C CENTRO DE INVESTIGACION Y DE ESTUDIOS AVANZADOS

DEL INSTITUTO POLITECNICO NACIONAL

UNIDAD ZACATENCO

DEPARTAMENTO DE MATEMATICAS

Sobre el Programa de Langlands

para Campos Numericos

T E S I S

Que presenta

MIGUEL ANGEL VALENCIA BUCIO

Para obtener el grado de

MAESTRO EN CIENCIAS

EN LA ESPECIALIDAD DE MATEMATICAS

Director de Tesis: Dr. Jose Martınez Bernal

Mexico D.F. Agosto 2014

C CENTER FOR RESEARCH AND ADVANCED STUDIES

OF THE NATIONAL POLYTECHNIC INSTITUTE

Campus Zacatenco

Department of Mathematics

On the Langlands Program

for Number Fields

T H E S I S

Submitted by

MIGUEL ANGEL VALENCIA BUCIO

To obtain the degree of

MASTER OF SCIENCE

IN THE SPECIALITY OF MATHEMATICS

Thesis advisor: Dr. Jose Martınez Bernal

Mexico City August 2014

Contents

Resumen i

Introduccion iii

Abstract i

Introduction iii

1 Abelian Class Field Theory 1

1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Abelian Class Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Artin L-functions for characters . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Tate’s Thesis and Hecke L-functions . . . . . . . . . . . . . . . . . . . . 9

1.5 The Langlands Program for n = 1 . . . . . . . . . . . . . . . . . . . . . . 11

2 Shimura-Taniyama Conjecture 15

2.1 Elliptic curves and modular forms . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Elliptic curves and Galois representations . . . . . . . . . . . . . . . . . . 20

2.3 Modular forms and automorphic representations . . . . . . . . . . . . . . 22

2.4 The Langlands Program for n = 2 . . . . . . . . . . . . . . . . . . . . . . 25

3 The Langlands Program 27

3.1 Galois representations and L-functions . . . . . . . . . . . . . . . . . . . 27

3.2 Automorphic representations . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3 The Langlands Program . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4 Reciprocity Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.5 What follows? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

References 39

A mis padres y a Vivi

Resumen

En este trabajo presentamos el Programa de Langlands para el caso de campos numericos,

el cual es actualmente uno de los problemas abiertos mas importantes en teorıa de

numeros. Tambien lo conectamos de manera directa con otros resultados bien conocidos

en teorıa de numeros y damos una potencial aplicacion concerniente a la factorizacion

de polinomios irreducibles sobre campos finitos a traves de algunos ejemplos.

Primero conectamos el Programa de Langlands con otros resultados que pueden

considerarse leyes de reciprocidad. En el capıtulo uno mostramos la relacion entre el

Programa de Langlands y otra correspondencia importante y bien conocida: La Ley

de Reciprocidad de Artin. Comenzamos enunciando el mapeo de Artin en el lenguaje

de la Teorıa de Campos de Clase y despues construimos la conexion por medio de las

L-funciones.

Por un lado, introducimos las L-funciones de Artin para caracteres, y mas general-

mente para representaciones. Desde su introduccion, Artin las utilizo para hallar una

“ley de reciprocidad no abeliana”. Por otro lado, presentamos las L-funciones de Hecke

definidas sobre el grupo de clases de ideles, como Tate las enunciara en su famosa tesis

doctoral.

Bajo este contexto, la Teorıa de Campos de Clase Abeliana se corresponde con el

Programa de Langlands para el caso n = 1 en alguna forma.

En el capıtulo dos se presentan las ideas basicas detras del Programa de Langlands

y se establece la conexion con otro resultado importante en teorıa de numeros: La

Conjetura de Shimura-Taniyama, ahora un teorema [2]. Este teorema asocia curvas

elıpticas, a traves de informacion relacionada con el numero de puntos modulo p, y

ii

formas modulares, a traves de los coeficientes de su serie de Fourier.

Comenzamos dando un panorama suficiente para enunciar la conjetura de Shimura-

Taniyama. Despues usamos la estructura de grupo de las curvas elıpticas para construir

representaciones bidimensionales de Gal(Q), a la vez que introducimos sus L-funciones.

Por otro lado, dada una forma modular, la reescribimos dentro de un espacio de fun-

ciones cuadrado-integrables sobre GL2(AQ) y la usamos para generar una representacion

natural de GL2(AQ), conocida como representacion automorfa.

Bajo esta aproximacion, la conjetura de Shimura-Taniyama puede verse como un

caso especial de la correspondencia de Langlands para el caso n = 2.

Finalmente, en el capıtulo tres, presentamos el Programa de Langlands en su version

completa. Definimos las representaciones de Galois, las representaciones automorfas,

y sus respectivas L-funciones. Hecho esto, enunciaremos la conjetura principal. Con-

cluımos con algunos ejemplos de descomposicion de polinomios y algunas observaciones.

Nuestro proposito es introducirnos en el Programa de Langlands y familiarizarnos

con algunas de las ideas, problemas y conjeturas en esta area, para una investigacion

futura.

—————————

Agradezco al Consejo Nacional de Ciencia y Tecnologıa (Conacyt) el apoyo otorgado

durante el desarrollo de este trabajo.

Introduccion

Descomponer polinomios sobre campos finitos es un problema interesante y difıcil en

teorıa de numeros. Esto esta sumamente relacionado con la factorizacion de polinomios,

ya que, por ejemplo, si la reduccion de un polinomio f(X) ∈ Z[X] sobre algun campo

finito es irreducible, entonces tambien lo es sobreQ. O bien, si f(X) modulo p se descom-

pone en factores lineales para casi todos los primos p, entonces tambien se descompone

en factores lineales sobre Q.

Surge la pregunta: ¿Para que primos p un polinomio dado f(X) es irreducible modulo

p? O mas generalmente ¿es posible clasificar los primos p, tal vez salvo un numero finito

de ellos, de acuerdo a la factorizacion de f(X) modulo p? El Teorema de Kummer-

Dedekind nos permite considerar esta pregunta dentro del marco de la Teorıa de Galois:

estudiar la factorizacion del polinomio f(X) ∈ Q[X] sobre campos finitos nos lleva a

estudiar la factorizacion de numeros primos sobre el campo de descomposicion de f(X).

Se sabe que la informacion aritmetica acerca de las raıces de f(X), aun trabajando

sobre campos finitos, esta contenida en su grupo de Galois sobre cada campo; ergo,

resulta natural pensar que el campo de descomposicion de f(X) sobre Q, a traves de

su grupo de Galois, tiene suficiente informacion como para clasificar los numeros primos

respecto a la factorizacion modulo p de dicho polinomio y de otros que tengan el mismo

campo de descomposicion.

Para contestar a esto se cuenta con herramientas como la Ley de Reciprocidad de

Artin, pero esta no alcanza para todos los polinomios. En esta direccon, el Programa o

Correspondencia de Langlands podrıa ser de utilidad [15, 33].

El Programa de Langlands consiste en una serie de conjeturas que conectan la

teorıa de numeros y el analisis armonico; fue propuesto por Robert Langlands en 1967, y

iv

recientemente ha sido desarrollado y generalizado por muchos investigadores; obteniendo

resultados profundos. Grosso modo, el programa predice una correspondencia entre dos

clases importantes de objetos asociados a un campo numerico. La primera concierne

a representaciones del grupo absoluto de Galois y del grupo de Weil de un campo

numerico. La segunda concierne a representaciones automorfas de cierto espacio de

Hilbert de tipo L2 sobre el anillo de adeles del campo. De manera mas precisa, y en

un contexto mas general, dado un campo numerico F y un grupo reductivo G sobre

F , el Programa de Langlands establece una correspondencia uno-a-uno entre clases de

equivalencia de: (cf. [8])

1. Representaciones automorfas irreducibles y temperadas de G(AF ), donde AF es el

anillo de adeles de F .

2. Homomorfismos W(F ) → LG, del grupo de Weil W(F ), al grupo reductivo LG

llamado el grupo dual de Langlands de G.

Ademas, dicha correspondencia es a traves de L-functions: las de la parte (1) tienen

informacion analıtica que se obtiene de los operadores de Hecke, y aquellas de la parte

(2) tienen informacion aritmetica dada por la estructura de la representacion.

Las propiedades de estas L-funciones en el plano complejo (continuacion meromorfa,

comportamiento en una lınea crıtica, localizacion de ceros, etc.) tienen informacion

aritmetica relevante.

En esta tesis solo consideramos el caso G = GLn, el grupo lineal general de orden

n y nos restringimos a las representaciones de Gal(F ). En esta situacion se tiene que

LG = GLn(C) para todo n, y el Programa de Langlands se reduce a una correspondencia

entre representaciones irreducibles n-dimensionales y representaciones automorfas de

GLn(AF ).

Abstract

In this dissertation we present the Langlands Program for the case of number fields,

which is to date one of the most important open problems in number theory. We also

connect it with other well-known number-theoretic results and we give a potential ap-

plication concerning to the factorization of polynomials over finite fields through some

examples.

First we connect the Langlands Program with other results that may be conside-

red reciprocity laws. In chapter one we show the relationship between the Langlands

Program and another important and well-known correspondence: The Artin Reciprocity

Law. We start describing the Artin map in the language of Class Field Theory and then

make the connection through L-functions.

On one hand, we introduce the Artin L-functions for characters; and more generally,

for representations. Since its introduction, Emil Artin used them to find a “nonabelian

reciprocity law”. On the other hand, we present the Hecke L-functions defined over the

idele class group, as Tate stated them in his famous Ph.D. thesis.

Under this background, Abelian Class Field Theory corresponds to the Langlands

Program for the case n = 1 in some sense.

The chapter two presents the basic ideas behind the Langlands Program, while makes

the connection with other important result in number theory: The Shimura-Taniyama

Conjecture, now a theorem [2]. This theorem associates elliptic curves, through data

related with the number of points modulo p, and modular forms, through the coefficients

of their Fourier series.

We start giving background enough to state the Shimura-Taniyama conjecture. Then

ii

we use the group structure of elliptic curves to construct two-dimensional representations

of Gal(Q), as well to introduce their L-functions.

By another side, given a modular form, we rewrite it into a space of square-integrable

functions on GL2(AQ) and use it to generate a natural representation of GL2(AQ), known

as the automorphic representation.

Under this approach, the Shimura-Taniyama conjecture may be seen as a special case

of the Langlands Program for the case n = 2.

Finally, in the chapter three, we present the Langlands Program in its full version.

We define Galois representations, automorphic representations, and their respective L-

functions. Then we state the main conjecture. To conclude we give some examples of

splitting polynomials and some remarks.

Our purpose in this thesis is to introduce ourselves to the Langlands Program and

to get familiarized with some of the ideas, problems and conjectures in this area, to the

end of future research.

—————————

I thank the Science and Technology National Council (Conacyt) for the financing

given during the development of this Thesis.

Introduction

Splitting polynomials over finite fields is an interesting and difficult problem in number

theory. This question is closely related with factorizing polynomials, because, for ins-

tance, if the reduction of a polynomial f(X) ∈ Z[X] over some finite field is irreducible,

then it is irreducible over Q. Or, if f(X) modulo p splits into linear factors for almost

all primes p, then it splits into linear factors over Q.

This raises the question: for which primes p a given polynomial f(X) is irreducible

modulo p? More generally, is it possible to classify all primes p, maybe with finitely many

exceptions, according with the factorization of f(X) modulo p? The Kummer-Dedekind

Theorem translates this question into the background of Galois theory: studying the

factorization of a polynomial f(X) ∈ Q[X] over finite fields implies studying the factor-

ization of prime numbers over the splitting field of f(X).

It is known that arithmetic information about the roots of f(X), when we even work

over finite fields, is contained in its Galois group over each field; ergo, it is natural to

think that the splitting field of f(X) over Q, through its Galois group, has enough

information about the classification of the prime numbers for this polynomial as well as

others with the same splitting field.

To answer this we have tools like the Artin Reciprocity Law, but these tools cannot

cover all the polynomials. In this direction, a promising machinery would be the Lang-

lands Program [15, 33].

Langlands Program is a series of deep conjectures connecting number theory and

harmonic analysis; it was introduced by Robert Langlands in 1967, and recently has

been developed and generalized by many other people. Roughly speaking, the program

predicts a correspondence between two important classes of objects arising from a num-

iv

ber field. The first one concerns to representations of the absolute Galois group and

the Weil group of a number field. The second one concerns to automorphic repre-

sentations of some Hilbert space of L2-type on the adele ring of this field. In a more

precise, and more general context, given a number field F and a reductive group G over

F , the Langlands Program establishes a one-to-one correspondence between equivalence

classes of: (cf. [8])

1. Irreducible tempered automorphic representations of G(AF ), where AF is the adele

ring of F .

2. Homomorphisms W(F ) → LG, from the Weil group W(F ), to a reductive group

LG called the Langlands dual group of G.

Moreover, the correspondence is through L-functions: that corresponding to (1) have

analytic information obtained from Hecke operators, and those corresponding to (2) have

arithmetic information given by the structure of the representation.

The properties of these L-functions in the complex plane (meromorphic continua-

tion, behavior in a critical line, localization of zeroes, etc.) have relevant arithmetical

information.

In this thesis we only consider the case G = GLn, the general linear group of order

n, and we restrict ourselves to representations of Gal(F ). In this case it results that

LG = GLn(C) for all n, and the Langlands Program reduces to a correspondence between

some irreducible n-dimensional representations and some automorphic representations

of GLn(AF ).

Chapter 1

Abelian Class Field Theory

The Langlands correspondence, roughly speaking, associates irreducible representations

of the Galois group of a given number field with automorphic forms on its adelic groups.

In this chapter we treat the case n = 1, that is, the abelian case [7, 15]. This case is

well-understood and it corresponds to class field theory [8].

Class field theory is considered as a reciprocity law for abelian polynomials, that is,

polynomials whose splitting field is an abelian extension of its ground field. Class field

theory had its beginning in the quadratic reciprocity law (known since Euler and Gauss)

and it can be stated in many forms [9, 19, 20, 24, 29]. However, when one speak about

the Langlands Program, and consider this case, it just consider some of the principal

ideas and the conclusions.

Our purpose in this chapter is to give a broader background about this connection.

For this, we first state the main theorem of class field theory: the Artin Reciprocity

Law; which is done after a preliminary section where all basic language is introduced,

see [18, 20, 31] for more details.

Then we introduce the main ideas behind the connection: the Artin L-functions

associated to characters of the group Gal(F ) and the Hecke L-functions attached to

idele class characters. These objects are important to stating the connection between

the Artin reciprocity law and the Langlands Program for n = 1.

2 1.1. PRELIMINARIES

1.1 Preliminaries

A number field is a finite extension of the rational numbers Q. A local number field

is a topological field that is a completion of a number field. There are two kinds of local

number fields, and they are determined by the places of its corresponding number field:

1. The archimedean places. It is well-known [18] that any number field is simple,

i.e., it is of the form F = Q(α) for some α ∈ F called a primitive element of F .

Let f(X) be the irreducible polynomial of α over Q. Then f(X) decomposes on

R as a product of irreducible polynomials ϕ(X); whose degree is one or two. For

each ϕ(X) we have the isomorphism

F ' Q[X]/f(X) → C

induced by the map g(X) 7→ g(ξ), with ξ a root of ϕ(X).

Therefore, we have an embedding of F into R or C, whether ϕ(X) is linear

or quadratic respectively. The archimedean place Fϕ associated to ϕ(X) is the

completion in C of the respective embedding associated, and it is isomorphic (as

a local field) to R or C.1 Note that this definition does not depend of the choice

of α.

2. The non-archimedean places. Let OF be the ring of integers of F , that is, the

ring of all elements of F whose irreducible polynomial lies in Z[X]. Let p be a

maximal ideal of OF . Since OF is a Dedekind domain, the localization Op of OF

at p is a discrete valuation ring.

Let Fp be the quotient field of Op. Then every x ∈ Fp is of the form x = uπk,

where π is a uniformizing parameter (i.e., a generator of the maximal ideal of

Op), u is a unit of Op and k ∈ Z. Fp has a canonical topological structure induced

by the absolute value

| · |p : uπk 7→ q−k,

where q is the cardinality of the residual field of Fp, i.e, the quotient field OF /p.

1It is possible that a complex number field has a real archimedean place and that a real numberfield has a complex archimedean place. We give the (clear) example Q(α), where α is a root of thepolynomial X3 − 2 = (X − 3

√2)(X2 + 3

√2X + 3

√4).

CHAPTER 1. ABELIAN CLASS FIELD THEORY 3

It is straightforward to prove that Fp is a complete field respect to the metric

induced by its absolute value. Moreover, if p lies above a rational prime p (i.e., if

p ⊆ p, where p is seen as an ideal of Z), then Fp is a finite extension of Qp, the

p-adic rationals; see [20].

Let us say a little bit about Galois theory. Given a number field or a local number

filed F , let Gal(F ) denote the absolute Galois group of F , that is, the group Gal(F/F ).

Being this a profinite group, it is a compact and totally disconnected topological group

[19, 20]. There exists a bijective correspondence between closed subgroups of Gal(F ) and

algebraic extensions of F . Moreover, this correspondence associates normal subgroups

to Galois extensions and open subgroups to finite extensions [18].

Let K/F be an arbitrary field extension, and let p be a prime ideal ofOF . The natural

extension of rings OK/OF gives us a prime ideal P lying above p, that is, pOK ⊆ P.

Let P be any fixed prime lying above p, and let DP be its decomposition group, i.e.,

DP = {σ ∈ Gal(K/F ) : σ(P) = P}.

Let kp = Fq be the residual field of Fp; ıdem for kP. We have the natural map DP →Gal(kP/kp). This is a surjective map and its kernel is called the inertia group of P|p.

The group Gal(kP/kp) is topologically cyclic and generated by the Frobenius map

φ : x 7→ xq. The inverse image of φ is called a Frobenius substitution for P, and it

is denoted by FrP. Note that FrP is well-defined up to conjugacy in Gal(K/F ). When

K/F is a Galois extension, we have that FrP does not depend of the choice of P, and

we may denote it as Frp (cf. [24]). If, further, p is also unramified at K (i.e., p is still

a prime ideal in OK), then we have that Frp is well-defined in Gal(K/F ).

We now introduce the notion of the adelic group. The restricted direct product

of a family of locally compact groups {Gα}α, with respect to the family of compact open

subgroups {Hα ≤ Gα}α, is defined as

∏α

′ Gα = {(xα) ∈∏α

Gα : xα ∈ Hα for almost all indexes α}.

Under this definition, it is more simple to define the following groups:

4 1.1. PRELIMINARIES

◦ The adele ring of a number field F is the restricted direct product over all places

of F ,

AF =∏ν

′ Fν ,

with respect to {Op : p is an archimedean place}; [20, Prop. 5.1, Lemma 5.2], [24]

◦ The idele class group is the restricted direct product

F ∗\A∗F =∏ν

′ F ∗ν

with respect to {O∗p : p is an archimedean place}; and

◦ The general linear group

GLn(AF ) =∏ν

′ GLn(Fν)

with respect to {GLn(Op) : p is an archimedean place}.

The last two groups are examples of adelic groups, that are nothing but groups of

the form G(AF ), where G is any algebraic group.

We finalize this section with some basic notions about representation theory. Let

G be a topological group. A representation of G is a continuous homomorphism

ρ : G → B(H), the group of bounded linear operators of a Hilbert space H (with

the operator norm structure). Two representations ρ and ψ, with respective Hilbert

spaces H and H ′, are equivalent if there exists an isomorphism Φ : H → H ′ such that

Φ◦ρ = ψ◦Φ. A representation ρ is said to be irreducible if it has no nontrivial invariant

closed subspaces, that is, if for any vector v 6= 0 in H the set {ρ(x)(v) : x ∈ G} is dense

in H. It is a fact that any representation of G may be decomposed as a direct sum of

irreducible representations. Moreover, any irreducible summand of ρ has attached an

invariant Hilbert subspace of H under ρ, inducing a decomposition of H as a direct sum

of invariant subspaces.

CHAPTER 1. ABELIAN CLASS FIELD THEORY 5

1.2 Abelian Class Field Theory

A basic problem in Galois theory is to give a survey of all the Galois extensions L of

a given field F . This raises the question: does there exist enough information in F ,

or even in an object related with F , to construct Gal(F )? Class Field Theory solves

this problem for abelian extensions [19, 24], and its main result is Artin’s Reciprocity

Law.

Let K/F be a finite field extension. K is a finite-dimensional vector space over F and

for every x ∈ K the map y 7→ xy is an endomorphism of K, whose matrix representation

over F we also denote by x. The norm map is defined as the map

NK : K∗ → F ∗, x 7→ det(x).

Theorem 1.2.1 (Artin Reciprocity Law, global case). Let F be a global field.

There exists a homomorphism, called the global Artin map,

AF : F ∗\A∗F → Gal(F )ab

such that:

1. For every abelian extension K/F , the composition AK/F , of the Artin map with

the canonical map Gal(F )ab → Gal(K/F ), is surjective with kernel NK(F ∗\A∗F ).

2. For every open subgroup H of F ∗\A∗F , of finite index, there exists an abelian ex-

tension K/F such that (F ∗\A∗F )/H ' Gal(K/F ).

There also exists an abelian class field theory for local fields [19], and is given by:

Theorem 1.2.2 (Artin Reciprocity Law, local case). Let F be a local field. There

exists a homomorphism, called the local Artin map,

AF : F ∗ → Gal(F )ab

such that:

1. For every abelian extension K/F , the composition AK/F , of the Artin map with

the canonical map Gal(F )ab → Gal(K/F ), is surjective with kernel NK(F ∗).

6 1.2. ABELIAN CLASS FIELD THEORY

2. For every open subgroup H of F ∗, of finite index, there exists an abelian extension

K/F such that F ∗/H ' Gal(K/F ).

Proofs of these fundamental theorems are given in [19, 24].

Remark 1.2.3. Let F be a number field and let p be a prime of F . Let jp : F ∗p → F ∗\A∗F

be the map x 7→ (1, . . . , 1, x, 1 . . . ). We have the relation between global and local Artin

maps

AFp = AF ◦ jp.

Let K/F be an abelian Galois extension of number fields, let P be a prime that lies

above p and let KP be the completion of K at P. Then we can identify any group

GP = Gal(KP/Fp) with a unique decomposition subgroup of Gal(K/F ) [24]. It can

be shown that the group GP is independent of P, and we denote it by Gp. Then we

have that AFp(KP) ⊆ Gp. We also have [24] that for any finite extension and for any

x ∈ F ∗\A∗F ,

AK/F (x) =∏

p

AKP/Fp(xp).

The latter claim is a consequence of the Hasse local-global principle, that states that

any property related to global fields can be obtained through similar properties related

to all the associated local fields. We justify this claim below in Chapter 3.

Remark 1.2.4. As we will see, in the Section 1.5, Gal(F ) has “a little of” characters;

moreover, it has “a little of” representations. The problem becomes more difficult for

higher dimensions because we need much more representations. In this way, there is

an important extension of Gal(F ) for any global field: its Weil group W(F )2 [26, 29].

This group satisfies the following properties:

1. There exists a continuous homomorphism ϕ : W(F ) → Gal(F ) with dense image.

2. If K/F is a finite extension of global fields, then W(F )/W(K) ' Gal(F )/Gal(K).

3. There exists an isomorphism ϑ : F ∗\A∗F →W(F )ab such that the map

F ∗\A∗F ϑ−→W(F )ab induced by ϕ−−−−−−−→ Gal(F )ab

is the Artin map.

2It is very intricate to define the Weil group of a global number field. Weil emphazised the necessityto redefine W(F ) in a more simple language [29].

CHAPTER 1. ABELIAN CLASS FIELD THEORY 7

4. Gal(F ) is isomorphic to the group of connected components of W(F ).

1.3 Artin L-functions for characters

It is well known that a lot of information about the rational primes is encoded in the

Riemann zeta function

ζ(s) =∞∑

n=1

1

ns,

which defines an holomorphic function on the half-plane <(s) > 1. It has an Euler

product

ζ(s) =∏

p prime

1

1− p−s

and satisfies the functional equation

ξ(s) = ξ(1− s), where ξ(s) = π−s/2Γ(s

2)ζ(s),

that defines a meromorphic continuation to the whole s-plane; Γ(s) denotes the Euler

gamma function. Analogue results are verified for the Dedekind zeta function of a

number field F , defined as

ζF (s) =∑

aEOF

1

N(a)s,

where N(a) is the index of the ideal a in OF . Note that the Riemann zeta function is

precisely the Dedekind zeta function for Q. In this section we introduce a generalization

of these functions, due to Artin, cf. [15, 19]. He introduced his L-functions to study

the structure of the Galois group for number fields, and his purpose was to develop a

nonabelian class field theory [4]. Although Artin defined his L-functions for any Galois

representation (see Chapter 3.1 below), in this section we only consider characters of

Gal(F ).

Remark 1.3.1. Let G be a profinite group and let ρ be a complex n-dimensional

representation of G, that is, a homomorphism ρ : G → GLn(C). We have that ρ

factorizes through a finite group. In fact, let V be a neighborhood of the identity matrix

“1” in GLn(C), containing no nontrivial subgroups of GLn(C); this neighborhood exists

because any matrix in GLn(C) that is close enough to 1 is of the form exp(A) for some

8 1.3. ARTIN L-FUNCTIONS FOR CHARACTERS

n × n matrix A over C. Let U = ϕ−1(V ). By definition of the Krull topology for

G [19, 20], there exists a normal subgroup N E G of finite index such that N ⊆ V .

Therefore, we may consider the well-defined representation ρ : G/N → C given by

ρ(xN) := ρ(x). It is clear that φ has finite image.

Let F be a number field, and let ρ be a complex character of Gal(F ), that is, a

representation ρ : Gal(F ) → C∗ ' GL1(C). Since Gal(F ) is profinite, we have that

ρ factorizes through a finite group, corresponding to a finite Galois extension K/F .

We know [20] that this extension is unramified for almost all primes p of F ; hence the

Frobenius map Frp is defined for almost all primes p.

Definition 1.3.2. We define the Artin L-function associated to the character ρ as

L(ρ, s) = LK/F (ρ, s) :=∏

p

1

1− ρ(Frp)N(p)−s,

where the product runs through all unramified primes p. We also define the local

L-factors of F for any prime p as

L(ρp, s) =

{1

1−ρ(Frp)N(p)−s , if p is unramified

1, otherwise.

We have that L(ρ, s) =∏

p L(ρp, s).

Remark 1.3.3. The Artin L-function satisfies the following properties [19]:

1. From the principal character ρ = 1 we recover the Dedekind zeta function

L(1, s) = ζF (s).

This is because we can choose K = F in the decomposition of ρ.

2. If we have a tower of fields F ⊆ K ⊆ E, then LE/F (ρ, s) = LK/F (ρ, s), viewing the

character ρ of Gal(K/F ) as a character of Gal(E/F ).

3. If F ⊆ K ⊆ E is a tower of fields, ρ is a character of Gal(E/K) and θ is the

induced character of Gal(E/F ) [19, p. 122], then LE/F (θ, s) = LE/K(ρ, s).

Artin gave a definition of L-function for any n-dimensional representation of Gal(F ),

and his definition also considers the ramified places of finite extensions. We will treat it

CHAPTER 1. ABELIAN CLASS FIELD THEORY 9

at Chapter 3. For the case n = 1 (that corresponds to taking characters) this definition

reduces to set, the corresponding L-factor to ramified places, L(ρp, s) = 1.

Artin also described L-factors for archimedean places. He proved that this “com-

plete” L-function for characters has a meromorphic continuation and a functional equa-

tion

L(ρ, s) = ε(ρ, s)L(ρ, 1− s),

where ε(ρ, s) is an entire function without zeroes and ρ is the dual character of ρ.

1.4 Tate’s Thesis and Hecke L-functions

In this section we introduce another generalization of Dedekind zeta functions, due to

Hecke [20, 25]. He considered continuous characters χ of the idele class group of a number

field F and was also able to establish the meromorphic continuation and the functional

equation of L(χ, s). However, Hecke had too many problems to establish the functional

equation because he used similar methods to those to obtain the Riemann functional

equation, and that was very complicated. Around 1950, Tate, following a suggestion of

his adviser, Artin, re-proved Hecke results in his Ph.D. thesis using harmonic analysis

in a much more straightforward way. We follow Tate’s basic ideas.

Definition 1.4.1. Let F be a local field. It admits an absolute value and a Haar measure

dx [20]. We define the measure

d∗x = cdx

|x|for a fixed and suitable c > 0. Let χ be a (continuous) character of F ∗.

We say that χ is unramified if χ(x) = 1 whether |x| = 1. If F is non-archimedean

with residual field Fq and uniformizing parameter π, we define [4, 20]

L(χ) =

{(1− q−sχ(π))−1, if χ is unramified

1, otherwise.

If F = C, then χ = rseinθ for some n ∈ Z, s ∈ C. We define [20, p. 244]

L(χ) = (2π)−(s+|n|/2)Γ(s +|n|2

).

10 1.4. TATE’S THESIS AND HECKE L-FUNCTIONS

For F = R, we have χ = | · |s or χ = sgn · | · |s, where sgn(x) = 1 or −1 whether x > 0

or x < 0. We define

L(χ) =

{π−s/2Γ( s

2), if χ = | · |s

π−(s+1)/2Γ( s+12

), if χ = sgn · | · |s.Finally, we also define the local L-factor attached to F and χ as L(χ, s) = L(χ| · |s).

For a given local field F , a Schwartz-Bruhat function defined on F is:

1. A C∞-complex-valued function f defined on F such that p(x)f(x) → 0 as |x| → ∞for any polynomial p(x) if F is archimedean.

2. A locally constant complex-valued function with compact support if F is non-

archimedean.

Let S(F ) denote the space of all Schwartz-Bruhat functions defined on F .

Given a function f ∈ S(F ), and given a nontrivial character ψ of F , the Fourier

transform of f is defined as

f(y) =

∫

F

f(x)ψ(xy)dx.

When f is well-defined, it lies in S(F ), however, it depends on the choice of ψ and dx.

We also define the local zeta function associated to f and the character χ of F ∗ as

ζ(f, χ) =

∫

F ∗

f(x)χ(x)d∗x.

This zeta function satisfies the functional equation [20, Thm. 7.2]

L(χ)ζ(f , χ) = ε(χ, dx) L(χ) ζ(f, χ),

where χ = χ−1| · | and the ε-factor is an entire factor of s that depends of a special

character ψ called the standard character [20, p. 253].

Definition 1.4.2. Let F be a number field and let χ : F ∗\A∗F be an idele class character.

We have that χ decomposes as a product of local characters

χ =∏ν

χν

CHAPTER 1. ABELIAN CLASS FIELD THEORY 11

where χν is a character of the local field Fν and χν is unramified for almost all places ν.

We define the Hecke L-function attached to F as

L(χ, s) = LF (χ, s) =∏ν

L(χν , s).

We have that L(χ, s) defines a holomorphic function on the half-plane <(s) > 1 [20,

Thm. 7.19]. We also have a functional equation of Hecke L-functions in terms of L- and

ε-factors:

L(χ, 1− s) = ε(χ, s)L(χ, s),

where ε(χ, s) =∏

ε(χν | · |s, dxν), this product running through all places ν. Thus, Hecke

L-functions admit a meromorphic continuation on the whole complex plane.

Remark 1.4.3. For the trivial character χ = 1, it holds that the Dedekind zeta function

of F is precisely LF (1, s) = ζF (s).

1.5 The Langlands Program for n = 1

Remark 1.5.1. Let G be a topological group. For any x, y ∈ G and any (continuous)

character ϕ : G → C it holds that ϕ([x, y]) = 0. Let G∗ be the closure of the com-

mutator subgroup [G,G]. Then we have that any character of G anihilates on G∗, and

Gab = G/G∗ is abelian. Therefore, studying the topological abelianization of a group is

equivalent to studying every nonzero characters of this group.

This remark implies two facts. Let F be a number field. On one side, we have the

topological group Gal(F )ab, and it is equivalent to the set of all complex characters ρ

of Gal(F ). Since any ρ factorizes through a finite subgroup of Gal(F ) (Remark 1.3.1),

any character ρ may be seen as the character of an abelian finite extension Kρ/F ,

and we may associate an Artin L-function LK/F (ρ, s) = L(ρ, s), which is well-defined by

Remark 1.3.3. On the other hand, the study of the idele class group F ∗\A∗F is equivalent

to studying all idele class characters. If χ is a such character, then we may associate to

it a Hecke L-function L(χ, s).

Now consider the Artin map

AF : F ∗\A∗F → Gal(F )ab.

12 1.5. THE LANGLANDS PROGRAM FOR N = 1

Let ρ be a character of Gal(F ). We have that the Artin map induces an idele class

character χ such that the diagram

Gal(F ) Gal(F )ab,

F ∗\A∗F C

π //

ρ

OOχ //

AQ

%%LLLLLLLLLL

where π is the canonical projection, is commutative. Then the Artin map sends Galois

characters to Hecke characters. Note that, since ρ has finite order, χ also has finite

order3.

What happened to their L-functions? First consider F = Q. Let p be any rational

prime and let Qab(p) be the maximal abelian extension of Q that is unramified at p. We

have that [7]

Gal(Qab(p)/Q) '∏

q 6=p prime

Z∗q,

which is also isomorphic to the group of connected components of Q∗\A∗Q/Z∗p. Deligne

[7] defined the Artin map in such a way that the Frobenius map Frp is sent to the

double coset of the adele (1, . . . , 1, p, 1, . . . )4. Under this definition we can claim that

the Artin map sends the adele (1, . . . , 1, p, 1, . . . ) to the Frobenius element Frp. This

convention can be easily stated for other number fields, and we have the association

(1, . . . , 1, πp, 1, . . . ) 7→ Frp, where πp is any uniformizing parameter of Fp.

Now let χ be an idele class character of F . We have that

χ(1, . . . , 1, πp, 1, . . . ) =∏

q

χq(xq),

where xq = 1 for any q 6= p, and xp = πp. That is, χ(1, . . . , 1, πp, 1, . . . ) = χp(πp). By

another side, any character ρ of Gal(F ) is uniquely determined by its values in every

Frp; this is because the group generated by all the Frp’s is dense in Gal(F ). Then we

have that through the Artin map

χp(πp) = ρ(Frp)

3We have a bijective correspondence between all characters of F ∗\A∗F of finite order and all Dirichletcharacters [20, pp. 237-238]

4As a matter of fact, Deligne used the inverse of the Frobenius map. He wanted reformulate theLanglands Program in a geometric language [7].

CHAPTER 1. ABELIAN CLASS FIELD THEORY 13

for almost all primes p (at least for all the unramified ones) if χ is the character associated

to ρ. Hence

L(ρp, s) = L(χp, s)

for almost primes p, and the L-functions are essentially the same.

Summarizing, we have the following:

Theorem 1.5.2. There exists a one-to-one correspondence between characters of Gal(F )

and characters of F ∗\A∗F ' GL1(F )\GL1(AF ) of finite order. The correspondence ρ 7→ χ

is given by χ = ρ ◦ AF , where AF is the Artin map, and it satisfies that

L(ρp, s) = L(χp, s)

for almost all primes p.

We finalize this section with a definition. Let L2(GL1(F )\GL1(AF )) be the space

of all square-integrable functions on GL1(F )\GL1(AF ). We have the regular right

representation of GL1(AF ) on L2(GL1(F )\GL1(AF )) given by the translation

x 7→ (R(x) : f 7→ fx),

where fx(y) = f(yx) for all y ∈ GL1(AF ). This representation decomposes as a direct

sum of irreducible unitary representations on L2(GL1(F )\GL1(AF )), called automor-

phic representations on GL1(AF ) [10, p. 99]. Note that

R(x) =⊕

χ

∫

GL1(F )\GL1(AF )

χ(x)dχ,

where the sum runs through all idele class characters. Therefore automorphic representa-

tions on GL1(AF ) are simply idele class characters. Since any character is an (irreducible)

one-dimensional representation, we may rewrite the Theorem 1.5.2 as follows:

Theorem 1.5.3 (Langlands Correspondence, n = 1). There exists a one-to-one

correspondence between one-dimensional representations of Gal(F ) and automorphic

representations on GL1(AF ) of finite image. The correspondence ρ 7→ π is given by

χ = ρ ◦ AF , where AF is the Artin map, and it satisfies

L(ρp, s) ≡ L(χp, s)

for almost all primes p.

Chapter 2

Shimura-Taniyama Conjecture

The Shimura-Taniyama conjecture, that establishes a connection between elliptic

curves and modular forms, was stated by the Japanese mathematicians Yutaka Taniyama

and Goro Shimura around 1955. French mathematician Andre Weil gave many examples

[30] supporting the conjecture1. Nowadays, this conjecture is a theorem (called the

modularity theorem), which was firstly shown for semistable curves by Andrew Wiles

[32] (and proving the Fermat’s Last Theorem in the process) and for the general case by

Ch. Breuil, B. Conrad, F. Diamond and R. Taylor [2].

In this chapter we show the relationship between the Shimura-Taniyama conjecture

and the Langlands Program. We start developing the necessary background to state the

conjecture and then we introduce the shift of paradigm, due to Langlands, to associate

an automorphic representation to each modular form. Finally, we present the Shimura-

Taniyama conjecture as a particular subcase of the case n = 2 of the Langlands Program.

2.1 Elliptic curves and modular forms

Definition 2.1.1. An elliptic curve is a complex curve given by an equation of the

form

E : y2 = x3 + ax + b

where its discriminant ∆E is nonzero2.

1Weil left this problem “as an exercise for the interested reader” [30]. That article contributed tospread the conjecture all over the world.

2A Weierstrass equation over Q is a cubic equation

E : y2 + a1xy + a3y = x3 + a2x2 + a4x + a6,

16 2.1. ELLIPTIC CURVES AND MODULAR FORMS

Remark 2.1.2. There exists a change of variable

x = u2x′ + r, y = u3y′ + su2x′ + t,

with u 6= 0, r, s, t ∈ Q suitably chosen, such that ∆E is as minimum as possible in the

sense that, if ∆E′ is the same curve under other such change of variable, then ∆E divides

∆E′ . Then, we may assume that the equation defining E has this property. Moreover,

we may suppose that a, b ∈ Z [6, p. 323].

We may associate an elliptic curve E with a complex torus as follows. Let z1, z2 ∈ Cbe such that z1/z2 6∈ R, and consider the lattice Λ = z1Z ⊕ z2Z. A complex torus is

simply a quotient of the form C/Λ for some lattice Λ. Note that all the meromorphic

functions defined over the torus C/Λ are in a 1-1 correspondence with all doubly periodic

functions of period (z1, z2).

For the lattice Λ we define the Weierstrass ℘-function as

℘(z) =1

z2+

∑

ω∈Λ,ω 6=0

(1

(z − ω)2− 1

ω2

), z ∈ C \ Λ.

We have that ℘ is meromorphic over C/Λ and satisfies the functional equation

℘′(z)2 = 4℘(z)3 − g2℘(z)− g3,

where g2 = 60∑

ω 6=01

ω4 and g3 = 140∑

ω 6=01

ω6 . We have a map C/Λ → C2 given by

z 7→ (℘(z), ℘′(z)).

This map is a bijection, and it extends to a map C→ E, where E is the corresponding

projective curve. Conversely, any elliptic curve arises in this way by a complex torus [6].

An operation can be defined on the elliptic curve E through lines and points. How-

ever, the correspondence with complex tori gives a more natural construction of this

operation. In fact, this operation is simply the lifting of the natural quotient operation

in an associated torus C/Λ over the elliptic curve [6, p. 34]. This lifting identifies the

zero of C/Λ with the “point at infinity” of the projective elliptic curve.

where all the ai’s are rational numbers. Is straightforward verifying [6, p. 310]) that every Weierstrassequation can be reduced to an elliptic curve y2 = x3 + ax + b.

CHAPTER 2. SHIMURA-TANIYAMA CONJECTURE 17

Given an elliptic curve E, we are interested in its reductions Ep modulo any rational

prime p. According with the image of E in Fp, we say that the reduction of E at p

is good or bad whether Ep defines or not an elliptic curve, that is, whether ∆Ep is or

not zero in Fp respectively. This classification of all primes (with respect to E) is more

subtle [6, p. 323], and is perfectly encoded by the conductor of E, an integer number

NE that satisfies

p|∆E ⇔ p|NE.

Moreover, E has a bad reduction only at any prime p dividing NE, and the kind of

reduction is given by the exponent of p in the factorization of NE.

We define for any p prime and any k ∈ N the number apk(E) = p + 1− |Epk |, where

|Epk | is the number of points of the reduction of E in the finite field Fpk . There is a

recursive expression for apk(E), but we avoid it. However, we can define an(E) for any

n ∈ N multiplicatively,

amn(E) = am(E)an(E) ∀m,n ∈ N, gcd(m, n) = 1.

This permit us define the Hasse-Weil L-function of E as

L(E, s) =∞∑

n=1

an(E)n−s =∏

p prime

(1− ap(E)p−s + 1E(p)p1−2s)−1,

where 1E is the trivial character modulo NE of E.

The other objects involved in the Shimura-Taniyama correspondence are modular

forms. But we need some technical language. Let SL2(Z) be the group of invertible

matrices with determinant one. We know that this group is isomorphic to the group

of Mobius transformations that leave invariant the upper halfplane H. Moreover, the

isomorphism is given by

(a bc d

)= γ → (fγ : z 7→ az + b

cz + d);

so that SL2(Z) acts on H.

18 2.1. ELLIPTIC CURVES AND MODULAR FORMS

Definition 2.1.3. A congruence subgroup at level N ∈ N of SL2(Z) is a subgroup

Γ that contains the subgroup

Γ(N) =

{(a bc d

)≡

(1 00 1

)mod N

}.

A weakly modular form of weight k ∈ Z with respect to Γ, or simply a weakly

Γ-modular form of weight k is a meromorphic function f : H → C such that

f(z) = (cz + d)−kf(γ(z)) =: f [γ]k(z)

for every γ ∈ Γ, z ∈ H.

For each congruence subgroup Γ there exists a minimum h ∈ N such that ( 1 h0 1 ) ∈ Γ.

This matrix is equivalent to the translation operator z 7→ z + h. Therefore, if f is a

weakly Γ-modular form of weight k, then f is hZ-periodic. The holomorphic map

z 7→ q = e2πiz

takes H to the punctured disc D \ {0}; then we may see the function f as a function

g : D \ {0} → C, g(q) = f(z).

If f is holomorphic in H, then g is also holomorphic on D \ {0} and it has a Laurent

expansion around 0. We say that f is holomorphic at ∞ if g extends holomorphically

to q = 0. In this case, f has a Fourier expansion

f(z) =∞∑

n=0

ane2πinz/h.

This observation motivates the following definition.

Definition 2.1.4. Given a congruence subgroup Γ and an integer k, a function f : H →C is a Γ-modular form of weight k if it is holomorphic, weakly Γ-modular of weight

k and if f [γ]k is holomorphic at ∞ for any γ ∈ SL2(Z). If in addition a0 = 0 in the

Fourier expansion of f [γ]k for all γ, then f is a Γ-cuspidal form of weight k.

Let Mk(Γ) and Sk(Γ) be the vector spaces of all Γ-modular and Γ-cuspidal forms of

weight k respectively. We are interested in the structure of these spaces when Γ = Γ1(N)

is the congruence subgroup

Γ1(N) =

{(a bc d

): a ≡ d ≡ 1, c ≡ 0 mod N

},

CHAPTER 2. SHIMURA-TANIYAMA CONJECTURE 19

and when Γ = Γ0(N), where

Γ0(N) =

{(a bc d

): c ≡ 0 mod N

}.

In both cases we have the decomposition, as a direct sum, [6, p. 119]

Mk(Γ) =⊕

χ

Mk(Γ, χ), Sk(Γ) =⊕

χ

Sk(Γ, χ),

where each sum runs through all Dirichlet characters χ modulo N and Mk(Γ, χ) is the

χ-eigenspace of Mk(Γ),

Mk(Γ, χ) =

{f : f [γ]k = χ(d)f for all γ =

(a bc d

)∈ Γ

},

and similarly for cuspidal forms. Note that Mk(Γ0(N)) ⊆ Mk(Γ1(N)), this is because

Γ0(N) ⊇ Γ1(N). The same inclusion holds for cuspidal forms and for χ-eigenspaces.

Finally, let N ∈ N fixed. We define the Hecke operators on Mk(Γ1(N)) as follows.

First, for any n ∈ N we define the operator [6]

〈n〉(f) =

{f [γ]k, if gcd(N, n) = 1;

0, if gcd(N, n) > 1.

Here, γ is any matrix in Γ0(N) such that its bottom right entry is congruent with n

modulo N . On the other hand, we also define the operator

Tp(f) = f [γp]k,

where γp = ( 1 00 p ) and p is a rational prime. For composite n ∈ N we define Tn =

∏Tpr

where n =∏

pr, and Tpr is defined inductively on r [6, p. 178]. There is an explicit

expression for Tp. We define a newform f ∈ Sk(Γ0(N)) as an eigenform (i.e., an

eigenvector) for all the Hecke operators 〈n〉 and Tn such that a1(f) = 1 [6, p. 195].

Note that, if f is a newform with Fourier expansion f(z) =∑∞

n=1 an(f)qn, then

Tn(f) = an(f) for all n ∈ N.

Each modular form f has attached a natural L-function

L(f, s) =∞∑

n=1

an(f)n−s,

20 2.2. ELLIPTIC CURVES AND GALOIS REPRESENTATIONS

where f(z) =∑∞

n=0 an(f)e2πinz is its Fourier expansion. If f is cuspidal, the series

L(f, s) converges and is holomorphic on the halfplane <(s) > k/2 + 1.

We have the following theorem [2, 6, 32]:

Theorem 2.1.5 (Shimura-Taniyama conjecture). Let E be an elliptic curve over

Q with conductor NE. Then there exists a newform f ∈ S2(Γ0(NE)) such that

L(E, s) = L(f, s).

2.2 Elliptic curves and Galois representations

Remark 2.2.1. Remember (Remark 1.3.1) that every finite-dimensional representation

of a profinite group G has finite image in C. Therefore, the image of a such representation

φ is contained in a number field K (and we may assume that this field is minimal under

inclusion). Let ` be a prime number, and let L be a prime of K lying above `. Consider

the completion KL of K at L. Then [20] KL is a finite extension of the `-adic fieldQ`, and

φ(G) is contained in KL. Hence, we may consider any finite-dimensional representation

of G as an `-adic representation.

Let E be an elliptic curve. For any N ∈ N, let E[N ] denote the set of N-torsion

points of E, that is, the set of all points z ∈ C/Λ, where C/Λ is an associated torus to

E, that Nz = 0 in C/Λ. That is, we say that a point x + Λ ∈ E is in E[N ] if the point

Nx ∈ C lies in Λ.

Definition 2.2.2. Let E be an elliptic curve, and let ` be a prime number. The `-adic

Tate module of E is the inverse limit

Ta`(E) = lim←

E[`n]

with respect to the directed system

E[`] E[`2] E[`3] . . .`oo `oo `oo

where the maps are multiplication by `.

CHAPTER 2. SHIMURA-TANIYAMA CONJECTURE 21

Remark 2.2.3. Let (Pn, Qn) be a basis of E[`n] for each n ∈ N. We can choose each

basis such that

` · Pn+1 = Pn, ` ·Qn+1 = Qn.

Note that each basis determines an isomorphism E[`n]∼−→ (Z/`nZ)2. This implies that

Ta`(E) ' Z2` .

Let n ∈ N. The field Q(E[`n]) is a Galois number field; hence we have the restriction

map Gal(Q) → Gal(Q(E[`n])/Q). These maps give us an injection

Gal(Q(E[`n])/Q) → Aut(E[`n]).

The maps are compatible in the sense that the diagram

Aut(E[`n]) Aut(E[`n+1])

Gal(Q)

ooÂÂ?

????

????

??

ÄÄÄÄÄÄ

ÄÄÄÄ

ÄÄÄÄ

is commutative for any n. That is, the Tate module is a Gal(Q)-module [6]. Since

Aut(E[`n])∼−→ GL2(Z/`nZ) implies Aut(Ta`(E))

∼−→ GL2(Z`), we have a continuous

homomorphism

ρE,` : Gal(Q) → GL2(Z`) ⊆ GL2(Q`).

That is, the Tate module of an elliptic curve induces a two-dimensional `-adic represen-

tation of Gal(Q), that is called the Galois representation associated to E.

A representation Gal(Q) → GLn(F ) is unramified at a prime p if its kernel contains

the inertia subgroup of every maximal ideal p ∈ Z lying above p, where Z is the ring of

algebraic integers. We have the following

Theorem 2.2.4. The representation ρE,` is unramified at any prime p that does not

divide `N , where N is the conductor of E. The characteristic polynomial of ρE,`(Frp)

for those p is

X2 − ap(E)X + p.

Moreover, ρE,` is irreducible.

22 2.3. MODULAR FORMS AND AUTOMORPHIC REPRESENTATIONS

Given an elliptic curve E, the L-function associated to the Galois representation

ρE,` is given by

L(ρ, s) =∏

p

[det(1− p−sρ(Frobp))]−1,

Note. We can associate to any elliptic curve a zeta function [13, Ch. 18]. This zeta

function and the L-series are related by the equality ζ(E, s)L(E, s) = ζ(s)ζ(s−1), where

ζ(s) is the Riemann zeta function. We also have the Birch and Swynnerton-Dyer

conjecture, roughly speaking, it says that the behavior of the meromorphic continuation

to L(E, s) on the line s = 1 has important information about the curve E.

We will see below that, multiplying by some special factors for those primes divid-

ing `N , the L-function associated to the representation ρE,` is the Artin L-function

associated to the representation ρ.

2.3 Modular forms and automorphic representations

In this section we associate automorphic forms to modular forms. First we have the

following theorem [10, 27]:

Theorem 2.3.1 (Strong Approximation). Let N ∈ N. The adelic group GL2(AQ)

can be decomposed as

GL2(AQ) ' GL2(Q)GL+2 (R) K0(N),

where GL+2 (R) is the set of all real matrices with positive determinant and K0(N) is

the subgroup of∏

p GL2(Zp), the product running through all primes p, of all matrices

whose lower left entry is in N Z, where Z is the direct limit of all Z/nZ with respect the

canonical maps Z/nZ→ Z/mZ when m|n.

Hence, every α ∈ GL2(AQ) can be (non-uniquely) written as α = γg∞κ, with γ ∈GL2(Q), g∞ ∈ GL+

2 (R) and κ ∈ K0(N).

There is a more general result [10], but this case is enough for our purposes.

Definition 2.3.2. Let f be a modular form of weight k at level N in Mk(Γ0(N), χ),

with χ a Dirichlet character modulo N . Recall that χ can be seen as a character of

CHAPTER 2. SHIMURA-TANIYAMA CONJECTURE 23

GL1(Q)\GL1(AQ), that we still denote as χ. The adelization of f is defined as the

map ϕf : GL2(AQ) → C defined by

ϕf (α) = [det(g∞)−1/2(ci + d)]−kf

(ai + b

ci + d

)ωχ(κ),

where α = γg∞κ is as in Theorem 2.3.1, g∞ = ( a bc d ) in GL+

2 (R) and ωχ denotes here the

evaluation of χ at the lower right entry of κ.

Note that ϕf is well defined (this is because the modularity of f); is continuous by

definition; is GL2(Q)-left invariant (ϕf (γα) = ϕf (α), ∀γ ∈ GL2(Q)); is smooth at the

infinite component; and is locally constant at each finite component3 of GL2(AQ) [27].

The function ϕf has another properties [10]:

1. (K-finiteness) For g∞ = ( cos θ sin θ− sin θ cos θ ) ∈ SO2(R), κ ∈ K0(N) and a ∈ GL2(AQ) we

have that

ϕf (γg∞κ) = ωχ(κ)e2πikθϕf (γ).

We can rewrite this assertion adelicly as follows: let K be the maximal compact

subgroup of GL2(AQ), which is isomorph to SO2(R)×∏p GL2(Zp). K-finiteness is

equivalent to the following property: the subspace span{R(α)ϕf : α ∈ K}, where

(R(α)ϕf )(β) = ϕf (αβ) for all β ∈ GL2(AQ) is the right regular representation

of GL2(AQ), is finite-dimensional.

2. (Action of the center) Let Z be the center of GL2(AQ). We have that Z ' A∗Q;

then we may see ωχ as a character of Z. Note also that

ϕf (zα) = ωχ(z)ϕf (α), ∀α ∈ GL2(AQ), z ∈ Z.

3. (Growth condition) There exists a real number A > 0 such that ϕf (α) = O(‖α‖A),

where ‖α‖ is the norm ‖α‖ =∏

ν | det(αν)|ν . Moreover, if f is cuspidal, ϕf is

bounded, that is, we may choose A = 0.

3A finite component of an adelic group G(AQ) ' ∏ ′G(Qν) is a component associated with anarchimedean place; the infinite component of G(AQ) is the component associated with the non-archimedean place. This extension can be extended to adelic groups associated with other numberfields.

24 2.3. MODULAR FORMS AND AUTOMORPHIC REPRESENTATIONS

4. (Cuspidality) If f is cuspidal, for any α ∈ GL2(AQ)∫

Q\AQ

ϕf

((1 x0 1

)α

)dµ(x) = 0.

Now, given a modular form f of weight k at level N and eigencharacter χ, define

the modular automorphic representation attached to f as the restriction of the

(unitary) right regular representation of GL2(AQ) on the closed subspace

Hf := span{R(a)ϕf : a ∈ GL2(AQ)}

of L20(ZGL2(Q)\GL2(AQ), ω) (see Section 4.2 below). Denote by πf such representation.

Note that [10] if f is a newform, or if f is an eigenform of almost all Hecke operators

Tp, then πf is irreducible.

Let πf be a modular automorphic form. Then [7, 10] it decomposes as a restricted

tensor product

πf = π∞ × (⊗

p

′πp),

where each πp is an irreducible representation of GL2(Qp) and π∞ is a module on the

Lie algebra gl2. We say that πp is unramified if πp has a non-zero GL2(Zp)-invariant

vector vp in its space of representation Vp. Note that πp is unramified for any prime

p - N .

Definition 2.3.3. Let N ∈ N, let f be a newform at level N and let p - N be a

prime number. The spherical Hecke algebra corresponding to p is the algebra Hp

of those locally constant functions f : GL2(Qp) → C of compact support and GL2(Zp)-

biinvariant,

f(xay) = f(a), ∀x, y ∈ GL2(Zp), a ∈ GL2(Qp),

with the product given by convolution,

(f ∗ g)(a) =

∫

GL2(Qp)

f(ab−1)g(b) db.

We have [7] that Hp is isomorphic to the free algebra in two generators H1,p, H2,p,

whose action on the invariant vector vp of πp is given by the formula

Hi,p · vp =

∫

M i2(Zp)

ξp(a) · vp da,

CHAPTER 2. SHIMURA-TANIYAMA CONJECTURE 25

where ξp : GL2(Zp) → End Vp is the representation homomorphism and M i2(Zp) are the

double cosets

M12 (Zp) = GL2(Zp)

(p 00 1

)GL2(Zp), M2

2 (Zp) = GL2(Zp)

(p 00 p

)GL2(Zp),

in GL2(Qp). The operators Hi,p are called the Hecke operators of GL2(Qp) [7, 27].

We have that Hi,p · vp is a GL2(Zp)-invariant vector for i = 1, 2; hence Hi,p · vp =

zi(πp)vp, where the zi(πp)’s are called the Hecke eigenvalues of πp.

Hecke eigenvalues zi(πp) permite us define the local L-factor associated to πp as

L(πp, s) := (1− p−sz1(πp)) (1− p−sz2(πp)). Define the Hecke L-function associated to

π = πf as

L(π, s) :=∏

p

L(πp, s).

2.4 The Langlands Program for n = 2

In the previous sections we associated two-dimensional representations of Gal(Q) to

elliptic curves, and we also saw how to attach a cuspidal form and an automorphic

representation of GL2(AQ). By another side, we have the Shimura-Taniyama conjecture,

that relates elliptic curves and cuspidal forms through L-functions. Call an elliptic

representation to the representation ρE,` of Gal(Q) associated to an elliptic curve E,

and call a new-automorphic representation to the automorphic form πf of GL2(AQ)

attached to a newform F . This correspondence can be translated to the new developed

paradigm, and we may state the Shimura-Taniyama conjecture, now a theorem as:

Theorem 2.4.1. There exists a one-to-one correspondence between elliptic Galois rep-

resentations ρ and new-automorphic representations π of GL(AQ). The correspondence

is such that

L(ρp, s) = L(πp, s)

for all primes p.

Here, we use the extension of the L-functions associated to each object. Note that

these L-functions are not the ones stated in the original conjecture; hence, the ε-factors

are not the ones either. However, the parameters involved in both types of L-functions

26 2.4. THE LANGLANDS PROGRAM FOR N = 2

are closely related, and we can consider they as the same. Therefore, the Shimura-

Taniyama conjecture implies the Theorem 2.4.1, and viceversa. That is, Theorems 2.1.5

and 2.4.1 are equivalent.

Note also that we may reformulate the problem in more general terms. We may

consider all two-dimensional `-adic representations of Gal(Q) and all automorphic rep-

resentations of GL2(AQ) that are realized in L2(GL2(Q)\GL2(AQ)). Moreover, we may

consider an arbitrary number field F instead of Q. In this case, we have the following

conjecture:

Conjecture 2.4.2 (Langlands Correspondence, n = 2). Let ρ be a two-dimensional

representation of Gal(F ). Then there exists an automorphic representation of GL2(AF )

that is realized in L20(GL2(F )\GL2(AF )). The association is such that

L(ρ, s) = L(π, s).

Moreover, the L- and ε-factors are equal for every prime p.

Remark 2.4.3. This conjecture is still unproven, even for Q, in contrast to the Shimura-

Taniyama conjecture, that was verified through Galois representations techniques. The

proof of the modularity theorem uses the shift of paradigm developed at [6].

The converse of the Conjecture 2.4.2 is false [7, p. 32]. This is because the archimedean

places are much more delicate than the non-archimedean ones; the group Gal(F ) acts

on the Lie algebra of each archimedean place. For the case F = Q, there are auto-

morphic representations whose archimedean factor is a representation of the principal

series of representations of gl2. For those automorphic representations there are no two-

dimensional Galois representations corresponding to them. This disadvantage is avoided

using the Weil group (cf. Remark 1.2.4) instead of the Galois group, because the former

has more representations.

Chapter 3

The Langlands Program

Let us summarize the main previous results. In the chapter one we study the Artin reci-

procity law, and we rewrite it as a correspondence, through L-functions and ε-factors,

between one-dimensional representations of Gal(F ) and automorphic representations of

GL1(AF ) that are constituents in the decomposition of the space L2(GL1(F )\GL1(AF ))

under the right action of GL1(AF ). On the other hand, we associate the Shimura-

Taniyama conjecture to a–conjectural–correspondence between two-dimensional repre-

sentations of Gal(F ) and some automorphic representations of GL2(A, F ) realized in the

space L2(GL2(F )\GL2(AF )).

In this chapter we treat the general case, that is, the case GLn. We will start intro-

ducing the Artin L-function attached to an n-dimensional representation ρ of Gal(F ).

We also state its meromorphic equation and its functional equation. Then we develop

the concept of automorphic representation and define its associated Hecke L-function,

with its meromorphic continuation and its functional equation. All this bring us to

the principal statement: The Langlands Program. We also give some remarks for fur-

ther analysis. Finally, we present some examples of reciprocity laws and remark their

relationship with the Langlands Program.

3.1 Galois representations and L-functions

Let F be a number field and let n ∈ N. Consider a n-dimensional representation ρ of

Gal(F ). Let p be a prime of F , and let Frp be any Frobenius substitution. We have that

ρ(Frp) is well-defined [4].

28 3.1. GALOIS REPRESENTATIONS AND L-FUNCTIONS

If V ' Cn is the vector space where Gal(F ) acts via ρ, let V Ip be the subspace of V

fixed by any element of the inertia subgroup Ip of P|p1,

σ(v) = v, ∀v ∈ V Ip , σ ∈ Ip.

Note that if p is unramified then V Ip = 0.

Definition 3.1.1. Under the above notation, the local L-factor (associated to ρ and

p) is defined as

L(ρp, s) :=1

(1−N(p)−sρ(σp))|V Ip

.

This definition works only for non-archimedean places. For the archimedean ones we

need some remarks. We have two cases:

1. For the complex places, the Galois group Gal(C) is the trivial group, and the

decomposition group DC is also the trivial group.

2. For the real places, the Galois group Gal(R) is of order 2, and it contains the trivial

automorphism and the complex conjugation σ. In this case, DR is isomorphic to

〈σ〉 ' Gal(R).

In any case, if ρ is a representation of Gal(F ), whether F = R or C, then ρ|Dν

decomposes into a direct sum of characters. In the complex case, ρ|Dν = 1 ⊕ · · · ⊕ 1 is

the direct sum of n trivial characters; in the real case, ρ decomposes as the direct sum

of n+ trivial characters and n− times the sign character. Note that n+ + n− = n.

Definition 3.1.2. Under the above notation, let

ΓR(s) = π−s/2Γ(s

2); ΓC(s) = 2(2π)−sΓ(s).

We define the local L-factor (associated to ρ and ν) as

L(ρν , s) =

{ΓR(s)

n+ ΓR(s + 1)n− , if ν = RΓC(s)

n, if ν = C.

1Here, P is any prime that lies above p. We have that the inertia subgroups of any two primes lyingabove p are isomorphic; hence we can define Ip unambiguously.

CHAPTER 3. THE LANGLANDS PROGRAM 29

Definition 3.1.3. Let F be a number field, and let ρ be a n-dimensional representation.

The Artin L-function associated to ρ is defined as

LF (ρ, s) = L(ρ, s) =∏ν

L(ρν , s),

where the product runs through all places of F .

Remark 3.1.4. For the case n = 1, the inertia subgroup of any prime is isomorphic to

0 or C. That is, if p is ramified then the unique vector fixed by the character ρ(σp) is the

vector 0. Therefore, the L-factor associated to p is by definition L(ρp, s) = 1, and this

definition corresponds with that given at the chapter one (except for the archimedean

places).

This function was defined by Artin in 1930 [4]. It has the following properties:

1. If ρ is such that tr(ρ) = 1, then the Artin L-function corresponds to the Dedekind

zeta function,

LF (ρ, s) = ζF (s).

2. If ρ1 and ρ2 are representations of Gal(F ), then ρ1 ⊕ ρ2 is also a representation of

Gal(F ) and

L(ρ1 ⊕ ρ2, s) = L(ρ1, s)L(ρ2, s).

3. Let E/F be a finite extension of number fields. Let ρ be a representation of Gal(E)

and let ψ be the representation of Gal(F ) induced by ρ. Then

L(ρ, s) = L(ψ, s).

Note that in general ρ and ψ does not have the same dimension.

Moreover, it has a meromorphic continuation on the whole complex plane and it satisfies

a functional equation

L(ρ, s) = ε(ρ, s)L(ρ, 1− s),

where ε(ρ, s) is an entire function that have no zeroes. Artin also conjectured [19] that

L(ρ, s) is entire if ρ is not the trivial representation.

30 3.2. AUTOMORPHIC REPRESENTATIONS

3.2 Automorphic representations

The Section 2.3 motivates the following definition. Let F be a number field, n ∈ N and

χ be a character of the center Z ' A∗F of GLn(AF ). Define the space

L20(ZGLn(F )\GLn(AF ), χ)

of all bounded square-integrable functions ϕ : GLn(F ) → C that are smooth at each

infinite component, locally constant at each finite component, and such that:

1. (K-finitness) The subspace span{R(a)ϕ : a ∈ K} of L20(ZGLn(F )\GLn(AF ), χ),

where K is the maximal compact subgroup of GLn(AF ), has finite dimension.

2. (Action of Z) For each z ∈ Z, ϕ(za) = χ(z)ϕ(a).

The following property characterizing L20(ZGLn(F )\GLn(AF ), χ) is the cuspidality. In

the section 2.3 the functions ϕf satisfy

∫

Q\AQ

ϕf

((1 x0 1

)α

)dµ(x) = 0.

That is, this integral on the unipotent radical of the parabolic subgroup P ≤ GL2(AQ)

is zero. Considering this, for the case n > 2, we have that the parabolic subgroup of

GLn(AF ) may be seen as the (non disjoint) union of the standard parabolic subgroups

P =⋃

n1,n2

Pn1,n2

where the union runs through all pairs (n1, n2) such that n1, n2 ∈ N and n1 + n2 = n.

For example, P3 = P1,2 ∪ P2,1.

Each Pn1,n2 has the structure Pn1,n2 '(

GLn1 Nn1,n2

0 GLn2

), where Nn1,n2 is the unipotent

radical of Pn1,n2 . Nn1,n2 is provided with the structure of a quotient measure. Then we

have the property:

3 (Cuspidality) For any pair n1, n2 as above and for any α ∈ GLn(AF )∫

Nn1,n2 (F )\Nn1,n2 (AF )

ϕ(uα)dµ(u) = 0.

CHAPTER 3. THE LANGLANDS PROGRAM 31

Note that GLn(AF ) acts on L20(ZGLn(F )\GLn(AF ), χ) from the right. Under this

action L20(ZGLn(F )\GLn(AF ), χ) decomposes into a direct sum of irreducible represen-

tations, called cuspidal automorphic representations of GLn(AF ).

Langlands showed that every irreducible cuspidal automorphic representation of

GLn(AF ) is either cuspidal or is induced from the tensor product π1 ⊗ π2 of two such

representations of GLn1(AF ) and GLn2(AF ).

We have that a cuspidal automorphic representation π of GLn(AF ) decomposes into

a restricted tensor product

π =⊗

p

′πp,

where for almost every prime p in F the irreducible representation πp of GLn(Fp) is

unramified, that is, there exists a nonzero vector vp stable under GLn(Op).

Definition 3.2.1. For each prime p of F where πp is unramified, let Hp be its corre-

sponding spherical Hecke algebra, i.e., the convolutive algebra of GLn(Op)-biinvariant

compactly supported functions on GLn(Fp). This algebra is isomorphic [7] to the com-

plex commutative algebra freely spanned by the Hecke operators Hp,i (i = 1, . . . , n−1),

H±1p,n. We know [7] that vp is an eigenvector of any Hecke operator Hp,i, with eigenvalue

zi(p). These eigenvalues are called the Hecke eigenvalues.

Given a cuspidal automorphic representation π, we define the Hecke L-function

corresponding to π as

L(π, s) =∏

p

[(N(p)−s − z1(p)) · · · (N(p)−s − zn(p))]−1.

There also exists a definition of L-functions for archimedean places which involves

the Euler gamma function [14, p. 404]. The complete L-function, that we also denote

as L(π, s), converges in some right half-plane; this theorem was obtained by Langlands

[15, p. 281]. Godement and Jacket defined a L-function L∗(π, s) for an automorphic

representation π in terms of an Euler product

L∗(π, s) =∏

all places ν

L∗(πν , s).

32 3.3. THE LANGLANDS PROGRAM

This function has a meromorphic continuation with possible singularities at s = 0 and

s = 1. Moreover, L∗(π, s) satisfies a functional equation

L∗(π, s) = ε∗(π, s)L∗(π, 1− s),

where ε∗(π, s) = N−s for some N ∈ Z [15, Thm. 8.7]. It is conjectured that

L∗(π, s) = L(π, s)

for all representations π; however, L∗(πν , s) = L(πν , s) for all unramified places ν.

3.3 The Langlands Program

In a similar way to that in the chapter two, a natural question is: can we connect

n-dimensional `-adic Galois representations with automorphic forms? This question is

part of some deep conjectures formulated by Robert Langlands in 1967, in a letter to

Andre Weil, after associate automorphic forms on GL2 to each modular form. These

conjectures are stated in [17] in a more precise form, and they all are known as the

Langlands Program. For now, we consider the following conjecture, which is a very

particular case of this program:

Conjecture 3.3.1. Let n ∈ N and let ` be a rational prime. There exists a corre-

spondence between irreducible representations π : Gal(Q) → GLn(F`) and automorphic

representations ρ of GLn(F )\GLn(AF ). This correspondence is such that

L(ρ, s) = L(π, s).

Moreover, the local L- and ε-factors are equal for all primes.

We must make some remarks before to continue:

1. The case n = 1 here and the case explained in the chapter one are different. At

chapter one we only consider complex characters (which may be seen as `-adic

characters since they have finite image). However, in the case n = 1 of Theo-

rem 3.3.1 we consider a wider class of representations: not every `-adic character

has finite image.

CHAPTER 3. THE LANGLANDS PROGRAM 33

2. The notion of automorphic representation does not exist in the case n = 1. This

is because GL1 have no parabolic subgroups.

3. Since Gal(F ) is compact, any complex representation factorizes through a group

of finite index. Such groups are open and normal in Gal(F ), and they correspond

to finite Galois extensions of F . In other words, complex representations describe

all Galois number fields, and we may restrict the Conjecture 3.3.1 to these cases,

seeing any complex representation of Gal(F ) as an `-adic representation.

4. This correspondence is not surjective. Indeed, there are automorphic representa-

tions which do not correspond to any Galois representation. Considering `-adic

representations of the Weil group W(F ), Langlands conjectured that the new cor-

respondence is surjective.

5. There exists a version of the Langlands correspondence for local fields: given

a local field F , there exists a correspondence between n-dimensional irreducible

representations of Gal(F ) and automorphic representations of GLn(F ) realized in

the space L20(GLn(F )). Nowadays, this correspondence is a theorem [14, 16, 22].

Using advanced machinery, Don Blasius and others have given more evidence for the

case n = 3 of the Langlands program. Many results are compiled in [3].

3.4 Reciprocity Laws

What is a reciprocity law? Wyman makes this question himself [33], and at the end of his

paper he concludes “I have to confess that I still do not know what a reciprocity law is,

or what one should be.” However, he shows many remarks and examples about splitting

polynomials in finite fields in his paper. We may say that a reciprocity law is a relation

that say us how a polynomial splits over finite fields, or in a more number-theoretical

language, how all prime ideals split in a number field. To ilustrate this better, let us

consider some examples.

First consider the most archetypical one [13, Thm. 5.2.1]:

Theorem 3.4.1 (Quadratic Reciprocity Law). Let p 6= q be odd rational primes.

Then (p

q

)(q

p

)= (−1)(p−1)(q−1)/4,

34 3.4. RECIPROCITY LAWS

where (ap) is the Legendre symbol.

Remark 3.4.2. There also exists a complementary law for q = −1 and q = 2. [13, p.

53]. For q = −1 we have that

(−1

p

)= ±1 if and only if p ≡ ±1 mod 4.

For q = 2 the law is

(2

p

)=

{1 if and only if p ≡ ±1 mod 8

−1 if and only if p ≡ ±3 mod 8.

If well this theorem states when the congruence x2 ≡ q mod p has a solution, the

Quadratic Reciprocity Law is stronger: it says us when an odd prime q splits in the

quadratic field Q(√

p). We have [31, p. 126] that a prime q 6= p splits in Q(√

p) if and

only if ( qp) = 1.

Let F be an abelian extension of Q, and let m ∈ N be such that F ⊆ Q(ζm). Let

a ∈ Z with gcd(a,m) = 1, and let (Fa) be the Artin symbol, i.e., the automorphism of

F obtained by restricting the automorphism ζm 7→ ζam of Q(ζm). Let IF,m be the kernel

of the map (F∗ ) : (Z/mZ)∗ → Gal(F/Q). We have this equivalent form of the Artin

Reciprocity Law for Q [9]:

Proposition 3.4.3. The sequence

1 → IF,m → (Z/mZ)∗ → Gal(F/Q) → 1

is exact.

Let p be a prime number. We have two cases:

1. Suppose that p ≡ 1 mod 4. Then Q(√

p) ⊆ Q(ζp). We have that IQ(√

p),p is

a subgroup of index two in the cyclic group (Z/pZ)∗; hence there is a unique

such subgroup, namely the squares (or quadratic residues) modulo p. If q 6= p

is a rational prime, (Q(√

p)

q) = ±1. Identifying IQ(

√p),p with the group of squares

modulo p and Gal(Q(√

p)/Q) with {±1}, we have that (Q(√

p)

q) = ( q

p).

CHAPTER 3. THE LANGLANDS PROGRAM 35

2. Suppose that p ≡ 3 mod 4. This case is more difficult.

We have that Q(√

p) ⊆ Q(ζ4p). Here, there are three subgroups of index two in

the group (Z/4pZ)∗: the subgroup G1 of squares modulo p, the subgroup G2 of all

unities modulo 4p that are 1 mod 4 and the subgroup IQ(√

p),p.

Since G1 ∩G2 ⊆ IQ(√

p),p, it contains all the squares modulo p that are ≡ 1 mod 4.

The other elements are not squares, and they are ≡ −1 mod 4. Now, let q 6= p

be a rational prime. Identifying again Gal(Q(√

p)/Q) with {±1}, if q is a square

modulo p, then (Q(√

p)

q) = 1 = ( q

p) if q ≡ 1 mod 4 and (

Q(√

p)

q) = −1 = −( q

p) if

q ≡ 3 mod 4.

In both cases, interchanging q and p, we conclude the proof of Theorem 3.4.1.

Quadratic Reciprocity Law implies another fact related with polynomials. Let fp(X) =

X2−p, where p is an odd prime number2. We can see [33, Thm. 2.3] that the polynomial

fp splits in Fq if q ∈ IQ(√

p),p. The unique “bad” prime is the prime p where fp ramifies.

This condition can be stated through the discriminant of fp, that is, 4p.

Other example is the Quartic Reciprocity Law, that is related with the equation

X4 − p for a prime p and its reduction modulo q. We have that the Galois group of

this polynomial over Q is isomorphic to the dihedral group D8; however, it is cyclic of

order 4 over the complex field Q(i). We actually consider a more general equation. Let

π ∈ Z[i] be an irreducible element in Z[i]3. We have [13] that for every α such that π - αthere exists a unique integer k modulo 4 that satisfies

α(Nπ−1)/4 ≡ ik mod π.

Here, Nπ is the norm of π, the cardinality of its quotient field. This motivates the

following definition: if π is a prime in Z[i] and π - α, the quartic residue character

is defined as (α

π

)≡ α(Nπ−1)/4 mod π.

2One may consider any other polynomial equivalent; we say two polynomials f, g ∈ Z[X] are equiv-alent if their splitting fields are equals. For example, the polynomials X2 − X − 1 and X2 − 5 areequivalent.

3The ring Z[i] is called the ring of the Gaussian integers. We know that Z[i] is an Euclidean domainand that a rational prime p is irreducible in Z[i] if and only if p ≡ 3 mod 4.

36 3.4. RECIPROCITY LAWS

We say that a prime element π = a+bi ∈ Z[i] is primary if a ≡ 1, b ≡ 0 or a ≡ 3, b ≡ 2,

both conditions modulo 4. We have [13]:

Theorem 3.4.4 (Quartic Reciprocity Law). Let π 6= λ be primary primes of Z[i].

Then (λ

π

)=

(π

λ

)(−1)

(Nπ−1)4

(Nλ−1)4 .

We cannot make an analysis as the one made for the quadratic reciprocity law4.

However, we can say that the quartic reciprocity law may be encoded in all characters

χ : Z[i]/αZ[i] → C of order 4 (α ∈ Z[i]), if such a characters exist.

Let us consider one last example: the polynomial φ(X) = X3 − X − 1. Its Galois

group is isomorphic to the symmetric group S3. We have four cases to consider all primes

[23]:

1. Let p 6= 23 be a prime that is not a square modulo 23. In this case, the reduction

of φ(x) factorizes as a product of a linear polynomial by a quadratic polynomial;

ergo φ(x) has just one root.

2. Let p 6= 23 be a prime such that p = a2 + ab + 6b2, where a, b ∈ Z. We have that p

is a square modulo 23, and the reduction of φ(X) modulo p has 3 distinct roots.

3. If p 6= 23 is a prime of the form p = 2a2 + ab + 3b2, then p is a square modulo 23

and the reduction of φ(X) modulo p is irreducible.

4. Finally, for p = 23, the reduction of φ(X) modulo p has two roots; one simple and

another double.

Let Np(φ) denote the number of roots of φ(X) mod p, and let ap = Np(φ)− 1. Note

that ap is the coefficient of qp in the Fourier series

f(q) = q

∞∏ν=1

(1− qν)(1− q23ν) =∞∑

ν=1

aνqν ,

4Indeed, the development of the quadratic reciprocity law given here is a consequence of theKronecker-Weber theorem, that says that any abelian field over Q is contained in some cyclotomicfield. There is no a similar result for other number fields.

CHAPTER 3. THE LANGLANDS PROGRAM 37

where q = e2πiτ , τ ∈ H. Then f(τ) is a newform of weight 1 at level 23 and character

the Legendre symbol ( ∗23

),

f(γτ) =

(d

23

)(cτ + d)f(τ), γ =

(a bc d

)∈ Γ0(23),

and it may be considered as a “reciprocity law” for the polynomial X3 −X − 1. Serre

also studied the polynomial X4 −X − 1 in [23] obtaining the expression

a2p =

( p

283

)+ Np(f)− 1

for every prime p 6= 283, where Np(f) denotes the number of roots of the reduction of f

modulo p.

These examples show an interesting fact: Langlands Program can be related to some

“reciprocity laws” for polynomials. That is, given a polynomial f(x), the Langlands

program may give a “pattern” (through automorphic representations) to describe those

primes p for which the reduction of f(x) modulo p splits completely. The key to find this

pattern [15] is putting into correspondence the L-function associated to each irreducible

representation of Gal(K/Q) (that extends to an irreducible representation of Gal(Q),

and contains encoded all the information about all local Galois groups) with a more

familiar object (as the Langlands Program postulates) and analysing it.

Note also that the Shimura-Taniyama conjecture may be considered as a “reciprocity

law” for elliptic polynomials in two variables.

3.5 What follows?

What does the Langlands Program mean? Roughly speaking, and, in accordance with

the Tannakian philosophy, Langlands correspondence says that the structure of the Ga-

lois group Gal(F ) of a number field is encoded in the structure of the family of GLn(AF )-

modules L20(GLn(F )\GLn(AF )); n ∈ N. If we understand the nature of automorphic

representations, then we can get all information about all irreducible polynomials. This

is because any irreducible polynomial f(X) ∈ Q[X] has associated its splitting field

(which is a Galois number field), and any Galois number field is attached to a family

38 3.5. WHAT FOLLOWS?

of irreducible polynomials in Q[X]. Of course, automorphic representations are very

intricate objects, and their properties, as well as their behavior, are unknown for us yet.

We finish this work with some remarks that may help to clarify or solve the problem.

First note that any automorphic representation π is perfectly determined by any

vector of the space associated to π; that is possible because π is irreducible. Moreover,

we conjecture that this associated vector vπ can be chosen such that it gives rise to

a more friendly function fπ. This may reformulate the Langlands Program in easier

terms: any `-adic irreducible representation ρ of Gal(F ) has attached, via L-functions,

a function fρ, and the association is as follows: if ρ 7→ π is the map of the Langlands

Program, then

fρ = fπ and L(fρ, s) = L(π, s).

We actually have seen, in the Chapter 1 (where n = 1), that these functions fρ are the

Dirichlet characters, and in the Chapter 2 (the modular case), that these fρ are the

cuspidal newforms.

Another idea is closely related to the local correspondence. If well we are not prepared

to manipulate the automorphic representations yet, we may try to analyse the space

L20(ZGLn(F )\GLn(AF )) through all spaces L2

0(GLn(Fν)), for the distinct places ν. Of

course, this implies that we also have to study the connection between the groups Gal(F )

and Gal(Fν) for any ν, as well as the connection for the Weil groups too. If we may

develop the necessary background, we may use the (now shown) local correspondence to

prove the global one.

Finally, we think that the information related to reciprocity laws is encoded, es-

sentially, in the behavior of the Artin L-functions related to the Galois group of any

polynomial. We have many questions about the Langlands Program and related sub-

jects, and we expect that some of the basic questions will be answered soon.

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