On the Functional Equation of the Artin L-functions. Robert P. Langlands

Transcription

1 On the unctional Equation of the Artin L-functions Robert P. Langls

2 Introduction TABLE O CONTENTS 0. Introduction 1. Weil Groups. The Main Theorem 3. The Lemmas of Induction 4. The Lemma of Uniqueness 5. A Property of λ-unctions 6. A iltration of the Weil Group 7. Consequences of Stickelberger s Result 8. A Lemma of Lamprecht 9. A Lemma of Hasse 10. The irst Main Lemma 11. Artin Schreier Equations 1. The Second Main Lemma 13. The Third Main Lemma 14. The ourth Main Lemma 15. Another Lemma 16. Definition of the λ-unctions 17. A Simplification 18. Nilpotent Groups 19. Proof of the Main Theorem 0. Artin L-unctions 1. Proof of the unctional Equation. Appendix 3. References

3 Introduction 3 O. Introduction In this paper I want to consider not just the L-functions introduced by Artin [1] but the more general functions introduced by Weil [15]. To define these one needs the notion of a Weil group as described in [3]. This notion will be explained in the first paragraph. or now a rough idea will suffice. If E is a global field, that is an algebraic number field of finite degree over the rationals or a function field over a finite field, C E will be the idéle class group of E. IfE is a local field, that is the completion of a global field at some place [16], archimedean or nonarchimedean, C E will be the multiplicative group of E. IfK/E is a finite Galois extension the Weil group W K/E is an extension of GK/E, the Galois group of K/E, byc K. It is a locally compact topological group. If E E K K/E is finite Galois W K/E may be regarded as a subgroup of W K/E.It is closed of finite index. If E K L there is a continuous map of W L/E onto W K/E. Thus any representation of W K/E may be regarded as a representation of W L/E. In particular the representations ρ 1 of W K1 /E ρ of W K /E will be called equivalent if there is a Galois extension L/E containing K 1 /E K /E such that ρ 1 ρ determine equivalent representations of W L/E. This allows us to refer to equivalence classes of representations of the Weil group of E without mentioning any particular extension field K. In this paper a representation of W K/E is understood to be a continuous representation ρ in the group of invertible linear transformations of a finite-dimensional complex vector space which is such that ρw is diagonalizable, that is semisimple, for all w in W K/E. Any one-dimensional representation of W K/E can be obtained by inflating a one-dimensional representation of W E/E = C E. Thus equivalence classes of one-dimensional representations of the Weil group of E correspond to quasi-characters of C E, that is, to continuous homomorphisms of C E into C. Suppose E is a local field. There is a stard way of associating to each equivalence class ω of one-dimensional representations a meromorphic function Ls, ω. Suppose ω corresponds to the quasi-character χ E.IfEis nonarchimedean ϖ E is a generator of the prime ideal P E of O E, the ring of integers in E, weset 1 Ls, ω = 1 χ E ϖ E ϖ E s if χ E is unramified. Otherwise we set Ls, ω =1.IfE = R χ E x =sgnx m x r with m equal to 0 or 1 we set Ls, ω =π 1 s + r + m s+r+m Γ. If E = C z E then, for us, z will be the square of the ordinary absolute value. If χ E z = z r z m z n where m n are integers such that m + n 0, mn =0, then Ls, ω =π s+r+m+n Γs + r + m + n

4 Introduction 4 It is not difficult to verify, we shall do so later, that it is possible, in just one way, to define Ls, ω for all equivalence classes so that it has the given form when ω is one-dimensional, so that Ls, ω 1 ω =Ls, ω 1 Ls, ω so that if E is a separable extension of E ω is the equivalence class of the representation of the Weil group of E induced from a representation of the Weil group of E in the class Θ then Ls, ω =Ls, Θ. Now take E to be a global field ω an equivalence class of representations of the Weil group of E. It will be seen later how, for each place v, ω determines an equivalence class ω v of representations of the Weil group of the corresponding local field E v. The product Ls, ω v v which is taken over all places, including the archimedean ones, will converge if the real part of s is sufficiently large. The function it defines can be continued to a function Ls, ω meromorphic in the whole complex plane. This is the Artin L-function associated to ω. It is fairly well-known that if ω is the class contragredient to ω there is a functional equation connecting Ls, ω L1 s, ω. The factor appearing in the functional equation can be described in terms of the local data. To see how this is done we consider separable extensions E of the fixed local field.ifψ is a non-trivial additive character of let ψ E/ be the non-trivial additive character of E defined by χ E,ψ E =χ E γ ψ E/ x =ψ S E/ x where S E/ x is the trace of x. We want to associate to every quasi-character χ E of C E every nontrivial additive character ψ E of E a non-zero complex number χ E,ψ E.IfEis nonarchimedean, if P m E is the conductor of χ E, if P n E is the largest ideal on which ψ E is trivial choose any γ with O E γ = P m+n E set U E ψ E χ 1 E αdα The right side does not depend on γ. If E = R, with m equal to 0 or 1, ψ E x =e πiux then α γ α U E ψ E γ χ E x =sgnx m x r χ E,ψ E =i sgnu m u r. χ 1 E αdα. If E = C, ψ C z =e 4πi Rewz, with m + n 0, mn =0then χ C z = z r z m z n χ C,ψ C =i m+n χ C w. The bulk of this paper is taken up with a proof of the following theorem. Theorem A

5 Introduction 5 Suppose is a given local field ψ a given non-trivial additive character of. It is possible in exactly one way to assign to each separable extension E of a complex number λe/,ψ to each equivalence class ω of representations of the Weil group of E a complex number εω, Ψ E/ such that i If ω corresponds to the quasi-character χ E then εω, ψ E/ = χ E,ψ E/. ii εω 1 ω,ψ E/ =εω 1,ψ E/ εω,ψ E/. iii If ω is the equivalence class of the representation of the Weil group of induced from a representation of the Weil group of E in the class θ then εω, ψ =λe/, ψ dim θ εθ, ψ E/. α s will denote the quasi-character x x s of C as well as the corresponding equivalence class of representations. Set εs, ω, ψ =ε α s 1 ω, ψ. The left side will be the product of a non-zero constant an exponential function. Now take to be a global field ω to be an equivalence class of representations of the Weil group of. Let A be the adéle group of let ψ be a non-trivial character of A/. or each place v let ψ v be the restriction of ψ to v. ψ v is non-trivial for each v almost all the functions εs, ω v,ψ v are identically 1 so that we can form the product v εs, ω v,ψ v. Its value will be independent of ψ will be written εs, ω. Theorem B The functional equation of the L-function associated to ω is Ls, ω =εs, ω L1 s, ω. This theorem is a rather easy consequence of the first theorem together with the functional equations of the Hecke L-functions. or archimedean fields the first theorem says very little. or nonarchimedean fields it can be reformulated as a collection of identities for Gaussian sums. our of these identities which we formulate as our four main lemmas are basic. All the others can be deduced from them by simple group-theoretic arguments. Unfortunately the only way at present that I can prove the four basic identities is by long involved, although rather elementary, computations. However Theorem A promises to be of such importance for the theory of automorphic forms group representations that we can hope that eventually a more conceptual proof of it will be found. The first the second, which is the most difficult, of the four main lemmas are due to Dwork [6]. I am extremely grateful to him not only for sending me a copy of the dissertation of Lakkis [9] in which a proof of these two lemmas is given but also for the interest he has shown in this paper.

6 Chapter 1 6 Chapter One. Weil Groups The Weil groups have many properties, most of which will be used at some point in the paper. It is impossible to describe all of them without some prolixity. To reduce the prolixity to a minimum I shall introduce these groups in the language of categories. Consider the collection of sequences S : C λ 1 G µ G of topogical groups where λ is a homeomorphism of C with the kernel of µ µ induces a homeomorphism of G/λC with G. Suppose λ S 1 : C 1 µ 1 G1 G1 is another such sequence. Two continuous homomorphisms ϕ ψ from G to G 1 which take C into C 1 will be called equivalent if there is a c in C 1 such that ψg =cϕgc 1 for all g in G. S will be the category whose objects are the sequences S Hom S0 S, S 1 will be the collection of these equivalence classes. S will be the category whose objects are the sequences S for which C is locally compact abelian G is finite; if S S 1 belong to S Hom S S, S 1 =Hom S0 S, S 1. Let P 1 be the functor from S to the category of locally compact abelian groups which takes S to C let P be the functor from S to the category of finite groups which takes S to G. We have to introduce one more category S 1,0. The objects of S 1 will be the sequences on S for which G c, the commutator subgroup of G, is closed. Moreover the elements of Hom S1 S, S 1 will be the equivalence classes in Hom S S, S 1 all of whose members determine homeomorphisms of G with a closed subgroup fo finite index in G 1. If S is in S 1 let V S be the topological group G/G c.ifφ Hom S1 S, S 1 let ϕ be a homeomorphism in the class Φ let G = ϕg. Composing the map G 1 /G c 1 G/Gc given by the transfer with the map G/G c G/G c determined by the inverse of ϕ we obtain a map Φ v : V S 1 V S which depends only on Φ. The map S V S becomes a contravariant functor from S 1 to the category of locally compact abelian groups. If S is the sequence C G G the transfer from G to C determines a homomorphism τ from G/G c to the group of G-invariant elements in C. τ will sometimes be regarded as a map from G to this subgroup. The category E will consist of all pairs K/ where is a global or local field K is a finite Galois extension of. HomK/, L/E will be a certain collection of isomorphisms of K with a subfield of L under which corresponds to a subfield of E. If the fields are of the same type, that is all global or all local we dem that E be finite separable over the image of.if is global E is local we dem that E be finite separable over the closure of the image of. I want to turn the map which associates to each K/ the group C K into a contravariant functor which I will denote by

7 Chapter 1 7 C.Ifϕ : K/ L/E E are of the same type let K 1 be the image of K in L let ϕ C be the composition of N L/K1 with the inverse of ϕ. If is global E is local let K 1 be the closure in L of the image of K. As usual C K1 may be considered a subgroup of the group of idèles of K. ϕ C is the composition of N L/K1 with the projection of the group of idèles onto C K. If K is given let E K be the subcategory of E whose objects are the extensions with the larger field equal to K whose maps are equal to the identity on K. Let C be the functor on E K which takes K/ to C. If is given let E have as objects the extensions with the smaller field equal to. Its maps are to equal the identity on. A Weil group is a contravariant functor W from E to S with the following properties: i P 1 W is C. ii P W is the functor G : L/ GL/. iii If ϕ GL/ HomL/, L/ g is any element of W L/, the middle group of the sequence W L/, whose image in GL/ is ϕ then the map h ghg 1 is in the class ϕ w. iv The restriction of W to E K takes values in S 1. Moreover, if K/ belongs to E K τ: W K/ /W c K/ C is a homeomorphism. inally, if ϕ : K/ K/E is the identity on K Φ=ϕ w then the diagram W K/ /WK/ c τ C Φ v ϕc WK/E /W c K/E τ C E is commutative if ψ : / K/ is the imbedding, ψ W is τ. Since the functorial properties of the Weil group are not all discussed by Artin Tate, we should review their construction of the Weil group pointing out, when necessary, how the functorial properties arise. There is associated to each K/ a fundamental class α K/ in H GK/, C K. The group W K/ is any extension of GK/ by C K associated to this element. We have to show, at least, that if ϕ : K/ L/E the diagram 1 C L W L/E GL/E 1 ϕ C ϕ G 1 C K W K/ GK/ 1 can be completed to a commutative diagram by inserting ϕ : W L/E W K/. The map ϕ C commutes with the action of GL/E on C L C K so that ϕ exists if only if ϕ C α L/E is the restriction ϕ G α K/ of ϕ K/ to GL/E. If this is so, the collection of equivalence classes to which ϕ may belong is a principal homogeneous space of H 1 GL/E, C K. In particular, if this group is trivial, as it is when ϕ G is an injection, the class of ϕ is uniquely determined. An examination of the definition of the fundamental class shows that it is canonical. In other words, if ϕ is an isomorphism of K L of E, then ϕ G α K/ =ϕ 1 α L/E = ϕ C α L/E. If K = L ϕ is the identity on K, the relation ϕ G α K/ =α L/E = ϕ C α L/E is one of the basic properties of the fundamental class. Thus in these two cases ϕ exists its class is unique.

8 Chapter 1 8 Now take K to be global L local. Suppose at first that K is contained in L, that its closure is L, that = K E. Then, by the very definition of α K/,ϕ G α K/ =ϕ C α L/E. More generally, if K 1 is the image of K in L, 1 the image of in E, we can write ϕ as ϕ 1 ϕ ϕ 3 where ϕ 3 : K/ K 1 / 1,ϕ : K 1 / 1 K 1 /K 1 E, ϕ 1 : K 1 /K 1 E L/E. ϕ 3 ϕ exist. If the closure of K 1 is L then ϕ 1 therefore ϕ = ϕ 3 ϕ ϕ 1 also exist. The class of ϕ is uniquely determined. Artin Tate show that WK/ c is a closed subgroup of W K/ that τ is a homeomorphism of W K/ /WK/ c C. Granted this, it is easy to see that the restriction of W to ξ K takes values in S 1. Suppose we have the collection of fields in the diagram with L K normal over L K normal over. Let α, β, ν be the imbeddings α : L/ L/K, β : L/ L/K, ν : L/ L/. L K K We have shown the existence of α, β, ν. It is clear that ν βw L/K is contained in αw L/K. Thus we have a natural map Let us verify that the diagram π : ν βw L/K / ν βw c L/K αw L/K/ αw c L/K. W L/K /W c ν βw L/K L/K / ν βw L/K C π αw L/K / αw K/K c W L/K/WL/K c τ τ N K /K C k A C k is commutative. To see this let W L/K be the disjoint union U r i=1 C Kh i. Then we can choose h i,g j, 1 i r, 1 j s so that W L/K is the disjoint union r i=1 s j=1 C K g j h i ν βh i = αh i. Using these coset representatives to compute the transfer one immediately verifies the assertion. We should also observe that the transitivity of the transfer implies the commutativity of the diagram W K/ /WK/ c τ C Φ v WK/ /WK/ c ϕc C

9 Chapter 1 9 if Φ is the class of an imbedding ϕ where ϕ is an imbedding K/ K/. We have still not defined ϕ W for all ϕ. However we have defined it when ϕ is an isomorphism of the two larger fields or when the second large field is the closure of the first. Moreover the definition is such that the third condition all parts of the fourth condition except the last are satisfied. The last condition of iv can be made a definition without violating i ii. What we do now is show that there is one only one way of extending the definition of ϕ W to all ϕ without violating conditions i or ii the functorial property. Suppose K L, K/ L/ are Galois, ψ is the imbedding L/ L/K. It is observed in Artin Tate that there is one only class of maps {θ} which make the following diagram commutative 1 W L/K /WL/K c ψw L/K / ψw L/K c W L/ / ψw L/K c W L/ / ψw L/K 1 τ θ 1 C K W K/ GK/ 1. The homomorphism on the right is that deduced from W L/ /W L/K = GL/ /GL/K GK/. Let ϕ, µ, ν be imbeddings ϕ : K/ L/, µ : K/K L/K, ν : K/ K/K. Then ψ ϕ = µ ν, so that ν µ = ϕ ψ. Moreover ν µ is the composition of the map τ : W L/K C K the imbedding of C K in W K/. Thus the kernel of ϕ contains ψwl/k c so that ϕ ψ restricted to W L/K /WL/K c must be τ the only possible choice for ϕ is, apart from equivalence, θ. To see that this choice does not violate the second condition observe that the restriction of τ to C L will be N L/K that ψ is the identity on C L. Denote the map θ : W L/ W K/ by θ L/K the map τ : W K/ C K by τ K/. It is clear that τ K/ θ L/K is the transfer from W L/ /WL/ c to ψw L/K / ψwl/k c followed by the transfer from ψw L/K / ψwl/k c to ψc L = C. By the transitivity of the transfer τ K/ θ L/K = τ L/. It follows immediately that if K L L all extensions are Galois the map θ L /K θ L/K θ L /L are in the same class. Suppose that ϕ is an imbedding K/ K / choose L so that K L L/ is Galois. Let ψ : K / L/, µ : K/ L/, ν : L/ L/ be imbeddings. Then ψ ϕ = ν µ so that µ ν = ϕ ψ. Ifα : L/ L/K, β : L/ L/K are the imbeddings then the kernel of ψ is ν βwl/k c which is contained in αw c L/K the kernel of µ. Thus there is only one way to define ϕ so that µ ν = ϕ ψ. The diagram W L/K /W c L/K β W L/ / βwl/k c ψ W K / ν ϕ W L/ / ν βw c L/K W L/ / αw c L/K µ W K/

10 Chapter 1 10 will be commutative. Since ψ β = τ L/K µ α = τ K/ diagram A shows that ϕ has the required effect on C K. To define ϕ W in general, we observe that every ϕ is the composition of isomorphisms, imbeddings of fields of the same type, a map K/ K / where K is global, K is local, K is the closure of K, = K. Of course the identity ϕ ψ W = ψ W ϕ W must be verified. I omit the verification which is easy enough. The uniqueness of the Weil groups in the sense of Artin Tate implies that the functor W is unique up to isomorphism. The sequence Sn, C : GLn, C id GLn, C 1 belongs to S 1.IfS : C G G belongs to S 1 then Hom S0 S, Sn, C is the set of equivalence classes of n-dimensional complex representations of G. Let Ω n S be the set of all Φ in Hom S0 S, Sn, C such that, for each ϕ Φ, ϕg is semi-simple for all g in G. Ω n S is a contravariant functor of S so is ΩS = n=1 Ω ns. On the category S 1, it can be turned into a covariant functor. If ψ : S S 1,ifΦ ΩS, if ϕ Φ, let ψ associate to Φ the matrix representations corresponding to the induced representation IndG 1,ψG,ϕ ψ 1. It follows from the transitivity of the induction process that Ω is a covariant functor of S 1. To be complete a further observation must be made. Lemma 1.1 Suppose H is a subgroup of finite index in G ρ is a finite-dimensional complex representation of H such that ρl is semi-simple for all h in H. If then σg is semi-simple for all g. σ =IndG, H,ρ H contains a subgroup H 1 which is normal of finite index in G, namely, the group of elements acting trivially on H\G. To show that a non-singular matrix is semi-simple one has only to show that some power of it is semi-simple. Since σ n g =σg n g n belongs to H 1 for some n we need only show that σg is semi-simple for g in H 1. In that case σg is equivalent to r i=1 ρg igg 1 i if G is the disjoint union r Hg i i=1 Suppose L/ K/ belong to E ϕ Hom E L/, K/. Since the maps of the class ϕ W all take W K/ onto W L/ the associated map ΩW L/ ΩW K/ is injective. Moreover it is independent of ϕ. IfL 1 / L / belong to E there is an extension K/ maps ϕ 1 Hom E L 1 /, K/, ϕ Hom E L /, K/. ω 1 in ΩW L 1 / ω in ΩW L / have the same image in ΩW K/ for one such K if only if they have the same image for all such K. If this is so we say that ω 1 ω are equivalent. The collection of equivalence classes will be denoted by Ω. Its members are referred to as equivalence classes of representations of the Weil group of.

11 Chapter 1 11 Let be the category whose objects are local global fields. If E are of the same type Hom,E consists of all isomorphisms of with a subfield of E over which E is separable. If is global E is local Hom,E consists of all isomorphisms of with a subfield of E over whose closure E is separable. Ω is clearly a covariant functor on. Let gl, loc be the subcategories consisting of the global local fields respectively. Suppose E are of the same type ϕ Hom,E. If ω ΩE choose K so that ω belongs to ΩW K/E. We may assume that there is an L/ an isomorphism ψ from L onto K which agrees with ϕ on. Then ψ W : W K/E W L/ is an injection. Let θ be the equivalence class of the representation σ =IndW L/,ψ W W K/E,ρ ψ 1 W with ρ in ω. I claim that θ is independent of K depends only on ω ϕ. To see this it is enough to show that if L L,L / is Galois, ψ is an isomorphism from L to K which agrees with ψ on L, ρ is a representation of W K / in ω the class of σ =IndW L /, ψ W W K /E, ρ ψ 1 w is also Θ. Suppose µ is a map from W K /E to W K/E associated to the imbedding K/E K /E ν is a map from W L / to W L/ associated to the imbedding L/ L /. We may suppose that ψ W µ = ν ψ W. The kernel of µ is W K c /K if, for simplicity of notation, W K /K is regarded as a subgroup of W K /E that of ν is WL c /L. Moreover ψ W W K c /K =W L c /L. Take ρ = ρ µ. Then σ acts on the space V of functions f on W K/ satisfying fvw ρψw 1hfw for v in ψ ww K/E. Let V be the analogous space on which σ acts. Then V = {f ν f V }. The assertion follows. Thus Ω is a contravariant functor on gl loc. After this laborious clumsy introduction we can set to work prove the two theorems. The first step is to reformulate Theorem A.

12 Chapter 1 Chapter Two. The Main Theorem It will be convenient in this paragraph at various later times to regard W K/E as a subgroup of W K/ if E K. If E L K we shall also occasionally take W L/E to be W K/E /W c K/L. If K/ is finite Galois, PK/ will be the set of extensions E /E with E E K P K/ will be the set of extensions in PK/ with the lower field equal to. Theorem.1 Suppose K is a Galois extension of the local field ψ is a given non-trivial additive character of. There is exactly one function λe/,ψ defined on P K/ with the following two properties i λ/,ψ =1. ii If E 1,...,E r,e 1,...,E s are fields lying between K,ifχ E i, 1 i r, is a quasi-character of C Ei,ifχ E j, 1 j s, is a quasi-character of C E j, if r i=1 IndW k/, W K/Ei,χ Ei is equivalent to then is equal to r s s j=1 IndW K/, W K/E j,χ E j i=1 χ E i,ψ Ei / λe i /, ψ j=1 χ E j,ψ E j / λe j /, ψ. A function satisfying the conditions of this theorem will be called a λ-function. It is clear that the function λe/, ψ of Theorem A when restricted to P K/ becomes a λ-function. Thus the uniqueness in this theorem implies at least part of the uniqueness of Theorem A. To show how this theorem implies all of Theorem A we have to anticipate some simple results which will be proved in paragraph 4. irst of all a λ-function can never take on the value 0. Moreover, if K L the λ-function on P K/ is just the restriction to P K/ of the λ-function on P L/. Thus λe/, ψ is defined independently of K. inally if E E E λe /E, ψ E =λe /E,ψ E /EλE /E, ψ E [E :E ]. We also have to use a form of Brauer s theorem [4]. If G is a finite group there are nilpotent subgroups N 1,...,N m, one-dimensional representations χ 1,...,χ m of N 1,...,N m respectively, integers n 1,...,n m such that the trivial representation of G is equivalent to m i=1 n iindg, N i,χ i.

13 Chapter 13 The meaning of this when some of the n i are negative is clear Lemma. Suppose is a global or local field ρ is a representation of W K/. There are intermediate fields E 1,...,E m such that GK/E i is nilpotent for 1 i m, one-dimensional representations χ Ei of W K/Ei, integers n 1,...,n m such that ρ is equivalent to m i=1 n iindw K/, W K/Ei,χ Ei. Theorem.1 Lemma. together imply the uniqueness of Theorem A. Before proving the lemma we must establish a simple well-known fact. Lemma.3 Suppose H is a subgroup of finite index in the group G. Suppose τ is a representation of G, σ a representation of H, ρ the restriction of τ to H. Then τ IndG, H,σ IndG, H,ρ σ. Let τ act on V σ on W. Then IndG, H, σ acts on X, the space of all functions f on G with values in W satisfying fhg =σh fg while IndG, H, ρ σ acts on Y, the space of all functions f on G with values in V W satisfying fhg =ρh σh fg. Clearly, V X Y have the same dimension. The map of V X to Y which sends v f to the function f g =τgv fg is G-invariant. If it were not an isomorphism there would be a basis v 1,...,v n of V functions f 1,...,f n which are not all zero such that This is clearly impossible. Σ n i=1 τgv i f i g 0. To prove Lemma. we take the group G of Brauer s theorem to be GK/. Let i be the fixed field of N i let ρ i be the tensor product of χ i, which we may regard as a representation of W K/i the restriction of ρ to W K/i. Then ρ ρ 1 m i=1n i IndW K/i,ρ i. This together with the transitivity of the induction process shows that in proving the lemma we may suppose that GK/ is nilpotent. We prove the lemma, with this extra condition, by induction on [K : ]. We use the symbol ω to denote an orbit in the set of quasi-characters of C K under the action of GK/. The restriction of ρ to C K is the direct sum of one-dimensional representations. If ρ acts on V let V ω be the space spanned by the vectors transforming under C K according to a quasi-character in ω. V is the direct sum of the

14 Chapter 14 spaces V ω which are each invariant under W K/. or our purposes we may suppose that V = V ω for some ω. Choose χ K in this ω let V 0 be the space of vectors transforming under C K according to χ K. Let E be the fixed field of the isotrophy group of χ K. V 0 is invariant under W K/E. Let σ be the representation of W K/E in V 0. It is well-known that ρ IndW K/, W K/E,σ. To see this one has only to verify that the space X on which the representation on the right acts V have the same dimension that the map f W K/E \W K/ ρg 1 fg of X into V which is clearly W K/ -invariant has no kernel. It is easy enough to do this. If E the assertion of the lemma follows by induction. If E = choose L containing so that K/L is cyclic of prime degree L/ is Galois. Then ρw K/L is an abelian group W c K/L is contained in the kernel of ρ. Thus ρ may be regarded as a representation of W L/. The assertion now follows from the induction assumption the concluding remarks of the previous paragraph. Now take a local field E a representation ρ of W K/E. Choose intermediate fields E 1,...,E m, one-dimensional representations χ Ei of W K/Ei, integers n 1,...,n m so that If ω is the class of ρ set ρ m i=1 n i IndW K/E, W K/Ei,χ Ei. εω, ψ E = m { χ E i, Ψ Ei /EλE i /E, Ψ E } n i. i=1 Theorem.1 shows that the right side is independent of the way in which ρ is written as a sum of induced representations. The first second conditions of Theorem A are clearly satisfied. If ρ is the representation above σ the representation IndW K/, W K/E,ρ then σ m i=1 n iindw K/, W K/Ei,χ Ei. Thus if ω is the class of σ εω,ψ = m { χ E i,ψ Ei / λe i /, ψ } n i i=1 while εω, ψ E/ = m { χ E i,ψ Ei / λe i /E, ψ E/ } n i. i=1 The third property follows from the relations dim ω =Σ m i=1 n i [E i : E] λe i /, ψ =λe i /E, ψ E/ λe/,ψ [E i:e]

15 Chapter 3 15 Chapter Three. The Lemmas of Induction In this paragraph we prove two simple but very useful lemmas. Lemma 3.1 Suppose K is a Galois extension of the local field. Suppose the subset A of PK/ has the following four properties. i or all E, with E K, E/E A. ii If E /E E /E belong to A so does E /E. iii If L/E belongs to PK/ L/E is cyclic of prime degree then L/E belongs to A. iv Suppose that L/E in PK/ is a Galois extension. Let G = GL/E. Suppose G = H C where H {1}, H C = {1}, C is a non-trivial abelian normal subgroup of G which is contained in every non-trivial normal subgroup of G. IfE is the fixed field of H if every E /E in P L/E for which [E : E] < [E : E ] is in A so is E /E. Then A is all of PK/. It is convenient to prove another lemma first. Lemma 3. Suppose K is a Galois extension of the local field E K. Suppose that the only normal = subfield of K containing E is K itself that there are no fields between E. Let G = GK/ let E be the fixed field of H. Let C be a minimal non-trivial abelian normal subgroup of G. Then G = HC, H C = {1} C is contained in every non-trivial normal subgroup of G. In particular if H = {1}, G = C is abelian of prime order. H is contained in no subgroup besides itself G contains no normal subgroup but {1}. Thus if H is normal it is {1} G has no proper subgroups is consequently cyclic of prime order. Suppose H is not normal. Since G is solvable it does contain a minimal non-trivial abelian normal subgroup C. Since C is not contained in H, H HC G = HC. Since H C is a normal subgroup = of G it is {1}. IfD is a non-trivial normal subgroup of G which does not contain C then D C = {1} D is contained in the centralizer Z of C. Then DC is also Z must meet H non-trivially. But Z H is a normal subgroup of G. This is a contradiction the lemma is proved. The first lemma is certainly true if [K : ]=1. Suppose [K : ] > 1 the lemma is valid for all pairs [K : ] with [K : ] < [K : ]. If the Galois extension L/E belongs to PK/ then A PL/E satisfies the condition of the lemma with K replaced by L by E. Thus, by induction, if [L : E] < [K : ], PL/E A. In particular if E /E is not in G then E = the only normal subfield of K containing E is K itself. If A is not PK/ then amongst all extensions which are not in G choose one E/ for which [E : ] is minimal. Because of ii there are no fields between E. Lemma 3., together with iii iv, show that E/ is in A. This is a contradiction. There is a variant of Lemma 3.1 which we shall have occasion to use. Lemma 3.3 Suppose K is a Galois extension of the local field. Suppose the subset A of P K/ has the following properties.

16 Chapter 3 16 i / A. ii If L/ is normal L = K then P L/ A. iii If E E K E/ belong to G then E / belong to A. iv If L/ in P K/ is cyclic of prime degree then L/ A. v Suppose that L/ in P K/ is Galois G = GL/. Suppose G = HC where H {1}, H C = {1}, C is a non-trivial abelian normal subgroup of G which is contained in every non-trivial normal subgroup. If E is the fixed field of H if every E / in P L/ for which [E : ] < [E : ] is in A so is E/. Then A is P K/. Again if A is not P K/ there is an E/ not in A for which [E : ] is minimal. Certainly [E : ] > 1. By ii iii, E is contained in no proper normal subfield of K there are no fields between E. Lemma 3. together with iv v lead to the contradiction that E/ is in A.

17 Chapter 4 17 Chapter our. The Lemma of Uniqueness Suppose K/ is a finite Galois extension of the local field ψ is a non-trivial additive character of. A function E/ λe/, ψ on P K/ will be called a weak λ-function if the following two conditions are satisfied. i λ/, Ψ =1. ii If E 1,...,E r,e 1,...,E s are fields lying between K,ifµ i, 1 i r, is a one-dimensional representation of GK/E i,ifν j, 1 j s, is a one-dimensional representation of GK/E j, if r IndGK/, GK/E i,µ i is equivalent to then is equal to s r s i=1 j=1 i=1 j=1 IndGK/, GK/E j,ν j χ Ei,ψ Ei / λe i /, ψ χ E j ψ E j / λe j /, ψ if χ Ei is the character of C Ei corresponding to µ i χ E j is the character of C E j corresponding to ν j. Supposing that a weak λ-function is given on P K/, we shall establish some of its properties. Lemma 4.1 i If L/ in P K/ is normal the restriction of λ,ψ to P L/ is a weak λ-function. ii If E/ belongs to P K/ λe/, ψ 0the function on P K/E defined by is a weak λ-function. λe /E, ψ E/ =λe /, ψ λe/, ψ [E :E] Any one-dimensional representation µ of GL/E may be inflated to a one-dimensional representation, again called µ, ofgk/e IndGK/, GK/E, µ is just the inflation to GK/ of IndGL/, GL/E, µ. The first part of the lemma follows immediately from this observation. As for the second part, the relation λe/e, ψ E/ =1

18 Chapter 4 18 is clear. If fields E i, 1 i r, are given as prescribed if E j, 1 j s, lying between E K representations µ i ν j is equivalent to then is equivalent to so that is equal to r s i=1 IndGK/E, GK/E i, µ i =ρ j=1 IndGK/E, GK/E j,ν j=σ r s r s Since ρ σ have the same dimension so that r i=1 IndGK/, GK/E i,µ i j=1 IndGK/, GK/E j, ν j i=1 χ E i,ψ Ei / λe 1 /, ψ A j=1 χ E j,ψ E j / λe j /, ψ. B Σ r i=1 [E i : E] =Σ s j=1 [E j : E] λe/, Ψ [Ei:E] = s λe/, ψ [E j : ]. i=1 j=1 Dividing A by the left side of this equation B by the right observing that the results are equal we obtain the relation needed to prove the lemma. If K/ is abelian SK/ will be the set of characters of C which are 1 on N K/ C K. Lemma 4. If K/ is abelian λk/, Ψ = µ SK/ µ,ψ. µ determines a one-dimensional representation of GK/ which we also denote by µ. The lemma is an immediate consequence of the equivalence of IndGK/, GK/K, 1 µ SK/ IndGK/, GK/, µ. Lemma 4.3 Suppose K/ is normal G = GK/. Suppose G = HC where H C = {1} C is a non-trivial abelian normal subgroup. Let E be the fixed field of H L that of C. Let T be a set of

19 Chapter 4 19 representatives of the orbits of SK/L under the action of G. Ifµ T let B µ be the isotropy group of µ let B µ = GK/L µ. Then [L µ : ] < [E : ] λe/, ψ = µ T µ,ψ Lµ / λl µ /, ψ. Here GK/L µ =GK/L GK/L µ GK/E µ is the character of C Lµ character of GK/L µ :g µg 1 if associated to the g = g 1 g, g 1 GK/L, g GK/L µ GK/E, We may as well denote the given character of GK/L µ by µ also. To prove the lemma we show that IndGK/, GK/E, 1 = σ is equivalent to IndGK/, GK/L µ, µ. µ T Since T has at least two elements it will follow that [E : ] = dim IndGK/, GK/E, 1 is greater than [L µ : ] = dim IndGK/, GK/L µ,µ. The representation σ acts on the space of functions on H\G. Ifν SK/L, that is, is a character of C, let ψ ν hc =νc if h H, c C. The set {ψ ν ν SK/L} is a basis for the functions on H\G. Ifµ T let S µ be its orbit; then V µ =Σ ν Sµ Cψ ν is invariant irreducible under G. Moreover, if g belongs to GK/L µ σgψ µ = µ gψ µ. Since dim V µ =[GK/, GK/L µ ] the robenius reciprocity theorem implies that the restriction of σ to V µ is equivalent to IndGK/, GK/L µ,µ. Lemma 4. is of course a special case of Lemma 4.3. Lemma 4.4 λe/,ψ is different from 0 for all E/ in P K/.

20 Chapter 4 0 The lemma is clear if [K : ]=1. We prove it by induction on [K : ]. Let G be the set of E/ in P K/ for which λe/, ψ 0. We may apply Lemma 3.3. The first condition of that lemma is clearly satisfied. The second follows from the induction assumption the first part of Lemma 4.1; the third from the induction assumption the second part of Lemma 4.1. The fourth fifth follow from Lemmas respectively. We of course use the fact that χ E, Ψ E, which is basically a Gaussian sum when E is non-archimedean, is never zero. or every E /E in PK/ we can define λe /E, ψ E/ to be λe /, ψ λe/, ψ [E :E]. Lemma 4.5 If E /E E /E belong to PK/ then λe /E, ψ E/ =λe /E,ψ E / λe /E, ψ E/ E :E ]. Indeed λe /E, ψ E/ =λe /, ψ λe/,ψ [E :E] which equals { λe /, ψ λe /, ψ [E :E ] }{ λe /, ψ [E :E ] λe/,ψ [E :E] } this in turn equals λe /E,ψ E / λe /E, ψ E/ [E :E ]. Lemma 4.6 If λ 1, Ψ λ, Ψ are two weak λ-functions on P K/ then for all E /E in PK/. λ 1 E /E, ψ E/ =λ E /E, ψ E/ We apply Lemma 3.1 to the collection G of all pairs E /E in PK/ for which the equality is valid. The first condition of that lemma is clearly satisfied. The second is a consequence of the previous lemma. The third fourth are consequences of Lemmas respectively. Since a λ-function is also a weak λ-function the uniqueness of Theorem.1 is now proved.

21 Chapter 5 1 Chapter ive. A Property of λ-unctions It follows immediately from the definition that if ψ E x =ψ Eβx then χ E,ψ E =χ Eβ χ E,ψ E. Associated to any equivalence class ω of representations of the Weil group of the field is a onedimensional representation or, what is the same, a quasi-character of C. It is denoted detω is obtained by taking the determinant of any representation in ω. Suppose ρ is in the class ω ρ is a representation of W K/. To find the value of the quasi-character detω at β choose w in W K/ so that τ K/ w = β. Then calculate detρw which equals detωβ. If E K the map τ = τ K/ can be effected in two stages. We first transfer W K/ /WK/ c into W K/E /WK/E c ; then we transfer W K/E/WK/E c into C K.IfW K/ is the disjoint union r W K/Ew i i=1 if w i w = u i ww j i then the transfer of w in W K/E /WK/E c is the coset to which w = r i=1 u iw belongs. Suppose σ is a representation of W K/E ρ =IndW K/, W K/E,σ. ρ acts on a certain space V of functions on W K/ if V i is the collection of functions in V which vanish outside of W K/E w i then V = r i=1 V i. We decompose the matrix of ρw into corresponding blocks ρ ji w. ρ ji w is 0 unless j = ji when ρ ji w =σu i, w. This makes it clear that if ι E/ is the representation of W K/ induced from the trivial representation of W K/E detρw = detι E/ w dim σ detσw or, if θ is the class of σ, detωβ ={det ι E/ β} dim θ {det θβ}. Lemma 5.1 Suppose is a local field E/ λe/,ψ ω εω, ψ E/ satisfy the conditions of Theorem A for the character ψ. Let ψ x =ψ βx with β in C.IfE/ λe/,ψ ω εω, ψ E/ satisfy the conditions of Theorem A for ψ then λe/,ψ =detι E/β λe/,ψ εω, ψ E/ =detωβ εω, ψ E/. Because of the uniqueness all one has to do is verify that the expressions on the right satisfy the conditions of the theorem for the character ψ. This can now be done immediately.

22 Chapter 6 Chapter Six. A iltration of the Weil Group In this paragraph I want to reformulate various facts found in Serre s book [1] as assertions about a filtration of the Weil group. Although some of the lemmas to follow will be used to prove the four main lemmas, the introduction of the filtration itself is not really necessary. It serves merely to unite in a form which is easily remembered the separate lemmas of which we will actually be in need. Let K be a finite Galois extension of the non-archimedean local field let G = GK/. Let O be the ring of integers in let p be the maximal ideal of O.Ifi 1is an integer let G i be the subgroup of G consisting of those elements which act trivially on O /p i+1.ifu 1is a real number i is the smallest integer greater than or equal to u set G u = G i. inally if u 1set ϕ K/ u = u 0 1 [G 0 : G t ] dt. The integr is not defined at -1 but that is of no consequence. ϕ K/ is clearly a piecewise linear, continuous, increasing map of [ 1, onto itself. The inverse function* ψ K/ will have the same properties. We take from Serre s book the following lemma. Lemma 6.1 If L K L/ is normal then ϕ K/ = ϕ L/ ϕ K/L ψ K/ = ψ K/L ψ L/. The circle denotes composition not multiplication. This lemma allows us to define ϕ E/ ψ E/ for any finite separable extension E/ by choosing a Galois extension L of which contains E setting ϕ E/ = ϕ L/ ψ L/E ψ E/ = ϕ L/E ψ L/ because if L is another such extension we can choose a Galois extension K containing both L L ϕ L/ ψ L/E = ϕ L/ ϕ K/L ψ K/L ψ L/E = ϕ K/ ψ K/E = ϕ L / ψ L /E ϕ L/E ψ L/ = ϕ L/E ϕ K/L ψ K/L ψ L/ = ϕ K/E ψ K/ = ϕ L /E ψ L /. Of course ψ E/ is the inverse of ϕ E/. Lemma 6. If E E E E /E is finite separable, ϕ E /E = ϕ E /E ϕ E /E ψ E /E = ψ E /E ψ E /E. * In this chapter ψ K/ does not appear as an additive character. Nonetheless, there is a regrettable conflict of notation.

23 Chapter 6 3 Each of these relations can be obtained from the other by taking inverses; we verify the second ψ E /E ψ E /E = ϕ L/E ψ L/E ϕ L/E ψ L/E = ϕ L/E ψ L/E = ψ E /E. It will be necessary for us to know the values of these functions in a few special cases. Lemma 6.3 i If K/ is Galois unramified ψ K u u. ii If K/ is cyclic of prime degree l if G = G t while G t+1 = {1} where t is a non-negative integer then ψ K/ u =u u t = t + lu t u t. These assertions follow immediately from the definitions. Lemma 6.4 Suppose K/ is Galois G = GK/ is a product HC where H {1}, H C = {1}, C is a non-trivial abelian normal subgroup of G which is contained in every non-trivial normal subgroup. i If K/ is tamely ramified so that G 1 = {1} then G 0 = C is a cyclic group of prime order l [G : G 0 ]=[H :1]divides l 1. If E is the fixed field of H, ψ E/ u =u for u 0 ψ E/ u =lu for u 0. ii If K/ is wildly ramified there is an integer t 1 such that C = G 1 =... = G t while G t+1 = {1}. [G 0 : G 1 ] divides [G 1 :1] 1 every element of C has order p or 1. If E is the fixed field of H L that of C ψ L/ u =u u 0 =[G 0 : G 1 ]u u 0 while ψ E/ u =u = t [G 0 : G 1 ] +[G 1 :1] u t [G 0 : G 1 ] u u t [G 0 : G 1 ] t [G 0 : G 1 ] We observed in the third paragraph that C must be its own centralizer. G 0 cannot be {1}. Thus C G 0. In case i G 0 is abelian thus G 0 = C. In both cases if l is a prime dividing the order of C the set of elements in C of order l or 1 is a non-trivial normal subgroup of G thus C itself. In case i C is cyclic thus of prime order l. Moreover, H which is isomorphic to G/G 0 is abelian, if h H, {c C hc = ch} is a normal subgroup of G hence {1} or C. Ifh 1it must be 1. Consequently each orbit of H in C {1} has [H :1]elements [H :1]divides l 1. In case ii G 1 is a non-trivial normal subgroup hence contains C. G 1 C are both p-groups. The centralizer of G 1 in C is not trivial. As a normal subgroup of G it contains C. Therefore it is C G 1 is contained in C which is its own centralizer. Since each G 1,i 1, is a normal subgroup of G,itis either C or {1}. Thus there is an integer t 1 such that G 1 = G t = C while G t+1 = {1}. Ifi 0 is an

24 Chapter 6 4 integer let UK i be the group of units of O K which are congruent to 1 modulo p i+1 1 K ; let U K = C K, if U 1is any real number let i be the smallest integer greater than or equal to u set UK u = U K i. If θ t is the map of G t /G t+1 into p t K /pt+1 K θ 0 the map of G 0 /G 1 into UK 0 /U K 1 introduced in Serre then, for g in G 0 h in C, θ t ghg 1 =θ 0 g t Θ t h. If h 1,ghg 1 = h if only if θ 0 g t =1 then g belongs to the centralizer of C, that is to G 1. Again C {1} is broken up into orbits, each with [G 0 : G 1 ] elements [G 0 : G 1 ] divides [G i :1] 1. Observe that t must be prime to [G 0 : G 1 ]. It follows immediately from the definitions that H u = H G u. In case i H 0 will be {1} ϕ K/E u will be identically u. Thus ψ E/ = ψ K/, from the definition, ψ K/ u =u if u 0 while ψ K/ u =[G 0 :1]u if u 0. In case ii, ϕ K/E u =u if u 0 ϕ K/E u = if u 0 while ψ K/ u =u if u 0 u [H 0 :1] = u [G 0 : G 1 ] ψ K/ u =[G 0 : G 1 ]u = t +[G 0 :1] t u [G 0 : G 1 ] t 0 u [G 0 : G 1 ] t [G 0 : G 1 ] u. The lemma follows. Lemma 6.5 or every separable extension E /E the function ψ E /E is convex, if u is an integer so is ψ E /Eu. All we have to do is prove that the assertion is true for all E /E in PK/ if is an arbitrary non-archimedean local field K an arbitrary Galois extension of it. To do this we just combine the previous three lemmas with Lemma 3.1. We are going to use the same method to prove the following lemma. Lemma 6.6 or every separable extension E /E any u 1 N E /E U ψ E /E u E U u E. We have to verify that the set G of all E /E in PK/ for which the assertion is true satisfies the conditions of Lemma 3.1. There is no problem with the first two. Lemma 6.7 E /E belongs to G if only if for every integer n 1 N E /E U ψ E /E n E U n E

25 Chapter 6 5 N E /E U ψ E /E n+1 E U n+1 E. If E /E belongs to G choose ε>0 so that ψ E /En + ε =ψ E /En+1. The smallest integer greater than or equal to n + ε is at least n +1so N E /E U ψ E /E n+1 E U n+ε E U n+1 E. Conversely suppose the conditions of the lemma are satisfied n<u<n+1. Since ψ E /En is an integer the smallest integer greater than or equal to ψ E /Eu is at least ψ E /En+1. Thus N E /E U ψ E /E u E N E /E U ψ E /E n+1 E U n+1 E = UE. u Lemma 6.8 If L/E is Galois then, for every integer n 1, N L/E U ψ L/En L UE n N L/E U ψ L/En+1 L U n+1 E. The assertion is clear if n = 1. A proof for the case n 0 L/E totally ramified is given in Serre s book. Since that proof works equally well for all L/E we take the lemma as proved. Lemma 6.9 Suppose K/ is Galois G = GK/. Suppose G = HC where H {1}, H C = {1}, C is a non-trivial abelian normal subgroup of G which is contained in every non-trivial normal subgroup of G. IfE is the fixed field of H N E/ U ψ E/ u U u for all u 1. E Let L be the fixed field of C. If K/ is tamely ramified K/E L/ are unramified so that ψ E/ = ψ K/L U v E = C E U v K, U v = C U v L for every v 1. Ifα belongs to C E, then delete N K/L α = N E/ α. Since K/L is Galois N E/ U ψ E/ u E C N K/L U ψ K/Lu L C U u L = U u. If K/ is not tamely ramified if n 1 0 m<[g 0 : G 1 ]. Thus p n E = E p[g 0:G 1 ]n m k U v E = G E U v K if 1 v 0 U v E = C E U [G 0:G 1 ]v K

26 Chapter 6 6 if v 0 or, more briefly, UE v = C E U ψ K/Ev K for all v 1. In the same way we find U v = C U ψ L/ v L for all v 1. Since K/L is normal N E/ U ψ E/ u C N K/L Lemma 6.6 now follows immediately. Lemma 6.10 E U ψ K/ u K C U ψ L/ u L = U u. a Suppose K/ is Galois G = GK/. Suppose t 1is an integer such that G = G t G t+1. Then ψ K/ u =u for u t. Moreover N K/ defines an isomorphism of C K /UK t with C /U t if 1 u t the inverse image of U u/u t is U K u /U K t. However the map of C K/U t+1 K into C /U t+1 defined by the norm is not surjective. b Suppose K/ is Galois G = GK/. Suppose s 1is an integer G = G s. If E K, ψ E/ u =u for u s N E/ defines an isomorphism of C E /UE s C /U s. If 1 u s the inverse image of U u/u s is U E u/u E s. If t = 1 the assertions of part a are clear. If t 0,K/ is totally ramified. The relation ψ K/ u =u for u t is an immediate consequence of the definition. Since the extension is totally ramified N K/ defines an isomorphism of U 1 K /U K 0 1 U /U 0. It follows from Proposition V.9 of Serre s book that if 0 n<tthe associated map UK n n+1 /UK U n n+1 /U is an isomorphism but that the map UK t t+1 /UK U t t+1 /U has a non-trivial cokernel. The first part of the lemma is an immediate consequence of these facts. To prove part b we first observe that there is a t s such that G = G t G t+1. It then follows from part a that the map N K/ determines an isomorphism of C K /UK s C /U s under which UK u /U K s U u/u s correspond if 1 u s. Let E be the fixed field of H. We have H s = H G s = H, so that N K/E determines on isomorphism of C K /UK s C E/UE s under which UK u /U K s U E u/u E s correspond if 1 u s. Moreover if u s, ψ K/ u =ψ K/E u =u so that ψ E/ u =u. Part b follows from these observations the relation N K/ = N E/ N K/E. If E is any non-archimedean local field u> 1 If α belongs to C E set U u E = v<uu v E. v E α =sup{u α U u E}. Then v E 1 =, but v E α is finite if α 1 α belongs to U v Eα E. If L K, τ K/,L/ will be any of the maps W K/ W L/ associated to the imbedding L/ K/. We abbreviate τ K/,/ to τ K/. If w belongs to W K/,σw is the image of w in GK/, E is the fixed field of σw, weset v K/ w =ϕ E/ v E τ K/E w.

27 Chapter 6 7 Note that we regard W K/E as a subgroup of W K/.Ifv 1 let W v K/ = {w v K/ w v}. We shall show that W v K/ is a normal subgroup of W K/. These groups provide a filtration of the Weil group, some of whose properties are established in the following lemmas. Lemma 6.11 If σ GK/ t =sup{u σ G u }, set v K/ σ =ϕ K/ t. Then v K/ σ =max{v K/ w σw =σ}. If σ =1both sides are infinite the assertion is clear. If σ 1let E be the fixed field of σ. If σw =σ, w belongs to W K/E v K/ w =ϕ E/ v K/E w. Also v K/ σ =ϕ E/ v K/E σ. Consequently it is sufficient to prove the lemma when = E. The set S = {τ K/ w σw =σ} is a coset of N K/ C K in C C is generated by N K/ C K together with any element of S. Moreover s =max{v β β S} is the largest integer such that S U s is not empty. Since G = G t G t+1 the preceding lemma shows that s = t = ϕ K/ t. Lemma 6.1 a or all w w 1 in W K/,v K/ w =v K/ w 1 v K/ w 1 ww 1 1 =v K/ w. b If E K w belong to W K/E then v K/ w =ϕ E/ v K/E w. c or all w in W K/,τ K/ w U v K/ w. The first two assertions follow immediately from the definitions the basic properties of the Weil group. I prove only the third. Let me first observe that if E K w W K/E, then τ K/ w =N E/ τ K/E w. To see this, choose a set of representatives w 1,...,w r for the cosets of C K in W K/E then a set of representatives v 1,...,v s for the cosets of W K/E in W K/. Let w i w = a i w ji with a i in C K ; then However v j w i w = v j a i v 1 j v j w ji so that τ K/ w = s j=1 r v ja i v 1 i=1 j τ K/E w = r a i. i=1 = s v jτ K/E wv 1 j=1 j = N E/ τ K/E w. In particular, if E is the fixed field of σw, τ K/E w is contained U ψ E/ v K/ w E contained in N E/ U ψ E/ v K/ w E U v K/ w. τ K/ w is

28 Chapter 6 8 Lemma 6.13 If u v belong to W K/ then v K/ uv min{v K/ u,v K/ v}. Let σ = σu let τ = σv. Because of the second assertion of the previous lemma we may assume that σ τ generate GK/. Let E be the fixed field of στ. If t = {min ψ K/ v K/ σ,ψ K/ v K/ τ} G = GK/ then G = G t G t+1. According to Lemma 6.11, if s =min{v K/ u,v K/ v}, then t ψ K/ s which, by Lemma 6.10, is therefore equal to s. Since τ K/ uv = τ K/ uτ K/ v, τ K/ uv lies in U s. On the other h τ K/ uv =N E/ τ K/E uv so that, by Lemma 6.10 again, τ K/E uv belongs to U s E v K/ uv ϕ E/ s =s. Thus the sets W x K/,x 1, give a filtration of W K/ by a collection of normal subgroups. The next sequence of lemmas show that the filtration is quite analogous to the upper filtration of the Galois groups. Lemma 6.14 or each x 1 the map τ K/,L/ takes G x K/ into Gx L/. If w belongs to W K/ let w = τ K/,L/ w. We must show that v L/ w v K/ w. Let σ = σw let σ = σ w. IfE is the fixed field of σ then E = E L is the fixed field of σ. Since v L/ w =ϕ E/ v L/E w v K/ w =ϕ E/ v K/E w we may suppose E =. Since τ K/ w =τ L/ w, Lemma 6.1 implies that τ L/ w lies in U v K/ w. Thus v L/ w =v τ L/ w v K/ w. Of course W / is C, if v 1, W v / = U v. Lemma 6.15

29 Chapter 6 9 or each v 1, τ K/ maps W v K/ onto U v. Since v 1 v implies W v K/ W v 1 K/ it is enough to prove the lemma when v = n is an integer. The lemma is clear if [K : ]=1; so we proceed by induction on [K : ]. If[K : ] > 1, choose an intermediate normal extension L so that [L : ] = l is a prime. Let G = GL/. Lemma 6.1 implies that W ψ L/ v K/L = W K/L WK/ v. There is an integer t 1such that G = G t G t+1 = {1}. It is shown in Chapter V of Serre s book that if n>t N L/ U ψ L/ n = U n. By induction τ K/L L W ψ L/ n K/L Since τ K/ w =N L/ τ K/L w if w is in W K/L, τ K/ W n K/ =U n = U ψ L/ n L. if n>t. Suppose σ generates G. Then V L/ σ =t. By Herbr s theorem there is a σ in GK/ with v K/ σ =t whose restriction to L is σ. By Lemma 6.11 there is a w in W K/ such that σ = σw v K/ w =t. Then τ K/ w lies in U t but not in N L/ C L. rom Serre s book again [ ] U t : N L/ U ψ L/ t = l so that U t is generated by τ K/ w N L/ U ψ L/ t L hence is contained in the image of WK/ t. To complete the proof of the lemma we have only to observe that Lemma 6.10 implies that if n t. Lemma 6.16 U n = U t N L/ U ψ L/ n L Suppose L K L/ K/ are Galois. Then, for each v 1, τ K/,L/ maps W v K/ onto W v L/. If [L : ] = 1 this is just the previous lemma so we proceed by induction on [L : ]. We have to show that if w belongs to W L/ there is a w in W K/ such that w = τ K/,L/ w v K/ w v L/ w. Let σ = σw let E be the fixed field of σ. IfE then, by the induction assumption, there is a w in W K/E such that τ K/E,L/E w =w v K/E w v L/E w. By Lemma 6.1, v K/ w v L/ w. Moreover, we may assume that τ K/E,L/E is the restriction to W K/E of τ K/,L/. Suppose E =. Then v L/ w =v τ L/ w. Choose w 1 in W K/ so that τ K/ w 1 = τ L/ w v K/ w 1 v τ L/ w. Let w 1 = τ K/,L/ w 1 set u = w 1 1 w. Certainly v L/ u v L/ w. Moreover, τ L/ u =1. Let L 1 L where L 1 / is cyclic of prime order. If u does not belong to W L/L1 the group C is generated by N L1 / C L1 τ L/ u, which is impossible since τ L/ u =1. Thus u belongs to W L/L1, as observed, there is a u in W K/L1 such that τ K/,L/ u =τ K/L1,L/L 1 u =u. Then τ K/,L/ uw 1 =w.