On the Local Artin Conductor F <Subscript>Artin </Subscript

Proc. Indian Acad. Sci. (Math. Sci.) Vol. 113, No. 2, May 2003, pp. 99–137. © Printed in India

On the local Artin conductor Artin (χ) of a character χ of Gal(E/K) – II: Main results for the metabelian case

KAZIMˆ ILHAN˙ IKEDA˙

Department of Mathematics, Istanbul Bilgi University, In¨on¨uCaddesi No. 28, Ku¸stepe, 80310 S¸i¸sli, Istanbul, Turkey E-mail: [email protected]

MS received 13 November 2001; revised 9 July 2002

Abstract. This paper which is a continuation of [2], is essentially expository in nature, although some new results are presented. Let K be a local field with finite residue class field κK . We first define (cf. Definition 2.4) the conductor f(E/K) of an arbitrary finite Galois extension E/K in the sense of non-abelian local class field theory as

[[nG]]+1 f(E/K) = pK ,

where nG is the break in the upper ramification filtration of G = Gal(E/K) defined by nG nG+δ G 6= G = 1, ∀δ ∈ R 0. Next, we study the basic properties of the ideal f(E/K) in OK in case E/K is a metabelian extension utilizing Koch–de Shalit metabelian local class field theory (cf. [8]). After reviewing the Artin character aG : G → C of G := Gal(E/K) and Artin representations AG : G → GL(V ) corresponding to aG : G → C, we prove that (Proposition 3.2 and Corollary 3.5)

(V ) n + dimC [ G/ker(ρ) 1] fArtin(χρ ) = pK ,

where χρ : G → C is the character associated to an irreducible representation ρ : G → GL(V ) of G (over C). The ﬁrst main result (Theorem 1.2) of the paper states that, if in particular, ρ : G → GL(V ) is an irreducible representation of G (over C) with metabelian image, then • Eker(ρ) K (n + ) [ : ] G/ker(ρ) 1 fArtin(χρ ) = pK ,

• where Gal(Eker(ρ)/Eker(ρ) ) is any maximal abelian normal subgroup of Gal(Eker(ρ)/K) ker(ρ) 0 containing Gal(E /K) , and the break nG/ker(ρ) in the upper ramification filtration of G/ker(ρ) can be computed and located by metabelian local class field theory. The proof utilizes Basmaji’s theory on the structure of irreducible faithful representations of finite metabelian groups (cf. [1]) and on metabelian local class field theory (cf. [8]). We then discuss the application of Theorem 1.2 on a problem posed by Weil on the construction of a ‘natural’ AG of G over C (Problem 1.3). More precisely, we prove in Theorem 1.4 that if E/K is a metabelian extension with Galois group G, then X N • AG ' [(E ) : K](nG/N + 1) N X × G ω ◦ π | , Ind −1 • N π−1((G/N)•) πN ((G/N) ) N [ω]∼∈VN

99 100 Kaˆzim I˙lhan I˙keda

where N runs over all normal subgroups of G, and for such an N, VN denotes the collection of all ∼-equivalence classes [ω]∼, where ‘∼’ denotes the equivalence relation on the set of all representations ω : (G/N)• → C× satisfying the conditions

• Inert(ω) ={δ∈G/N : ωδ = ω}=(G/N)

and \ ker(ωδ) =h1G/N i, δ

where δ runs over R((G/N)•\(G/N)), a ﬁxed given complete system of representatives • of (G/N) \(G/N), by declaring that ω1 ∼ ω2 if and only if ω1 = ω2,δ for some δ ∈ R((G/N)•\(G/N)). Finally, we conclude our paper with certain remarks on Problem 1.1 and Problem 1.3.

Keywords. Local fields; higher-ramification groups; local Artin conductor; metabelian local class field theory; non-abelian local class field theory; local Langlands correspondence for GL(n).

1. Introduction This paper is the natural continuation of [2]. However, for the sake of completeness, we have included all the necessary results from our previous companion article, so that one can directly study this work without assuming any background material from [2]. Let K be a local field with finite residue class field OK /pK =: κK of qK elements, where as usual, OK stands for the ring of integers in K with the unique maximal ideal pK . Let ν : K → Z ∪ {∞} denote the corresponding normalized exponential valuation on K (normalized by νν(K×) = Z), and eν the unique extension of ν to a fixed seperable closure Ksep of K. For any sub-extension L/K in Ksep/K, let eν L be the normalized form of the valuation eν |L on L. Let E be a finite Galois extension over K, and ρ : Gal(E/K) → GL(V ) an irreducible and finite-dimensional representation (over C) of the Galois group Gal(E/K) of the extension E/K. Let Eker(ρ) ={x∈E:σ(x) = x,∀σ ∈ ker(ρ)} be the fixed-field of ker(ρ), and for the time being, assume that Eker(ρ) is an abelian extension over K (equivalently, im(ρ) is an abelian subgroup of GL(V ) by the fundamental Galois duality). Observe that, under this assumption, the representation ρ : Gal(E/K) → GL(V ) is one-dimensional. In fact, the representation (ρ, V ) of Gal(E/K) factors through

ρ (E/K) / Gal Q 7GL(V ) QQQ nnn QQQ nnn QQQ nnn π QQ( nnn ρe Gal(E/K)/ker(ρ)

∼ with an isomorphism Gal(E/K)/ker(ρ) −−→ Gal(Eker(ρ)/K) deﬁned by the fundamental Galois duality, where π : Gal(E/K) → Gal(E/K)/ker(ρ) is the canonical mapping, and ρe : Gal(E/K)/ker(ρ) → GL(V ) is deﬁned by ρe(σ ker(ρ)) = ρ(σ) for every σ ∈ Gal(E/K). Note that, the homomorphism ρe : Gal(E/K)/ker(ρ) → GL(V ) is an irreducible representation of the group Gal(E/K)/ker(ρ) in V over C,as(ρ, V ) is Local Artin conductor – II 101 an irreducible representation of Gal(E/K). Recall that, Eker(ρ)/K is assumed to be an abelian extension. Thus, dimC(Vρe) = 1, since an irreducible representation of an abelian group should be one-dimensional. Hence, (ρ, V ) is a one-dimensional representation of Gal(E/K), as the representation spaces Vρ and Vρe coincide. It is then well-known that (cf. Proposition 11.6 of chapter VII of [9], or §3)

ker(ρ) fArtin(χρ) = f(E /K), (1) where f(Eker(ρ)/K) is the conductor of the abelian extension Eker(ρ)/K defined in the sense of abelian local class field theory (which will be reviewed in the next section), and fArtin(χρ) is the local Artin conductor of the character χρ : Gal(E/K) → C associated to the representation (ρ, V ) of Gal(E/K) (which will be reviewed in §3). This identity is very important. In fact, we note that, this identity apart from its own beauty, plays the key role in proving the Artin conjecture in the abelian case of Artin L- functions and in establishing the functional equation satisfied by Artin L-functions. It is then natural to ask, more generally, the following problem:

Problem 1.1. How can we describe class ﬁeld theoretically the local Artin conductor fArtin(χρ) of the character χρ : Gal(E/K) → C associated to an irreducible representation (ρ, V ) of the Galois group Gal(E/K) of the extension E/K without imposing any condition on the representation (ρ, V ) of Gal(E/K)?

Of course, in this generality, full solution of Problem 1.1 does not seem to be possible for the time being, but utilizing a result of Henniart and Tunnell, we can partially answer Problem 1.1. That is, we can describe the local Artin conductor fArtin(χρ) of the character χρ : Gal(E/K) → C associated to an irreducible representation ρ : Gal(E/K) → GL(V ) in terms of the ‘conductor’f(Eker(ρ)/K) of the Galois extension Eker(ρ)/K (defined in §2 as Definition 2.4 following an idea of Sen in [10]) as 1 ker(ρ) ordpK (fArtin(χρ)) = ordpK f(E /K) , dimC(V ) where, for x ∈ R,[[x]] denotes the integer part of the number x (cf. Corollary 3.5). This equation, however, has two gaps: First of all, the above equation does not give any explicit information about the dimension of the irreducible representation (ρ, V ) of Gal(E/K) in terms of the extension E/K. What is also missing in this equation is the class field theoretic information hidden in the ‘conductor’ f(Eker(ρ)/K) of the Galois extension Eker(ρ)/K. In fact, the definition of f(Eker(ρ)/K) involves the integer part of a certain break ker(ρ) nGal(E/K)/ker(ρ) of the upper ramification filtration of the Galois group Gal(E /K) (cf. Definition 2.4). Since Eker(ρ)/K is an n-abelian extension (cf. §4) for some 1 ≤ n ∈ Z, in order to compute and locate nGal(E/K)/ker(ρ), we need Hasse–Arf-type theorems for n- abelian extensions, for every 1 ≤ n ∈ Z. If, on the other hand, the representation (ρ, V ) of Gal(E/K) has metabelian image, that is, if the image im(ρ) ⊂ GL(V ) of the representation (ρ, V ) of Gal(E/K) is a metabelian group (i.e., the 2nd-commutator subgroup im(ρ)(2) of im(ρ) is trivial, or equivalently, if Eker(ρ)/K is a metabelian extension); utilizing Koch–de Shalit theory (cf. [8]) and the representation theory of finite metabelian groups developed by Basmaji (cf. [1]), it is possible to fill these two gaps, and hence it is ker(ρ) possible to describe fArtin(χρ) in terms of the ‘conductor’ f(E /K) of the metabelian extension Eker(ρ)/K defined in the sense of Koch–de Shalit metabelian local class field 102 Kaˆzim I˙lhan I˙keda

theory (which is the specialization of Definition 2.4 to metabelian extensions and which is defined explicitly in §5). More precisely, the first main result of this paper is the following:

Theorem 1.2. Let E be a ﬁnite Galois extension over K, and ρ : Gal(E/K) → GL(V ) be an irreducible and ﬁnite-dimensional representation (over C) of the Galois group Gal(E/K) of the extension E/K. ker(ρ) If E /K is a metabelian extension, then the Artin conductor fArtin(χρ) is given explicitly by

(ρ)• Eker K (n (ρ) + ) [ : ] Gal(Eker /K) 1 fArtin(χρ) = pK , (2)

n = n (Eker(ρ)/K) where Gal(Eker(ρ)/K) : is the break in the upper ramification filtration of Gal (ρ) n (ρ) n+δ defined by Gal(Eker /K) 6= Gal(Eker /K) = 1, ∀δ ∈ R> , which can be located 0 • by Koch–de Shalit metabelian local class field theory, and Gal(Eker(ρ)/Eker(ρ) ) is any maximal abelian normal subgroup of Gal(Eker(ρ)/K) containing Gal(Eker(ρ)/K)0.

In order to prove this theorem, we have organized our paper as follows: Sections 2 and 3 are dedicated to a quick review of the higher ramification groups and the Artin (A ,V ) (E/K) representation Gal(E/K) AGal(E/K) of the Galois group Gal of the extension E/K. In particular, following an idea of Sen in [10], we introduce the conductor f(E/K) of any finite Galois extension E/K in Definition 2.4. We then review, in §4, the metabelian local class field theory following [8]. In §5, we give the interpretation of Definition 2.4 in the case of a finite metabelian (= 2-abelian) extension E/K in the sense of metabelian local class field theory, following the main theorem of higher-ramification theory in metabelian local class field theory (cf. Lemma 5.5), and in Proposition 5.6, again assuming E/K is finite metabelian, give an explicit description of the exponent of the conductor f(E/K) in terms of the exponent of the conductor f(E/E•), where G• = Gal(E/E•) is any maximal abelian normal subgroup of G = Gal(E/K) containing G0. In §6, we compute the Artin conductor fArtin(χρ) of the character χρ : Gal(E/K) → C associated to a given irreducible representation (ρ, V ) of Gal(E/K) under the assumption that Eker(ρ)/K is a 2-abelian extension using Basmaji’s theory (cf. [1]) and the basic properties of Artin conductors, and get eq. (2), which completes the proof of Theorem 1.2. In §7, as an application of Theorem 1.2, we partially answer the following open problem posed by Weil.

Problem 1.3 (Weil). Determine a ‘natural’ AGal(E/K) of Gal(E/K) over C.

In fact, we prove the following theorem in §7.

Theorem 1.4. Let E be a ﬁnite metabelian extension over K. For any N G Gal(E/K), let Gal(EN /K)• = Gal(EN /(EN )•) denote a maximal abelian normal subgroup of Gal(EN /K) containing the 1st commutator subgroup Gal(EN /K)0 of Gal(EN /K). Let I(Gal(EN /K)•)o denote the set of all one-dimensional representations

N • × ω : Gal(E /K) → C

satisfying

N • Inert(ω) = Gal(E /K) Local Artin conductor – II 103 and \ (ω ) =h i, ker δ 1Gal(EN /K) δ∈Gal((EN )•/K)

N • × − where ωδ : Gal(E /K) → C is the representation defined by ωδ : x 7→ ω(δxδ 1) ∼ for x ∈ Gal(EN /K)• and δ ∈ Gal((EN )•/K) −−→ Gal(EN /K)/Gal(EN /K)•. Now, N • o define an equivalence relation ‘∼’onI(Gal(E /K) ) by declaring ω1 ∼ ω2 in case N • ω1 = ω2,δ for some δ ∈ Gal((E ) /K). Let VN denote the set of ∼-equivalence classes [I(Gal(EN /K)•)o/ ∼]. Let A (E/K) → GL(V ) Gal(E/K) : Gal AGal(E/K) be an Artin representation of Gal(E/K) over C. Then, X A ' (EN )• K (n + ) Gal(E/K) [ : ] Gal(EN /K) 1 N X × G ω ◦ π | , Ind −1 N • N π−1( (EN /K)•) πN (Gal(E /K) ) N Gal [ω]∼∈VN N (E/K) n where runs over all normal subgroups of Gal . Here, Gal(EN /K) is the number defined in Theorem 1.2. Hence it can be computed by metabelian local class field theory; N πN : Gal(E/K) → Gal(E /K) is the canonical mapping, and [ω]∼ runs over all VN , N • where [ω]∼ denotes the ∼-equivalence class of ω ∈ I(Gal(E /K) )o.

The proof utilizes Theorem 1.2 and Basmaji’s theory. Note that, if E/K is in particular A ' anP abelian extension in Theorem 1.4, then we get the well-known identity that Gal(E/K) (n + )χ χ (E/K) → C× χ Gal(Eker(χ)/K) 1 , where : Gal runs over all one-dimensional representations of G over C. Finally, in §8, we conclude our paper with a set of complementary remarks on Prob- lem 1.1 and Problem 1.3.

2. Review of higher-ramiﬁcation groups Main references for this section are [7, 9, 10, 12]. For a ﬁnite seperable extension L/K, and for any σ ∈ HomK (L, Ksep), introduce

iL/K (σ ) := min {eν L(σ (y) − y)}, y∈OL put sep γt := # σ ∈ HomK (L, K ) : iL/K (σ ) ≥ t + 1 for −1 ≤ t ∈ R, and deﬁne the function ϕL/K : R≥−1 → R≥−1 (the Hasse–Herbrand transition function of the extension L/K)by (R u γt γ dt, 0 ≤ u ∈ R, ϕL/K (u) := 0 0 u, −1 ≤ u ≤ 0. 104 Kaˆzim I˙lhan I˙keda

It is well-known that, ϕL/K : R≥−1 → R≥−1 is a monotone-increasing and piecewise- ≈ linear function, and induces a homeomorphism R≥−1 −−→ R ≥−1. Let ψL/K : R≥−1 → R≥−1 be the mapping which is inverse to the function ϕL/K : R≥−1 → R≥−1. From now on, unless otherwise stated, assume that E is a non-trivial ﬁnite Galois extension over K with Galois group Gal(E/K) =: G. Under this assumption, OE is monogenic over OK , that is, OE = OK [x] for some x ∈ OE (cf. Lemma 10.4 of chapter II of [9]). Therefore

iE/K(σ ) = eν E (σ(x)−x) for every σ ∈ G. In fact, it sufﬁces to prove that, for any σ ∈ Gal(E/K),

eν E (σ(x)−x) ≤eνE (σ(y)−y) P y ∈ O y= s αxi α ∈ O i = ,...,s for every E.Now,let i=0 i with i K for 0 . Then ! Xs Xs Xs i i i i σ(y)−y = σ αix − αix = αi(σ (x) − x ), i=0 i=0 i=1 P i i j k and as σ(x) −x = (σ (x) − x) 0≤j,k∈Z σ(x) x , for 1 ≤ i ∈ Z, it follows that j+k=i−1   Xs  X   j k σ(y)−y = αi (σ (x) − x) σ(x) x  i=1 0≤j,k∈Z j+k=i−1 = (σ(x)−x)z for some z ∈ OE, which proves the desired inequality.

Remark 2.1. Note that, iE/K : Gal(E/K) → Z ∪ {∞} does not depend on the choice of the generator x ∈ OE over OK . ¤

The subgroup Gu of G deﬁned by

Gu ={σ ∈G: iE/K(σ ) ≥ u + 1} for −1 ≤ u ∈ R is called the uth ramification group of G (in the lower numbering), and has γ {G } G order u. The family u u∈R≥−1 induces a filtration on , called the lower ramification G {G } G filtration of . A break in the filtration u u∈R≥−1 of is defined to be any number u ∈ R≥−1 satisfying Gu 6= Gu+ε for every 0 <ε∈R. Observe that

Gj+ε = Gj+1 (3) for −1 ≤ j ∈ Z and 0 <ε≤1. In fact

Gj+ε ={σ ∈G|iE/K(σ ) ≥ (j + ε) + 1}

={σ ∈G|iE/K(σ ) ≥ j + 2}=Gj+1 Local Artin conductor – II 105 for −1 ≤ j ∈ Z and 0 <ε≤1, since iE/K(σ ) ∈ Z ∪ {∞} for every σ ∈ G. It is also well-known that, for 0 ≤ j ∈ Z, there exists an injection

j j+1 Gj /Gj+1 ,→ U (E)/U (E) (4) defined by σ(π ) σ 7→ E πE for σ ∈ Gj , where πE is a fixed prime element in E. This embedding does not depend on the choice of the prime element πE in E. In this case, Hasse–Herbrand transition function ϕE/K : R≥−1 → R≥−1 of the Galois extension E/K is defined by (R u 1 (G G ) dt, 0 ≤ u ∈ R, ϕE/K(u) = 0 0: t u, −1 ≤ u ≤ 0.

Note that, for 0

ϕ (u) G E/K = Gu for −1 ≤ v,u ∈ R, where Gv is called the vth upper ramification group of G. A break {Gv} G v ∈ R in the upper filtration v∈R≥−1 of is defined to be any number ≥−1 satisfying Gv 6= Gv+δ for every 0 <δ∈R.

Remark 2.2. Observe that u ∈ R {G } G (i) If 0 ≥−1 is not a break in the lower ramification filtration u u∈R≥−1 of , G = G <ε ∈R G = G that is, if u0 u0+ε0 for some 0 0 ; then u0 u0+ε for every 0 ≤ ε ≤ ε0. Since ϕE/K : R≥−1 → R≥−1 is a monotone-increasing and piecewise- linear function, ϕE/K ([u, u + ε0]) = [v0,v0 + δ0], where ϕE/K(u0) = v0 and v v +δ ϕE/K(u0 +ε0) = v0 +δ0 for some 0 <δ0 ∈R, and therefore G 0 = G 0 for every 0 ≤ δ ≤ δ0. So, ϕE/K(u0) = v0 is not a break in the upper ramification filtration {Gv} G v∈R≥−1 of . (ii) Note that, following the same lines of reasoning, the converse of (i) is also true: That v ∈ R {Gv} G is, if 0 ≥−1 is not a break in the upper ramification filtration v∈R≥−1 of , ψ (v ) ∈ R {G } then E/K 0 ≥−1 is not a break in the lower ramification filtration u u∈R≥−1 of G. Thus, combining (i) and (ii): 106 Kaˆzim I˙lhan I˙keda

v = ϕ (u ) ∈ R {Gv} (iii) 0 E/K 0 ≥−1 is a break in the upper ramification filtration v∈R≥−1 of G iff ψE/K(v0) = u0 ∈ R≥−1 is a break in the lower ramification filtration {G } G ¤ u u∈R≥−1 of . Basic properties of lower and upper ramification filtrations on G. Suppose that K ⊆ F ⊆ E is a sub-extension of E/K, let Gal(E/F ) = H . (i) The lower numbering on G passess well to the subgroup H of G in the sense that

Hu = H ∩ Gu

for −1 ≤ u ∈ R, (ii) and if furthermore, H G G, the upper numbering on G passes well to the quotient G/H as (G/H )v = GvH/H

for −1 ≤ v ∈ R; but the lower numbering on G passes via ‘ϕE/F -action’ to the quotient G/H as

(G/H )ϕE/F (u) = GuH/H

for −1 ≤ u ∈ R, (iii) (transitivity of the Hasse–Herbrand function). If F/K is a Galois sub-extension of E/K, then

ϕE/K = ϕF/K ◦ϕE/F

and

ψE/K = ψE/F ◦ ψF/K.

u ` Let BE/K (resp. BE/K) be the set of all numbers v ∈ R≥−1 which occur as breaks in the upper (resp. lower) ramification filtration of G. Then, by eq. (3), ` (iv) BE/K is a finite subset of Z ∩ R≥−1, and by Remark 2.2, u ` ` u (v) ψE/K(BE/K) ⊆ BE/K and ϕE/K(BE/K) ⊆ BE/K. u The description of the set BE/K is far more interesting, and as we will observe in §4, much more involved: u (vi) (Hasse–Arf theorem). If E/K is an abelian extension, then BE/K is a finite subset of Z ∩ R≥−1. ¤

Remark 2.3. Note that, if E/K is an abelian extension, and 0 ≤ n ∈ Z, then Gv = Gn for every n − 1

v n Proof. To prove this, assume that there exists v0 ∈ (n − 1,n) such that G 0 6= G . By Hasse–Arf theorem, this chosen v0 is not a break in the upper ramiﬁcation ﬁltration v v0 v0+δ0 {G }v∈R≥− of G.SoG =G for some 0 <δ0 ∈R. Now, by Remark 2.2(ii), the 1 v v δ v + δ

So by eq. (3), there exists an integer k such that ψE/K(v0) k ψE/K(v1) (Reason: G 6= G ψ (v ), ψ (v ) Note that, ψE/K(v0) ψE/K(v1). So, if the open interval E/K 0 E/K 1 is a k,k + k G = subset of some closed interval [ 1] for some integer , then by eq. (3), ψE/K(v0) G ψE/K(v1), a contradiction.), which must be a break in the lower ramification filtration of G G 6= G ϕ (k) since ψE/K(v0) ψE/K(v1). Thus, E/K must be a break in the upper ramification filtration of G by Remark 2.2(iii). Since G is assumed to be abelian, ϕE/K(k) is an integer, contradicting the fact that (v0,v1) ⊂ (n − 1,n). Thus, v1 must be in S. Therefore, n − 1

Recall that, for the abelian extension E/K, the norm-residue symbol

× (?, E/K) : K → Gal(E/K)

of abelian local class field theory maps the higher-unit group U n(K) (which is defined by n × n 0 U (K) ={x∈K :x≡1 (mod pK )} for 1 ≤ n ∈ Z and U (K) = U(K)) onto the nth upper ramification group Gn of G for 0 ≤ n ∈ Z. Hence, combining with Remark 2.3, for x ∈ K×,

x ∈ N(E/K)Un(K) ⇐⇒ (x, E/K) ∈ Gv

for every n − 1

mG f(E/K) = pK

m in OK , where 0 ≤ mG ∈ Z is the minimal power of pK satisfying U G (K) ⊆ N(E/K), m or equivalently, 0 ≤ mG ∈ Z is the minimal integer satisfying G G = 1.

DEFINITION2.4 (Conductorf(E/K)ofaﬁniteGaloisextensionE/K)

Let E/K be a ﬁnite Galois extension (with Galois group G). Deﬁne the real number nG by

x+δ nG = inf{x ∈ R≥−1 : G = 1, ∀δ>0}.

The conductor f(E/K) of E/K (in the sense of ‘non-abelian’ local class ﬁeld theory1)is deﬁned to be the ideal

[[nG]]+1 f(E/K) = pK

in OK , where [[nG]] denotes the ‘integer-part’ of the number nG.

1Note that, E/K is a finite n-abelian extension. So, modulo n-abelian local class field theories (1 ≤ n ∈ Z)in the sense of Koch–de Shalit (yet to be constructed for 3 ≤ n ∈ Z !), we can locate and compute nG by n-abelian Hasse–Arf theorem. This is the reason for us to call f(E/K) as the conductor of the extension E/K in the sense of ‘non-abelian’ local class field theory. 108 Kaˆzim I˙lhan I˙keda

Remark 2.5. Observe that, for the extension E/K, the number nG introduced in Def- inition 2.4 is a jump in the ramification filtration of E/K in upper numbering, and n + 0 ≤ [[nG]] + 1 ∈ Z is the minimal integer satisfying G[[ G]] 1 = 1. If, in particular, E/K is abelian, then nG is, by Hasse–Arf theorem, the maximal integer satisfying n n + G G 6= 1. Thus, [[nG]] + 1 = nG + 1 is the minimal integer satisfying G[[ G]] 1 = 1, that is [[nG]] + 1 = mG. So, Definition 2.4 coincides with the well-known definition of conductor of E/K in case E/K is abelian. ¤

Lemma 2.6. GnG is an abelian normal subgroup of G.

nG Proof. In fact, since G = GψE/K(nG) and nG is a jump in the upper ramification filtration of E/K, it follows that ψE/K(nG) is a jump in the lower ramification filtration of E/K and ψE/K(nG) ∈ Z≥−1 by Remark 2.2(iii) and eq. (3). Now, assume that ψE/K(nG) ≥ 0 (that is, equivalently, assume that nG ≥ 0). Then by eq. (4), there exists an injection nG ψE/K(nG) ψE/K(nG)+1 G /GψE/K(nG)+1 ,→ U (E) U (E), which is defined by σ(π ) σ 7→ E πE n for σ ∈ G G , where πE is any chosen and fixed prime element in E. The assertion now follows. In fact, ϕE/K : R≥−1 → R≥−1 is a piecewise-linear, strictly increasing function. So

ϕE/K([ψE/K(nG), ψE/K(nG) + 1]) = [nG,nG +δ0]

nG+δ0 nG for a unique 0 <δ0 ∈R. Therefore, GψE/K(nG)+1 = G = 1, which proves that G is an abelian subgroup of G.IfψE/K(nG) =−1, then necessarily nG =−1 and therefore ∼ the inertia G0 is trivial, proving that G −−→ Gal(κE/κK ), which is a cyclic group. ¤

Basic properties of lower and upper ramification filtration on G (continuation). If E/K is an infinite Galois extension with Galois group Gal(E/K) = G (which is a topological {Gv} G group under the Krull topology), define the upper ramification filtration v∈R≥−1 on by the projective limit

Gv = (F/K)v : lim←− Gal K ⊆ F ⊂ E deﬁned over the transition morphisms

F 0 tF v (G/Gal(E/F ))v ←−−− G/Gal(E/F 0) kk can. GvGal(E/F )/Gal(E/F ) ←−−−− G v Gal(E/F 0)/Gal(E/F 0) induced from (ii), as K ⊆ F ⊆ F 0 ⊆ E runs over all finite Galois extensions F and F 0 over K inside E. Observe that (vii) G−1 = G and G0 is the inertia group of G, Local Artin conductor – II 109 T (viii) Gv =h1i, v∈R≥−1 (ix) Gv is a closed subgroup of G (with respect to the Krull topology) for −1 ≤ v ∈ R. In this setting, a number −1 ≤ v ∈ R is said to be a break in the upper ramification filtration v {G }v∈R≥− of G,ifvis a break in the upper filtration of some finite quotient G/H for 1 u some H G G. As introduced previously, let BE/K be the set of all numbers v ∈ R≥−1, which occur as breaks in the upper ramification filtration of G. Then, Bu ⊆ Z ∩ R (x) (Hasse–Arf theorem). Kab/K ≥−1, and the final important result, in the spirit of Koch–de Shalit local class field theory, is u (xi) BKsep/K ⊆ Q ∩ R≥−1. ¤

3. Artin representation AGal(E/K) of Gal(E/K) Main references for this section are [9, 11, 12]. Now, again assume that E/K is a ﬁnite Galois extension with Galois group Gal(E/K) = G. Introduce a complex-valued function aG : G → C on G by  −f(E/K)iE/K(σ ), σ 6= 1  a (σ ) = P G  f(E/K) iE/K(τ), σ = 1  τ∈G τ6=1 − for every σ ∈ G. Note that aG : G → C is a class function on G, that is, aG(δσ δ 1) = − − aG(σ ) for every δ,σ ∈ G since iE/K(δσ δ 1) = iE/K(σ ) whenever δσδ 1 6= 1. In fact −1 −1 iE/K δσδ = eνE((δσ δ )(x) − x)

−1 −1 =eνE(δ(σ (δ (x)) − δ (x))), and since eν E (δ(y)) =eνE(y) for every y ∈ OE, −1 −1 −1 iE/K(δσ δ ) = eν E(σ (δ (x)) − δ (x)) = iE/K(σ ), − − as OK [x] = OE = δ 1OE = OK [δ 1(x)], and by Remark 2.1. Note that, this argument also proves that Gu G G, for every −1 ≤ u ∈ R. Let X(G) be the vector space of all class functions G → C on G. The mapping (?, ?)G : X(G) × X(G) → C deﬁned by X (f, g) = 1 f(σ)g(σ) G E K [ : ] σ∈G for every f, g ∈ X(G) is a Hermitian inner-product on X(G) (note that #(G) = [E : K] in this particular case), and the set B(G) of all irreducible characters χ : G → C of G forms an orthonormal basis of the vector space X(G). Hence, the dimension of X(G) over C is equal to the number of conjugacy classes r(G) in G. Any class function g : G → C on G is a C-linear combination X g = fχ (g)χ χ 110 Kaˆzim I˙lhan I˙keda

of irreducible characters χ : G → C of G, where the uniquely deﬁned χ-coordinate fχ (g) ∈ C of the function g : G → C, which is called the Fourier coefﬁcient of g : G → C at χ : G → C,isgivenby

fχ(g) = (g, χ)G.

In the remaining of this section we will sketch the proof of the fact that fχ (aG) ∈ Z≥0, for every irreducible character χ : G → C of G. Thus, the class function aG : G → C is in

fact a character, called the ‘Artin character’of G, of a representation AG : G → GL(VAG ), called an ‘Artin representation’ of G over C.

Digression: Regular representation RG of G. Recall that,P the regular representation RG : G → GL (C[G]) is defined by the G-module C[G] ={ g∈Gxgg|xgP∈C,∀g∈G}, whereP the action of G on C[G] is definedP by the left-multiplication (h, g∈G xgg) 7→ g∈G xg(hg) for every h ∈ G and g∈G xgg ∈ C[G]. So, RG : G → GL (C[G]) is defined explicitly by

RG : h 7→ Th,

where Th : C[G] → C[G] is the linear isomorphism deﬁned by X X Th : xgg 7→ xg(hg) g∈G g∈G P for every g∈G xgg ∈ C[G], and for every h ∈ G. Note that G is the canonical C-basis of C[G]. So C[G]isa#(G) = γ−1 =: γ -dimensional vector space over C. Fixing an ordering on the group G, for any h ∈ G, the matrix Mh = [Th]G ∈ GLγ (C) corresponding to Th ∈ GL(C[G]) is Mh = (Mh;g,t)g,t∈G, where the (g, t) component Mh;g,t of Mh

is Mh;g,t = δh,gt−1 for every g,t ∈ G. Thus, the character χRG = rG : G → C of G associated to the regular representation RG : G → GL (C[G]) is ( #(G) = γ, if h = 1G; rG(h) = 0, if h 6= 1G

for every h ∈ G. Note that ( ) X M X X C[G] = C g xgg | xg = 0 , g∈G g∈G g∈G P P P where both C[G]1 = C g∈G g and C[G]0 ={ g∈Gxgg| g∈Gxg =0}are G- invariant subspaces of C[G]. Thus, the regular representation RG : G → GL (C[G]) of G is the direct sum of the trivial representation 1G : G → GL (C[G]1) and the subrepresentation UG : G → GL(C[G]0), called the augmentation representation of G, that is,

RG = 1G ⊕ UG.

The character rG : G → C associated to the regular representation (RG, C[G]) of G is then given by the sum = χ + χ , rG 1G UG Local Artin conductor – II 111

G → C G → where χ1G : is the character associated to the trivial representation 1G : GL (C[G]1) of G and χUG = uG : G → C is the character associated to the augmentation G → GL C G (h) = χ (h) + (h) representation UG : ( [ ]0) of G. Hence, rG 1G uG for every h ∈ G, proving that ( #(G) − 1 = γ − 1, if h = 1G uG(h) = −1, if h 6= 1G for every h ∈ G. ¤

Let Gi be the ith-ramiﬁcation group of G (in the lower numbering), uGi : Gi → C the character associated to the augmentation representation UGi : Gi → GL (C[Gi]0) of Gi over C for −1 ≤ i ∈ Z. It is then well-known that X γi G aG = Ind (uG ), (6) γ Gi i 0≤i∈Z 0 where γi = #(Gi) for every 0 ≤ i ∈ Z; and for any subgroup H of G,

H H ResH (aG) = ν K (d(E /K))rH + f(E /K)aH , (7) where d(EH /K) denotes the discriminant of the extension EH /K. For any class function g : G → C on G, set (aG,g)G =: fArtin(g). The complex number fArtin(g) is then given by X γ f (g) = i (g(h i) − g(G )), Artin γ 1G i (8) 0≤i∈Z 0 P 1 γi = (Gi) g(Gi) = g(h) ≤ i ∈ Z where # and γi h∈Gi for 0 . In fact, by the Frobenius reciprocity law

(g, G ( )) = ( (g), ) IndGi uGi G ResGi uGi Gi 1 X = ResGi (g)(h)uGi (h), γi h∈Gi and since (

γi − 1, if h = 1Gi uGi (h) = −1, if h 6= 1Gi it follows that   X G 1   (g, Ind (uG ))G = ResG (g)(1G )(γi − 1) − ResG (g)(h) Gi i γ i i i i h6= 1Gi 1 X = ResGi (g)(1Gi ) − ResGi (g)(h) γi h∈Gi = g(h1Gi) − g(Gi), 112 Kaˆzim I˙lhan I˙keda

which yields the desired equality. In particular, if χρ : G → C is the character of a representation (ρ, V ) of G over C, then X γ f (χ ) = i (χ (h i) − χ (G )) Artin ρ γ ρ 1G ρ i 0≤i∈Z 0 X γi = ( (V ) − (V Gi )). γ dimC dimC 0≤i∈Z 0

G Since χρ(h1Gi) = dimC(V ) and χρ(Gi) = dimC(V i ), where

G V i ={v∈V |ρ(g)(v) = v,∀g ∈ Gi}

is a G-invariant subspace of V for 0 ≤ i ∈ Z, which follows from the fact that Gi G G for −1 ≤ i ∈ Z. Observe that

G G V i ⊆ V i0 (9)

if i ≤ i0 with 0 ≤ i, i0 ∈ Z. The following proposition generalizes a theorem of Henniart and Tunnell (cf. p. 103 of [12e], and for the one-dimensional case, Proposition 11.3 of chapter VII of [9]). However, before stating the proposition, we ﬁrst introduce some notation.

Notation 3.1. Let ρ : G → GL(V ) be a non-trivial representation of G over C. Let −1 ≤ j ∈ Z be maximal among all integers −1 ≤ i satisfying

Gi * ker(ρ).

Then, the set of integers I(ρ) =[0,j]∩Z={0,...,j}has a disjoint decomposition G I(ρ) = I(ρ)k 1≤k≤s

deﬁned by a sequence −1 = i0 i1 ··· is =j so that I(ρ)k = [ik−1 +1,ik]∩Z= {ik−1 +1,...,ik}for 1 ≤ k ≤ s is the maximal interval satisfying

Gi + G V k−1 1 =···=V ik.

Note that, the existence of such a disjoint collection of intervals I(ρ)k for 1 ≤ k ≤ s follows from eq. (9). Therefore, for the representation (ρ, V ) of G over C, there exists an s-step ﬂag

Gi + Gi Gi + Gi V 0 1 =···=V 1 (···(V s−1 1 =···=V s

G G of V j constructed as above. Let dk = dimC(V i ) for i ∈ I(ρ)k,1≤k≤s. ¤

PROPOSITION3.2

Let χρ : G → C be the character associated to a non-trivial representation (ρ, V ) of G over C. Let −1 ≤ j ∈ Z be maximal among all integers −1 ≤ i satisfying

Gi * ker(ρ). Local Artin conductor – II 113

Then, following Notation 3.1, X fArtin(χρ) = dimC(V ) ϕE/K(j) + 1 − dk[ϕE/K(ik) − ϕE/K(ik−1)]. 1≤k≤s

Moreover, (ρ) (i) ϕE/K(j) is a jump in the ramification filtration of Eker /K in upper numbering and ϕE/K(j) = nG/ker(ρ) (cf. Definition 2.4), ker(ρi + ) (ii) ϕE/K(ik) is a jump in the ramification filtration of E k 1 /K in upper numbering G ik+1 for 1 ≤ k ≤ s (here, ρik+1 : G → GL(V ) is the subrepresentation of (ρ, V ) G corresponding to the G-invariant subspace V ik+1 in V for 1 ≤ k ≤ s) and ϕE/K(ik) = nG/ (ρ ) ker ik+1 . Furthermore, if (ρ, V ) is an irreducible representation of G over C, then we get the result of Henniart and Tunnell, namely

fArtin(χρ) = dimC(V )[ϕE/K(j) + 1].

Proof. If j

f (χρ) = dimC(V )[ϕE/K(j) + 1] Artin   X X γ X γ X γ − d  i − i  − d i k γ γ 1 γ 1 k≤s 0≤i≤ik 0 0≤i≤ik− 0 0≤i≤i 0 X 1 1 = dimC(V )[ϕE/K(j) + 1] − dk[ϕE/K(ik) − ϕE/K(ik−1)] 1≤k≤s

again by eq. (5), which completes the proof of the desired equality. 114 Kaˆzim I˙lhan I˙keda

(i) Now, let E0 = Eker(ρ) be the fixed field of ker(ρ) = H . As usual, let ρe : G/H → can. GL(V ) be the non-trivial faithful representation of G/H defined by ρ : G −−−−→ ρe G/H −−→ GL(V ). By Herbrand’s theorem G H/H = (G/H ) . j ϕE/E0(j)

G = GϕE/K(j) (G/H ) = (G/H )ϕE0/K (ϕE/E0(j)) = (G/H )ϕE/K(j) As j and ϕE/E0(j) , in terms of the upper ramiﬁcation groups: GϕE/K(j)H/H = (G/H )ϕE/K(j).

Now, ρ(Ge jH/H) 6= 1 and ρ(Ge j+εH/H) = ρ(Ge j+1H/H) = 1 for every 0 <ε∈Rby the choice of the integer j and by eq. (3). Thus

Gj H/H 6= Gj+εH/H, 0 for every 0 <ε∈R. Thus, ϕE/K(j) is a jump in the ramiﬁcation ﬁltration of E /K in upper numbering, since GϕE/K(j)H/H = (G/H )ϕE/K(j) 6= (G/H )ϕE/K(j)+δ = GϕE/K(j)+δH/H,

for every 0 <δ∈R,asϕE/K : R≥−1 → R≥−1 is a continuous and strictly increasing function. Moreover, since ρ(Ge j+εH/H) = ρ(Ge j+1H/H) = 1 for every 0 <ε∈Rby the choice of the integer j and by eq. (3), it follows that Gj+ε ⊆ H for every 0 <ε∈R. That is, Gj+εH/H =h1G/H i for every 0 <ε∈R, proving that ϕ (j)+δ (G/H ) E/K =h1G/H i

for every 0 <δ∈R. Hence, ϕE/K(j) = nG/H following the notation of Deﬁnition 2.4. G ik+1 (ii) Consider the subrepresentation ρik+1 : G → GL(V ) of the representation (ρ, V ) G G ik ik+1 of G. Clearly Gik+1 ⊆ ker(ρik+1) and Gik * ker(ρik+1) since V ( V . Thus ik is the maximal integer satisfying Gik * ker(ρik+1). Now, the proof follows from the same lines of reasoning of part (i). G Finally, since Gu G G for every −1 ≤ u ∈ R, V u is a G-invariant subspace of V . Therefore, dk = 0 for k = 1,...,s, since (ρ, V ) is assumed to be an irreducible representation of G over C. Now, the result of Henniart and Tunnell follows. ¤

For a representation (ρ, V ) of G over C, in order to prove that fArtin(χρ) ∈ Z≥0,it sufﬁces to prove this integrality result for any one-dimensional representation χ : G → C× of G over C. In fact, by Brauer induction theorem, there exist certain type of subgroups (called the elementary subgroups) Hi of G and one-dimensional representations χi : Hi → × C of Hi over C such that the character χρ : G → C of G is a certain linear combination of the form X χ = n G (χ ), ρ i IndHi i i

where ni ∈ Z, and i runs over a ﬁnite set. Thus, by Frobenius reciprocity law ! X X f (χ ) = a , n G (χ ) = n (a , G (χ )) Artin ρ G i IndHi i i G IndHi i G X i G i

= ni(ResHi (aG), χi)Hi , i Local Artin conductor – II 115 and by eq. (7), X ν Hi Hi fArtin(χρ) = ni(νν(d(E /K))rHi + f(E /K)aHi ,χi)Hi Xi X ν Hi Hi = niνν(d(E /K))(rHi ,χi)Hi + nif(E /K)(aHi ,χi)Hi Xi X i ν Hi Hi = niνν(d(E /K))χi(1Hi ) + nif(E /K)fArtin(χi). i i

So, if fArtin(χi) ∈ Z≥0 for every i, then fArtin(χρ) ∈ Z. Moreover, γ0fArtin(χρ) ≥ 0by eq. (6) and so fArtin(χρ) ≥ 0. Thus, let χ : G → C× be a one-dimensional representation of G over C. The fact that fArtin(χ) ∈ Z≥0 is then a direct consequence of Proposition 3.2 stated as follows: COROLLARY3.3 Let χ : G → C× be a non-trivial one-dimensional representation of G over C, −1 ≤ j ∈ Z the maximal integer satisfying Gj * ker(χ). Then ker(χ) fArtin(χ) = ϕE/K(j) + 1 = ordpK (f(E /K)), ker(χ) where ϕE/K(j) = nG/ker(χ) is a jump in the ramification filtration of E /K in upper numbering, and ϕE/K(j) ∈ Z≥−1. Proof. The proof of the first equality follows from the main identity of Proposition 3.2. In fact, following Notation 3.1,

fArtin(χ) = ϕE/K(j) + 1 G G since in this case dim(Vχ ) = 1 and any i ∈ I(χ) ={0,...,j}satisﬁes V i = V j = 0 by eq. (9). The second equality follows from part (i) of Proposition 3.2 and by Remark 2.5 applied to the abelian extension Eker(ρ)/K. ¤ Remark 3.4. If χ : G → C× is the trivial representation of G over C, then the main equality of Corollary 3.3 remains true by setting j =−1. In fact, fArtin(χ) = (χ, aG)G = ker(χ) 0 by eq. (8). So fArtin(χ) = ϕE/K(−1) + 1. On the other hand, E = K, since ker(χ) = G. Thus, the main equality in Corollary 3.3 follows immediately. ¤ The following corollary can be viewed as a partial answer (i.e., answer modulo n-abelian local class ﬁeld theories (1 ≤ n ∈ Z), and modulo explicit information on the dimensions of irreducible representations of G over C) to Problem 1.1. COROLLARY3.5 Let ρ : G → GL(V ) be an irreducible representation of G over C. Then, 1 ker(ρ) fArtin(χρ) = ordpK (f(E /K)). dimC(V )

Proof. Let −1 ≤ j ∈ Z be the maximal integer satisfying Gj 6⊆ ker(ρ). Then, by Henniart–Tunnell’s theorem fArtin(χρ) = dimC(V ) ϕE/K(j) + 1 , where ϕE/K(j) = nG/ker(ρ), by part (i) of Proposition 3.2. Combining with Deﬁnition 2.4, the proof now follows. ¤ 116 Kaˆzim I˙lhan I˙keda

However, in case Eker(ρ)/K is a metabelian extension, utilizing the results of [1] and [8], we have a complete and detailed answer to Problem 1.1.

4. Metabelian local class field theory Main references for this section are: [3, 4, 7, 8]. Digression: n-abelian extensions over a field F . The main reference for this part is §2 of [4]. Recall that, a group G is called n-abelian (resp. ‘strict’ n-abelian), if the nth-commutator (n) (n− ) (n− ) (n) (m) subgroup G := [G 1 ,G 1 ]ofGis h1Gi (resp. G =h1Giand G 6= h1Gi for every integer 0 ≤ m n), that is, G is a solvable group with derived length dl(G) ≤ n (resp. dl(G) = n) for 1 ≤ n ∈ Z. (Notational convention: G(0) = G.) Now, an extension Q over any field F is called n-abelian (resp. ‘strict’ n-abelian), if it is a Galois extension with an n-abelian (resp. ‘strict’ n-abelian) Galois group Gal(Q/F ). Note that, the derived series for Gal(Q/F ) =: G is then given by (n) (n− ) ( ) ( ) h1i=G ≤ G 1 ≤···≤G 1 ≤G 0 =G,

G(i) (i) and passing to the ﬁxed ﬁeld Q = Qi of G for i = 0,...,n, the derived series for G transforms to the chain of sub-extensions in Q/F ,

Qn = Q ⊇ Qn−1 ⊇···⊇Q1 ⊇Q0 =F

with Qi/Qi−1 abelian extension for i = 1,...,n, called the ‘canonical chain of sub- extensions in Q/F ’ throughout the text. Moreover, if Q/F is ‘strict’ n-abelian, then the derived series for G is of the form (n) (n− ) ( ) ( ) h1i=G G 1 ··· G 1 G 0 =G,

where the inclusions are strict in this case, because if G(i) = G(i−1) for some i ∈ {1,...,n−1}, then G(i) = [G(i−1),G(i−1)] = [G(i),G(i)] = G(i+1), and hence G(n−1) = (n− ) h1Gi, which contradicts the assumption that G 1 6= h1Gi. The corresponding chain of sub-extensions in Q/F obtained by Galois duality is then of the form

Qn = Q ) Qn−1 ) ···)Q1 )Q0 =F

with Qi/Qi−1 non-trivial abelian extension (i = 1,...,n). Note that, any m-abelian extension over F is n-abelian for m ≤ n. Fix a seperable closure F sep of F (when it is convenient, instead of F sep, sometimes the n notation F will be used to denote the separable closure of F ), and let F (ab) denote the maximal n-abelian extension of F inside F sep (which exists! cf. [4]). This concludes the digression on n-abelian extensions over F . ¤ In [8], Koch and de Shalit have constructed class field theory for metabelian (that is, 2-abelian) extensions of a local field K, which induces the abelian local class field theory when restricted to the abelian extensions of K. Koch–de Shalit local class field theory is indeed an ‘approximation’ of the non-abelian local class field theory2.

2The local Langlands correspondence for GL(n) (cf. [14]) and Koch–de Shalit local class ﬁeld theory are closely related to each other. We plan to investigate the relationship between the local Langlands correspondence for GL(n) and Koch–de Shalit local class ﬁeld theory in a seperate paper. Local Artin conductor – II 117

The main idea in [8] is the following: as before, let K be a local field with finite residue field κK of qK elements. Let κK be a fixed separable closure of κK . Fix an extension φ ∈ Gal(Ksep/K) of the Frobenius automorphism φK over K. That is, fix a Lubin–Tate splitting φ over K. It is then well-known that, depending on the choice of φ, there exists a unique norm-compatible set of primes

0 Lφ ={πL ∈L: K⊆L⊂Kφs.t[L:K]<∞},

where Kφ denotes the ﬁxed ﬁeld of φ in Ksep, which has a canonical extension to a norm- compatible set of primes

gnr sep Lφ ={πL ∈L : K ⊆ L ⊂ K s.t[L:K]<∞},

called the Lubin–Tate labelling over K attached to the Lubin–Tate splitting φ over K. Here, Lgnr denotes the completion of Lnr . For a ﬁnite extension L over K which is pointwise ﬁxed by φ, there exists a unique Lubin–Tate formal power series fL ∈ OL[[X]] (belonging gnr 0 to πL for L , where πL chosen from Lφ) satisfying certain properties (cf. 0.3 in [8]), and a unique Lubin–Tate formal group law FL ∈ OLgnr [[X, Y ]] attached to fL. Let [u]fL : FL → FL be the unique endomorphism of FL over OLgnr of the form [u]fL = uX + (higher-degree terms) ∈ XOLgnr [[X]] for u ∈ OL, and let

{u}fL = [u]fL (mod πL).

d × d ∼ Let K× denote the profinite completion of K , and fix the isomorphism K× −−→ U(K)× b νa d× Z defined by a = uaπK 7→ (ua,νa)for a ∈ K , where πK is the prime element in K 0 chosen uniquely from Lφ. For 1 ≤ d ∈ Z, consider the topological group ( ) ξ φd {u } G[2](K, φ) = (a, ξ) ∈ Kd× × κ X × = a fK d : K [[ ]] : ξ X

under the law of composition deﬁned by

φ−νa (a, ξ)(b, ψ) = (ab, ξ(ψ ◦{ua}fK))

[2] for (a, ξ), (b, ψ) ∈ Gd (K, φ), and with a basis of neighborhoods of the underlying topology

[2] (i,j) [2] i j Gd (K, φ) ={(a, ξ) ∈ Gd (K, φ) : a ∈ U (K), ξ ≡ 1 (mod X )}

[2] for 0 ≤ i, j ∈ Z. Note that, Gd (K, φ) is a non-trivial topological group for every 1 ≤ d ∈ Z, by [6]. G[2](K, φ) = G [2](K, φ) Let : lim←− d , where the projective limit is deﬁned over the tran- d [2] [2] 0 0 sition morphisms Gd0 (K, φ) → Gd (K, φ) for every 1 ≤ d,d ∈ Z with d|d , which is deﬁned by   Y  di  (a, ξ) 7→ a, ξφ  d0 0≤i d 118 Kaˆzim I˙lhan I˙keda

[2] [2] (i,j) for (a, ξ) ∈ Gd0 (K, φ). Note that, these transition morphisms send Gd0 (K, φ) to [2] (i,j) 0 0 Gd (K, φ) for every 1 ≤ d,d ∈ Z with d|d , which in return deﬁnes a subgroup G[2](K, φ)(i,j) = G [2](K, φ)(i,j) G[2](K, φ) lim←− d of . d Let L be a φ-compatible extension over K, φ0 = φf (L/K) (cf. 0.4 in [8]), and M G[2](L, φ0) −−−−→φ,L/K G [2](K, φ)

the ‘2-abelian’ norm map which is a canonical morphism defined via ‘Coleman theory’ 0 (cf. 1.5 in [8]). As a notation, put Mφ(L/K) := Mφ,L/K (G[2](L, φ )), which is a closed subgroup of finite index in G[2](K, φ), and for an infinite extension F/K which is a union of φ-compatible extensions L over K (such anTF will be called as an infinite φ-compatible extension over K), define Mφ(F/K) := Mφ(L/K) where L runs over all φ- K⊆L⊂F compatible sub-extensions in F/K (so Mφ(F/K) is a closed subgroup of G[2](K, φ)). Define the map φ Mφ -compatible extensions −−−→ closed subgroups over K of G[2](K, φ) by

F/K 7→ Mφ(F/K)

for every φ-compatible extension F/K. The metabelian local class field theory of Koch–de Shalit states the following: Theorem 4.1. (Metabelian local class field theory). Let K be a local field with finite residue field κK of qK elements. Fix an extension φ ∈ Gal(Ksep/K) of the Frobenius nr automorphism φK ∈ Gal(K /K) over K, that is, fix a Lubin–Tate splitting φ over K. (i) (2-abelian reciprocity map). There exists a topological isomorphism ( ,K) G[2](K, φ) ? φ (K(ab)2 /K) −−−−−→∼ Gal

called the ‘2-abelian’ local Artin map in this text, which depends only on the choice of φ, (ii) (Existence). There exists an order-preserving bijection M 2-abelian extensions −−−→φ closed subgroups over K of G[2](K, φ) deﬁned by − (ab)2 1 E/K 7→ Mφ(E/K) = Gal(K /E), K φ

for any 2-abelian extension E/K. Note that, the closed subgroup Mφ(E/K) of G[2](K, φ) is of ﬁnite index if and only if E/K is a ﬁnite metabelian extension, and if this is the case, then [2] [E : K] = G (K, φ) : Mφ(E/K) , Local Artin conductor – II 119

(iii) (2-abelian reciprocity map – Continuation –). For any 2-abelian extension E over K, the surjective morphism

2 K(ab) (?,K)φ res. [2] / (ab)2 E / / (E/K) G (K, φ) ∼ Gal(K /K) Gal5 5

(?,K)φ |E

induces the canonical topological isomorphism

(?,E/K)φ [2] / G (K, φ)/Mφ(E/K) ∼ Gal(E/K),

where Mφ(E/K) is the closed normal subgroup of G[2](K, φ) deﬁned as in part (ii) , (iv) (Functoriality).IfK0 is a ﬁnite φ-compatible extension over K, then the square

( ,K0) ? φ0 (ab)2 G[2](K0,φ0)−−−−−→ Gal(K0 /K0) Mφ,K0/K ↓↓resK (?,K)φ 2 G[2](K, φ) −−−−−→ Gal(K(ab) /K)

is commutative. Therefore, the 2-abelian norm map Mφ φ-compatible extensions / closed subgroups , over K of G[2](K, φ)

deﬁned via Coleman theory, has a natural extension to Mφ 2-abelian extensions / closed subgroups , over K of G[2](K, φ)

deﬁned as in part (ii), in the sense that the triangle 2-abelian extensions overO K Q( O QQQ O QQQMφ QQQ O QQQ O ( ( O O closed subgroups O G[2](K, φ) O m6 of O mmm O mmm mmm O mm Mφ m φ-compatible extensions over K 2 Gal-cl is commutative, where the vertical arrow is deﬁned by K0 K0 ∩ K(ab) , which is known to be a 2-abelian extension over K (cf. [4]), for any φ-compatible extension K0 over K, 120 Kaˆzim I˙lhan I˙keda

(v) (Ramiﬁcation theory). The breaks in the upper numbering of the ramiﬁcation groups for 2-abelian extensions over K occur at r = 0 and at rational numbers of the form

i qK − 1 − j ui,j = i − i i−1 qK − qK

i−1 i with 1 ≤ i ∈ Z and j ∈ Z such that qK ≤ j

i − 1 ≤ ui,j−1

then

[2] (i,j) r α ∈ Mφ(E/K)G (K, φ) ⇐⇒ (α, E/K)φ ∈ Gal(E/K) .

The 2-abelian local class ﬁeld theory is related with the abelian local class ﬁeld theory as follows: if E/K is a 2-abelian extension, then pr1 Mφ(E/K) = N(E/K). (10)

The abelian extension over K, which is class field to N(E/K) is the maximal abelian sub-extension of E/K. Let K = E0 ⊆ E1 ⊆ E2 = E be the ‘canonical chain of sub- extensions in E/K’ as introduced in the beginning of this section. Note that, E1/K is the ab maximal abelian sub-extension of E/K, that is, E1 = E ∩ K . In fact, to simplify the 0 notation, setting G = Gal(E/K) as usual, G = Gal(E/E1).IfF/K is any abelian sub- extension of E/K, and N = Gal(E/F ), which is a closed and normal subgroup of G, then G/N = Gal(F/K) is an abelian group, proving that G0 ⊆ N. Thus, passing to the fixed fields, E1 ⊇ F , which completes the proof of the maximality of E1/K among all abelian ab sub-extensions F/K in E/K, which also shows that E1 = E ∩ K . Therefore, pr1 Mφ(E/K) = N(E1/K). (11)

Lemma 4.2. (Koch–de Shalit)

[2] (i,j) i pr1(G (K, φ) ) = U (K)

i−1 i for 1 ≤ i ∈ Z and j ∈{qK ,...,qK −1}.

So, combining eq. (11) and Lemma 4.2,

Lemma 4.3. Let E/K be a ﬁnite 2-abelian extension. [2] (i,j) i−1 i (i) If α ∈ Mφ(E/K)G (K, φ) for 1 ≤ i ∈ Z and j ∈{qK ,...,qK −1}, then i i pr1(α) ∈ N(E/K)U (K) = N(E1/K)U (K) for 1 ≤ i ∈ Z. r (ii) If (α, E/K)φ ∈ Gal(E/K) for ui,j−1

Remark 4.4. Note that, the metabelian local class ﬁeld theory depends on a choice of Lubin–Tate splitting φ over K. Let LT K denote the set of all Lubin–Tate splittings over nr K. Then, there exists a bijection between LT K and IK (the inertia group Gal(Ksep/K ) Local Artin conductor – II 121

of K). We can define such a bijection by fixing a Lubin–Tate splitting φc over K, and −1 defining the map LT K → IK by φ 7→ φ ◦ φc for every φ ∈ LT K . Now, consider the − set 0c := γφcγ 1 : γ ∈IK . Clearly, 0c ⊆ LT K . Moreover, there exists a bijection − between IK and 0c defined by γ 7→ γφcγ 1 for every γ ∈ IK . Note that, this defines a I 0 γ φ γ −1 = γ φ γ −1 γ ,γ ∈ I bijection between K and c, since if 1 c 1 2 c 2 for some 1 2 K, then γ −1γ = φn n ∈ Z hφ i (Ksep/K) hφ i 2 1 c for some as the normalizer of c in Gal is c itself (cf. 0.2 in [8]). Thus, there exist bijections LT K ↔ IK ↔ 0c. Observe that, even though 0c and LT K have the same cardinality and 0c ⊆ LT K , it may happen in general that 0c 6= LT K . However, there exists a collection {φi}i∈I of Lubin–Tate splittings over K (indexed over a set I), and a partition of LT K as G LT K = 0i, i

where i runs over this indexing set I. −1 Now, if φ,φ0 ∈ 0i for some i ∈ I, that is, if φ0 = eσφeσ for some eσ ∈ IK ; then in view of [3], + −1 eσ (α), K = eσ | (ab)2 (α, K)φ eσ | (ab)2 φ0 K K

[2] + [2] [2] for every α ∈ G (K, φ), where eσ : G (K, φ) → G (K, φ0) is a certain natural isomorphism of topological groups, which depends on eσ (cf. Lemma 5.2 of [3]). It is now easy to verify that the following statements are equivalent for a ﬁnite 2-abelian extension E/K and for a given r ∈ R satisfying i − 1 ≤ ui,j−1

iφ,φ G[2](K, φ) 0 / G[2](K, φ ) OOO oo 0 OOO ooo OOO ooo ( ,K) O o ( ,K)φ ? φ O' owow ? 0 2 Gal(K(ab) /K)

commutative. Then, (i (α), K) = (α, K) , φ,φ0 φ0 φ

for every α ∈ G[2](K, φ). It is now easy to verify the equivalence of the following statements for a ﬁnite 2-abelian extension E/K and for a given r ∈ R satisfying i − 1 ≤ ui,j−1

r (ii) (α, E/K)φ ∈ Gal(E/K) , [2] (i,j) (iii) iφ,φ (α) ∈ Mφ (E/K)G (K, φ0) , 0 0 r (iv) iφ,φ (α), K ∈ Gal(E/K) . 0 φ0 So, ramiﬁcation theory of metabelian extensions does not depend on the choice of Lubin– Tate splitting over K. ¤

5. Conductor f(E/K) of a finite metabelian extension E/K Main reference for this section is [2]. Let E/K be a finite Galois extension with Galois group G, and H a normal subgroup of G. Let H = Gal(E/F ), where F/K is the Galois sub-extension of E/K defined as the fixed field of H.

Lemma 5.1. For any −1 ≤ u, t ∈ R, if Gal(E/K)u ⊆ Gal(E/F ) and Gal(E/F )t = 1, then Gal(E/K)max{u,t} = 1.

Proof. If σ ∈ Gmax{u,t}, then iE/K(σ ) ≥ max{u, t}+1≥u+1, and therefore σ ∈ H , since Gu ⊆ H. Moreover,

iE/F (σ ) = min {eν E (σ(y)−y)}=iE/K(σ ) ≥ max{u, t}+1≥t+1. y∈OE

Thus, σ ∈ Ht , which proves that Gmax{u,t} = 1. ¤

PROPOSITION5.2 For any −1 ≤ v,w,z ∈ R, (i) Gal(E/K)v = 1 H⇒ Gal(F/K)v = 1, (ii) Gal(E/K)v = 1 H⇒ Gal(E/F )ψF/K(v) = 1, (iii) Gal(F/K)w = 1 and Gal(E/F )z = 1 H⇒ Gal(E/K)max{w,ϕF/K(z)} = 1.

Proof. (i) For any −1 ≤ v ∈ R, (G/H )v = GvH/H = 1, if Gv = 1. v (ii) For any −1 ≤ v ∈ R,ifG =GψE/K(v) = 1, then HψE/K(v) = H ∩ GψE/K(v) = 1, H = H = H ψF/K(v) = and therefore ψE/K(v) ψE/F (ψF/K(v)) 1. w w w (iii) For any −1 ≤ w ∈ R,if(G/H ) = G H/H = 1, then G = GψE/K(w) ⊆ H .For z −1≤z∈R,H =HψE/F (z) = 1. Thus, by Lemma 5.1, Gmax{ψE/K(w),ψE/F (z)} = 1. Now, note that

ϕE/K(max{ψE/K(w), ψE/F (z)}) = max{ϕE/K ◦ ψE/K(w), ϕE/K ◦ ψE/F (z)} = max{w, ϕF/K ◦ϕE/F ◦ ψE/F (z)} = max{w, ϕF/K(z)}

by the transitivity of the Hasse–Herbrand functions and by the fact that Hasse– Herbrand functions are monotone increasing. Thus, Gmax{w,ϕF/K(z)} = 1. ¤ Local Artin conductor – II 123

From now on, assume that E is a ﬁnite 2-abelian extension over K. Let E0 = K ⊆ E1 ⊆ E2 = E be the ‘canonical chain of sub-extensions in E/K’ as introduced in the 0 beginning of §4. Hence, G = Gal(E/K) and G = H = Gal(E/E1) with F = E1.Asan immediate consequence of Proposition 5.2, we have the following: COROLLARY5.3 For any −1 ≤ v ∈ R, v Gal(E/K) = 1 H⇒ v max{ordpK (f(E1/K)) − 1,

ϕE /K (ordpE (f(E/E )) − 1)}. 1 1 1 Proof. By Proposition 5.2 parts (i) and (ii), for any −1 ≤ v ∈ R,ifGv = 1, then v ψE /K (v) (G/H ) = H 1 = v (f(E /K)) − ψ (v) 1 and 1. Therefore, ordpK 1 1 and E1/K ordpE (f(E/E )) − 1 by Remark 2.3 and by the deﬁnition of the conductor of an abelian 1 1 extension. Thus

v max{ordpK (f(E /K)) − 1,ϕE /K (ordpE (f(E/E )) − 1)}, 1 1 1 1 which completes the proof. ¤ Recall that, the conductor f(E/K) (in the sense of ‘non-abelian’ local class field theory) [[nG]]+1 is defined in §2 (look at Definition 2.4) to be the ideal pK in OK , where nG is the real number defined by x+δ ∀ nG = inf{x ∈ R≥−1 : G = 1, δ>0}. By Remark 2.5, it follows that, the number nG is a jump in the ramification filtration of E/K in upper numbering, and 0 ≤ [[nG]] + 1 ∈ Z is the minimal integer satisfying G[[nG]]+1 = 1. As E/K is in particular a finite 2-abelian extension, it follows from Koch– de Shalit theory (look at Theorem 4.1(v)) that, either nG = 0or i qK −1−j nG =i− =ui,j i i−1 qK −qK i−1 i for some 1 ≤ i ∈ Z and j ∈{qK ,...,qK −1}. Remark 5.4. Observe that, for 0

i(r)−1 i(r)−1 i(r) j(r)−qK +1 Then, there exists a unique j(r) ∈{qK ,...,qK − 1} such that r ∈ Ir . That is,  1 i(r)−1 Ir = (u i(r)−1 ,u i(r)−1], if j(r) = qK ;  i(r)−1,qK −1 i(r),qK r ∈  i(r)−1  j(r)−qK +1 i(r)−1 i(r) Ir = (ui(r),j (r)−1,ui(r),j (r)], if qK j(r)≤ qK −1.

¤ 124 Kaˆzim I˙lhan I˙keda

Following Remark 5.4, there is the following reformulation of Deﬁnition 2.4 for ﬁnite metabelian extensions.

i−1 i Lemma 5.5. Let (i0,j0)be the minimal pair in the set {(i, j) ∈ Z×Z : qK ≤ j ≤ qK −1} endowed with lexicographic ordering (induced from that of Z × Z) with the property

[2] (i,j) G (K, φ) ⊆ Mφ(E/K).

(Note that, the pair (i0,j0) does not depend on the choice of the Lubin–Tate splitting φ over K.) Then,

 i − i u , q 0 1 j ≤ q 0 − ,  i0,j0−1 if K 0 K 1 nG =  i −1 u i − , q 0 = j . 0 1 if K 0 i0−1,qK −1 The conductor f(E/K) of the ﬁnite metabelian extension E/K (in the sense of metabelian local class ﬁeld theory) is then given by the ideal

i0 f(E/K) = pK

in OK .

i−1 Proof. In fact, let (i0,j0)be the minimal pair in the set {(i, j) ∈ Z × Z : qK ≤ j ≤ i qK −1} endowed with lexicographic ordering (induced from that of Z×Z) with the property

[2] (i,j) G (K, φ) ⊆ Mφ(E/K).

Then, by metabelian local class ﬁeld theory (cf. part (v) of Theorem 4.1), Gr = 1 for ui ,j − i −1 i i −1 u r G 0 0 1 6= q 0 j ≤ q 0 − j = q 0 i0,j0−1 , while 1, whenever K 0 K 1. In case 0 K , u i −1 r i −1,q 0 −1 G = u i − r G 0 K 6= then 1 for 0 1 and 1. Therefore, i0−1,qK −1

 i − i u , q 0 1 j ≤ q 0 − ,  i0,j0−1 if K 0 K 1 nG =  i −1 u i − , q 0 = j . 0 1 if K 0 i0−1,qK −1 Now, following Remark 5.4,  i0−1 i0 i0, if qK j0 ≤ qK − 1, i(n ) = G   i0−1 i0 − 1, if qK = j0,

where nG i(nG) in the ﬁrst case, while nG = i(nG) in the second. Therefore, [[nG]] = i i0 − 1, proving that [[nG]] + 1 = i0 is the minimal integer satisfying G 0 = 1. Thus,

i0 f(E/K) = pK .

Note that, by Remark 4.4, (i0,j0) (and thereby nG, [[nG]],...) does not depend on the choice of the Lubin–Tate splitting φ over K. ¤ Local Artin conductor – II 125

i Observe that, if the pair (i0,j0)is chosen as in Lemma 5.5, then by Lemma 4.3, U 0 (K) ⊆ i N(E1/K). Therefore, Gal(E1/K) 0 = 1, which is a special case of Proposition 5.2(i). Now, assume that E/K is a ‘strict’2-abelian extension with Galois group G as introduced in the beginning of §4. Let G• be a maximal abelian normal subgroup of G with G0 ⊆ G•. • n As further notation, let E• = EG and EnG = EG G .

= E

abelian ext. E•EnG t KK tt KK tt KK tt KKK ØtØt K E• EnG . VJVJ JJ sss JJ ss JJ ss J sss E• ∩ EnG abelian ext. ! K • n • n • n Note that, E E G ( E. In fact, if E E G = E, then the extensions E (E ∩ E G ) and (E• ∩ EnG ) K are abelian, and therefore Gal(E (E• ∩ EnG )) is an abelian nor- 0 • • • mal subgroup of G containing G and G . Now, the maximality of G yields that G = Gal(E (E• ∩ EnG )). Hence, E• ∩ EnG = E• and so E• ⊆ EnG . This leads to a contradiction, since E = EnG and GnG = 1. Therefore, E•EnG ( E. This result also proves that Gal(E/E•EnG ) is a non-trivial group. So, G• ∩ GnG is a non-trivial intersection.

PROPOSITION5.6 Keeping the above notation,

ψE•/K (nG) = nG• , where nG• ∈ Z.

Proof. Note that

(G•)µ = (G•) = G• ∩ G ψE/E•(µ) ψE/E•(µ) = G• ∩ GϕE/K(ψE/E•(µ)) = G• ∩ GϕE•/K (µ).

• • • µ• • n Now, let µ ∈ R≥− such that ϕE•/K (µ ) = nG. Then, (G ) = G ∩ G G 6= 1 and 1 • • µ +δ • nG+δ for any 0 <δ0 ∈R, there exists 0 <δ∈Rsuch that (G ) 0 = G ∩ G = 1, • • proving that µ = ψE•/K (nG) = nG• . Finally, since G is abelian, by Hasse–Arf theorem, nG• ∈ Z. ¤

ker(ρ) 6. Computation of Artin(χρ) in case E /K is a metabelian extension Main references for this section are [1] and [2]. 126 Kaˆzim I˙lhan I˙keda

Now, we are in a position to prove the main theorem (Theorem 1.2). As before, let E/K be a ﬁnite Galois extension, and ρ : Gal(E/K) → GL(V ) be an irreducible representation of the Galois group Gal(E/K) =: G of the extension E/K. Note that, the representation ∼ ρe: Gal(Eker(ρ)/K) −−→ G/ker(ρ) → GL(V )

is an irreducible faithful representation of Gal(Eker(ρ)/K).Now,ifeg:G/ker(ρ) → C is any class function on G/ker(ρ), and g is the corresponding class function on G deﬁned by

canonical map eg g : G −−−−−−−−−→ G/ker(ρ) −−→ C,

then fArtin(eg) = fArtin(g). In particular, for g = χρ,wehaveeg=χρe. Therefore,

fArtin(χρ) = fArtin(χρe).

Hence, in order to compute explicitly the local Artin conductor fArtin(χρ) of an irreducible representation ρ : G → GL(V ) in the case Eker(ρ)/K is a metabelian extension, we have to study irreducible faithful representations of finite metabelian groups. This is the content of the following digression. Digression: Representation theory of finite metabelian groups. The main reference for this part is [1]. Theorem. (Basmaji). Let M be a finite metabelian group, and assume that M has an irreducible faithful representation. Let M• denote a maximal abelian normal subgroup of M containing M0. Then, (i) all irreducible faithful representations ρ : M → GL(V ) have the same degree d = (M : M•). Let ρ : M → GL(V ) be an irreducible faithful representation of M in V (over C). Then, M M (ii) ρ : M → GL(V ) is equivalent to the representation IndM• ω : M → GL(IndM• Vω) • × • induced from a one-dimensional representation ω : M → C of M in C = Vω. Of course, the assumption on the existence of faithful irreducible representations of M seems to be a strong assumption, but there are many metabelian groups with this property. In fact, if M is metacyclic, then M has faithful irreducible representations (over C)iff Z(M) is a cyclic group. ¤ Now, for simplicity, fix the following notation: let F = Eker(ρ), M = Gal(F/K), and ab H = Gal(F/F1) where F1 = F ∩ K , the maximal abelian sub-extension in F over K. Since M is assumed to be a metabelian group, H = M0. Consider the irreducible faithful representation ρe : M → GL(V ). Then, there exists a one-dimensional representation ω : M• → C×, such that M ρe' IndM• ω, where M• is a maximal abelian normal subgroup of M containing M0. Set M• = Gal(F/F •), where F •/K is a minimal abelian sub-extension of F/K. Thus, by Frobenius reciprocity law

f (χρe) = (χρe,aM)M =(χ M ,aM)M Artin IndM• ω M =(IndM• χω,aM)M =(χω, ResM• (aM ))M• , Local Artin conductor – II 127

and by eq. (7),

• • fArtin(χρe) = (χω,ννK(d(F /K))rM• + f(F /K)aM• )M• • • = ν K (d(F /K)) + f(F /K)fArtin(χω),

where d(F •/K) is the discriminant and f(F•/K) the residue-class degree of the extension • • • • • F /K. Note that, ν K (d(F /K)) = f(F /K)ννF • (D(F /K)), where D(F /K) denotes the different of the extension F •/K. Thus,

• • fArtin(χρe) = f(F /K)[ν F • (D(F /K)) + fArtin(χω)]. (12)

It is well-known that (for instance, see §2 of chapter III of [9]), X∞ • • ν F • (D(F /K)) = (#(M/M )i − 1). i=0 Thus, combining with eq. (12) " # X∞ • • fArtin(χρe) = f(F /K) (#(M/M )i − 1) + fArtin(χω) . i=0 By Corollary 3.3 and Proposition 5.6,

• f (χ ) = (f(F/F )) = n • + = ψ • (n ) + , Artin ω ordpF • M 1 F /K M 1 P M • × since ResM• (ρ) = ResM• IndM• ω = δ∈R(M•\M) ωδ (where ωδ : M → C is the • − • one-dimensional representation of M deﬁned by ωδ : x 7→ ω(δxδ 1) for x ∈ M and for δ ∈ R(M•\M) a ﬁxed complete system of representatives of M•\M) is an injection. (M/M•)v = (M/M•) = v n Moreover, ψF •/K (v) 1if M by Proposition 5.2(i). Therefore (note that, ψF •/K (nM ) ∈ Z by Proposition 5.6),  ψF •X/K (nM ) •  • fArtin(χρe) = f(F /K) #(M/M )i i=0   −(ψF •/K (nM ) + 1) + fArtin(χω)

ψF •X/K (nM ) • • = f(F /K) #(M/M )i. i=0 Now, by eq. (5), • • fArtin(χρe) = f(F /K)#(M/M )0 ϕF •/K ψF •/K (nM ) + 1 • = [F : K](nM + 1),

• • since (M/M )0 is the inertia group of M/M . Thus, we have the desired result, namely

• [F :K](nM +1) fArtin(χρ) = pK , 128 Kaˆzim I˙lhan I˙keda

which, combining with the results of §5 and Corollary 3.5, completes the proof of Theo- rem 1.2.3

Remark 6.1. Note that, by Corollary 3.3, fArtin(χρ) ∈ Z≥0 for any given representation ρ : Gal(E/K) → GL(V ). Thus, following the notation of Theorem 1.2, the exponent • Eker(ρ) K (n + ) f (χ ) [ : ] Gal(Eker(ρ)/K) 1 of the Artin conductor Artin ρ of the representation (ρ, V ) of Gal(E/K) is a non-negative integer. Therefore, setting

num(n ker(ρ) ) n = Gal(E /K) , Gal(Eker(ρ)/K) (n ) den Gal(Eker(ρ)/K) (n ), (n ) ∈ Z < (n ) where num Gal(Eker(ρ)/K) den Gal(Eker(ρ)/K) ,0 den Gal(Eker(ρ)/K) , and are (n ) | Eker(ρ)• K relatively prime, it then follows that den Gal(Eker(ρ)/K) [ : ]. That is, the n (V ) ¤ denominator of Gal(Eker(ρ)/K) divides dimC .

7. Artin representation AGal(E/K) of Gal(E/K) (continuation) Main reference for this section is [11]. From now on, unless otherwise stated (as in Notation 7.1, in the following digression and in Remark 7.2), we assume that E/K is a ﬁnite metabelian extension of the local ﬁeld K with Galois group Gal(E/K) = G. In this section, we prove Theorem 1.4. That is,

we construct a ‘natural’Artin representation AG : G → GL(VAG ) of G over C, utilizing Theorem 1.2, which settles Problem 1.3 in the case G is a metabelian Galois group.

Notation 7.1. For a ﬁnite group G, let B(G) ={χ}denote the set of all irreducible characters of G, which is a ﬁnite set of r(G) elements, where r(G) denotes the number of conjugacy classes in G. If there is no fear of confusion, we let B(G) = B. Let ρχ be

an irreducible representation of G corresponding to χ (so in our notation, χ = χρχ ), and Vρχ =: Vχ the representation space of G corresponding to the representation ρχ of G over C, for χ ∈ B. ¤

The irreducible representation ρχ : G → GL(Vχ ) of G has metabelian image for χ ∈ B (ρ ) since G is assumed to be metabelian. Let Eχ = Eker χ , Mχ = G/ker(ρχ ) = Gal(Eχ /K) • which is a metabelian group, and let Mχ denotes any maximal abelian normal subgroup 0 of Mχ containing the 1st commutator subgroup Mχ of Mχ for χ ∈ B. Then, the triangle

ρχ G / GL(Vχ ) @@ v; @@ vv @@ vv πχ @ vv @ vv ρeχ Mχ

3Let π denote a prime element of O nr , where Knr denotes the unique unramified extension of degree d over Kd d i (i,j) K. Consider the abelian extension Kd,i of K which is class field to hN nr (π)iU (K), and let K be the Kd /K d j abelian extension of Kd,i which is class field to U (Kd,i). Note that, if E/K is a finite metabelian extension, i (i,j) then there exists an i ∈ Z≥0 and an integer 0 ≤ j

Mχ ρeχ ' Ind • ωχ , Mχ

• × • where ωχ : Mχ → C is a one-dimensional representation of Mχ satisfying

• Inert(ωχ ) ={δ∈Mχ :ωχ,δ = ωχ}=Mχ, by Mackey irreducibility criterion and \ (ω ) =h •i, ker χ,δ 1Mχ • δ∈R(Mχ \Mχ )

Mχ T by the fact that ker(Ind • ωχ ) = • ker(ωχ,δ). Mχ δ∈R(Mχ \Mχ ) Digression: On induced representations. Here, all through this digression, G denotes an arbitrary group, and N a normal subgroup of G. For simplicity, we set G := G/N.

(i) Let ξ : N → GL(Vξ ) be any finite-dimensional representation of N over C. Recall G G that, the G-module IndN Vξ is defined by IndN Vξ ={f :G→Vξ :f(τx) = G ξ(τ)f(x),∀τ ∈ N,∀x ∈ G}, and the G-action on IndN Vξ by (σ, f ) 7→ σ (f ), where G σ (f )(x) := f(xσ)for every σ, x ∈ G and f ∈ IndN Vξ . Now, assume that (G : N) < ∞. Let R(N\G) ={δ1 =1G,...,δm}be a complete system of representatives of N\G, and B ={ε1,...,εn}a basis of Vξ over C. Note G that, any f ∈ IndN Vξ is completely determined by the values f(δ1),...,f(δm)in Vξ . In fact, for x ∈ G, there exists a unique τ ∈ N and a unique δ ∈ R(N\G) satisfying x = τδ. Thus, f(x) = f(τδ) = ξ(τ)f(δ), proving that f is completely determined by the values f(δ1),...,f(δm). For 1 ≤ i ≤ m and 1 ≤ j ≤ n, define G fij ∈ IndN Vξ by

fij (δi) = 0; ...;fij (δi) = εj ; ...;fij (δm) = 0.

G Then, {fij :1≤i≤m;1≤j≤n}is a basis of IndN Vξ over C. So, G dimC(IndN Vξ ) = mn = (G : N)dimCVξ .

(ii) Assume that (G : N) < ∞. Then, for any ﬁnite-dimensional representation ξ : N → GL(Vξ ) of N over C, \ G ker IndN ξ = ker(ξs). s∈R(N\G)

G In fact, if σ ∈ G such that (Ind ξ)(σ ) = σ = 1 G , then σ(f ) = f for every N IndN Vξ G f ∈ IndN Vξ . Hence, in particular, σ(fij ) = fij for every 1 ≤ i ≤ (G : N) =: m and 1 ≤ j ≤ dimCVξ =: n. Therefore, in view of part (i), σ(fij )(δk) = fij (δk) for every 1 ≤ k ≤ m.Now,

σ(fij )(δk) = fij (δkσ), 130 Kaˆzim I˙lhan I˙keda

and setting δkσ = σkδk0 for σk ∈ N and δk0 ∈ R(N\G) (both of them unique!),

σ (fij )(δk) = fij (δkσ) = fij (σkδk0 ) ( 0 ξ(σk)(εj ), if k = i, = ξ(σ )f (δ 0 ) = k ij k 0 ξ(σk)(0) = 0, if k 6= i.

On the other hand, ( εj , if k = i, fij (δk) = 0, if k 6= i.

i0 = i ξ(σ )(ε ) = ε δ σδ−1 = σ ξ(δ σδ−1)(ε ) = ε Thus, and i j j . Hence, i i T i, and i i j j . ξ (σ ) = ξ(δ σδ−1) = σ ∈ (ξ ) Thus, δi i i 1Vξ , proving that 1≤i≤m ker δi . For the reverse −1 inclusion, if σ ∈ ker(ωδi ) for every 1 ≤ i ≤ m, then ω(δiσδi )(εj ) = εj for every −1 1 ≤ i ≤ m and 1 ≤ j ≤ n. Let σi = δiσδi , for 1 ≤ i ≤ m. Then, δiσ = σiδi and

σ (fij )(δi) = fij (δiσ) = fij (σiδi) = ξ(σi)εj = εj = fij (δi),

and for k 6= i,

σ (fij )(δk) = fij (δkσ) = fij (σkδk) = ξ(σk)0 = 0 = fij (δk),

for every 1 ≤ i 6= k ≤ m and 1 ≤ j ≤ n. So, σ(fij ) = fij for every i and j, proving G that σ ∈ ker(IndN ξ). (iii) Now, consider a representation ρ¯ : G¯ → GL(V ) of G¯ in V over C. Then, there exists a representation ρ : G → GL(V ) of G in V over C, which is deﬁned by π ρ¯ ρ : G −−→ G¯ −−→ GL(V ), where π : G → G¯ denotes the canonical map (so N ⊆ ker(ρ)). If ρ¯ ' G¯ ω IndH¯ for some normal subgroup H¯ of G¯ satisfying (G¯ : H)¯ < ∞and for some represen- ¯ ¯ tation ω : H → GL(Vω) of H in Vω over C, then ρ ' G (ω ◦ π | ). Indπ−1(H)¯ π−1(H)¯

G¯ G ( ω) ◦ π ' (ω ◦ π | − ) In fact, it suffices to prove that IndH¯ Indπ−1(H)¯ π 1(H)¯ . Note that, π G → G¯ π G¯ V −−→∼ the canonical map : defines a linear isomorphism ∗ : IndH¯ ω G G¯ Vω◦π| π∗ f 7→ f ◦π f ∈ Vω ={f G¯ → Indπ−1(H)¯ π−1(H)¯ defined by : for every IndH¯ : ¯ ¯ Vω :f(τx) = ω(τ)f(x),∀τ ∈ H,∀x ∈ G}. Note that, surjectivity follows from the assumption that (G¯ : H)¯ < ∞and by observing that (G : π −1(H))¯ = (G¯ : H)¯ . Moreover, π ◦ ( G¯ ω) ◦ π (g) = G (ω ◦ π | ) (g) ◦ π ∗ [ IndH¯ ] [Indπ−1(H)¯ π−1(H)¯ ] ∗

g ∈ G f ∈ G¯ V for every . In fact, choose IndH¯ ω. Then, computing the left-hand side {π ◦ ( G¯ ω) ◦ π (g)}(f ) ={( G¯ ω) ◦ π (g)(f )}◦π, ∗ [ IndH¯ ] [IndH¯ ] Local Artin conductor – II 131

and therefore { ( G¯ ω) ◦ π (g)(f ) ◦ π}(x) = ( G¯ ω) ◦ π (g)(f )(π(x)) [ IndH¯ ] [ IndH¯ ] = f(π(x)π(g))

for every x ∈ G. Now, computing the right-hand side { G (ω ◦ π | ) (g) ◦ π }(f ) [Indπ−1(H)¯ π−1(H)¯ ] ∗ = G (ω ◦ π | ) (g)(f ◦ π), [Indπ−1(H)¯ π−1(H)¯ ]

and therefore { G (ω ◦ π | ) (g)(f ◦ π)}(x) = (f ◦ π)(xg) [Indπ−1(H)¯ π−1(H)¯ ] = f(π(x)π(g))

for every x ∈ G. Thus

{π ◦ ( G¯ ω) ◦ π (g)}(f ) ∗ [ IndH¯ ] ={ G (ω ◦ π | ) (g) ◦ π }(f ) [Indπ−1(H)¯ π−1(H)¯ ] ∗

f ∈ G¯ V for every IndH¯ ω. That is

( G¯ ω) ◦ π ' G (ω ◦ π | ), IndH¯ Indπ−1(H)¯ π−1(H)¯

which completes the proof. ¤

Hence, ρ = ρe ◦ π ' G (ω ◦ π | ) χ χ χ Ind −1 • χ χ π−1(M• ) (13) πχ (Mχ ) χ χ for χ ∈ B.

Remark 7.2. For an arbitrary group G, let Irr(G) (resp. Irr(G, N) for a normal subgroup N of G) denote the set of all irreducible ﬁnite-dimensional representations of G over C (resp. the set of all irreducible ﬁnite-dimensional representations of G over C with kernel N). Now, let I(G) denote the set [Irr(G)/']of'-equivalence classes [ρ]ofρ∈ Irr(G). Therefore, I(G) ={[ρχ]:χ∈B(G)}. Observe that, for ρ,ψ ∈ Irr(G),if ρ'ψ, then ker(ρ) = ker(ψ). Let ker([ρ]) = ker(ρ). For a normal subgroup N of G, let I(G, N) = [Irr(G, N)/']. So, the set I(G, N) consists of those [ρ]inI(G) with kernel N. In particular, I(G, h1Gi) is the set of '-equivalence classes [ρ]ofρ∈Irr(G, h1Gi) (= the set of all irreducible faithful representations of G over C). Note that G Irr(G) = Irr(G, N), (14) N and G I(G) = I(G, N), (15) N 132 Kaˆzim I˙lhan I˙keda where N runs over all normal subgroups of G. Note that, it may turn out that Irr(G, N) = ∅ and I(G, N) = ∅ for some N G G. Observe that, for any N G G, the canonical map πN : G → G/N induces a bijection

πN,∗ : I(G/N, h1G/N i) → I(G, N) deﬁned by

πN,∗ :[ρe]7→ [ρe◦ πN ] for every ρe∈ I(G/N, h1G/N i). ¤

For N G G, the quotient group G/N is a metabelian group. Let (G/N)• denote a (ﬁxed) maximal abelian normal subgroup of G/N containing the 1st commutator subgroup (G/N)0 of G/N. Then, since irreducible representations of an abelian group are all one- dimensional, • • × Irr((G/N) ) ={ω:(G/N) → C }, and • • × I((G/N) ) ={ω:(G/N) → C },

• as for one-dimensional representations (ω1, C) and (ω2, C) of (G/N) , ω1 ' ω2 if and • 0 • only if ω1 = ω2. Now, consider the subset I((G/N) ) of I((G/N) ) deﬁned by • • × • I((G/N) )0 = ω : (G/N) → C : Inert(ω) = (G/N) ,  \  (ω ) =h i , ker δ 1G/N  δ∈R(M•\M) where M denotes here the quotient group G/N and R(M•\M)denotes a complete system • • • × of representatives of M \M.Forδ∈R(M \M), the representation ωδ : M → C of • − • M is deﬁned by ωδ : x 7→ ω(δxδ 1) for x ∈ M .

Lemma 7.3. The mapping

G/N • 0 Ind(G/N)• : I((G/N) ) → I(G/N, h1G/N i) (16) deﬁned by h i G/N G/N Ind(G/N)• : ω 7→ Ind(G/N)• ω , for every ω ∈ I((G/N)•)0 is a surjection. Deﬁne an equivalence relation ‘∼’on • 0 • I((G/N) ) by ω1 ∼ ω2 if and only if ω1 = ω2,δ for some δ ∈ R(M \M), where M = G/N. Then, • 0 (i) for ω1,ω2 ∈I((G/N) ) , G/N G/N Ind(G/N)• ω1 ' Ind(G/N)• ω2 ⇐⇒ ω 1 ∼ ω 2 , Local Artin conductor – II 133

(ii) the surjection eq. (16) induces a well-deﬁned bijection

• 0 [I((G/N) ) / ∼] =: VN → I(G/N, h1G/N i)

deﬁned by h i G/N [ω]∼ 7→ Ind(G/N)• ω ,

• where [ω]∼ denotes the ∼-equivalence class of ω ∈ I((G/N) )0.

Proof. Note that, the surjectivity of this map follows immediately from Basmaji’s theory. It is also straightforward to observe that ‘∼’ is an equivalence relation on I((G/N)•)0.

(i) The proof of this part follows from the Frobenius reciprocity law. In fact, for ω1,ω2 ∈ I((G/N)•)0, note that G/N G/N Ind(G/N)• ω1 ' Ind(G/N)• ω2, if and only if G/N G/N χ = χ G/N = χ G/N = χ . Ind(G/N)• ω1 Ind(G/N)• ω2 (17) Ind(G/N)• ω1 Ind(G/N)• ω2

G/N G/N Since both Ind(G/N)• ω1 and Ind(G/N)• ω2 are irreducible representations of G/N,by Mackey’s irreducibility criterion, eq. (17) is equivalent to G/N G/N Ind • χω , Ind • χω = 1. (G/N) 1 (G/N) 2 G/N Now, by the Frobenius reciprocity law, eq. (17) is equivalent to G/N = χω , (G/N)• • χω 1 1 Res Ind(G/N) 2 • (G/N)

= χ ,χ G/N ω1 • ω Res(G/N) Ind(G/N)• 2 • ! (G/N) X P = χω ,χ = (χω ,χω ) • , 1 ω2,δ 1 2,δ (G/N) • δ∈R(M \M) (G/N)• δ∈R(M•\M)

(χ ,χ ) • = ω = ω which is the case if and only if ω1 ω2,δ (G/N) 1, that is, 1 2,δ, for some δ ∈ R(M•\M), completing the proof. (ii) The proof follows immediately from part (i). ¤ So, there exists the following chain of bijections

G/N Ind • π • 0 (G/N) N,∗ [I((G/N) ) / ∼] = VN −−−−−−−→ I (G/N, h1G/N i) −−−−→ I (G, N) (18) deﬁned by G/N G/N [ω]∼ 7→ [Ind(G/N)• ω] 7→ [Ind(G/N)• ω ◦ πN ] = G (ω ◦ π | ) , [Ind −1 • N π−1((G/N)•) ] (19) πN ((G/N) ) N 134 Kaˆzim I˙lhan I˙keda

• for [ω]∼ ∈ [I((G/N) )0/ ∼]. Note that, the equality in the right-hand side follows from eq. (13). Now, we are in a position to prove Theorem 1.4. An Artin representation AG of G over C can be deﬁned generally by X AG = ordpK (fArtin(χρ))ρ. [ρ]∈I(G)

So, by eq. (15), X X AG = ordpK (fArtin(χρ))ρ, (20) N [ρ]∈I(G,N) where N runs over all normal subgroups of G. Thus, combining eq. (20) with Theorem 1.2, X X N • AG = [(E ) : K](nG/N + 1)ρ N ρ ∈I(G,N) X [ ] X N • = [(E ) : K](nG/N + 1) ρ, N [ρ]∈I(G,N) where N runs over all normal subgroups of G. Observe that X X ρ ' G (ω ◦ π | ), Ind −1 • N π−1((G/N)•) (21) πN ((G/N) ) N [ρ]∈I(G,N) [ω]∼∈VN

• where VN = [I((G/N) )0/ ∼] for every N G G, by eqs. (18) and (19). Hence, X N • AG ' [(E ) : K](nG/N + 1) N X × G (ω ◦ π | ), Ind −1 • N π−1((G/N)•) πN ((G/N) ) N [ω]∼∈VN where N runs over all normal subgroups of G, which completes the proof of Theorem 1.4. The following is then an easy consequence of Theorem 1.4.

COROLLARY7.4 (DimensionformulaforAG) Let E/K be a ﬁnite metabelian extension with Galois group G. Then X • ker(ρ) 2 dimC(AG) = [E : K] (nG/ker(ρ) + 1). [ρ]∈I(G)

Proof. Note that, by eqs. (21) and (22), X X N • dimC(AG) = [(E ) : K](nG/N + 1) dimC(ρ) N ρ ∈I(G,N) X [ ] N • = [(E ) : K](nG/N + 1) N X × ( G (ω ◦ π | )), dimC Ind −1 • N π−1((G/N)•) πN ((G/N) ) N [ω]∼∈VN Local Artin conductor – II 135

where N runs over all normal subgroups of G. Now, by eq. (13), for any ω ∈ [ω]∼ ∈ VN , ( G (ω ◦ π | )) = ( G/N ω) dimC Ind −1 • N π−1((G/N)•) dimC Ind(G/N)• πN ((G/N) ) N • = (G/N : (G/N) ) N • = [(E ) : K].

Thus, X X N • N • dimC(AG) = [(E ) : K](nG/N + 1) [(E ) : K] N ω ∈V X X [ ]∼ N N • 2 = [(E ) : K] (nG/N + 1) N ω ∈V X [ ]∼XN N • 2 = [(E ) : K] (nG/N + 1) N ρ ∈I(G,N) X[ ] ker(ρ) • 2 = [(E ) : K] (nG/ker(ρ) + 1), [ρ]∈I(G) by eqs. (15), (18) and (19), which completes the proof. ¤

8. Conclusion We conclude our paper with the following set of remarks. Remark 8.1. (On problem 1.1). Note that, the general solution of Problem 1.1 of describing the local Artin conductor fArtin(χρ) of the character χρ : Gal(E/K) → C associated to an irreducible representation (ρ, V ) of Gal(E/K) requires the construction of non- abelian local class field theory over K al´` a Koch–de Shalit (that is, ‘n-abelian’ local class field theories over K for each n ≥ 1), and a detailed study of finite-dimensional complex representations of finite solvable groups (i.e., generalized Basmaji theory)4. In fact, if ρ : Gal(E/K) → GL(V ) is any irreducible representation of Gal(E/K) =: G (over C), then by Proposition 3.2,

dimC(V )[nG/ker(ρ)+1] fArtin(χρ) = pK , where G/ker(ρ) is an n-abelian group, for some 1 ≤ n ∈ Z. Thus, assuming that we have generalized Koch–de Shalit theory for Galois extensions over K with corresponding Galois groups having degrees of solvability n, where 3 ≤ n ∈ Z (for instance, for 3-abelian local class field theory, look at [5]), the number nG/ker(ρ) can be located and computed by this n-abelian local class field theory. Also, assuming that we have generalized Basmaji’s theory5, the dimension of the representation space V can be computed in terms of the extension E/K. Therefore, following the same lines of this paper, we would have solution to Problem 1.1. ¤ 4The study of representation theory of finite solvable groups over C seems to be closely related with the following theorem of Dade: if S is any solvable group, then there exists an M-group T , such that S can be embedded into T . 5The following well-known theorem (cf. [11]) may be a good starting point to investigate finite-dimensional representations of finite n-abelian groups (over C). Let G be a finite n-abelian group. Let ρ : G → GL(V ) be an irreducible representation of G (over C). Then, either (i) there exists a proper subgroup H of G containing G0, G and an irreducible representation σ : H → GL(Vσ ) such that ρ ' IndH σ; or (ii) the representation ResG0 (ρ) of G0 is isotypic. 136 Kaˆzim I˙lhan I˙keda

Remark8.2 (OnWeil’sproblem).LetE/KbeafiniteGaloisextensionwithGalois group G. (i) In the direction of Weil’s problem, the first important result is obtained by Serre (cf. ` [13]), where he proved that there exists a representation AG of G over Q` (here ` is a prime number satisfying ` 6= qK = char(κK )) whose character is the Artin character aG of G. (ii) Recall, from §7, that an Artin representation AG of G over C can be defined by X AG = ordpK (fArtin(χρ))ρ. [ρ]∈I(G)

Now, by eq. (15), X X AG = ordpK (fArtin(χρ))ρ, N [ρ]∈I(G,N)

where N runs over all normal subgroups of G. Let d denote the derived length of the solvable group G. Any subgroup of G is solvable and of derived length bounded from above by d. Now, partition the set {N} of all normal subgroups of G as G G G {N}=N1 N2 ··· Nd,

where Ni denotes the set of all normal subgroups of G of derived length i for 1 ≤ i ≤ d. Thus, X X X AG = ordpK (fArtin(χρ))ρ 1≤i≤d N∈Ni [ρ]∈I(G,N) [1] [2] [d] = AG + AG +···+AG ,

[i] where AG is the representation of G deﬁned by X X [i] AG = ordpK (fArtin(χρ))ρ N∈Ni [ρ]∈I(G,N)

for 1 ≤ i ≤ d. [1] [2] Note that, we can compute AG and AG explicitly following Theorem 1.4. However, to [i] compute AG for 2 i ≤ d, we need the full solution of Problem 1.1 as well as generalized Basmaji’s theory. ¤

Acknowledgement This work has been started at Institut de Mathématiques, ‘Théorie des Groupes, Représentationset Applications’, UniversitéParis 7, and completed at the School of Mathematics of Tata Institute of Fundamental Research, while the author was holding a TIFR-Visiting Fellowship. The author wishes to express his gratitude to both of the institutions for their hospitality and support. Local Artin conductor – II 137

References [1] Basmaji B G, Monomial representations and metabelian groups, Nagoya Math. J. 35 (1969) 99–107 [2] Ikeda K I, On the Artin conductor fArtin(χ) of a character χ of Gal(E/K) I: Basic definitions, Tr. J. Math. 23 (1999) 519–530 [3] Ikeda K I, On the metabelian local Artin map I: Galois conjugation law, Tr. J. Math. 24 (2000) 25–58 [4] Ikeda K I, A note on totally-ramified, n-abelian extensions of a local field, Acta Math. Vietnam (submitted) [5] Ikeda K I, Non-abelian local class field theory I: Construction of 3-abelian local class field theory (preprint) [6] Ikeda K I and Ikeda M G, Two lemmas on formal power series, Tr. J. Math. 23 (1999) 435–440 [7] Iwasawa K, Local class field theory (1986) (Oxford, NewYork:Oxford University Press) [8] Koch H and de Shalit E, Metabelian local class field theory, J. Reine Angew. Math. 478 (1996) 85–106 [9] Neukirch J, Algebraic number theory (1999) (Berlin, Heidelberg, New York: Springer- Verlag) [10] Sen S, Ramification in p-adic Lie extensions, Invent. Math. 17 (1972) 44–50 [11] Serre J-P, Représentationslinéairesdes groupes finis (1971) (Paris: Hermann) [12] Serre J-P, Corps locaux (Paris: Hermann) (1962) [12e] Serre J-P, Local fields (English translation of [12] by M J Greenberg) GTM (1979) (Berlin, Heidelberg, New York: Springer-Verlag) vol. 67 [13] Serre J-P, Sur la rationalitédes représentationsd’Artin, Ann. Math. 72 (1960) 405–420 [14] Tate J, Number theoretic background in: Automorphic Forms, Representations, L-functions. Proc. Symp. Pure Math., Am. Math. Soc. (1979) (Providence, Rhode Island, USA) vol. 33, pp. 3–26

On the Local Artin Conductor F &lt;Subscript&gt;Artin &lt;/Subscript

On the Local Artin Conductor F <Subscript>Artin </Subscript