Local Galois representations of Swan conductor one

We construct the local Galois representations over the complex field whose Swan conductors are one by using etale cohomology of Artin-Schreier sheaves on affine lines over finite fields. Then, we study the Galois representations, and give an explicit description of the local Langlands correspondences for simple supercuspidal representations. We discuss also a more natural realization of the Galois representations in the etale cohomology of Artin-Schreier varieties.


Introduction
Let K be a nonarchimedean local field.Let n be a positive integer.The existence of the local Langlands correspondence for GL n (K ), proved in [Laumon et al. 1993] and [Harris and Taylor 2001], is one of the fundamental results in the Langlands program.However, even in this fundamental case, an explicit construction of the local Langlands correspondence has not yet been obtained.One of the most striking results in this direction is the result of Bushnell and Henniart [2005a;2005b;2010] for essentially tame representations.On the other hand, we don't know much about the explicit construction outside essentially tame representations.
We discuss this problem for representations of Swan conductor 1.The irreducible supercuspidal representations of GL n (K ) of Swan conductor 1 are equivalent to the simple supercuspidal representations in the sense of Adrian and Liu [2016] (see [Gross and Reeder 2010;Reeder and Yu 2014]).Such representations are called "epipelagic" in [Bushnell and Henniart 2014].
Let p be the characteristic of the residue field k of K .If n is prime to p, the simple supercuspidal representations of GL n (K ) are essentially tame.Hence, this case is covered by the work of Bushnell and Henniart.See also [Adrian and Liu 2016].It is discussed in [Kaletha 2015] to generalize the construction of the local Langlands correspondence for essentially tame epipelagic representations to other reductive groups.
In this paper, we consider the case where p divides n.In this case, the simple supercuspidal representations of GL n (K ) are not essentially tame.Moreover, if n is a power of p, the irreducible representations of the Weil group W K of Swan conductor 1, which correspond to the simple supercuspidal representations via the local Langlands correspondence, cannot be induced from any proper subgroup.Such representations are called primitive (see [Koch 1977]).For simple supercuspidal representations, we have a straightforward characterization of the local Langlands correspondence given in [Bushnell and Henniart 2014].Further, Bushnell and Henniart study the restriction to the wild inertia subgroup of the Langlands parameters for the simple supercuspidal representations explicitly.Actually, the restriction to the wild inertia subgroup already determines the original Langlands parameters up to character twists, but we need additional data, which appear in Bushnell and Henniart's characterization, to pin down the correct Langlands parameters.On the other hand, the construction of the irreducible representations of W K of Swan conductor 1 is a nontrivial problem.What we will do in this paper is • to construct the irreducible representations of W K of Swan conductor 1 without appealing to the existence of the local Langlands correspondence, and • to give a description of the Langlands parameters themselves for the simple supercuspidal representations.
Let ℓ be a prime number different from p.For the construction of the irreducible representations of W K of Swan conductor 1, we use étale cohomology of an Artin-Schreier ℓ-adic sheaf on ‫ށ‬ 1 k ac , where k ac is an algebraic closure of k.It will be possible to avoid usage of geometry in the construction of the irreducible representations of W K of Swan conductor 1.However, we prefer this approach, because • we can use geometric tools such as the Lefschetz trace formula and the product formula of Deligne-Laumon to study the constructed representations, and • the construction works also for ℓ-adic integral coefficients and mod ℓ coefficients.
A description of the local Langlands correspondence for the simple supercuspidal representations is discussed in [Imai and Tsushima 2022] in the special case where n = p = 2.Even in the special case, our method in this paper is totally different from that in [Imai and Tsushima 2022].
We explain the main result.We write n = p e n ′ , where n ′ is prime to p.We fix a uniformizer ϖ of K and an isomorphism ι : ‫ޑ‬ ℓ ≃ ‫.ރ‬ Let L ψ be the Artin-Schreier ‫ޑ‬ ℓ -sheaf on ‫ށ‬ 1 k ac associated to a nontrivial character ψ of ‫ކ‬ p .Let π : ‫ށ‬ 1 k ac → ‫ށ‬ 1 k ac be the morphism defined by π(y) = y p e +1 .Let ζ ∈ µ q−1 (K ), where q = |k|.We put Then we can define a natural action of W E ζ on H 1 c ‫ށ(‬ 1 k ac , π * L ψ ).Using this action, we can associate a primitive representation τ n,ζ,χ ,c of W E ζ to ζ ∈ µ q−1 (K ), a character χ of k × and c ∈ ‫ރ‬ × .We construct an irreducible representation τ ζ,χ ,c of Swan conductor 1 as the induction of τ n,ζ,χ ,c to W K .
We can associate a simple supercuspidal representation π ζ,χ ,c of GL n (K ) to the same triple (ζ, χ, c) by type theory.Any simple supercuspidal representation can be written in this form uniquely (see [Imai and Tsushima 2018, Proposition 1.3]).
In Section 1, we recall a general fact on representations of a semidirect product of a Heisenberg group with a cyclic group.In Section 2, we give a construction of the irreducible representations of W K of Swan conductor 1.To construct a representation of W K which naturally fits a description of the local Langlands correspondence, we need a subtle character twist.Such a twist appears also in the essentially tame case in [Bushnell and Henniart 2010], where it is called a rectifier.Our twist can be considered as an analogue of the rectifier.We construct the representations of W K using geometry, but we give also a representation theoretic characterization of the constructed representations.In Section 3, we give a construction of the simple supercuspidal representations of GL n (K ) using the type theory.
In Section 4, we state the main theorem and recall a characterization of the local Langlands correspondence for simple supercuspidal representations given in [Bushnell and Henniart 2014].The characterization consists of the three equalities of (i) the determinant and the central character, (ii) the refined Swan conductors, and (iii) the epsilon factors.
In Section 5, we recall some general facts on epsilon factors.In Section 6, we recall facts on Stiefel-Whitney classes, multiplicative discriminants and additive discriminants.We use these facts to calculate Langlands constants of wildly ramified extensions.In Section 7, we recall the product formula of Deligne-Laumon.In Section 8, we show the equality of the determinant and the central character using the product formula of Deligne-Laumon.
In Section 9, we construct a field extension which we call an imprimitive field.In Section 10, we show the equality of the refined Swan conductors.We see also that the constructed representations of W K are actually of Swan conductor 1.
In Section 11, we show the equality of the epsilon factors.It is difficult to calculate the epsilon factors of irreducible representations of W K of Swan conductor 1 directly, because primitive representations are involved.However, we know the equality of the epsilon factors up to p e -th roots of unity if n = p e , since we have already checked the conditions (i) and (ii) in the characterization.Using this fact and p ∤ [T u ζ : E ζ ], the problem is reduced to study an epsilon factor of a character.Next we reduce the problem to the case where the characteristic of K is p and k = ‫ކ‬ p .At this stage, it is possible to calculate the epsilon factor if p ̸ = 2.However, it is still difficult if p = 2, because the direct calculation of the epsilon factor involves an explicit study of the Artin reciprocity map for a wildly ramified extension with a nontrivial ramification filtration.This is a special phenomenon in the case where p = 2.We will avoid this difficulty by reducing the problem to the case where e = 1.In this case, we have already known the equality up to sign.Hence, it suffices to show the equality of nonzero real parts.This is easy, because the difficult study of the Artin reciprocity map involves only the imaginary part of the equality.
In Appendix, we discuss a construction of irreducible representations of W K of Swan conductor 1 in the cohomology of Artin-Schreier varieties.This geometric construction incorporates a twist by a "rectifier".We see that the "rectifier" parts come from the cohomology of Artin-Schreier varieties associated to quadratic forms.The Artin-Schreier varieties which we use have origins in studies of Lubin-Tate spaces in [Imai and Tsushima 2017;2021].
Notation.Let A ∨ denote the character group Hom ‫ޚ‬ (A, ‫ރ‬ × ) for a finite abelian group A. For a nonarchimedean local field K , let • O K denote the ring of integers of K , • p K denote the maximal ideal of O K , • v K denote the normalized valuation of K which sends a uniformizer of K to 1, • ch K denote the characteristic of K , • G K denote the absolute Galois group of K , • W K denote the Weil group of K , • I K denote the inertia subgroup of W K , • P K denote the wild inertia subgroup of W K , and we put U m K = 1 + p m K for any positive integer m.

Representations of finite groups
First, we recall a fact on representations of Heisenberg groups.Let G be a finite group with center Z .We assume: (i) The group G/Z is an elementary abelian p-group.
(ii) For any g ∈ G \ Z , the map c g : ] is surjective.Remark 1.1.The map c g in (ii) is a group homomorphism.Hence, Z is automatically an elementary abelian p-group.
Let ψ ∈ Z ∨ be a nontrivial character.
Proposition 1.2.There is a unique irreducible representation ρ ψ of G such that ρ ψ | Z contains ψ.Moreover, we have (dim ρ ψ ) 2 = [G : Z ] and we can construct ρ ψ as follows: Take an abelian subgroup G The claims other than the construction of ρ ψ is Proposition 8. 3.3 in [Bushnell and Fröhlich 1983].Note that if an abelian subgroup G 1 of G satisfies the conditions in the claim, then G 1 /Z is a maximal totally isotropic subspace of G/Z under the pairing Hence the construction follows from the proof of [Bushnell and Fröhlich 1983, Proposition 8.3.3].□ Next, we consider representations of a semidirect product of a Heisenberg group with a cyclic group.Let A ⊂ Aut(G) be a cyclic subgroup of order p e + 1 where e = 1 2 (log p [G : Z ]).We assume: (3) The group A acts on Z trivially.
(4) For any nontrivial element a ∈ A, the action of a on G/Z fixes only the unit element.
We consider the semidirect product A ⋉ G by the action of A on G.
Lemma 1.3.There is a unique irreducible representation ρ ′ ψ of A ⋉ G such that ρ ′ ψ | G ≃ ρ ψ and tr ρ ′ ψ (a) = −1 for every nontrivial element a ∈ A. Proof.The claim is proved in the proof of Lemma 22.2 in [Bushnell and Henniart 2006] if Z is cyclic and ψ is a faithful character.In fact, the same proof works also in our case.□ Corollary 1.4.There exists a unique representation for every nontrivial element a ∈ A. Further, the representation ρ ′ ψ | G is irreducible.Proof.First we show the existence.We take the representation ρ ′ ψ in Lemma 1.3.Then ρ ′ ψ has a central character equal to ψ by Proposition 1.2.This shows the existence.
We show the uniqueness and the irreducibility of ρ ′ ψ | G .Assume that ρ ′ ψ satisfies the condition in the claim.Take an irreducible subrepresentation ρ ψ of ρ ′ ψ | G .Then ρ ψ satisfies the condition of Proposition 1.2.Hence, dim ρ ψ = p e .Then we have

Galois representations
2A. Swan conductor.Let K be a nonarchimedean local field with residue field k.
Let p be the characteristic of k.Let f be the extension degree of k over ‫ކ‬ p .We put q = p f .Let Art K : K × − → ∼ W ab K be the Artin reciprocity map, which sends a uniformizer to a lift of the geometric Frobenius element.
2B. Construction.We construct a group Q which acts on a curve C over an algebraic closure of k.By using this action of Q and Frobenius action, we construct a representation of a semidirect product Q ⋊ ‫ޚ‬ in étale cohomology of C. Then we use the representation of Q ⋊ ‫ޚ‬ to construct a representation of a Weil group.
We fix an algebraic closure K ac of K .Let k ac be the residue field of K ac .Let n be a positive integer.We write n = p e n ′ with ( p, n ′ ) = 1.Throughout this paper, we assume that e ≥ 1.Let be the group whose multiplication is given by Remark 2.1.The construction of the group Q has its origin in a study of the automorphism of a curve C defined below.We can check that the above multiplication gives a group structure on Q directly, but it's also possible to show this by checking that the inclusion from Q to the automorphism group of C defined below is compatible with the multiplications.
Note that |Q| = p 2e+1 ( p e + 1).Let Q ⋊ ‫ޚ‬ be a semidirect product, where m ∈ ‫ޚ‬ acts on Q by (a, b, c) → (a p −m , b p −m , c p −m ).We put Let C be the smooth affine curve over k ac defined by We define a right action of Q ⋊ ‫ޚ‬ on C by (by) p i + c, a(y + b p e ) , (x, y) Fr(1) = (x p , y p ).
We consider the morphisms Then we have the fiber product where π ′ and h ′ are the natural projections to the first and second coordinates respectively.Let g = ((a, b, c), m) ∈ Q ⋊ ‫.ޚ‬We consider the morphism Let ℓ be a prime number different from p. Then we have a natural isomorphism We take an isomorphism ι : ‫ޑ‬ ℓ ≃ ‫.ރ‬We sometimes view a character over ‫ރ‬ as a character over ‫ޑ‬ ℓ by ι.Let ψ ∈ ‫ކ‬ ∨ p .We write L ψ for the Artin-Schreier ‫ޑ‬ ℓ -sheaf on ‫ށ‬ 1 k ac associated to ψ, which is equal to F(ψ) in the notation of [Deligne 1977, Sommes trig. 1.8(i)].Then we have a decomposition h * ‫ޑ‬ ℓ = ψ∈‫ކ‬ ∨ p L ψ .This decomposition gives canonical isomorphisms The isomorphisms c g and (2-3) induce c g,ψ : Let τ ψ be the representation of Q ⋊ ‫ޚ‬ over ‫ރ‬ defined by H 1 c ‫ށ(‬ 1 k ac , π * L ψ ) and ι.For θ ∈ µ p e +1 (k ac ) ∨ , let K θ be the smooth Kummer ‫ޑ‬ ℓ -sheaf on ‫އ‬ m,k ac associated to θ .We view µ p e +1 (k ac ) × ‫ކ‬ p as a subgroup of Q by (a, c) → (a, 0, c).
Lemma 2.2.We have a natural isomorphism which is compatible with the actions of µ p e +1 (k ac ) × ‫ކ‬ p where Proof.By the projection formula, we have natural isomorphisms Further, we have since π is a finite étale µ p e +1 (k ac )-covering over ‫އ‬ m,k ac .Therefore, we have be the natural immersions.From the exact sequence we have the exact sequence Note that H 0 ({0}, i * π * L ψ ) ≃ ψ.By (2-4), we have isomorphisms We know that for any θ ∈ µ p e +1 (k ac ) ∨ by the proof of [Imai and Tsushima 2017, Lemma 7.1] (see [Imai and Tsushima 2023, (2.3)]).Since the composition of and (2-6) is compatible with the actions of µ p e +1 (k ac ) × ‫ކ‬ p , it factors through an isomorphism Then the claim follows from (2-5), (2-6) and (2-7). □ for a ∈ µ p e +1 (k ac ).For an integer m and a positive odd integer m ′ , let m m ′ denote the Jacobi symbol.For an odd prime p, we set We have ϵ( p) 2 = −1 p .We define a representation τ n of Q ⋊ ‫ޚ‬ as the twist of τ ψ 0 by the character (2-9) The value of this character is related to a quadratic Gauss sum.A geometric origin of this character is given in We take a uniformizer ϖ of K .We choose an element ϕ (2-11) given by the multiplication.Let Frob p : k × → k × be the inverse of the p-th power map.We consider the following composition: We will see that τ ζ,χ ,c is an irreducible representation of Swan conductor 1 in Proposition 10.8.This Galois representation τ ζ,χ ,c is our main object in this paper.We will study several invariants associated to this, for example, its determinant and epsilon factor.
is surjective.The claim follows from the surjectivity of Tr ‫ކ‬ p e ‫ކ/‬ p .□ By this lemma, we can apply the results from Section 1 to our situation with G = Q 0 , Z = F and A = µ p e +1 (k ac ), where the action of µ p e +1 (k ac ) on Q 0 is given by the embedding and the conjugation.Let τ 0 denote the unique representation of Q characterized by We have a decomposition L θ such that a ∈ µ p e +1 (k ac ) acts on L θ by θ (a), since the both sides of (2-15) have the same character as representations of µ p e +1 (k ac ).For a positive integer m dividing p e +1, we consider µ m (k ac ) ∨ as a subset of µ p e +1 (k ac ) ∨ by the dual of the surjection We simply write Q for the subgroup Proof.This follows from Corollary 1.4, equation (2-14) and Lemma 2.5.□ For any odd prime p, we have (2-16) by Gauss.
Lemma 2.7.We have Proof.By the Lefschetz trace formula, we have where Fr p is the geometric p-th power Frobenius morphism.Since where we use (2-16) in the last equality.□ We assume p = 2 in this paragraph.We take b , where we use Tr ‫ކ‬ 2 2e ‫ކ/‬ 2 (b 0 ) = 1 at the third equality.We put Lemma 2.8.We assume that p = 2. Then we have Tr τ ψ 0 (g −1 ) = −2.
Proof.We note that For y ∈ k ac satisfying y 2 + b 2 e 0 = y, we take x y ∈ k ac such that x 2 y − x y = y 2 e +1 .We take y 0 ∈ k ac such that y 2 0 + b 2 e 0 = y 0 .Then, by the Lefschetz trace formula and (2-19), we have where we change a variable by y = y 0 + z at the second equality, and use In particular, τ ψ 0 does not depend on the choice of ℓ and ι.

Representations of general linear algebraic groups
3A. Simple supercuspidal representation.Let π be an irreducible supercuspidal representation of GL n (K ) over ‫.ރ‬ Let ε(π, s, ) denote the Godement-Jacquet local constant of π with respect to the nontrivial character : K → ‫ރ‬ × .We simply write ε(π, ) for ε π, 1 2 , .By [Godement and Jacquet 1972, Theorem 3.3(4)], there exists an integer sw(π ) such that We put Sw(π) = max{sw(π ), 0}, which we call the Swan conductor of π. 3B.Construction.In the following, we construct a smooth representation Then I is a hereditary O K -order (see [Bushnell and Kutzko 1993, (1.1)]).Let P denote the Jacobson radical of the order I.We put Then, L ζ is a totally ramified extension of K of degree n.
We put ϕ ζ,n = n ′ ϕ ζ and We define a character ζ,χ ,c : where tr means the trace as an element of M n (K ).We put Then, π ζ,χ ,c is a simple supercuspidal representation of GL n (K ), and every simple supercuspidal representation is isomorphic to π ζ,χ ,c for a uniquely determined Proof.This follows from [Bushnell and Henniart 1999, Section 6.1, Lemma 2 and Section 6.3, Proposition 1].□

Local Langlands correspondence
Our main theorem is the following.To prove this theorem, we recall a characterization of the local Langlands correspondence for epipelagic representations due to Bushnell-Henniart.Recall that for any tamely ramified character χ of W K .This property determines the coset γ τ, U 1 K uniquely.Definition 4.3.Let τ be an irreducible smooth representation of W K such that sw(τ ) ≥ 1.We take γ τ, as in Lemma 4.2.We put which we call the refined Swan conductor of τ with respect to .
For an irreducible supercuspidal representation π of GL n (K ), let ω π denote the central character of π.

General facts on epsilon factors
In this section, we recall some general facts on epsilon factors.
For a finite separable extension L over K , we put L = • Tr L/K and let denote the Langlands constant which is independent of s, where 1 is the trivial representation of W L (see [Bushnell and Henniart 2006, Section 30.4]).
Proposition 5.1.Let τ be a finite dimensional smooth representation of W K such that τ | P K is irreducible and nontrivial.Let L be a tamely ramified finite extension of K .Then we have Proof.This is proved by the same arguments as in [Bushnell and Henniart 2006, Proposition 48.3].□ Proposition 5.2.Let τ be a finite dimensional smooth representation of W K such that τ | P K does not contain the trivial character.
(2) Let L be a tamely ramified finite extension of K .Then we have Proof.This is [Bushnell and Henniart 2006, Theorem 48.1(2) .
(2) We have Proof.For a finite Galois extension L of K , let ψ L/K denote the Herbrand function of L/K and Gal(L/K ) i denote the lower numbering i-th ramification subgroup of Gal(L/K ) for i ≥ 0 (see [Serre 1968, Chapter IV]).We use the following lemmas to calculate the refined Swan conductor of a character of a Weil group.
Lemma 5.4.Let m be a positive integer dividing f .Let h be a positive integer that is prime to p and less than p m v K ( p)/( p m − 1).Let L be a Galois extension of K defined by x p m − x = 1/ϖ h .Then we have Proof.Take an integer l such that lh ≡ 1 mod p m .Then we have Hence, for σ ∈ Gal(L/K ) and i ≥ 0, we have σ ∈ Gal(L/K ) i if and only if The right-hand side of (5-1) is h + 1 if σ ̸ = 1.Hence the first claim follows.The second claim follows from the first claim.□ Lemma 5.5.Let L be a totally ramified finite abelian extension of K .Let m ≥ 1.
(1) We have (2) We take α ∈ K and Let ϖ L be a uniformizer of L. Then we have Proof.
where we take z x ∈ k ac such that z p x − z x = x.Then we have the second claim, since Stiefel-Whitney class and discriminant 6A.Stiefel-Whitney class.Let R(W K , ‫)ޒ‬ be the Grothendieck group of finitedimensional representations of W K over ‫ޒ‬ with finite images.For V ∈ R(W K , ‫,)ޒ‬ we put V ‫ރ‬ = V ⊗ ‫ޒ‬ ‫ރ‬ and define ε(V ‫ރ‬ , ) by the additivity using the epsilon factors in Section 2A.For V ∈ R(W K , ‫,)ޒ‬ we define the i-th Stiefel-Whitney class where the first map is induced by ‫ޚ2/ޚ‬ → K ac,× , m → (−1) m and the second isomorphism is the invariant map.
Theorem 6.1 [Deligne 1976, Théorème 1.5].Assume that V ∈ R(W K , ‫)ޒ‬ has dimension 0 and determinant 1.Then we have In particular, we have ε( 6B. Discriminant.Let L be a finite separable extension of K .We put as the discriminant of the quadratic form Tr L/K (x 2 ) on L. For a ∈ K × /(K × ) 2 , let {a} ∈ H 1 (G K , ‫)ޚ2/ޚ‬ and κ a ∈ Hom(W K , {±1}) be the elements corresponding to a under the natural isomorphisms and Saito 2010, Proposition 6.5].Let m be the extension degree of L over K .We take a generator a of L over K .Let f (x) ∈ K [x] be the minimal polynomial of a.We put D = f ′ (a) ∈ L. Then we have 6B2.Additive discriminant.We put P m (x) = x m − x for any positive integer m.We assume that ch K = 2. Definition 6.3 [Bergé and Martinet 1985, Définition 2.7].Let m be the extension degree of L over K .Let f (x) ∈ K [x] be the minimal polynomial of a generator of L over K .We have a decomposition f (x) = 1≤i≤m (x − a i ) over the Galois closure of L over K .We put which we call the additive discriminant of L over K .Theorem 6.4 [Bergé and Martinet 1985, Théorème 2.7].Let L ′ be the subextension of K ac over K corresponding to Ker δ L/K .Then the extension L ′ over K corresponds to d + L/K ∈ K/P 2 (K ) by the Artin-Schreier theory.

Product formula of Deligne-Laumon
We recall a statement of the product formula of Deligne-Laumon.In this paper, we need only the rank one case, which is proved in [Deligne 1973, Proposition 10.12.1], but we follow the notation from [Laumon 1987].
7A. Local factor.We consider a triple (T, F, ω) which consists of the following.
• The affine scheme T = Spec O K T where O K T is the ring of integers in a local field K T of characteristic p whose residue field contains k.
7B. Product formula.Let X be a geometrically connected proper smooth curve over k of genus g.Let F be a constructible ‫ޑ‬ ℓ -sheaf on X .Let Frob q ∈ G k be the geometric Frobenius element.We put Let rk(F) be the generic rank of F.
Theorem 7.1 [Laumon 1987, Théorème 3.2.1.1].Let ω be a nonzero meromorphic 1-form on X .Then we have where |X | is the set of closed points of X , and X (x) is the completion of X at x.
Lemma 8.1.We have Proof.By Lemma 2.4, we have Hence, we obtain the claim.□ We view θ 0 defined in (2-8) as a character of Q by (a, b, c) → θ 0 (a).Recall that τ 0 is the representation of Q defined in (2-14).
Lemma 8.2.We have det τ 0 = θ 0 .Proof.By Lemma 8.1, it suffices to show det τ 0 = θ 0 on µ p e +1 (k ac ).By Lemma 2.2 and Lemma 2.5, we have for a ∈ µ p e +1 (k ac ).Hence, the claim follows. □ For a ∈ k × , let a k denote the quadratic residue symbol of k defined by a k = −1 if a is not square in k.Lemma 8.3.Let m be a positive integer that is prime to p.We take an m-th root ϖ 1/m of ϖ , and put L = K (ϖ 1/m ).

Imprimitive field
In this section, we construct a field extension 9A. Construction of character.Here we construct subgroups R ⊂ Q ′ ⊂ Q ⋊ ‫ޚ‬ and a character φ n of R. Later (see Section 9B) we will see that ޚ‬ Let e 0 be the positive integer such that e 0 ∈ 2 ‫ގ‬ and e/e 0 is odd.Proof.This follows from (2-9) and Lemma 9.1.□ Let n 0 be the biggest integer such that 2 n 0 divides p e 0 + 1.We take r ∈ k ac such that r 2 n 0 = −1.We define a subgroup R 0 of Q 0 by (2) If p = 2, then the action of g on Q ⋊ ‫ޚ‬ by conjugation stabilizes R 0 .
Lemma 9.4.We have Proof.We write ψn for φ n | R 0 .We know that First, we consider the case where p is odd.The claim for general f follows from the claim for f = 1 by the restriction.Hence, we may assume that f = 1.
If ψ ∈ R ∨ 0 satisfies ψ| F = ψ 0 , then we have 2, and obtain an injective homomorphism ψ → τ n | R 0 as representations of R 0 by Frobenius reciprocity.Hence we have a decomposition The ψn -component in (9-6) is the unique component that is stable by the action of ((1, 0, 0), 2e 0 ), since the homomorphism : R] = p e .Next we consider the case where p = 2. Then it suffices to show that by (2-9) and Proposition 2.9.We have a decomposition Let ψ′ n be the twist of ψn by the character Then only the ψn -component and the ψ′ n -component in (9-6) are stable by the action of ((1, b 0 , c 0 ), 1), since the image of the homomorphism is equal to Ker Tr ‫ކ‬ 2 e ‫ކ/‬ 2 .The action of Fr(e) permutes the ψn -component and the ψ′ n -component.Hence, g acts on the ψ′ n -component by φ n (g) times Hence we have We use the notations from equation (2-10).We set Let K ur be the maximal unramified extension of K in K ac .Let K u ⊂ K ur be the unramified extension of degree N over K .Let k N be the residue field of K u .For a finite field extension L of K in K ac , we write L u for the composite field of L and K u in K ac .For a ∈ k ac , we write â ∈ O K ur for the Teichmüller lift of a.We put Then we have δ Proposition 9.5.We have Proof.This follows from Lemma 9.4.□ Remark 9.6.Our imprimitive field is different from that in [Bushnell and Henniart 2014, Section 5.1].In our case, T u ζ need not be normal over K .This choice is technically important in our proof of the main result.

9C. Study of character.
Here we study the character ξ n,ζ in detail.
Assume that ch K = p and f = 1 in this subsection.We will use results in this subsection to compute the epsilon factor of ξ n,ζ later after a reduction to the case where ch K = p and f = 1.By (2-10), (9-8), (9-9) and ch K = p, we have that 9C1.Odd case.Assume p ̸ = 2.We put (9-10) Since r p e 0 +1 = −1 and ( p e + 1)/( p e 0 + 1) is an odd integer, we have r p e +1 = −1.Then we have (9-11) We put where n σ is as before (2-12).
by (9-2).Hence, we see that We put Proof.We can check the first claim easily.We show the second claim.We use P m in Section 6B2.We have (9-16) Hence, we have Hence, by using e−1 i=1 t 2 i = 1 − t and t ∈ ‫ކ‬ 2 e , we have where we use (9-17) at the second equality and (9-13) at the third one.□ We take , which is a cyclic extension of M ′u ζ of order 4 by Lemma 9.8.Lemma 9.9.The character ξ We have .

Refined Swan conductor
Let K ⊂ K ur be the unramified extension of K u generated by µ p 4 pe −1 (K ur ).For a finite field extension L of K in K ac , we write L for the composite field of L and K in K ac .We write By equations (9-8) and (9-9), we can take ) by Krasner's lemma.Lemma 10.1.(1) We have For any finite extension M of K , we write ψ M for the composite ψ K • Tr M/K .
Lemma 10.3.We have rsw , and regard it as a character of M ′× ζ .By (2-12), Lemmas 5.5(1) and Lemma 10.1, the restriction of ξn,ζ to U 2 where the isomorphism Gal( then have where we use Lemmas 5.5(2) and 10.2 at the last equality.Since we have The claim follows from (10-4) and Proposition 5.3(1).□ Lemma 10.4.We have rsw( By Proposition 5.2(1), we may assume that χ = 1, c = 1.By Proposition 9.5 and Lemma 10.3, we have Since T u ζ is a tamely ramified extension of E ζ , we have by Proposition 5.2(2).The claim follows from (10-5) and (10-6).□ Lemma 11.2.We have Proof.Let K (0) and K ( p) be nonarchimedean local fields of characteristic 0 and p respectively.Assume that the residue fields of K (0) and K ( p) are isomorphic to k.We take uniformizers ϖ (0) and ϖ ( p) of K (0) and K ( p) respectively.We define T u ζ,(0) similarly as T u ζ starting from K (0) .We use similar notations also for other objects in the characteristic zero side and the positive characteristic side.We have the isomorphism of algebras, where ξ 0 , ξ 1 ∈ k.
Therefore, we may assume that n = p e .By Lemmas 11.1 and 11.5, it suffices to show that ε By this, Lemmas 11.1 and 11.4, it suffices to show that This follows from Lemma 11.2 and Proposition 11.7.□ We set ϖ M ′u ζ = δ −1 ζ .Proposition 11.7.Assume that n = p e .Then we have Proof.First, we reduce the problem to the positive characteristic case.Assume that ch K = 0. Take a positive characteristic local field K ( p) whose residue field is isomorphic to k.We define M ′u ζ,( p) similarly as M ′u ζ starting from K ( p) .We use similar notations also for other objects in the positive characteristic side.Then we have the isomorphism Hence, the problem is reduced to the positive characteristic case by [Deligne 1984, Proposition 3.7.1].We may assume K = ‫ކ‬ q ((t)).We put K ⟨1⟩ = ‫ކ‬ p ((t)).We define M ′u ζ,⟨1⟩ similarly as M ′u ζ starting from K ⟨1⟩ .We use similar notations also for other objects in the K ⟨1⟩ -case.We put f ) ) f ′ by Proposition 5.1.By (11-5), the problem is reduced to the case where f = 1.In this case, the claim follows from Lemmas 11.11 and 11.16.□ 11B.Special cases.We assume that n = p e , ch K = p and f = 1 in this subsection.

ζ
and φ n | R .By the local class field theory, we regard ξ n,ζ as a character of M ′u ζ × .