Class field theory viewed as a Langlands correspondence

Transcription

1 Class field theory viewed as a Langlands correspondence Milo Bogaard June 30, 2010 Bachelorscriptie wiskunde Begeleiding: dr. Ben Moonen KdV Instituut voor wiskunde Faculteit der Natuurwetenschappen, Wiskunde en Informatica Universiteit van Amsterdam

2 Abstract In this thesis we formulate the main results of class field theory for local fields and for number fields. This theory gives a description of the Galois group of the maximal abelian extension. A corollary of these theorems is the Langlands correspondence for n = 1. This is a bijection between 1-dimensional representations of the absolute Galois group and certain 1-dimensional representations of a topological group which is the multiplicative group in the local case and the idèle class group in the number field case. In the number field case we attach a complex function, called the Artin L-series, to a representation and using this bijection and theorems on representations of the idèle class group we show that the Artin L-series of a 1-dimensional representation has a holomorphic extension to the complex plane. For the first chapter we follow section 23 of [1] and for the second chapter we follow chapter VII of [5] Gegevens Titel: Class field theory viewed as a Langlands correspondence Auteur: Milo Bogaard, Milo.Bogaard@student.uva.nl, Begeleider: dr. Ben Moonen Tweede beoordelaar: dr. Jochen Heinloth Einddatum: June 30, 2010 Korteweg de Vries Instituut voor Wiskunde Universiteit van Amsterdam Science Park 904, 1098 XH Amsterdam

3 Contents Introduction 2 1 Local theory Local fields Absolute values Extensions of local fields Local class field theory Frobenius automorphisms The reciprocity homomorphism The reciprocity law The Weil group L-functions and ε-factors Characters The Haar-integral L-functions and ε-factors Calculation of the ε-factor The local Langlands correspondence for GL(1) Global theory Number fields Absolute values on number fields Some Galois theory Galois representations Artin L-series The conjectures of Dedekind and Artin The idèle class group Construction Galois module structure Global class field theory Hecke characters The global correspondence

4 Introduction In the first chapter of this thesis we consider class field theory for a local field K. This theory describes the maximal abelian extension K ab of K and its Galois group G(K ab K). Class field theory gives an isomorphism between a dense subgroup of G(K ab K), called the Weil group, and the multiplicative group of K. We then consider continuous 1-dimensional representations, which are called characters. The isomorphism between the Weil group and K gives a bijection between the characters of the groups. This bijection is called the local Langlands correspondence for GL(1). In the case of a number field F class field theory takes the form of an isomorphism between G(F ab F ) and a quotient of the unit group of the adèle ring. The adèle ring is a topological ring specifically constructed for this purpose. As characters of the absolute Galois group of a number field factor though the abelianization this gives a description of the characters of the absolute Galois in terms of specific characters of the idèle class group. For both types of representations we can define a complex function, in the Galois case called Artin L-series and in the idèle case called Hecke L-series. We will see that this bijection respects the L-series. Thus we can derive properties of the Artin L-series from properties of Hecke L-series. Furthermore we know that Artin L-functions have nice properties with respect to representations induced from subgroups, so we can use Brauer s theorem on induced representations to answer some questions about higher dimensional representations. As a result for a Galois extension L F of number fields of degree n we obtain an identity ζ L (s) n ζ F (s) = h, n where ζ L and ζ F are the Dedekind zeta functions of L and F are h is some holomorphic function. This shows ζ L(s) ζ F is holomorphic, which was already (s) conjectured by Richard Dedekind. 2

5 Chapter 1 Local theory The goal of this chapter is to relate 1-dimensional representations of the absolute Galois group of a local field to those of of the multiplicative group of this field. The link between these types of representations is given by local class field theory which gives an isomorphism between the multiplicative group of the field and the Weil group which is a dense subgroup of the absolute Galois group. We will see that the Weil group has more representations and it is easy to see which ones come from representations of the absolute Galois group. 1.1 Local fields We recall some basic properties of local fields. For the proofs the reader is referred to [5]. The notation introduced in this section will be used throughout the first chapter Absolute values Local fields can be defined in several ways, the most convenient way is using properties of an absolute value. Definition 1.1. An absolute value on a field K is a function : K R such that for all x, y K 1. x 0 and x = 0 x = 0, 2. xy = x y, 3. x + y x + y. 3

6 An absolute value is called nonarchimedean if also 3 x + y max{ x, y }. It is called discrete if the image of K in R >0 is a discrete subspace. If K is a field with nonarchimedean absolute value then we denote by o K the set {x K : x 1}, which is called the valuation ring of K. The terminology is justified by the following proposition. Proposition 1.2. The subset o K of K is a local ring, its group of units is U K := {x K : x = 1} and its unique maximal ideal is p := {x K : x < 1}. If is discrete the ideal p is principal and all the ideals of o K have the form p n. We make some conventions regarding notation. A generator of p will usually be denoted by π and is called a prime element. The quotient o K /p is called the residue field. The indices denoting the field will usually be dropped when no confusion is possible. Now local fields can be defined. Definition 1.3. A local field is a field that is complete for a discrete absolute value with a finite residue field. From now on in this chapter K will be used to denote a local field. For n > 0 the subgroups 1 + p n of U are called the higher units and are denoted by U (n). The group U is sometimes denoted by U (0). The following results on the structure of the multiplicative group of K are used later. Proposition 1.4. With the notation as above, for n > 0 one has isomorphisms of groups p n /p n+1 = o/p and U (n) /U (n+1) = o/p. Furthermore, U/U (n) = (o/p n ). Proposition 1.5. For a given choice π of prime element the map Z U K given by (n, u) π n u is a isomorphism of topological groups if Z is given the discrete topology and U the subspace topology inherited from K. For a given x = π n u K the number n Z is independent of the choice of π and is denoted by v K (x). It is called the p-adic valuation of x. 4

7 1.1.2 Extensions of local fields We will use the notation G(k k) for the Galois group of an extension of fields k k. For a finite extension of fields L K the norm map N L K : L K is defined by N L K (x) = det(x) where x is considered as a linear map L L of K vector spaces given by y xy. Let K be a local field and Ω an algebraic closure. The following theorem shows that finite extensions of local fields are also local fields. Theorem 1.6. If K is a local field for the absolute value K and L a finite extension of degree n then the map L : L R given by y L = n N L K (y) K is an absolute value which makes L a local field. Furthermore any extension of to L which makes L a local field is equal to L. If b is the maximal ideal of the valuation ring of L then o L /b is a finite extension of o K /p. Now theorem 1.6 and proposition 1.2 show that there exist e L K, f L K N such that b e L K = pol and f L K = [o L /b : o K /p]. The number e L K is called the ramification index and f L K the inertia index. Again, in situations where no confusion is possible we omit the indices indicating the field. The ramification and inertia index satisfy the following relations. Proposition 1.7. If M L K are extensions of local fields one has 1. e M K = e M L e L K, 2. f M K = f M L f L K, 3. [L : K] = e L K f L K. A finite extension such that e L K = 1 is called unramified and an infinite algebraic extension is called unramified if all finite subextensions are unramified. Unramified extensions are important in local class field theory so we prove some facts here. Proposition 1.8. If K is a local field with residue field κ and L an algebraic extension obtained from K by adjoining a primitive n-th root of unity ζ such that n is not divisible by char(κ) then 1. The extension L K is unramified and of degree f, where f is the least natural number such that (#κ) f 1 mod n. 5

8 2. The Galois group G(L K) is canonically isomorphic to G(µ κ) where µ is the residue field of L. 3. The ring of integers of L is o[ζ] and po[ζ] is its maximal ideal. Proof. (1) Let ζ L be a primitive n-th root of unity generating L and let f be its minimum polynomial. Because f divides X n 1, Gauss s lemma gives f o[x]. Now f κ[x] is irreducible, because if ḡ h = f then ḡ and h are relatively prime because ḡ h divides x n 1 and x n 1 has n distinct roots in an algebraic closure of κ. Now Hensels lemma shows that if ḡ and h are not constant f is not irreducible. If the identification κ = F q is made, then the residue field of L is the smallest extension F q f of F q containing the n th roots of unity. As F is a cyclic group of order q f 1 the field F q f q f contains there roots if and only if n q f 1. Part (1) now follows as n q f 1 if and only if q f 1 mod n. (2) As any element of G(L K) maps the valuation ring and its maximal ideal to themselves, it defines an element of G(µ κ). It is clear that this assignment is a homomorphism. As the maximal ideal is mapped to itself the action of the Galois group commutes with quotienting out the maximal ideal, so the action of G(L K) on the roots of f is the same as the action on the roots of f. As distinct elements of G(L K) act differently on the roots of f they also act differently on the roots of f, so this mapping is injective and as the proof of part (1) shows that #G(L K) = #G(µ κ) it is an isomorphism. (3) We know that the residue field of L is generated by ζ as an extension of the residue field of K. This shows o L = o K [ζ] + po L as o K modules. Nakayama s lemma now shows that o L = o K [ζ]. The compositum of all unramified extension of K is called the maximal unramified extension of K and is denoted by K nr. Any finite subextension of K nr is unramified and K nr is not contained in a larger unramified extension. Proposition 1.9. Let K be a local field contained in an algebraic closure Ω then: 1. The extension K nr K is obtained by adjoining to K all roots of unity ζ n with n prime to p. 2. There are canonical isomorphisms G(K nr K) = G(κ κ) = Ẑ. 3. We have [L K nr : K] = f L K for every finite extension L K. 4. The extension LK nr L is the maximal unramified extension of L. 6

9 Proof. Let K be the extension of K obtained by adjoining to K all roots of unity ζ n with n prime to p. We first prove that for any finite extension L K we have [L K : K] = f L K. Let µ be the residue field of L and κ the residue field of K. Then µ is generated by a primitive n-th roof of unity ζ with gcd(n, char(κ)) = 1. By Hensels lemma the polynomial x n 1 factors into linear terms over L. Let ζ be the root of x n 1 which is mapped to ζ under the quotient map o L µ and let f be its minimum polynomial over K. By proposition 1.8 the degree of f is equal to the degree of µ κ. As K(ζ) K this shows [L K : K] f L K. As L K is unramified [L K : K] = f L K K, so part 2 of proposition proposition 1.7 shows [L K : K] = f L K. This implies a finite extension is unramified if and only if it is contained in K, therefore K = K nr, which proves part (1) and part (3) The infinite extension K can be constructed as a tower of finite extensions. The Galois group is then the projective limit of the Galois groups of the finite extensions. This gives an isomorphism with the Galois group of some extension of F q. By construction this extension contains all n th roots of unity with order prime to p and therefore is F q. The isomorphism G(F q F q ) = Ẑ is well known. The extension LK nr is exactly the extension of L obtained by adjoining to L all n-th roots of unity such that n is relatively prime to the characteristic of the residue field of K. As this is equal to the characteristic of the residue field of L this follows by applying part (1) to L. Note K nr not a local field as it contains a primitive root of unity for every prime q p and thus the residue field would contain such a root and thus be infinite. Furthermore as K is the splitting field of a set of polynomials K nr is a normal extension of K. 1.2 Local class field theory Now we give a brief overview of local class field theory. The goal of this theory is to describe the abelian extensions of a local field and their Galois groups. We will use this description to understanding 1-dimensional representations of arbitrary Galois groups, as these factor through an abelian quotient group, which is the Galois group of an abelian extension Frobenius automorphisms Throughout this section L will be a finite Galois extension of a local field K contained in an algebraic closure Ω. Let G = G(Ω K) be the absolute Galois 7

10 group of K. We identify K nr with the maximal unramified extension of K in Ω and define d K : G G(K nr K) to be the quotient map. If κ is the residue field of K then the Frobenius ϕ K over K is the element of G(K nr K) which corresponds to the map x x #κ in G( κ κ) under the isomorphism in proposition 1.9. If L is unramified it is a subextension of K nr by proposition 1.9 so we have a quotient map G(K nr K) G(L K). We define the Frobenius element ϕ L K of G(L K) to be the image of ϕ K under this map. Finally, to a finite Galois extension L K we attach a semigroup Frob(L nr K) which consists of all elements of G(L nr K) such that the restriction to K nr is a positive integer power of ϕ K. We note that d K factors through G(L nr K), so by abuse of notation we will also denote the induced map by d K. The following propositions give some basic properties of Frob(L nr K). Proposition For σ Frob(L nr K) with fixed field Σ one has: 1. [Σ : K] <, 2. Σ nr = L nr, 3. σ = ϕ Σ. Proof. (1) By definition d( σ) = ϕ j K for some j N >0, which has fixed field Σ K nr, so [Σ K nr : K] = j <. This shows that the inertia index of a finite subextension is bounded by j. As any finite subextension of L nr over L is unramified the ramification index of a finite subextension of Σ is bounded by the ramification index of L. By proposition 1.7 this implies that the degree of any finite subextension of Σ is bounded, thus [Σ : K] <. (3) By the previous point G(Σ nr Σ) = Ẑ, thus the canonical surjection G(L nr Σ) G(Σ nr Σ) gives a continuous and open surjection Ẑ Ẑ which is injective as 1 generates Ẑ. By the fundamental theorem of Galois theory G(L nr Σ nr ) is isomorphic to the kernel of this map and thus trivial so L nr = Σ nr. (4) Let τ κ be the residue field of Σ. As d( σ) = ϕ j K the map induced on κ is the map x x j #κ. As j = [Σ K nr : K] the residue field τ of Σ has j #κ elements. Thus σ is the Frobenius of Σ. Proposition If L K is a finite Galois extension and σ G(L K) then σ can be extended to an element of Frob(L nr K). 8

11 Proof. Let σ G(L K) and let ϕ G(L nr K) be such that d K (ϕ) = ϕ K. The restriction of σ to L K nr is a power of the Frobenius automorphism, so σ L K nr = ϕ n L K for some n N. By Galois theory the map nr G(L nr K nr ) = G(L L K nr ), given by τ τ L is an isomorphism. Thus there is a τ G(L nr K nr ) such that τ L = σϕ n L. Now define σ = τϕ n. Then by construction σ L = τϕ n L = σϕ n ϕ n = σ and d K ( σ) = ϕ n, so σ is the required lift The reciprocity homomorphism For a finite Galois extension L K we will now construct a map from G(L K) to a quotient of K, called the reciprocity homomorphism. Note that as K is abelian this map will factor trough the Galois group of an abelian extension. This construction is done in several steps. For the proofs at each step see [5]. For the infinite extension L nr K we define the subgroup N L nr K(L nr ) of K as the intersection of the subgroups N M K M, where M ranges over all finite subextensions L nr. First we define a map from the Frob(L nr K) to to the group K /N L nr K(L nr ) Proposition For a finite Galois extension of local fields L K the map given by r L nr K : Frob(L nr K) K /N L nr K(L nr ) r L nr K(σ) = N Σ K (π Σ ) mod N L nr K(L nr ), where Σ is the fixed field of σ and π Σ is a prime element of Σ, is a well defined multiplicative map. By proposition 1.11 an element σ G(L K) can be extended to an element of G(L nr K), so we can state the following theorem. Theorem For a finite Galois extension of local fields L K the map given by r L K : G(L K) K /N L K L r L K (σ) = N Σ K (π Σ ) mod N L K L, where Σ is the fixed field of an extension σ of σ to L nr and π Σ is a prime element of Σ, is a well defined homomorphism. In the case of an unramified extension the reciprocity homomorphism can be given explicitly. Proposition If L K is unramified then the reciprocity map is given by r L K (ϕ L K ) = π mod N L K L, where ϕ L K is the Frobenius of L K and π a prime element of K.In this case it is an isomorphism. 9

12 1.2.3 The reciprocity law We can now state the main theorems of class field theory. referred to [5] for the proofs. The reader is Theorem For finite Galois extensions L K the reciprocity homomorphism induces an isomorphism G(L K) ab K /N L K L. As G(L K) ab K /N L K L is an isomorphism the map K G(L K) ab obtained by composing the quotient map with the inverse of r L K is a surjective homomorphism with kernel N L K L. This map is called the norm residue symbol and is denoted by (, L K). The reciprocity map satisfies several functorial relations. To describe them we need a group theoretic notion, the transfer homomorphism. For a subgroup H of G of finite index define a map Ver: G ab H ab as follows. First let R be a set of representatives of the left cosets of H. For every x G and r R there is a unique x r in H such that xr = r x r with r R. We define Ver( x) = r R x r for x G ab. It is not very difficult to verify that this does not depend on choice of representative x or coset representatives R and that Ver is a homomorphism. Note that the next theorem could also be given in terms of the reciprocity homomorphism. Theorem Let L K and L K be finite Galois extensions such that K K and L L. The following diagrams are commutative: 1. (K ) (,L K ) G(L K ) ab N K K K (,L K) G(L K) ab, where G(L K ) ab G(L K) ab is given by restriction. 2. Let also σ G and the map G(L K) ab G(σ(L) σ(k)) ab be given by conjugation then K (,L K) G(L K) ab σ(k ) (,σ(l) σ(k)) G(σ(L) σ(k)) ab ) is commutative. 10

13 3. If M L K are Galois extensions then is commutative. L (,M L) G(M L) K (,M K) Ver G(M K) ab The following theorem is called the existence theorem as the hard part of the proof is to show that for a given open subgroup of K there is an abelian extension whose group of norms is exactly this subgroup. Theorem The map L N L K L gives an inclusion reversing bijection between finite extensions of K and open subgroups of K of finite index, which satisfies the following rules L 1 L 2 N L1 KL 1 N L2 KL 2, and N L1 L 2 K(L 1 L 2 ) = N L1 KL 1 N L2 KL 2 N L1 L 2 K(L 1 L 2 ) = N L1 KL 1N L2 KL The Weil group The results of the previous section can also be stated in a slightly different way. We start by noting that for all finite algebraic extensions L and M of K the diagram K (,L M K) G(L M K) ab K (,M K) G(M K) ab. is commutative by theorem 1.16 so for all a K the automorphisms (a, L K) and (a, M K) agree on L M. Thus the map (a, K ab K) defined by (a, K ab K)(x) = (a, K(x) K)(x) is an automorphism of K ab and the map (, K ab K): K G(K ab K) is a homomorphism. To formulate the reciprocity law in terms of the map (, K ab K) we now need the Weil group. Definition The Weil group W K of a local field K is the inverse image of Z under the map d K : G G(K nr K) = Ẑ. 11

14 Let I K denote the kernel of d K. This group is called the inertia group and consists of all elements of G which act trivially on K nr. By definition W K = n Z ψ n I K, where ψ n is some element of d 1 K (ϕn K ). This gives a topology on the Weil group by defining a subset to be open if and only if the intersections with ψ n I K are open for every n Z. This topology is finer than the subspace topology inherited from G, because if U is open in G then U I K is open in I K so U W K is open in W K. Proposition The map (, K ab K) gives an isomorphism of topological groups K WK ab and image of the unit group is the inertia group. Proof. Let a = π n u then (a, K ab K) K nr = (a, K nr K). Restricting to a finite subextension L gives (a, K nr K) L = (a, L K). As L is unramified prop 1.14 gives (a, L K) = ϕ n L K = (ϕ K) n L where ϕ L K and ϕ K are the Frobenius elements. This implies (a, K nr K) = ϕ n K, thus the image of (, Kab K) is in WK ab. The same argument shows that a is mapped to the inertia group if and only if a is a unit. If (x, K ab K) is the identity map then for every abelian extension L we have x N L K L, so by theorem 1.17 we have x = 1 as the open subgroups form a neighborhood basis of 1. If π K is a prime element we have (π, K nr K) = ϕ K, so the image of π in WK ab, which we call ϕ K, is in d 1 K (ϕ K). There are isomorphisms K (π) U K and W K (ϕ K ) I K, the first is given by proposition 1.5 and the second is given by the fact that the exact sequence 0 I K W K Z 0 splits and the topology on (ϕ K ) is discrete by definition. This gives a sequence (,K (π) U ab K) K (ϕ d K) I K Z where the maps are the identity on the Z components and the last map is projection on the first coordinate. It is thus sufficient to show that U K (,K ab K) I K is a isomorphism of topological groups. Theorem 1.17 shows shows that the subgroups of finite index in U K correspond with those of I K and by the reciprocity theorem the quotients must be isomorphic. Thus for any open subgroup H of finite index in U K there is an exact sequence 0 U K /H I K /(H, K ab K) 0. Now proposition 2.7 of chapter IV of [5] gives lim U K /H = lim K/(H, K ab K). As both U K and I K are profinite and H respective (H, K ab K) range over all open subgroups of finite index this is an isomorphism U K = IK. 12

15 1.3 L-functions and ε-factors The reciprocity isomorphism can be used to get information about 1-dimensional representations of the Weil group as it is in principle simpler to analyze representations of K than representations of the Weil group. This is because the group K is the multiplicative group of a normed field which makes it possible to apply techniques from analysis Characters For technical reason we will also be interested in 1-dimensional representations of K as additive group so the following definition is useful. Definition A locally profinite group is a topological group G such that every open neighborhood of 1 contains a compact open subgroup. The following lemma shows that this concept generalizes K and K. Lemma For a local field K the additive group K and multiplicative group K are locally profinite. Proof. The valuation ring o is compact and open in K. For K note that p n = {x K : x < q n }, so the p n for n Z form a neighborhood basis of 0 of compact subgroups. For K note that the 1 + p n for n 1 are compact open in K and also in K as they do not contain 0. As the p n form a neighborhood basis of 0 the 1 + p n form a basis of neighborhoods for 1 in K. Now we define characters, which are essentially the same as 1-dimensional representations as C = GL(1, C). Definition A character for a locally profinite group G is a continuous homomorphism G C. A character is unitary if its image is in S 1. The characters form a group for multiplication, denoted by Ĝ. The following characterization is sometimes useful. Proposition For a homomorphism ψ : G C equivalent are 1. ψ is a character; 2. ker(ψ) is open. If ψ is a character and G is the union of its compact open subgroups the image of ψ is contained in the unit circle. 13

16 Proof. If ψ 1 (1) is open and g G then gψ 1 (1) = ψ 1 (ψ(g)), so all inverse images are open, in particular those of the open sets. Conversely let ψ be continuous and U an open neighborhood of 1 C, then ψ 1 (U) is open and thus contains a compact open subgroup H. If U is small it contains no nontrivial subgroups C so then H ker(ψ), so ker(ψ) = g ker(ψ) gh, which is open. The second assertion follows as the image of a compact subgroup is in the unit circle. If K is a local field then the subgroups p n for n N >0 form a neighborhood basis of 0 so the kernel of each character ψ of the additive group of K must contain such a group. The level of ψ is defined to be the smallest positive integer n such that p n ker(ψ). For the multiplicative group the subgroups 1 + p n form form a neighborhood basis of the unit element a basis so for a character χ the level is defined to be the smallest nonnegative integer such that 1 + p n+1 ker(χ). A character χ of K is called unramified if U K ker(χ) or equivalently if it factors trough the valuation homomorphism. The following proposition shows that the character group of K is fully determined by any nontrivial character. The construction of such an object can be found in [2]. Proposition Let ψ ˆK be of level d > 0 then: 1. For a K the map x ψ(ax), denoted by aψ, is in ˆK. If a 0 then aψ ˆK has level d v K (a). 2. The map a aψ is a group isomorphism K = ˆK. Proof. The map x ψ(ax) is a continuous homomorphism, so a character. If p is generated by π then p d ker(ψ) is equivalent to π d ker(ψ) so the level of aψ is d v K (a). The map a aψ clearly is an injective group homomorphism. Let θ ˆK be a character of level l. The character π d l ψ has level l so agrees with θ on p l. These characters define a character of p l 1 /p l. For u, u U K the characters uπ d l ψ and u π d l ψ agree on p l 1 /p l if and only if they agree mod p, thus there is a u 1 U K such that if u 1 π d l ψ = θ on p l 1. Continuing inductively there are u j U K such that u j π d l ψ = θ on p l j. As u j u k mod p k this defines an element u U K such that uπ d l ψ = θ. Now we give some examples of characters of Q p. Note that these can be generalized to arbitrary local fields of characteristic 0. Example The map Q p C given by x x p is clearly a character and is unramified. 14

17 To find a ramified character consider the maps Q p U Qp given by x x x and the quotient map U Qp U Qp /1 + p which are continuous homomorphisms. As U Qp U Qp /1 + p = (Z/pZ) any 1 dimensional representation of (Z/pZ) gives by composition a character of Q p The Haar-integral For simplicity from now on G is assumed to be abelian, as this is the only case needed. The proofs are slightly easier in the abelian case as there is no distinction between right and left measures. The reader familiar with Haar measures can safely skip this section. What is defined here is not actually a Haar measure but something simpler which integrates only functions of the following type. Definition For a locally profinite group G let Cc (G) be the C-vector space of functions Φ : G C which are locally constant and have compact support. We have: Lemma For Φ : G C equivalent are: 1. Φ C c (G); 2. Φ is a finite linear combinations of characteristic functions of cosets gh for some open compact subgroup H. Proof. (2) (1) is clear. (1) (2). For every x in the support of Φ there is a compact open subgroup H x such that Φ is constant on xh x. By compactness finitely many of these, call them H 1,..., H n, cover the support, so the support is a union of cosets of H 1... H n. By compactness Φ can only be nonzero on finitely many cosets, so it is a linear combination of characteristic functions of the cosets of H 1... H n. Lemma The space Cc (K) is spanned by the characteristic functions of the sets a + p n for a K and n Z. Proof. As the subgroups p n for n > 0 form a neighborhood basis of 0 in K every compact open subgroup is a finite union of cosets of some p n, so this follows from lemma Definition A Haar integral on G is a non-zero translation invariant linear functional I : Cc (G) C such that I(f) R 0 if f(g) R 0 for all g G. 15

18 A Haar integral only allows us to calculate integrals of functions in Cc but this is sufficient for our purposes. (G), Proposition Up to multiplication with a constant G has a unique Haar integral. Proof. Let {H i } i I be the directed system of compact open subgroups of G and let X be the set of translates of the H i and their finite unions. Claim. There exists a finitely additive and translation invariant function m : X R. This function is unique up to a constant. Proof of claim: Let m(h i ) := c for some i I and c R. By additivity c the function m has to take the value [H i :H i H j on H ] i H j for any j I and by translation invariance the value c [H j:h i H j ] [H i :H i H j on H ] j. Defining m(h j ) to be this value and defining the other values on X by additivity and translation invariance it is clear that m is a finitely additive and translation invariant function m : X R. As noted for given c the function m is determined on all compact open subgroups and therefore determined uniquely on X, so m is unique up to choice of c, which proves the claim. Let f Cc (G), then by lemma 1.27 there is a compact open subgroup H and elements g i G, for 1 i n such that the support of Φ is the union of the cosets g i H. Defining fdm = n G i=0 f(g i) m(h) gives a Haar integral. This integral is unique up to a constant as is gives a function m by integration the indicator functions and this function is unique up to a constant L-functions and ε-factors In this section we define two invariants attached to a character of K. The first is the simple looking L-function. Definition Let χ be a character of K and s a complex variable then we define the L-function L(χ, s) by L(χ, s) = { 1 1 χ(π)q s if χ is unramified 1 otherwise Note that this is well defined as χ(π) does not depend on the choice of π and that an unramified character is determined by its L-function as it is defined by its value at π. For the moment this is all that is necessary concerning the L-function. We will see it again when we look at L-functions in the global case. 16

19 The second invariant will be found by comparing a zeta functions with the zeta function obtained by Fourier transforming the arguments. To define the zeta function first some work needs to be done. Definition Let ψ be a nontrivial character of K and let Φ Cc (F ). The Fourier transform ˆΦ relative to ψ is the function defined by ˆΦ(x) = Φ(y)ψ(xy)dµ(y). F Note that for fixed x the function y Φ(y)ψ(xy) has the same support as Φ and is locally constant, so the integral is defined. If we fix Haar measures µ on K and µ on K we have: Proposition Let Φ C c (K) then: 1. The function ˆΦ is in C c (K). 2. There is a constant c(ψ, µ) R >0 such that for all Φ C c (K) and x K ˆΦ(x) = cφ( x). 3. For a given ψ there is a unique measure µ ψ such that c(ψ, µ ψ ) = 1. This measure satisfies µ ψ (o) = q l/2 where l is the level of ψ. 4. For a K one has µ aψ = a 1/2 µ ψ. Proof. By linearity of the Fourier transform it suffices to check the statements for a basis of Cc (K). Let l be the level of ψ and Φ j the characteristic function of p j. Suppose v K (x) < l j, then ψ(xy), as a character of p j, factors trough p j /p l vk(x) and is nontrivial on this group so Φ(y)ψ(xy)dµ(y) = µ(p l vk(x) ) ψ(xy). F y p j /p l v K (x) This sum is the inner product of the character ψ with the trivial character so by theorem 2.3 of [7] it is zero. If on the other hand v K (x) l j then ψ(xy) is the trivial character on p j, so for x p l j the value of ˆΦ j is µ(p l j ) = µ(o)q l j, where q is the characteristic of the residue field. Applying this twice gives ˆΦj (x) = µ(o) 2 q l Φ j ( x), thus points (1) and (2) hold for Φ j. For a translate of Φ j, a function x Φ j (x a) the Fourier transform becomes Φ j (y a)ψ(xy)dµ(y) = Φ j (y)ψ(x(y + a))dµ(y + a) = aψ(x)ˆφ(x). F F 17

20 Again applying the Fourier transform gives ψ(ay)ˆφ(y)ψ(xy)dµ(y) = ˆΦ(y)ψ((x + a)y)dµ(y) = ˆΦ(x + a), F F so the translates also satisfy the (2) and as these form a basis this proves (2) for C c (K) with c = µ(o) 2 q j. Point (3) requires µ(o) 2 q j and as c(ψ, bµ) = b 2 c(ψ, µ) putting µ(o) = q l/2 suffices. Point (4) follows directly from (3) and proposition 1.24 The unique measure in 3 is called the self dual measure for ψ. Let e m be the characteristic function for p n p n+1. Since p is compact open and p n+1 is a compact open subgroup of finite index e m is of compact support. It follows that for Φ C c (K) the function Φ m = e m Φ is in C c (K ) as 0 is not in the support. Definition Let µ be a Haar measure on K, χ a character of K, Φ C c (K ) and q the order of the residue field. For a complex variable s define a function ζ(φ, χ, s) = m Z z m (q s ) m, where z m = F Φ m (x)χ(x)dµ (x). As a compact subset of K is bounded the support of Φ it is contained in p k for some k Z. For m < k the function e m Φ is constant zero so then z m = 0. This shows that ζ(φ, χ, s) is a formal power series in q s. The following lemma will show that it is also a rational function in q s. For a K and Φ Cc (K) denote by aφ the function x Φ(a 1 x). The relation for the coefficients of the zeta function is z m (aφ, χ) = Φ(a 1 x)χ(x)dµ (x) π m U K = χ(a) Φ(x)χ(x)dµ (x) π m v K (a) U K = χ(a)z m vk (a)(φ, χ), so ζ(aφ, χ, s) = χ(a)x v K(a) ζ(φ, χ, s). Denote Z(χ, s) = {ζ(φ, χ, s) : Φ C c (K )}. Lemma If χ is a character of K then Z(χ, s) = L(χ, s)c[q s, q s ]. 18

21 Proof. If Φ(0) = 0 the restriction of Φ to K is in Cc (K ) so then only finitely many terms are nonzero and ζ(φ, χ, s) C[q s, q s ]. As χ(x) x s is locally constant choosing Φ to be a sufficiently small neighborhood of 1 implies ζ(φ, χ, s) C. Thus by linearity of the map Φ ζ(φ, χ, s) we have {ζ(φ, χ, s) : Φ C c (K )} = C[q s, q s ], where Cc (K ) is identified with a subset of Cc (K) by defining the value in 0 to be 0. Now let Φ 0 be the characteristic function of o then ζ(φ 0, χ, s) = χ(π m )(q s ) m χ(x)dµ m 0 U (x). K If χ is unramified U K χ(x)dµ (x) = µ (U K ) so ζ(φ 0, χ, s) = µ (U K ) m 0 χ(π) m (q s ) m = µ (U K )L(χ, s) as the power series is a geometric series for sufficiently large s. If χ is not unramified χ UK is a character of a compact group and so the kernel is open and χ factors through a finite abelian group G, so χ(x)dµ (x) = µ (ker(χ UK )) χ(g) = 0. U K g G As Φ 0 and C c (K) span C c (K) this proves the lemma. The following lemma is the main part of the proof of the next proposition. Lemma Let Λ be set of all maps λ : Cc all Φ Cc (F ) and a F we have (F ) C(q s ) such that for λ(aφ) = χ(a)x vk(a) λ(φ). Then Λ is a vector space of dimension 1 over C(q s ). Proof. As the map defined by Φ ζ(φ, χ, s) is in Λ it is at least 1 dimensional. Now choose n 0 such that UK n ker(χ) and let Φ k be the characteristic function of UK k. The map Λ C(q s ) given by λ λ(φ n ) is linear. To prove the lemma it is sufficient to prove it is injective. Suppose λ(φ n ) = 0 for some λ Λ. Then for k n and a UK n we have λ(φ k ) = q n k λ(φ n ) = 0. Every element of Cc (K ) is a linear combination of these elements so it is sufficient to prove this for Φ which are the characteristic function of a neighborhood of 0. For such a Φ the function Φ(ax) Φ(x) is in Cc (K ) and λ(φ(ax) Φ(x)) = 0, so Φ(ax) = Φ(x) for all a K. For an a such that χ(a) 1 this implies Φ = 0. 19

22 Note that ζ depends up to a constant on the choice of µ but the invariant defined in the next proposition does not. If χ is a character let ˇχ denote the dual character. Proposition Let ψ be a nontrivial character of K. For Φ Cc (K) let ˆΦ denote the Fourier transform of Φ for the Haar measure on K which is self dual with respect to ψ. If χ is a character of K then there is a unique c(χ, ψ, q s ) C(q s ) such that ζ(ˆφ, ˇχ, 1 s) = c(χ, ψ, q s )ζ(φ, χ, s) for all Φ C c (K). Proof. The map Φ ζ(φ, χ, s) is in Λ so by lemma 1.36 it suffices to show that the map Φ ζ(ˆφ, ˇχ, 1 s) is in Λ. We have âφ(x) = Φ(ay)ψ(xy)dµ(x) F = Φ(y)ψ(a 1 xy)dµ(a 1 x) F = a (a 1 ˆΦ)(x), so by translation invariance of µ ζ(ˆφ, ˇχ, 1 s) = a ˆΦ(a 1 y)χ 1 (x)dµ (x) F = a χ 1 (a) ˆΦ(y)χ 1 (ax)dµ (ax) F = a χ F 1 ˆΦ(y)χ 1 (x)dµ (x) = a χ 1 ζ(ˆφ, ˇχ, 1 s), as required. Definition For a character χ of K the ε-factor is defined by ɛ(χ, s, ψ) = c(χ, ψ, q s L(χ, s) ) L(ˇχ, 1 s), where ψ is a nontrivial character of K and s is a complex variable. 20

23 1.3.4 Calculation of the ε-factor For this section character ψ of K is fixed and the measure on K is taken to be self dual for ψ. Equation 1.1 is called Tate s local functional equation. Theorem The function ε(χ, s, ψ) satisfies the equation and it has the form for some n(χ, ψ) Z. ε(χ, s, ψ)ε(ˇχ, 1 s, ψ) = χ( 1) ε(χ, s, ψ) = q ( 1 2 s)n(χ,ψ) ε(χ, 1 2, ψ), Proof. By proposition 1.33 we have ˆΦ(x) = Φ( x), so the coefficients of the function ζ(ˆφ, χ, s) are z m (ˆΦ, χ) = ˆΦ(x)χ(x)dµ F (x) = Φ( x)χ( x)dµ (x) = χ( 1)z m (Φ, χ), F thus ζ(ˆφ, χ, s) = χ( 1)ζ(Φ, χ, s). Twice applying prop 1.37 and the definition of the ε-factor gives ζ(ˆφ, χ, s) = ε(χ, s, ψ)ε(ˇχ, 1 s, ψ)ζ(φ, χ, s). This proves the functional equation for the ε-factor. By proposition 1.37 ζ(ˆφ, χ 1, 1 s) L(χ 1, 1 s) ζ(φ, χ, s) = ε(χ, s, ψ) L(χ, s) (1.1) and by proposition 1.35 ζ(ˆφ,χ 1,1 s) C[q s, q s ]. The quotient ζ(φ,χ,s) can L(χ 1,1 s) L(χ,s) be made constant by choice of Φ as in the proof of proposition 1.35: If χ is unramified choose Φ such that ζ(φ, χ, s) = 1 and if χ is ramified choosing Φ characteristic on o gives ζ(φ, χ, s) = L(χ, s). This proves ε(χ, s, ψ),ε(ˇχ, 1 s, ψ) C[q s, q s ]. By the functional equation the ε-factor is a unit in the ring C[q s, q s ]. All units of this ring are of the form aq ns for a C and n Z and thus ε(χ, 1, ψ) = aq n 2 and aq ns = q n ( 1 2 )n ε(χ, 1, ψ), thus choosing 2 2 n(χ, ψ) = n suffices. 21

24 It is also possible to give an explicit formula for the ε-factor, the rest of this section will be used to derive it. Lemma For a K one has ɛ(χ, s, aϕ) = χ(a) a s 1 2 ɛ(χ, s, ϕ). Proof. The function ζ(ˆφ, ˇχ, 1 s) change by a factor χ(a 1 ) a 1 2 s as for the Fourier transform one has ˆΦ aψ (x) = K Φ(y)ψ(axy)dµ aψ(y) = a 1/2 a 1 ˆΦψ (x), so by the formula ζ(aφ, χ, s) = χ(a)x v K(a) ζ(φ, χ, s) and the definition of ε the lemma holds. Proposition If χ is unramified and ψ has level one ε(χ, s, ψ) = q s 1 2 χ(π) 1. Proof. The proof of lemma 1.35 shows that if Φ is the characteristic function of o one has ζ(φ, χ, s) = L(χ, s). On the other side ˆΦ(x) = q 1/2 Φ(π 1 x), as in the proof of proposition 1.33, so ζ(ˆφ, ˇχ, s) = q 1/2 K Φ(πx)χ(x) 1 x s d x = q 1/2 s χ(π) 1 K Φ(x)χ(x) 1 x s d x = q 1/2 s χ(π) 1 L(χ 1, s), so ε(χ, s, ψ) = ζ(ˆφ,ˇχ,s) L(χ 1,1 s) = qs 1 2 χ(π) 1. Theorem Let χ be a ramified character of level n 0, let ψ ˆK be of level one, and a K such that v K (a) = n, then ε(χ, s, ψ) = q (ns 1 2 ) r R ˇχ(ax)ψ(ax), where R ranges over a system of coset representatives of U K /U n+1 K. Proof. It is sufficient to calculate this for one Φ. Choosing the indicator function on U n+1 K then χφ = Φ so ζ(φ, χ, s) = Φ(x)χ(x) x dµ (x) = F µ (U n+1 K ). As in proposition 1.33 ˆΦ(y) = q 1 2 n 1 ψ(y) for y p n and zero elsewhere. This shows z m (ˆΦ, χ) = 0 for m < n. For m > n + 1 choose an z p n such that χ(1 + z) 1, this is possible as the level of χ is n. Now z m (ˆΦ, χ) = q 1 2 n 1 p m p m+1 ψ(y)ˇχ(y)dµ (y). 22

25 by translation invariance this integral equals ψ(y(1 + z))ˇχ(y(1 + z))dµ (y) = ˇχ(1 + z) ψ(y)ˇχ(y)dµ (y) p m p m+1 p m p m+1 as yz p. As ˇχ(1 + z) 1 here z m vanishes. Calculating z n gives q 1 2 n 1 ψ(x)ˇχ(x)dµ (x). Considering the levels of ψ and χ the p n p n+1 integrand is constant on cosets of U n+1 K, so ζ(ˆφ, ˇχ, 1 s) = (q s ) n µ (U n+1 K )q 1 2 n 1 ˇχ(ax)ψ(ax). r R The formula for ε(χ, s, ψ) follows from equation The local Langlands correspondence for GL(1) The second theorem in this section is a simple consequence of the class field theory of the previous chapter. First of all we like to know that that the invariants defined so far completely determine the characters. The formulation of the theorem will, considering the proof, be strange, as not all assumptions are used. It turn out that this form is the correct form for the higher dimensional cases, so it is formulated in that fashion. Theorem 1.43 (converse theorem). If θ 1 and θ 2 are characters of K such that for some nontrivial character ψ of K we have L(χθ 1, s) = L(χθ 2, s) and ε(χθ 1, s, ψ) = ε(χθ 2, s, ψ) for all characters χ of K then θ 1 = θ 2. Proof. Let θ 1 and θ 2 satisfy the assumptions of the theorem, then L(θ 1 θ2 1, s) = L(θ 2 θ2 1, s) = 1, so in particular θ 1 q s 1 θ2 1 is unramified. If for given a, b C 1 we have that = 1 for all s C then a = b, so θ 1 aq s 1 bq s 2 θ2 1 (π) = 1. As unramified characters are determined by χ(π) this proves θ 1 θ2 1 is the trivial character, so θ 1 = θ 2. Let W K be the Weil group of K and let r : W K K be the reciprocity map. As the reciprocity map induces an isomorphism WK ab = K each character ρ of W K factors through K, so ρ = χ r for a unique character χ of K. Thus for a character ρ = χ r of W K we define L(ρ, s) = L(χ, s), ε(ρ, s, ϕ) = ε(χ, s, ϕ). Let ch(g) be the set of characters of a group G. The preceding discussion can be summarized by the following theorem. 23

26 Theorem For ψ ˆK nontrivial there is a unique bijection β : ch(w K ) ch(k ) such that L(χβ(ρ), s) = L(χ ρ, s) and ε(χβ(ρ), s, ψ) = ε(χ ρ, s, ψ) for all ρ ch(w K ) and χ ch(k ). If χ is a character of G( K K) then χ WK is a character of the Weil group and χ is uniquely determined by χ WK as W K is dense in G( K K). The characters of W K which are the restriction of a character of a character of G( K K) are those with finite image so we have: Corollary. The map χ β(χ WK ) is a bijection between ch(g( K K)) and the unitary characters of K. 24

27 Chapter 2 Global theory The purpose of this chapter is to give a bijection similar to the one given in section 1.3.5, but now for number fields. It will again rely on class field theory. In the number field case class field theory gives, for a number field F, an isomorphism between G(F ab F ) and a quotient of the idèle class group, which is a topological group specifically constructed for this purpose. This isomorphism will give a bijection between the characters. On both sides of the bijection we can define L-functions which are infinite products of L-functions similar to those considered in section As the absolute Galois group of a number field is a profinite group any continuous representation factors through the Galois group of a finite extension of F. Thus we only need to consider representations of Galois groups of finite extensions of number fields. Some of the L-functions for the characters of the absolute Galois group are familiar, namely the Dedekind zeta functions. Following [7] we give the proof of the Aramata-Brauer theorem which solves part of an old conjecture of Dedekind. 2.1 Number fields We will of course needs some results from number theory, mainly concerning absolute values and the relation between Galois groups and ramification. Throughout this chapter F will be a number field, meaning a finite extension of the rational numbers. Proofs of the results stated can be found in [5]. 25

28 2.1.1 Absolute values on number fields Recall the basics of absolute values as defined in section We now have to introduce some new terminology. Two absolute values 1 and 2 on F are called equivalent if there is an r R >0 such that x r 1 = x 2 for all x F. An equivalence class of nontrivial absolute values is called a place of F. If L is an extension of F and p is a place of L then we say p lies over a place p of F if the restriction of the absolute values in p are in p. The equivalence classes of nonarchimedean absolute values are called finite, the others infinite. Write p if p is an infinite place and p if p is a finite place. If is an absolute value which belongs to a place p then the completion of F with respect to only depends on p and is denoted by F p. For the Archimedean places we have: Theorem 2.1. If a field k is complete with respect to an archimedean valuation k then there is an isomorphism σ from k to either R or C and an s (0, 1] satisfying x k = σ(x) s for all x k, where is the standard absolute value on R or C. This shows that there are only finitely many infinite places and F p = R or F p = C if p is infinite. We call p real if Fp = R and complex if Fp = C. We denote the ring of integers of F by o F. The finite places correspond one on one with the prime ideals of o F and if p is a prime ideal the place it induces will also be denoted by p. Also the completion of F with respect to a finite place is a local field. A infinite place is called normalized if it is the composition of the standard absolute value of C with an embedding. A finite place is called normalized if in the completion the generator of the maximal ideal has norm 1 where #κ κ is the residue field. We will use p for the normalized representative of p. We will need the following results on the behavior of the places of a number field. Theorem 2.2 (Approximation theorem). If p 1,..., p n are pairwise distinct places of F and a 1,..., a n F then for every ε > 0 there exists an x F such that x a i pi < ε for all 1 i n. Theorem 2.3 (Product formula). For a F the norm we have a p = 1 for almost all places p and a p = 1. p 26

29 2.1.2 Some Galois theory We need the results of this section for the proofs later. The results on the Galois groups of completions are used to define an action of the Galois group on the idèle class group and the results on ramification are necessary to prove the basic properties of Artin L series. For the rest of this section L F is a Galois extension of number fields and G = G(L F ). Also let p be a prime ideal of the ring of integers of F and let o L be the ring of integers of L. If P is a prime ideal of o L over p and σ G then σp also is a prime ideal of o L over p. The decomposition group G P of P is defined by G P = {σ G : σp = P} and the inertia group of P is is the normal subgroup of G P defined by I P = {σ G : σx x mod P for all x o L }. As L is dense in L P and an element σ G P is continuous for the topology induced by the absolute value we get a map G P G(L P F p ) by defining for σ G P a map σ : L P L P. The map σ acts on x L P by writing x = lim n x n and defining σ(x) = lim n σ(x). Because of the following proposition we will often identify σ with σ and G P with G(L P F p ). Proposition 2.4. The map G P G(L P F p ) is an isomorphism and the image of I P is the inertia group as defined for G(L P F p ). The following proposition shows that it is visible in the Galois group if L F is unramified at p. Proposition 2.5. The extension L F is unramified at p if and only if I P = 1 for some P above p Let κ(p) and κ(p) be the residue fields of L P and F p. We see that the Galois group behave as expected with respect to the residue fields. Proposition 2.6. The extension κ(p) κ(p) is normal and G(κ(P) κ(p)) = G P /I P. The Frobenius ϕ P G P /I P is defined as the element which acts on κ(p) by x x #κ(p). Not that for unramified extensions this corresponds to the Frobenius element of G(L P F p ). Lemma 2.7. If L M F are Galois extensions and P P p are prime ideals then the maps G P G P, I P I P and G P /I P G P /I P induced by the quotient map G(L F ) G(M F ) are surjective and the image of ϕ P is ϕ P. 27

30 The next proposition shows that in a Galois extension all ideals over a given ideal p are essentially equivalent. Proposition 2.8. If P and P are prime ideals of o L lying over p then there is a σ G such that σp = P. For this σ we have G P = σg P σ 1 and I P = σi P σ 1. Furthermore if τ is such that τ = ϕ P mod I P then στσ 1 = ϕ P mod I P. This proposition shows that if L F is unramified at p then the elements of G which occur as the Frobenius of a prime ideal P over p form a conjugacy class of G. Thus if L F is an abelian extension, then for a given prime ideal p there is a uniquely determined Frobenius element in G(L F ) for the prime ideals over p. The following propositions describe the splitting of p in the fixed fields of the decomposition and inertia groups of a prime ideal P over p. We fix e and f as the ramification index and inertia index of P over p. Proposition 2.9. Let P be the prime ideal of L G P below P then 1. The ideal P is the only prime ideal of the ring of integers of L above P. 2. The ramification index of P in L is e and the inertia index is f. 3. The inertia and ramification index of P over F are 1. For the next proposition we make a further assumption on G. It could also be stated in terms of the extension L L G P. Proposition Assume G = G P. There is one prime ideal P in L I P which lies over p. This ideal has ramification index e and inertia index 1 over p. Furthermore the ramification index of P over P is 1 and the inertia index is f. Let M be a subextension of L F with Galois group H G and let q 1,...q k be the prime ideals of the ring of integers of M which lie over p. Lemma Let τ i G be an element such that τ i (P M) = q i. The the system {τ i } k i=1 forms a system of coset representatives of G P \G/H. Denote the Frobenius element of H P /J P by γ P. Lemma We have H P = G P H and J P = I P H where J P is the inertia group. Furthermore we have γ P = ϕ [G:H] P. 28