An Extension of Deligne-Henniart's Twisting Formula and Its Applications

AN EXTENSION OF DELIGNE-HENNIART’S TWISTING FORMULA AND ITS APPLICATIONS

SAZZAD ALI BISWAS

Abstract. Let F/Qp be a non-Archimedean local ﬁeld, and GF be the absolute Galois group of F . Let ρ1 and ρ2 be two ﬁnite dimensional complex representations of GF . Let ψ be a nontrivial additive character of F . Then question is:

What is the twisting formula for the root number W (ρ1 ⊗ ρ2, ψ)?

In general, answer of this question is not known yet. But if one of ρi(i = 1, 2) is one- dimensional with “suﬃciently” large conductor, then in [13] Deligne gave a twisting formula

for W (ρ1 ⊗ ρ2, ψ). Later, in [12], Deligne and Henniart give a general twisting formula for a zero dimensional virtual representation twisted by a ﬁnite dimensional representation of

GF . In this paper, ﬁrst we extend Deligne’s twisting formula for Heisenberg representation of dimension prime p, then we further extend Deligne-Henniart’s result. Finally, we give two very important applications of our twisting formula – invariant formula of local root numbers for U-isotropic Heisenberg representations and a converse theorem in the Galois side.

1. Introduction

Let F be a non-Archimedean local field of characteristic zero, i.e., a finite extension of Qp, where p is a prime. Let GF be the absolute Galois representation of F . Let ψ be a nontrivial additive character of F . For a given multiplication character χ : F × → C× of F , we have explicit formula for the root number W (χ, ψ) (cf. [35]). If χ1 and χ2 are two unramified characters of F × and ψ is a nontrivial additive character of F , we have

(1.1) W (χ1χ2, ψ) = W (χ1, ψ)W (χ2, ψ).

Further, let χ1 be ramiﬁed and χ2 unramiﬁed then (cf. [35], (3.2.6.3))

a(χ1)+n(ψ) (1.2) W (χ1χ2, ψ) = χ2(πF ) · W (χ1, ψ).

Here a(χ1) (resp. n(ψ)) is the conductor of χ1 (resp. ψ). We also have twisting formula of local root numbers by Deligne (cf. [13], Lemma 4.16) under some special condition and which arXiv:2101.08118v1 [math.NT] 20 Jan 2021 is as follows (for proof, see Corollary 3.2 (2) of [5]): Let χ1 and χ2 be two multiplicative characters of a local ﬁeld F such that a(χ1) > 2 · a(χ2). × Let ψ be an additive character of F . Let yχ1,ψ be an element of F such that

χ1(1 + x) = ψ(yχ1,ψx)

a(χ1) −n(ψ) for all x ∈ F with valuation νF (x) > 2 (if a(χ1) = 0, yχ1,ψ = πF ). Then −1 (1.3) W (χ1χ2, ψ) = χ2 (yχ1,ψ) · W (χ1, ψ).

2010 Mathematics Subject Classiﬁcation. 11S37; 20C15, 11R39) Keywords: Local ﬁelds, Galois representations, local root numbers, Converse theorems. 1 2 BISWAS

For characters without any restriction, we also have general twisting formula for characters (cf. Theorem 3.5 on p. 592 of [3]). Moreover, if a ﬁnite dimensional Galois representation ρ is twisted by an unramiﬁed char- −sνF (x) acter ωs(x) := qF we have the following twisting formula (cf. [35] (3.4.5)):

(1.4) W (ρωs, ψ) = W (ρ, ψ) · ωs(cρ,ψ) for any c = cρ,ψ such that νF (c) = a(ρ) + n(ψ)dim(ρ), where a(ρ) is the Artin conductor of the representation ρ.

Let ρ1 and ρ2 be two arbitrary ﬁnite dimensional representations of GF . Now question is:

Is there any explicit formula for W (ρ1 ⊗ ρ2, ψ)?

The answer is: NO. But under some special conditions–when any of ρi(i = 1, 2) is one dimensional with suﬃciently large conductor, then Deligne gives an explicit formula for W (ρ1⊗

ρ2, ψ) (cf. [13], Subsection 4.1):

Let ρ1 = ρ be a ﬁnite dimensional representation of GF and let ρ2 = χ be any nontrivial character of F ×. For each χ there exists an element c ∈ F × such that

χ(1 + y) = ψ(cy) for suﬃciently small y.

For all χ with suﬃciently large conductor, we have the following formula:

(1.5) W (ρ ⊗ χ, ψ) = W (χ, ψ)dim(ρ) · det(ρ)(c−1).

But for arbitrary characters (specially characters with smaller conductors), the equation (1.5) is not true. If ρ is a minimal U-isotropic Heisenberg representations, then in this paper, we give an extension of the above result (1.5) of Deligne (cf. Theorem 1.2 below). Moreover, by the construction (cf. [13], [36]) of local root numbers, we can attach local root numbers for any virtual representations of GF . So, now if we deﬁne a zero-dimensional virtual representation ρ0 := ρ − dim(ρ) · 1GF , where 1GF is the trivial representation of GF , by the representation ρ, then from above equation (1.5) we have

−1 (1.6) W (ρ0 ⊗ χ, ψ) = det(ρ0)(c ).

Further in [12], Deligne and Henniart generalize the above result (1.6) (see Section 4 of [12]), in which χ is replaced by an arbitrary ﬁnite dimensional representation ρ of GF , and the condition on the conductor of χ becomes a condition on the Artin conductor of ρ.

Theorem 1.1 (Deligne-Henniart’s Twisting formula, Theorem 4.6, [12]). Let ρ be a × virtual representation of GF (without moderate component). There exists an element γ ∈ F j(ρ)/2−1 uniquely determined modulo UF , such that for any virtual representation ρ0 of GF of dimension zero with verifying β(ρ0) < j(ρ)/2, then we have

(1.7) W (ρ0 ⊗ ρ, ψ) = det(ρ0)(γ).

Here νF (γ) = a(ρ) + dim(ρ) · n(ψ), and j(ρ) is the jump of ρ. β(ρ0) is the maximum jump among all the components of ρ0. AN EXTENSION OF DELIGNE-HENNIART’S TWISTING FORMULA AND ITS APPLICATIONS 3

In this article, we ﬁrst generalize Deligne’s twisting formula (1.5) for U-isotropic minimal

Heisenberg representation ρ0 of GF of dimension prime to p and twisted by an arbitrary character χF with conductor a(χF ) > 2. In our following twisting formula the conductor of χF needs not be suﬃciently large, but the Deligne’s twisting formula (1.5) is only true for those characters with suﬃciently large conductors.

Theorem 1.2. Let ρ = ρ(Xη, χK ) = ρ0 ⊗ χfF be a Heisenberg representation of the absolute Galois group GF of a non-archimedean local field F/Qp of dimension m with gcd(m, p) = 1, where ρ0 = ρ0(Xη, χ0) is a minimal conductor Heisenberg representation of GF and χfF : × × × GF → C corresponds to χF : F → C by class field theory. If a(χF ) > 2, then we have m (1.8) W (ρ, ψ) = W (ρ0 ⊗ χfF , ψ) = W (χF , ψ) · det(ρ0)(c), × where ψ is a nontrivial additive character of F , and c := c(χF , ψ) ∈ F , satisfies a(χF ) −1 a(χF )−[ 2 ] χF (1 + x) = ψ(c x) for all x ∈ PF . Then by using Theorem 1.2 above and Deligne-Henniart’s theorem 1.1 we have the following twisting formula for the representation σ ⊗ ρm, where σ is an arbitrary finite dimensional representation of GF and ρm is a U-isotropic representation of dimension m prime to p.

Theorem 1.3. Let σ be an arbitrary finite dimension complex representation of GF and let ψ be a nontrivial additive character of F . Let ρm = ρm(Xη, χK ) = ρ0 ⊗ χfF be a Heisenberg representation of the absolute Galois group GF of a non-archimedean local field F/Qp of dimension m with gcd(m, p) = 1, where ρ0 = ρ0(Xη, χ0) is a minimal conductor Heisenberg × × × representation of GF and χfF : GF → C corresponds to χF : F → C by class field theory. If a(χF ) > 2 and j(ρm) > 2 · β(σ), then we have

dim(σ⊗ρm) dim(σ) (1.9) W (σ ⊗ ρm, ψ) = det(σ)(γ) · W (χF , ψ) · det(ρ0)(c ).

× Here c ∈ F is same as in Theorem 1.2 and νF (γ) = a(ρm) + m · n(ψ).

Finally, in Section 6, we also give two very important applications of Theorem 1.3, and they are: (i) Invariant formula of root number for U-isotropic Heisenberg representations; and (ii) a local converse theorem in the Galois side. By the dimension theorem (cf. Theorem 4.1), we know that the dimension of a Heisenberg r representation ρ of GF is of the form dim(ρ) = p · m, where r > 0 and gcd(m, p) = 1, m|(qF −1). And a U-isotropic representation ρ can be expressed as ρ = ρp ⊗ρm, where ρp and r ρm are U-isotropic representations of GF of dimension p and m respectively. Therefore for giving invariant formula W (ρ, ψ), we can use Theorem 1.3, and we have the following result.

Theorem 1.4 (Invariant Formula). Let ρ be a U-isotropic Heisenberg representation of GF r of the form ρ = ρp ⊗ ρm with dim(ρp) = p (r > 1), and dim(ρm) = m, and gcd(m, p) = 1. Let ψ be an nontrivial additive character of F . If the jump: j(ρm) > 2 · j(ρp), then we have

dim(ρ) pr W (ρ, ψ) = W (ρp ⊗ ρm, ψ) = det(ρp)(γ) · W (χF , ψ) det(ρ0)(c ).

Here χF , c, ρ0 are same as in Theorem 1.2, and νF (γ) = a(ρm) + m · n(ψ). 4 BISWAS

By using Theorem 1.3, we give a converse theorem in the Galois side.

Theorem 1.5 (Converse Theorem in the Galois side). Let ρm = ρ0 ⊗ χfF be a U- isotopic Heisenberg representation of GF of dimension prime to p. Let ψ be a nontrivial additive character of F . Let ρ1, ρ2 be two ﬁnite dimensional complex representations of GF with

det(ρ1) ≡ det(ρ2), and j(ρm) > 2 · max{β(ρ1), β(ρ2)}. If

W (ρ1 ⊗ ρm, ψ) = W (ρ2 ⊗ ρm, ψ), × × then ρ1 ≡ ρ2 or ρ1 ≡ ρ2 ⊗ µ, where µ : F → C is an unramiﬁed character whose order divides dim(ρi), i = 1, 2.

2. Preliminaries and Notation Let F be a non-archimedean local ﬁeld of characteristic zero, i.e., a ﬁnite extension of the

field Qp (field of p-adic numbers), where p is a prime. Let K/F be a finite extension of the field F . Let eK/F be the ramification index for the extension K/F and fK/F be the residue degree of the extension K/F .

Let OF be the ring of integers in the local field F and PF = πF OF the unique prime ideal in OF and πF a uniformizer, i.e., an element in PF whose valuation is one, i.e., νF (πF ) = 1. i Let UF = OF − PF be the group of units in OF . Let PF = {x ∈ F : νF (x) > i} and for i > 0 i i 0 × define UF = 1 + PF (with proviso UF = UF = OF ). We also let a(χ) be the conductor of × × nontrivial character χ : F → C , i.e., a(χ) is the smallest integer > 0 such that χ is trivial a(χ) on UF . We say χ is unramified if the conductor of χ is zero and otherwise ramified. The conductor of any nontrivial additive character ψ of the field F is an integer n(ψ) if −n(ψ) −n(ψ)−1 ψ is trivial on PF , but nontrivial on PF . Let GF := Gal(F /F ) (resp. WF ) be the absolute Galois (resp. Weil group) of the field F , where F is an absolute algebraic closure of F . 2.1. Ramification break. Let K/F be a Galois extension of F and G be the Galois group of the extension K/F . For each i > −1 we define the i-th ramification subgroup of G (in the lower numbering) as follows:

Gi = {σ ∈ G| vK (σ(α) − α) > i + 1 for all α ∈ OK }. An integer t is called a ramification break or jump for the extension K/F or the ramification groups {Gi}i>−1 if Gt 6= Gt+1. We also know that there is a decreasing filtration (with upper numbering) of G and which is defined by the Hasse-Herbrand function Ψ = ΨK/F as follows: u G = GΨ(u), where u ∈ R, u > −1. Since by the definition of Hasse-Herbrand function, Ψ(−1) = −1, Ψ(0) = 0, we have G−1 = 0 G−1 = G, and G = G0. Thus a real number t > −1 is called a ramification break for K/F i or the filtration {G }i>−1 if Gt 6= Gt+ε, for all ε > 0. AN EXTENSION OF DELIGNE-HENNIART’S TWISTING FORMULA AND ITS APPLICATIONS 5

When G is abelian, it can be proved (cf. Hasse-Arf theorem, [31], p. 91) that the ramification breaks for G are integers. But in general, the set of ramification breaks of a Galois group of a local fields is countably infinite and need not consist of integers..

Deﬁnition 2.1 (Artin and Swan conductor). Let G be a ﬁnite group and R(G) be the complex representation ring of G. For any two representations ρ1, ρ2 ∈ R(G) with characters

χ1, χ2 respectively, we have the Schur’s inner product: 1 X < ρ , ρ > =< χ , χ > := χ (g) · χ (g). 1 2 G 1 2 G |G| 1 2 g∈G Let K/F be a finite Galois extension with Galois group G := Gal(K/F). For an element g ∈ G different from identity 1, we define the non-negative integer (cf. [32], Chapter IV, p. 62)

iG(g) := inf{νK(x − g(x))| x ∈ OK}.

By using this non-negative (when g 6= 1) integer iG(g) we deﬁne a function aG : G → Z as follows: P aG(g) = −fK/F · iG(g) when g 6= 1, and aG(1) = fK/F g6=1 iG(g). P Thus from this deﬁnition we can see that g∈G aG(g) = 0, hence < aG, 1G >= 0. It can be proved (cf. [32], p. 99, Theorem 1) that the function aG is the character of a linear representation of G, and that corresponding linear representation is called the Artin representation

AG of G. Similarly, for a nontrivial g 6= 1 ∈ G, we deﬁne (cf. [34], p. 247)

−1 × X sG(g) = inf{νK(1 − g(x)x )| x ∈ K }, sG(1) = − sG(g). g6=1

And we can deﬁne a function swG :G → Z as follows:

swG(g) = −fK/F · sG(g)

It can also be shown that swG is a character of a linear representation of G, and that corresponding representation is called the Swan representation SWG of G. From [33], p. 160 , we have the relation between the Artin and Swan representations (cf. [34], p. 248, equation (6.1.9))

G G (2.1) SWG = AG + IndG0 (1) − Ind{1}(1),

G0 is the 0-th ramification group (i.e., inertia group) of G. Now we are in a position to define the Artin and Swan conductor of a representation ρ ∈ R(G). The Artin conductor of a representation ρ ∈ R(G) is defined by

aF (ρ) :=< AG, ρ >G=< aG, χ >G, where χ is the character of the representation ρ. Similarly, for the representation ρ, the Swan conductor is:

swF(ρ) :=< SWG, ρ >G=< swG, χ >G . For more details about Artin and Swan conductor, see Chapter 6 of [34] and Chapter VI of [32]. 6 BISWAS

From equation (2.1) we obtain

(2.2) aF (ρ) = swF(ρ) + dim(ρ)− < 1, ρ >G0 . Moreover, from Corollary of Proposition 4 on p. 101 of [32], for an induced representation Gal(K/F) ρ := IndGal(K/E)(ρE) = IndE/F(ρE), we have (2.3) aF (ρ) = fE/F · dE/F · dim(ρE) + aE(ρE) .

We apply this formula (2.3) for ρE = χE of dimension 1 and then conversely

aF (ρ) (2.4) a(χE) = − dE/F , fE/F where dE/F is the exponent of the diﬀerent of the extension E/F . So if we know aF (ρ) then we can compute the conductor a(χE) of χE.

Deﬁnition 2.2 (Jump for a representation). Now let ρ be an irreducible representation of G. For this irreducible ρ we deﬁne jump for ρ as follows:

j(ρ) := max{i | ρ|Gi 6≡ 1}.

0 Now if ρ is a ramified irreducible representation of G, then ρ|I 6≡ 1, where I = G = G0 is the inertia subgroup of G. Thus from the definition of j(ρ) we can say, if ρ is irreducible, then we always have j(ρ) > 0, i.e., ρ is nontrivial on the inertia group G0. Then from the definitions of Swan and Artin conductors, and equation (2.2), when ρ is irreducible, we have the following relations

(2.5) swF(ρ) = dim(ρ) · j(ρ), aF(ρ) = dim(ρ) · (j(ρ) + 1). From the Theorem of Hasse-Arf (cf. [32], p. 76), if dim(ρ) = 1, i.e., ρ is a character of

G/[G, G], we can say that j(ρ) must be an integer, then swF(ρ) = j(ρ), aF(ρ) = j(ρ) + 1.

Moreover, by class ﬁeld theory, ρ corresponds to a linear character χF , hence for linear character χF , we can write

j(χF ) := max{i | χF| i 6≡ 1}, UF because under class ﬁeld theory (under Artin isomorphism) the upper numbering in the

ﬁltration of Gal(Fab/F) is compatible with the ﬁltration (descending chain) of the group of units UF .

Similarly, one can deﬁne jump for any virtual representation of GF . Let ρ be a virtual representation of GF . We denote j(ρ) (resp. β(ρ)) the lower (resp. upper) bound of j(ρi), when ρi runs over all the components of ρ. That is, if n X ρ = niρi, i then

j(ρ) 6 {j(ρ1), ··· , j(ρn)} 6 β(ρ). Further, j(ρ) > α means that the components of ρ do not have any non-null vector ﬁxed by α α GF GF : ρ = 0. And β(ρ) < β (we then have β > 0) means that ρ comes by inﬂation from a β virtual representation of GF /GF . AN EXTENSION OF DELIGNE-HENNIART’S TWISTING FORMULA AND ITS APPLICATIONS 7

2.2. Heisenberg Representations. Let G be a proﬁnite group (in our case, G = GF ). An irreducible representation ρ of G is called a Heisenberg representation if it represents commutators by scalar matrices. Therefore higher commutators are represented by 1 (see [39]). We can see that the linear characters of G are Heisenberg representations as the degenerate special case. To classify Heisenberg representations we need to mention two invariants of an irreducible representation ρ ∈ Irr(G):

(1) Let Zρ be the scalar group of ρ, i.e., Zρ ⊆ G and ρ(z) = scalar matrix for every z ∈ Zρ. If V/C is a representation space of ρ we get Zρ as the kernel of the composite map

ρ π × (2.6) G −→ GLC(V ) −→ P GLC(V ) = GLC(V )/C E,

where E is the unit matrix and denote ρ := π ◦ ρ. Therefore Zρ is a normal subgroup of G.

(2) Let χρ be the character of Zρ which is given as ρ(g) = χρ(g) · E for all g ∈ Zρ.

Apparently χρ is a G-invariant character of Zρ which we call the central character of ρ. Let A be a proﬁnite abelian group. Then we know that (cf. [41], p. 124, Theorem 1 and Theorem 2) the set of isomorphism classes PI(A) of projective irreducible representations (for projective representation, see [11], §51) of A is in bijective correspondence with the set of continuous alternating characters Alt(A). If ρ ∈ PI(A) corresponds to X ∈ Alt(A) then Ker(ρ) = Rad(X) and [A : Rad(X)] = dim(ρ)2, where Rad(X) := {a ∈ A| X(a, b) = 1, for all b ∈ A}, the radical of X. Let A := G/[G, G], so A is abelian. We also know from the composite map (2.6) ρ is a projective irreducible representation of G and Zρ is the kernel of ρ. Therefore modulo commutator group [G, G], we can consider that ρ is in PI(A) which corresponds an alternating character X of A with kernel of ρ is Zρ/[G, G] = Rad(X). We also know that

[A : Rad(X)] = [G/[G, G] : Zρ/[G, G]] = [G : Zρ]. Then we observe that q dim(ρ) = dim(ρ) = [G : Zρ]. Let H be a subgroup of A, then we deﬁne the orthogonal complement of H in A with respect to X H⊥ := {a ∈ A : X(a, H) ≡ 1}. An isotropic subgroup H ⊂ A is a subgroup such that H ⊆ H⊥ (cf. [37], p. 270, Lemma 1(v)). And when isotropic subgroup H is maximal, we call H is a maximal isotropic for X. Thus when H is maximal isotropic we have H = H⊥. Let C1G = G, Ci+1G = [CiG, G] denote the descending central series of G. Now assume that every projective representation of A lifts to an ordinary representation of G. Then by I. Schur’s results (cf. [11], p. 361, Theorem 53.7) we have (cf. [41], p. 124, Theorem 2):

(1) Let A ∧Z A denote the alternating square of the Z-module A. The commutator map ∼ 2 3 ˆ (2.7) A ∧Z A = C G/C G, a ∧ b 7→ [ˆa, b] 8 BISWAS

is an isomorphism.

(2) The map ρ → Xρ ∈ Alt(A) from Heisenberg representations to alternating characters on A is surjective.

3. U-isotropic Heisenberg representations

Let F/Qp be a local field, and F be an algebraic closure of F . Denote GF = Gal(F/F) the absolute Galois group for F /F . We know that (cf. [29], p. 197) each representation ρ : GF → GL(n, C) corresponds to a projective representation ρ : GF → GL(n, C) → P GL(n, C). On the other hand, each projective representation ρ : GF → P GL(n, C) can be lifted to a ab representation ρ : GF → GL(n, C). Let AF = GF be the factor commutator group of GF . Define FF × := lim(F ×/N ∧ F ×/N) ←− where N runs over all open subgroups of finite index in F ×. Denote by Alt(F×) as the set of all alternating characters X : F × × F × → C× such that [F × : Rad(X)] < ∞. Then the local × reciprocity map gives an isomorphism between AF and the profinite completion of F , and induces a natural bijection

∼ × (3.1) PI(AF) −→ Alt(F ), where PI(AF) is the set of isomorphism classes of projective irreducible representations of AF . By using class ﬁeld theory from the commutator map (2.7) (cf. p. 125 of [41]) we obtain

× ∼ (3.2) c : FF = [GF ,GF ]/[[GF ,GF ],GF ].

× Let K/F be an abelian extension corresponding to the norm subgroup N ⊂ F and if WK/F denotes the relative Weil group, the commutator map for WK/F induces an isomorphism (cf. p. 128 of [41]):

× × × × (3.3) c : F /N ∧ F /N → KF /IF K , where × × × KF := {x ∈ K | NK/F (x) = 1}, i.e., the norm-1-subgroup of K , × 1−σ × × IF K := {x | x ∈ K , σ ∈ Gal(K/F)} < KF , the augmentation with respect to K/F . Taking the projective limit over all abelian extensions K/F the isomorphisms (3.3) induce:

(3.4) c : FF × ∼ lim K×/I K×, = ←− F F where the limit on the right side refers to norm maps. This gives an arithmetic description of Heisenberg representations of the group GF .

Theorem 3.1 (Zink, [38], p. 301, Corollary 1.2). The set of Heisenberg representations ρ of

GF is in bijective correspondence with the set of all pairs (Xρ, χρ) such that: × (1) Xρ is a character of FF , × × (2) χρ is a character of K /IF K , where the abelian extension K/F corresponds to the × radical N ⊂ F of Xρ, and × (3) via (3.3) the alternating character Xρ corresponds to the restriction of χρ to KF . AN EXTENSION OF DELIGNE-HENNIART’S TWISTING FORMULA AND ITS APPLICATIONS 9

Given a pair (X, χ), we can construct the Heisenberg representation ρ by induction from

GK := Gal(F/K) to GF :

p × (3.5) [F : N] · ρ = IndK/F(χ), where N and K are as in (2) of the above Theorem 3.1 and where the induction of χ (to be considered as a character of GK by class ﬁeld theory) produces a multiple of ρ. From [F × : N] = [K : F ] we obtain the dimension formula:

(3.6) dim(ρ) = p[F× : N], where N is the radical of X. × × × Let K/E be an extension of E, and χK : K → C be a character of K . In the × following lemma, we give the conditions of the existence of characters χE ∈ Ec such that

χE ◦ NK/E = χK , and the solutions set of this χE.

× × Lemma 3.2. Let K/E be a finite extension of a field E, and χK : K → C . × × (i) The existence of characters χE : E → C such that χE ◦ NK/E = χK is equivalent to × KE ⊂ Ker(χK). (ii) In case (i) is fulfilled, we have a well defined character

−1 × (3.7) χK/E := χK ◦ NK/E : NK/E → C ,

× × on the subgroup of norms NK/E := NK/E(K ) ⊂ E , and the solutions χE such that × χE ◦NK/E = χK are precisely the extensions of χK/E from NK/E to a character of E .

× Proof. (i) Suppose that an equation χK = χE ◦NK/E holds. Let x ∈ KE , hence NK/E(x) = 1. Then

χK (x) = χE ◦ NK/E(x) = χE(1) = 1.

× So x ∈ Ker(χK), and hence KE ⊂ Ker(χK). × × × Conversely assume that KE ⊂ Ker(χK). Then χK is actually a character of K /KE . Again × × ∼ × \× × ∼ we have K /KE = NK/E ⊂ E , hence K /KE = N\K/E. Now suppose that χK corresponds −1 to the character χK/E of NK/E. Hence we can write χK ◦ NK/F = χK/E. Thus the character × × × χK/E : NK/E → C is well defined. Since E is an abelian group and NK/E ⊂ E is a × subgroup of finite index (by class field theory) [K : E], we can extend χK/E to E , and χK is of the form χK = χE ◦ NK/E with χE|NK/E = χK/E. (ii) If condition (i) is satisfied, then this part is obvious. If χE is a solution of χK = χE ◦NK/E, −1 × with χK/E := χK ◦ NK/E : NK/E → C , then certainly χE is an extension of the character χK/E. −1 Conversely, if χE extends χK/E, then it is a solution of χK = χE ◦ NK/E with χK ◦ NK/E = × χK/E : NK/E → C .

Remark 3.3. Now take Heisenberg representation ρ = ρ(X, χK ) of GF . Let E/F be any extension corresponding to a maximal isotropic for X. In this Heisenberg setting, from × × Theorem 3.1(2), we know χK is a character of K /IF K , and from the ﬁrst commutative 10 BISWAS

× × × diagram on p. 302 of [38] we have NK/E : KF /IF K → EF /IF NK/E. Thus in the Heisenberg setting, we have more information than Lemma 3.2(i), that χK is a character of

N × × × K/E × (3.8) K /KE IF K −−−→NK/E/IF NK/E ⊂ E /IF NK/E, and therefore χK/F is actually a character of NK/E/IF NK/E, or in other words, it is a × Gal(E/F)-invariant character of the Gal(E/F)-module NK/E ⊂ E . And if χE is one of σ the solution of Lemma 3.2(ii), then the complete solutions is the set {χE | σ ∈ Gal(E/F)}. We know that W (χE, ψ ◦ TrK/E) has the same value for all solutions χE of χE ◦

NK/E = χK , which means for all χE which extend the character χK/E. −1 Moreover, from the above Lemma 3.2, we also can see that χE|NK/E = χK ◦ NK/E.

Let ρ = ρ(X, χK ) be a Heisenberg representation of GF . Let E/F be any extension corresponding to a maximal isotropic for X. Then by using the above Lemma 3.2, we have the following lemma.

Lemma 3.4. Let ρ = ρ(Z, χρ) = ρ(Gal(L/K), χK) be a Heisenberg representation of a ﬁnite local Galois group G = Gal(L/F), where F is a non-archimedean local ﬁeld. Let H = Gal(L/E) be a maximal isotropic for ρ. Then we obtain

σ (3.9) ρ = IndE/F(χE) for all σ ∈ Gal(E/F),

× × where χE : E /IF NK/E → C with χK = χE ◦ NK/E. Moreover, for a ﬁxed base ﬁeld E of a maximal isotropic for ρ, this construction of ρ is independent of the choice of this character χE.

Deﬁnition 3.5 (U-isotropic). Let F be a non-archimedean local ﬁeld. Let X : FF × → C× be an alternating character with the property

X(ε1, ε2) = 1, for all ε1, ε2 ∈ UF .

× In other words, X is a character of FF /UF ∧ UF . Then X is said to be the U-isotropic. These X are easy to classify:

Lemma 3.6 (cf. Section 2.4 of [6]). Fix a uniformizer πF and write U := UF . Then we obtain an isomorphism

∼ × Ub = FF\/U ∧ U, η 7→ Xη, ηX ← X between characters of U and U-isotropic alternating characters as follows:

a b b −a (3.10) Xη(πF ε1, πF ε2) := η(ε1) · η(ε2) , ηX (ε) := X(ε, πF ),

× where a, b ∈ Z, ε, ε1, ε2 ∈ U, and η : U → C . Then #η #η Rad(Xη) =< πF > ×Ker(η) =< (πFε) > ×Ker(η), does not depend on the choice of πF , where #η is the order of the character η, hence × ∼ #η ∼ F /Rad(Xη) =< πF > / < πF > ×U/Ker(η) = Z#η × Z#η.

Therefore all Heisenberg representations of type ρ = ρ(Xη, χ) have dimension dim(ρ) = #η. AN EXTENSION OF DELIGNE-HENNIART’S TWISTING FORMULA AND ITS APPLICATIONS 11

× ∼ Remark 3.7. From Proposition 5.2(i) on p. 50 of [40], we know that FF /UF ∧ UF = UF , ∼ × hence we have UcF = FF \/UF ∧ UF . From the above Lemma 3.6 we have: #η Rad(Xη) =< πF > ×Ker(η). i Therefore we can conclude that a U-isotropic character X = Xη has UF contained in its i radical if and only if η is a character of UF /UF . From the above Lemma 3.6 we know that the dimension of a U-isotropic Heisenberg rep- × ∼ resentation ρ = ρ(Xη, χ) of GF is dim(ρ) = #η, and F /Rad(Xη) = Z#η × Z#η, a direct product of two cyclic (bicyclic) groups of the same order #η.

In the following lemma, we give an equivalent condition for U-isotropic Heisenberg representation.

Lemma 3.8. Let GF be the absolute Galois group of a non-archimedean local ﬁeld F . For a

Heisenberg representation ρ = ρ(Z, χρ) = ρ(X, χK ) the following are equivalent: (1) The alternating character X is U-isotropic. (2) Let E/F be the maximal unramiﬁed subextension in K/F . Then Gal(K/E) is maximal isotropic for X. × (3) ρ = IndE/F(χE) can be induced from a character χE of E (where E is as in (2)).

Proof. This proof follows from the above Lemma 3.6. First, assume that X is U-isotropic, i.e., X ∈ FF\×/U ∧ U. We also know that Ub ∼= × FF\/U ∧ U. Then X corresponds a character of U, namely X 7→ ηX . Then from Lemma 3.6 × ∼ we have F /Rad(X) = Z#ηX × Z#ηX , i.e., product of two cyclic groups of same order. Since K/F is the abelian bicyclic extension which corresponds to Rad(X), we can write: ∼ × NK/F = Rad(X), Gal(K/F) = F /Rad(X).

Let E/F be the maximal unramiﬁed subextension in K/F . Then [E : F ] = #ηK because the order of maximal cyclic subgroup of Gal(K/F) is #ηX . Then fE/F = #ηX , hence fK/F = 2 eK/F = #ηX because fK/F · eK/F = [K : F ] = #ηX and Gal(K/F) is not cyclic group. Now we have to prove that the extension E/F corresponds to a maximal isotropic for X.

Let H/Z be a maximal isotropic for X, hence [GF /Z : H/Z] = #ηX , hence H/Z = Gal(K/E), i.e., the maximal unramiﬁed subextension E/F in K/F corresponds to a maximal isotropic subgroup, hence

ρ(X, χK ) = IndE/F(χE), for χE ◦ NK/E = χK . Finally, since E/F is unramiﬁed and the extension E corresponds a maximal isotropic sub- × group for X, we have UF ⊂ NE/F , hence UF ⊂ NK/F and X|U×U = 1 because UF ⊂ F ⊂ NK/E. This shows that X is U-isotropic.

Corollary 3.9. The U-isotropic Heisenberg representation ρ = ρ(Xη, χ) can never be wild because it is induced from an unramiﬁed extension E/F , but the dimension dim(ρ(Xη, χ)) = #η can be a power of p.

The representations ρ of dimension prime to p are precisely given as ρ = ρ(Xη, χ) for characters η of U/U 1. 12 BISWAS

1 Proof. This is clear from the above lemma 3.6 and the fact: |U/U | = qF − 1 is prime to p. We know that the dimension dim(ρ) = p[K : F] = p[F× : Rad(X)]. If this is prime to p then 1 1 K/F is tame and UF ⊆ Rad(X). But U/U is cyclic, hence X is then U-isotopic.

Lemma 3.10. Let ρ = ρ(X, χK ) be a Heisenberg representation of the absolute Galois group GF of a non-archimedean local ﬁeld F/Qp. Then following are equivalent: (1) dim(ρ) is prime to p.

(2) dim(ρ) is a divisor of qF − 1. 1 (3) The alternating character X is U-isotropic and X = Xη for a character η of UF /UF , i.e., a(η) = 1. (4) The abelian extension K/F which corresponds to Rad(X) is tamely ramiﬁed. Proof. (1) implies (2): From Corollary 3.9 we know that all Heisenberg representations of dimensions prime to p, are U-isotropic representations of the form ρ = ρ(Xη, χ), where η : 1 × UF /UF → C , and the dimensions dim(ρ) = #η. Thus if dim(ρ) is prime to p, then dim(ρ) = #η is a divisor of qF − 1.

(2) implies (3): If dim(ρ) is a divisor of qF − 1, then gcd(p, dim(ρ)) = 1. Then from Corollary 1 3.9, the alternating character X is U-isotropic and X = Xη for a character η ∈ U\F /UF . (3) implies (4): We know that q q × × dim(ρ) = [F : Rad(Xη)] = [F : NK/F] = #η. Here since K/F is abelian, we have dim(ρ)2 = [K : F]. Again since #η = dim(ρ) is a divisor of qF − 1, hence K/F is tamely ramified. 1 × (4) implies (1): If K/F is tamely ramified, then we can write UF ⊂ NK/F ⊂ F , and hence × × 1 F /NK/F is a quotient group of F /UF . Therefore if K/F is the abelian tamely ramified × 1 extension and NK/F = Rad(X), then X must be an alternating character of F /UF . We also × 1 know that F =< πF > × < ζ > ×UF , where ζ is a root of unity of order qF − 1. This × 1 × 1 a b implies F /UF =< πF > × < ζ >. So each element x ∈ F /UF can be written as x = πF ·ζ , a1 b1 a2 b2 × 1 where a, b ∈ Z. We now take x1 = πF ζ , x2 = πF ζ ∈ F /UF , where ai, bi ∈ Z(i = 1, 2), then

a1 b1 a2 b2 X(x1, x2) = X(πF ζ , πF ζ )

a1 b2 b1 a2 = X(πF , ζ ) · X(ζ , πF )

a1 b2 b1 a2 = χρ([πF , ζ ]) · χρ([ζ , πF ]). But this implies XqF −1 ≡ 1 because ζqF −1 = 1, which means that X is actually an al- × ×(qF −1) 1 ternating character on F /(F UF ), and therefore GF /GK is actually a quotient of × ×(qF −1) 1 1 F /(F UF ). We also know that UF is a pro-p-group and therefore

1 1 qF −1 × UF = (UF ) ⊂ F .

× ×(qF −1) 1 2 Thus the cardinality of F /(F UF ) is (qF − 1) because × ×(qF −1) 1 ∼ ∼ F /(F UF ) = Z/(qF − 1)Z× < ζ >= ZqF −1 × ZqF −1.

Therefore dim(ρ) divides qF − 1. Hence dim(ρ) is prime to p. AN EXTENSION OF DELIGNE-HENNIART’S TWISTING FORMULA AND ITS APPLICATIONS 13

Remark 3.11. We let Kη|F be the abelian bicyclic extension which corresponds to Rad(Xη):

∼ × NKη/F = Rad(Xη), Gal(Kη/F) = F /Rad(Xη).

Then we have fKη|F = eKη|F = #η and the maximal unramiﬁed subextension E/F ⊂ Kη/F corresponds to a maximal isotropic subgroup, hence

ρ(Xη, χ) = IndE/F(χE), for χE ◦ NKη/E = χ.

× × × We recall here that χ : Kη /IF Kη → C is a character such that (cf. Theorem 3.1(3)) × × × × χ| × ↔ X , with respect to (K ) /I K ∼ F /Rad(X ) ∧ F /Rad(X ). (Kη )F η η F F η = η η × × In particular, we see that (K ) /I K is cyclic of order #η and χ| × must be a faithful η F F η (Kη )F character of that cyclic group.

4. Computation of Artin conductors, Swan conductors, dimension theorem The Artin conductors and Swan conductors are the main ingredients of the Deligne- Henniart’s twisting formula (cf. Theorem 1.1) and Deligne’s formula 1.5. Therefore for U-isotropic Heisenberg representations, we need to compute them explicitly and in this section we do all the necessary computation. In the following theorem we give a general dimension formula of a Heisenberg representation.

Theorem 4.1 (Dimension). Let F/Qp be a local ﬁeld and GF be the absolute Galois group n 0 of F . If ρ is a Heisenberg representation of GF , then dim(ρ) = p · d , where n > 0 is an 0 integer and where the prime to p factor d must divide qF − 1.

Proof. By the deﬁnition of Heisenberg representation ρ we have the relation

[[GF ,GF ],GF ] ⊆ Ker(ρ).

Then we can consider ρ as a representation of G := GF /[[GF ,GF ],GF ]. Since [x, g] ∈

[[GF ,GF ],GF ] for all x ∈ [GF ,GF ] and g ∈ GF , we have [G, G] = [GF ,GF ]/[[GF ,GF ],GF ] ⊆ Z(G), hence G is a two-step nilpotent group. We know that each nilpotent group is isomorphic to the direct product of its Sylow subgroups. Therefore we can write

G = Gp × Gp0 , where Gp is the Sylow p-subgroup, and Gp0 is the direct product of all other Sylow subgroups.

Therefore each irreducible representation ρ has the form ρ = ρp ⊗ ρp0 , where ρp and ρp0 are irreducible representations of Gp and Gp0 respectively. We also know that ﬁnite p-groups are nilpotent groups, and direct product of a ﬁnite number of nilpotent groups is again a nilpotent group. So Gp and Gp0 are both two-step nilpotent group because G is a two-step nilpotent group. Therefore the representations ρp and ρp0 are both Heisenberg representations of Gp and Gp0 respectively.

Now to prove our assertion, we have to show that dim(ρp) can be an arbitrary power of p, whereas dim(ρp0 ) must divide qF − 1. Since ρp is an irreducible representation of p-group

Gp, so the dimension of ρp is some p-power. 14 BISWAS

Again from the construction of ρp0 we can say that dim(ρp0 ) is prime to p. Then from

Lemma 3.10 dim(ρp0 ) is a divisor of qF − 1. This completes the proof.

Remark 4.2. (1). Let VF be the wild ramification subgroup of GF . We can show that ρ|VF is irreducible if and only if Zρ = GK ⊂ GF corresponds to an abelian extension K/F which is 1 × totally ramified and wildly ramified (cf. [38], p. 305). If N := NK/F (K ) is the subgroup of 1 × norms, then this means that N · UF = F , in other words,

× 1 1 1 F /N = N · UF /N = UF /N ∩ UF , where N can be also considered as the radical of Xρ. So we can consider the alternating 1 × character Xρ on the principal units UF ⊂ F . Then q p × 1 1 dim(ρ) = [F : N] = [UF :N ∩ UF],

1 is a power of p, because UF is a pro-p-group. n Here we observe: If ρ = ρ(X, χK ) with ρ|VF stays irreducible, then dim(ρ) = p , n > 1 and K/F is a totally and wildly ramiﬁed. But there is a big class of Heisenberg representations ρ such that dim(ρ) = pn is a p-power, but which are not wild representations (see the Deﬁnition 3.5 of U-isotropic).

(2). Let ρ = ρ(X, χK ) be a Heisenberg representation of the absolute Galois group GF of dimension d > 1 which is prime to p. Then from above Lemma 3.10, we have d|qF − 1. For this representation ρ, here K/F must be tame if Rad(X) = NK/F (cf. [31], p. 115).

By using the equation (2.3) in our Heisenberg setting, we have the following proposition.

Proposition 4.3. Let ρ = ρ(Z, χρ) = ρ(X, χK ) be a Heisenberg representation of the absolute Galois group GF of a ﬁeld F/Qp of dimension m. Let E/F be any subextension in K/F corresponding to a maximal isotropic subgroup for X. Then

aF (ρ) = aF (IndE/F(χE)), m · aF(ρ) = aF(IndK/F(χK)). As a consequence we have

a(χK ) = eK/E · a(χE) − dK/E.

In particular a(χK ) = a(χE) if K/E is unramiﬁed.

Proof. We know that ρ = IndE/F(χE) and m · ρ = IndK/F(χK). By the deﬁnition of Artin conductor we can write

aF (dim(ρ) · ρ) = dim(ρ) · aF(ρ) = m · aF(IndE/F(χE)).

1 GF Group theoretically, if ρ|VF = IndH (χH)|VF is irreducible, then from Section 7.4 of [33], we can say

GF = H · VF . Here H = GL, where L is a certain extension of F , and VF = GFmt where Fmt/F is the maximal tame extension of F . Therefore GF = H · VF is equivalent to F = L ∩ Fmt that means the extension

L/F must be totally ramified and wildly ramified, and [GF : H] = [L : F ] = |VF |. We know that the wild ramification subgroup VF is a pro-p-group (cf. [31], p. 106). Then dim(ρ) is a power of p. AN EXTENSION OF DELIGNE-HENNIART’S TWISTING FORMULA AND ITS APPLICATIONS 15

Since K/E/F is a tower of Galois extensions with [K : E] = m = eK/EfK/E, we have the transitivity relation of diﬀerent (cf. [32], p. 51, Proposition 8)

DK/F = DK/E ·DE/F . Now from the deﬁnition of diﬀerent of a Galois extension, and taking K-valuation we obtain:

(4.1) dK/F = dK/E + eK/E · dE/F . Now by using equation (2.3) we have: (4.2) m · aF (IndE/F(χE)) = m · fE/F dE/F + a(χE) = m · fE/F · dE/F + eK/E · fK/F · a(χE), and (4.3) aF (IndK/F(χK)) = fK/F · dK/F + a(χK) = fK/F · dK/F + fK/F · a(χK). By using equation (4.1), from equations (4.2), (4.3), we have

a(χK ) = eK/E · a(χE) − dK/E.

And when K/E is unramified, i.e., eK/E = 1 and dK/E = 0, hence a(χK ) = a(χE). Definition 4.4 (Jump for alternating character). For each X ∈ FF[× we define jump as follows:  0 when X is trivial (4.4) j(X) := max{i | X|UUi 6≡ 1} when X is nontrivial, where UU i ⊆ FF × is a subgroup which under (3.2) corresponds

i i GF ∩ [GF ,GF ]/GF ∩ [[GF ,GF ],GF ] ⊆ [GF ,GF ]/[[GF ,GF ],GF ].

Remark 4.5. Let ρ = ρ(Xρ, χK ) be a minimal conductor (i.e., a representation with the smallest Artin conductor) Heisenberg representation for Xρ of the absolute Galois group GF . From Theorem 3 on p. 125 of [41], we have q × (4.5) swF(ρ) = dim(ρ) · j(Xρ) = [F : Rad(Xρ)] · j(Xρ).

Moreover if ρ0 = ρ0(X, χ0) is a minimal representation corresponding X, then all other

Heisenberg representations of dimension dim(ρ) are of the form ρ = χF ⊗ ρ0 = (X, (χF ◦ × × NK/F )χ0), where χF : F → C . Then we have (cf. [38], p. 305, equation (5))

p × (4.6) swF(ρ) = swF(χF ⊗ ρ0) = [F : Rad(X)] · max{j(χF), j(X)}.

For minimal conductor U-isotopic Heisenberg representation we have the following proposition.

Proposition 4.6. Let ρ = ρ(Xη, χK ) be a U-isotropic Heisenberg representation of GF of minimal conductor. Then we have the following conductor relation

j(Xη) = j(η), swF(ρ) = dim(ρ) · j(Xη) = #η · j(η),

aF (ρ) = swF(ρ) + dim(ρ) = #η(j(η) + 1) = #η · aF(η). 16 BISWAS

Proof. From [41], on p. 126, Proposition 4(i) and Proposition 5(ii), and U ∧ U = U 1 ∧ U 1, we see the injection U i ∧ F × ⊆ UU i induces a natural isomorphism i ∼ i i U ∧ < πF >= UU /UU ∩ (U ∧ U) for all i > 0.

n n−1 n Now let j(Xη) = n − 1, hence Xη|UU = 1 but Xη|UU 6= 1. This gives Xη|U ∧<πF > = 1

n−1 but Xη|U ∧<πF > 6= 1. Now from equation (3.10) we can conclude that η(x) = 1 for all x ∈ U n but η(x) 6= 1 for x ∈ U n−1. Hence

j(η) = n − 1 = j(Xη). Again from the deﬁnition of j(χ), where χ is a character of F ×, we can see that j(χ) = a(χ)−1, i.e., a(χ) = j(χ) + 1. From equation (4.5) we obtain:

swF(ρ) = dim(ρ) · j(Xη) = #η · j(η), since dim(ρ) = #η and j(Xη) = j(η). Finally, from equation (2.2) for ρ (here < 1, ρ >G0 = 0), we have

(4.7) aF (ρ) = swF(ρ) + dim(ρ) = #η · j(η) + #η = #η · (j(η) + 1) = #η · aF(η).

Now by combining Proposition 4.3 with Proposition 4.6, we get the following result.

Lemma 4.7. Let ρ = ρ(Xη, χK ) be a U-isotopic Heisenberg representation of minimal conductor of the absolute Galois group GF of a non-archimedean local field F . Let K = Kη correspond to the radical of Xη, and let E1/F be the maximal unramified subextension, and E/F be any maximal cyclic and totally ramified subextension in K/F . Let m denote the order of η. Then ρ is induced by χE1 or by χE respectively, and we have

(1) aE(χE) = m · a(η) − dE/F ,

(2) aE1 (χE1 ) = a(η), × (3) and for the character χK ∈ Kd,

aK (χK ) = m · a(η) − dK/F .

Moreover, aE(χE) = aK (χK ).

Proof. Proof of these assertions follows from equation (2.3) and Proposition 4.6. When ρ =

IndE/F(χE), where E/F is a maximal cyclic and totally ramiﬁed subextension in K/F , from equation (2.3) we have

aF (ρ) = m · a(η) using Proposition 4.6, = fE/F · dE/F · 1 + aE(χE) , since ρ = IndE/F(χE) = 1 · dE/F + aE(χE) . because E/F is totally ramiﬁed, hence fE/F = 1. This implies aE(χE) = m · a(η) − dE/F .

Similarly, when ρ = IndE1/F(χE1 ), where E1/F is the maximal unramiﬁed subextension in

K/F , hence fE1/F = m and dE1/F = 0, by using equation (2.3) we obtain aE1 (χE1 ) = a(η). AN EXTENSION OF DELIGNE-HENNIART’S TWISTING FORMULA AND ITS APPLICATIONS 17

Again from Proposition 4.3 we have

aK (χK ) = m · a(χE1 ) − dK/E1 = m · a(η) − dK/F . Finally, since E/F is a maximal cyclic totally ramiﬁed implies K/E is unramiﬁed and therefore

dE/F = dK/F , and hence aE(χE) = aK (χK ).

Remark 4.8. Assume that we are in the dimension m = #η prime to p case. Then from 1 Corollary 3.9, η must be a character of U/U (for U = UF ), hence

a(η) = 1 aF (ρ0) = m.

Therefore in this case the minimal conductor of ρ is m, hence it is equal to the dimension of ρ. From the above Lemma 4.7, in this case we have

aE1 (χE1 ) = a(η) = 1.

And K/F, E/F are tamely ramiﬁed of ramiﬁcation exponent eK/F = m, hence

aE(χE) = aK (χK ) = m · a(η) − dK/F = m − (eK/F − 1) = m − (m − 1) = 1.

Thus we can conclude that in this case all three characters (i.e., χE1 , χE, and χK ) are of conductor 1.

In the general case aE1 (χE1 ) = a(η) and

aE(χE) = aK (χK ) = m · a(η) − d, where d = dE/F = dK/F , conductors will be diﬀerent.

In general, if ρ = ρ0 ⊗χF , where ρ0 is a ﬁnite dimensional minimal conductor representation × of GF , and χF ∈ Fc , then we have the following result.

Lemma 4.9. Let ρ0 be a ﬁnite dimensional representation of GF of minimal conductor. Then we have

(4.8) aF (ρ) = dim(ρ0) · aF(χF),

× a(ρ0) where ρ = ρ0 ⊗ χF = ρ(Xη, (χF ◦ NK/F )χ0) and χF ∈ Fc with a(χF ) > dim(ρ) .

Proof. From equation (2.5) we have aF (ρ0) = dim(ρ0) · (1 + j(ρ0)). By the given condition ρ0 is of minimal conductor. So for representation ρ = ρ0 ⊗ χF , we have

aF (ρ) = aF (ρ0 ⊗ χF ) = dim(ρ0) · (1 + max{j(ρ0), j(χF)})

= dim(ρ0) · max{1 + j(χF), 1 + j(ρ0)}

= dim(ρ0) · max{a(χF), 1 + j(ρ0)}

= dim(ρ0) · aF(χF), 18 BISWAS because by the given condition

a(ρ0) dim(ρ0) · (1 + j(ρ0)) a(χF ) > = = 1 + j(ρ0). dim(ρ0) dim(ρ0)

Proposition 4.10. Let ρ = ρ(X, χK ) be a Heisenberg representation of dimension m of the absolute Galois group GF of a non-archimedean local ﬁeld F . Then m|aF (ρ) if and only if:

X is U-isotropic, or (if X is not U-isotropic) aF (ρ) is with respect to X not the minimal conductor.

Proof. From the above Lemma 4.9 we know that if ρ is not minimal, then aF (ρ) is always a multiple of the dimension m. So now we just have to check for minimal conductors. In the U-isotropic case the minimal conductor is multiple of the dimension (cf. Proposition 4.6).

Finally, suppose that X is not U-isotropic, i.e., X|U∧U = X|U 1∧U 1 6≡ 1, because U ∧ U = U 1∧U 1 (see the Remark on p. 126 of [41]). We also know that UU i = (UU i∩U 1∧U 1)×(U i∧ <

πF >) (cf. [41], p. 126, Proposition 5(ii)). In Proposition 5 of [41], we observe that all the i 1 1 jumps v in the ﬁltration {UU ∩ (U ∧ U )}, i ∈ R+ are not integers with v > 1. This shows that j(X) is also not an integer, hence aF (ρ0) is not multiple of the dimension. This implies the conductor aF (ρ) is not minimal.

For minimal conductor Heisenberg representation, we have the following theorem.

Proposition 4.11. Let ρ = ρ(Xη, χK ) be a Heisenberg representation of the absolute Galois group GF of a non-archimedean local ﬁeld F/Qp of dimension m prime to p. Then it is of minimal conductor aF (ρ) = m if and only if ρ is a representation of GF /VF , where VF is the subgroup of wild ramiﬁcation.

Proof. By the given condition, the dimension dim(ρ) = m is prime to p. Then from Lemma

3.10 we can conclude that K/F is tamely ramiﬁed with fK/F = eK/F = m (cf. Remark

3.11)and hence dK/F = eK/F − 1 = m − 1. Then from the conductor formula (2.3) we can easily see that a(ρ) = m is minimal if and only if a(χK ) = 1.

Further, for some extension L/K, if NL/K = Ker(χK), then by class ﬁeld theory we can conclude: L/K is tamely ramiﬁed if and only if a(χK ) = 1.

Now suppose that ρ is a Heisenberg representation of G := GF /VF of dimension m prime to p. This implies VF ⊂ Ker(ρ) = Ker(χK) = NL/K, where L/K is some tamely ramiﬁed extension. Then a(χK ) = 1, hence a(ρ) = m is minimal.

Conversely, when conductor a(ρ) = m is minimal, we have a(χK ) = 1. By class field theory this character χK determines an extension L/K such that NL/K = Ker(χK). Since a(χK ) = 1, here L/K must be tamely ramified, hence L/F is tamely ramified. This means VF sits in the kernel Ker(ρ) = GL , therefore ρ is actually a representation of GF /VF . AN EXTENSION OF DELIGNE-HENNIART’S TWISTING FORMULA AND ITS APPLICATIONS 19

5. Proof of Theorem 1.2 and Theorem 1.3

Theorem 5.1. Let ρ = ρ(Xη, χK ) = ρ0 ⊗ χfF be a Heisenberg representation of the absolute Galois group GF of a non-archimedean local ﬁeld F/Qp of dimension m with gcd(m, p) = 1, where ρ0 = ρ0(Xη, χ0) is a minimal conductor Heisenberg representation of GF and χfF : × × × GF → C corresponds to χF : F → C by class ﬁeld theory. If a(χF ) > 2, then we have

m (5.1) W (ρ, ψ) = W (ρ0 ⊗ χfF , ψ) = W (χF , ψ) · det(ρ0)(c), × where ψ is a nontrivial additive character of F , and c := c(χF , ψ) ∈ F , satisﬁes a(χF ) −1 a(χF )−[ 2 ] χF (1 + x) = ψ(c x) for all x ∈ PF .

Proof. Step-1: By the given conditions,

ρ = ρ0 ⊗ χχgF , where ρ0 is a minimal conductor Heisenberg representation of GF of dimension m prime to × × × 2 p and χfF : GF → C corresponds to χF : F → C by class ﬁeld theory. And here × ×m × χ : F /F → C such that ρ0 = ρ0 ⊗ χe. 1 Let ζ be a (qF − 1)-st root of unity. Since UF is a pro-p-group and gcd(p, m) = 1, we have × ×m 1 m m 1 ∼ (5.2) F /F =< πF > × < ζ > ×UF / < πF > × < ζ > ×UF = Zm × Zm, that is, a direct product of two cyclic group of same order. Hence F ×/F ×m ∼= F ×\/F ×m. ×m m m 1 × ×m ∼ Since F =< πF > × < ζ > ×UF , and F /F = Zm × Zm, we have a(χ) 6 1 and #χ × ×m × is a divisor of m for all χ ∈ F \/F . Now if we take a character χF of F conductor > 2, × ×m hence a(χF ) > 2a(χ) for all χ ∈ F \/F . Then by using Deligne’s formula (1.3) we have

m m m m (5.3) W (χχF , ψ) = χ(c) · W (χF , ψ) = W (χF , ψ) ,

× where c ∈ F with νF (c) = a(χF ) + n(ψ), satisﬁes

−1 × χF (1 + x) = ψ(c x), for all x ∈ F with 2νF (x) > a(χ).

Now from Proposition 4.11 we can consider the representation ρ0 as a representation of ¯ G := Gal(Fmt/F), where Fmt/F is the maximal tamely ramiﬁed subextension in F /F . Then we can write

ρ0 = IndE/F(χE,0), ρ = IndE/F(χE), where E/F is a cyclic tamely ramiﬁed subextension of K/F of degree m and

χE := χE,0 ⊗ (χF ◦ NE/F ).

Step-2: Let G be a ﬁnite group and R(G) be the character ring provided with the tensor product as multiplication and the unit representation as unit element. Then for any zero

2 2 × ×m We also know that there are m characters of F /F such that ρ0 ⊗χe = ρ0 (cf. [38], p. 303, Proposition 1.4). So we always have:

ρ = ρ0 ⊗ χfF = ρ0 ⊗ χχgF , × ×m × where χ ∈ F \/F , and χfF : GF → C corresponds to χF by class ﬁeld theory. 20 BISWAS dimensional representation π ∈ R(G) we have (cf. Theorem 2.1(h) on p. 40 of [7]):

X G (5.4) π = nH IndH(χH − 1H), H6G where nH ∈ Z (cf. Proposition 2.24 on p. 48 of [7]) and χH ∈ Hb. Moreover, from Theorem 2.1 (k) of [7] we know that nH 6= 0 only if Z(G) 6 H and χH |Z(G) = χZ , where Z(G) is the center of G and χZ is the center character.

Now we will use this above formula (5.4) for the representation ρ0 − m · 1F and we get r X (5.5) ρ0 − m · 1F = niIndEi/F(χEi − 1Ei ), i=1 where Ei/F are intermediate ﬁelds of K/F and for nonzero ni, we have the relation

χ0 = χEi ◦ NK/Ei .

3 Since a(χ0) = 1 and K/Ei are cyclic tamely ramiﬁed , we have a(χEi ) = 1 for all i = 1, 2, ··· , r. Then from equation (5.5) we have r Y ni × (5.6) det(ρ0) = (χEi |F ) . i=1 Step-3: From equation (5.5) we also can write r X (5.7) ρ ⊗ χχ − m · χχ = n Ind (χ θ − θ ), 0 gF F i Ei/F Ei i i i=1 where θi := χχF ◦ NEi/F for all i ∈ {1, 2, ··· , r}. Since a(χF ) > 2, and Ei/F are tamely ramiﬁed, the conductors a(θi) > 2 for all i ∈ {1, 2, ··· , r}. Then from equation (5.7) we can write r m Y ni W (ρ, ψ) = W (χχF , ψ) · W (χEi θi − θi, ψEi ) i=1 r ni m Y W (χEi θi, ψEi ) = W (χχF , ψ) · W (θ , ψ )ni i=1 i Ei r m Y ni (5.8) = W (χχF , ψ) · χEi (ci) , i=1 × where ψEi = ψ ◦ TrEi/F and ci ∈ Ei such that

a(θi) y a(θi)−[ 2 ] θi(1 + y) = ψE ( ), for all y ∈ P . i ci Ei

Moreover, here Ei/F are tamely ramiﬁed extensions, then from Lemma 18.1 of [8] on p. 123, we have ∼ a(χF) NEi/F (1 + y) = 1 + TrEi/F(y) (mod PF ),

3 The subﬁelds Ei ⊆ K are related to Boltje’s approach by Gal(K/Ei) = Hi/Z(G) and the fact that

χHi extends the character χZ which translates via class ﬁeld theory to χEi ◦ NK/Ei = χK . Moreover, X = χZ ◦ [−, −] and χZ extendible to Hi means that Gal(K/Ei) = Hi/Z(G) must be isotropic for X, hence in our situation K/Ei must be a cyclic extension of degree dividing m. AN EXTENSION OF DELIGNE-HENNIART’S TWISTING FORMULA AND ITS APPLICATIONS 21

a(χ )−[ a(χF) ] a(θ )−[ a(θi) ] and Tr (y) ∈ P F 2 . Therefore for all y ∈ P i 2 we can write (cf. Proposition Ei/F F Ei 18.1 on p. 124 of [8]):

θi(1 + y) = χχF ◦ NEi/F (1 + y) = χF (1 + TrEi/F(y)) Tr (y) y = ψ( Ei/F ) = ψ ( ), c Ei c

a(χF ) x a(χF )−[ 2 ] where c := c(χF , ψ) for which χF (1 + x) = ψ( c ) for x ∈ PF . This variﬁes that × the choice ci(θi, ψEi ) = ci(χχF , ψEi ) = c(χF , ψ) ∈ F is right for applying Tate-Lamprecht formula (cf. [5]). Then by using equation (5.6) in equation (5.8) we have:

m (5.9) W (ρ, ψ) = W (χχF , ψ) · det(ρ0)(c).

Finally, by using equation (5.3) from equation (5.9) we can write

m W (ρ, ψ) = W (χχF , ψ) · det(ρ0)(c) m = W (χF , ψ) · det(ρ0)(c).

Now we are in a position to prove our Theorem 1.3.

Theorem 5.2. Let σ be an arbitrary finite dimension complex representation of GF and let ψ be a nontrivial additive character of F . Let ρm = ρm(Xη, χK ) = ρ0 ⊗ χfF be a Heisenberg representation of the absolute Galois group GF of a non-archimedean local field F/Qp of dimension m with gcd(m, p) = 1, where ρ0 = ρ0(Xη, χ0) is a minimal conductor Heisenberg × × × representation of GF and χfF : GF → C corresponds to χF : F → C by class field theory. If a(χF ) > 2 and j(ρm) > 2 · β(σ), then we have

dim(σ⊗ρm) dim(σ) (5.10) W (σ ⊗ ρm, ψ) = det(σ)(γ) · W (χF , ψ) · det(ρ0)(c ).

× Here c ∈ F is same as in Theorem 1.2 and νF (γ) = a(ρm) + m · n(ψ).

Proof. The proof of the above assertion is the combination of Theorem 1.2, and Deligne- Henniart Theorem 1.1. Now from the representation σ, we deﬁne a zero dimensional virtual representation as follows:

σ0 := σ − dim(σ) · 1GF . Then

(5.11) det(σ0) = det(σ − dim(σ) · 1GF ) = det(σ).

Now we want to use Theorem 1.1 for the representation σ0. It can be seen that β(σ0) =

β(ρ − dim(σ) · 1GF ) = β(σ). Hence σ0 satisﬁes the condition: j(ρm) > 2 · β(σ) = 2 · β(σ0). 22 BISWAS

Therefore we can use Theorem 1.1 for the representation σ0, hence we can write

W (σ0 ⊗ ρm, ψ) = det(σ0)(γ)

= W ((σ − dim(σ) · 1GF ) ⊗ ρm, ψ) − dim(σ) = W (σ ⊗ ρm, ψ) · W (ρm, ψ) . This gives

dim(σ) dim(σ) (5.12) W (σ ⊗ ρm, ψ) = det(σ0)(γ) · W (ρm, ψ) = det(σ)(γ) · W (ρm, ψ) . Now we use Theorem 1.2 in equation (5.12) and we obtain

dim(σ⊗ρm) dim(σ) W (σ ⊗ ρm, ψ) = det(σ)(γ) · W (χF , ψ) · det(ρ0)(c ).

Remark 5.3. Suppose ρ is any arbitrary ﬁnite dimensional Galois representation and W (ρ, ψ) and W (ρ ⊗ χ, ψ) are explicitly known. Then by above method, under some conditions on jump of σ one can give explicit twisting formulas for W (σ ⊗ ρ, ψ) and W (σ ⊗ ρ0, ψ), where ρ0 := ρ ⊗ χ as the above proof.

6. Applications 6.1. Invariant root number formula for Heisenberg representations. By the construction for Heisenberg representations ρ = ρ(X, χK ) of GF , we can write

ρ = IndE/F (χE), where K/E/F is ﬁxed ﬁeld of a maximal isotropic subgroup H = Gal(F /E) of ρ. Then root number of ρ is

(6.1) W (ρ, ψ) = λE/F (ψ) · W (χE, ψ).

Here λE/F (ψ) := W (IndE/F (1E), ψ), the Langlands λ-function for the extension E/F (cf. [1], [2]). Since for a given Heisenberg representation ρ, the maximal isotropic subgroups for ρ are not unique, there will be many maximal isotropic subgroups H for ρ, hence their fixed fields E. Suppose that for a Heisenberg representation ρ, we have two different maximal isotropic subgroups H1, and H2 of G. Let E1 and E2 be the fixed fields of H1 and H2 respectively. Then we can write

(1) ρ = IndE1/F (χE1 ), (2) ρ = IndE2/F (χE2 ). Then

(6.2) W (ρ, ψ) = λE1/F (ψ) · W (χE1 , ψ) = λE2/F (ψ) · W (χE2 , ψ).

Now if we notice equation (6.2), the right hand side depends on Ei (i = 1, 2). But root number of ρ is unique for its equivalence classes. Therefore, to give explicit formula for

W (ρ, ψ), one needs to give a formula (invariant) which is independent of the choice of Ei.

For a Heisenberg representation ρ of GF dimension prime to p, one can see explicit invariant formula for W (ρ, ψ) in [6]. AN EXTENSION OF DELIGNE-HENNIART’S TWISTING FORMULA AND ITS APPLICATIONS 23

Example 6.1. Let ρ = ρ(X, χK ) be a 2 - dimensional Heisenberg representation of GF , where F/Qp and p 6= 2 (see Example (7.1) in Appendix for explicit description for 2-dimensional ∼ Heisenberg representations). Then we will have Gal(K/F ) = Z/2Z × Z/2Z. Therefore, there are three maximal isotropic subgroups Hi(i = 1, 2, 3) for ρ, and hence Ei/F (i = 1, 2, 3) are three quadratic extensions of F . It can be proved (cf. [1], Lemma 4.7 on pp. 191-192) that

λEi/F (ψ) are not same. Similarly, it also can be proved that W (χEi , ψ) are also not same. Therefore we simply cannot use equation (6.2) for giving explicit formula for W (ρ, ψ).

Let ρ be a U-isotropic Heisenberg of GF . Then the alternating character X = Xρ cor- × responds to a character η : UF → C of the group of units. Then we have the unique decomposition of η as follows: 0 η = ηp · η ,

0 where η is of order prime to p and the order of ηp is a power of p. Correspondingly:

0 0 0 X = Xp · X , where ηp ↔ Xp, η ↔ X .

For the representation ρ this means:

0 (6.3) ρ = ρp ⊗ ρ

r 0 0 0 0 where dim(ρp) = p (r > 0), and dim(ρ ) =: m , and gcd(m , p) = 1, m |(qF − 1). Moreover expression (6.3) is not unique because

0 −1 0 ρp ⊗ ρ = ρpω ⊗ ωρ

× 0 0 for any character ω of F . Thus for appropriate ω we may assume that aF (ρ ) = m is minimal and aF (ρp) = mpa, where a > aF (ηp). Then here we can use Theorem 1.3 to give 0 0 an invariant formula for W (ρ, ψ) = W (ρp ⊗ ρ ) when j(ρ ) > 2 · j(ρp).

Theorem 6.2 (Invariant Formula). Let ρ be a U-isotropic Heisenberg representation of GF r of the form ρ = ρp ⊗ ρm with dim(ρp) = p (r > 1), and dim(ρm) = m, and gcd(m, p) = 1. Let ψ be an nontrivial additive character of F . If the jump: j(ρm) > 2 · j(ρp), then we have

dim(ρ) pr W (ρ, ψ) = W (ρp ⊗ ρm, ψ) = det(ρp)(γ) · W (χF , ψ) det(ρ0)(c ).

Here χF , c, ρ0 are same as in Theorem 1.2, and νF (γ) = a(ρm) + m · n(ψ).

Proof. The idea of the proof is same as Theorem 1.3. Here just replace σ by ρp. And another important thing is that the representation ρp is Heisenberg, hence irreducible. Therefore β(ρp) = j(ρp). Remark 6.3. Since maximal isotropic subgroups for ρ are not unique, giving invariant for det(ρ) we cannot simply use Gallagher’s result (cf. Theorem 30.1.6 of [28]):

G G (6.4) det(ρ)(g) = det(IndH (χH ))(g) = ∆H (g) · χH (TG/H (g)), for all g ∈ G,

G G where ∆H is the determinant of IndH (1H ), and TG/H is the transfer map from G to H. For invariant formula of det(ρ), one can see Theorem 5.1 and Theorem 5.1.A of [4]. 24 BISWAS

6.2. Converse theorem in the Galois side. It is well-known the answer of the following question: How to construct a modular form from a given Dirichlet series with ’nice’ properties (e.g., analytic continuation, moderate growth, functional equation), i.e., starting with the series ∞ X an L(s) = , ns n=1 under what conditions is the function ∞ X 2πinz f(z) = ane n=1 a modular form for some Fuchsian group? The answer of this question is known as the classical converse theorem in number theory (cf. [16], [20], [42]). The classical converse theorems establish a one-to-one correspondence between “nice” Dirichlet series and automorphic functions. Traditionally, the converse theorems have provided a way to characterize Dirichlet series associated to modular forms in terms of their analytic properties. The modern version of the classical converse theorems are stated in terms of automorphic representations instead of modular forms. Again we know that via the Langlands local correspondence that automorphic representations are associated with the Galois representations. Therefore one can ask the following questions: (a). Is there any converse theorems for automorphic representations (automrphic side of the converse theorem)? (b). Similarly, is there any converse theorem for Galois representations (Galois side of the converse theorem)? The answer of (a) is YES. For local converse theorems in the automorphic side, here we L refer [26]. Let G be a reductive group over a p-adic local ﬁeld F/Qp. Let G be the Langlands dual group of G, which is a semi-product of the complex dual group G∨ and the absolute L Galois group GF := Gal(F/F). Let φ : WF × SL2(C) → G be a continuous homomorphisms, and which is admissible. The G∨-conjugacy class of such a homomorphism φ is called a local Langlands parameter. Let Φ(G/F ) be the set of local Langlands parameters and let Π(G/F ) be the set of equivalence classes of irreducible admissible representations of G(F ). The local Langlands conjecture (cf. [17], [18], [19], [23], [30]) for G over F asserts that for each local Langlands parameter φ ∈ Φ(G/F ), there should be a ﬁnite subset Π(φ), which is called the local L-packet attached to φ such that the set {Π(φ)|φ ∈ Φ(G/F )} is a partition of Π(G/F ), among other required properties. The map φ 7→ Π(φ) is called the local Langlands correspondence or local Langlands reciprocity law for G over F .

Remark 6.4 (γ-factors). We deﬁne the local γ-factors as follows(cf. [25]):

V V L(1 − s, π1 × π2 ) (6.5) γ(s, π1 × π2, ψ) := W (s, π1 × π2, ψ) · . L(s, π1 × π2)

On the WF ×SL2(C) side, one deﬁnes the γ-factor in the same way [35]. For more information about local factors, we refer [26], [14], [15]. AN EXTENSION OF DELIGNE-HENNIART’S TWISTING FORMULA AND ITS APPLICATIONS 25

(a) Converse theorem in the automorphic side: Roughly, the local converse theorem is to find the smallest subcollection of twisted local γ-factors γ(s, π × τ, ψ) which classifies the irreducible admissible representations π up to isomorphism. But this is usually not the case in general. From the local Langlands conjecture, one may expect a certain subcollection of local γ-factors classifies the irreducible representation π up to L-packet. On the other hand, if the irreducible admissible representations under consideration have additional structures, then one may still expect a certain subcollection of local γ-factors classifies the irreducible representation π up to equivalence.

For GLn(F ), we have the following theorem.

Theorem 6.5 (Jacquet-Liu, 2016, [24]). Let π1, π2 be irreducible generic representations of GLn(F ). Suppose that they have the same central character. If

γ(s, π1 × τ, ψ) = γ(s, π2 × τ, ψ) as functions of the complex variable s, for all irreducible generic representations τ of GLr(F ) n ∼ with 1 6 r 6 [ 2 ], then π1 = π2. Remark 6.6. The above Jacquet-Liu’s theorem was ﬁrst conjectured by Jacquet (cf. Conjec- ture 1.1 of [27]). On p. 170 of [8] one also can see converse theorem for GL2(F ) in terms of L- and -factors. For further details about local converse theorems, one can see [22], [9], [10].

(b) Converse theorem in the Galois side:

So far in the Galois side, we do not have any converse theorem as like GLn side except Volker Heiermann’s [21] work, because we do not have general twisting formula for local root numbers for Galois representations. Therefore, to translate local converse theorems to Galois side, we need general twisting formula. By using Theorem 1.3, we have the following converse theorem in the Galois side.

Theorem 6.7 (Converse Theorem in the Galois side). Let ρm = ρ0 ⊗ χfF be a U- isotopic Heisenberg representation of GF of dimension prime to p. Let ψ be a nontrivial additive character of F . Let ρ1, ρ2 be two ﬁnite dimensional complex representations of GF with

det(ρ1) ≡ det(ρ2), and j(ρm) > 2 · max{β(ρ1), β(ρ2)}. If

W (ρ1 ⊗ ρm, ψ) = W (ρ2 ⊗ ρm, ψ),

× × then ρ1 ≡ ρ2 or ρ1 ≡ ρ2 ⊗ µ, where µ : F → C is an unramiﬁed character whose order divides dim(ρi), i = 1, 2.

Proof. By the given condition j(ρm) > 2 · max{β(ρ1), β(ρ2)}, then by using Theorem 1.3, we can write

(6.6) W (ρ1 ⊗ ρm, ψ) = W (ρ2 ⊗ ρm, ψ) =⇒

dim(ρ1) dim(ρ2) det(ρ1)(γ)W (ρm, ψ) = det(ρ2)(γ)W (ρm, ψ) . 26 BISWAS

Again by using Theorem 1.2, from equation (6.6), we have (6.7) dim(ρ1⊗ρm) dim(ρ1) dim(ρ2⊗ρm) dim(ρ2) det(ρ1)(γ)·W (χF , ψ) ·det(ρ0)(c ) = det(ρ2)(γ)·W (χF , ψ) ·det(ρ0)(c ).

× Since det(ρ1) ≡ det(ρ2) on F , and χF is arbitrary character of a(χF ) > 2, then from above equation (6.7), we can conclude that dim(ρ1) = dim(ρ2). This gives: ∼ × × Case-1: ρ1 = ρ2 ⊗ µ, where µ : GF → C is a character of GF with , hence µ can be considered a character of F × (via class ﬁeld theory).

Case-2: ρ1 ≡ ρ2, and it is one of our assertions.

When we are in Case-1, by the given assumption det(ρ1) ≡ det(ρ2) = det(ρ1 ⊗ µ) = dim(ρ1) det(ρ1) · µ , then the order of µ must be a divider of dim(ρ1) = dim(ρ2). Now we are left to prove that µ is unramiﬁed, and which follows from the given condition

W (ρ1 ⊗ ρm, ψ) = W (ρ2 ⊗ ρm, ψ). This completes the proof.

7. Appendix Example 7.1 (Explicit description of Heisenberg representations of dimension prime to p). Let F/Qp be a local ﬁeld, and GF be the absolute Galois group of F . Let ρ = ρ(X, χK ) be a Heisenberg representation of GF of dimension m prime to p. Then from 1 × Corollary 3.9 the alternating character X = Xη is U-isotropic for a character η : UF /UF → C . p × Here from Lemma 3.6 we can say m = [F : Rad(Xη)] = #η divides qF − 1. 1 1 m 1 ×m Since UF is a pro-p-group and gcd(m, p) = 1, we have (UF ) = UF ⊂ F , and therefore

× ×m ∼ F /F = Zm × Zm, is a bicyclic group of order m2. So by class ﬁeld theory there is precisely one extension K/F ∼ × ×m such that Gal(K/F) = Zm × Zm and the norm group NK/F := NK/F (K ) = F . 1 1 ∼ 1 We know that UF /UF is a cyclic group of order qF − 1, hence U\F /UF = UF /UF . By the 1 given condition m|(qF − 1), hence UF /UF has exactly one subgroup of order m. Then number 1 of elements of order m in UF /UF is ϕ(m), the Euler’s ϕ-function of m. In this setting, we 1 ∼ × 1 1 have η ∈ U\F /UF = FF \/UF ∧ UF with #η = m. This implies that up to 1-dimensional 1 × character twist there are ϕ(m) representations corresponding to Xη where η : UF /UF → C is of order m. According to Corollary 1.2 of [38], all dimension-m-Heisenberg representations of GF = Gal(F/F) are given as

(1H) ρ = ρ(Xη, χK ),

× × × where χK : K /IF K → C is a character such that the restriction of χK to the subgroup × KF corresponds to Xη under the map (3.3), and

× ×m × ×m ∼ × × (2H) F /F ∧ F /F = KF /IF K , AN EXTENSION OF DELIGNE-HENNIART’S TWISTING FORMULA AND ITS APPLICATIONS 27 which is given via the commutator in the relative Weil-group WK/F (for details arithmetic description of Heisenberg representations of a Galois group, see [38], pp. 301-304). The condition (2H) corresponds to (3.3). Here the above Explicit Lemma ?? comes in. Here due to our assumption both sides of (2H) are groups of order m. And if one choice

χK = χ0 has been ﬁxed, then all other χK are given as

(7.1) χK = (χF ◦ NK/F ) · χ0,

× for arbitrary characters of F . For an optimal choice χK = χ0, and order of χ0 we need the following lemma.

× Lemma 7.2. Let K/F be the extension of F/Qp for which Gal(K/F) = Zm × Zm. The KF × and IF K are as above. Then the sequence

N 1 × 1 × 1 × K/F 1 ×m (7.2) 1 → UK KF /UK IF K → UK /UK IF K −−−→ UF /UF → UF /UF ∩ F → 1 is exact, and the outer terms are both of order m, hence inner terms are both cyclic of order qF − 1.

×m × × ×m ∼ Proof. The sequence is exact because F = NK/F (K ) is the group of norms, and F /F = 4 Zm × Zm implies that the right hand term is of order m. By our assumption the order of × × KF /IF K is m. Now we consider the exact sequence

1 × 1 × × × 1 × 1 × (7.3) 1 → UK ∩ KF /UK ∩ IF K → KF /IF K → UK KF /UK IF K → 1.

1 Since the middle term has order m, the left term must have order 1, because UK is a pro- p-group and gcd(m, p) = 1. Hence the right term is also of order m. So the outer terms of the sequence (7.2) have both order m, hence the inner terms must have the same order 1 1 1 qF − 1 = [UF : UF ], and they are cyclic, because the groups UF /UF and UK /UK are both cyclic.

We now are in a position to choose χK = χ0 as follows: × 1 × (1) we take χ0 as a character of K /UK IF K , 1 × 1 × (2) we take it on UK KF /UK IF K as it is prescribed by the above Explicit Lemma ??, in particular, χ0 restricted to that subgroup (which is cyclic of order m) will be faithful. 1 × (3) we take it trivial on all primary components of the cyclic group UK /UK IF K which Qn ai are not pi-primary, where m = i=1 pi . (4) we take it trivial for a ﬁxed prime element πK .

Under the above optimal choice of χ0, we have

4Since gcd(m, p) = 1, we have

×m 1 m m 1 m 1 UF · F = (< ζ > ×UF )(< πF > × < ζ > ×UF ) =< πF > × < ζ > ×UF , where ζ is a (qF − 1)-st root of unity. Then

×m ×m ×m m 1 m m 1 ∼ UF /UF ∩ F = UF · F /F =< πF > × < ζ > ×UF / < πF > × < ζ > ×UF = Zm.

×m Hence |UF /UF ∩ F | = m. 28 BISWAS

νp(n) Lemma 7.3. Denote νp(n) := as the highest power of p for which p |n. The character χ0 must be a character of order

Y νl(qF −1) mqF −1 := l , l|m which we will call the m-primary part of qF − 1, so it determines a cyclic extension L/K of degree mqF −1 which is totally tamely ramiﬁed, and we can consider the Heisenberg representation ρ = (X, χ0) of GF = Gal(F/F) is a representation of Gal(L/F), which is of order 2 m · mqF −1.

Proof. By the given conditions, m|qF − 1. Therefore we can write

Y νl(qF −1) Y νp(qF −1) Y νp(qF −1) qF − 1 = l · p = mqF −1 · p , l|m p|qF −1, p-m p|qF −1, p-m

Q νl(qF −1) where l, p are prime, and mqF −1 = l|m l . From the construction of χ0, πK ∈ Ker(χ0), hence the order of χ0 comes from the restriction 1 to UK . Then the order of χ0 is mqF −1, because from Lemma 7.2, the order of UK /UK IF K is qF − 1. Since order of χ0 is mqF −1, by class ﬁeld theory χ0 determines a cyclic extension L/K of degree mqF −1, hence × NL/K (L ) = Ker(χ0) = Ker(ρ).

This means GL is the kernel of ρ(X, χ0), hence ρ(X, χ0) is actually a representation of ∼ GF /GL = Gal(L/F).

Since GL is normal subgroup of GF , hence L/F is a normal extension of degree [L : F ] = 2 2 [L : K] · [K : F ] = mqF −1 · m . Thus Gal(L/F) is of order m · mqF −1.

Moreover, since [L : K] = mqF −1 and gcd(m, p) = 1, L/K is tame. By construction we × have a prime πK ∈ Ker(χ0) = NL/K(L ), hence L/K is totally ramiﬁed extension. Lemma 7.4. (Here L, K, and F are the same as in Lemma 7.3) Let F ab/F be the maximal abelian extension. Then we have

L ⊃ L ∩ F ab ⊃ K ⊃ F, {1} ⊂ G0 ⊂ Z(G) ⊂ G = Gal(L/F),

ab 0 where [L : L ∩ F ] = |G | = m and [L : K] = |Z(G)| = mqF −1.

Proof. Let F ab/F be the maximal abelian extension. Then we have

L ⊃ L ∩ F ab ⊃ K ⊃ F.

Here L ∩ F ab/F is the maximal abelian in L/F . Then from Galois theory we can conclude

Gal(L/L ∩ Fab) = [Gal(L/F), Gal(L/F)] =: G0.

Since Gal(L/F) = GF/Ker(ρ), and [[GF ,GF ],GF ] ⊆ Ker(ρ), from relation (3.3) we have

0 ∼ × × G = [GF ,GF ]/Ker(ρ) ∩ [GF, GF] = [GF, GF]/[[GF, GF], GF] = KF /IFK .

From the Heisenberg property of ρ, we have [[GF ,GF ],GF ] ⊆ Ker(ρ), hence Gal(L/F) = 0 0 GF/Ker(ρ) is a two-step nilpotent group. This gives [G ,G] = 1, hence G ⊆ Z := Z(G). Thus G/Z is abelian. Moreover, here Z is the scalar group of ρ, hence the dimension of ρ is:

dim(ρ) = p[G : Z] = m

Therefore the order of Z is mqF −1 and Z = Gal(L/K). Remark 7.5 (Special case: m = 2, hence p 6= 2). Now if we take m = 2, hence p 6= 2, and choose χ0 as the above optimal choice, then we will have mqF −1 = 2qF −1 = 2-primary factor of the number qF − 1, and Gal(L/F) is a 2-group of order 4 · 2qF −1. When qF ≡ −1 (mod 4), qF is of the form qF = 4l − 1, where l > 1. So we can write qF − 1 = 2(2l − 1). Since 2l − 1 is always odd, therefore when qF ≡ −1 (mod 4), the order of χ0 is 2qF −1 = 2. Then Gal(L/F) will be of order 8 if and only if qF ≡ −1 (mod 4), i.e., if and only if i 6∈ F . And if qF ≡ 1 (mod 4), then similarly, we can write qF − 1 = 4m for some integer m > 1, hence 2qF −1 > 4. Therefore when qF ≡ 1 (mod 4), the order of Gal(L/F) will be at least 16.

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Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, Copenhagen 2100, Denmark Email address: [email protected], [email protected]