國 立 交 通 大 學
應用數學系
碩
士
論
文
伽羅瓦表現與模型式
Galois Representations and Modular Forms
研 究 生：黃彥璋
指導老師：楊一帆 教授
伽羅瓦表現與模型式
Galois Represetations and Modular Forms
研 究 生：黃彥璋 Student：
Yan-Jhang Huang
指導教授：楊一帆 教授 Advisor：
Professor Yi-Fan 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 2010
Hsinchu, Taiwan, Republic of China
伽羅瓦表現與模型式
學生：黃彥璋 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
我們想要探討的問題，是如何去尋找一個簡單的方法來討論一個特別
的函數x
4-2 在模掉一個質數p之後解的個數。這是一個跟Hecke L-函數、伽
羅瓦群、群的表現有關的應用問題。
更進一步來說，我們可利用這個多項式解空間的伽羅瓦表現來找出一
個 weight 為 12，level 為 256 的 cusp form。
Galois Representations and Modular Forms
Student：Yan-Jhang Huang Advisor：Professor Yi-Fan Yang
Department of Applied Mathematics
National Chiao Tung University
Degree of Master
ABSTRACT
The problem we want to discuss in this thesis is trying to find a simple description “How
the polynomial splits modulo a prime p for a special polynomial x
4-2.” This is an application
of Hecke L-function, the Galois theorem and the group representation. We will try to connect
them by some well-known knowledge, and use them to solve the problem in our discussion.
Moreover, we will use the Galois representation of the splitting filed of the polynomial
x
4-2 to construct a cusp form of weight 1 with level 256.
誌 謝
首先，在我完成我的碩士論文之後，最要感謝的是我的指導教授楊一
帆老師。感謝他帶我走入這個研究領域，並且給予我許多的學習機會。也
引導我走向更高深、更有趣的數學殿堂。
另外我也要感謝潘戍衍老師以及王千真老師在百忙之中抽空擔任我的
口試委員。
接著我要感謝凃芳婷學姊與林家銘同學。在我研究各方面問題遇上瓶
頸時，他們兩位總是能夠提供我所需要的協助。
最後，我要感謝我的家人，因為有你們的栽培，我才能夠順利完成學
業。
黃彥璋
謹誌于交通大學
2010年7月
iii
目錄
中文提要………
i
英文提要………
ii
誌謝………
iii
目錄………
iv
1.Introduction………
1
1.1 Examples From Serre's Paper………
1
1.1.1 Case n = 2: x
2-x- 1………
1
1.1.2 Case n = 3: x
3-x- 1………
2
1.1.3 Case n = 4: x
4-x- 1………
2
1.1.4 A Small Table Of Np(f) for f = x
n-x-1………
3
1.2 Abstract………
3
2.Basic Knowledge………
5
2.1 Number Fields………
5
2.2 Representations, Characters And Artin L-functions………
6
2.3 Adeles And Ideles………
8
2.4 Hecke Characters………
9
2.5 Modular Forms………
10
2.6 Hecke Operators………
12
3.The detail of Serre’s examples………
15
3.1 Use Cyclic Group To Determine Np(f) ………
15
3.2 Relation Between Np(f) And Legendre Symbols………
16
4. Main Results………
17
4.1 Galois Group Of f Is Isomorphic To D4………
17
4.2 Np(f) Is A Class Function Of Frobenius………
18
4.3 Construct A Cusp Form………
21
Bibliography ……… 25
Chapter 1
Introduction
In 2003, Jean-Pierre Serre gave a paper “On A Theorem Of Jordan”, which interests us. The paper has a part which is talking about the number of roots for a given polynomial f in Z/pZ. Let Np(f ) denotes the number of zeros of f in Z/pZ. In the paper, Serre gave three special examples,
all of them has form f (x) = xn− x − 1 for n = 2, 3, 4. He related the N
p(f ) to the coefficients of
theta series.
1.1
Examples From Serre’s Paper
In this section, we talk about the special case of f (x) = xn− x − 1 for n = 2, 3, 4, and we relate
the numbers Np(f ) to the coefficients of theta series.
1.1.1
Case n = 2: x
2− x − 1
The discriminant of f = x2− x − 1 is 5. The polynomial f has a double root modulo 5, hence
N5(f ) = 1. For p 6= 5, we have
Np(f ) =
(
2, if p ≡ ±1 mod 5 0, if p ≡ ±2 mod 5. If one defines a power series F (q) =P∞
n=0anqn by F = q − q 2 − q3+ q4 1 − q5 = q − q 2− q3+ q4+ q6− q7− q8+ · · · = ∞ X n=1 n 5 qn, the above formula can be restated as
Np(f ) = ap+ 1 for all primes p,
where ap is the p-th term coefficient in the L-function ∞ X n=1 an ns = Y p 1 −p 5 p−s −1 , 1
2 CHAPTER 1. INTRODUCTION which is analytic continued to the complex plane.
1.1.2
Case n = 3: x
3− x − 1
The discriminant of f = x3− x − 1 is −23. The polynomial f has a double root and a simple root
mod 23, hence N23(f ) = 2. For p 6= 23, one has,
Np(f ) =
(
0 or 3, if (23p) = 1 1, if (23p) = −1.
Moreover, in the ambiguous case where (23p) = 1, p can be written either as x2+ xy + 6y2 or as
2x2+ xy + 3y2
with x, y ∈ Z; in the first case, one has Np(f ) = 3; in the second case, one has
Np(f ) = 0. (The smallest p of the form x2+ xy + 6y2is 59 = 52+ 5 × 2 + 6 × 22, hence N59(f ) = 3.)
Let us define a power series F =P∞
n=0anqn by the formula F = q ∞ Y k=1 (1 − qk)(1 − q23k) = 1 2 X x,y∈Z qx2+xy+6y2− X x,y∈Z q2x2+xy+3y2 = q − q2− q3+ q6+ q8− q13− q16+ q23− q24+ · · · .
This is a modular form of weight 1 on Γ0(23) with character −23n . The formula for Np(f ) given
above can be reformulated as,
Np(f ) = ap+ 1 for all primes p.
Note that the coefficients of F are multiplicative, one has amm0 = amam0 if m and m0 are relatively
prime. And the associated Dirichlet series is
∞ X n=1 an ns = Y p 1 −ap ps + p 23 1 p2s −1 . (This equation comes from [6].)
1.1.3
Case n = 4: x
4− x − 1
The discriminant of f (x) = x4− x − 1 is −283. The polynomial f has two simple roots and a
double root modulo 283, hence N283(f ) = 3. If p 6= 283, one has,
Np(f ) =
0 or 4, if p can be written as x2+ xy + 71y2 1, if p can be written as 7x2+ 5xy + 11y2
0 or 2 if (283p ) = −1.
A complete determination of Np(f ) can be obtained via a newform of weight 1 and level 283
as follows F = ∞ X n=1 anqn = q +√−2q2−√−2q3− q4−√−2q5+ 2q6− q7− q9+ 2q10+ q11+ · · · .
1.2. ABSTRACT 3 One has,
Np(f ) = 1 + a2p− (
p
283) for all primes p 6= 283.
1.1.4
A Small Table Of N
p(f ) for f = x
n− x − 1
In the end of this section, we give a small table of Np(f ) for f (x) = xn− x − 1, n = 2, 3, 4.
p n = 2 n = 3 n = 4 2 0 0 0 3 0 0 0 5 1 1 0 7 0 1 1 11 2 1 1 13 0 0 1 17 0 1 2 19 2 1 0 23 0 2 1 .. . ... ... ... 59 2 3 1 .. . ... ... ... 83 0 1 4
1.2
Abstract
The problem we want to discuss in this thesis is trying to find a simple description “How the polynomial splits modulo a prime p for a special polynomial x4− 2.” This is an application of Hecke L-function, the Galois theorem and the group representation. We will try to connect them by some well-known knowledge,and use them to solve the problem in our discussion.
In the further chapters, we will introduce some background, and explain the detail of the example for f (x) = x3− x − 1 , and in the last chapter, we will pick a special polynomial to be
our main subject “f (x) = x4− 2”. Whose spliting field is L = Q(√42, i), and the Galois group
Gal(L/Q) is isomorphic to D4. And for this case, we try to construct a weight 1 cusp form, says
that if the cusp form be written as a Fourier expansion, then the coefficients of prime terms are just the same as the prime terms of an Artin L-function associated with the Galois group Gal(L/Q).
Chapter 2
Basic Knowledge
2.1
Number Fields
Let K be a number field, and L be a Galois extension of K. Let OL, OK denote the ring of integers
in L, K respectively. Let p be a prime of OK, P be a prime of OL lying over p.
Definition 2.1.1. The decomposition group of P is
D(P|p) = {σ ∈ GAL(L/K) : σ(P) = P} , and the inertia group of P is
E(P|p) = {σ ∈ GAL(L/K) : σ(α) ≡ α mod P} ∀α ∈ OL.
Theorem 2.1.2 ([7]). Let L be a Galois extension of K, a prime p is ramified in L if and only if pdivides the discriminant of L.
And for the prime is unramified in L, there is a special proposition for them.
Proposition 2.1.3 ([7]). Let L be a Galois extension of K, p be an unramified prime lies under Pin L. Then there exists a unique automorphism σ ∈ Gal(L/K) such that
σ(a) ≡ aN (p) mod P for all a ∈ OL.
We give a definition and a notation for this special automorphism.
Definition 2.1.4. The special element σ is called to be the Frobenius automorphism of P over p. Obviously, we have Frob(P|p) ∈ D(P|p). We denote it by Frob(P|p).
Proposition 2.1.5 ([7]). Assume that L is a Galois extension of K and p is unramified in L, Let P1 and P2 be two primes of OL lying over p. Suppose that σ ∈ Gal(L/K) maps P1 to P2. Then
we have
Frob(P1|p) = σFrob(P2|p)σ−1.
6 CHAPTER 2. BASIC KNOWLEDGE From the above proposition, if Gal(L/K) is abelian, then the Frobenius automorphism depends only on p. In this case, the Frobenius automorphism σ will satisfy
σ(a) ≡ aN (p) mod pOL
for all a ∈ OL. Thus, the proposition lead to the following definition.
Definition 2.1.6. Assume that Gal(L/K) is abelian. For a prime p in K unramified in L, define the Artin symbol by
L/K p
= Frob(P|p)
where P is any prime in L lying over p. Let TL/Kdenote he multiplicative group of fractional ideals
generated by primes of K unramified in L. Then the Artin map FrobL/K : TL/K -Gal(L/K) is the group homomorphism defind by
FrobL/K(pe11· · · p ek k ) = k Y i=1 L/K pi ei
where pi are primes of K unramified in L and eiare integers.
Proposition 2.1.7 ([7]). Each automorphism σ of L in D(P|p) induces an automorphism σ : OL/P → OL/P of the field OL/P that fixes Z/pZ pointwise and if we let γ : OL → OL/P be
the canonical homomorphism γ(α) = α + P, then σ ◦ γ = γ ◦ σ Since σ ∈ D(P|p), the property σ(P) = P implies σ is defined by σ(a + P) = σ(a) + P ∀a ∈ OL.
More precisely, in the Proposition 2.1.7 we say that σ fixes Z/pZ pointwise means there exists an embedding i : Z/pZ → OL/P, defined by i(a + pZ) = a + P, such that α + P contained in the
image of i.
2.2
Representations, Characters And Artin L-functions
In our discussion, L-function plays an important role. Before we discuss L-functions, we need to introduce representations and characters.
Definition 2.2.1. Let V be a vector space over a field F and GL(V ) be the group of isomorphisms of V onto itself. A representation of a group G in V is a group homomorphism ρ from G to GL(V ). The dimension of V is called the degree of ρ.
Now, Let F be a field and G be a finite group. Consider the set
FG = X g∈G cgg : cg∈ F
of all formal linear combinationsP cgg with all but finitely many cgequal to 0. With the obvious
addition and scalar multiplication, it becomes a vector space over F. Then the algebraic structure FG given above is the group algebra of G over F.
2.2. REPRESENTATIONS, CHARACTERS AND ARTIN L-FUNCTIONS 7 Definition 2.2.2. Let G be a group and V be a vector space over a field F. Then V is a module over the group algebra FG or simply an FG-module if there is a group action of G on V such that the group action respects the linearity of the vector spaces.
And we say a vector subspace W of V is an FG-submodule if gw ∈ W for all g ∈ G and all w ∈ W .
Next, we will introduce restrictions and induced representation.
If H is a subgroup of a finite group G, then the restriction of an representation of G to H is automatically a representation of H. Conversely, given a representation of H, there are many ways to construct representations of G.
Definition 2.2.3. Let H be a subgroup of a finite group G, and V be an FG-module. The FH-module V , obtained by restricting the action on V of G to H, is the restriction of V to H, and is denoted by ResGHV or simply ResV if it is clear which groups are involved. Equivalently, if ρ : G → GL(V ) is a representation of G, then ρH: H → GL(V ) defined by ρH(h) = ρ(h), ∀ h ∈ H
is the restriction of ρ to H.
Next, let U be an FH-submodule of FH. The action of G on U can be taken just as the ordinary multiplication. Then the FG-module (FG)U = {ru : r ∈ FG, u ∈ U } is the induced module of U , and we denote it by IndGHU or simply IndU . The representation associated to IndU is the induced representation.
Definition 2.2.4. Let G be a finite group, and V be a vector space over C. Given a representation ρ : G → GL(V ), the function χ : G → C defined by χ(g) = trace(ρ) is called the character of the representation ρ. Similarly we have the definition of restriction, denote by Resχ, and induced character, denoted by Indχ.
Now, we can define the Artin L-function associated to a representation.
Definition 2.2.5. Let G be the Galois group of the Galois extension L/K, and ρ be a represen-tation of G over Q. Then we define Artin L-function as follows.
L(s, χ) = Y p 1 det(I − ρ(Frob(p))p−s) = ∞ X n=1 an ns
where χ is the character of the representation ρ.
Proposition 2.2.6 ([1]). Let L/K be a Galois extension of number field. then the following equalities were only up to a finite number of Euler factor.
1. If χ1 and χ2 are characters of Gal(L/K), then
L(s, χ1+ χ2, L/K) = L(s, χ1, L/K) × L(s, χ2, L/K)
2. Let M be an intermediate subfield and H = Gal(L/M ) < Gal(L/K). If χ is a character of H, then
L(s, Indχ, L/K) = L(s, χ, L/M )
Theorem 2.2.7 ([3]). Let G be a finite group. Then for each character χ of G, there exist integers ni and subgroups Hi of G that are either abelian groups or p-groups such that
χ =XniIndψi,
where ψi are characters of Hi.
8 CHAPTER 2. BASIC KNOWLEDGE
2.3
Adeles And Ideles
Definition 2.3.1. A field K is a global field if K is a number field or a function field of one variable over a finite field. And a local field is a locally compact topological field with respect to a non-discrete topology.
It is easy to see, that a local field arises naturally as completions of a global field.
Definition 2.3.2. Let K be a global field and consider the set of all embeddings of K into local fields L such that the image of K is dense in L. Two such embeddings i : K → L and i0 : K → L0 are said to be equivalent if there exists a continuous isomorphism f : L → L0 such that i0 = f ◦ i. An equivalence class is called a place of K.
A place is denoted by v, and the corresponding embedding and local field are denoted by iv
and Kv, respectively.
If K is a number field, we say a place of K is infinite if it is either a real embedding or a pair of complex-conjugate embeddings. And a place of K is called to be finite if it is non-Archimedean place.
Definition 2.3.3. Let K be a global field. For each finite place v, consider a locally compact space Kv and its valuation ring Rv. Then the restricted product
AK =
( (xv) ∈
Y
v∈V
Kv : xv ∈ Rv for all but finitely many finite places
)
is the adele ring of K, and the elements of AK are called adeles.
Let K be a global field. For each finite place v, consider a locally compact space Kv∗ and its unit group R∗v. Then the restricted product
IK =
( (xv) ∈
Y
v∈V
Kv∗ : xv ∈ R∗v for all but finitely many finite places
)
is the idele group of K, and the elements of IK are called ideles.
Now, we consider the subgroup I1K of IK,where
I1K = {x ∈ IK : kxk = 1}
contains all the elements of modulus 1. In different situation modulus have several definitions. Here we define modulus || · || as
||x|| = |x|, if K = R x2 1+ x22, if K = C and x = x1+ ix2 q−n, if K is non-Archimedean and x ∈ πnR∗
where the non-Archimedean might be Qp, p is a prime, or Fq[[T ]] of formoal Laurent series, q = pk
is a prime power. And, we called the factor group IK/K∗ the idele class group. For example, if
K = Q, we have IQ/Q∗∼= R +
× I1 Q/Q
2.4. HECKE CHARACTERS 9
2.4
Hecke Characters
Definition 2.4.1. Let K be a number field. A Hecke character or a Gr¨ossencharakter is a character of the idele class group IK/K∗.
Definition 2.4.2. Let χ be a Hecke character on IK/K∗. Define a formal product
m(χ) =Y
v
vnv
1. nv= 1, if v is a real place and χv(−1) = −1.
2. nv= 0, if v is a complex place or if v is a real place and χv(−1) = 1.
3. nv= 0, if χv(xv) = 1 for all xv ∈ R∗v.
4. nv = ev, else, where ev is the smallest positive integer such that 1 + πevvRv is contained in
the kernel of χv and πv∈ IK/K∗.
The formal product m(χ) is called the modulus of the Hecke character.
Definition 2.4.3. Let K be a number filed and χ be a Hecke character. Then we define the associated Hecke L-function by
L(s, χ) = Y
v is finite:χv(R∗v)=1
1
1 − χv(πv)N (v)−s
A Hecke character is also a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class of L-functions larger than Dirichlet L-functions, and a natural setting for the Dedekind zeta-functions and certain others which have functional equations analogous to that of the Riemann zeta-function.
For each Hecke L-function, we can give the following functional equation.
Theorem 2.4.4 ([8]). Let χ be a Hecke character on IK/K∗ and L(s, χ) be its Hecke L-function.
Let r+1 be the number of real places v with χv(−1) = 1 and r1− be the number of real places v with
χv(−1) = −1. Let r2 be the number of complex places. Set
Z+ R(s) = π −s/2Γ(s/2), Z− R(s) = π −(s+1)/2Γ((s + 1)/2), Z C(s) = (2π) 1−sΓ(s), and ZK(s, χ) = (dKdχ)s/2ZR+(s)r + 1Z− R(s) r1−Zr2 CL(s, χ)
where dK is the absolute value of the discriminant of K. Then ZK(s) has an analytic continuation
to the whole complex plane, except for two simple poles at s = 0 and s = 1 in the case χ is trivial on IK/K∗. Moreover it satisfies the functional equation
ZK(s, χ) = (−i)r
− 1 τ (χ)
d1/2K
ZK(1 − s, χ−1)
where τ (χ) is the Gaussian sum associated to χ.
At last, we come back to the Artin L-function. From the definition of an Artin L-function, it is not clear whether it has an analytic continuation to the whole complex plane. But after we introduce the Hecke L-function, and by Proposition 2.2.6, Theorem 2.6.4. We see that every Artin L-function can be written as a product of finitely many Hecke L-functions. And the products is taken over all places including the archimedean ones. So Artin L-function can be meromorphic continued to the whole plane.
10 CHAPTER 2. BASIC KNOWLEDGE
2.5
Modular Forms
Definition 2.5.1. We called SL(2, Z) or its subgroup of finite index a modular group.
Now, we give SL(2, Z) a group action on upper half-plane H = {τ = x + iy : x ∈ R, y > 0} by the linear fractional transformation
γτ =aτ + b cτ + d , for τ ∈ H, γ = a b c d ∈ SL(2, Z).
The linear fractional transformations are rigid motions of the hyperplane, and they move points in distinct ways. And an element γ ∈ SL(2, Z) has fixed points, then we give definitions of those γ and fixed points as follows.
Definition 2.5.2. An element γ ∈ SL(2, Z) is parabolic if it has one fixed point, hyperbolic if it has two distinct fixed points on P1
(R), elliptic if it has a pair of conjugate complex numbers as fixed points. A point in P1
(Q) fixed by a parabolic element is called a cusp, and a point in the upper half-plane fixed by an elliptic element is called an elliptic point.
Now, we change our objective to those subgroups of SL(2, Z) with finite index. Definition 2.5.3. Let Γ be a discrete subgroup of SL(2, Z). If Γ contains the subgroup
Γ(N ) = γ ∈ SL(2, Z) : γ ≡ ±1 00 1 mod N
for some positive integer N , then Γ is a congruence subgroup. The smallest such positive integer N is the level of Γ. The group Γ(N ) is called the principal congruence subgroup of level N .
The following congruence subgroup Γ0(N ) = a b c d ∈ SL(2, Z) : c ≡ 0 mod N
is also called the Hecke congruence subgroups.
Since they are subgroups of SL(2, Z), we want to ask what indices of them in SL(2, Z) are. Theorem 2.5.4 ([8]). [SL(2, Z) : Γ(N )] = N3Y p|N 1 − 1 p2 , [Γ0(N ) : Γ(N )] = N × φ(N )
where φ is the Euler function.
Now, we try to discuss how many inequivalent elliptic points and cusps are there in Γ0(N ),
Theorem 2.5.5 ([8]). For N > 2, we have,
1. The number of inequivalent elliptic points of Γ0(N ) of order 2
v2(Γ0(N )) = 0, if 4|N Q p|N (1 + −1p ), if 4 - N.
2.5. MODULAR FORMS 11 2. The number of inequivalent elliptic points of Γ0(N ) of order 3
v3(Γ0(N )) = 0, if 9|N Q p|N (1 + −3p ), if 9 - N.
3. The number of inequivalent cusps of Γ0(N )
v∞=
X
0<d|N
φ((d, N/d))
where (p·) is the Legendre symbol and φ is the Euler function.
And for each modular group Γ we give a theorem to calculate the genus g as follows.
Theorem 2.5.6 ([8]). Let Γ is a modular group, and the index [SL(2, Z) : Γ] = m. Let v2, v3, v∞
be the number of inequivalent elliptic point of order 2, order 3, and cusps, respectively. Then the genus g of Γ is given by the formula:
g = 1 + m 12− v2 4 − v3 3 − v∞ 2 .
Definition 2.5.7. A function f is said to be a modular form of weight k on Γ if it satisfies the following conditions,
1. f is holomorphic in the upper half-place H. 2. f ( aτ + bcτ + d) = (cτ + d)kf (τ ) for every γ =a b
c d
in the modular group Γ. 3. f is holomorphic at every cusps.
Moreover, if f vanishes at every cusp, then the function f is a cusp form of weight k. And for convenience, let Mk(Γ) denotes the space which contains all modular forms of wright k on Γ, and
Sk(Γ) denote the space which contains all cusp forms of weight k with respect to Γ.
If a modular form f be written as a Fourier expansion and an be the Fourier coefficients, then
we put L(s, f ) = ∞ X n=1 ann−s.
L(s, f ) converges absolutely and uniformly for <(s) > 1+k/2, then we called L(s, f ) the L-function associated with f .
For each L-function associated with a modular form f we also have a functional equation as follow.
Theorem 2.5.8 ([8]). For N > 0 be the level of the modular form f , we let ΛN(s, f ) = 2π √ N −s Γ(s)L(s, f ), then the following functional equation will hold
ΛN(s, f ) = ΛN(k − s, g)
12 CHAPTER 2. BASIC KNOWLEDGE Now, we are going to define the divisor of a holomorphic in H.
Definition 2.5.9. For a nonzero holomorphic function f , we define the divisor of f by div(f ) =X
a
va(f )a
where a runs over all elliptic points and cusps, and va(f ) denotes the order of f at a.
Then for the divisor of holomorphic in H, we define the degree of the divisor as follows theorem. Theorem 2.5.10 ([8]). Let k be an odd integer. Assume -1 is not contained in a modular group Γ. For a nonzero element f , f is an meromorphic form of weight k with respect to Γ, then we have, deg(div(f )) = k(g − 1) +k 2 X a (1 − 1 ea ) where a ∈ Γ\H∗, ea is the ramification index of a, g is genus of f .
2.6
Hecke Operators
Now, we try to introduce Hecke operator. Let G be a group, and Γ and Γ0 are two subgroups of G. We say that Γ and Γ0 are commensurable if
[Γ : Γ ∩ Γ0] < ∞ and [Γ0: Γ ∩ Γ0] < ∞. Definition 2.6.1. For N ∈ N, if α ∈ GL+
(2, Z), and Γ0(N ) and α−1Γ0(N )α are commensurable.
The double coset Γ0(N )αΓ0(N ) is a finite union of right coset,
Γ0(N )αΓ0(N ) = h [ i=1 Γ0(N )αi, where αi ∈ GL+(2, Z), h = Γ0(N ) : Γ0(N )T α−1Γ0(N )α
Then we define a linear operator [Γ0(N )αΓ0(N )] on all f ∈ Mk(Γ0(N )) by
f | [Γ0(N )αΓ0(N )]k=
X f |αi.
Then we call the linear operator [Γ0(N )αΓ0(N )] as a Hecke operator.
Definition 2.6.2. For each divisor d of N , let id be the map
id: (Sk(Γ0(N d−1)) × (Sk(Γ0(N d−1))) → Sk(Γ0(N ))
given by
(f, g) 7→ f + g[αd]k.
The subspace of oldforms at level N is Sk(Γ0(N ))old=
X
p|N p is prime
ip(Sk(Γ0(N p−1)) × Sk(Γ0(N p−1)))
and the subspace of newforms at level N is the orthogonal complement with respect to the Petersson inner product,
2.6. HECKE OPERATORS 13 Definition 2.6.3. We say a nonzero f ∈ Mk(Γ0(N )) is an Hecke eigenform if it is an eigenform
for Hecke operators. And we say the eigenform f =P∞
n=0anq
n is normalized if a
1= 1. Moreover
a newform is a normalized eigenform in Sk(Γ0(N ))new.
Then, for a normalized eigenform f , there is a theorem such that L(s, f ) has an Euler product expansion.
Theorem 2.6.4 ([6]). Let f =P∞
n=0anqn, q = e2πiτ be a modular form with a character χ. The
following are equivalent.
1. f is a normalized eigenform.
2. L(s, f ) has an Euler product expansion L(s, f ) =Y
p
(1 − app−s+ χ(p)pk−1−2s)−1,
Chapter 3
The Detail Of Serre’s Examples
In this chapter, we will use some basic knowledge that we introduced in Chapter 2 to explain Serre’s example f (x) = x3− x − 1.
First, we give some precise result corresponding to the table we gave in Chapter 1. Using Maple, we can easily check that
f (x) = x3− x − 1 ≡ x3+ x + 1 mod 2 (x2+ 2x + 3)(x − 2) mod 5 (x − 10)2(x − 3) mod 23 (x − 4)(x − 13)(x − 42) mod 59 .. .
For convenience, follows primes appear in this chapter do not equal to 23.
3.1
Use Cyclic Group To Determine N
p(f )
In this section, we try to determine Np(f ) in algebraic number theory. We give the following
theorem to help us determine Np(f ) from the order of Frob(P|p).
Theorem 3.1.1. Let L be the splitting field of f (x) = x3− x − 1 over Q, and the Galois group Gal(L/Q) is identified with S3. If
1. Frob(P|p) = e, then Np(f ) = 3.
2. Frob(P|p) ∈ {(12)}, then Np(f ) = 1.
3. Frob(P|p) ∈ {(123)}, then Np(f ) = 0.
where the permutation (123) means that the Frob(P|p) acts three roots in L transitively. Similarly, if Frob(P|p) ∈ {(12)}, then we say Frob(P|p) has order 2 and fixes a root in L.
Proof. Assume that f has three distinct roots, names α1, α2, α3 in L.
Note that Frob(P|p) ∈ D(P|p), then by Proposition 2.1.7, then for all a ∈ OL we have an
automorphism
Frob(P|p) : OL/P → OL/P is defined by Frob(P|p)(a + P) = Frob(P|p)(a) + P.
16 CHAPTER 3. THE DETAIL OF SERRE’S EXAMPLES And a canonical homomorphism
γ : OL→ OL/P is defined by γ(a) = a + P.
Such that Frob(P|p) ◦ γ = γ ◦ Frob(P|p), and Frob(P|p) fixes Z/pZ pointwise.
If Frob(P|p) = e, that says Frob(P|p) fixes α1, α2, α3. We have Frob(P|p)(αi) = αi for
i = 1, 2, 3. Then
Frob(P|p)(γ(αi)) = Frob(P|p)(αi+ P) = Frob(P|p)(αi) + P = αi+ P,
so we have Frob(P|p) fixes αi+P, that means there is an embedding i : Z/pZ → OL/P is defined by i(a+
pZ) = a + P for all a ∈ Z. And αi+ P is contained in the image of i. Thus, there exist an element
ai ∈ Z such that i(ai+ pZ) = αi+ P.
Now, we will claim that ai+ P are roots of f in Z/pZ. Consider the norm of f (ai+ P), easy
to check that will be divided by p for i = 1, 2, 3. Thus, if Frob(P|p) = e, then Np(f ) = 3.
We can show the other two cases by the same argument.
3.2
Relation Between N
p(f ) And Legendre Symbols
From above section, we know the relation between Np(f ) and the order of Frobenius automorphism.
Now we can use that to describe the Serre’s example precisely. For p 6= 23, one has,
Np(f ) =
(
0 or 3 if 23p = 1 1 if 23p = −1. Assume 23p
= −1. Consider that Q(√−23) ⊂ L, and p 23
= −1 says that p is inert in Q(
√
−23), so Frob(P|p) has order 2. Thus, Np(f ) = 1.
For 23p = 1, in Serre’s article, we have that p can be written either as x2+ xy + 6y2 or 2x2+ xy + 3y2 with x, y ∈ Z, in the first case, one has Np(f ) = 3, another case, Np(f ) = 0. Now,
we give a Theorem to describe this statement completely.
Theorem 3.2.1 ([5]). Let m be an integer, m ≡ 1 mod 4. Then there is a monic irreducible polynomial fm(x) ∈ Z[x] of degree h(m) (Where h(m) denotes the ideal class number of K =
Q( √
m).) such that if an odd prime p divides neither n nor the discriminant of fm(x), then
p = x2+ xy +1 − dK
4 y2 ⇔ dK
p
= 1 and fm(x) ≡ 0 mod p has a integer solution.
Where, dK is the discriminant of Q(
√
m). Furthermore, fm(x) may be taken to be the minimal
polynomial of a real algebraic integer α for which L = K(α) is the Hilbert class field. And fm(x)
is said to be the Hilber class polynomial.
In Theorem 3.2.1, fortuitously, the Hilbert class polynomial is same as f (x) = x3− x − 1. Since the field L is a cubic cyclic extension of the quadratic field K = Q(√−23), it is unramified, and since h(−23) = 3. So L is the Hilbert class field of K.
That is why if p can be written as x2+ xy + 6y2 then N
p(f ) = 3. The other case, if p =
2x2+ xy + 3y2
for some x, y ∈ Z, since the prime is not inert in Q(√−23), and it will not satisfy Theorem 3.2.1, so in this case Np(f ) = 0.
Chapter 4
Main Results
Theorem 4.0.2. Let Np(f ) denotes the number of roots for a given polynomial f (x) = x4− 2 in
Z/pZ. Then we have Np(f ) = ap+ 1 +
2 p
, where ap is the Fourier coefficient of prime terms in
a cusp form of weight 1 on Γ0(256). The cusp form we find is
F (τ ) = 1 2 X m,n∈Z qm2+64n2− X m,n∈Z q4m2+4mn+17n2 = q + q9− 2q17− q25− 2q41+ q49+ 2q73+ · · ·
4.1
Galois Group Of f Is Isomorphic To D
4Now, we claim that the Galois group Gal(L/Q) of the polynomial f (x) = x4− 2 is isomorphic to
D4. The simplest way to describe the Galois group of f is write down all the automorphisms of
Q(4 √ 2, i). id : (√4 2 7→√4 2 i 7→ i τ : (√4 2 7→√4 2 i 7→ −i σ : (√4 2 7→√42i i 7→ i στ : (√4 2 7→√42i i 7→ −i σ2: (√4 2 7→ −√4 2 i 7→ i σ 2τ : (√4 2− 7→√4 2 i 7→ −i σ3: (√4 2 7→ −√4 2i i 7→ i σ 3τ : (√4 2 7→ −√4 2i i 7→ −i.
It is easy to check that our statement is true. For the Dihedral group of order 8, we give the following character table.
18 CHAPTER 4. MAIN RESULTS e σ2 τ, σ2τ σ, σ3 στ, σ3τ χ1 1 1 1 1 1 χ2 1 1 −1 1 −1 χ3 1 1 1 −1 −1 χ4 1 1 −1 −1 1 χ5 2 −2 0 0 0
4.2
N
p(f ) Is A Class Function Of Frobenius
Similarly, the discriminant of f (x) = x4− 2 is −2048, thus the only prime p of Z ramified in Q(4
√
2, i) is 2. We consider the case p = 2 specially. An easy compution gives x4− 2 ≡ x4 mod 2.
Thus, N2(f ) = 1.
After that, we will start to compute another Np(f ) for p 6= 2. There is a Theorem just like
Theorem 3.1.1.
The following theorem gives the notation in D4as permutations, but we use the notations σ, τ
to denote the mapping in D4. There is a question. How to connect two kinds symbol? Since σ
acts on√4
2, √4
2i, −√4
2, −√4
2i transitively, then we note that √4
2,√4
2i, −√4
2, −√4
2i as 1, 2, 3, 4, respectively. So σ corresponds to (1234), and the others have following relations.
στ, σ3τ corresponds to {(12)(34)} , τ, σ2τ corresponds to {(12)} , σ, σ3 corresponds to {(1234)} , σ2 corresponds to (13)(24), e corresponds to e. So we can give a theorem as follows.
Theorem 4.2.1. Let L be the splitting field of f (x) = x4− 2, and the Galois group Gal(L/Q) is
isomorphic to D4. If 1. Frob(P|p) = e, then Np(f ) = 4 2. Frob(P|p) = (13)(24), then Np(f ) = 0 3. Frob(P|p) ∈ {(1234)}, then Np(f ) = 0 4. Frob(P|p) ∈ {(12)}, then Np(f ) = 2 5. Frob(P|p) ∈ {(12)(34)}, then Np(f ) = 0
Proof. Use same argument as Theorem 3.1.1.
Now we try to connect Np(f ) and Frob(P|p). Before that, we need to determine that what
conjugacy classes are those Frob(P|p) contained in D4.
Lemma 4.2.2. For the prime
4.2. NP(F ) IS A CLASS FUNCTION OF FROBENIUS 19
2. p ≡ 5 mod 8, Frob(P|p) ∈σ, σ3 .
3. p ≡ 7 mod 8, Frob(P|p) ∈τ, σ2τ .
Before we prove this Lemma, we need two lemmas. Lemma 4.2.3 ([7]). a p ≡ a(p−1)/2 mod p. Lemma 4.2.4 ([7]). 2 p ≡ ( 1 if p ≡ ±1 mod 8 −1 if p ≡ ±3 mod 8. Now, we can start our proof.
Proof of Lemma 4.2.2. For p ≡ 3 mod 8, assume that p splits as P1, · · · , Pk in Q(4
√
2, i). Then the Frobenius automorphism of P over p, Frob(P|p) will satisfy following equation.
Frob(P|p)(√42) ≡ (√42)N (p) mod P, Frob(P|p)(i) ≡ iN (p) mod P.
First, we consider how Frob(P|p) acts on i, from the above equation, we have Frob(P|p)(i) ≡ i8k+3 mod P, for some k ∈ Z, It is easy to check that Frob(P|p) sends i to −i.
Next, we observe that how Frob(P|p) acts on √4
2. Similarly, since Frob(P|p) ∈ Gal(L/Q), so it sends √4
2 to its conjugate element √4
2, √4
2i, −√4
2, −√4
2i, so we let be the fourth root of unity such that Frob(P|p)(√4
2) = √4
2. Then, we can rewrite the equation as follows, ·√42 ≡ (√42)p mod P, where ∈ {±1, ± i} .
thus, ≡ 2(p−1)/4 mod P,
Since p ≡ 3 mod 8, by Lemma 4.2.3 and Lemma 4.2.4, we have, −1 ≡ 2
p
≡ 2(p−1)/2 mod p. Thus, we know that p|(2(p−1/2)+ 1) so P|(2(p−1)/2+ 1). Then we have,
2(p−1)/2≡ −1 mod P, thus, 2(p−1)/4≡ ±i mod P. Finially, we conclude that ≡ ±i mod P. That says Frob(P|p)(√4
2) = ±√4
2i. Thus,we have that for p ≡ 3 mod 8, Frob(P|p) ∈στ, σ3τ .
Next, for p ≡ 5 mod 8 and p ≡ 7 mod 8, we can give the proof by same argument as the case p ≡ 3 mod 8.
20 CHAPTER 4. MAIN RESULTS So, for the prime p not congruence to 1 modulo 8, we have known that Np(f ) precisely. How
about the prime p congruence to 1 modulo 8? Unfortunately, we can not decide what conjugacy classes will contain the Frob(P|p) precisely by Lemma 4.2.2. But, we have another way to determine the Frob(P|p) case by case.
Now, we compute an example p = 3 with another algorithm, we using the norm of p to determine the Frob(P|p).
Example 4.2.5. We consider the prime p = 3 in Z. Although, we do not know how the prime ideal (3) splits in Q(√4
2, i). But, without lost of generality, we can assume that (3) splits as P1P2...Pk
in Q(√4
2, i). And that no matter which P we pick, the Frob(P|3) will satisfy Frob(P|3)(√4
2) ≡ (√4
2)3 mod P and Frob(P|3)(i) ≡ (i)3 mod P.
Since Frob(P|3) ∈ Gal(L/Q), then there also exist a fourth root of unity such that Frob(P|3)(√42) =
(√42). Then, we just need to determine that = ±1 or ± i precisely.
From above two equations we have, √4
2 ≡ (√4
2)3 mod P (ie. ≡√2 mod P). and we know that P|(3), so NQL(√2 − ) must dividing by 3. So has to be ±i. Thus we can make sure that Frob(P|3) sends√4
2 to ±√4
2i.
Next, we want to know that, how Frob(P|3) acts on i. By the same argument, we have Frob(P|3)(i) ≡ i3 mod P. Again, from above equation, and we have Frob(P|3)(i) = ±i since it
contained in the Galois group. Obviously, it has to be −i. Thus, we have Frob(P|3) : (√4 2 7→ ±√42i i 7→ −i. Then Frob(P|3) ∈στ, σ3τ .
Now we have another algorithm, and we know that really works. So we can try to determine the Frobenius automorphism of P over p ≡ 1 mod 8.
Example 4.2.6. For p = 17, we also assume that (17) splits as P1P2...Pk in Q(4
√
2, i). Then the Frob(P|17) will satisfy that
Frob(P|17)(i) ≡ i17 mod P. Easy to check that Frob(P|p)(i) = i. We also have
Frob(P|17)(√4
2) ≡ (√4
2)17 mod P, as above, there exist a fourth root of unity such that Frob(P|17)(√4
2) = √4
2. Then we rewrite the equation as
√42 ≡ (√42)17 mod P. Obviously, = −1, that says Frob(P|p)(√4
2) = −√4
2. Thus, Frob(P|p) = σ2.
Using the above algorithm we can determine that, for p = 17, 41, 73, the Frob(P|p) are σ2, σ2, e respectively. Although, this way looks like very inefficient, but it can determine all kinds of prime.
4.3. CONSTRUCT A CUSP FORM 21 Now, we want to find the relation between Np(f ) and ap. Note ap = χ5(Frob(P|p)), then we
have ap= 2, if Frob(P|p) = e −2, if Frob(P|p) = σ2 0, else.
From Frob(P|p) = e, we have Np(f ) = ap+ 2. And Frob(P|p) = σ2 tells us Np(f ) = ap+ 2.
Similarly Frob(P|p) ∈τ, σ2τ gives Np(f ) = ap+ 2. Then we have
Np(f ) = ap+ 1 +
2 p
= χ5(Frob(P|p)) + χ1(Frob(P|p)) + χ3(Frob(P|p)).
That is our first result.
4.3
Construct A Cusp Form
We start from the Artin L-function, L(s, χ5) = Y Frob(p)=e 1 (1 − p−s)2 × Y Frob(p)=σ2 1 (1 + p−s)2 × Y Frob(p)∈{τ,τ σ2} 1 (1 − p−2s)× Y Frob(p)∈{σ,σ3} 1 (1 + p−2s)× Y Frob(p)∈{τ σ,τ σ3} 1 (1 − p−2s)× Y p is ramified ∗ = ∞ X n=1 an ns
Those ramified primes are not particularly important in our consideration. Thus, by the Propo-sition 2.2.6 and Theorem 2.6.4. We have
L(s, χ5) = L(s, Indψ, L/Q) = L(s, ψ, L/K)
where K is Q(i), χ5= Indψ, and ψ should be a Hecke character corresponding to Q(i). The Hecke
character ψ gives values as following table.
Frob(p) e σ σ2 σ3 ψ(p) 1 i −1 −i
Now, we try to find a modular form F such that L(s, F ) = L(s, ψ). So F should be written as P∞
n=1anqn, where q = e2πiτ, that says F is a cusp form. Next, we try to explain that why the
cusp form F with weight 1 of level 256.
From Theorem 2.5.8, the Dirichlet series associated with F , L(s, F ) will satisfy the functional equation 2π √ N −s Γ(s)L(s, F ) = 2π√ N −(k−s) Γ(k − s)L(k − s, G) (4.1)
where k and N are the weight and level of F . And from Theorem 2.4.4, the Hecke L-function L(s, ψ) will satisfy the functional equation
2π pdKdψ !−s Γ(s)L(s, ψ) = 2π pdKdψ !−(1−s) Γ(1 − s)L(1 − s, ψ−1). (4.2)
22 CHAPTER 4. MAIN RESULTS where is a root of unity and dK is the discriminant of K takes absolute value, dψ is the norm of
modulus of ψ. Remember that, our goal is finding a modular form F such that L(s, F ) = L(s, ψ). Compare (4.1) and (4.2) we have weight k = 1 and level N = dK× dψ.
In the Hecke L-function L(s, ψ, L/K), ψ is a Hecke character of modulus (1 + i)k, since the
only prime ramifies in OL is 2. And ψ takes values ±i, so we have a property, that (Z[i]/m)∗ has
a non-unit element of order 4. Apply the definition of modulus, in this case m = (1 + i)k for some k ∈ N.
Note that, if k=1, then |(Z[i]/m)∗| = 1.
If k=2, then |(Z[i]/m)∗| = 2. ie. (Z[i]/m)∗ ∼= Z2 If k=3, then |(Z[i]/m)∗| = 4. ie. (Z[i]/m)∗ ∼= Z4 If k=4, then |(Z[i]/m)∗| = 8. ie. (Z[i]/m)∗ ∼= Z4× Z2
If k=5, then |(Z[i]/m)∗| = 16. ie. (Z[i]/m)∗∼= Z
4× Z2× Z2
If k=6, then |(Z[i]/m)∗| = 32. ie. (Z[i]/m)∗∼= Z
4× Z4× Z2
so the modulus m is (1 + i)6
Thus, we can conclude that the level of cusp form here is 256.
Now, we know the level of cusp form is 256, then we can use the algorithm in [5], there are four primitive quadratic forms
m2+ 64n2, 5m2± 2mn + 13n2, 4m2+ 4mn + 17n2
might be a part of the exponent of cusp form. Consider the series
a X m,n∈Z qm2+64n2+ b X m,n∈Z q5m2+2mn+13n2 +c X m,n∈Z q5m2−2mn+13n2+ d X m,n∈Z q4m2+4mn+17n2 for some a, b, c, d ∈ Q, where q = e2πiτ. And the expansion of each series are
X m,n∈Z qm2+64n2 = 1 + 2q + 2q4+ 2q9+ · · · X m,n∈Z q5m2+2mn+13n2= 1 + 2q5+ 2q13+ 2q16+ 4q20+ 2q29+ 2q37+ · · · X m,n∈Z q4m2+4mn+17n2= 1 + 2q4+ 2q16+ 4q17+ 4q25+ 2q36+ 4q41+ · · · Now, we consider a modular form,
F (τ ) = 1 2 X m,n∈Z qm2+64n2− X m,n∈Z q4m2+4mn+17n2 = q + q9− 2q17− q25− 2q41+ q49+ 2q73+ · · ·
Fortunately, for the first 73 terms, the coefficients of prime terms are just equal to ap for each
prime p.
But, how can we make sure that every terms after 73 are also equal? In fact, we only need to check the degree of divisor terms. Since the degree of divisor means the summation of orders of
4.3. CONSTRUCT A CUSP FORM 23 cusps, elliptic points (here, we only have cusps), so the order of i∞ is equal or less then the degree of divisor.
In our discussion, the notation a in the Theorem 2.5.10 are just cusps, thus, 1/a are all zeros, then the summation in the degree of divisor is just the number of inequivalent cusps. Using Theorem 2.5.5, we know that there has 24 inequivalent cusps. And we use Theorem 2.5.6 to compute the genus of F . Before that, we need Theorem 2.5.4 to evaluate the index m = 384, and by Theorem 2.5.5 we have v2= 0, v3= 0, v∞= 24. Thus the genus g equals 21.
Now, from Theorem 2.5.10 we can compute the degree of divisor of F is 32. So we only need to compare first 32 terms. We compare first 73 terms already, then we can make sure the cusp form’s Fourier coefficients are equal to ap for every prime terms.
Moreover, since an are Hecke eigenvalues and from the functional equation we have F has level
256. Thus, F is a newform. Then the Theorem 2.6.4 can help us to write L(s, F ) as an Euler product. L(s, F ) =Y p 1 − app−s+ −2 p p−2s −1 .
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