowledg(cid:0)ements.(cid:1)The authorsgreatlythankProfessorYe Tianforsuggestingthis problemandhis
persistent encouragement.
2. The Modular Curve and Heegner points
2.1. The Modular Curve X (N). Let K be an imaginary quadratic field with discriminant D. Let
K
N =N N2 be a positive integer such that pN if and only if p is inert in K. Let c be another positive
0 1 | 1
integer coprime to N. Take an embedding K ֒ M (Q) which is admissible in the sense that
2
→
K M (Z)=K R (N )= .
2 0 0 c
∩ ∩ O
Denote by R= +N R (N ) an order of M (Q). Then R has discriminant N with R K = .
c 1 0 0 2 c
O ∩ O
Let Γ (N) = R SL (Z) and X (N) be the modular curve over Q with level Γ (N). It’s well
K × 2 K K
∩
known that X(N N )(C) parameterizes (E,(Z/N N )2 E[N N ])) where E is an elliptic curve over
0 1 0 1 0 1
≃
C. By [9], it parameterizes (E,(Z/N )2 E[N ],(Z/N )2 E[N ])). Then X (N) parameterizes
0 0 1 1 K
≃ ≃
(E,C,α:(Z/N )2 E[N ])), where C is a cyclic subgroupof E[N ] of orderN , andα is an H-orbitof
1 1 0 0
a basis of E[N ]) w≃here H := ( /N ) GL (Z/N Z). Precisely, the class of z H in X (N)
1 K 1 K × 2 1 K
O O ⊂ ∈
corresponds to the triple
1 z
C/Z z+Z, ,H N1 .
· N0 1 !!
D E N1
Lemma 2.1. If m is a positive integer and (m,cND ) = 1, then for any invertible fractional ideals
K
a, of , satisfying 1a/a Z/N Z, there exist a Z-basis u,v of a and g GL+(Q), such that
N Ocmu v N− ≃ 0 { } ∈ 2
1a=Z +Zv, =g 1h , and K gRg 1 = .
− − 0 − cm
N N u ∩ O
0
3
Proof. Consider the curve morphism X (N) X (N ) over Q, which is the forgetful functor in the
K 0 0
→
moduliaspect(E,C,[α]) (E,C). a, definesaHeegnerpoint(C/a, 1a/a)onX (N ). Thenexists
− 0 0
7→ N N
a b u v
aZ-basis u,v ofaandg = GL+(Q), 1a=Z ,v , =g 1h ,andK gR\(N )g 1 =
{ } c d ∈ 2 N− hN i u − 0 ∩ 0 0 −
(cid:18) (cid:19) 0
. Different choice of basis (u,v) make g differ an element in Γ (N )=R (N ) and a scalar. So we
cm 0 0 0 0 ×
O
have to prove there exists g Γ (N ), such that K gg R\(N )g′ 1g 1 = .
′ 0 0 ′ 0 0 − − cm
∈ ∩ O
In fact, we choose g′ ∈ Γ0(N0), such that gg′ ∈ ℓ|N1Kℓ×(1 + N1M2(Zℓ)), embedding gg′ into
a b
GL (Q ). Note that if g 1 = ′ ′ , and mQultiply a scalar if necessary, we may assume
ℓ|N1 2 ℓ − (cid:18)c′ d′(cid:19)
Qg−1 M2(Z), then Z(a′h0+b′)+Z(c′h0+d) and a belong to a same class in Pic( cm), hence
∈ O
[ : a]
(detg−1)m−1 =[Ocm :a]:= O[acm: Ocm∩a] ;
cm
O ∩
ThereforethereexistsanintegerN whoseprimefactorsareprimefactorsofN ,s.t. ℓ∤det(N g 1), ℓN .
1′ 1 1′ − ∀ | 1
NtSho1′eXNn1′t∈gh−e1reℓ∈|NeGx1iLKst2ℓs×(Z(g1ℓ′)+,∈∀NℓΓ|1N0M(1N.20(BZ),yℓX)s)t.r∈oTnhQgeaℓ|ppNrp1orOoofxK×iims,ℓ(ca1otmi+onpN,leΓ1t0eM(dN2(0Z)ℓQ))ℓ,|Nsu1cOhK×t,ℓh(a1t+gN′X1M=2(ZNℓ1′)g)−=1,Qthp|uNs1gGgL′2=(cid:3)(Zℓ),
DefinitQion 2.2. Let a, and u,v,g be as in theabove lemma. A Heegner point on X (N) of conductor
K
N
cm is a triple
v
P = C/a,N−1a/a,H NNu11!!.
v
Remark 2.3. This point corresponds to the point H.
u ∈
Let h be the point H K×. The order = Z+Z̟ where ̟ = cD+c√D. Denote by x y
0 Oc c c 2 z w ∈
(cid:18) (cid:19)
GL (Q) the image of ̟ under the fixed embedding K ֒ M (Q). Since K is a field, z =0.
2 c 2
→ 6
Lemma 2.4. K =Q+Qh0 and Oc =Z+Zyh−01.
c2(D2 D) x y
Proof. We have x+w =Dc,xw yz = − . As h is fixed by ,
− 4 0 z w
(cid:18) (cid:19)
xh +y
0 =h and zh2+(w x)h y =0.
zh +w 0 0 − 0−
0
Hence
(x w)+c√D 2z (x w) c√D
h0 = − 2z ∈K\Q and h−01 = (x w)+c√D = − 2y− ,
−
so
Dc+c√D
yh−01 =−w+ 2 .
(cid:3)
Lemma 2.5. Let a=Z+Z·h−01 and N−1 =Z+Z·N0−1h−01. Then End(a)={x∈K :xa⊂a}=Oc,
and End( 1)= .
− c
N O
Proof. Let (a+bh−01)a⊂a. Then it is equivalent to
(a+bh−01)∈a and (a+bh−01)h−01 ∈a.
The first condition implies a,b∈Z, then the second one is equivalent to bh−02 ∈a. But
bh−02 =y−1b((w−x)h−01+z)∈a.
The conditionR K = tells (w x,z,y)=1,so the aboveconditionimplies b yZ,whichis exactly
c
End(a)= . Th∩e asserOtion for −1 is the same, noticing that N z. ∈ (cid:3)
c − 0
O N |
Clearly, a and are invertible ideals of , and 1/a Z/N Z. Summing up:
c − 0
N O N ≃
4
Proposition2.6. LetK ֒→M2(Q)beanadmissibleembeddingandh0 ∈HK×. Denotebya=Z+Z·h−01
and N−1 =Z+Z·N0−1h−01, then
y
P = C/a, 1/a,H N1
N− yh−01!!
N1
is a Heegner point on X (N) of conductor c.
K
Example 2.1. NowweconstructanadmissibleembeddingK ֒ M (Q)asfollowing. SinceℓN implies
2 0
→ |
that ℓ is split in K, there exists an integral ideal N of such that /N Z/N , which implies
0 K K 0 0
O O ≃
Z+N = . Then there exists n Z and m N such that
0 K 0
O ∈ ∈
D+√D
=n+m.
2
Take trace and norm, we get
D2 D
D =2n+(m+m¯) and − =n2+n(m+m¯)+mm¯
4
Since mm¯ N N =N and it is an integer, so mm¯ =N b for some b Z. Let a=D 2n, we see
0 0 0 K 0
∈ O ∈ −
D =a2 4N b
0
−
It’s easy to check that (a,b,N ) = 1. Given an integer c such that (c,N) = 1, we let the embedding
0
i :K B be given by
c
−→
D+√D ✤ // D2+a −c−1 , or Dc+√Dc2 ✤ // Dc2+ac −1 .
2 (cid:18)N0bc D−2a(cid:19) 2 (cid:18)N0bc2 Dc2−ac(cid:19)
a+√D
We can see this embedding is normal in the sense of [18], and h = .
0
2N bc
0
The modular curve X (N) depends on the admissible embedding K ֒ M (Q). However, we will
K 2
→
provethat,allthosemodularcurvesgivenbyadmissibleembeddingsareisomorphicoverQ. Leti:K ֒
→
M (Q)beanadmissibleembedding,H =i( /N ) GL (Z/N Z),andH betheupper-triangular
2 K 1 K 2 1 0
O O ⊂
matrices in GL2(Z/N0Z), then H = p|N1Hp, where Hp ⊂GL2(Z/pordpN1Z). Then
XQ (N)=X(N N )/(H H ).
K 0 1 0
×
Ifi′ isanotheradmissibleembedding,andforanyp|N1,Hp andHp′ areconjugateinGL2(Z/pordpN1Z),
then we obviously have
X(N N )/(H H ) X(N N )/(H H )
0 1 0 0 1 ′ 0
× ≃ ×
In fact, Hp is the image of following kind of morphism:Z×p2 →GL2(Zp)→GL2(Zp/pordp(N1)Zp), then
the following lemma implies that Hp and Hp′ are conjugate in GL2(Z/pordpN1Z).
Lemma 2.7. For any two embeddings ϕ :Z M (Z ),i=1,2, there exists g GL (Z ), such that
i p2 2 p 2 p
→ ∈
ϕ =g 1ϕ g.
1 − 2
Remark 2.8. If we change Z to Q , and Z to Q , this lemma is well-known.
p p p2 p2
Proof. Consider V = Z Z , with a natural action of M (Z ). Via ϕ , we view V as anZ -module,
p p 2 p i p2
⊕
denoted by V ,i= 1,2. Since Z is a discrete valuation ring and V are torsion free, so V ,V are both
i p2 i 1 2
free Z -module of rank 1, so there exists an isomorphism g :V V of Z -module, this isomorphism
p2 1 2 p2
→
corresponds to an element of GL (Z ), also denoted by g. g is an isomorphism of Z -modules means
2 p p2
that
gϕ (x)=ϕ (x)g, x Z .
1 2 p2
∀ ∈
(cid:3)
Lemma 2.9. Let ζ be a primitive N -th root of unity. Then the cusp of X (N) is defined over
N1 1 ∞ K
Q(ζ ).
N1
5
Proof. In the adelic language, we have the following complex uniformization
XK(N)(C)=GL2(Q)+ GL2(Z)/R× cusps
\H× ∪{ }
where R=R Z and the cusps are
⊗Z b b
GL (Q) P1(Q) GL (Z)/R .
b b 2 +\ × 2 ×
The cusps are all defined over Qab. By [16, pp.507], if we let r : Q /Q Gal(Qab/Q) be the Artin
b b × × →
x 0
map, then r(x) Gal(Qab/Q) acts on the cusps by left multiplication the matrix . Since
∈ b 0 1
(cid:18) (cid:19)
α β
Q /Q Z ,thenifx Z suchthatr(x) [ ,1]=[ ,1],thereexists GL+(Q), suchthat
× × ≃ × ∈ × · ∞ ∞ 0 γ ∈ 2
(cid:18) (cid:19)
α β x 0
b b R , whbich implies
0 γ 0 1 ∈ ×
(cid:18) (cid:19)(cid:18) (cid:19)
γ Zb, αx Z , β N Z for all p, and αx γ(modN ) for all pN .
∈ ×p p ∈ ×p ∈ 1 p p ≡ 1 | 1
Then α=γ = 1, and x 1(modN ). So the definition field of [ ,1] corresponds to
p 1
± ≡ ∞
Q×/Q×Z×(N1) (1+N1Zp),
pY|N1
via class field theory, which is Q(ζ )b. (cid:3)
N1
In the following, we fix the embedding i :K ֒ M (Q).
c 2
→
1 0
Atkin-Lehner operator. Take j = , then kj = jk for all k K where k is the Galois
a 1 ∈
(cid:18) − (cid:19)
conjugation of k. For each p N , let
0
|
0 1
w = GL (Q )
p pordpN 0 ∈ 2 p
(cid:18) (cid:19)
be the local Atkin-Lehner operator. Define
w =j(N0) w GL (Q) M (Z).
p 2 2
· ∈ ∩
pY|N0
b b
Since w normalizes R , it acts on X (N).
× K
For each pN , write N = pkm with (p,m) = 1. Similar to the proof of lemma (2.1), we can choose
0 0
|
b pk 1
u,v Z such that pku+mv =1, let g = GL+(Q) M (Z), such that g 1w U, so
∈ N0v pku ∈ 2 ∩ 2 − p ∈
(cid:18)− (cid:19)
1 g−1h0
wpP =[h0,wp]=[g−1h0,1]= C/Z·g−1h0+Z, N0 ,H N11 !!
D E N1
Modify it,we getwpP = C/Zh−01+Z·pk, v+ hm−01 ,H pkNphk1−01!!. However,a=Zh−01+Z·pk =pk
D E N1
is an invertible ideal of O dividing N. Suppose N =am, let N′ =am, then N′−1a/a= v+ hm−01 .
In another words, consider the quotient map ξ : XK(N) X0(N0) induced by ΓKD(N) ΓE0(N0),
→ ⊂
which is defined over Q, then the above argument says that ξ w = w ξ, where the action of w on
p p p
◦ ◦
X (N ) is defined by [6, pp.90].
0 0
To study the action of w on X (N), we can prove the following lemma:
K
Lemma 2.10. There exists t K and u R such that w =t ju.
0 × × 0
∈ ∈
Proof. For pN , let k=ord N 1, K =(Q +Q (√D)) . Let x,y Q , then
| 0 p 0 ≥b p× bp p × ∈ p
2pky x+ay
(x+y√D)jp−1wp = pk−(x+ay) a(x+ay) 2N by
0
(cid:18)− − (cid:19)
So we choose ord y = k ord 2,x+ay Z and such that a(x+ay) Z . Then let t =x+y√D.
For p∤N , letpt =−1.−Thenpsuch choi∈ce opf t works. ∈ ×p 0,p (cid:3)
0 0,p 0
6
Remark 2.11. By Shimura reciprocity law, if we use [x] [x] to denote the complex conjugation on
7→
X (N)(C), then
K
[h ,g]=[h ,jg], g GL (A ).
0 0 2 f
∀ ∈
Lemma 2.10 in fact tells that the action of w is the composition of a Galois action and the complex
conjugation.
Hecke correspondence. Let ℓ∤N be a prime, the Hecke correspondence on X is defined by
U
x x modC
1 1 i
T E,C,H = E/C ,(C+C )/C ,H
ℓ i i i
x x modC
(cid:18) (cid:18) 2(cid:19)(cid:19) i (cid:18) (cid:18) 2 i(cid:19)(cid:19)
X
wherethesumistakenoverallcyclicsubgroupsC ofE oforderℓ,α isgivenby(Z/N Z)2 α E[N ]
i i 1 1
−→ ≃
(E/C )[N ]. E[N ] (E/C )[N ] is because (ℓ,N)=1. This is just by definition.
i 1 1 i 1
≃
2.2. Gross-Zagier Formula. Let E be anelliptic curve of conductor N, K be animaginary quadratic
field of discriminant D and χ be a ring class character over K of conductor c. Assume
E,K,χ satisfy the condition ( )
∗
Then we can write N =N N2, where pN is inert in K if and only if pN .
0 1 | | 1
Embed K in M (Q) by i . There is a modular parametrization f :X (N) E mapping [ ] to the
2 c K
→ ∞
identity of E. If f ,f are two such morphisms, then there exist integers n ,n such that n f = n f .
1 2 1 2 1 1 2 2
Let h0 be the point in H fixed by K×, then Oc = Z+Zh−01, N = Z+ZN0−1h−01 is an invertible ideal
of such that / =Z/N Z. Consider the following Heegner point on X (N) of conductor c
Oc N Oc ∼ 0 K
1
P = C/Oc,N/Oc,H hN−011!!∈XK(N)(Hc).
N1
Form the cycle:
P (f)= f(Pσ)χ(σ) E(H ) C.
χ c Z
∈ ⊗
σ∈GaXl(Hc/K)
Theorem 2.12 (Explicit Gross-Zagierformula [3]). We have the following equation
8π2(φ,φ) h (P (f))
L(1,E,χ)=2 µ(N,D) Γ0(N) K χ (GZ)
′ −
· u2c D degf
| | b
here φ is the normalized newform associated to E, µ(N,pD) is the number of prime factors of (N,D),
u=[OK× :Z×], hK is the Neron-Tate height pairing over K and the Petersson inner product
b (φ,φ) = φ(x+iy)2dxdy.
Γ0(N) | |
ZΓ0(N)\H
2.3. Euler System. Let S be a finite set of primes containing the prime factors of 6cND , N de-
K S
note the set of integers any of whose prime divisors is not in S. For any ℓ,m N with ℓ a prime
S
∈
and ℓ ∤ m, let P = (C/a , 1a /a ,α ) be a Heegner point of conductor cm. Let P =
m m Nm− m m m mℓ
(C/amℓ,Nm−ℓ1amℓ/amℓ,αmℓ), such that Nmℓ =Nm∩Omℓ,amℓ =a∩Omℓ, αmℓ is the composition
(Z/N1Z)2 −α→m N1−1am/am −∼→N1−1amℓ/amℓ.
Theorem 2.13. Then we have that [H : H ] = (ℓ+1)/u if ℓ is inert in K and (ℓ 1)/u if ℓ is
mℓ m m m
−
split and
T P , if ℓ is inert in K,
u Pσ = ℓ m
mσ∈Gal(XHmℓ/Hm) mℓ ((Tℓ− w|ℓFrobw)Pm, if ℓ is split in K,
where T is the Hecke correspondence, Frob iPs the Frobenius at wℓ in Gal(H /K), and u = 1 if
ℓ w m m
|
m6=1 and u1 =[OK× :Z×].
This theorem is proved in general by [10, Proposition 4.8] or [20, Theorem 3.1.1].
7
2.4. Waldspurger Formula and Gross Points. Let φ= ∞n=1anqn be a newform of weight 2, level
Γ (N), normalized such that a =1. Let K be an imaginary quadratic field of discriminant D and χ a
0 1
P
ring class characteroverK ofconductor c. Let L(s,φ,χ) be the Rankin-Selberg convolutionof φ and χ.
Assume that (c,N)=1. Denote by Sthe set ofprimes pN satisfying one of the followingconditions:
|
p is inert in K with ord (N) odd;
p
•
pD, ord (N) = 1 and χ([p]) = a where p is the prime of above p and [p] is its class in
p p K
• | O
Pic( );
c
O
pD, ord (N) 2 and the local root number of L(s,φ,χ) at p equals η ( 1) where η is the
p p p
• | ≥ − −
quadratic character for K /Q .
p p
ThenthesignofL(s,φ,χ)is+1. LetB bethedefinitequaternionalgebradefinedoverQramifiedexactly
atprimes inS . Fix anembedding fromK inB. Let Rbe anorderin B withdiscriminantN and
∪{∞}
R K = . Denote by Rˆ =R Zˆ and U =Rˆ which is an open compact subgroupof Bˆ . Consider
c Z × ×
the∩ShimuOrasetX =B Bˆ /⊗U whichis afinite set. A pointinX representedby x Bˆ is denoted
U × × U ×
\ ∈
by [x]. Note that for p(D,N), K normalizes U and then K acts on X by right multiplication. Let
| p× p× U
C[X ]0 = f C[X ] f(x)=0 .
U U
( ∈ (cid:12) )
(cid:12)(cid:12)xX∈XU
Foreachp N,thereareHeckecorrespondencesT and(cid:12) S . Inthis case,B issplitwhile U ismaximal.
6| p (cid:12) p p p
Then the quotient B /U can be identified with Z -lattices in Q2. Then for any [x] X ,
p× p p p ∈ U
S [x]:=[x(p)s ], T [x]:= [x(p)h ]
p p p p
Xhp
where if x corresponds to a lattice Λ, then s is the lattice pΛ and the set h is the set of sublattices
p p p
{ }
Λ of Λ with [Λ:Λ]=p. There is then a line V(φ,χ) of C[X ]0 characteristized as following
′ ′ U
for any p N, T acts on V(φ,χ) by a and S acts trivially;
p p p
• 6|
for any p(D,N) with ord (N) 2, K acts on V(φ,χ) by χ .
• | p ≥ p× p
Let f be a nonzero vector in V(φ,χ) and consider the period
P (f)= f(σ)χ(σ)
χ
σ∈GaXl(Hc/K)
where the embedding of K to B induces a map
Gal(H /K)=K Kˆ / X .
c ×\ × Oc× −→ U
Theorem 2.14 (Explicit Waldspurger formula [3]). We have the following equation
b
8π2(φ,φ) P (f)2
L(1,φ,χ)=2−µ(N,D) Γ0(N)| χ | (Wald)
· u2c D f,f
| | h i
where the pairing p
f,f = f(x)2w([x]) 1
−
h i | |
[xX]∈XU
and w([x]) is the order of the finite group (B xUx 1)/ 1 .
× −
∩ {± }
There is an analogue to Heegner points, the so called Gross points. Let S be a set of finite places of
Q containing all places dividing 6cND. Let N denote the set of integers whose prime divisors are not
S
in S.
Definition 2.15. Let m N . A point x K Bˆ /U is called a Gross Point of conductor cm, if
S m × ×
∈ ∈ \
x Ux 1 Kˆ = .
m −m ∩ × Oc×m
EachelementinK K / actsonx byleftmultipication. ThisinducesanactionofGal(H /K)
b ×\ × Oc×m m cm
on x , also called the Galois action.
m
For each prime ℓ NbS, fibx an isomorphism βℓ : Bℓ ∼ M2(Qℓ), such that βℓ(Uℓ) = GL2(Zℓ), and,
∈ →
under this isomorphism, we have
a 0
β (K )= :a,b Q , if ℓ is split in K;
• ℓ ℓ 0 b ∈ ℓ
(cid:26)(cid:18) (cid:19) (cid:27)
a bδ
β (K )= :a,b Q , where δ Z Z 2, if ℓ is inert in K.
• ℓ ℓ b a ∈ ℓ ∈ ×p\ ×p
(cid:26)(cid:18) (cid:19) (cid:27)
8
For m N , define x B by
S m ×
∈ ∈
b (xm)ℓ = βℓ−1(cid:18)ℓor0dℓm 10(cid:19) ℓ|m
1 ℓ∤m

Then the image of xm in K× B×/U, stilldenoted by xm, is a Gross point of conductor cm.
\
Theorem 2.16. For any ℓ,m N with ℓ a prime and ℓ∤m, we have that
b∈ S
T [x ], if ℓ is inert in K,
ℓ m
u [σ.x ]=
m mℓ
σ∈Gal(HXcmℓ/Hcm) ((Tℓ− w|ℓFrobw)[xm], if ℓ is split in K,
where the equality holds as divisors on X , with FProb and u the same as Theorem 2.13.
U w m
The proof is the same as the norm relation of Heegner points on Shimura curves. One can refer to
[10, Proposition 4.8] or [20, Theorem 3.1.1].
3. Quadratic Twists of X (36)
0
ThemodularcurveX (36)hasgenusoneanditscusp[ ]isrationaloverQsothatE =(X (36),[ ])
0 0
∞ ∞
isanellipticcurvedefinedoverQ. TheellipticcurveE hasCMbyQ(√ 3)andhasminimalWeierstrass
−
equation
y2 =x3+1.
Note that its Tamagawa numbers are c = 3,c = 2 and E(Q) = Z/6Z is generated by the cusp
2 3 ∼
[0]=(2,3), we use T to denote the non-trivial2-torsionpoint in the following. Denote by Lalg(E,s) the
algebraic part of L(E,s). Then Lalg(E,1)=1/6.
Fora non-zerointegerm, letE(m) :y2 =x3+m3 the quadratictwistof E by the fieldQ(√m). Then
E(m) and E( 3m) are 3-isogenous to each other.
−
Lemma 3.1. Let D Z be a fundamental discriminant of a quadratic field. Then the sign for the
∈
functional equation of E(D), denoted by ǫ(E(D)), is
( 1)# pD,p=2,3,
{ | ∞}
−
where D means that D <0.
∞|
Proof. For each D, denote by K =Q(√D), then
L(s,E )=L(s,E)L(s,E(D))
K
whereL(s,E )isthebasechangeL-functionanditsufficestodeterminethesignofL(s,E ). Notethat
K K
thelocalcomponentsofthecuspidalautomorphicrepresentationforEatplaces2and3aresupercuspidal
with conductor 2, then by [19, Proposition3.5],the localrootnumber for the base change L-function at
places 2 (resp. 3) is negative if and only if 2D (resp. 3D). Meanwhile, the local root number at is
| | ∞
positive if and only if D is positive and for any place not dividing 6 it is positive. Summing up, the
result holds. ∞ (cid:3)
3.1. The Waldspurger Formula. Let B be the definite quaternion algebra over Q ramified at 3, ,
∞
then we know that
B =Q+Qi+Qj+Qk, i2 = 1,j2 = 3,k=ij = ji.
− − −
Let OB =Z[1,i,(i+j)/2,(1+k)/2] of B. The unit group OB× of OB equals to
1, i, (i+j)/2, (i j)/2, (1+k)/2, (1 k)/2 .
{± ± ± ± − ± ± − }
Let K = Q(√ 3) and η : Q /Q 1 is the quadratic character associated to K. Embed K ֒ B
× ×
− →{± } →
by sending √ 3 to k, which induces an embedding K ֒ B .
× ×
− →
Let π =⊗vπv be the aubtomorphic representation of BA× corresponding to E via the modularity of E
and the Jacquet-Langlands correspondence. Let =b bbe an order of B defined as following. If
R pRp ×
p = 2, then = +2 . If p = 3, then = +λ where λ B is a uniformizer of
R2 OK,2 OB,2 R3 QOK,3 OB,3 ∈ ×
B ; for example, we may choose λ=k, which is also a uniformizer of K . Forbp 6, = . Denote
3 3 p B,p
6| R O
by U = . Then U is an open compact subgroup of B .
× ×
R
The local components of π have the following properties:
π is trivial; b
• ∞
9
×
πp is unramified if p=2,3, , i.e. πOB,p is one dimensional;
• × 6 ∞ ×
πOK,2 is one dimensional and πOK,3 is two dimensional.
• 2 3
The first two properties are standard, while the last property comes from [3, proposition 3.8]. Then πU
is a representation of B3× with dimension 2. As K3×-modules, πU =Cχ+⊕Cχ− where χ+ is the trivial
character of K3× and χ is the nontrivial quadratic unramified character on K3×.
This representation−πU is naturally realized as a subspace of the space of the infinitely differentiable
complex-valuedfunctionsC (B B /Q ). ThespaceπU iscontainedinthespaceC (B B /Q U)
∞ × × × ∞ × × ×
\ \
and is perpendicular to the spectrum consisting of characters (the residue spectrum). In fact, we have
the following more detailed propositbion:b b b
Proposition 3.2. (1) πU has an orthonormal basis f , f under the Petersson inner product de-
+
−
fined by
f 2= f(g)2dg
k k ZB×\Bb×/Qb×| |
with the Tamagawa measure Vol(B B /Q )=2.
× × ×
\
(2) Moreover, f (resp. f ) is the function on B B /Q U, supported on those g B with
+ × × × ×
χ (g) = +1 (resp. = −1), valued in 0b, 1bwith to\tal mass zero, where χ is the com∈position of
0 0
− ±
the following morphisms: b b b
det η
B B /Q U Q 1.
× × × ×
\ −→ −→±
(3) For any t∈K3×, π(t)f+ =χ+(t)fb+ anbd π(t)f−b=χ−(t)f−.
Since the class number of B with respect to is 1 by [24, pp. 152], one has
B
O
(3)
B× =B×OB× =B×B3×OB× .
Therefore,
b b b
(3) (6)
B×\B×/Q×U =B×\B×B3×OB× /U2U3OB× =H\B3×OB×,2/U2U3,
whereH =B×∩B3×OB×b(3) =bOB×λZ ⊂B3×OB×,2 abndthelastbinclusionisgivenbythediagonalembedding.
Lemma 3.3. The dobuble coset H\OB×,2/U2 is trivial and H ∩U2\B3×/U3 =OB×,3/U3.
Proof. The proof is elementary. Firstly, we prove that H\OB×,2/U2 is trivial. Recall that U2 =OK×,2(1+
2M (Z )). As GL (Z )/(1+2M (Z ))=GL (F ),the claimfollowsfromthatfor anyg GL (F ), one
2 2 2 2 2 2 2 2 2 2
∈
may find h∈H and u∈OK×,2 such that g ≡hu(mod2Z2). For the second claim, note that
1+k
H U = k, 1, ,
2
∩ h − 2 i
For any x ∈ B3×, x−1(1+λ)x = 1+x−1λx ∈ U3 where λ is any uniformizer of B3×. In particular, the
aHct∩ioUn2o\fBH3×/∩UU32=oHn B∩3×U/2U\(3Bi3×s/eUqu3)al=toOtB×h,e3/aUc3t.ion of the group generated by some uniformizer. Henc(cid:3)e
If we denote Z the integer ring for the unramified quadratic extension field of Q , then
9 3
OB×,3 =Z×9(1+λZ9);U3 =OK×,3(1+λOB,3)=µ2(1+3Z9)(1+λZ9)
where µ = 1 . Hence
2
{± }
H\B3×OB×,2/U2U3 ←∼−H ∩U2\B3×/U3
←∼−OB×,3/U3
←∼−Z×9/µ2(1+3Z9)∼=Z/4Z,
and we can identify C (B B /Q U) with C[Z/4Z].
∞ × × ×
\
The image of B×\Bˆ×/U under the norm map is Q×+\Qˆ×/NrU. If p 6= 3, NrUp = Z×p while if
NrU =1+3Z . Therefore, bybthebapproximation theorem,
3 3
Q×+\Qˆ×/NrU =Z×3/NrU3 =Z×3/1+3Z3.
In paticular, the cardinality of Q×+\Qˆ×/NrU is 2. Forms in C∞(B×\B×/Q×U) of the form µ◦Nr for
some Hecke character µ correspond to characters on C[Z/4Z] of order dividing 2. Sum up, we obtain
10 b b