On Atkin and Swinnerton-Dyer congruence relations (3)

0.23 MB·English

by  Ling Long

ON ATKIN AND SWINNERTON-DYER CONGRUENCE
RELATIONS (3)
7
0
0
LINGLONG
2

n
a Abstract. Intheprevioustwopaperswiththesametitle([LLY05]byW.C.
J Li,L.Long,Z.Yangand[ALL05]byA.O.L.Atkin,W.C.Li,L.Long),theau-
0 thorshavestudiedspecialfamiliesofcuspformsfornoncongruence arithmetic
subgroups. It was found that the Fourier coefficients of these modular forms
1
atinfinitysatisfythree-termAtkinandSwinnerton-Dyercongruencerelations

which are the p-adic analogue of the three-term recursions satisfied by the
]
T coefficients ofclassicalHeckeeigenforms.
N In this paper, we first consider Atkin and Swinnerton-Dyer type congru-
ences which generalize the three-term congruences above. These weaker con-
.
h gruences aresatisfiedbycuspformsforspecialnoncongruence arithmeticsub-
t groups. Then we will exhibit an infinite familyof noncongruence cuspforms,
a eachofwhichsatisfiesthree-termAtkinandSwinnerton-Dyertypecongruences
m
foralmosteveryprimep. Finally,wewillstudyaparticularspaceofnoncon-
[ gruencecuspforms. Wewillshowthattheattachedl-adicSchollrepresentation
isisomorphictothel-adicrepresentation attached toaclassicalautomorphic

1 form. Moreover, foreach ofthe fourresidueclassesof oddprimesmodulo12
v there is a basis so that the Fourier coefficients of each basis element satisfy
0 three-term Atkin and Swinnerton-Dyer congruences in the stronger original
1 sense.
3
1
0
7
1. Introduction
0
/ Thispaperisacontinuationoftwopreviouspaperswiththe sametitle: [LLY05]
h
t byW.C.Li,L.Long,Z.Yangand[ALL05]byA.O.L.Atkin,W.C.Li,L.Long. Here,
a
we continue to explore modular forms for noncongruence arithmetic subgroups.
m
The serious study of these functions was initiated in the late 1960’s by Atkin and
:
v Swinnerton-Dyer [ASD71] and further developed by Scholl [Sch85, Sch86, Sch88,
i etc]. In particular, under a general assumption Scholl has established a system
X
of compatible l-adic representations of the absolute Galois group attached to each
r
a space of noncongruence cuspforms. In [LLY05, ALL05], two intricate cases have
been exhibited in which noncongruence cuspforms are related closely to classical
congruenceautomorphicformsasfollows: Fromthep-adicpointofview,ineachof
thesecasesthereisasimultaneous(orsemi-simultaneous)basisforalmostallprimes
p such that each basis function is an “eigenform” for the “p-adic” Hecke operators
T (c.f. Section 2.6). Moreover,the “traces” of these T are Hecke eigenvalues (up
p p
toatmostanidealclasscharacter). Fromthe representationpointofview,ineach
case the associated l-adic Scholl representationρ can be decomposed into a direct
l
sum of 2-dimensional subrepresentations,when ρ is restricted to a suitable Galois
l
subgroup. Furthermore, each subrepresentation is shown to be isomorphic to an
1991 Mathematics Subject Classification. 11F30,11F11.
The author was supported inpartby anNSF-AWM mentoring travel grant for women. She
wouldliketothankthePennsylvaniaStateUniversityforhostinghervisitduringMay-June2006.
1
2 LINGLONG
l-adic representation attached to a classical congruence automorphic form. In this
paper,weintend to giveamore generaldiscussiononthe firstaspectanda further
discussion on the second one.
Inthispaper,wecontinuetoconsidermodularformsfornoncongruencecharacter
groups(followingthenotationofA.O.L.Atkin). Tobeprecise,acharactergroupis
anarithmeticsubgroupΓwhichisnormalinitscongruenceclosureΓ0 (thesmallest
congruencesubgroupwhichcontainsΓ)withabelianquotient. Forexample,thelat-
ticegroupsstudiedbyRankin[Ran67]andallgroupsin[LLY05,ALL05,KL06]are
character groups and most of them are noncongruence. The construction of these
groupsendowsthecorrespondingmodularformswithspecialarithmeticproperties.
This paperis organizedinthefollowingway: Section2isdevotedtothe proper-
tiesofmodularformsforcharactergroups. InSection3,westudythearithmeticof
weight3cuspforms fora specialfamily ofnoncongruencecharactergroupsdenoted
by Γ . In particular, we will prove the following theorem
n
Theorem 1.1. For every positive integer n and almost every prime p, the space
S (Γ )ofweight3cuspformsforΓ hasarationalbasisindependentofpsuchthat
3 n n
for every basis element a(n)wn, there exist a natural number r depending
n 1
on p and n and two algPebra≥ic numbers A(p) and B(p) with A(p) 2r pr and
| | ≤ r
B(p) p2r such that for any positive integer n (cid:0) (cid:1)
| |≤
a(npr) A(p)a(n)+B(p)a(n/pr)
− (1)
(pn)2
is integral at p.
The congruence relation (1) is weaker but more general than those Atkin and
Swinnerton-Dyer relations obtained in [LLY05, ALL05].
In Section 4, we investigate the arithmetic properties of weight 3 cuspforms for
Γ in the above family. This case is very similar to the one obtained in [ALL05].
6
The main result is
Theorem 1.2. Let ρ :Gal(Q/Q) Aut(W ) be the l-adic Scholl represen-
3,l,6 3,l,6
→
tations attached to weight 3 cuspforms for Γ . Then ρ are isomorphic up to
6 3,l,6
semisimplification to the l-adic representations attached to a classical congruence
automorphic form. Moreover, for every prime p ∤ 6l, the space of weight 3 cusp-
forms has a basis, depending onthe congruence classof p modulo 12, suchthat for
each basis element a(n)wn and all n N
n 1 ∈
≥
P
a(np) A(p)a(n)+B(p)a(n/p)
− (2)
(pn)2
is integral at some place in Z[i] above p where A(p) and B(p) are two algebraic
numbers with A(p) 2p and B(p) p2. Moreover, A(p) and B(p) can be
| | ≤ | | ≤
obtained from the coefficients of a congruence cuspform.
In this paper, we use Γ to denote a finite index subgroup of SL (Z) such that
2
I / Γ and use Γ0 to denote its congruence closure. Let X denote the compact
2 Γ
− ∈
modular curve for Γ. We assume all modular curves considered here are defined
over Q. Let M denote the field of meromorphic modular functions for Γ, i.e. the
Γ
field of rational functions on X . In particular, M is a finite extension of M .
Γ Γ Γ0
Let k be a positive integer. By S (Γ) we mean the space of weight k holomorphic
k
cuspforms for Γ. In the sequel, let ω = e2πi/n and L = Q(ω ). For any field
n n n
ON ATKIN AND SWINNERTON-DYER CONGRUENCE RELATIONS (3) 3
K, let G =Gal(K/K). Unless otherwise mentioned, we follow all other notation
K
used in [ALL05].
2. Modular forms for character groups
2.1. Character groups.
Definition 2.1. An arithmetic subgroup Γ is called a character group of another
arithmetic subgroup Γ0 of if Γ is normal in Γ0 with abelian quotient.
Since the quotient group Γ0/Γ acts on the C-vector space S (Γ) by the stroke
k
operator , by the representationof finite abelian group, S (Γ) can be decomposed
k
|
into a direct sum of representation subspaces parameterized by the characters of
Γ0/Γ.
2.2. Character groups with cyclic Γ0/Γ. By the Fundamental Theorem of Fi-
nite Abelian Groups, we will focus on character groups with Γ0/Γ cyclic in the
sequel. Assume Γ is a character group and Γ0/Γ = ζΓ = Z where n = [Γ0 :Γ].
h i ∼ n
By the representation theory of finite cyclic groups,
S (Γ)= S (Γ0,χj),
k k
1Mj n
≤ ≤
where χ(ζΓ)=ω and χj (1 j n) are the characters of Γ0/Γ. Here S (Γ0,χj)
n k
≤ ≤
consists of functions f in S (Γ) such that f = ωjf. Thus, when (j,n) > 1,
k |ζ n
f S (Γ0,χj) is a modular form for an intermediate group sitting between Γ and
k
∈
Γ0. So, S (Γ0,χj) consists of “old” forms for supergroups of Γ; while
(j,n)>1 k
LS (Γ0,χj) consists of “new” forms orthogonal to those ones in S (Γ)old.
(j,n)=1 k k
ALccordingly, we denote these two spaces by S (Γ)old and S (Γ)new respectively.
k k
Under the Petersson inner product, we have
S (Γ)=S (Γ)new S (Γ)old.
k k k
M
2.3. Special symmetry. Since ζ normalizes Γ, it induces an order n cyclic cov-
ering map from the modular curve X of Γ to the modular curve X of Γ0. We
Γ Γ0
use ζ againto denote such an involution and assume its minimal field of definition
to be Q(ω ). Moreover,we assume that as a simple finite extensionof M , M is
n Γ0 Γ
generatedby t with minimal polynomial(t )n t for some t M with rational
n n Γ0
− ∈
Fourier coefficients at infinity. Then X satisfies a symmetry
Γ
ζ :t ω 1t . (3)
n 7→ n− n
Such a map ζ is defined over L =Q(ω ).
n n
2.4. l-adic Scholl representations. Assume dim S (Γ) = d. For any prime
C k
number l, let ρ : G Aut(W ) be the l-adic Scholl representation attached
k,l Q k,l
→
to S (Γ) which is unramified outside of a few primes [Sch85]. Here W is a 2d-
k k,l
dimensional Q -vector space. Denoted by N the product of all ramifying primes
l
of ρ except l together with all prime divisors of n. Let p ∤ Nl be a prime and
k,l
F bethe canonicalFrobeniusconjugacyclassinthe quotientofthe decomposition
p
group at p by the inertia group at p. Scholl has also shown that the characteristic
polynomial of ρ (F ) for any p ∤ Nl has integral coefficients and the eigenvalues
k,l p
have the same absolute value p(k 1)/2.
−
4 LINGLONG
When Γ is a character group as above. The map ζ (3) endows W Q (ω )
k,l l n
⊗
withanorderninvolution. Underourassumptions,W Q (ω )decomposesinto
k,l l n
⊗
eigenspaces of ζ. Let be such an eigenspace with eigenvalue ωj. Let
Wj n
Wnew = and Wold = .
k,j Wj k,j Wj
M M
(j,n)=1 (j,n)>1
The space Wnew is simply denoted by Wnew when there is no ambiguity.
k,l
Lemma 2.1. Assume p∤Nl is a prime. In that case
F ζ =ζpF , and ζF =F ζpˆ,
p p p p
where pˆdenotes the inverse of p in (Z/nZ) .
×
Proof. Modulo the prime ideal (p), we have
F ζ(t ) = F (ω 1t )=ω ptp =ζpF (t ).
p n p n− n n− n p n
ζF (t ) = ω 1tp =(ω pˆt )p =F ζpˆ(t ).
p n n− n n− n p n
(cid:3)
Consequently,foranyw ,ζ(F w)=F ζpˆw =ωpˆ(F w).ThusF .
∈Wj p p n p pWj ⊆Wjpˆ
By iteration, F permutes and
p j
W
F = .
p j jpˆ
W W
tiveSliyn,ce the group generated by all Fp with p ∤ Nl acts on {Wj}j∈(Z/nZ)× transi-
dim =δ , (4)
Ql(ωn)Wj p
where δ is an integer independent of j in (Z/nZ) . Therefore
p ×
dim Wnew =δ φ(n), (5)
Ql(ωn) p·
where φ(n) is the Euler number of n. For any p∤n, use O (p) to denote the order
n
of p in the multiplicative group (Z/nZ) . Let r =O (p). Moreover,we have
× n
r
= (6)
j,p jpˆm
L W
mM=1
is invariant under F .
p
Corollary 2.2. The spaces Wnew and Wold are invariant under each F when
k,l k,l p
p∤Nl. Therefore as a Gal(Q/Q)-module, W Q (ω )=Wnew Wold.
k,l⊗ l n k,l k,l
L
Proof. For any p ∤ Nl, (pˆ,n) = 1, and (j,n) = (jpˆ,n). The assertions follow natu-
rally. (cid:3)
Intheremainingpartofthissection,wewillfocusonWnew andletρnew :G
k,l k,l Q →
Aut(Wnew).
k,l
ON ATKIN AND SWINNERTON-DYER CONGRUENCE RELATIONS (3) 5
Aswehaveextendedthescalarfieldtoincludeω ,thereexistsabasis ofWnew
n
B
under which the matrix of ζ on Wnew is a diagonal matrix of the following block
form
ωnj1Iδp 0 0 ··· 0 0
 0 ωnj2Iδp 0 ··· 0 0 
ζ = , (7)
 ·0·· ·0·· ·0·· ······ ωnjφ(·n·)−·1Iδp ·0·· 
 0 0 0 0 ωjφ(n)I 
 ··· n δp
where j runs through the set (Z/nZ) .
i ×
If B is anoperatoronavectorspaceL,let Char(L,B)(T)denotethe character-
istic polynomial of B on L with variable T.
Lemma 2.3. Let p ∤ Nl be a prime number and j (Z/nZ) . Let r = O (p).
× n
∈
Then
Char( ,F )(T) Q (ω )[Tr].
j,p p l n
L ∈
Proof. Assume that under the matrix of F on Wnew is (E ) where
p i,j 1 i,j φ(n)
E are δ δ matrices. TBhe commutativity F ζ = ζpF in Lem≤ma≤2.1 and (7)
i,j p p p p
×
imply that E = (0) are all-zero matrices unless i ( p)+j = 0 mod n. So
the matrix ofi,(jF )n reδspt×rδipcted to consists of block fo·rm−s with diagonal blocks
p j,p
L
(0) unlessrn. ThereforethetraceofFn on is0unlessrn. Consequently
theδpc×hδapracteristic| polynomial of F on ips in tLerj,mps of Tr. | (cid:3)
p j,p
L
Lemma 2.4. Char( ,F )(T)=Char( ,Fr)(Tr).
Lj,p p Wj p
Proof. ByLemma2.3,ifαisasolutionofChar( ,F )(T)=0,soisω α. Hence,
j,p p r
L
all the roots of Char( ,F )(T) are ωjα .
OntheotherhandLCjh,par(p ,Fr)(T){corinmci}d1e≤sjw≤irt,1h≤tmh≤eδcpharacteristicpolynomial
of the δp×δp matrix rm=1WEjp−mpj,p−m+1j. Since the order of the product does not
effect the characteristQic polynomial,
Char( ,Fr)(T)= Char( ,Fr)(T) r = (T αr )r.
Lj,p p Wj p − m
(cid:0) (cid:1) 1≤Ym≤δp
Hence Char( ,Fr)(T)= (T αr ). It follows
Wj p 1≤m≤δp − m
Q
Char( ,Fr)(Tr)= (Tr αr )=Char( ,F )(T).
Wj p − m Lj,p p
1≤Ym≤δp
(cid:3)
Proposition 2.5. The polynomial Char(Wnew,F )(T) is in Z[Tr] with all roots
p
having the same absolute value p(k 1)/2.
−
Proof. By Lemma 2.3 and Scholl’s Theorem which says Char(W ,F )(T) Z[T]
k,l p
∈
anditseigenvalueshavethesameabsolutevaluep(k 1)/2,itsufficestoshowChar(Wnew,F )(T)
− p
∈
Z[T]. Let Γ′ be an intermediate group Γ Γ′ Γ0. By our assumption, XΓ′ has
⊂ ⊂
a model over Q. Let W (Γ) be the l-adic Scholl representation space associated
k,l ′
with S (Γ). Then Char(W (Γ),F )(T) Z[T]. When m=[Γ0 :Γ] is a prime,
k ′ k,l ′ p ′
∈
Char(Wold(Γ),F )(T)=Char(W (Γ0),F )(T) Z[T]
k,l ′ p k,l p ∈
6 LINGLONG
and hence
Char(Wkn,elw(Γ′),Fp)(T)∈Z[T].
By induction, for any divisor mn, Char( ,F )(T) Z[T]. In par-
| j,(j,n)=n/mWj p ∈
ticular, L
Char(Wnew,F )(T)=Char( ,F )(T) Z[Tr].
k,l p Wj p ∈
M
j,(j,n)=n/n
(cid:3)
2.5. Induced representations. As a consequence of the above discussions we
have for any p∤Nl
Tr(ρnew(F ))=0 if p=1 mod n.
k,l p 6
SincetheimagesofallFrobeniuselementswilldetermineanyGaloisrepresentation
up to semisimplification, this implies that the character of ρnew is invariant under
k,l
any twisting by Dirichlet characters of modulus divisible by n. Let as above,
j
W
L = Q(ω ), and ρ : G Aut( ) for any j (Z/nZ) . The group G is
n n j Ln → Wj ∈ × Ln
an index φ(n) subgroup of G and is a δ -dimensional representation space of
Q j p
W
G . Applying a result in [Ser77, Prop. 19], we obtain the following result.
Ln
Proposition 2.6. Let j (Z/nZ) . We have
×
∈
ρnew =IndGQ ρ .
k,l GQ(ωn) j
2.6. p-adic spaces and Atkin and Swinnerton-Dyer type congruences. Let
S (Γ,Z ) = S (Γ) Z , and V be the p-adic Scholl space attached to S (Γ) with
k p k p k
⊗
S (Γ,Z ) as a subspace [Sch85]. These p-adic spaces are endowed with the action
k p
of the Frobenius morphism, which is denoted by F. The operator ζ also acts on V
and its order is still n. Like before, we can define V to be the ω -eigenspace of ζ
j j
on V (over Q (ω )) as well as Vnew and Vold similarly.
p n
In[ASD71],AtkinandSwinnerton-Dyerobservedthatsomenoncongruencemod-
ular forms satisfy three-term Hecke-like recursions in a p-adic sense. Their obser-
vations have been verified by Cartier for weight 2 cases [Car71] and Scholl [Sch85]
for all 1-dimensional cases. Here, we will first define Atkin and Swinnerton-Dyer
type congruence relations.
Definition 2.2. Let K be a number field and f = a(n)wn a weight k mod-
n 1
ular form. The function f is said to satisfy the AtPkin≥and Swinnerton-Dyer type
congruence relationat p given by polynomial T2d+A T2d 1+ +A K[T]if
1 − 2d
··· ∈
for all n Z,
∈
a(npd)+A a(npd 1)+ +A a(n)+ +A a(n/pd)
1 − d 2d
··· ···
(np)k 1
−
is integral at some place of K above the prime number p.
To examine three-term Atkin and Swinnerton-Dyer congruence relations com-
putationally, Atkin uses “p-adic” Hecke operators which we will explain. For any
prime p∤N, define the p-adic Hecke operator to be T =U +s(p) pk 1V which
p p − p
·
acts on a weight k form f = a(n)wn as follows:
P
f|Tp = (a(np)+s(p)pk−1a(n/p))wn.
Xn
ON ATKIN AND SWINNERTON-DYER CONGRUENCE RELATIONS (3) 7
Here, s(p) is an algebraic number with absolute value 1. It is called a “p-adic”
Hecke operator as the nth coefficient of f is only determined up to modulo
|Tp
some suitable power of p depending on n. In general, it is totally nontrivial to
guess the right s(p) values so that T has “eigen” forms in S (Γ). Moreover s(p)
p k
are rarely Dirichlet characters. Unlike the classical Hecke operators which can be
diagonalizedsimultaneously,theseoperatorsareaprioridefinedovervariousp-adic
fields. It is clearly exceptionalthat the space S (Γ) has a simultaneous eigen basis
k
for all “p-adic” Hecke operators T , which is the case in [LLY05].
p
2.7. Atkin and Swinnerton-Dyer type congruences satisfied by modular
formsfornoncongruencecharacter groups. ApplyingTheorem5.6in[Sch85],
we have
Theorem 2.7. [Scholl, [Sch85]] For any prime p∤Nl.
Char(V,F)(T)=Char(W,F )(T) Z[T].
p
∈
Proposition2.8(Fangetal[F+05]). LetGbeafinitegroupofautomorphismsof
theellipticmodularsurfaceπ : X associatedwithΓ. Letχbeanirreducible
Γ Γ
E →
character of G and if V is a representation of G, let Vχ denote the χ-isotypical
subspace of V. Let K/Q be the field of definition of the representation whose
character is χ and let λ be a place of K above l and ℘ be a place of K above p.
Finally, we let r to be the smallest positive integer such that (F )r G . Then
p K
∈
Char((V K )χ,Fr)(T)=Char((W K )χ,(F )r)(T). (8)
℘ λ p
⊗ ⊗
In our case, we use G=Γ0/Γ which is generatedby ζΓ. Then (V K )χj =V
℘ j
⊗
and (W K )χj = .
λ j
⊗ W
Corollary 2.9. If j (Z/nZ) , then dim V =δ where δ is even.
∈ × Qp(ωn) j p p
Corollary 2.10.
Char(Vnew,F)(T)=Char(Wnew,F )(T) Z[T].
p
∈
Moreover, the roots of these monic polynomials have the same absolute value
p(k 1)/2.
−
Theorem 2.11. Assume Γ is a character group of Γ0 with Γ0/Γ =< ζΓ >=
∼
Z , X has a model defined over Q, and the action of ζ on X is defined over
n Γ Γ
Q(ω ). Let p ∤ Nl be a prime, j (Z/nZ) , and r = O (p). Assume the space
n × n
∈
V has a basis consisting of forms with Fourier coefficients in L . Let H (T) =
j n j
Char( ,F )(T) L [Tr]. For every f V with coefficients in L , f satisfies an
j p n j n
L ∈ ∈
Atkin and Swinnerton-Dyer type congruence at p given by H (T).
j
Proof. It follows from Scholl’s arguments in [Sch85]. (cid:3)
In particular, when δ =2, we have
p
H (T)=T2r A (p)Tr+B (p), (9)
j j j
−
where A (p) 2r p(k 1)r/2, B (p) = p(k 1)r. The corresponding three-term
| j | ≤ r − | j | −
recursion is weaker(cid:0)th(cid:1)an the original three-term Atkin and Swinnerton-Dyer con-
gruence relation when r>1.
8 LINGLONG
3. Atkin and Swinnerton-Dyer type congruences satisfied by
cuspforms for Γ
n
In this section, we will use the results obtained in the previous section to derive
three-termAtkinandSwinnerton-Dyertypecongruencerelationssatisfiedbyweight
3 cuspforms for Γ .
n
3.1. The groups Γ . In [LLY05, ALL05], the following family of noncongruence
n
character group is considered.
Let
1 0
Γ1(5)= γ SL (Z) γ mod 5 .
{ ∈ 2 ≡(cid:18) 1(cid:19) }
(cid:12) ∗
Itisanindex12congruencesubgroupof(cid:12)SL (Z)with4cusps , 2,0, 5/2. This
2
∞ − −
group has four generators γ ,γ ,γ ,γ (each stabilizes one cusp as indicated
2 0 5/2
∞ − −
by the subscripts) subject to one relation γ γ γ γ = I . Let ϕ be the
2 0 5/2 2 n
∞ − −
homomorphism
Γ1(5) C
×
→
γ ω
n
∞ 7→
γ ω 1
−2 7→ n−
γ ,γ 1.
0 5/2
− 7→
The kernel Γ of ϕ is an index n noncongruence character group of Γ1(5). When
n n
n=1,5, Γ is a noncongruence subgroup.
n
6
Let E ,E be the weight 3 Eisenstein series for Γ1(5) as in [LLY05]. A Haupt-
1 2
modul for Γ1(5) is t = EE21 which generates MΓ1(5) and tn = qn EE12 is a Haupt-
modul for Γ . Such a group Γ has two distinguished normalizers in SL (Z):
n n 2
1 5 2 5
ζ = , A= − . In particular
(cid:18)0 1(cid:19) (cid:18)1 2(cid:19)
−
1
E =E , E = E , t = . (10)
1 A 2 2 A 1 A
| | − | −t
We use the following algebraic model for the elliptic modular surface asso-
EΓn
ciated to the group Γ to investigate the roles of the above operators.
n
1+12(tn t3n)+14t2n+t4n
: y2 =tn(x3 n− n n n x
EΓn n − 48t2n
n (11)
1+18(tn t5n)+75(t2n+t4n)+t6n
+ n− n n n n ).
864t3n
n
The actions of A and ζ on are:
EΓn
y ω
2n
A(x,y,t )=( x, , ); (12)
n
− t t
n n
ζ(x,y,t )=(x,y,ω 1t ). (13)
n n− n
Therefore, A and ζ are defined over Q(ω ) and Q(ω ) and have order 4 and n
2n n
respectively. In fact, the A map used in [LLY05] which is defined over Q is a
derivation of the A map here.
Following the notation used before, let
ρ :G Aut(W )
k,l,n Q k,l,n
→
be the l-adic Scholl representation attached to S (Γ ).
k n
ON ATKIN AND SWINNERTON-DYER CONGRUENCE RELATIONS (3) 9
3.2. The space S (Γ ). When n>1, k =3,j (Z/nZ) ,
3 n ×
∈
δ =dim =dim V =2.
p Ql(ωn)Wj Qp(ωn) j
For any positive integer n, let h[jn] = qn E1n−jE2j ∈ Z[1/n][w]],w = e2πiz/5n. By
[ALL05,Prop2.1],S3(Γn)=hh[jn]ijn=−11. Itisstraightforwardtoverifythath[jn] ∈Vj.
According to Theorem 2.11, we have
Theorem 3.1. For every basis element h[n] of S (Γ ) and any prime p > n, h[n]
j 3 n j
satisfies a three-term Atkin and Swinnerton-Dyer type congruence relation at p
given by Char( ,Fr)(Tr) as in Lemma 2.4 where r =O (p).
Wj p n
Toachievethisresult,weonlyneedtheoperatorζ. Whenweconsiderinaddition
the A map, we will obtain the following result using [Ser77, Prop. 19].
Let ρnew :G Aut(Wnew) be the l-adic representationattached to the cusp-
3,l,n Q → 3,l,n
forms genuinely belonging to Γ as before. When n=2,3,4,let B be A, A, and ζ;
n
let K be Q(i),Q(√ 3), and Q(i) respectively. The action of B on Wnew satisfies
− 3,l,n
B2 = 1. Decompose Wnew according to ( i)-eigenspaces of B. As a Galois G -
− 3,l,n ± K
module, Wnew = Wnew Wnew . Let ρnew : G Aut(Wnew ). Then
we have 3,l,n 3,l,n,i ⊕ 3,l,n,−i 3,l,n,±i K → 3,l,n,±i
Proposition 3.2. When n=2,3,4,
ρnew =IndGQ ρnew .
3,l,n GK 3,l,n,±i
Remark 3.3. The above theorem provides a different perspective for the results
in [LLY05]. When n = 2, each ρnew is a one-dimensional representation, hence
3,l,n, i
a character of G . Naturally it c±orresponds to a cuspform η(4z)6 with com-
Q(i)
plex multiplication ([LLY05, section 8]). When n = 3, ρnew is induced from a
3,l,3
2-dimensional representation of G . Such a point of view was suggested by
Q(√ 3)
−
J.P. Serre as one of his comments on [LLY05]. A similar result also holds for the
n=6 case when another derivation of A, called A (14) is used.
e
4. S (Γ )
3 6
When k =3, dim Wnew =2φ(n). Accordingly, only when n=2,3,4, or 6
Ql(ωn) 3,l,n
do we have dim Wnew 4. Since the first three cases have been handled in
Ql(ωn) 3,l,n ≤
[LLY05, ALL05], we will now treat the n = 6 case here. The current case under
consideration shares a lot of similarities with the case n = 4 studied in [ALL05].
To avoid duplication, we will refer the readers to [ALL05] for some arguments. In
this case, we use the following action which is a variation of A:
y i
A(x,y,t ) = ( x, , ). (14)
6
− t t
6 6
e
We use A here since it is defined over a smaller field Q(i) while A is defined over
Q(e2πi/12).
e
Associated with S (Γ ) is a 10-dimensionall-adic Scholl representation [Sch85]
3 6
ρ :Gal(Q/Q) Aut(W ).
3,l,6 3,l,6
→
Inthiscase,wecanpickN =6(c.f. Section4.2). Asbefore,thespaceW
3,l,6⊗Ql
Q (ω ) decomposes naturally into eigenspaces of ζ. Denoted by the Q (ω )-
l 6 j l 6
W
eigenspace of ζ with eigenvalue ωj. Similar to the discussion in Section 5.3 of
6
10 LINGLONG
[ALL05], (resp. ) is isomorphic to the l-adic Scholl representation
3 2 4
W W ⊕W
associated with weight 3 cuspforms for Γ (resp. Γ ) tensoring with Q (ω ). The
2 3 l 6
remaining piece , denoted by Wnew (or simply Wnew), is a representation
W1⊕W5 3,l,6
space of Gal(Q/Q) (c.f. Cor. 2.2). By Prop. 2.5, for every Frobenius element F
p
where p∤6,
Char(Wnew,F )(T) Z[T].
3,l,6 p ∈
Let V be the p-adic Scholl space attached to S (Γ ) with the action of Frobe-
3 6
nius F. We will drop the superscript [6] below for convenience. Decompose V
accordingly into Vold Vnew such that
⊕
h , h , h Vold and Vnew =V V ,
2 3 4 1 5
∈ ⊕
where h V , and h V .
1 1 5 5
∈ ∈
By consideringh =E t n j andthe actions ofζ andAonE ,E , andt , we
j 2 n − 1 2 n
·
obtain
e
h ζ =ωjh , h A= in jh .
j| n j j| − − n−j
It follows e
ζAζ =A. (15)
A straightforwardcomputation revealsethate
FAF 1 =ζ3(p 1)/2A. (16)
− −
e e
4.1. Numerical data. In this section, we will indicate how the results were dis-
coverednumerically.
4.1.1. p-adic side. Since h (resp. h and h ) has been discussedin [LLY05] (resp.
3 2 4
[ALL05]), we will investigate three-term Atkin and Swinnerton-Dyer congruences
for h and h only. Using the “p-adic” Hecke operators mentioned in Section 2.6,
1 5
A.O.L. Atkin observed that for any odd prime p>3,
h =c h , h =c h , when p=1(mod 6) with eigenvalues c and
• 1|Tp · 1 5|Tp 2· 5
c .
2
h = c h , h = c h , when p = 5(mod 6), where c = c if
• 1|Tp · 5 5|Tp 2 · 1 2
p = 1 mod 4 (resp. c = c if p = 1 mod 4). Hence h h (resp.
2 1 5
− − ±
h ih ) are eigenfunctions with eigenvalues √c c .
1 5 2
± ± ·
Here, s(p)=1, if p=1 mod 12 and s(p)= 1 otherwise.
−
In table 1, we list constants c canonically modified as follows:
when p=1 mod 12, we divide c by ( 3)(p 1)/4 mod p= 1.
−
• − ±
Otherwise we divide c by a canonical square root c determined as follows:
1
when p=5 mod 12, c = √ 1= ( 3)(p 1)/4 mod p;
1 −
• ± − − −
when p=7 mod 12, c = √ 3=( 3)(p+1)/4 mod p;
1
• ± − −
when p=11 mod 12, c = √3=( 3)(p+1)/4 mod p.
1
• ± −