Sobre el programa de Langlands para campos numéricos

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C

CENTRO DE INVESTIGACI ´ON Y DE ESTUDIOS AVANZADOS DEL INSTITUTO POLIT´ECNICO NACIONAL

UNIDAD ZACATENCO

DEPARTAMENTO DE MATEM ´

ATICAS

Sobre el Programa de Langlands

para Campos Num´

ericos

T E S I S

Que presenta

MIGUEL ´

ANGEL VALENCIA BUCIO

Para obtener el grado de

MAESTRO EN CIENCIAS

EN LA ESPECIALIDAD DE MATEM ´

ATICAS

Director de Tesis: Dr. Jos´e Mart´ınez Bernal

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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. Jos´e Mart´ınez Bernal

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Contents

Resumen i

Introducci´on 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 forn = 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 forn = 2 . . . 25

3 The Langlands Program 27 3.1 Galois representations andL-functions . . . 27

3.2 Automorphic representations . . . 30

3.3 The Langlands Program . . . 32

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Resumen

En este trabajo presentamos el Programa de Langlands para el caso de campos num´ericos, el cual es actualmente uno de los problemas abiertos m´as importantes en teor´ıa de n´umeros. Tambi´en lo conectamos de manera directa con otros resultados bien conocidos en teor´ıa de n´umeros y damos una potencial aplicaci´on concerniente a la factorizaci´on de polinomios irreducibles sobre campos finitos a trav´es de algunos ejemplos.

Primero conectamos el Programa de Langlands con otros resultados que pueden considerarse leyes de reciprocidad. En el cap´ıtulo uno mostramos la relaci´on 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 Teor´ıa de Campos de Clase y despu´es construimos la conexi´on por medio de las L-funciones.

Por un lado, introducimos las L-funciones de Artin para caracteres, y m´as general-mente para representaciones. Desde su introducci´on, Artin las utiliz´o para hallar una “ley de reciprocidad no abeliana”. Por otro lado, presentamos lasL-funciones de Hecke definidas sobre el grupo de clases de ideles, como Tate las enunciara en su famosa tesis doctoral.

Bajo este contexto, la Teor´ıa de Campos de Clase Abeliana se corresponde con el Programa de Langlands para el caso n= 1 en alguna forma.

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formas modulares, a trav´es de los coeficientes de su serie de Fourier.

Comenzamos dando un panorama suficiente para enunciar la conjetura de Shimura-Taniyama. Despu´es usamos la estructura de grupo de las curvas el´ıpticas para construir representaciones bidimensionales de Gal(Q), a la vez que introducimos susL-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 representaci´on natural de GL2(AQ), conocida como representaci´on automorfa.

Bajo esta aproximaci´on, la conjetura de Shimura-Taniyama puede verse como un caso especial de la correspondencia de Langlands para el caso n= 2.

Finalmente, en el cap´ıtulo tres, presentamos el Programa de Langlands en su versi´on completa. Definimos las representaciones de Galois, las representaciones automorfas, y sus respectivas L-funciones. Hecho ´esto, enunciaremos la conjetura principal. Con-clu´ımos con algunos ejemplos de descomposici´on de polinomios y algunas observaciones.

Nuestro prop´osito es introducirnos en el Programa de Langlands y familiarizarnos con algunas de las ideas, problemas y conjeturas en esta ´area, para una investigaci´on futura.

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Introducci´

on

Descomponer polinomios sobre campos finitos es un problema interesante y dif´ıcil en teor´ıa de n´umeros. Esto est´a sumamente relacionado con la factorizaci´on de polinomios, ya que, por ejemplo, si la reducci´on de un polinomio f(X) Z[X] sobre alg´un campo finito es irreducible, entonces tambi´en lo es sobreQ. O bien, sif(X) m´odulopse descom-pone en factores lineales para casi todos los primosp, entonces tambi´en se descompone en factores lineales sobre Q.

Surge la pregunta: ¿Para qu´e primospun polinomio dadof(X) es irreducible m´odulo p? O m´as generalmente ¿es posible clasificar los primosp, tal vez salvo un n´umero finito de ellos, de acuerdo a la factorizaci´on de f(X) m´odulo p? El Teorema de Kummer-Dedekind nos permite considerar esta pregunta dentro del marco de la Teor´ıa de Galois: estudiar la factorizaci´on del polinomio f(X) Q[X] sobre campos finitos nos lleva a estudiar la factorizaci´on de n´umeros primos sobre el campo de descomposici´on de f(X). Se sabe que la informaci´on aritm´etica acerca de las ra´ıces de f(X), a´un trabajando sobre campos finitos, est´a contenida en su grupo de Galois sobre cada campo; ergo, resulta natural pensar que el campo de descomposici´on de f(X) sobre Q, a trav´es de su grupo de Galois, tiene suficiente informaci´on como para clasificar los n´umeros primos respecto a la factorizaci´on m´odulop de dicho polinomio y de otros que tengan el mismo campo de descomposici´on.

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 direcc´on, el Programa o Correspondencia de Langlands podr´ıa ser de utilidad [15, 33].

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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 num´erico. La primera concierne a representaciones del grupo absoluto de Galois y del grupo de Weil de un campo num´erico. La segunda concierne a representaciones automorfasde cierto espacio de Hilbert de tipo L2 sobre el anillo de adeles del campo. De manera m´as precisa, y en un contexto m´as general, dado un campo num´erico 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 deG(AF), dondeAF es el

anillo de adeles deF.

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

llamado el grupo dual de Langlandsde G.

Adem´as, dicha correspondencia es a trav´es de L-functions: las de la parte (1) tienen informaci´on anal´ıtica que se obtiene de los operadores de Hecke, y aquellas de la parte (2) tienen informaci´on aritm´etica dada por la estructura de la representaci´on.

Las propiedades de estasL-funciones en el plano complejo (continuaci´on meromorfa, comportamiento en una l´ınea cr´ıtica, localizaci´on de ceros, etc.) tienen informaci´on aritm´etica relevante.

En esta tesis s´olo consideramos el caso G = GLn, el grupo lineal general de orden

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

LG= GL

n(C) para todon, y el Programa de Langlands se reduce a una correspondencia

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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 ArtinL-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 HeckeL-functions defined over the id`ele 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 modulop, and modular forms, through the coefficients of their Fourier series.

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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.

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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 psplits into linear factors for almost all primes p, then it splits into linear factors overQ.

This raises the question: for which primes p a given polynomial f(X) is irreducible modulop? More generally, is it possible to classify all primesp, maybe with finitely many exceptions, according with the factorization off(X) modulop? 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 off(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].

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num-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 ad`ele ring of this field. In a more precise, and more general context, given a number field F and a reductive group Gover F, the Langlands Program establishes a one-to-one correspondence between equivalence classes of: (cf.[8])

1. Irreducible tempered automorphic representations ofG(AF), whereAF is the ad`ele

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= GL

n(C) for alln, and the Langlands Program reduces to a correspondence between

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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 ad`elic 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.

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1.1

Preliminaries

Anumber field is a finite extension of the rational numbersQ. Alocal 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 theplacesof 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 formF =Q(α) for someα F called a primitive elementof 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 eachϕ(X) we have the isomorphism

F Q[X]/f(X)C

induced by the map g(X)7→g(ξ), withξ 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

atp 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 ofOp andk ∈Z. Fphas a canonical topological structure induced by the absolute value

| · |p:uπk 7→q−k,

where q is the cardinality of the residual field ofFp, i.e, the quotient fieldOF/p.

1

It is possible that a complex number field has a real archimedean place and that a real number field has a complex archimedean place. We give the (clear) example Q(α), where α is a root of the polynomialX3

−2 = (X−√3

2)(X2

+√3

2X+√3

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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 primep (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 filedF, let Gal(F) denote the absolute Galois group ofF, that is, the group Gal(F /F). Being this aprofinite 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].

LetK/F be an arbitrary field extension, and letpbe a prime ideal ofOF. The natural

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

LetP be any fixed prime lying abovep, and letDPbe 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 thead`elic group. Therestricted 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

Y

α

G

α ={(xα)∈

Y

α

Gα :xα ∈Hα for almost all indexesα}.

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◦ Thead`ele ring of a number fieldF is the restricted direct product over all places of F,

AF =

Y

ν

F

ν,

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

◦ The id`ele class groupis the restricted direct product

F∗\A∗F =Y

ν

F

ν

with respect to{O

p:p is an archimedean place}; and

◦ The general linear group

GLn(AF) =

Y

ν

GL

n(Fν)

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

The last two groups are examples ofad`elic 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 Hsuch that

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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 isArtin’s Reciprocity Law.

LetK/F be a finite field extension. Kis a finite-dimensional vector space overF and for everyxK the mapy7→xyis an endomorphism ofK, whose matrix representation over F we also denote by x. The norm map is defined as the map

NK :K∗ →F∗, x7→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 N

K(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 N

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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. LetF be a number field and letpbe a prime ofF. Letjp:F∗

p →F∗\A∗F

be the mapx7→(1, . . . ,1, x,1. . .). We have the relation between global and local Artin maps

AFp =AF ◦jp.

LetK/F be an abelian Galois extension of number fields, let Pbe 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

xF∗\A

F,

AK/F(x) =

Y

p

AKP/Fp(xp).

The latter claim is a consequence of theHasse 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. IfK/F is a finite extension of global fields, thenW(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 → Wϑ (F)ab −−−−−−−→induced byϕ Gal(F)ab is the Artin map.

2

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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) =

X

n=1 1 ns,

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

ζ(s) = Y

pprime 1 1p−s

and satisfies the functional equation

ξ(s) =ξ(1s), 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) =

X

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 forQ. 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, letV be a neighborhood of the identity matrix “1” in GLn(C), containing no nontrivial subgroups of GLn(C); this neighborhood exists

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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 Frpis defined for almost all primes p.

Definition 1.3.2. We define the Artin L-function associated to the characterρ as L(ρ, s) =LK/F(ρ, s) :=

Y

p

1

1ρ(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 thatL(ρ, s) = Q

pL(ρp, s).

Remark 1.3.3. The ArtinL-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 chooseK =F in the decomposition ofρ.

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

characterρ 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).

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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(ˇρ,1s),

where ε(ρ, s) is an entire function without zeroes and ˇρis 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 charactersχof the id`ele class group of a number field F and was also able to establish the meromorphic continuation and the functional equation ofL(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. LetF 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 suitablec > 0. Let χbe 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 fieldFq and uniformizing parameter π, we define [4, 20]

L(χ) = (

(1q−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]

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

orx <0. We define

L(χ) = (

π−s/2Γ(s

2), if χ=| · |s π−(s+1)/2Γ(s+1

2 ), 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 fieldF, aSchwartz-Bruhat functiondefined on F is:

1. AC∞-complex-valued functionf defined onF such thatp(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.

LetS(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

ˆ f(y) =

Z

F

f(x)ψ(xy)dx.

When ˆf is well-defined, it lies in S(F), however, it depends on the choice of ψ and dx. We also define thelocal zeta function associated to f and the character χ of F∗ as

ζ(f, χ) = Z

F∗

f(x)χ(x)d∗x.

This zeta function satisfies the functional equation [20, Thm. 7.2] L(χ)ζ( ˆf ,χ) =ˇ ε(χ, dx)L( ˇχ)ζ(f, χ),

where ˇχ = 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. LetF be a number field and letχ:F∗\A

F be an id`ele class character.

We have that χ decomposes as a product of local characters χ=Y

ν

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CHAPTER 1. ABELIAN CLASS FIELD THEORY 11

whereχν is a character of the local field Fν andχν is unramified for almost all places ν.

We define the Hecke L-functionattached to F as L(χ, s) = LF(χ, s) =

Y

ν

L(χν, s).

We have thatL(χ, s) defines a holomorphic function on the half-plane(s)>1 [20, Thm. 7.19]. We also have a functional equation of HeckeL-functions in terms ofL- and ε-factors:

L( ˇχ,1s) =ε(χ, s)L(χ, s), whereε(χ, s) =Q

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

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

Remark 1.4.3. For the trivial characterχ= 1, it holds that the Dedekind zeta function of F is preciselyLF(1, s) = ζF(s).

1.5

The Langlands Program for

n

= 1

Remark 1.5.1. LetG 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/Gis 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 anyρ 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 ArtinL-functionLK/F(ρ, s) = L(ρ, s), which is well-defined by

Remark 1.3.3. On the other hand, the study of the id`ele class groupF∗\A

F is equivalent

to studying all id`ele class characters. If χ is a such character, then we may associate to it a Hecke L-functionL(χ, s).

Now consider the Artin map

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Let ρ be a character of Gal(F). We have that the Artin map induces an id`ele class character χsuch that the diagram

Gal(F) Gal(F)ab,

F∗\A

F C π / / ˜ ρ O O χ / / AQ % % L L L L L L L L L L

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 letQab(p) be the maximal abelian extension ofQ that is unramified atp. We

have that [7]

Gal(Qab(p)/Q) Y

q6=pprime

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 ad`ele (1, . . . ,1, p,1, . . .)4. Under this definition we can claim that

the Artin map sends the ad`ele (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 πpis any uniformizing parameter of Fp.

Now let χ be an id`ele class character of F. We have that χ(1, . . . ,1, πp,1, . . .) =Y

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)

3

We have a bijective correspondence between all characters ofF∗\A

F of finite order and all Dirichlet

characters [20, pp. 237-238]

4

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CHAPTER 1. ABELIAN CLASS FIELD THEORY 13

for almost all primesp(at least for all the unramified ones) ifχis the character associated toρ. 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 ofF∗\A

F ≃GL1(F)\GL1(AF)of finite order. The correspondenceρ7→χ

is given by χ=ρAF, 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(A

F)) 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

x7→(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(A

F)), called

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

R(x) = M

χ

Z

GL1(F)\GL1(AF)

χ(x)dχ,

where the sum runs through all id`ele class characters. Therefore automorphic representa-tions on GL1(AF) are simply id`ele 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=ρAF, where AF is the Artin map, and it satisfies

L(ρp, s)L(χp, s)

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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 Andr´e 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 casen= 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.

1

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

2

AWeierstrass equationoverQis a cubic equation

E :y2+a1xy+a3y=x 3

+a2x 2

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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, bZ [6, p. 323].

We may associate an elliptic curveE with a complex torus as follows. Letz1, z2 C

be such that z1/z2 6∈ R, and consider the lattice Λ = z1Zz2Z. 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 torusC/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 + X

ω∈Λ,ω6=0 µ

1 (zω)2 −

1 ω2

, z C\Λ.

We have that ℘ is meromorphic overC/Λ and satisfies the functional equation ℘′(z)2 = 4℘(z)3g2℘(z)g3,

where g2 = 60P

ω6=0ω14 and g3 = 140

P

ω6=0 ω16. We have a map C/Λ →C2 given by

z 7→(℘(z), ℘′(z)).

This map is a bijection, and it extends to a map CE, whereE 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 curveE 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 theai’s are rational numbers. Is straightforward verifying [6, p. 310]) that every Weierstrass

equation can be reduced to an elliptic curvey2

=x3

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CHAPTER 2. SHIMURA-TANIYAMA CONJECTURE 17

Given an elliptic curveE, 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 pin the factorization of NE.

We define for anyp prime and any kN 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

nN 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) =

X

n=1

an(E)n−s =

Y

pprime

(1ap(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 M¨obius transformations that leave invariant the upper halfplane H. Moreover, the isomorphism is given by

µ a b c d

(fγ :z 7→

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Definition 2.1.3. A congruence subgroup at level N N of SL2(Z) is a subgroup Γ that contains the subgroup

Γ(N) = ½µ a b c d ¶ ≡ µ 1 0 0 1 ¶ mod N ¾ .

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 functionf :H → Csuch 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 hNsuch that (1 h

0 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 ishZ-periodic. The holomorphic map

z7→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 thatf isholomorphic at ifg extends holomorphically toq = 0. In this case, f has a Fourier expansion

f(z) =

X

n=0

ane2πinz/h.

This observation motivates the following definition.

Definition 2.1.4. Given a congruence subgroup Γ and an integerk, a functionf :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, thenf is a Γ-cuspidal form of weight k.

LetMk(Γ) andSk(Γ) be the vector spaces of all Γ-modular and Γ-cuspidal forms of

weightk respectively. We are interested in the structure of these spaces when Γ = Γ1(N) is the congruence subgroup

Γ1(N) = ½µ

a b c d

:ad1, c 0 mod N ¾

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CHAPTER 2. SHIMURA-TANIYAMA CONJECTURE 19

and when Γ = Γ0(N), where Γ0(N) =

½µ a b c d

:c0 mod N ¾

.

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

Mk(Γ) =

M

χ

Mk(Γ, χ), Sk(Γ) =

M

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) =

½

f :f[γ]k=χ(d)f for all γ =

µ a b c d ¶ ∈Γ ¾ ,

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, letN Nfixed. We define theHecke operatorsonMk(Γ1(N)) as follows. First, for any nN we define the operator [6]

hni(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 =QTpr

where n = Q

pr, and T

pr 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 hni and Tn such that a1(f) = 1 [6, p. 195].

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

n=1an(f)qn, then Tn(f) = an(f) for all n∈N.

Each modular formf has attached a natural L-function

L(f, s) =

X

n=1

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where f(z) = P∞

n=0an(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 groupGhas finite image inC. Therefore, the image of a such representation φ is contained in a number fieldK (and we may assume that this field is minimal under inclusion). Let ℓ be a prime number, and let Lbe a prime of K lying aboveℓ. Consider

the completionKLofKatL. Then [20]KLis a finite extension of theℓ-adic fieldQℓ, and

φ(G) is contained inKL. Hence, we may consider any finite-dimensional representation of G as an ℓ-adic representation.

Let E be an elliptic curve. For any N N, let E[N] denote the set of N-torsion points ofE, that is, the set of all points z C/Λ, whereC/Λ is an associated torus to E, that N z = 0 inC/Λ. That is, we say that a point x+ ΛE is in E[N] if the point N xC lies in Λ.

Definition 2.2.2. LetE be an elliptic curve, and let ℓbe a prime number. The ℓ-adic Tate module of E is the inverse limit

Taℓ(E) = lim

← E[ℓ

n]

with respect to the directed system

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CHAPTER 2. SHIMURA-TANIYAMA CONJECTURE 21

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

basis such that

·Pn+1 =Pn, ℓ·Qn+1 =Qn.

Note that each basis determines an isomorphism E[ℓn](Z/ℓnZ)2. This implies that Taℓ(E)≃Z2ℓ.

Letn N. The fieldQ(E[ℓn]) is a Galois number field; hence we have the restriction

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

Gal(Q(E[ℓn])/Q)Aut(E[ℓn]). The maps are compatible in the sense that the diagram

Aut(E[ℓn]) Aut(E[ℓn+1]) Gal(Q)

o o   ? ? ? ? ? ? ? ? ? ? ? Ä Ä ÄÄÄÄ ÄÄÄÄ ÄÄÄÄ

is commutative for any n. That is, the Tate module is a Gal(Q)-module [6]. Since Aut(E[ℓn]) GL2(Z/ℓnZ) implies Aut(Ta

ℓ(E)) −→∼ GL2(Zℓ), we have a continuous

homomorphism

ρE,ℓ: Gal(Q)→GL2(Zℓ)⊆GL2(Qℓ).

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) isunramifiedat a primepif its kernel contains

the inertia subgroup of every maximal ideal pZlying above p, where Z is the ring of algebraic integers. We have the following

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

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

for those p is

X2a

p(E)X+p.

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Given an elliptic curve E, the L-function associated to the Galois representation ρE,ℓ is given by

L(ρ, s) =Y

p

[det(1p−sρ(Frobp))]−1,

Note. We can associate to any elliptic curve a zeta function [13, Ch. 18]. This zeta function and theL-series are related by the equalityζ(E, s)L(E, s) = ζ(s)ζ(s1),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

toL(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 ℓN, the L-function associated to the representation ρE,ℓ 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 ad`elic 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 Q

pGL2(Zp), the product running through all primes p, of all matrices

whose lower left entry is in NZˆ, whereis the direct limit of all Z/nZ with respect the canonical maps Z/nZZ/mZ when m|n.

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

GL2(Q), g GL2+(R) and κK0(N).

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

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CHAPTER 2. SHIMURA-TANIYAMA CONJECTURE 23

GL1(Q)\GL1(AQ), that we still denote as χ. The ad`elization of f is defined as the map ϕf : GL2(AQ)→C defined by

ϕf(α) = [det(g∞)−1/2(ci+d)]−kf

µ ai+b ci+d

ωχ(κ),

whereα=γgκis as in Theorem 2.3.1,g= (a b

c d) in GL+2(R) and ωχ denotes here the

evaluation ofχ 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 functionϕf has another properties [10]:

1. (K-finiteness) For g = ( cosθsinθ

−sinθcosθ) ∈SO2(R), κ∈ K0(N) and a ∈ GL2(AQ) we have that

ϕf(γg∞κ) =ωχ(κ)e2πikθϕf(γ).

We can rewrite this assertion ad`elicly as follows: let K be the maximal compact subgroup of GL2(AQ), which is isomorph to SO2(R)×Q

pGL2(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 allβ ∈GL2(AQ) is theright 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 ofZ. Note also that

ϕf(zα) =ωχ(z)ϕf(α), ∀α∈GL2(AQ), z∈Z.

3. (Growth condition) There exists a real numberA >0 such thatϕf(α) = O(kαkA),

where kαk is the norm kαk = Q

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

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

3

A finite component of an ad`elic group G(AQ) ≃ Q′G(Qν) is a component associated with an

archimedean place; the infinite component of G(AQ) is the component associated with the

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4. (Cuspidality) If f is cuspidal, for any αGL2(AQ) Z

Q\AQ

ϕf µµ 1 x 0 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)}

ofL2

0(ZGL2(Q)\GL2(AQ), ω) (see Section 4.2 below). Denote byπf 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∞×(

O

p

π

p),

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, (fg)(a) =

Z

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 vectorvp of πp is given by the formula

Hi,p·vp =

Z

Mi 2(Zp)

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CHAPTER 2. SHIMURA-TANIYAMA CONJECTURE 25

where ξp : GL2(Zp)→EndVp is the representation homomorphism and M2i(Zp) are the

double cosets

M21(Zp) = GL2(Zp)

µ p 0 0 1

GL2(Zp), M22(Zp) = GL2(Zp)

µ p 0 0 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 thezi(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 theHecke L-function associated to

π =πf as

L(π, s) :=Y

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,ℓ 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.

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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 ℓ-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). Letρbe a two-dimensional representation of Gal(F). Then there exists an automorphic representation of GL2(AF)

that is realized in L2

0(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 forQ, 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].

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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(A

F)).

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

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IfV Cn is the vector space where Gal(F) acts viaρ, let VIp be the subspace ofV

fixed by any element of the inertia subgroup Ipof P|p1, σ(v) =v, v VIp, σ

∈Ip.

Note that if pis unramified then VIp = 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

(1N(p)−sρ(σp))| VIp

.

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 groupDC 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

i ≃Gal(R).

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

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=R

ΓC(s)n, if ν=C.

1

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CHAPTER 3. THE LANGLANDS PROGRAM 29

Definition 3.1.3. LetF be a number field, and letρbe an-dimensional representation. The Artin L-function associated to ρ is defined as

LF(ρ, s) =L(ρ, s) =

Y

ν

L(ρν, s),

where the product runs through all places ofF.

Remark 3.1.4. For the case n= 1, the inertia subgroup of any prime is isomorphic to 0 orC. That is, if pis 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. Ifρis such that tr(ρ) = 1, then the Artin L-function corresponds to the Dedekind zeta function,

LF(ρ, s) = ζF(s).

2. Ifρ1 and ρ2 are representations of Gal(F), then ρ1ρ2 is also a representation of Gal(F) and

L(ρ1 ρ2, s) =L(ρ1, s)L(ρ2, s).

3. LetE/F be a finite extension of number fields. Letρbe a representation of Gal(E) and letψ be the representation of Gal(F) induced by ρ. Then

L(ρ, s) = L(ψ, s).

Note that in generalρ 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(ˇρ,1s),

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3.2

Automorphic representations

The Section 2.3 motivates the following definition. Let F be a number field, n Nand χ be a character of the centerZ 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 L02(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 L2

0(ZGLn(F)\GLn(AF), χ) is the cuspidality. In

the section 2.3 the functions ϕf satisfy

Z

Q\AQ

ϕf µµ 1 x 0 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 ≃

µ

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 anyα GLn(AF)

Z

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

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CHAPTER 3. THE LANGLANDS PROGRAM 31

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

actionL2

0(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 representationπ of GLn(AF) decomposes into

a restricted tensor product

π =O 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-spondingspherical 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 theHecke operatorsHp,i(i= 1, . . . , n−1),

H±1

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

zi(p). These eigenvalues are called theHecke eigenvalues.

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

L(π, s) = Y p

[(N(p)−sz

1(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) = Y all placesν

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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∗(ˇπ,1s),

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∗

ν, 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 ℓ-adic Galois representations with automorphic forms? This question is

part of some deep conjectures formulated by Robert Langlands in 1967, in a letter to Andr´e 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 ℓ be a rational prime. There exists a corre-spondence between irreducible representations π: Gal(Q)GLn(Fℓ) and automorphic

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

L(ρ, s) =L(π, s).

Moreover, the local L- andε-factors are equal for all primes. We must make some remarks before to continue:

Figure

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Referencias

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