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國 立 交 通 大 學

應用數學系

非同餘子群的模型式的同餘性質

Atkin and Swinnerton-Dyer Congruences

Associated to Fermat Curves

研 究 生:林易萱

指導老師:楊一帆 教授

(2)

非同餘子群的模型式的同餘性質

Atkin and Swinnerton-Dyer Congruences

Associated to Fermat Curves

研 究 生:林易萱 Student:Yi-Hsuan Lin

指導教授:楊一帆 Advisor:Yifan Yang

國 立 交 通 大 學

應用數學系

碩 士 論 文

A Thesis

Submitted to Department of Applied Mathematics

College of Science,

National Chiao Tung University

in Partial Fulfillment of the Requirements

for the Degree of

Master

In

Applied Mathematics

June 2011

(3)

非同餘子群的模型式的同餘性質

學生:林易萱 指導老師:楊一帆教授

國立交通大學應用數學系(研究所)碩士班

摘 要

眾所周知的,費馬曲線

是一個與特殊線性群

( )的有限指數子群

相關聯的模曲線,當n不等於1, 2, 4, 8時,

是一個非同餘子群。現在令費馬曲線的虧

格為g,scholl的定理告訴我們,

上權為2的尖點型式與由此曲線相關聯的Tate模所建構

出的2g維l進數伽羅瓦表現會滿足Atkin and Swinnerton-Dyer同餘。

在這篇論文中,我們將會分解伽羅瓦表現,然後給一個更加精確的 Atkin and

Swinnerton-Dyer 同餘。我們將會解決 的情況。

(4)

Atkin and Swinnerton-Dyer Congruences

Associated to Fermat Curves

Student: Yi-Hsuan Lin

Advisor: Yifan Yang

Department (Institute) of Applied Mathematics

National Chiao Tung University

Abstract

It is known that each Fermat curve

is the modular curve associated to some

subgroup

of

( ) of finite index. Moreover if then

is a

noncongruence subgroup. Let g be the genus of the Fermat curve, by Scholl’s theorem,

cuspforms of weight 2 on

, together with the 2g-dimensional l-adic Galois representations

coming from the Tate module associate this curve, satisfy the Atkin and Swinnerton-Dyer

congruence.

In this thesis, we decompose this Galois representation and give a more precise Atkin and

Swinnerton-Dyer congruence. The case

will be completely worked out.

(5)

誌 謝

遙記當年第一次接觸數論,是大三那年在楊一帆老師的基礎數論課堂上,那時的我,

被簡單而漂亮的質數所吸引,或許是這種吸引力,引領著我進入了這個領域。

這篇論文的完成,要感謝所有授我知識的老師,尤其是我的指導教授楊一帆老師,

不僅僅啟發我在數論上的興趣以及傳授我課業上所需要的知識,更因老師的平易近人個

性,讓我們在生活各方面能與老師共同分享。

不能因為要感謝的人太多,就總是謝天,但若有遺漏到誰,麻煩請連絡我,讓我向

您當面致謝以表歉意。首要感謝學姊芳婷無論是對學業或生活上的關心以及帶給我們歡

樂的學習氣氛、學長家瑋為我排憂解難,營造學習氣氛、以及學長耀漢時常督促我們要

努力用功〃再來要感謝遠在英國的摯友廷蓉,時常幫助我解決英文上的困難。還有感謝

光祥陪伴我到清大修代數課程,使我不至於孤軍奮戰。也要感謝在碩士生生涯這兩年間

與我共同學習共同歡樂的定國、冠緯、敏豪、劭芃、聲華、權益,還有已經畢業的葉彬、

文昱等等,是你們豐富了我的人生。

還要感謝吾姊宛儀以及姊夫子濤,雖然你們已經移居香港,但仍時關心我的學業以

及生活。最後我要感謝我的父母、祖母、外祖母以及眾多長輩們,沒有你們的支持,在

這條道路上我無法走得安穩,我衷心感念,

林易萱

謹誌于交通大學

2011年6月

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中 文 提 要

i

英 文 提 要

ii

誌 謝

iii

目 錄

iv

1 Introducti on

1

2 Revi ew of modul ar forms on congruence s ubgroups

2

2.1 Modular forms and cus p form s

2

2.2 Hecke operators

5

2.3 P et ersson inner product

9

2.4 Oldforms and Newform s

1 0

2.5 Hecke ei genform s

11

3 At kin and S winnerton -D yer congruences for noncongruence sub groups

12

3.1 Noncongruence s ubgroups

13

3.2 Atki n and Swinnert on -D yer congruence

13

4 At kin and S winnerton -D yer congruences associ at ed t o Ferm at curves

17

4.1 Ferm at curve

17

4.2 C as e

18

5 References

24

(7)

1.

Introduction

The Fourier coefficients of a normalized newform f = P

n≥1an(f )qn of

weight k, level N , and character χ on a congruence subgroup, where q = e2πiτ, satisfy the recursive relation

anp(f ) − ap(f )an(f ) + χ(p)pk−1an/p(f ) = 0 (1.0.1)

for all prime p, p - N . For noncongruence subgroups, the recursive relation in (1.0.1) no longer holds. Nevertheless, Atkin and Swinnerton-Dyer observed other congruence relations which are introduced in Chapter 3.

Recall that the Fermat curve Fn = xn+ yn = 1 is the modular curve with

genus g =(n−1)(n−2)2 associated to the modular subgroup Γn= * 1 2 0 1 !n , 1 0 2 1 !n , Γ(2)0 +

where Γ(2)0 denotes the commutator subgroup of Γ(2). When n 6= 1, 2, 4, 8, Γn

is noncongruence. Cusp forms of weight 2 can be obtained by differential forms and the parametrization (x, y) = (√n

1 − λ, √n

λ), where λ = θ2(τ )4

θ3(τ )4, described in

section 4.1. By Scholl’s theorem, they satisfy the Atkin and Swinnerton-Dyer congruence with a characteristic polynomial of degree 2g. However, this means that even in the case of the smallest odd prime 3, we are required to figure out at least 32gterms in cusp forms. Therefore, this calculation is not a simple task.

In order to reduce difficulty, in Chapter 4, we decompose Scholl’s 2g-dimensional Galois representations into pieces for the case n = 6, and give a more precise Atkin and Swinnerton-Dyer congruence.

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2.

Review of modular forms on congruence

sub-groups

§ 2.1 Modular forms and cusp forms

The modular group is the group of 2 × 2 matrices with integer entries and determinant 1, SL2(Z) = ( a b c d ! : a, b, c, d ∈ Z, ad − bc = 1 ) The modular group is generated by the two matrices

1 1 0 1 ! and 0 −1 1 0 ! .

Each element of the modular group is also viewed as an automorphism (in-vertible self-map) of the Riemann sphere bC = C ∪ {∞}, the fractional linear transformation a b c d ! (τ ) = aτ + b cτ + d, τ ∈ bC.

This is understood to mean that if c 6= 0 then −d/c maps to ∞ and ∞ maps to a/c, and if c = 0 then ∞ maps to ∞. The identity matrix I and its negative −I both give the identity transformation, and more generally each pair ±γ of matrices in SL2(Z) gives a single transformation. The group of transformations

defined by the modular group is generated by the maps described by the two matrix generators,

τ 7→ τ + 1 and τ 7→ −1/τ The upper half plane is

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The formula Im(γ(τ )) = Im(τ ) |cτ + d|2, γ = a b c d ! ∈ SL2(Z)

shows that if γ ∈ SL2(Z) and τ ∈ H then also γ(τ ) ∈ H, i.e., the modular group

maps the upper half plane back to itself. In fact the modular group acts on the upper half plane, meaning that I(τ ) = τ where I is the identity matrix and (γγ0)(τ ) = γ(γ0(τ )) for all γ, γ0 ∈ SL2(Z) and τ ∈ H.

Let N be a positive integer. The principal congruence subgroup of level N is Γ(N ) = ( a b c d ! ∈ SL2(Z) : a b c d ! ≡ 1 0 0 1 ! (mod N ) ) In particular Γ(1) = SL2(Z). Being the kernel of the natural homomorphism

SL2(Z) → SL2(Z/N Z), the subgroup Γ(N ) is normal in SL2(Z). In fact the

map is a surjection, inducing an isomorphism

SL2(Z)/Γ(N ) ˜−→ SL2(Z/N Z).

This shows that [SL2(Z) : Γ(N )] is finite for all N . Specifically, the index is

[SL2(Z) : Γ(N )] = N3 Y p|N  1 − 1 p2 

where the product is taken over all prime divisors of N .

Definition 2.1. A subgroup Γ of SL2(Z) is a congruence subgroup if Γ(N ) ⊂

Γ for some N ∈ N, in which case Γ is a congruence subgroup of level N . If Γ does not contain Γ(N ) for any N , then we say Γ is a noncongruence subgroup.

Every congruence subgroup Γ has finite index in SL2(Z). Besides the

prin-cipal congruence subgroups, the most important congruence subgroups are Γ0(N ) = ( a b c d ! ∈ SL2(Z) : a b c d ! ≡ ∗ ∗ 0 ∗ ! (mod N ) )

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(where “∗” means “unspecified”) and Γ1(N ) = ( a b c d ! ∈ SL2(Z) : a b c d ! ≡ 1 ∗ 0 1 ! (mod N ) ) satisfying Γ(N ) ⊂ Γ1(N ) ⊂ Γ0(N ) ⊂ SL2(Z)

Two pieces of notation are essential before we continue. For any matrix γ = a b

c d !

∈ SL2(Z) define the factor of automorphy j(γ, τ ) ∈ C for τ ∈ H

to be

j(γ, τ ) = cτ + d

and for γ ∈ SL2(Z) and any integer k define the weight-k operator [γ]k on

functions f : H −→ C by

(f [γ]k)(τ ) = j(γ, τ )−kf (γ(τ )), τ ∈ H

Since the factor of automorphy is never zero or infinity, if f is meromorphic then f [γ]k is also meromorphic and has the same zeros and poles as f .

Definition 2.2. Let Γ be a congruence subgroup of SL2(Z) and let k be an

integer. A function f : H −→ C is a modular form of weight k with respect to Γ if

(1) f is holomorphic,

(2) f is weight-k invariant under Γ,

(3) f [α]k is holomorphic at ∞ for all α ∈ SL2(Z).

If in addition,

(4) a0= 0 in the Fourier expansion of f [α]k for all α ∈ SL2(Z),

then f is a cusp form of weight k with respect to Γ. The modular forms of weight k with respect to Γ are denoted Mk(Γ), the cusp forms Sk(Γ).

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§ 2.2 Hecke operators

Let Γ1 and Γ2 be congruence subgroups of SL2(Z). Then Γ1 and Γ2 are

subgroups of GL+2(Q), the group of 2 × 2 matrices with rational entries and positive determinant. For each α ∈ GL+2(Q) the set

Γ1αΓ2= {γ1αγ2 : γ1∈ Γ1, γ2∈ Γ2}

is a double coset in GL+2(Q). Under a definition to be developed in this section, such double cosets transform modular forms with respect to Γ1 into

modular forms with respect to Γ2.

The group Γ1 acts on the double coset Γ1αΓ2 by left multiplication,

par-titioning it into orbits. A typical orbit is Γ1β with representative β = γ1αγ2,

and the orbit space Γ1\Γ1αΓ2 is thus a disjoint union S Γ1βj for some choice

of representatives βj. The next two lemmas combine to show that this union is

finite.

Lemma 2.3 ([2] Lemma 5.1.1). Let Γ be a congruence subgroup of SL2(Z) and

let α be an element of GL+2(Q). Then α−1Γα ∩ SL

2(Z) is again a congruence

subgroup of SL2(Z).

Lemma 2.4 ([2] Lemma 5.1.2). Let Γ1 and Γ2 be congruence subgroups of

SL2(Z), and let α be an element of GL+2(Q). Set Γ3= α−1Γ1α ∩ Γ2, a subgroup

of Γ2. Then left multiplication by α,

Γ2−→ Γ1αΓ2 given by γ27→ αγ2,

induces a natural bijection from the coset space Γ3\Γ2to the orbit space Γ1\Γ1αΓ2.

In concrete terms, {γ2,j} is a set of coset representatives for Γ3\Γ2 if and only

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We say that two subgroups H1 and H2 of a group G are commensurable,

if the indices [H1: H1∩ H2] and [H2: H1∩ H2] are finite.

Theorem 2.5. Any two congruence subgroups Γ1 and Γ2 of SL2(Z) are

com-mensurable.

Proof. First we know [SL2(Z) : Γ(N )] = N3

Y p|N  1 − 1 p2  is finite. Consider for any subgroups Γ1, Γ2 of SL2(Z), take N1, N2 ∈ N such that Γ(N1) ⊂ Γ1

and Γ(N2) ⊂ Γ2, and let N3= lcm(N1, N2), then we have

Γ(N3) ⊂ Γ(N1) ∩ Γ(N2) ⊂ Γ1∩ Γ2

which implies [SL2(Z) : Γ(N3)] ≥ [Γ1 : Γ(N3)] ≥ [Γ1 : Γ(N1) ∩ Γ(N2)] ≥

[Γ1: Γ1∩ Γ2]

Similarly, we can prove [Γ2: Γ1∩ Γ2] is finite.

In particular, since α−1Γ1α ∩ SL2(Z) is a congruence subgroup of SL2(Z)

by Lemma 2.3, the coset space Γ3\Γ2 in Lemma 2.4 is finite and hence so is

the orbit space Γ1\Γ1αΓ2. With finiteness of the orbit space established, the

double coset Γ1αΓ2 can act on modular forms.

Now for β ∈ GL+2(Q) and k ∈ Z, and τ ∈ H, extend the formula j(β, τ ) = cτ + d to β ∈ GL+2(Q), and extend the weight-k operator to GL+2(Q) which called the weight-k β operator by the rule

(f [β]k)(τ ) = (detβ)k−1j(β, τ )−kf (β(τ )), for f : H → C

Definition 2.6. For congruence subgroups Γ1 and Γ2 of SL2(Z) and α ∈

GL+2(Q), the weight-k Γ1αΓ2operator takes functions f ∈ Mk(Γ1) to

f [Γ1αΓ2]k=

X

j

f [βj]k

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Now we introduces two operators on Mk(Γ1(N )). Consider the map Γ0(N ) −→ (Z/N Z)∗ taking a b c d ! to d (mod N )

is a surjective homomorphism with kernel Γ1(N ). This shows that Γ1(N ) is

normal in Γ0(N ) and induces an isomorphism

Γ0(N )/Γ1(N ) ˜−→(Z/NZ)∗ where

a b c d

!

to d (mod N )

To define the first type of Hecke operator, take any α ∈ Γ0(N ), set Γ1 =

Γ2 = Γ1(N ), and consider the weight-k double coset operator [Γ1αΓ2]k. Since

Γ1(N ) C Γ0(N ) this operator translating each function f ∈ Mk(Γ1(N )) to

f [Γ1αΓ2]k = f [α]k, α ∈ Γ0(N ),

again in Mk(Γ1(N )). Thus the group Γ0(N ) acts on Mk(Γ1(N )), and since its

subgroup Γ1(N ) acts trivially, this is really an action of the quotient (Z/N Z)∗.

The action of α determined by d (mod N ) and denoted hdi, is hdi : Mk(Γ1(N )) −→ Mk(Γ1(N ))

given by

hdif = f [α]k for any α =

a b c δ

!

∈ Γ0(N ) with δ ≡ d (modN )

This type of Hecke operator is also called a diamond operator. Now we are going to define the second type of Hecke operator, again Γ1 = Γ2 = Γ1(N ),

but now α = 1 0 0 p

!

, where p is a prime, we define a weight-k double coset operator

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is given by Tpf = f [Γ1(N ) 1 0 0 p ! Γ1(N )]k.

The double coset here is Γ1(N ) 1 0 0 p ! Γ1(N ) = ( γ ∈ M2(Z) : γ ≡ 1 ∗ 0 p ! (mod N ), detγ = p ) , so in fact 1 0 0 p !

can be replaced by any matrix in this double coset in the definition of Tp.

Proposition 2.7 ([2] Proposition 5.2.4). Let d and e be elements of (Z/N Z)∗,

and let p and q be prime. Then (1)hdiTp= Tphdi

(2)hdihei = heihdi = hdei (3)TpTq = TqTp

Now we can extend the definitions of hdi and Tpto hni and Tnfor all n ∈ Z+.

For n ∈ Z+

with (n, N ) = 1, hni is determined by n (mod N ). For n ∈ Z+

with (n, N ) > 1, define hni = 0, the zero operator on Mk(Γ1(N )). The mapping

n 7→ hni is totally multiplicative.

To define Tn, set T1 = 1 (the identity operator); Tp is already defined for

primes p. For prime powers, define inductively

Tpr= TpTpr−1− pk−1hpiTpr−2, for r ≥ 2,

and note that inductively on r and s starting from Proposition 2.7(c), TprTqs =

TqsTpr for distinct primes p and q. Extend the definition multiplicatively to Tn

for all n, Tn= Y Tpei i where n = Y pei i

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so that the Tn all commute by Proposition 2.7 and

Tnm= TnTm if (n, m) = 1.

Theorem 2.8 ([2] Proposition 5.3.1). Let f ∈ Mk(Γ1(N )) have Fourier

expan-sion f (τ ) = ∞ X m=0 am(f )qm where q = e2πiτ.

Then for all n ∈ Z+, T

nf has Fourier expansion

(Tnf )(τ ) = ∞ X m=0 am(Tnf )qm where am(Tnf ) = X d|(m,n) dk−1amn/d2(hdif ). (2.8.1) In particular, if f ∈ Mk(N, χ) then am(Tnf ) = X d|(m,n) χ(d)dk−1amn/d2(f ). (2.8.2)

§ 2.3 Petersson inner product

In this section, we make the space of cusp forms Sk(Γ) into an inner product

space, the integral in the following definition is well defined and convergent. Definition 2.9. Let Γ ⊂ SL2(Z) be a congruence subgroup. The Petersson

inner product, h, iΓ: Sk(Γ) × Sk(Γ) −→ C, is given by hf, giΓ= 1 VΓ Z X(Γ) f (τ )g(τ )(Im(τ ))kdµ(τ ). where VΓ is the volume of X(Γ) and dµ(τ ) = dxdyy2 for τ = x + iy.

This product is linear in f , conjugate linear in g, Hermitiansymmetric, and positive definite. The normalizing factor 1/VΓ ensures that if Γ

0

⊂ Γ then h, i0Γ= h, iΓ on Sk(Γ).

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§ 2.4 Oldforms and Newforms

So far the theory has all taken place at one generic level N . This section begins results that move between levels, taking forms from lower levels M |N up to level N , mostly with M = N p−1 where p is some prime factor of N .

Lemma 2.10. If M |N then Sk(Γ1(M )) ⊂ Sk(Γ1(N ))

Proof. If M |N , we have Γ1(N ) ⊂ Γ1(M ) since for any γ ∈ Γ1(N ), write γ =

   k1N + 1 ∗ k2N k3N + 1  

, and write N = lM for some integer l, then γ =    k1lM + 1 ∗ k2lM k3lM + 1   , hence r ∈ Γ1(M ).

Now if f is a modular form with respect to Γ1(M ), it is also a modular form

with respect to Γ1(N ) since Γ1(N ) ⊂ Γ1(M ).

Lemma 2.11. For any h factor of N/M , let αh=

   h 0 0 1   , so that (f [αh]k)(τ ) = hk−1

f (hτ ) for f : H −→ C. The linear map [αh]ktakes Sk(Γ1(M )) to Sk(Γ1(N )),

lifting the level from M to N . Proof. Let γ =    aN + 1 b cN dN + 1   ∈ Γ1(N ). We have hγτ = (aN + 1)(hτ ) + hb (cN/h)(hτ ) + dN + 1 =    aN + 1 hb cN/h dN + 1   (hτ ) By h is a factor of N/M , we have γ0 =    aN + 1 hb cN/h dN + 1    is in Γ1(M ). Therefore f (hγτ ) = f (γ0(hτ )) = (cN τ + dN + 1)kf (hτ ). This shows g(τ ) = f (hτ ) is a cusp form on Γ1(N ).

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Combining preceding two lemmas, it is natural to distinguish the part of Sk(Γ1(N )) coming from lower levels.

Definition 2.12. For each divisor d of N , let id be the map

id: (Sk(Γ1(N d−1)))2−→ Sk(Γ1(N ))

given by

(f, g) 7→ f + g[αd]k.

The subspace of oldforms at level N is Sk(Γ1(N ))old=

X

p|N prime

ip((Sk(Γ1(N d−1)))2)

and the subspace of newforms at level N is the orthogonal complement with respect to the Petersson inner product,

Sk(Γ1(N ))new= (Sk(Γ1(N ))old)⊥.

§ 2.5 Hecke eigenforms

In this section, we will show if f ∈ M(N, χ) is a normalized eigenform, then its Fourier coefficients will satisfy the recursive relation apr(f ) = ap(f )apr−1(f )−

χ(p)pk−1apr−2(f ) for all p prime and r ≥ 2.

Definition 2.13. Let f be a non-vanishing modular form. If f is a simutaneous eigenfunction for all Hecke operator Tn, then we say f is a Hecke eigenform.

If the Fourier expansion of f has leading coefficient 1, then f is normalized. Definition 2.14. Let χ be a Dirichlet character modulo N , we define the χ-eigenspace of Mk(Γ1(N )) by

Mk(N, χ) = {f ∈ Mk(Γ1(N )) : f [γ]k= χ(dγ)f f or all γ ∈ Γ0(N )} ,

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Theorem 2.15. Let f ∈ Mk(N, χ). Then f is a normalized eigenform if and

only if its Fourier coefficients satisfy the conditions (1) a1(f ) = 1,

(2) apr(f ) = ap(f )apr−1(f ) − χ(p)pk−1apr−2(f ) for all p prime and r ≥ 2,

(3) amn(f ) = am(f )an(f ) when (m, n) = 1.

Proof. The only if part is follows from the definition of Tn. Now we prove the

other way. Suppose f satisfies the three conditions. Then f is normalized, and to be an eigenform for all the Hecke operators it need only satisfy am(Tpf ) =

ap(f )am(f ) for all p prime and m ∈ Z+. If p - m then formula (2.8.2) gives

am(Tpf ) = apm(f ) and by the third condition this is ap(f )am(f ) as desired.

On the other hand, if p|m write m = prm0

with r ≥ 1 and p - m0. This time am(Tpf ) = apr+1m0(f ) + χ(p)pk−1apr−1m0(f ) by formula (2.8.2)

= (apr+1(f ) + χ(p)pk−1apr−1(f ))am0(f ) by the third condition

= ap(f )apr(f )am0(f ) by the second condition

= ap(f )am(f ) by the third condition.

3.

Atkin and Swinnerton-Dyer congruences for

noncongruence subgroups

Last section we have develop some properties of the modular forms for con-gruence subgroups. Given a cuspidal normalized newform g =P

n≥1an(g)q n,

where q = e2πiτ, of weight k ≥ 2 level N and character χ, the Fourier coefficients

of g satisfy the recursive relation

anp(g) − ap(g)an(g) + χ(p)pk−1an/p(g) = 0 (3.0.1)

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The following sections will introduce the substitution of the recursive relation for noncongruence subgroups.

§ 3.1 Noncongruence subgroups

Let f = P

n≥n0anw

n be the modular form with coefficients a

n in a fixed

number field. According to Hecke operators, a basis consisting of forms with integral coefficients exists in each space of holomorphic congruence modular forms. Consequently, for every congruence holomorphic modular form with algebraic coefficients, the sequence {an} has bounded denominators in the sense

that there exists an algebraic number M such that M an is algebraic integral

for all n. Therefore, the sequence {bn} having unbounded denominators implies

g =P

n≥n0bnw

n is noncongruence.

Some other distinctions between congruence and noncongruence subgroups are demonstrated in [5].

§ 3.2 Atkin and Swinnerton-Dyer congruence

Before we state the Atkin and Swinnerton-Dyer congruences conjecture, let us introduce a model of a modular curve over Q.

Let H be the upper half plane {τ ∈ C : Im(τ ) > 0}, and H∗ denotes the compactified half plane H ∪ P1

(Q).

Definition 3.1. Let Γ be a subgroup of SL2(Z) of finite index. Consider the

compactified quotient space Γ\H∗, and the canonical map Γ\H∗→ Γ(1)\H∗.

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(1)a nonsingular projective curve V /Q; (2)a finite morphism π : V → P1Q; (3)a point e ∈ V (Q); and

(4)an isomorphism φ : Γ\H∗−→V (C) such that φ(i∞) = e and the diagram˜

Γ\H∗ ' φ  // Γ(1)\H∗ ' j  V (C) πC // P1(C)

commutes (where here j is the usual modular invariant of level 1).

As explained in [1][6][7], there exists a subfield L of K, an element κ ∈ K with κµ∈ L, where µ is the width of the cusp ∞, and a positive integer M such

that κµ is integral outside M and S

k(Γ) has a basis consisting of M -integral

forms. Here a form f of Γ is called M -integral if in its Fourier expansion at the cusp ∞

f (τ ) =X

n≥1

an(f )qn/mu,

the Fourier coefficients an(f ) can be written as κncn(f ) with cn(f ) lying in the

ring OL[1/M ], where OL denotes the ring of integers of L.

Conjecture 3.2. (Atkin and Swinnerton-Dyer congruences). Suppose that the modular curve XΓ has a model over Q in the sense of Definition 3.1. There

exist a positive integer M and a basis of Sk(Γ) consisting of M -integral forms

fj, 1 ≤ j ≤ d, such that for each prime p not dividing M , there exists a

nonsin-gular d × d matrix (λi,j) whose entries are in a finite extension of Qp, algebraic

integers Ap(j), 1 ≤ j ≤ d, with |σ(Ap(j))| ≤ 2p(k−1)/2 for all embeddings σ ,and

characters χj unramified outside M so that for each j the Fourier coefficients

of hj := Piλi,jfi satisfy the congruence relation

ordp(anp(hj) − Ap(j)an(hj) + χj(p)pk−1an/p(hj)) ≥ (k − 1)(1 + ordpn)

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for all n ≥ 1; or equivalently, for all n ≥ 1,

(anp(hj) − Ap(j)an(hj) + χj(p)pk−1an/p(hj))/(np)k−1

is integral at all places dividing p.

In other words, the recursive relation (3.0.1) on Fourier coefficients of modu-lar forms for congruence subgroups is replaced by the congruence relation (3.2.1) for forms of noncongruence subgroups.

Theorem 3.3 (Scholl). Suppose that XΓhas a model over Q as before. Attached

to Sk(Γ) is a compatible family of 2d-dimensional l-adic representations ρlof the

Galois group Gal( ¯Q/Q) unramified outside lM such that for primes p > k + 1 not dividing M l, the following hold.

(1) The characteristic polynomial Hp(T ) =

X

0≤r≤2d

Br(p)T2d−r

of ρl(F robp) lies in Z[T ] and is independent of l, and its roots are algebraic

integers with absolute value p(k−1)/2;

(2) For any M -integral form f in Sk(Γ), its Fourier coefficients an(f ), n ≥ 1,

satisfy the congruence relation

ordp(anpd(f ) + B1(p)anpd−1(f ) + ... + B2d−1(p)an/pd−1(f ) + B2d(p)an/pd(f ))

≥ (k − 1)(1 + ordpn)

for n ≥ 1.

Remark 3.4. When k = 2, the 2d-dimensional representation of Gal( ¯Q/Q) can be presented explicitly by considering the Tate module of the Jacobian of XΓ (See [9] for the definition of Tate module).

Definition 3.5. The two forms f and g above are said to satisfy the Atkin and Swinnerton-Dyer congruence relations if, for all primes p not dividing M N and for all n ≥ 1,

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is integral at all places dividing p.

The following are two examples satisfy the Atkin and Swinnerton-Dyer con-gruence relations.

Example 3.6. For the noncongruence subgroup Γ711 studied in [1], the space

S4(Γ711) is 1-dimensional. Let f be a nonzero 14-integral form in S4(Γ711).

Scholl proved in [8] that there is a normalized newform g of weight 4 level 14 and trivial character such that f and g satisfy the Atkin and Swinnerton-Dyer congruence relations.

Example 3.7. An another example is demonstrated in [4]. Let Γ be the index 3 noncongruence subgroup of Γ1(5) such that the widths at two cusps ∞ and

−2 are 15.

(1) Then XΓhas a model over Q, κ = 1, and the space S3(Γ) is 2-dimensional

with a basis consisting of 3-integral forms f+(τ ) = q1/15+ iq2/15− 11 3 q 4/15− i16 3 q 5/154 9q 7/15+ i71 9 q 8/15 +932 81 q 10/15+ O(q11/15), f−(τ ) = q1/15− iq2/15− 11 3 q 4/15+ i16 3 q 5/15 −4 9q 7/15 − i71 9 q 8/15 +932 81 q 10/15+ O(q11/15),

(2) The 4-dimensional l-adic representation ρl of Gal( ¯Q/Q) associated to S3(Γ) constructed by Scholl is modular. More precisely, there are two

cuspidal newforms of weight 3 level 27 and character χ−3 given by

g+(τ ) = q − 3iq2− 5q4+ 3iq5+ 5q7+ 3iq8+ 9q10+ 15iq11− 10q13− 15iq14

− 11q16− 18iq17− 16q19− 15iq20+ 45q22+ 12iq23+ O(q24),

g−(τ ) = q + 3iq2− 5q4− 3iq5+ 5q7− 3iq8+ 9q10− 15iq11− 10q13+ 15iq14

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such that over the extension by joining√−1, ρldecomposes into the direct

sum of the two λ-adic representations attached to g+ and g−, where λ is

a place of Q(i) dividing l.

(3) f+ and g+ (resp. f− and g−) satisfy the Atkin and Swinnerton-Dyer

congruence relations.

4.

Atkin and Swinnerton-Dyer congruences

as-sociated to Fermat curves

§ 4.1 Fermat curve

For a positive integer n, let Fn denote the Fermat curve xn+ yn = 1 of

degree n. There are some properties of Fermat curves.

Lemma 4.1. For n ≥ 1, the genus of Fn is (n − 1)(n − 2)/2, and for n ≥ 3, a

basis for the space of holomorphic 1-form is ωi,j=

xidx

yj+2, 0 ≤ i ≤ j ≤ n − 3.

As shown in [10], we have following two lammas.

Lemma 4.2. The Fermat curve Fn is the modular curve associated to the group

Γn generated by    1 2 0 1    n ,    1 0 2 1    n , Γ(2)0, where Γ(2)0 denotes the commutator subgroup of Γ(2).

Moreover, let θ2(τ ) = X n∈Z q(2n+1)2/8, θ3(τ ) = X n∈Z qn2/2, θ4(τ ) = X n∈Z (−1)nqn2/2, and λ = θ4

2/θ34. Then the Fermat curve xn+ yn = 1 is parameterized by (x, y) =

(√n

1 − λ, √n

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Lemma 4.3. If n 6= 1, 2, 4, 8, then Γn is a noncongruence subgroup.

Let ζ = e2πi/n and µ

n be the group of nth root of unity. The group G =

µn× µn acts on Fn by (ζi, ζj) : (x, y) 7→ (ζix, ζjy). Let

σ : (x, y) 7→ (ζx, y), τ : (x, y) 7→ (x, ζy).

Assume that H is a subgroup of G. We consider the quotient curve Fn/H. The

pullbacks of holomorphic 1-forms on Fn/H will be holomorphic 1-forms on Fn

that are invariant under the action of H. Say, ωi,j = xidx/yj+2 is invariant

under the action of H. Using the parameterization given in Lemma 4.2, we get a cusp form fi,j= xiqdx/dq yj+2 = ∞ X k=1 akqk/2n

on Γn. On the other hand, we may consider the L-function L(s, Fn/H), i.e.,

the L-function of the Galois representation ρFn/H of Gal( ¯Q/Q) attached to the

algebraic curve Fn/H. (We assume for the moment that Fn/H is always defined

over Q for all H and n).

§ 4.2 Case x

6

+ y

6

= 1

Noticing that λ = 16q1/2+ · · · , we slightly modify the Fermat curve and

consider the curve

xn+ 16yn= 1

instead(so that the cusp form fi,j = xiqdx/yj+2dq has rational Fourier

coeffi-cient). We shall still let Fn denote this curve. Also we let

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where ζ = e2πi/n. Note that a differential form xidx/yj+2 is fixed by σaτb if and only if

(i + 1)a − (j + 2)b ≡ 0 mod 6.

The following table lists the subgroup Hi,j of G = µ6 × µ6 that fixes ωi,j.

ω0,0 ω0,1 ω1,1 ω0,2 ω1,2 ω2,2 ω0,3 ω1,3 ω2,3 ω3,3

hσ2τ i 3τ i 3τ2i 4τ i 2τ, σ3i 2τ3i 5τ i 5τ2i hστ3i hστ2i

We now work out the equations for the curves F6/Hi,j.

Lemma 4.4. We have

group differential forms equation hσ2τ i ω 0,0, ω1,2 v2= u6+ 1 hσ3τ i ω 0,1 v2= u3− 1 hσ4τ i ω 0,2 v2= u3+ 4 hσ5τ i ω 0,3, ω1,2 v2= u6− 1 hστ2i ω 1,2, ω3,3 v2= u6+ 1 hσ3τ2i ω 1,1 v2= u3+ 1 hσ5τ2i ω 1,3 v2= u3+ 16 hσ2τ3i ω 2,2 v2= u3+ 4 hστ3i ω 2,3 v2= u3− 16 hσ2τ, σ3i ω 1,2 v2= u3+ 1

Proof. Here we prove the case hσ2τ i. Consider both of xy4 and y6 are fixed by hσ2τ i, and the mapping (x, y) 7→ (xy4, y6) is 6-to-1. Thus, xy4and y6generate

the subfield of the function field of F6 that is fixed by hσ2τ i and an equation

for F6/hσ2τ i is given by the relation

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between U = xy4 and V = y6. Now the curve U6 = V4− 16V5 is birationally

equivalent to v2= u6+ 1 with the birational maps u = 2V U , v = V2(8V − 1) U3 , U = u2 4(u3− v), V = u3 8(u3− v)

This proves the case hσ2τ i.

Remark 4.5. We can compute the genus of F6/H using the Riemann-Hurwitz

formula. Taking H = hσ2τ i for example. For the affine part of F

6, the covering

F6→ F6/H is unramified at those points of F6where Pj(ζ2jx, ζjy), j = 0, . . . , 5

are 6 distinct points. If y 6= 0, then the six points are distinct. At those points, the covering is unramified. On the other hand, if y = 0, then P0= P3, P1= P4,

P2 = P5. The covering is ramified at those points with ramification index 2.

There are totally 6 such points (ζk, 0), k = 0, . . . , 5. Thus, the contribution from the affine part to the total branch number is 6. The infinity part of F6

consist of 6 points Qj= (ζj+1/2: 1 : 0) We have

σ2τ (Qj) = (ζj+5/2: ζ : 0) ∼ (ζj+3/2: 1 : 0) = Qj+1

Therefore, the covering is unramified at the 6 infinity points, and the total branch number is 6. By the Riemann-Hurwitz formula, if g is the genus of F6/hσ2τ i,

then

10 − 1 = 6(g − 1) +6 2.

Hence, we conclude that the genus of F6/hσ2τ i is 2 and the subspace of

dif-ferential 1-forms on F6 that are invariant under hσ2τ i should have dimension

2.

Theorem 4.6. The genus of Fn/H for a cyclic subgroup H = hσaτbi of µn×µn

with a, b are relative primes is

g = n − da− db− d(a−b) 2 + 1 where dx is the greatest common divisor of x and n.

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Proof. By the Riemann-Hurwitz formula, we only need to verify the total branch number B is n(da+ db+ d(a−b)− 3).

For the affine part of Fn, the covering Fn → Fn/H is unramified at those

point of Fn where Pj = (ζajx, ζbjy), j = 0, . . . , n − 1 are n distinct points. If

x 6= 0 and y 6= 0, since a, b are relative primes, we know the n points are distinct. At those points, the covering is unramified. On the other hand, if x = 0, then P0 = Pn/db = . . . = P(db−1)n/db, P1 = Pn/db+1 = . . . = P(db−1)n/db+1, . . .,

Pn/db−1 = Pn/db+n/db−1 = . . . = P(db−1)n/db+n/db−1. The covering is ramified

at those points with ramification index db. There are totally n such points.

Similarly, we can determine the case y = 0. Thus, the contribution from the affine part to the total branch number is n(da− 1) + n(db− 1). The infinity part

of Fn consist of n points Qj = (ζj+1/2: 1 : 0) We have

σaτb(Qj) = (ζj+a+1/2: ζb: 0) ∼ (ζj+(a−b)+1/2: 1 : 0) = Qj+(a−b)

Replaces a − b by a − b mod n if necessary. Therefore, the ramification index of the covering is d(a−b), and the total branch number of the infinity part is

n(da−b − 1). Sum up the total branch numbers of the affine part and the

infinity part, we have B = n(da+ db+ d(a−b)− 3).

Lemma 4.7. The L-functions for the curves in Lemma 4.4 are equation L-function v2= u3+ 16 L(s, f27) v2= u3+ 1 L(s, f36) v2= u3+ 4 L(s, f 108) v2= u3− 1 L(s, f 36⊗ χ−4) v2= u3− 16 L(s, f27⊗ χ−4) v2= u6+ 1 L(s, f 36)2 v2= u6− 1 L(s, f 36)L(s, f36⊗ χ−4) Here f27(τ ) = η(3τ )2η(9τ )2, f36(τ ) = η(6τ )4

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Remark 4.8. The modular forms f27, f36, f108 have the following description

in terms of Hecke characters.

Let K = Q(√−3) and ζ = e2πi/6. The ring of integers O

K is Z + Zζ. Let

m = 3 and define χ as follows. If a + bζ ∈ OK is not relatively prime to 3, we

let χ(a + bζ) = 0. For each a + bζ in OK relatively prime to m, there exists

a unique integer j with 0 ≤ j < 6 such that a + bζ ≡ ζj mod m. We set

χ(a + bζ) = ζ−j(a + bζ). That is,

(a, b) mod 3 (0, 1) (0, 2) (1, 0) (1, 2) (2, 0) (2, 1) χ(a + bζ)/(a + bζ) ζ5 ζ2 1 ζ −1 ζ4 Then f27(τ ) = 1 6 X a+bζ∈OK χ(a + bζ)qa2+ab+b2. For f36, we let m = 2 √

−3 and define χ as follows. If a + bζ ∈ OK is not

relatively prime to m, we set χ(a + bζ) = 0. For each a + bζ in OK that is

relatively prime to 2√−3, there exists a unique integer j with 0 ≤ j < 6 such that a + bζ ≡ ζj mod m. We set χ(a + bζ) = ζ−j(a + bζ) Then

f36(τ ) = 1 6 X a+bζ∈OK χ(a + bζ)qa2+ab+b2.

Proof. The only parts that requires a proof are v2 = u6+ 1 and v2 = u6− 1.

Here we consider the case v2 = u6− 1. Let x = u2 and y = v. Then we have

v2 = u3− 1. In other words, we have a two-fold cover from v2 = u6− 1 to

y2= x3− 1. Likewise, let x = −1/u2and y = v/u3. We have y2= x3+ 1. Then

L(s, v2− u6+ 1) = L(s, f

36)L(s, f36⊗ χ−4).

Theorem 4.9. The cusp forms fi,j= xiy−j−2qdx/dq satisfy the ASD

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fi,j L-function f0,0 L(s, f36) f0,1 L(s, f36⊗ χ−4) f1,1 L(s, f36) f0,2 L(s, f108) f1,2 L(s, f36) f2,2 L(s, f108) f0,3 L(s, f36⊗ χ−4) f1,3 L(s, f27) f2,3 L(s, f27⊗ χ−4) f3,3 L(s, f36) In fact, we find f0,0(τ ) = f36(2τ /3), f1,2(τ ) = f36(τ /3), f3,3(τ ) = f36(τ /6), f0,3(τ ) = f36⊗ χ−4(τ /6). Also, f0,1(2τ ) = q + 4 3q 310 9 q 540 81q 7553 243q 93740 729 q 11+ · · · , f1,1(2τ ) = q − 4 3q 310 9 q 5+40 81q 7553 243q 9+3740 729 q 11+ · · · , f0,2(3τ ) = q + 8 3q 44 9q 7320 81 q 10154 243q 133328 729 q 16+ · · · , f2,2(3τ ) = q − 8 3q 44 9q 7+320 81 q 10154 243q 13+3328 729 q 16+ · · · , f1,3(6τ ) = q + 4 3q 746 9 q 13472 81 q 19+1985 243 q 25+3532 729 q 31+ · · · , f2,3(6τ ) = q − 4 3q 746 9 q 13+472 81 q 19+1985 243 q 253532 729 q 31+ · · · .

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5.

References

[1] A.O.L. Atkin, H.P.F. Swinnerton-Dyer, Modular forms On non-congruence subgroups, Combinatorics (Proceedings of the Symposium on Pure Mathe-matics, Vol. XIX, University of California, Los Angeles, CA, 1968), Amer-ican Mathematical Society, Providence, RI, 1971, pp. 1-25.

[2] F. Diamond J. Shurman, A First Course in Modular Forms, Springer, 2005. [3] K.-I Hashimoto, L. Long, Y. Yang, Jacobsthal identity for Q(√2), Forum

Mathematicum, doi:10.1515/FORM.2011.102.

[4] W.-C. W. Li, L. Long, Z. Yang, On Atkin-Swinnerton-Dyer congruence relations, J. Number Theory 113 (2005), no. 1, 117-148.

[5] L. Long, The finite index subgroups of the modular group and their modular forms,Fields institute Communications, American Mathematical Society Volume 54 (2008), pp 83-102.

[6] A.J. Scholl, Modular forms and de Rham cohomology; AtkinVSwinnerton-Dyer congruences, Invent. Math. 79 (1985) 49-77.

[7] A.J. Scholl, Modular forms on noncongruence subgroups. S´eminaire de Th´eorie des Nombres, Paris 1985-86, Progress in Mathematics, Vol. 71, Birkh¨auser, Boston, MA, 1987, pp. 199-206.

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[8] A.J. Scholl, The l-adic representations attached to a certain non-congruence subgroup, J. Reine Angew. Math. 392 (1988) 1-15.

[9] J.H. Silverman, The arithmetic of elliptic curves, volume 106 of Graduate Texts in Mathematics. Springer, Dordrecht, second edition, 2009.

[10] T. Yang, Cusp form of weight 1 associated to Fermat curves, Duke Math J. 83 (1996), 141-156.

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