Extendible functions and local root numbers: Remarks on a paper of R. P. Langlands

This paper refers to Langlands' big set of notes devoted to the question if the (normalized) local Hecke–Tate root number Δ=Δ(E,χ)$\Delta =\Delta (E,\chi )$ , where E is a finite separable extension of a fixed nonarchimedean local field F, and χ a quasicharacter of E×$E^\times$ , can be appropriately extended to a local ε‐factor εΔ=εΔ(E,ρ)$\varepsilon _\Delta =\varepsilon _\Delta (E,\rho )$ for all virtual representations ρ of the corresponding Weil group WE$W_E$ . Whereas Deligne has given a relatively short proof by using the global Artin–Weil L‐functions, the proof of Langlands is purely local and splits into two parts: the algebraic part to find a minimal set of relations for the functions Δ, such that the existence (and unicity) of εΔ$\varepsilon _\Delta$ will follow from these relations; and the more extensive arithmetic part to give a direct proof that all these relations are actually fulfilled. Our aim is to cover the algebraic part of Langlands' notes, which can be done completely in the framework of representations of solvable profinite groups, where two modifications of Brauer's theorem play a prominent role.


Introduction
We begin from sketching the background of Langlands' paper [L] where for convenience we refer to [T3], where amongst others that background (for [L] and [D]) has been explained.Thus let F be a global field and ϕ : W F → G F the absolute Weil-and Galois group over F .Depending on an F -homomorphism i ν : F → F ν we have (for all places ν of F ) the local-global relationship connecting the absolute Weil-and Galoisgroups resp., where C F = I F /F × is the idele class group and the left square is given by class field theory.On F × ν and C F we have the normalized absolute values | | Fν and | | resp., which pull back to absolute values ω ν and ω on W Fν and W F resp. such that ω ν = ω • θ ν .Let now (ρ, V ) be a representation of the global Weil group W F , and let V ν be the action of W Fν on V which is obtained via θ ν .Then we have (V ⊗ ω s ) ν = V ν ⊗ ω s ν for all s ∈ C, and the Artin-Weil L-function where the product is over all places ν of F with local factors L((V ⊗ ω s ) ν ) as defined by E.Artin ( [T3],(3.3)).On the infinite product L(V ⊗ ω s ) we know (loc.cit.(3.5.3)):Theorem: The product converges for s in some right half-plane and defines a function L(V, s) which is meromorphic in the whole s-plane and satisfies the functional equation: where ε(V, s) := ε(V ⊗ ω s ), and V ∨ is the contragredient of V.
For 1-dimensional V corresponding to a quasicharacter χ of the idele-class-group C F this result was proved by Hecke and in modern version by Tate [T1].More precisely where now, due to loc.cit., we have a decomposition of ε(χ, s) = ε(χ| | s ) as a product of local factors which is obtained as follows: Consider A F the ring of F -adeles and dx the Tamagawa measure on the additive group A F .Then (see [W],VII, §2) depending on a fixed non-trivial additive character ψ of A F /F we obtain a decomposition dx = ν dx ν where dx ν is the Haar measure on F ν which is selfdual with respect to the local component And for the pair (ψ ν , dx ν ) we have Tate's local functional equation [T3],(3.2.1) which defines ε(χ ν | | s Fν , ψ ν , dx ν ) for the local components χ ν : The problem was, to establish a similar product decomposition also for general (virtual) representations of W F .Since a local functional equation for higherdimensional V ν is not available (Tate's approach rests on class field theory W ab Fν ∼ = F × ν ) the idea of [L] was (purely locally) to extend χ ν → ε(χ ν , ψ ν , dx ν ) to a function V ν ∈ R(W Fν ) → ε(V ν , ψ ν , dx ν ) on the Grothendieck group R(W Fν ) of virtual representations, which for field extensions E w |F ν is inductive in dimension zero (see below).Then via [T3],(2.3.5)V → ν ε(V ν , ψ ν , dx ν ) is inductive in dimension zero with respect to global extensions E|F , and is an extension of χ → ε(χ) = ν ε(χ ν , ψ ν , dx ν ).Because V → ε(V ) has this property too (it is actually inductive in the unconditional sense), the two extensions of χ → ε(χ) must agree.From now we restrict to the local problem with respect to the nonarchimedean completions F ν , that means F is now a nonarchimedean local field, ψ a nontrival character of F + and χ a quasicharacter of F × .And corresponding to the above decomposition of the Tamagawa measure, [L] restricts to functions ε(χ, ψ) := ε(χ, ψ, dx) where dx is the ψ-selfdual Haar measure on F + .Then Tate's local functional equation for the 1-dimensional case (loc.cit.)yields if we take (as [L] does) the normalization ∆(χ, ψ) := ε(χ| | 1/2 F , ψ).By class field theory we have W ab E ∼ = E × for all finite extensions E|F and [L] considers the function for all finite separable extensions E|F and quasicharacters χ E of E × .Problem: Is it possible to extend this to a function which is defined for all E|F and virtual representations ρ , and moreover The answer is yes, and going backwards we take ε( , ψ ν ) to obtain the decomposition of the global ε(V, s).The paper [L] splits into two parts: • Algebraic part: Find a minimal set of relations for the functions ∆ = ∆(χ E , ψ E|F ) such that the existence of a (uniquely determined) extension ε ∆ = ε ∆ (ρ E , ψ E|F ) will follow from these relations.
• Arithmetic part: Give a direct proof that all these relations are actually fulfilled.
In this paper we will cover only the algebraic part, but from a purely group theoretical point of view which presents the arguments in a more conceptual way.Instead of local Weil groups W F we will consider solvable profinite groups Ω as for instance the absolute Galois groups G F .This comes close because we have the continuous embedding W F ⊂ G F with dense image.Thus let Ω be a solvable profinite group.By a subgroup H of Ω, denoted H ≤ Ω, we usually understand an open subgroup, or, equivalently, a closed subgroup of finite index.Besides that we will consider the closed commutator subgroups [H, H] ≤ Ω which in general will not be open anymore.By definition we have Ω = lim ← −N Ω/N as a projective limit over the open normal subgroups.Together with N also [N, N] is normal in Ω and therefore also (1) where R(G) denotes the Grothendieck group of virtual continuous representations of G and R(Ω/ [N,N ] ) ֒→ R(Ω) is naturally embedded.Instead of pairs (E, χ E ) as above we consider now: R 1 (≤ Ω) ∋ (H, χ H ) the set of pairs such that H ≤ Ω and χ H : H → C × is a continuous 1-dimensional character, for short we will write: χ H ∈ H * .
From that point of view the question is, when a function ∆ = ∆(H, χ) on R 1 (≤ Ω) can be extended to a function F = F (H, ρ) on R(≤ Ω) := H≤Ω R(H) such that Our main tool will be Brauer's Theorem and it's variations: The induction map (H, χ H ) → Ind Ω H (χ H ) ∈ R(Ω) is actually a map which depends only on [H, χ H ] := Ω-conjugacy class of (H, χ H ) and induces the Brauer map (2) Ind Ω H (χ H ), where R + (≤ Ω) is the free abelian group, generated by the Ω-conjugation classes [H, χ H ]. Then we have: (see [S], section 10.5 as an easy accessible reference) Brauer's Theorem 1: Let Ω be a profinite group and ρ a virtual representation of Ω.Then there are pairs (H i , χ i ) consisting of open subgroups H i and 1-dimensional characters χ i ∈ H * i , and integers n i such that Consequently ϕ Ω : R + (≤ Ω) → R(Ω) is a surjective homomorphism of additive groups, and actually ( [D],1.9, see below) it is a homomorphism of commutative rings.Brauer's Theorem 2: (see [S],exercise 10.6.or [D],Prop.1.5)If ρ ∈ R(Ω) is virtual of dimension 0 then it admits a Brauer presentation: If Ω is finite, the groups H i can be taken elementary, in particular nilpotent, subgroups of Ω.For profinite groups this fails but anything else is maintained because R(Ω) = lim − →N R(Ω/N) for open normal subgroups N, yields a reduction to the finite case.In order to simplify the chapters 15-19 of [L] and to present them in a more conceptual way by going down to the purely group theoretical background, the manuscript [K] had been built on these two theorems and on the description of Ker(ϕ Ω ) as in [D], §1.But at a certain point this approach failed, because it didn't take care of the role of relative Weilgroups W E|F = W F /[W E , W E ] in the original paper [L].From the purely group theoretical point of view using relative Weil groups means to approach R(Ω) via (1) and to use the following two more modifications of Brauer's theorem: If N ≤ Ω is an open normal subgroup, then we will write R 1 (N ≤ Ω) for the set of pairs (H, χ H ) such that H ≥ N, hence χ H is actually a character of H/ such that any virtual representation ρ of Ω/ [N,N ] can be presented as integral combination of monomial representations , where all H i are open subgroups containing N.
Brauer's Theorem 4: ) is virtual of dimension 0 then it admits a Brauer presentation: where all occurring subgroups H i contain N.
Remark: Since ϕ N ≤Ω is nothing else than the restriction of ϕ Ω , it may happen that we sometimes omit the N. Similar as for Brauer 1 and 2, the case of profinite groups can be reduced to the case of finite ones.Some further remarks we will add in section 4 where Ω/ [N,N ] will be replaced by a relative Weil group The easiest example is the case N = Ω where ϕ Ω≤Ω : R ) is the identity because both sides represent the free Z-module over Ω * .Proofs of Brauer 3 and 4 will be given in Appendix 1.
Now we return to the question when a function ∆ = ∆(H, χ) on R 1 (≤ Ω) can be extended to a function F = F (H, ρ) on R(≤ Ω) := H≤Ω R(H) as above.Langlands' approach in [L], as we interpret it here, is, for all finite subquotients Ω ′ /N ′ of Ω to consider the restrictions ∆ ′ of ∆ to R 1 (N ′ ≤ Ω ′ ) = H; N ′ ≤H≤Ω ′ H * and to prove in an inductive way that ∆ ′ can be extended to Thus fixing a natural number n we assume that ∆ extends from R 1 (N ′ ≤ Ω ′ ) onto R(N ′ ≤ Ω ′ ) for all subquotients such that (Ω ′ : N ′ ) < n, and ask for conditions to conclude that ∆ will then extend also for the cases where (Ω ′ : N ′ ) = n.(Of course it can happen that the assumption (Ω ′ : N ′ ) = n is empty, then we may go directly to n + 1.)And since the arguments will not depend on taking Ω or Ω ′ as our absolute group, we come down to the problem of extending ∆ from R 1 (N ≤ Ω) onto R(N ≤ Ω), if we assume this for all proper subquotients Ω ′ /N ′ of Ω/N.As a general criterion which is motivated by [L], Theorem 2.1 we have: 1.4 Proposition: Let (Ω, N) be a profinite group and an open normal subgroup, and ∆ a function on R 1 (N ≤ Ω) with values in the multiplicative abelian group A.
) if and only if for all subgroups H ≤ Ω containing N there is a function such that: (c1) .λ Ω Ω (∆) = 1, and (c2) Any relation of the form According to Brauer 3 and 4 the extension , is then well defined and uniquely determined.
Now in order to check 1.4 (c2) we need generators for the kernel of the restricted Brauer map ϕ N ≤Ω as in (3).Specifying such a set of generators for the case that Ω is (pro)solvable is the main result of section 2. Actually this is a variation of [D], §1 where the case Ω finite and N = {1} has been dealt with.
Then in section 3 we proceed in three steps: Step 1: H/N (∆) where we will work in the finite quotient G = Ω/N.
Step 2: Step 3: Specifying three types of special relations (c2) which have still to be verified in order to ensure that ∆ extends from R 1 (N ≤ Ω) onto R(N ≤ Ω) if the extension for proper subquotients N ′ ≤ Ω ′ is already known to exist.More precisely these three types of relations occurring in the Main Theorem 3.1 are: (1) a Davenport-Hasse relation relative to subquotients B/K of Ω which are cyclic of prime order ℓ, (2) a Heisenberg identity relative to subquotients The idea to produce special relations is to consider the case N = C of a commutative normal subgroup, hence ϕ C≤Ω : R + (C ≤ Ω) ։ R(Ω), and to establish a projector The projector Φ C is obtained as a corollary to Lemma 2.2 which essentially goes back to [L],15.1 and [D],1.11resp.
Concerning steps 1 and 2 we will basically follow [L],chapt.16,and then our step 3 will cover chap.19.
Altogether this covers the purely algebraic part of [L].We will finish with a short introduction to the arithmetic part, where Ω = G F is the absolute Galois group of a p-adic field F (or the corresponding Weil group W F ⊂ G F ) and ∆(H, χ) = ∆(E, χ E ) is (as in (*) above) the local Hecke-Tate root number for E|F a finite extension and χ E : E × → C × a quasicharacter.Then the relations 3.( 1) -(3) turn into identities 4.( 1) -(3) for these local root numbers which are sufficient to ensure the existence of local ε-factors for higherdimensional representations.Verifying these arithmetic relations is the other main topic of [L] which is not touched here.In section 4 we do nothing else than translating the algebraic relations 3.( 1) -( 3) into the arithmetic relations of [L] and adding some first simplifications which occur in the arithmetic context.

The notion of extendible functions
The following approach is a generalization which includes the corresponding considerations in [T2] and in [L],chap.2as well.
Let (Ω, N) be a pair consisting of a profinite group Ω and an open normal subgroup N (equivalently: N is closed and of finite index).We denote by the set of all pairs (H, ρ) where: Since we consider equivalence classes, the group Ω/N acts on R(N ≤ Ω) by means of We will also consider the case N = {1} where we will write: R(≤ Above we have considered already Therefore by the very definition we have: where . More general we will write H = H/ [N,N ] for all H ≥ N. Also note that for N ≤ N ′ ≤ Ω ′ ≤ Ω where N ′ is a normal subgroup of Ω ′ , we have natural embeddings where in each case the first embedding is by inflation contravariant to and the second embedding considers R(N ≤ Ω ′ ) as the subset of pairs (H, ρ) such that H ≤ Ω ′ .

And the direct limit for
In [T2] and also in [K] only the absolute versions R(≤ Ω), R 1 (≤ Ω) and the limits with respect to Ω = lim ← −N Ω/N have been dealt with.But for the aims of this paper it is necessary two follow [L], who works with relative Weil-groups, more closely.From the group theoretical point of view, working with relative Weil groups means to consider the limit In the following we will focus to the relative case where a lower bound N ≤ Ω will be fixed.But obviously all considerations take over to the case N = {1}.
Suppose we have a function ∆ defined on R 1 (N ≤ Ω) taking values in some multiplicative abelian group A with properties (1) , where 1 H denotes the trivial representation of H.
(!) From now, when speaking of functions ∆ on R 1 (N ≤ Ω) we will always mean functions with the properties (1) and (2).
1.1 Definition: We will say that the function ( 1.2 Uniqueness: If ∆ is extendible, then the extension F satisfying (3) and (4) will be uniquely determined.
and therefore we may apply Brauer 4 which gives us: for groups H i ≥ N.And using (1) and the property (4) this implies: which computes F (G, ρ) in terms of the original ∆ on R 1 (N ≤ Ω).Uniqueness follows because for any other extension F ′ we could do the same computation.qed.
1.3 λ-Function: Let ∆ be a function on R 1 (N ≤ Ω) which extends to F on R(N ≤ Ω).
Then, in the situation (4) but for (H, ρ) ∈ R(N ≤ Ω) of arbitrary dimension, we will have: (5) where by definition: Indeed, we consider the virtual representation ρ − dim(ρ) ) and then due to (4) we have ) and the property (3), this rewrites as for subgroups H ≤ G between N and Ω, is Langlands' λ-function.(We need not to refer here to the extension F because it is unique.) We mention four properties of the λ-functions which are attached to extendible functions ∆ and which easily follow from the definitions.
(For subquotients G/H ′ of Ω/N the functions ∆ G/H ′ and F G/H ′ are defined via the inflation map; see the remark behind 1.4 and (F3) of section 2.) In particular this applies if , and the extensions will be uniquely determined.Moreover we have: if G/H is an abelian subquotient of Ω/N, because Ind G H (1) = χ∈(G/H) * χ and therefore (3) applies.
Actually the existence of a λ-function λ G H (∆) with appropriate behaviour can be turned into a criterion for the extendibility of ∆.Compared to [K], Lemma 3.2, we deal here with the refined problem of extending ∆ from R 1 (N ≤ Ω) to R(N ≤ Ω) which is motivated by [L],Theorem 2.1.
1.4 Proposition (Criterion): (i) Let (Ω, N) be a profinite group and an open normal subgroup, and ∆ a function on R 1 (N ≤ Ω) with values in the multiplicative abelian group A. Then ∆ is extendible to F on R(N ≤ Ω) = H; N ≤H≤Ω R(H/ [N,N ] ) if and only if for all subgroups H ≤ Ω containing N there is a function which is defined on the set U N (H) of subgroups of H which contain N, such that: Moreover the functions U ∈ U N (H) → λ H U (∆) are uniquely determined by these requirements and satisfy the tower relation: (ii): U/N (∆) ∈ A, such that (c1), (c2) are fulfilled in the particular case H = Ω, (and then using λ H U (∆) := λ Ω U (∆)λ Ω H (∆) −(H:U ) ).
If ∆ extends to F on R(N ≤ Ω) = H≥N R(H), then the restriction F 0 of F down to R(≤ Ω/N) = H≥N R(H/N) will be an extension of ∆ 0 and therefore , where the relations (c2) are requested only for characters χ of U/N.Proof of the Criterion 1.4 (i): If ∆ is extendible to F , then, as we see from (5), the definition λ H U (∆) := F (H, Ind H U (1)) will have the requested properties.Conversely suppose that for all H ≤ Ω containing N the functions λ H U (∆) together with the displayed properties (c1), (c2), do exist.Let now ρ ∈ R(H) be a virtual representation.Then by Brauer 3 there are subgroups U i ≥ N of H, characters χ i ∈ U * i and multiplicities n i ∈ Z such that: Then we define: ) is independent of the choice of the presentation (10) of ρ.And obviously the relation (3) will hold.It remains to show (5).Let G be a subgroup of Ω containing H and consider (H, ρ) ∈ R(N ≤ Ω) as before.Then (10) implies: Hence by definition Now dividing this by (11) we obtain , thus we see that in order to prove (5) it is sufficient to verify the property (7): To establish that property we remark: Remark 1.For a fixed H ≤ G the function U ∈ U N (H) → λ H U (∆) is by the requirements (c1), (c2) uniquely determined.Indeed, by Brauer 2 with respect to the group H/N, we have a relation: and using the property (1) of ∆ this comes down to: Thus the λ-function can be computed in terms of the original ∆, hence it will be uniquely determined if it exists.
Remark 2. For N ≤ H ≤ G ≤ Ω consider the function This will be a function satisfying (c1), (c2): because for any relation . But then we obtain: This finishes the proof of the first part of the Proposition.Finally we prove that the criterion 1.4 (ii), is already sufficient: If H ≤ Ω is any other subgroup containing N then we may define: The arguments from remark 2 above show that U ∈ U N (H) → λ H U (∆) ∈ A has the properties (c1), (c2) too, and remark 1 shows that it is uniquely determined by these properties.Thus we have the full set of functions λ H U (∆) at our disposal, and this is sufficient as we have seen already.qed.
Still we note the following compatibility which justifies the further approach: we consider the embeddings (e1), (e) from above, and we assume that Our intention is to go in the opposite direction: 1.6 Further approach to the extension problem: We are given a function ∆ on R 1 (≤ Ω) (that means we assume (1), ( 2)) and we want criteria for ∆ being extendible to a function F on R(≤ Ω).The idea is to consider the restrictions of ∆ to the subsets R 1 (N ′ ≤ Ω ′ ) for finite subquotients Ω ′ /N ′ of Ω, and then via criterion 1.4 to search for conditions to be put on ∆ such that an extension onto R(N ′ ≤ Ω ′ ) can be obtained.This will be done in an inductive way that means for any natural number n we assume that ∆ can always be extended if (Ω ′ : N ′ ) < n, and then we are going to prove that the criteria of Theorem 3.1 are sufficient to obtain the extensions of ∆ also for subquotients of index n.In particular we get then extensions for all open normal subgroups N ≤ Ω, and these extensions will fit together to the extended function F on R(≤ Ω).The induction begins from cases ) is the free Z-module over Ω ′ * .Therefore in this case the extension F of ∆ is already determined by property (3) and obviously does exist.Condition (4) plays no role because for N ′ = Ω ′ we have no induction such that we are left with (c1). .

The kernel of the Brauer map and its generating relations
To apply our criterion 1.4 we need a generating system of relations (c2).Here we follow essentially Deligne [D], §1, but with some modifications because we have to deal with pairs (Ω, N).
As already explained we consider the Brauer maps (2), (3) of the introduction: where ϕ N ≤Ω is the restriction of ϕ Ω .(Therefore the extra notation ϕ N ≤Ω will be sometimes omitted.)In view of 1.4, (c2) what we need are generators of Ker(ϕ for all open normal subgroups N ≤ Ω.Then, in order to verify the Criterion, it will be enough to check these generating relations. A) First of all we recall the multiplication in the Brauer maps ϕ Ω and ϕ N ≤Ω into homomorphisms of commutative rings: (1) where the brackets in the middle indicate the pairs which will occur and which have still put together into equivalence classes, such that ], because [., .]denotes Ω-equivalence classes.Therefore writing also the right side of (1) as a sum of equivalence classes it is then only over a subset of pairs (g 1 , g 2 ) representing the different equivalence classes, which means that g because (see [S],7.3): and a symmetric calculation for turns into a subring, and ϕ Ω restricts to a ring homomorphism ϕ N ≤Ω .
Thus Ker(ϕ N ≤Ω ) is actually an ideal in R + (N ≤ Ω), and our aim in this section is to determine a set of generators for Ker(ϕ N ≤Ω ) considered as additive group.The main result will be Theorem 2.7.
It is easy to adapt this to the restricted maps ϕ N ≤Ω which we leave to the reader.
which is induced by the inclusion R 1 (≤ Ω ′ ) ⊆ R 1 (≤ Ω) and takes Ω ′ -classes into Ω-classes, is called induction.This is justified by the commutative diagram: → Ω be a continuous homomorphism with finite cokernel.Then we define the corresponding restriction u * : R + (≤ Ω) → R + (≤ Ω ′ ) by: where an Ω-orbit is mapped to a sum of Ω ′ -orbits, and we will write ), as we see from [S], 7.3: ) is only one element, and we obtain: 2.1 Proposition: where For the proof we may assume that ρ = [H 1 , χ 1 ] is a generator of R + (≤ Ω) and then use (F2) where [H, χ] is replaced by [H 1 , χ 1 ] and u * = Res Ω H .This brings us down to the twist operation (2) on the level of H.
If Ω ′ is a subquotient of Ω which means if Ω ′ u և Ω ′′ ≤ Ω, then our functoriality properties fit together to a commutative diagram Moreover, since Ker(ϕ) is always an ideal and taking into account the character twist operation (2), we see from the diagram that C) Our next aim is to produce elements ρ ∈ Ker(ϕ Ω ) in a systematic way.
Here we restrict to the case where Then we are going to define a projector , and together with ϕ Ω also the restriction ϕ C≤Ω is surjective.
• We begin from [H, χ] ∈ R + (≤ Ω) and produce a certain decomposition of Ind Ω H (χ) which depends on C: H acts by conjugation on C which induces an action µ ∈ C * → µ h = h −1 µh, that means: is stable under that action, and for µ ∈ S the function χµ: But we want χµ to be a character, in particular it should be trivial on commutators.Since But this is also sufficient that means the restriction χµ ∈ (H µ C) * is actually a character.And conjugation by g ∈ Ω turns it into the character (χµ In particular: Finally let T (χ) ⊂ S = S(χ) be a system of representatives for the orbits 2.2 Lemma: (for (i) see: [D] 1.11., [L] 15.1 has the version for χ ≡ 1.) Depending on a commutative normal subgroup C ≤ Ω and with notations as we have fixed them, we will have the relation where on the right side the induction is via groups In the particular case where H ≥ C the relation is simply the identity because T (χ) = {χ| C } is then a single element and HC = H.And for χ ≡ 1 we obtain: Ind where H ′ = Stab H (µ) is for all µ ∈ S = S(χ) the projective kernel of Ind HC H (χ), µ 0 ∈ S is any fixed constituent, and ν ′ ∈ (HC/H) * is the extension of ν ∈ (C/H ∩ C) * for a system of representatives ν with respect to the equivalence relation: Proof: (i) Since H µ C ⊆ HC, we may rewrite our assertion as Ind HC HµC (χµ)   , so it is sufficient to prove the Lemma in the case Ω = HC.The Ω-module Ind HC H (χ) is spanned by complex functions f : HC → C, such that Because of Ω = HC and S = S(χ) we have #S = (C : H ∩ C) = (HC : H) = (Ω : H), and the elements µ ∈ S give rise to a distinguished basis {f µ := χµ} µ∈S for that space of functions: f µ (hc) := χ(h)µ(c) is well defined as we have seen above already.
The function f µ : HC → C extends the character µ and each of the functions f µ has the property Thus we obtain: We see that the action of HC on the basis {f µ } µ∈S is monomial that means up to scalar factors the basis is permuted: where gµg −1 = hµh −1 , because the action of HC on S ⊆ C * factors to an action of is a decomposition into HC-submodules.Now the assertion (i) will follow from HµC (χµ) by verifying the character identity and therefore the inner action of H on C/H ∩ C will be trivial and such that µ h−1 ∈ C * is for all µ ∈ S the same character.Therefore independently of µ we will have Sublemma: Assume H ≤ HC normal and let

and this implies:
Now returning to the proof of (ii) we use that S = µ 0 • (C/H ∩ C) * , for any fixed constituent µ 0 ∈ S, and equivalence µ 0 ν 1 ∼ µ 0 ν 2 with respect to the inner action of H, means: is in the image of the Sublemma-map.This turns relation (i) into relation (ii).qed.

Corollary: (The projector Φ C ):
(i) Via Lemma 2.2 we obtain (for a fixed abelian normal subgroup C ≤ Ω) a well defined additive homomorphism which is actually a projector: Φ 2 C = Φ C , and such that with respect to the Brauer map: χµ is the sum of all extensions of χ, and: Ind , where Ind Ω Ω ′ means to replace Ω ′ -conjugacy classes by the corresponding Ω-conjugacy classes.
Proof: (i): From the proof of the previous Lemma it is easy to see that [H, χ] → [HC, χµ] is a well defined map of Ω-conjugacy classes which does not depend on the choice of a representative µ ∈ T (χ).Thus Φ C ([H, χ]) is well defined and extends to arguments in R + (≤ Ω) which by definition is the free Z-module over the conjugacy classes [H, χ].And in Lemma 2.2 we have mentioned already that Φ Therefore the last displayed term rewrites as: where , and we may proceed: (ii) is obvious.
(iii): If C ′ ≥ C are both central in Ω then this follows from (ii) because extending χ ∈ H * to HC ′ is the same as first extending it from H to HC and then from HC to (HC)C ′ = HC ′ .The general case is less obvious.By definition we have: where Since C ′ is commutative too, we have the surjective restriction map: which is compatible with H-conjugation.Thus we are left to show that: As to the left side, by definition we have: where and (see before 2.2 above): χµ = χ on H µ ∩ C ′ ≤ H, and χµ = µ on C, we conclude: and together with H ∩ C ′ ≤ H µ ′ we obtain: such that all inclusions are equalities.Therefore for µ ∈ S C (χ) we will have: This proves (iii), and the assertion (iv) is easy to see.qed.
As an immediate consequence for arbitrary normal subgroups we note: any open normal subgroup, then we denote ) * , and this implies where χ ∈ Ω * runs over the extensions of χ.

D)
A list of basic relations.
2.5 Basic relations: Now we give a list of basic relations ρ ∈ Ker(ϕ , the members of which will actually be generators.Some additional comments will follow below.We begin by presenting those generators which refer directly to the group Ω: II For Z < Ω a normal subgroup and η ∈ Z * with the properties -the commutator induces an isomorphism [., .]: III For H < Ω a (proper) maximal subgroup which is not normal, that means K := g∈Ω gHg −1 = H and therefore Ω/K is a non-degenerate type-III-group (as defined in Appendix 2), let C < Ω be the uniquely determined normal subgroup such that: and for characters χ ∈ Ω * put: referring to the inner action of H on (C/K) * and µ ′ the trivial extension of µ ∈ (C/K) * onto the stabilizer H µ C, which implies Similarly, for all open subgroups B ≤ Ω we may consider elements referring now to subgroups K, Z, H of B, and the complete set of generators consists of : where Ind Ω B : R + (≤ B) → R + (≤ Ω) is the enlargement of conjugacy classes which according to (F1) takes Ker(ϕ B ) into Ker(ϕ Ω ).
2.5/N Basic relations ρ ∈ Ker(ϕ N ≤Ω ): If moreover N ≤ Ω is a fixed open normal subgroup then we consider the subsets I/N, II/N, III/N consisting of those generators which are in R + (N ≤ Ω), hence they are in Ker(ϕ Ω ) ∩ R + (N ≤ Ω) = Ker(ϕ N ≤Ω ).This means that all occurring subgroups K, Z, H, B, and for III/N also K = g∈B gHg −1 must contain N.

Definition:
We let R(≤ Ω) ⊆ Ker(ϕ Ω ) and R(N ≤ Ω) ⊆ Ker(ϕ N ≤Ω ) be the additive subgroups which are generated by the above elements of types I -III and I/N -III/N resp.This means that: Remarks: 2.6.1 If Ω is abelian then the types II and III will not occur because they refer to non-abelian groups.If Ω is nilpotent then types III will not occur because in a nilpotent group B maximal subgroups H < B will always be normal subgroups.Moreover in accordance with (3) above, we have: 2.6.2Twisting basic relations with a character X ∈ Ω * will stabilize this set and therefore the additive groups R(≤ Ω) and R(N ≤ Ω) will be modules over the character ring ).And type preserving we have: 2.6.3 (Induction) E) The main result of this section, and reducing it to the case of a central extension:

Theorem:
If Ω is a solvable profinite group then the inclusions R(≤ Ω) ⊆ Ker(ϕ Ω ) and R(N ≤ Ω) ⊆ Ker(ϕ N ≤Ω ) are actually equalities. Proof: the equality R(≤ Ω) = Ker(ϕ Ω ) will follow if the N-equalities hold for all open normal subgroups N ≤ Ω.We begin by reducing this to cases where N is in the center of Ω.
And as partial sums we have: Let and the Theorem equivalently means that the inclusions R(N ≤ Ω) ξ ⊆ Ker(ϕ ξ ) are equalities for all ξ ∈ N * /Ω.
Proof: We consider the group extension N ֒→ Ω ։ Ω/N with abelian kernel and let ρ ∈ R(Ω) be an irreducible representation that means a generator.Then it is well known that ρ determines an orbit ξ(ρ) ∈ N * /Ω and, for a fixed µ ∈ ξ(ρ), an irreducible representation ρ µ ∈ R(Ω µ ) such that Ind Ω Ωµ (ρ µ ) = ρ.Therefore the right hand vertical is a bijection.As to the left hand vertical consider , where we may assume for all i that Ind The direct sum on the right contains the term Ind Ωµ H i (χ i ) for s ∈ H i Ω µ and additional components for H i sΩ µ = H i Ω µ .And restricting further to the normal subgroup N ≤ H s i ∩ Ω µ we see that those other components are disjoint from Ind Ωµ H i (χ i ) because s / ∈ Ω µ .Therefore from (*) we may conclude: which gives us a uniquely determined preimage in Ker(ϕ µ ).
Then Ω-conjugation shrinks to Ω µ -conjugation.qed.As a reformulation for a commutative N = C we note: 2.10 Reformulation: Let C ≤ Ω be an abelian normal subgroup and assume ρ = Then for any fixed µ ∈ C * we will have: and fixing one representative µ in each conjugacy class ξ ∈ C * /Ω we recover ρ = µ Ind Ω Ωµ (ρ µ ).Proof: (i) In terms of Lemma 2.9 we will have Ker(ϕ µ ) = R(N ≤ Ω µ ) µ if we assume equality for cases ξ = µ, and therefore 2.9 and 2.6.3 yield According to (i) we may in (6) and (7) restrict to the partial sums for N/ F) Proving 2.7 in the case of a central extension.
2.12 Proposition: If (Ω, Z) is a pair consisting of a solvable profinite group Ω and an open subgroup Z ≤ Ω which is central in Ω, then R(Z ≤ Ω) ⊆ Ker(ϕ Z≤Ω ) is always an equality.
We begin from the case Z = {1} where Ω is finite and prove in three steps: 2.12.(i) (Revised version of [D],1.13,1.14 Step 1: The case where Ω is commutative.
If Ω is commutative then the projector Φ for all generators [H, χ] ∈ R + (≤ Ω).We do this by induction over (Ω : H).If (Ω : H) = ℓ is a prime then this is true by definition 2.6.Otherwise consider H < H ′ < Ω where (H ′ : H) = ℓ is again a prime.Then we have where the first term is in R(I, ≤ Ω), and Step 2: ([L], chapt.18)The case where Ω is a finite nilpotent group.
In this case Ω has a nontrivial center Z = {1}, and we argue by induction over (Ω : Z), where (Ω : Z) = 1 has been done in step 1.Thus let Z be a proper subgroup.
Then we consider ρ ∈ Ker(ϕ Ω ), and write ρ = Φ Z (ρ) + (ρ − Φ Z (ρ)).As to the second summand it is enough to consider Thus we are left to show that ρ ∈ Ker(ϕ Ω ) implies ρ ′ := Φ Z (ρ) ∈ R(≤ Ω).To this end we choose Ω > C > Z such that C is a commutative normal subgroup of Ω, and (C : Z) = ℓ is a prime, and C/Z is in the center of Ω/Z.Such a C exists because Ω is nilpotent.(Consider Z 2 /Z = center of Ω/Z, and the commutator map Z 2 /Z ∧ Z 2 /Z → Z. Then C/Z of prime order inside Z 2 /Z will do it).Then we consider Here we use 2.10 for Φ Therefore for all µ we have Φ C (ρ) µ ∈ Ker(ϕ Ω µ ) = R(≤ Ω µ ) by induction assumption.By inflation we are then in R(≤ Ω µ ) and induction brings us to Φ C (ρ ′ ) = Φ C (ρ) ∈ R(≤ Ω).Finally we need to handle the difference ρ ′ − Φ C (ρ ′ ) ∈ R + (Z ≤ Ω), and here it is enough to see: 2.13 Lemma: For generators [H, χ] ∈ R + (Z ≤ Ω) we always have: If HC = Ω then (HC : Z) < (Ω : Z), hence Ker(ϕ HC ) = R(≤ HC) by induction assumption.Now we assume HC = Ω.Then H is a normal subgroup of Ω, of index (Ω : H) | (C : Z) = ℓ.Since the case Ω = H ≥ C is trivial, we are left with the case where H < Ω is normal of index ℓ and H ∩ C = Z which means C/Z ∼ → Ω/H.In particular H is normalized by C such that due to 2.2(ii) and 2.3 we have where H ′ = H µ for all µ ∈ S(χ), is the projective kernel of Ind HC H (χ) = Ind Ω H (χ).And ν ′ ∈ (Ω/H) * is the natural extension of ν ∈ (C/Z) * , where the ν are running over a system of representatives for the cokernel of (11) Since (C/Z) * is of order ℓ, a prime, we have only two cases to deal with: Case 1: The map (11) is surjective, that means (H : H ′ ) = ℓ, and the cokernel of ( 11) is trivial, hence can be represented by the trivial character ν ≡ 1.Then we obtain: where (Ω : Thus we see that H and H ′ C are both normal of index ℓ in Ω and H ∩ H ′ C = H ′ (H ∩ C) = H ′ .This gives us: , Ω] is in the kernel of our representation (because H ′ is the projective kernel), • χ| because otherwise χ could be extended to a character χ ∈ Ω * and Ind Ω H (χ) = χ ⊗ Ind Ω H (1), hence the projective kernel would be H ′ = H.
Altogether we see that in case 1 we obtain: Case 2: H = H ′ that means the image of (11) is trivial, hence (C/Z) * is the cokernel and C/Z This ends the proof of 2.13 and of Step 2.
We prove this by induction over #Ω.Thus for proper subgroups H < Ω we will have R(≤ H) = Ker(ϕ H ) which is an ideal in R + (≤ H), and this will imply that: Moreover, applying (F2) we obtain ϕ Now let Z be the center of Ω.By Step 2 we can assume that Ω is not nilpotent hence it might happen that Z = {1}.
If Z = {1}, we use Brauer 1 for Ω/Z: There are nilpotent subgroups H i ⊇ Z and characters χ i of H i /Z such that • Together with Ω also Ω/Z is not nilpotent, but the H i /Z are nilpotent, hence they are proper subgroups of Ω/Z which ensures that we obtain here a non-trivial relation.
Since we do induction over the order of Ω, we may assume Ker(ϕ Ω/Z ) = R(≤ Ω/Z), and by inflation we obtain ) is an ideal, in particular ρσ ∈ R(≤ Ω) for all ρ ∈ Ker(ϕ Ω ).And by Proposition 2.1 we have: as we have seen already, and therefore which proves our assertion in the case Z = {1}.
If Z = {1} then let C be a minimal commutative normal subgroup of Ω which exists because Ω is solvable.And for ρ ∈ Ker(ϕ Ω ) write again: Thus by 2.10 we obtain where µ ∈ C * runs over a system of representatives for the conjugacy classes C * /Ω.
Thus we are left with the following variant of 2.13: 2.14 Lemma: For generators [H, χ] ∈ R + (≤ Ω) we always have: Since we argue by induction on #Ω, we may as in 2.13 restrict to the particular case where HC = Ω.Thus assume HC = Ω, hence H ∩ C is normal in Ω and therefore H ∩ C = {1} or = C, because C was minimal.The case C ≤ H is trivial such that we have to consider only the case where Suppose there is a nontrivial subgroup because Ker(ϕ Ω/H 1 ) = R(≤ Ω/H 1 ) by induction assumption.Thus we are left with the case where H < Ω is maximal and H does not contain normal subgroups of Ω = H ⋉ C.But then Ω > H is a type-III-group (as we see from Lemma 3 of Appendix 2) and by definition 2.6 we have then (12) in R(III, ≤ Ω) This ends the proof of 2.12.(i).
To complete the proof of 2.12, we are left with the assertion: 2.12.(ii):R(Z ≤ Ω) = Ker(ϕ Z≤Ω ) in all cases where Z is a non-trivial open subgroup which is contained in the center of the solvable profinite group Ω.
Proof: We prove this by induction over (Ω : Z).For Z = Ω the group is abelian and we obtain an isomorphism On the other hand: where χ ′ ∈ (HC) * runs over the extensions of χ and where χ is any of these extensions.
In case b), as in step 2 above, we take Z < C ≤ Z 2 such that (C : Z) = ℓ, and then we can argue in the same way: Indeed due to 2.10 we have Φ In case c) the quotient Ω/Z has trivial center, hence it is not nilpotent.Therefore we can argue precisely the same way as in the case Z = {1} of step 3 above: 2.15 Lemma: will follow.
The assumptions hold, because we do induction on (Ω : Z) and because Ω/Z is finite resp.This finishes the proof of 2.12.(ii) and of Theorem 2.7 .

A criterion for the extendibility of functions
Now we come to the goal of this paper.We want to prove the following criterion 3.1 for the extendibility of functions, in the sense of Definition 1.1.The criterion makes it more precise how to use Theorem 2.7 in order to see that a function ∆ on R 1 (≤ Ω) with values in a multiplicative abelian group, extends to a function F on R(≤ Ω).As already explained in 1.6 we are going to see by induction, that for all subquotients II. Heisenberg identity: Let Z < B be normal with abelian quotient B/Z ∼ = Z/ℓ × Z/ℓ, where ℓ is a prime and such that the commutator induces an isomorphism is required, (where µ ′ denotes the trivial extension of µ onto the stabilizer H µ C).
Remarks: For the degenerate case H = K, C = B the type-III-relation turns into type-I.
Proof of Theorem 3.1 (Later we will take for ∆ the local Hecke-Tate root number and then we will translate the requirements I.-III.from groups to fields, relations which have to be confirmed in order to verify the existence of local Artin root numbers.) In the situation I we have the relation and together with In the situation II let (H, η H ) and (H ′ , η H ′ ) be two isotropic pairs related to (Z, η).Then using Lemma 2.1 for the commutative normal subgroup , which rewrites as (2): Finally in the situation III we have HµC (χµ ′ ), which in particular will hold for the trivial character χ ≡ 1. Therefore: , which recovers the condition (3).
Now, conversely, we assume that ∆ is a function on R 1 (≤ Ω) which satisfies the relations (1) -( 3) and we want to show that ∆ is extendible to a function F on R(≤ Ω).
The basic idea (implicit in [L]) is to proceed by induction as follows: Let N ′ ≤ Ω ′ be any finite subquotient of Ω and let ∆ ′ on R 1 (N ′ ≤ Ω ′ ) be the restriction of ∆.Then, using Criterion 1.4, we will check by induction on (Ω ′ : N ′ ) that ∆ ′ extends to ).The way of induction has been sketched already in 1.6.And we remark once more that without loss of generality we may restrict to quotients N ≤ Ω because the same arguments will work if we replace Ω by a subgroup Ω ′ and N by a normal subgroup N ′ ≤ Ω ′ .Then we can go to (Ω ′ : N ′ ) → ∞, in particular (Ω : N) → ∞, and this will prove the Theorem.Thus we restrict to prove that the relations (1) -( 3) imply extendibility of ), if we assume this for all proper subquotients Ω ′ /N ′ of Ω/N.As already explained in the introduction we will argue in three steps (where concerning steps 1 and 2 we follow [L], chapt.16,and then step 3 will cover [L], chap.19): Step 1: where H = U/N, and T is a set of representatives for (C/H∩C) * /H, and we use this for the Definition 3.2: Let U ∈ U N (Ω) be subgroups containing N and let ∆ 0 on R 1 (≤ G) be the restriction of ∆.Then, using the data of (4), define: HµC/C (∆ 0 ) has been defined by induction.
• A priori U ∈ U N (Ω) → λ Ω U (∆) depends on C ≤ G, but we are going to see that it can be used to extend the function ∆ N = ∆| R 1 (N ≤Ω) , and therefore it is actually independent.
• In the particular case where H ≥ C, Lemma 2.2 comes down to the identity , where T consists only of the trivial character of C, and therefore our definition comes down to • If G = Ω/N is cyclic of prime order ℓ then the only possibility is C = G and therefore: where ∆ N ′ is the restriction of ∆ to R 1 (N ′ ≤ Ω) and the right sides are given by induction assumption.
And λ HµC/C (∆ 0 ), because we may apply 1.(6) for the proper quotient G/C.Therefore the definition 3.2 is left unchanged if we replace the representatives µ ∈ T by other representatives µ h .(ii) is the same argument as in (i).We have to compare (4) with the corresponding formula for U g = g −1 Ug ։ H g = g −1 Hg.Then we obtain T g = (C/H g ∩ C) * /H g with representatives µ g , and (H g ) , where the second equality follows because N ′ G /C ≤ G/C is a normal subgroup, and for the quotient G/C we assume property 1.( 8 G and C commute and therefore the inner action of N ′ G on C * is trivial.Thus we have N ′ G ≤ H µ for all µ ∈ (C/H ∩ C) * , and the relation (4) rewrites as: (5) And by assumption ∆ Step 2: We are going to prove that H ∈ U(G) → λ G H (∆ 0 ) satisfies 1.(7) too.Lemma 3.4: (N-Tower lemma) Let N ≤ Ω be a normal subgroup, ∆ on R 1 (N ≤ Ω), and assume that Then: Thus we are left with the assertion . At first we reduce this to the case where H is a complement of the fixed abelian normal subgroup C < G. Consider (7) By induction we have because here we can argue modulo C.And by definition we have Now substituting (8) on the right side of (9) and then using (7) we conclude: by induction.This we may substitute on the right side of (10), and in the resulting equality we may use (10) in the particular case where H ′ = H.Then we obtain our assertion (6) in this case.Thus we are left with the case G = HC.Then H ∩ C is a normal subgroup of G, because it is normalized by H and is a subgroup of the abelian group C. Since C has been chosen minimal, abelian, normal in G the only possibilities are: semidirect.This is the case we are going to proceed with.But before going further we need the following: which is given by Lemma 2.2 (in the particular case H ′ ∩ C = {1}) and where is the extension of µ by 1. Then this will imply:

Note that the case
which is an immediate consequence of definition 3.2 for H = {1}. Proof: ).This we may apply to

Now we may apply (10) with H
where the second equality follows from the first replacing H ′ by H ′ µ C.And the identity (G ′ : Thus multiplying (13) by this last equality and then using the equalities (14) will give us (12) in the case where and in (11) we have . Therefore ( 11) is now the same as (4) tensored by the character χ of G. Since H ∩ C = {1} we have Lemma 2.2(i) with S(χ) = C * , T (χ) = C * /H.Moreover, using Lemma 3 of Appendix 2 we have: Now we can finish the proof of Lemma 3.5.(in the case H ′ = H) by using our assumption (3).Indeed ∆ on R 1 (≤ Ω) induces ∆ 0 on R 1 (≤ Ω/N), and our type-IIIgroup G = H ⋉ C is actually a quotient of G = Ω/N.Therefore the relation (3) of ∆ for type-III-groups which are subquotients of Ω will induce a similar relation of ∆ 0 for type-III-groups which are quotients of G. Thus referring to (3) we may replace ∆, H < B, K, and C, such that HC = B, H ∩ C = K, and χ ∈ B * by: ∆ 0 , H < G, K, and C, such that H C = G, H ∩ C = K, and χ ∈ G * , which yields G/K = (H/K) ⋉ C a type-III-group, where we may identify C/K = C, and turns our assumption (3) into: Here we have used that the inner action of Finally, by our definition of λ G H (∆ 0 ) the equality (15) (which holds by assumption!) turns into (12) for H ′ = H, if we multiply it by µ∈C * /H λ G HµC (∆ 0 ).qed.
We come back to the proof of Lemma 3.4.where we were left to show that: satisfies the conditions (c1) and (c2).In other words we show that the function ∆ Thus we turn to the condition (c2) of Proposition 1.4, where we consider ∆ 0 on R 1 (≤ H).Assume a relation (16) as defined above.First of all from (16) we see: i n i (H : Thus in order to verify (17) it is enough to see that (16) will imply (18 For each pair (H ′ i , χ i ) we may consider the subgroup i /C and to this situation apply Lemma 3.5.Then we obtain relations of type (11) and (12) resp.which we will write as: where {µ ij } j ⊂ C * are representatives for the cosets C * /H ′ i and where and χ i is understood to be trivial on C, and µ ′ ij extends µ ij hence is non-trivial on C.And (12) applied to H ′ = H ′ i reads now as follows: (20) Implementing this into (18) the assertion turns into (21) On the other hand, applying Ind G H to (16) and using (19) we obtain: where all ).Now we may use our assumption that ∆ is extendible from R 1 ( C ≤ Ω) onto R( C ≤ Ω) because (Ω : C) < (Ω : N).Therefore, interpreting (22) as a representation of Ω/[ C, C] we obtain: Here we consider ]. Therefore we may replace ∆ by ∆ 0 , where λ Ω Step 3: (Using the generators of Ker(ϕ N ≤Ω ) and the N-tower Lemma 3.4.)So far we have seen that the function ∆ on R 1 (N ≤ Ω) gives rise to ∆ 0 on R 1 (≤ Ω/N) and a function where G := Ω/N, which satisfies the tower lemma.Finally in step 3 we want to verify that the definition (So far we have only used the induction assumption that ∆ extends from R 1 ( C ≤ Ω) onto R( C ≤ Ω).Now the assumptions (1) -(3) will come in.)For the extension onto R(N ≤ Ω) we will use the criterion 1.4 that means we have to verify (c2) for H = Ω, which comes down to check the generators I/N, II/N, III/N of the kernel of the Brauer map Type I/N: The basic relation ρ (K, B, χ) refers to for χ ∈ B * , where B/K is cyclic of prime order ℓ.Here our assumption (1) together with: µ∈(B/K) * ∆(B, µ) = λ B K (∆), (because B/K is cyclic of prime order ℓ and a subquotient of G = Ω/N, thus by induction we have 1.( 9) at our disposal), rewrites as and multiplying this by λ Ω B (∆) ℓ and using the N-tower relation 3.4 we obtain which verifies the condition 1.4 (c2) for the generating relations of type I/N.
Type II/N: Here we have to verify that Now multiplying this by λ Ω B (∆) ℓ and using the N-tower lemma 3.4 we obtain (23).
4. Recovering Theorem 3.1 for the case of local root numbers.
So far, beginning from a profinite solvable group Ω we have dealt with the question as to when a function ∆ = ∆(H, χ) on R 1 (≤ Ω) can be extended to a function where H runs over the open subgroups of Ω and χ ∈ H * is replaced by any virtual representation ρ ∈ R(H).Now we come to the arithmetic application: As our base field we fix a non-archimedean local field F with finite residue field of characteristic p and consider Ω = G F the absolute Galois group over F or Ω = W F ⊂ G F the absolute Weil group.Thus fixing a separable closure F |F we obtain • R 1 (≤ G F ) the set of all pairs (E, χ) where E|F is a finite subextension that means G E ≤ G F is an open subgroup and χ ∈ G * E is via class field theory a continuous character of E × which is of finite order.
• R 1 (≤ W F ) is the set of all pairs (E, χ) which now means that W E ≤ W F is an open subgroup of finite index and χ ∈ W * E is via class field theory a quasicharacter of Let ψ F be a fixed non-trivial additive character of F , and take as additive characters for the extension fields E|F.Then in both cases Ω = G F or Ω = W F our function on R 1 (≤ Ω) to begin with is: which is Tate's local root number assigned to the Gauss sum for the pair (χ, ψ E|F ).More precisely q E = #κ E is the order of the residue field, a hence for computing ∆(E, χ) we need ν E (c) = a E (χ) − ℓ(ψ E|F ), and the value ∆(E, χ) will not change if we modify c by a unit factor.In terms of [L], p.5 the definition (0) rewrites as where the second equality follows if we consider as a disjoint union of cosets and then use that χ , where we have used P a = {0}, 1+P a = {1} and ℓ((1−y)ψ) > 1 if y / ∈ 1+P a−1 .Therefore (0) and [L] agree.On the other hand [T3],(3.2.6) implies that: From the explicit definition (0) it is obvious that: • ∆(E, 1) = 1 and ∆(E, χ) = ∆(E g , χ g ) for all g ∈ G F , where E g = g −1 (E) and χ g = χ • g, hence the basic properties 1.(1), 1.
(2) to be expected of a function ∆ are fulfilled.
Since the absolute Galois group G F is profinite solvable, our Theorem 3.1 directly applies to the question if On the other hand W F is not profinite but comes as the extension of Z with the profinite group W 0 F = G 0 F = inertia subgroup.Thus we have a continuous embedding W F ֒→ G F with dense image and identical commutator subgroups According to [D] §4.10 every irreducible smooth representation of W F comes up to unramified twist as the restriction of an irreducible representation of G F .Therefore the tensor product of irreducibles will be semisimple such that R(W F ) becomes a commutative ring, and the restriction of representations induces an injection of rings R(G F ) ֒→ R(W F ). Also the induction is an open subgroup of finite index.(For these and further details one may also consult 28. of the textbook [BH]).Because of |N E|F | s F = |.|s E the inductions Ind E|F will commute with unramified twisting such that the modifications Brauer 3 and Brauer 4 apply to Ω = W F as well and refer now to the relative Weil groups for subfields E i ⊆ K.
The projector Φ C of Corollary 2.3 turns now into a projector and the relative criterion 1.4 deals with extending a function ∆ from R ).Our modest aim in this last section is to translate the sufficient conditions 3.(1) -3.(3) into conditions for the local root numbers ∆(E, χ) = ∆(χ, ψ E|F ).Thus we have to translate pairs (H, χ) for H ≤ Ω and χ ∈ H * into pairs (E, χ) for finite extensions E|F and continuous quasicharacters χ : E × → C × .First of all, in all of 3.( 1) -(3) we have ingredients B ≤ Ω which means we go from our base field F to some finite extension F ′ |F such that B = W F ′ ≤ Ω = W F .Then up to this change of base field the conditions look all the same.Expecting now that the verification of 3.(1) -(3) will not depend on taking F ′ or F as our base field, we will restrict here to expressing the conditions relative to the base field F. In other words we restrict to cases B = Ω, F ′ = F, and χ ∈ B * = Ω * is via class field theory nothing else than a quasicharacter of F × .Then: in order to check the extendibility of ∆(E, χ) to local root numbers ε ∆ (E, ρ) for all ρ ∈ R(W E ) we have to verify the following arithmetic conditions: (1) Davenport Hasse = [L], First Main Lemma: Let K|F be a cyclic extension of prime degree ℓ and let S(K|F ) be the characters of for all quasicharacters χ of F × .
(2) Heisenberg identity = [L], Second Main Lemma: Let K|F be an abelian extension of type G K|F ∼ = Z/ℓ×Z/ℓ for some prime number ℓ, and let η : Consider subextensions E|F of degree ℓ inside K|F and quasicharacters η E of E × such that η E • N K|E = η.Then: the expression ∆(E, η E )• µ∈S(E|F ) ∆(F, µ) does not depend on the choice of (E, η E ).
(3) Generalized Davenport Hasse = [L], Third and Fourth Main Lemma: Let E|F be an extension which is minimal (in the sense that there are no proper intermediate fields), and let K|F be the Galois closure of E|F.We assume here K = E because otherwise we are in the situation of (1).Then G K|F is a type-III-group and we have a unique normal subextension L|F such that C = G K|L is abelian, and moreover: for all quasicharacters χ of F × and characters µ of L × /N K|L (K × ) ∼ = G K|L where we restrict to a system of representatives with respect to the action of the maximal subgroup Remark: From the displayed formula we may separate the trivial character µ 0 ≡ 1 such that F µ 0 = F and µ ′ 0 is the trivial character of F × .Thus for µ = µ 0 we obtain the factors ∆(F, 1) = 1 on the left side, and ∆(F, χ) on the right.
A detailed translation of 3.(1) -3.(3) (for the special case B = Ω) into the arithmetic conditions (1) -(3) uses only standard facts from local class field theory.The underlying relations in R(W F ) are the following:

Unramified twisting
Here we want to see that the arithmetic conditions (1)-(3) are stable under unramified twisting, that means if (1) or (3) are valid for a certain χ ∈ F × * , then also for the unramified twists χ|.| s F and similarly if (2) holds for a certain η ∈ (K × /I F K × ) * then also for the unramified twists η|.| s K .Thus from each equivalence class of quasicharacters modulo unramified twisting we need to check only one representative and for this we may choose a character of finite order.The argument is similar as in [T2], end of §2.Because of the explicit formula (0) and using that χ and χ|.| s E have the same restriction to U E we obtain: , for all quasicharacters χ of E × .Now the argument is based on the following Lemma: For finite separable extensions E|F and semisimple virtual representations ρ E ∈ R(W E ), the homomorphic function As a generalization of a E (χ) from above we have used here the exponential Artin conductor a E (ρ E ).(In terms of [S],chap.VI, §2 it is the intertwining number f (ρ E ) with the Artin representation).
Proof: Obviously it is enough to deal with the case E = F. Then using the well known formulas (cf.[S] p.101 below, and chap.III Prop.7 resp.): where d K|F denotes the differental exponent, we obtain: qed.Now for the invariance of (1) and (3) resp.under unramified twisting we check the quotient relations which via (4) rewrite as resp.But due to the Lemma these equalities are direct consequences of the underlying relations (R1), (R3) in R(W F ).
Similarly concerning (2) we have to check that N(E, η E ) s does not depend on the choice of (E, η E ).Again this is clear from the Lemma because Ind E|F (η E ) does not depend on that choice.
Once the extendibility of ∆ has been verified the Lemma will imply that (4) extends to = N(E, ρ) s for all ρ ∈ R(W E ).
If now µ is of odd order then it can be recovered as a power of µ 2 , hence ∆(E, µ −1 ) = ∆(E, µ) −1 in that case.Therefore if ℓ = 2, or if (in case (3)) the Galois closure of E|F is of degree [K : F ] odd, then we are left with the arithmetic conditions: if we take the system of representatives µ for S(K|L)/H such that it is stable under µ → µ −1 .Actually we are free to choose such a system as we like, because our function ∆ has the invariance-property 1.(2).Note here that all µ are of odd order and the acting group H is of odd order too.Therefore we will have µ = µ −1 and moreover µ, µ −1 cannot be H-equivalent, if µ ≡ 1.
Langlands [L] divides the condition (3) into two subcases: the Third and the Fourth Main Lemma of tame and of wild ramification resp.which are dealt with in chapters 13 and 14 of loc.cit.As an introduction we consider Type-III-groups as local Galois groups For the group theoretical background we rely on Appendix 2. And we keep the notations K ⊇ L, E ⊇ F from (3).Thus let be a type-III-group realized as local Galois group, where we exclude now the degenerate case H = {1}, which means G = C is cyclic of prime order ℓ.Equivalently E|F is a minimal extension which is not Galois and K|F is its Galois closure.Due to Lemma 2 of A2 this actually means that E|F is anti-Galois in the sense that the roots of a minimal polynomial generate pairwise different extensions E i |F precisely one of them: E|F .Then E|F cannot be unramified and it cannot have mixed ramification because this would imply existence of an intermediate field.Thus the only cases are: • E|F is totally tamely ramified of prime degree [E : F ] = ℓ = p.Then L|F is the inertia subfield in K|F , the extension K|L is cyclic and totally ramified of degree ℓ and [K : E] = [L : F ] is the order of q F ∈ (Z/ℓ) × .The degenerate case happens for q F ≡ 1 (mod ℓ).Considering C = G K|L և U L /U 1 L as a 1-dimensional F ℓ -space we see that the inner action Int : G L|F → GL F ℓ (C) ∼ = F × ℓ will be injective.It can be interpreted as the map σ i → q i F (mod ℓ), where σ ∈ G L|F denotes the Frobenius.
• E|F is totally wildly ramified of prime power degree [E : F ] = p s .Then L|F is the maximal tamely ramified subextension in K|F and C = G K|L և U L /(U L ) p is a simple F p [G L|F ]-module which is realized as quotient of U L /(U L ) p .And the inner action Int : G L|F → GL Fp (C) must be faithful.More precisely with respect to the lower numeration of ramification subgroups we have: where U t L ⊆ (U L ) p .[L]p.172 puts q := #C = [E : F ] = dim(Ind E|F (χ • N E|F )).Proof: If E|F is tamely ramified then it is well known that the Galois closure K|F will be obtained by adjoining sufficiently many roots of unity of order prime to p that means K|E will be an unramified extension.Let F 0 |F be the inertia subfield of K|F.From Lemma 1 of A2 the inclusion C = G K|L ≤ G K|F 0 will follow because C is contained in every non-trivial normal subgroup.Therefore we have L|F 0 .Moreover K|F 0 is normal and totally tamely ramified, hence it will be cyclic.Then of course C = G(K|F 0 ) because it is the only abelian normal subgroup, hence L = F 0 and [K : L] = [E : F ] = ℓ must be a prime because C is cyclic and minimal.Therefore the Galois closure of E|F is obtained by adjoining the ℓ-th roots of unity that means [L : F ] must be the order of q F ∈ (Z/ℓ) × , which implies q L ≡ 1 (mod ℓ).If E|F is totally wildly ramified then let F 1 |F be the maximal tamely ramified subextension in K|F .Again C = G K|L ≤ G K|F 1 will follow because C is contained in every nontrivial normal subgroup.Thus C and G K|F 1 are both normal p-subgroups in G = G K|F .But from Lemma 1 of A2 we see that C is the only normal p-subgroup in G and therefore L = F 1 .Thus we have: C = C 1 = G 1 and this implies G i ≤ C for i ≥ 1 and therefore: Lemma 2: Let G > H be a type-III-group and assume that it is also type-III with respect to G > H ′ .Then we will have H ′ = c 0 Hc −1 0 for some c 0 ∈ C, and in the non-degenerate case H = {1} the conjugates {cHc −1 | c ∈ C} will be all different.
And without loss of generality we may restrict here to Ω ′ = Ω, because all arguments are left unchanged if we replace Ω by an open subgroup Ω ′ .Theorem 3.1: Let Ω be a solvable profinite group and let ∆ = ∆(H, χ) be a function on R 1 (≤ Ω) with values in a multiplicative abelian group A. Then ∆ is extendible to F on R(≤ Ω) = H≤Ω R(H) if and only if the following three conditions are fulfilled for all open subgroups B ≤ Ω : I. Davenport-Hasse: Let K < B be normal of prime index ℓ and χ ∈ B * .Then Ind B K (χ K ) = µ∈(B/K) * χµ, and we should have: (1) And replacing G by the proper quotients G/N ′ G and G/C resp. the relation 1.(8) holds by induction.Finally comparing this to our definition 3.2 we see that λ by definition, which turns our last equality into the asserted equality (21) and ends the proof of the N-Tower lemma 3.4.qed.Remark: The relation (22) is not concerned with representations of G/C.Therefore the induction (G : C) < #G with respect to ∆ 0 does not work; instead we have to use the condition C > N > [ C, C].On one hand G = Ω/N is a quotient of Ω/[ C, C], and on the other hand ∆ is extendible from R 1 ( C ≤ Ω) onto R( C ≤ Ω) because (Ω : C) < (Ω : N).
further to R 1 (C ≤ B) we may write:λ B/C HµC/C (∆ 0 ) = λ BHµC (∆).The only case where we cannot apply the induction assumption is the case B = Ω, K = N, but then the definition 3.2 applies because C/K is the uniquely determined abelian normal subgroup in B/K.Therefore multiplying (3) by µ λ B HµC (∆) we conclude: the group W F admits representations which are not semisimple.But our aim is to extend the local root numbers ∆(E, χ) only to semisimple representations of the Weil groups.Therefore from now we think of • R(W F ) := the free Z-module over the equivalence classes of irreducible smooth representations of W F .
the expression ∆(E, η E ) does not depend on the choice of (E, η E ), (3odd)∆(E, χ • N E|F ) = µ∈S(K|L)/H ∆ F µ , (χ • N Fµ|F ) • µ ′ , because C is minimal normal.And C = C t implies C = C t because lower and upper numeration for C = G K|L agree up to the first jump, and moreover C t+1 = {1} such that by class field theory C must be a quotient of U t L /U t+1 L .Again by class field theory this must be a morphism of F p [G L|F ]-modules where the action of G L|F on C is via conjugation.Finally C must be simple because a proper F p [G L|F ]submodule C 0 is nothing else than an abelian normal subgroup of G = G K|F .And from Lemma 1 of A2 we also see that the inner action must be faithful.In the tame case this yields quite a simplification of (3).Indeed S(K|L) consists of the trivial character µ 0 and (ℓ − 1) characters µ = µ 0 which are faithful becauseL × /N K|L (K × ) ∼ = G K|L = C is cyclic of prime order ℓ.And H = G K|E ∼ = G L|F ֒→ F ×ℓ is cyclic too and acts (via conjugation) faithful on C and on the dual C * resp.Thus: s ∈ H, µ = µ 0 ∈ C * and µ s = µ, implies (µ ν ) s = µ ν for all powers of µ, and this implies s = 1.This means we have:H µ = {1}, E µ = K and F µ = L, hence µ ′ = µ, if µ = µ 0 .Now taking into account the last remark of (3) we obtain the arithmetic condition: (see[L], Lemma 13.3 plus remark at the end of p.163)(3 tame) ∆(E, χ • N E|F ) • [µ] ={µ 0 } ∆(L, µ) = ∆(F, χ) • [µ] ={µ 0 } ∆(L, (χ • N L|F )µ)should hold for any character χ of F × , if E|F is ramified of prime degree ℓ = p, and L|F is the inertia subfield of its Galois closure K|F.The products are over representatives for the ℓ−1[L:F ] different H ∼ = G L|F -orbits [µ] of non-trivial characters µ ∈ S(K|L) where each orbit is of length #H = [L : F ].And the product on the left vanishes if [K : F ] = ℓ•[L : F ] is odd.. in G, and actually there exists only one such C.In particular, C will be contained in every non-trivial normal subgroup of G. (The degenerate case is H = {1} and C = G.) Therefore C must be ℓ-elementary:C ∼ = Z/ℓ × • • • × Z/ℓfor a certain prime ℓ = ℓ(G) such that (G : H) = #C is a power of ℓ.At the same time C is the unique normal ℓ-subgroup of G, hence the ℓ-Sylow-subgroups of G and of H must have intersection C and {1} resp.And C is its own centralizer that meansh ∈ H → Int(h) ∈ GL F ℓ (C) is an embedding by inner automorphisms which induces G = H ⋉ C ֒→ GL F ℓ (C) ⋉ C.C can be also characterized as C = F (G) the Fitting subgroup (=maximal nilpotent normal subgroup) of G.This follows because the Fitting subgroup will be abelian if the Frattini subgroup is trivial.And it shows that in the non-degenerate case G is not nilpotent.Proof:Let C be a non-trivial commutative normal subgroup.We will haveH • C = G because H is maximal and cannot contain the normal subgroup C.But H ∩ C is normal in HC, hence H ∩ C = {1} and G = H ⋉ C is semidirect.Thus we have seen that all nontrivial commutative normal subgroups must have the same order #C = (G : H), and this implies that C must be unique.Then it is also clear that C is the absolutely minimal normal subgroup in the solvable group G, henceC ∼ = Z/ℓ × • • • × Z/ℓfor a certain prime ℓ = ℓ(G).Let B be an ℓ-subgroup of G which is normal.Then the center Z(B) = {1} is a normal abelian subgroup of G, hence Z(B) = C and B = (B ∩ H) ⋉ C is actually a direct product because C is the center of B. Therefore B ∩ H is not only normal in H but also in G = H ⋉ C. However by the very definition H cannot contain normal subgroups of G, and therefore B ∩ H = {1} and B = C. Obviously C is then the intersection of all ℓ-Sylow groups P of G. On the other hand the ℓ-Sylow groups P H come as P H = H ∩ P for all P and therefore the intersection of all P H is H ∩ C = {1}.Similarly the centralizer C of C has the form C = ( C ∩ H) × C where C ∩ H is normal in G, hence C ∩ H = {1}.qed.
B/[Z, B]which are 2-step nilpotent and such that B/Z ∼ = Z/ℓ × Z/ℓ is bicyclic for some prime ℓ, and[B, B]/[Z, B] B/K = H/K ⋉ C/K where H < B is a subgroup which is maximal but not normal, and K = b∈B bHb −1 is the corresponding normal subgroup.Actually this means that B/K is a type-III-group as it is explained in Appendix 2.
Further comments: In II the group H/Z is cyclic, hence H/Z ∧ H/Z = {1}, and therefore[H, H] ≤ [Z, Ω], whereas [H, Ω] = [Ω, Ω] because Ω/Z = H/Z × H ′ /Z.Thus η H is nontrivial on [H, Ω]/[Z, Ω]which means that the Ω-conjugates of η H are all different, hence Ind Ω H (η H ) will be irreducible.This also means that the conjugacy class [H, η H ] Ω consists of all extensions of η onto H > Z. Nevertheless ρ(Z, η) is not a single element because we can form it for each pair (H 1 , H 2 ) of intermediate groups Z < H i < Ω.In III the normal subgroup C < Ω will usually not be abelian.Nevertheless Φ C([H, χ H and η : Z/[Z, B] → C × is a fixed character which is nontrivial on [B, B]/[Z, B], and η H i ∈ H * i is any extension of η.From 1.(9) we know that λ B H i (∆) = µ∈(B/H i ) * ∆(B, µ) because B/H i is cyclic of prime order ℓ and is a subquotient of G = Ω/N.Therefore our assumption (2) rewrites as: