An extension of Deligne-Henniart's twisting formula and its applications
AAN EXTENSION OF DELIGNE-HENNIART’S TWISTING FORMULAAND ITS APPLICATIONS
SAZZAD ALI BISWAS
Abstract.
Let F/ Q p be a non-Archimedean local field, and G F be the absolute Galoisgroup of F . Let ρ and ρ be two finite dimensional complex representations of G F . Let ψ be a nontrivial additive character of F . Then question is: What is the twisting formula for the root number W ( ρ ⊗ ρ , ψ ) ? In general, answer of this question is not known yet. But if one of ρ i ( i = 1 ,
2) is one-dimensional with “sufficiently” large conductor, then in [13] Deligne gave a twisting formulafor W ( ρ ⊗ ρ , ψ ). Later, in [12], Deligne and Henniart give a general twisting formula fora zero dimensional virtual representation twisted by a finite dimensional representation of G F . In this paper, first we extend Deligne’s twisting formula for Heisenberg representationof dimension prime p , then we further extend Deligne-Henniart’s result.Finally, we give two very important applications of our twisting formula – invariant formulaof local root numbers for U-isotropic Heisenberg representations and a converse theorem inthe Galois side. Introduction
Let F be a non-Archimedean local field of characteristic zero, i.e., a finite extension of Q p ,where p is a prime. Let G F be the absolute Galois representation of F . Let ψ be a nontrivialadditive character of F . For a given multiplication character χ : F × → C × of F , we haveexplicit formula for the root number W ( χ, ψ ) (cf. [35]). If χ and χ are two unramifiedcharacters of F × and ψ is a nontrivial additive character of F , we have(1.1) W ( χ χ , ψ ) = W ( χ , ψ ) W ( χ , ψ ) . Further, let χ be ramified and χ unramified then (cf. [35], (3.2.6.3))(1.2) W ( χ χ , ψ ) = χ ( π F ) a ( χ )+ n ( ψ ) · W ( χ , ψ ) . Here a ( χ ) (resp. n ( ψ )) is the conductor of χ (resp. ψ ). We also have twisting formula oflocal root numbers by Deligne (cf. [13], Lemma 4.16) under some special condition and whichis as follows (for proof, see Corollary 3.2 (2) of [5]): Let χ and χ be two multiplicative characters of a local field F such that a ( χ ) (cid:62) · a ( χ ) .Let ψ be an additive character of F . Let y χ ,ψ be an element of F × such that χ (1 + x ) = ψ ( y χ ,ψ x ) for all x ∈ F with valuation ν F ( x ) (cid:62) a ( χ )2 (if a ( χ ) = 0 , y χ ,ψ = π − n ( ψ ) F ). Then (1.3) W ( χ χ , ψ ) = χ − ( y χ ,ψ ) · W ( χ , ψ ) . Mathematics Subject Classification. a r X i v : . [ m a t h . N T ] J a n 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-acter ω s ( x ) := q − sν F ( x ) F we have the following twisting formula (cf. [35] (3.4.5)):(1.4) W ( ρω s , ψ ) = W ( ρ, ψ ) · ω s ( c ρ,ψ )for any c = c ρ,ψ such that ν F ( c ) = a ( ρ ) + n ( ψ )dim( ρ ), where a ( ρ ) is the Artin conductor ofthe representation ρ .Let ρ and ρ be two arbitrary finite dimensional representations of G F . Now question is: Is there any explicit formula for W ( ρ ⊗ ρ , ψ ) ? The answer is: NO . But under some special conditions –when any of ρ i ( i = 1 ,
2) is one dimensional with sufficiently large conductor , then Deligne gives an explicit formula for W ( ρ ⊗ ρ , ψ ) (cf. [13], Subsection 4.1): Let ρ = ρ be a finite dimensional representation of G F and let ρ = χ be any nontrivialcharacter 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 − ) . 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, wegive 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 localroot numbers for any virtual representations of G F . So, now if we define a zero-dimensional virtual representation ρ := ρ − dim( ρ ) · G F , where 1 G F is the trivial representation of G F ,by the representation ρ , then from above equation (1.5) we have(1.6) W ( ρ ⊗ χ, ψ ) = det( ρ )( 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 G F , and thecondition 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 avirtual representation of G F (without moderate component). There exists an element γ ∈ F × uniquely determined modulo U j ( ρ ) / − F , such that for any virtual representation ρ of G F ofdimension zero with verifying β ( ρ ) < j ( ρ ) / , then we have (1.7) W ( ρ ⊗ ρ, ψ ) = det( ρ )( γ ) . Here ν F ( γ ) = a ( ρ ) + dim( ρ ) · n ( ψ ) , and j ( ρ ) is the jump of ρ . β ( ρ ) is the maximum jumpamong all the components of ρ . N 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 ρ of G F of dimension prime to p and twisted by an arbitrarycharacter χ F with conductor a ( χ F ) (cid:62)
2. In our following twisting formula the conductor of χ F needs not be sufficiently large , but the Deligne’s twisting formula (1.5) is only true forthose characters with sufficiently large conductors. Theorem 1.2.
Let ρ = ρ ( X η , χ K ) = ρ ⊗ (cid:102) χ F be a Heisenberg representation of the absoluteGalois group G F of a non-archimedean local field F/ Q p of dimension m with gcd ( m, p ) = 1 ,where ρ = ρ ( X η , χ ) is a minimal conductor Heisenberg representation of G F and (cid:102) χ F : G F → C × corresponds to χ F : F × → C × by class field theory. If a ( χ F ) (cid:62) , then we have (1.8) W ( ρ, ψ ) = W ( ρ ⊗ (cid:102) χ F , ψ ) = W ( χ F , ψ ) m · det( ρ )( c ) , where ψ is a nontrivial additive character of F , and c := c ( χ F , ψ ) ∈ F × , satisfies χ F (1 + x ) = ψ ( c − x ) for all x ∈ P a ( χ F ) − [ a ( χF )2 ] F . Then by using Theorem 1.2 above and Deligne-Henniart’s theorem 1.1 we have the followingtwisting formula for the representation σ ⊗ ρ m , where σ is an arbitrary finite dimensionalrepresentation of G F and ρ m is a U -isotropic representation of dimension m prime to p . Theorem 1.3.
Let σ be an arbitrary finite dimension complex representation of G F and let ψ be a nontrivial additive character of F . Let ρ m = ρ m ( X η , χ K ) = ρ ⊗ (cid:102) χ F be a Heisenbergrepresentation of the absolute Galois group G F of a non-archimedean local field F/ Q p ofdimension m with gcd ( m, p ) = 1 , where ρ = ρ ( X η , χ ) is a minimal conductor Heisenbergrepresentation of G F and (cid:102) χ F : G F → C × corresponds to χ F : F × → C × by class field theory.If a ( χ F ) (cid:62) and j ( ρ m ) > · β ( σ ) , then we have (1.9) W ( σ ⊗ ρ m , ψ ) = det( σ )( γ ) · W ( χ F , ψ ) dim ( σ ⊗ ρ m ) · det( ρ )( c dim( σ ) ) . Here c ∈ F × is same as in Theorem 1.2 and ν F ( γ ) = a ( ρ m ) + m · n ( ψ ) . Finally, in Section 6, we also give two very important applications of Theorem 1.3, andthey 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 Heisenbergrepresentation ρ of G F is of the form dim ( ρ ) = p r · m , where r (cid:62) gcd ( m, p ) = 1, m | ( q F − U -isotropic representation ρ can be expressed as ρ = ρ p ⊗ ρ m , where ρ p and ρ m are U -isotropic representations of G F of dimension p r and m respectively. Therefore forgiving 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 G F of the form ρ = ρ p ⊗ ρ m with dim( ρ p ) = p r ( r (cid:62) , and dim( ρ m ) = m , and gcd ( m, p ) = 1 . Let ψ be an nontrivial additive character of F . If the jump: j ( ρ m ) > · j ( ρ p ) , then we have W ( ρ, ψ ) = W ( ρ p ⊗ ρ m , ψ ) = det( ρ p )( γ ) · W ( χ F , ψ ) dim( ρ ) det( ρ )( c p r ) . Here χ F , c , ρ are same as in Theorem 1.2, and ν F ( γ ) = a ( ρ m ) + m · n ( ψ ) . 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 ρ m = ρ ⊗ (cid:102) χ F be a U -isotopic Heisenberg representation of G F of dimension prime to p . Let ψ be a nontrivialadditive character of F . Let ρ , ρ be two finite dimensional complex representations of G F with det( ρ ) ≡ det( ρ ) , and j ( ρ m ) > · max { β ( ρ ) , β ( ρ ) } . If W ( ρ ⊗ ρ m , ψ ) = W ( ρ ⊗ ρ m , ψ ) , then ρ ≡ ρ or ρ ≡ ρ ⊗ µ , where µ : F × → C × is an unramified character whose orderdivides dim( ρ i ) , i = 1 , . Preliminaries and Notation
Let F be a non-archimedean local field of characteristic zero, i.e., a finite extension of thefield Q p (field of p -adic numbers), where p is a prime. Let K/F be a finite extension of thefield F . Let e K/F be the ramification index for the extension
K/F and f K/F be the residuedegree of the extension
K/F .Let O F be the ring of integers in the local field F and P F = π F O F the unique prime idealin O F and π F a uniformizer, i.e., an element in P F whose valuation is one, i.e., ν F ( π F ) = 1.Let U F = O F − P F be the group of units in O F . Let P iF = { x ∈ F : ν F ( x ) (cid:62) i } and for i (cid:62) U iF = 1 + P iF (with proviso U F = U F = O × F ). We also let a ( χ ) be the conductor ofnontrivial character χ : F × → C × , i.e., a ( χ ) is the smallest integer (cid:62) χ is trivialon U a ( χ ) F . 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 ψ is trivial on P − n ( ψ ) F , but nontrivial on P − n ( ψ ) − F .Let G F := Gal ( F /F ) (resp. W F ) 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 groupof the extension K/F . For each i (cid:62) − i -th ramification subgroup of G (in thelower numbering) as follows: G i = { σ ∈ G | v K ( σ ( α ) − α ) (cid:62) i + 1 for all α ∈ O K } .An integer t is called a ramification break or jump for the extension K/F or the ramifi-cation groups { G i } i (cid:62) − if G t (cid:54) = G t +1 . We also know that there is a decreasing filtration (with upper numbering) of G and which isdefined by the Hasse-Herbrand function Ψ = Ψ
K/F as follows: G u = G Ψ( u ) , where u ∈ R , u (cid:62) − . Since by the definition of Hasse-Herbrand function, Ψ( −
1) = − , Ψ(0) = 0, we have G − = G − = G , and G = G . Thus a real number t (cid:62) − K/F or the filtration { G i } i (cid:62) − if G t (cid:54) = G t + ε , for all ε > . N 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 theramification breaks for G are integers. But in general, the set of ramification breaks of aGalois 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 thecomplex representation ring of G . For any two representations ρ , ρ ∈ R ( G ) with characters χ , χ respectively, we have the Schur’s inner product: < ρ , ρ > G = < χ , χ > G := 1 | G | (cid:88) g ∈ G χ ( g ) · χ ( 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) i G ( g ) := inf { ν K (x − g(x)) | x ∈ O K } . By using this non-negative (when g (cid:54) = 1) integer i G ( g ) we define a function a G : G → Z asfollows: a G ( g ) = − f K/F · i G ( g ) when g (cid:54) = 1, and a G (1) = f K/F (cid:80) g (cid:54) =1 i G ( g ).Thus from this definition we can see that (cid:80) g ∈ G a G ( g ) = 0, hence < a G , G > = 0. It can beproved (cf. [32], p. 99, Theorem 1) that the function a G is the character of a linear represen-tation of G , and that corresponding linear representation is called the Artin representation A G of G .Similarly, for a nontrivial g (cid:54) = 1 ∈ G , we define (cf. [34], p. 247) s G ( g ) = inf { ν K (1 − g(x)x − ) | x ∈ K × } , s G (1) = − (cid:88) g (cid:54) =1 s G (g) . And we can define a function sw G : G → Z as follows:sw G (g) = − f K / F · s G (g)It can also be shown that sw G is a character of a linear representation of G , and that corre-sponding representation is called the Swan representation SW G of G .From [33], p. 160 , we have the relation between the Artin and Swan representations (cf.[34], p. 248, equation (6.1.9))(2.1) SW G = A G + Ind GG (1) − Ind G { } (1) ,G 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 a F ( ρ ) := < A G , ρ > G = < a G , χ > G , where χ is the character of the representation ρ . Similarly, for the representation ρ , the Swanconductor is: sw F ( ρ ) := < SW G , ρ > G = < sw G , χ > G . For more details about Artin and Swan conductor, see Chapter 6 of [34] and Chapter VI of[32].
BISWAS
From equation (2.1) we obtain(2.2) a F ( ρ ) = sw F ( ρ ) + dim( ρ ) − < , ρ > G . Moreover, from Corollary of Proposition 4 on p. 101 of [32], for an induced representation ρ := Ind Gal(K / F)Gal(K / E) ( ρ E ) = Ind E / F ( ρ E ), we have(2.3) a F ( ρ ) = f E/F · (cid:0) d E/F · dim( ρ E ) + a E ( ρ E ) (cid:1) . We apply this formula (2.3) for ρ E = χ E of dimension 1 and then conversely(2.4) a ( χ E ) = a F ( ρ ) f E/F − d E/F , where d E/F is the exponent of the different of the extension
E/F . So if we know a F ( ρ ) thenwe can compute the conductor a ( χ E ) of χ E . Definition 2.2 ( Jump for a representation).
Now let ρ be an irreducible representationof G . For this irreducible ρ we define jump for ρ as follows: j ( ρ ) := max { i | ρ | G i (cid:54)≡ } . Now if ρ is a ramified irreducible representation of G , then ρ | I (cid:54)≡
1, where I = G = G isthe inertia subgroup of G . Thus from the definition of j ( ρ ) we can say, if ρ is irreducible,then we always have j ( ρ ) (cid:62)
0, i.e., ρ is nontrivial on the inertia group G . Then from thedefinitions of Swan and Artin conductors, and equation (2.2), when ρ is irreducible , we havethe following relations(2.5) sw F ( ρ ) = dim( ρ ) · j( ρ ) , a F ( 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 sw F ( ρ ) = j( ρ ) , a F ( ρ ) = j( ρ ) + 1.Moreover, by class field theory, ρ corresponds to a linear character χ F , hence for linearcharacter χ F , we can write j ( χ F ) := max { i | χ F | U iF (cid:54)≡ } , because under class field theory (under Artin isomorphism) the upper numbering in thefiltration of Gal(F ab / F) is compatible with the filtration (descending chain) of the group ofunits U F .Similarly, one can define jump for any virtual representation of G F . Let ρ be a virtualrepresentation of G F . We denote j ( ρ ) (resp. β ( ρ )) the lower (resp. upper) bound of j ( ρ i ),when ρ i runs over all the components of ρ. That is, if ρ = n (cid:88) i n i ρ i , then j ( ρ ) (cid:54) { j ( ρ ) , · · · , j ( ρ n ) } (cid:54) β ( ρ ) . Further, j ( ρ ) (cid:62) α means that the components of ρ do not have any non-null vector fixed by G αF : ρ G αF = 0. And β ( ρ ) < β (we then have β >
0) means that ρ comes by inflation from avirtual representation of G F /G βF . N EXTENSION OF DELIGNE-HENNIART’S TWISTING FORMULA AND ITS APPLICATIONS 7
Heisenberg Representations.
Let G be a profinite group (in our case, G = G F ). Anirreducible representation ρ of G is called a Heisenberg representation if it represents com-mutators 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 degeneratespecial case. To classify Heisenberg representations we need to mention two invariants of anirreducible representation ρ ∈ Irr(G):(1) Let Z ρ be the scalar group of ρ , i.e., Z ρ ⊆ G and ρ ( z ) = scalar matrix for every z ∈ Z ρ . If V / C is a representation space of ρ we get Z ρ as the kernel of the compositemap(2.6) G ρ −→ GL C ( V ) π −→ P GL C ( V ) = GL C ( V ) / C × E, where E is the unit matrix and denote ρ := π ◦ ρ . Therefore Z ρ is a normal subgroupof G .(2) Let χ ρ be the character of Z ρ which is given as ρ ( g ) = χ ρ ( g ) · E for all g ∈ Z ρ .Apparently χ ρ is a G -invariant character of Z ρ 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 andTheorem 2) the set of isomorphism classes PI(A) of projective irreducible representations (forprojective representation, see [11], §
51) of A is in bijective correspondence with the set ofcontinuous alternating characters Alt(A). If ρ ∈ PI(A) corresponds to X ∈ Alt(A) thenKer( ρ ) = Rad(X) and [ A : Rad(X)] = dim( ρ ) ,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 aprojective irreducible representation of G and Z ρ is the kernel of ρ . Therefore modulo com-mutator group [ G, G ], we can consider that ρ is in PI(A) which corresponds an alternatingcharacter X of A with kernel of ρ is Z ρ / [ G, G ] = Rad(X). We also know that[ A : Rad(X)] = [G / [G , G] : Z ρ / [G , G]] = [G : Z ρ ] . Then we observe that dim( ρ ) = dim( ρ ) = (cid:113) [G : Z ρ ] . Let H be a subgroup of A , then we define the orthogonal complement of H in A withrespect to X H ⊥ := { a ∈ A : X ( a, H ) ≡ } . An isotropic subgroup H ⊂ A is a subgroup such that H ⊆ H ⊥ (cf. [37], p. 270, Lemma1(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 C G = G , C i +1 G = [ C i G, G ] denote the descending central series of G . Now assumethat 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(2.7) A ∧ Z A ∼ = C G/C G, a ∧ b (cid:55)→ [ˆ a, ˆ b ] BISWAS is an isomorphism.(2) The map ρ → X ρ ∈ Alt(A) from Heisenberg representations to alternating characterson A is surjective.3. U -isotropic Heisenberg representations Let F/ Q p be a local field, and F be an algebraic closure of F . Denote G F = Gal(F / F) theabsolute Galois group for
F /F . We know that (cf. [29], p. 197) each representation ρ : G F → GL ( n, C ) corresponds to a projective representation ρ : G F → GL ( n, C ) → P GL ( n, C ).On the other hand, each projective representation ρ : G F → P GL ( n, C ) can be lifted to arepresentation ρ : G F → GL ( n, C ). Let A F = G abF be the factor commutator group of G F .Define F F × := lim ←− ( F × /N ∧ F × /N )where N runs over all open subgroups of finite index in F × . Denote by Alt(F × ) as the set ofall alternating characters X : F × × F × → C × such that [ F × : Rad(X)] < ∞ . Then the localreciprocity map gives an isomorphism between A F and the profinite completion of F × , andinduces a natural bijection(3.1) PI(A F ) ∼ −→ Alt(F × ) , where PI(A F ) is the set of isomorphism classes of projective irreducible representations of A F .By using class field theory from the commutator map (2.7) (cf. p. 125 of [41]) we obtain(3.2) c : F F × ∼ = [ G F , G F ] / [[ G F , G F ] , G F ] . Let
K/F be an abelian extension corresponding to the norm subgroup N ⊂ F × and if W K/F denotes the relative Weil group, the commutator map for W K/F induces an isomorphism (cf.p. 128 of [41]):(3.3) c : F × /N ∧ F × /N → K × F /I F K × , where K × F := { x ∈ K × | N K/F ( x ) = 1 } , i.e., the norm-1-subgroup of K × , I F K × := { x − σ | x ∈ K × , σ ∈ Gal(K / F) } < K × F , 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 : F F × ∼ = lim ←− K × F /I F K × , where the limit on the right side refers to norm maps. This gives an arithmetic descriptionof Heisenberg representations of the group G F . Theorem 3.1 (Zink, [38], p. 301, Corollary 1.2) . The set of Heisenberg representations ρ of G F is in bijective correspondence with the set of all pairs ( X ρ , χ ρ ) such that:(1) X ρ is a character of F F × ,(2) χ ρ is a character of K × /I F K × , where the abelian extension K/F corresponds to theradical N ⊂ F × of X ρ , and(3) via (3.3) the alternating character X ρ corresponds to the restriction of χ ρ to K × F . N 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 G K := Gal(F / K) to G F :(3.5) (cid:112) [ F × : N ] · ρ = Ind K / F ( χ ) , where N and K are as in (2) of the above Theorem 3.1 and where the induction of χ (tobe considered as a character of G K by class field theory) produces a multiple of ρ . From[ F × : N ] = [ K : F ] we obtain the dimension formula: (3.6) dim( ρ ) = (cid:112) [F × : N] , where N is the radical of X .Let K/E be an extension of E , and χ K : K × → C × be a character of K × . In thefollowing lemma, we give the conditions of the existence of characters χ E ∈ (cid:99) E × such that χ E ◦ N K/E = χ K , and the solutions set of this χ E . Lemma 3.2.
Let
K/E be a finite extension of a field E , and χ K : K × → C × .(i) The existence of characters χ E : E × → C × such that χ E ◦ N K/E = χ K is equivalent to K × E ⊂ Ker( χ K ) .(ii) In case (i) is fulfilled, we have a well defined character (3.7) χ K/E := χ K ◦ N − K/E : N K/E → C × , on the subgroup of norms N K/E := N K/E ( K × ) ⊂ E × , and the solutions χ E such that χ E ◦ N K/E = χ K are precisely the extensions of χ K/E from N K/E to a character of E × .Proof. (i) Suppose that an equation χ K = χ E ◦ N K/E holds. Let x ∈ K × E , hence N K/E ( x ) = 1.Then χ K ( x ) = χ E ◦ N K/E ( x ) = χ E (1) = 1 . So x ∈ Ker( χ K ), and hence K × E ⊂ Ker( χ K ).Conversely assume that K × E ⊂ Ker( χ K ). Then χ K is actually a character of K × /K × E . Againwe have K × /K × E ∼ = N K/E ⊂ E × , hence (cid:92) K × /K × E ∼ = (cid:92) N K/E . Now suppose that χ K correspondsto the character χ K/E of N K/E . Hence we can write χ K ◦ N − K/F = χ K/E . Thus the character χ K/E : N K/E → C × is well defined. Since E × is an abelian group and N K/E ⊂ E × is asubgroup of finite index (by class field theory) [ K : E ], we can extend χ K/E to E × , and χ K is of the form χ K = χ E ◦ N K/E with χ E | N K/E = χ K/E . (ii) If condition (i) is satisfied, then this part is obvious. If χ E is a solution of χ K = χ E ◦ N K/E ,with χ K/E := χ K ◦ N − K/E : N K/E → C × , then certainly χ E is an extension of the character χ K/E .Conversely, if χ E extends χ K/E , then it is a solution of χ K = χ E ◦ N K/E with χ K ◦ N − K/E = χ K/E : N K/E → C × . (cid:3) Remark . Now take Heisenberg representation ρ = ρ ( X, χ K ) of G F . Let E/F be anyextension corresponding to a maximal isotropic for X . In this Heisenberg setting, fromTheorem 3.1(2), we know χ K is a character of K × /I F K × , and from the first commutative diagram on p. 302 of [38] we have N K/E : K × F /I F K × → E × F /I F N K/E . Thus in the Heisenbergsetting, we have more information than Lemma 3.2(i), that χ K is a character of(3.8) K × /K × E I F K × N K/E −−−→ N
K/E /I F N K/E ⊂ E × /I F N K/E , and therefore χ K/F is actually a character of N K/E /I F N K/E , or in other words, it is aGal(E / F)-invariant character of the Gal(E / F)-module N K/E ⊂ E × . And if χ E is one ofthe solution of Lemma 3.2(ii), then the complete solutions is the set { χ σE | σ ∈ Gal(E / F) } . We know that W ( χ E , ψ ◦ Tr K / E ) has the same value for all solutions χ E of χ E ◦ N K/E = χ K , which means for all χ E which extend the character χ K/E .Moreover, from the above Lemma 3.2, we also can see that χ E | N K/E = χ K ◦ N − K/E .Let ρ = ρ ( X, χ K ) be a Heisenberg representation of G F . Let E/F be any extensioncorresponding to a maximal isotropic for X . Then by using the above Lemma 3.2, we havethe following lemma. Lemma 3.4.
Let ρ = ρ ( Z, χ ρ ) = ρ (Gal(L / K) , χ K ) be a Heisenberg representation of afinite 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 (3.9) ρ = Ind E / F ( χ σ E ) for all σ ∈ Gal(E / F) , where χ E : E × /I F N K/E → C × with χ K = χ E ◦ N K/E .Moreover, for a fixed base field E of a maximal isotropic for ρ , this construction of ρ isindependent of the choice of this character χ E . Definition 3.5 ( U-isotropic).
Let F be a non-archimedean local field. Let X : F F × → C × be an alternating character with the property X ( ε , ε ) = 1 , for all ε , ε ∈ U F . In other words, X is a character of F F × /U F ∧ U F . 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 π F and write U := U F . Then weobtain an isomorphism (cid:98) U ∼ = (cid:92) F F × /U ∧ U , η (cid:55)→ X η , η X ← X between characters of U and U -isotropic alternating characters as follows: (3.10) X η ( π aF ε , π bF ε ) := η ( ε ) b · η ( ε ) − a , η X ( ε ) := X ( ε, π F ) , where a, b ∈ Z , ε, ε , ε ∈ U , and η : U → C × . Then Rad(X η ) = < π η F > × Ker( η ) = < ( π F ε ) η > × Ker( η ) , does not depend on the choice of π F , where η is the order of the character η , hence F × / Rad(X η ) ∼ = < π F > / < π η F > × U / Ker( η ) ∼ = Z η × Z η . Therefore all Heisenberg representations of type ρ = ρ ( X η , χ ) have dimension dim( ρ ) = η . N EXTENSION OF DELIGNE-HENNIART’S TWISTING FORMULA AND ITS APPLICATIONS 11
Remark . From Proposition 5.2(i) on p. 50 of [40], we know that
F F × /U F ∧ U F ∼ = U F ,hence we have (cid:99) U F ∼ = (cid:92) F F × /U F ∧ U F . From the above Lemma 3.6 we have:Rad(X η ) = < π η F > × Ker( η ) . Therefore we can conclude that a U-isotropic character X = X η has U iF contained in itsradical if and only if η is a character of U F /U iF .From the above Lemma 3.6 we know that the dimension of a U-isotropic Heisenberg rep-resentation ρ = ρ ( X η , χ ) of G F is dim( ρ ) = η , and F × / Rad(X η ) ∼ = Z η × Z η , a directproduct of two cyclic (bicyclic) groups of the same order η .In the following lemma, we give an equivalent condition for U-isotropic Heisenberg repre-sentation. Lemma 3.8.
Let G F be the absolute Galois group of a non-archimedean local field F . For aHeisenberg representation ρ = ρ ( Z, χ ρ ) = ρ ( X, χ K ) 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 maximalisotropic for X .(3) ρ = Ind E / F ( χ E ) can be induced from a character χ E 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 ∈ (cid:92) F F × /U ∧ U . We also know that (cid:98) U ∼ = (cid:92) F F × /U ∧ U . Then X corresponds a character of U , namely X (cid:55)→ η X . Then from Lemma 3.6we have F × / Rad(X) ∼ = Z η X × Z η X , 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: N K/F = Rad(X) , Gal(K / F) ∼ = F × / Rad(X) . Let
E/F be the maximal unramified subextension in
K/F . Then [ E : F ] = η K becausethe order of maximal cyclic subgroup of Gal(K / F) is η X . Then f E/F = η X , hence f K/F = e K/F = η X because f K/F · e K/F = [ K : F ] = η X 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 [ G F /Z : H/Z ] = η X , hence H/Z = Gal(K / E),i.e., the maximal unramified subextension
E/F in K/F corresponds to a maximal isotropicsubgroup, hence ρ ( X, χ K ) = Ind E / F ( χ E ), for χ E ◦ N K/E = χ K .Finally, since E/F is unramified and the extension E corresponds a maximal isotropic sub-group for X , we have U F ⊂ N E/F , hence U F ⊂ N K/F and X | U × U = 1 because U F ⊂ F × ⊂N K/E . This shows that X is U-isotropic. (cid:3) Corollary 3.9.
The U-isotropic Heisenberg representation ρ = ρ ( X η , χ ) can never be wild be-cause it is induced from an unramified extension E/F , but the dimension dim( ρ (X η , χ )) = η can be a power of p. The representations ρ of dimension prime to p are precisely given as ρ = ρ ( X η , χ ) for char-acters η of U/U . Proof.
This is clear from the above lemma 3.6 and the fact: | U/U | = q F − p .We know that the dimension dim( ρ ) = (cid:112) [K : F] = (cid:112) [F × : Rad(X)]. If this is prime to p then K/F is tame and U F ⊆ Rad(X). But
U/U is cyclic, hence X is then U -isotopic. (cid:3) Lemma 3.10.
Let ρ = ρ ( X, χ K ) be a Heisenberg representation of the absolute Galois group G F of a non-archimedean local field F/ Q p . Then following are equivalent:(1) dim( ρ ) is prime to p .(2) dim( ρ ) is a divisor of q F − .(3) The alternating character X is U -isotropic and X = X η for a character η of U F /U F ,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 ofdimensions prime to p , are U-isotropic representations of the form ρ = ρ ( X η , χ ), where η : U F /U F → C × , and the dimensions dim( ρ ) = η . Thus if dim( ρ ) is prime to p , then dim( ρ ) = η is a divisor of q F − ρ ) is a divisor of q F −
1, then gcd ( p, dim( ρ )) = 1. Then from Corollary3.9, the alternating character X is U-isotropic and X = X η for a character η ∈ (cid:92) U F /U F .(3) implies (4): We know thatdim( ρ ) = (cid:113) [F × : Rad(X η )] = (cid:113) [F × : N K / F ] = η. Here since
K/F is abelian, we have dim( ρ ) = [K : F]. Again since η = dim( ρ ) is a divisorof q F −
1, hence
K/F is tamely ramified.(4) implies (1): If
K/F is tamely ramified, then we can write U F ⊂ N K/F ⊂ F × , and hence F × / N K/F is a quotient group of F × /U F . Therefore if K/F is the abelian tamely ramifiedextension and N K/F = Rad(X), then X must be an alternating character of F × /U F . We alsoknow that F × = < π F > × < ζ > × U F , where ζ is a root of unity of order q F −
1. Thisimplies F × /U F = < π F > × < ζ > . So each element x ∈ F × /U F can be written as x = π aF · ζ b ,where a, b ∈ Z . We now take x = π a F ζ b , x = π a F ζ b ∈ F × /U F , where a i , b i ∈ Z ( i = 1 , X ( x , x ) = X ( π a F ζ b , π a F ζ b )= X ( π a F , ζ b ) · X ( ζ b , π a F )= χ ρ ([ π a F , ζ b ]) · χ ρ ([ ζ b , π a F ]) . But this implies X q F − ≡ ζ q F − = 1, which means that X is actually an al-ternating character on F × / ( F × ( q F − U F ) , and therefore G F /G K is actually a quotient of F × / ( F × ( q F − U F ) . We also know that U F is a pro-p-group and therefore U F = ( U F ) q F − ⊂ F × . Thus the cardinality of F × / ( F × ( q F − U F ) is ( q F − because F × / ( F × ( q F − U F ) ∼ = Z / ( q F − Z × < ζ > ∼ = Z q F − × Z q F − . Therefore dim( ρ ) divides q F − . Hence dim( ρ ) is prime to p . (cid:3) N EXTENSION OF DELIGNE-HENNIART’S TWISTING FORMULA AND ITS APPLICATIONS 13
Remark . We let K η | F be the abelian bicyclic extension which corresponds to Rad(X η ) : N K η /F = Rad(X η ) , Gal(K η / F) ∼ = F × / Rad(X η ) . Then we have f K η | F = e K η | F = η and the maximal unramified subextension E/F ⊂ K η /F corresponds to a maximal isotropic subgroup, hence ρ ( X η , χ ) = Ind E / F ( χ E ) , for χ E ◦ N K η / E = χ. We recall here that χ : K × η /I F K × η → C × is a character such that (cf. Theorem 3.1(3)) χ | ( K × η ) F ↔ X η , with respect to ( K × η ) F /I F K × η ∼ = F × / Rad(X η ) ∧ F × / Rad(X η ) . In particular, we see that ( K × η ) F /I F K × η is cyclic of order η and χ | ( K × η ) F must be a faithfulcharacter 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 sec-tion 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/ Q p be a local field and G F be the absolute Galois groupof F . If ρ is a Heisenberg representation of G F , then dim( ρ ) = p n · d (cid:48) , where n (cid:62) is aninteger and where the prime to p factor d (cid:48) must divide q F − .Proof. By the definition of Heisenberg representation ρ we have the relation[[ G F , G F ] , G F ] ⊆ Ker( ρ ) . Then we can consider ρ as a representation of G := G F / [[ G F , G F ] , G F ]. Since [ x, g ] ∈ [[ G F , G F ] , G F ] for all x ∈ [ G F , G F ] and g ∈ G F , we have [ G, G ] = [ G F , G F ] / [[ G F , G F ] , G F ] ⊆ 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 sub-groups. Therefore we can write G = G p × G p (cid:48) , where G p is the Sylow p -subgroup, and G p (cid:48) is the direct product of all other Sylow subgroups.Therefore each irreducible representation ρ has the form ρ = ρ p ⊗ ρ p (cid:48) , where ρ p and ρ p (cid:48) areirreducible representations of G p and G p (cid:48) respectively.We also know that finite p -groups are nilpotent groups, and direct product of a finitenumber of nilpotent groups is again a nilpotent group. So G p and G p (cid:48) are both two-stepnilpotent group because G is a two-step nilpotent group. Therefore the representations ρ p and ρ p (cid:48) are both Heisenberg representations of G p and G p (cid:48) respectively.Now to prove our assertion, we have to show that dim( ρ p ) can be an arbitrary power of p ,whereas dim( ρ p (cid:48) ) must divide q F −
1. Since ρ p is an irreducible representation of p -group G p , so the dimension of ρ p is some p -power. Again from the construction of ρ p (cid:48) we can say that dim( ρ p (cid:48) ) is prime to p . Then fromLemma 3.10 dim( ρ p (cid:48) ) is a divisor of q F − (cid:3) Remark . (1). Let V F be the wild ramification subgroup of G F . We can show that ρ | V F isirreducible if and only if Z ρ = G K ⊂ G F corresponds to an abelian extension K/F which istotally ramified and wildly ramified (cf. [38], p. 305). If N := N K/F ( K × ) is the subgroup ofnorms, then this means that N · U F = F × , in other words, F × /N = N · U F /N = U F /N ∩ U F , where N can be also considered as the radical of X ρ . So we can consider the alternatingcharacter X ρ on the principal units U F ⊂ F × . Thendim( ρ ) = (cid:112) [F × : N] = (cid:113) [U : N ∩ U ] , is a power of p , because U F is a pro-p-group.Here we observe: If ρ = ρ ( X, χ K ) with ρ | V F stays irreducible, then dim( ρ ) = p n , n (cid:62) K/F is a totally and wildly ramified. But there is a big class of Heisenberg representations ρ such that dim( ρ ) = p n is a p -power, but which are not wild representations (see the Definition3.5 of U-isotropic). (2). Let ρ = ρ ( X, χ K ) be a Heisenberg representation of the absolute Galois group G F ofdimension d > p . Then from above Lemma 3.10, we have d | q F −
1. Forthis representation ρ , here K/F must be tame if Rad(X) = N K / 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, χ ρ ) = ρ ( X, χ K ) be a Heisenberg representation of the absoluteGalois group G F of a field F/ Q p of dimension m . Let E/F be any subextension in
K/F corresponding to a maximal isotropic subgroup for X . Then a F ( ρ ) = a F (Ind E / F ( χ E )) , m · a F ( ρ ) = a F (Ind K / F ( χ K )) . As a consequence we have a ( χ K ) = e K/E · a ( χ E ) − d K/E . In particular a ( χ K ) = a ( χ E ) if K/E is unramified.Proof.
We know that ρ = Ind E / F ( χ E ) and m · ρ = Ind K / F ( χ K ). By the definition of Artinconductor we can write a F (dim( ρ ) · ρ ) = dim( ρ ) · a F ( ρ ) = m · a F (Ind E / F ( χ E )) . Group theoretically, if ρ | V F = Ind G F H ( χ H ) | V F is irreducible, then from Section 7.4 of [33], we can say G F = H · V F . Here H = G L , where L is a certain extension of F , and V F = G F mt where F mt /F is themaximal tame extension of F . Therefore G F = H · V F is equivalent to F = L ∩ F mt that means the extension L/F must be totally ramified and wildly ramified, and [ G F : H ] = [ L : F ] = | V F | . We know that the wildramification subgroup V F is a pro-p-group (cf. [31], p. 106). Then dim( ρ ) is a power of p . N 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 = e K/E f K/E , we have thetransitivity relation of different (cf. [32], p. 51, Proposition 8) D K/F = D K/E · D
E/F . Now from the definition of different of a Galois extension, and taking K -valuation we obtain:(4.1) d K/F = d K/E + e K/E · d E/F . Now by using equation (2.3) we have:(4.2) m · a F (Ind E / F ( χ E )) = m · f E / F (cid:0) d E / F + a( χ E ) (cid:1) = m · f E / F · d E / F + e K / E · f K / F · a( χ E ) , and(4.3) a F (Ind K / F ( χ K )) = f K / F · (cid:0) d K / F + a( χ K ) (cid:1) = f K / F · d K / F + f K / F · a( χ K ) . By using equation (4.1), from equations (4.2), (4.3), we have a ( χ K ) = e K/E · a ( χ E ) − d K/E . And when
K/E is unramified, i.e., e K/E = 1 and d K/E = 0, hence a ( χ K ) = a ( χ E ). (cid:3) Definition 4.4 ( Jump for alternating character).
For each X ∈ (cid:91) F F × we define jump asfollows:(4.4) j ( X ) := X is trivialmax { i | X | UU i (cid:54)≡ } when X is nontrivial , where U U i ⊆ F F × is a subgroup which under (3.2) corresponds G iF ∩ [ G F , G F ] /G iF ∩ [[ G F , G F ] , G F ] ⊆ [ G F , G F ] / [[ G F , G F ] , G F ] . Remark . Let ρ = ρ ( X ρ , χ K ) be a minimal conductor (i.e., a representation with thesmallest Artin conductor) Heisenberg representation for X ρ of the absolute Galois group G F .From Theorem 3 on p. 125 of [41], we have(4.5) sw F ( ρ ) = dim( ρ ) · j(X ρ ) = (cid:113) [F × : Rad(X ρ )] · j(X ρ ) . Moreover if ρ = ρ ( X, χ ) is a minimal representation corresponding X , then all otherHeisenberg representations of dimension dim( ρ ) are of the form ρ = χ F ⊗ ρ = ( X, ( χ F ◦ N K/F ) χ ), where χ F : F × → C × . Then we have (cf. [38], p. 305, equation (5))(4.6) sw F ( ρ ) = sw F ( χ F ⊗ ρ ) = (cid:112) [F × : Rad(X)] · max { j( χ F ) , j(X) } . For minimal conductor U-isotopic Heisenberg representation we have the following propo-sition.
Proposition 4.6.
Let ρ = ρ ( X η , χ K ) be a U-isotropic Heisenberg representation of G F ofminimal conductor. Then we have the following conductor relation j ( X η ) = j ( η ) , sw F ( ρ ) = dim( ρ ) · j(X η ) = η · j( η ) , a F ( ρ ) = sw F ( ρ ) + dim( ρ ) = η (j( η ) + 1) = η · a F ( η ) . Proof.
From [41], on p. 126, Proposition 4(i) and Proposition 5(ii), and U ∧ U = U ∧ U , wesee the injection U i ∧ F × ⊆ U U i induces a natural isomorphism U i ∧ < π F > ∼ = U U i /U U i ∩ ( U ∧ U )for all i (cid:62) j ( X η ) = n −
1, hence X η | UU n = 1 but X η | UU n − (cid:54) = 1. This gives X η | U n ∧ <π F > = 1but X η | U n − ∧ <π F > (cid:54) = 1. Now from equation (3.10) we can conclude that η ( x ) = 1 for all x ∈ U n but η ( x ) (cid:54) = 1 for x ∈ U n − . Hence j ( η ) = n − j ( X η ) . Again from the definition of j ( χ ), where χ is a character of F × , we can see that j ( χ ) = a ( χ ) − a ( χ ) = j ( χ ) + 1.From equation (4.5) we obtain:sw F ( ρ ) = dim( ρ ) · j(X η ) = η · j( η ) , since dim( ρ ) = η and j ( X η ) = j ( η ). Finally, from equation (2.2) for ρ (here < , ρ > G = 0),we have(4.7) a F ( ρ ) = sw F ( ρ ) + dim( ρ ) = η · j( η ) + η = η · (j( η ) + 1) = η · a F ( η ) . (cid:3) Now by combining Proposition 4.3 with Proposition 4.6, we get the following result.
Lemma 4.7.
Let ρ = ρ ( X η , χ K ) be a U-isotopic Heisenberg representation of minimal con-ductor of the absolute Galois group G F of a non-archimedean local field F . Let K = K η correspond to the radical of X η , and let E /F be the maximal unramified subextension, and E/F be any maximal cyclic and totally ramified subextension in
K/F . Let m denote the orderof η . Then ρ is induced by χ E or by χ E respectively, and we have(1) a E ( χ E ) = m · a ( η ) − d E/F ,(2) a E ( χ E ) = a ( η ) ,(3) and for the character χ K ∈ (cid:100) K × , a K ( χ K ) = m · a ( η ) − d K/F . Moreover, a E ( χ E ) = a K ( χ K ) .Proof. Proof of these assertions follows from equation (2.3) and Proposition 4.6. When ρ =Ind E / F ( χ E ), where E/F is a maximal cyclic and totally ramified subextension in
K/F , fromequation (2.3) we have a F ( ρ ) = m · a ( η ) using Proposition 4 . , = f E/F · (cid:0) d E/F · a E ( χ E ) (cid:1) , since ρ = Ind E / F ( χ E )= 1 · (cid:0) d E/F + a E ( χ E ) (cid:1) . because E/F is totally ramified, hence f E/F = 1. This implies a E ( χ E ) = m · a ( η ) − d E/F .Similarly, when ρ = Ind E / F ( χ E ), where E /F is the maximal unramified subextension in K/F , hence f E /F = m and d E /F = 0, by using equation (2.3) we obtain a E ( χ E ) = a ( η ). N EXTENSION OF DELIGNE-HENNIART’S TWISTING FORMULA AND ITS APPLICATIONS 17
Again from Proposition 4.3 we have a K ( χ K ) = m · a ( χ E ) − d K/E = m · a ( η ) − d K/F . Finally, since
E/F is a maximal cyclic totally ramified implies
K/E is unramified andtherefore d E/F = d K/F , and hence a E ( χ E ) = a K ( χ K ) . (cid:3) Remark . Assume that we are in the dimension m = η prime to p case. Then fromCorollary 3.9, η must be a character of U/U (for U = U F ), hence a ( η ) = 1 a F ( ρ ) = 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 a E ( χ E ) = a ( η ) = 1 . And
K/F, E/F are tamely ramified of ramification exponent e K/F = m , hence a E ( χ E ) = a K ( χ K ) = m · a ( η ) − d K/F = m − ( e K/F −
1) = m − ( m −
1) = 1 . Thus we can conclude that in this case all three characters (i.e., χ E , χ E , and χ K ) are ofconductor 1.In the general case a E ( χ E ) = a ( η ) and a E ( χ E ) = a K ( χ K ) = m · a ( η ) − d, where d = d E/F = d K/F , conductors will be different.In general, if ρ = ρ ⊗ χ F , where ρ is a finite dimensional minimal conductor representationof G F , and χ F ∈ (cid:99) F × , then we have the following result. Lemma 4.9.
Let ρ be a finite dimensional representation of G F of minimal conductor. Thenwe have (4.8) a F ( ρ ) = dim( ρ ) · a F ( χ F ) , where ρ = ρ ⊗ χ F = ρ ( X η , ( χ F ◦ N K/F ) χ ) and χ F ∈ (cid:99) F × with a ( χ F ) > a ( ρ )dim( ρ ) .Proof. From equation (2.5) we have a F ( ρ ) = dim( ρ ) · (1 + j( ρ )). By the given condition ρ is of minimal conductor. So for representation ρ = ρ ⊗ χ F , we have a F ( ρ ) = a F ( ρ ⊗ χ F ) = dim( ρ ) · (1 + max { j( ρ ) , j( χ F ) } )= dim( ρ ) · max { χ F ) , ρ ) } = dim( ρ ) · max { a( χ F ) , ρ ) } = dim( ρ ) · a F ( χ F ) , because by the given condition a ( χ F ) > a ( ρ )dim( ρ ) = dim( ρ ) · (1 + j( ρ ))dim( ρ ) = 1 + j ( ρ ) . (cid:3) Proposition 4.10.
Let ρ = ρ ( X, χ K ) be a Heisenberg representation of dimension m of theabsolute Galois group G F of a non-archimedean local field F . Then m | a F ( ρ ) if and only if: X is U -isotropic, or (if X is not U -isotropic) a F ( ρ ) is with respect to X not the minimalconductor.Proof. From the above Lemma 4.9 we know that if ρ is not minimal, then a F ( ρ ) is always amultiple of the dimension m . So now we just have to check for minimal conductors. In theU-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 ∧ U (cid:54)≡
1, because U ∧ U = U ∧ U (see the Remark on p. 126 of [41]). We also know that U U i = ( U U i ∩ U ∧ U ) × ( U i ∧ <π F > ) (cf. [41], p. 126, Proposition 5(ii)). In Proposition 5 of [41], we observe that all thejumps v in the filtration { U U i ∩ ( U ∧ U ) } , i ∈ R + are not integers with v >
1. This showsthat j ( X ) is also not an integer, hence a F ( ρ ) is not multiple of the dimension. This impliesthe conductor a F ( ρ ) is not minimal. (cid:3) For minimal conductor Heisenberg representation, we have the following theorem.
Proposition 4.11.
Let ρ = ρ ( X η , χ K ) be a Heisenberg representation of the absolute Galoisgroup G F of a non-archimedean local field F/ Q p of dimension m prime to p . Then it is ofminimal conductor a F ( ρ ) = m if and only if ρ is a representation of G F /V F , where V F is thesubgroup of wild ramification.Proof. By the given condition, the dimension dim( ρ ) = m is prime to p . Then from Lemma3.10 we can conclude that K/F is tamely ramified with f K/F = e K/F = m (cf. Remark3.11)and hence d K/F = e K/F − m −
1. Then from the conductor formula (2.3) we caneasily see that a ( ρ ) = m is minimal if and only if a ( χ K ) = 1.Further, for some extension L/K , if N L/K = Ker( χ K ), then by class field theory we canconclude: L/K is tamely ramified if and only if a ( χ K ) = 1 . Now suppose that ρ is a Heisenberg representation of G := G F /V F of dimension m primeto p . This implies V F ⊂ Ker( ρ ) = Ker( χ K ) = N L / K , where L/K is some tamely ramifiedextension. Then a ( χ K ) = 1, hence a ( ρ ) = m is minimal.Conversely, when conductor a ( ρ ) = m is minimal, we have a ( χ K ) = 1. By class field theorythis character χ K determines an extension L/K such that N L/K = Ker( χ K ). Since a ( χ K ) = 1,here L/K must be tamely ramified, hence
L/F is tamely ramified. This means V F sits in thekernel Ker( ρ ) = G L , therefore ρ is actually a representation of G F /V F . (cid:3) N EXTENSION OF DELIGNE-HENNIART’S TWISTING FORMULA AND ITS APPLICATIONS 19 Proof of Theorem 1.2 and Theorem 1.3Theorem 5.1.
Let ρ = ρ ( X η , χ K ) = ρ ⊗ (cid:102) χ F be a Heisenberg representation of the absoluteGalois group G F of a non-archimedean local field F/ Q p of dimension m with gcd ( m, p ) = 1 ,where ρ = ρ ( X η , χ ) is a minimal conductor Heisenberg representation of G F and (cid:102) χ F : G F → C × corresponds to χ F : F × → C × by class field theory. If a ( χ F ) (cid:62) , then we have (5.1) W ( ρ, ψ ) = W ( ρ ⊗ (cid:102) χ F , ψ ) = W ( χ F , ψ ) m · det( ρ )( c ) , where ψ is a nontrivial additive character of F , and c := c ( χ F , ψ ) ∈ F × , satisfies χ F (1 + x ) = ψ ( c − x ) for all x ∈ P a ( χ F ) − [ a ( χF )2 ] F .Proof. Step-1:
By the given conditions, ρ = ρ ⊗ (cid:103) χχ F , where ρ is a minimal conductor Heisenberg representation of G F of dimension m prime to p and (cid:102) χ F : G F → C × corresponds to χ F : F × → C × by class field theory. And here χ : F × /F × m → C × such that ρ = ρ ⊗ (cid:101) χ .Let ζ be a ( q F − U F is a pro-p-group and gcd ( p, m ) = 1, we have(5.2) F × /F × m = < π F > × < ζ > × U F / < π mF > × < ζ > m × U F ∼ = Z m × Z m , that is, a direct product of two cyclic group of same order. Hence F × /F × m ∼ = (cid:92) F × /F × m .Since F × m = < π mF > × < ζ > m × U F , and F × /F × m ∼ = Z m × Z m , we have a ( χ ) (cid:54) χ is a divisor of m for all χ ∈ (cid:92) F × /F × m . Now if we take a character χ F of F × conductor (cid:62) a ( χ F ) (cid:62) a ( χ ) for all χ ∈ (cid:92) F × /F × m . Then by using Deligne’s formula (1.3) we have(5.3) W ( χχ F , ψ ) m = χ ( c ) m · W ( χ F , ψ ) m = W ( χ F , ψ ) m , where c ∈ F × with ν F ( c ) = a ( χ F ) + n ( ψ ), satisfies χ F (1 + x ) = ψ ( c − x ) , for all x ∈ F × with 2 ν F ( x ) (cid:62) a ( χ ) . Now from Proposition 4.11 we can consider the representation ρ as a representation of G := Gal(F mt / F), where F mt /F is the maximal tamely ramified subextension in ¯ F /F . Thenwe can write ρ = Ind E / F ( χ E , ) , ρ = Ind E / F ( χ E ) , where E/F is a cyclic tamely ramified subextension of
K/F of degree m and χ E := χ E, ⊗ ( χ F ◦ N E/F ) . Step-2:
Let G be a finite group and R ( G ) be the character ring provided with the tensorproduct as multiplication and the unit representation as unit element. Then for any zero We also know that there are m characters of F × /F × m such that ρ ⊗ (cid:101) χ = ρ (cf. [38], p. 303, Proposition1.4). So we always have: ρ = ρ ⊗ (cid:102) χ F = ρ ⊗ (cid:103) χχ F , where χ ∈ (cid:92) F × /F × m , and (cid:102) χ F : G F → C × corresponds to χ F by class field theory. dimensional representation π ∈ R ( G ) we have (cf. Theorem 2.1(h) on p. 40 of [7]):(5.4) π = (cid:88) H (cid:54) G n H Ind GH ( χ H − H ) , where n H ∈ Z (cf. Proposition 2.24 on p. 48 of [7]) and χ H ∈ (cid:98) H . Moreover, from Theorem2.1 (k) of [7] we know that n H (cid:54) = 0 only if Z ( G ) (cid:54) H and χ H | Z ( G ) = χ Z , where Z ( G ) is thecenter of G and χ Z is the center character.Now we will use this above formula (5.4) for the representation ρ − m · F and we get(5.5) ρ − m · F = r (cid:88) i =1 n i Ind E i / F ( χ E i − E i ) , where E i /F are intermediate fields of K/F and for nonzero n i , we have the relation χ = χ E i ◦ N K/E i . Since a ( χ ) = 1 and K/E i are cyclic tamely ramified , we have a ( χ E i ) = 1 for all i =1 , , · · · , r . Then from equation (5.5) we have(5.6) det( ρ ) = r (cid:89) i =1 ( χ E i | F × ) n i . Step-3:
From equation (5.5) we also can write(5.7) ρ ⊗ (cid:103) χχ F − m · χχ F = r (cid:88) i =1 n i Ind E i / F ( χ E i θ i − θ i ) , where θ i := χχ F ◦ N E i /F for all i ∈ { , , · · · , r } . Since a ( χ F ) (cid:62)
2, and E i /F are tamelyramified, the conductors a ( θ i ) (cid:62) i ∈ { , , · · · , r } . Then from equation (5.7) we canwrite W ( ρ, ψ ) = W ( χχ F , ψ ) m · r (cid:89) i =1 W ( χ E i θ i − θ i , ψ E i ) n i = W ( χχ F , ψ ) m · r (cid:89) i =1 W ( χ E i θ i , ψ E i ) n i W ( θ i , ψ E i ) n i = W ( χχ F , ψ ) m · r (cid:89) i =1 χ E i ( c i ) n i , (5.8)where ψ E i = ψ ◦ Tr E i / F and c i ∈ E × i such that θ i (1 + y ) = ψ E i ( yc i ) , for all y ∈ P a ( θ i ) − [ a ( θi )2 ] E i .Moreover, here E i /F are tamely ramified extensions, then from Lemma 18.1 of [8] on p. 123,we have N E i /F (1 + y ) ∼ = 1 + Tr E i / F (y) (mod P a( χ F )F ) , The subfields E i ⊆ K are related to Boltje’s approach by Gal(K / E i ) = H i / Z(G) and the fact that χ H i extends the character χ Z which translates via class field theory to χ E i ◦ N K/E i = χ K . Moreover, X = χ Z ◦ [ − , − ] and χ Z extendible to H i means that Gal(K / E i ) = H i / Z(G) must be isotropic for X , hencein our situation K/E i must be a cyclic extension of degree dividing m . N EXTENSION OF DELIGNE-HENNIART’S TWISTING FORMULA AND ITS APPLICATIONS 21 and Tr E i / F (y) ∈ P a( χ F ) − [ a( χ F)2 ]F . Therefore for all y ∈ P a ( θ i ) − [ a ( θi )2 ] E i we can write (cf. Proposition18.1 on p. 124 of [8]): θ i (1 + y ) = χχ F ◦ N E i /F (1 + y ) = χ F (1 + Tr E i / F (y))= ψ ( Tr E i / F (y) c ) = ψ E i ( yc ) , where c := c ( χ F , ψ ) for which χ F (1 + x ) = ψ ( xc ) for x ∈ P a ( χ F ) − [ a ( χF )2 ] F . This varifies thatthe choice c i ( θ i , ψ E i ) = c i ( χχ F , ψ E i ) = c ( χ F , ψ ) ∈ F × is right for applying Tate-Lamprechtformula (cf. [5]).Then by using equation (5.6) in equation (5.8) we have:(5.9) W ( ρ, ψ ) = W ( χχ F , ψ ) m · det( ρ )( c ) . Finally, by using equation (5.3) from equation (5.9) we can write W ( ρ, ψ ) = W ( χχ F , ψ ) m · det( ρ )( c )= W ( χ F , ψ ) m · det( ρ )( c ) . (cid:3) 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 G F and let ψ be a nontrivial additive character of F . Let ρ m = ρ m ( X η , χ K ) = ρ ⊗ (cid:102) χ F be a Heisenbergrepresentation of the absolute Galois group G F of a non-archimedean local field F/ Q p ofdimension m with gcd ( m, p ) = 1 , where ρ = ρ ( X η , χ ) is a minimal conductor Heisenbergrepresentation of G F and (cid:102) χ F : G F → C × corresponds to χ F : F × → C × by class field theory.If a ( χ F ) (cid:62) and j ( ρ m ) > · β ( σ ) , then we have (5.10) W ( σ ⊗ ρ m , ψ ) = det( σ )( γ ) · W ( χ F , ψ ) dim ( σ ⊗ ρ m ) · det( ρ )( c dim( σ ) ) . Here c ∈ F × is same as in Theorem 1.2 and ν F ( γ ) = a ( ρ m ) + 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 asfollows: σ := σ − dim( σ ) · G F . Then(5.11) det( σ ) = det( σ − dim( σ ) · G F ) = det( σ ) . Now we want to use Theorem 1.1 for the representation σ . It can be seen that β ( σ ) = β ( ρ − dim( σ ) · G F ) = β ( σ ). Hence σ satisfies the condition: j ( ρ m ) > · β ( σ ) = 2 · β ( σ ). Therefore we can use Theorem 1.1 for the representation σ , hence we can write W ( σ ⊗ ρ m , ψ ) = det( σ )( γ )= W (( σ − dim( σ ) · G F ) ⊗ ρ m , ψ )= W ( σ ⊗ ρ m , ψ ) · W ( ρ m , ψ ) − dim( σ ) . This gives(5.12) W ( σ ⊗ ρ m , ψ ) = det( σ )( γ ) · W ( ρ m , ψ ) dim( σ ) = det( σ )( γ ) · W ( ρ m , ψ ) dim( σ ) . Now we use Theorem 1.2 in equation (5.12) and we obtain W ( σ ⊗ ρ m , ψ ) = det( σ )( γ ) · W ( χ F , ψ ) dim( σ ⊗ ρ m ) · det( ρ )( c dim( σ ) ) . (cid:3) Remark . 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 onjump of σ one can give explicit twisting formulas for W ( σ ⊗ ρ, ψ ) and W ( σ ⊗ ρ (cid:48) , ψ ), where ρ (cid:48) := ρ ⊗ χ as the above proof. 6. Applications
Invariant root number formula for Heisenberg representations.
By the con-struction for Heisenberg representations ρ = ρ ( X, χ K ) of G F , we can write ρ = Ind
E/F ( χ E ) , where K/E/F is fixed field of a maximal isotropic subgroup H = Gal ( F /E ) of ρ . Then rootnumber of ρ is(6.1) W ( ρ, ψ ) = λ E/F ( ψ ) · W ( χ E , ψ ) . Here λ E/F ( ψ ) := W ( Ind
E/F (1 E ) , 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 arenot 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 isotropicsubgroups H , and H of G . Let E and E be the fixed fields of H and H respectively.Then we can write (1) ρ = Ind E /F ( χ E ) , (2) ρ = Ind E /F ( χ E ) . Then(6.2) W ( ρ, ψ ) = λ E /F ( ψ ) · W ( χ E , ψ ) = λ E /F ( ψ ) · W ( χ E , ψ ) . Now if we notice equation (6.2), the right hand side depends on E i ( i = 1 , 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 E i .For a Heisenberg representation ρ of G F dimension prime to p , one can see explicit invariantformula for W ( ρ, ψ ) in [6]. N EXTENSION OF DELIGNE-HENNIART’S TWISTING FORMULA AND ITS APPLICATIONS 23
Example 6.1.
Let ρ = ρ ( X, χ K ) be a 2 - dimensional Heisenberg representation of G F , where F/ Q p and p (cid:54) = 2 (see Example (7.1) in Appendix for explicit description for 2-dimensionalHeisenberg representations). Then we will have Gal ( K/F ) ∼ = Z / Z × Z / Z . Therefore, thereare three maximal isotropic subgroups H i ( i = 1 , ,
3) for ρ , and hence E i /F ( i = 1 , ,
3) arethree quadratic extensions of F . It can be proved (cf. [1], Lemma 4.7 on pp. 191-192) that λ E i /F ( ψ ) are not same. Similarly, it also can be proved that W ( χ E i , 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 G F . Then the alternating character X = X ρ cor-responds to a character η : U F → C × of the group of units. Then we have the uniquedecomposition of η as follows: η = η p · η (cid:48) , where η (cid:48) is of order prime to p and the order of η p is a power of p . Correspondingly: X = X p · X (cid:48) , where η p ↔ X p , η (cid:48) ↔ X (cid:48) . For the representation ρ this means:(6.3) ρ = ρ p ⊗ ρ (cid:48) where dim( ρ p ) = p r ( r (cid:62) , and dim( ρ (cid:48) ) =: m (cid:48) , and gcd ( m (cid:48) , p ) = 1, m (cid:48) | ( q F − not unique because ρ p ⊗ ρ (cid:48) = ρ p ω − ⊗ ωρ (cid:48) for any character ω of F × . Thus for appropriate ω we may assume that a F ( ρ (cid:48) ) = m (cid:48) is minimal and a F ( ρ p ) = m p a , where a (cid:62) a F ( η p ). Then here we can use Theorem 1.3 to givean invariant formula for W ( ρ, ψ ) = W ( ρ p ⊗ ρ (cid:48) ) when j ( ρ (cid:48) ) > · j ( ρ p ). Theorem 6.2 (Invariant Formula) . Let ρ be a U -isotropic Heisenberg representation of G F of the form ρ = ρ p ⊗ ρ m with dim( ρ p ) = p r ( r (cid:62) , and dim( ρ m ) = m , and gcd ( m, p ) = 1 . Let ψ be an nontrivial additive character of F . If the jump: j ( ρ m ) > · j ( ρ p ) , then we have W ( ρ, ψ ) = W ( ρ p ⊗ ρ m , ψ ) = det( ρ p )( γ ) · W ( χ F , ψ ) dim( ρ ) det( ρ )( c p r ) . Here χ F , c , ρ are same as in Theorem 1.2, and ν F ( γ ) = a ( ρ m ) + m · n ( ψ ) .Proof. The idea of the proof is same as Theorem 1.3. Here just replace σ by ρ p . And anotherimportant thing is that the representation ρ p is Heisenberg, hence irreducible. Therefore β ( ρ p ) = j ( ρ p ). (cid:3) Remark . Since maximal isotropic subgroups for ρ are not unique, giving invariant fordet( ρ ) we cannot simply use Gallagher’s result (cf. Theorem 30.1.6 of [28]):(6.4) det( ρ )( g ) = det( Ind GH ( χ H ))( g ) = ∆ GH ( g ) · χ H ( T G/H ( g )) , for all g ∈ G, where ∆ GH is the determinant of Ind GH (1 H ), and T G/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]. Converse theorem in the Galois side.
It is well-known the answer of the followingquestion:
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 L ( s ) = ∞ (cid:88) n =1 a n n s , under what conditions is the function f ( z ) = ∞ (cid:88) n =1 a n e πinz 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 haveprovided a way to characterize Dirichlet series associated to modular forms in terms of theiranalytic properties.The modern version of the classical converse theorems are stated in terms of automorphicrepresentations instead of modular forms. Again we know that via the Langlands local corre-spondence 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 theconverse theorem)?(b). Similarly, is there any converse theorem for Galois representations (Galois side of theconverse theorem)?
The answer of (a) is
YES . For local converse theorems in the automorphic side, here werefer [26]. Let G be a reductive group over a p-adic local field F/ Q p . Let L G be the Langlandsdual group of G , which is a semi-product of the complex dual group G ∨ and the absoluteGalois group G F := Gal(F / F). Let φ : W F × SL ( C ) → L 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 letΠ(
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 foreach local Langlands parameter φ ∈ Φ( G/F ), there should be a finite subset Π( φ ), which iscalled the local L -packet attached to φ such that the set { Π( φ ) | φ ∈ Φ( G/F ) } is a partitionof Π( G/F ), among other required properties.
The map φ (cid:55)→ Π( φ ) is called the local Langlandscorrespondence or local Langlands reciprocity law for G over F .Remark γ -factors) . We define the local γ -factors as follows(cf. [25]):(6.5) γ ( s, π × π , ψ ) := W ( s, π × π , ψ ) · L (1 − s, π V × π V ) L ( s, π × π ) . On the W F × SL ( C ) side, one defines the γ -factor in the same way [35]. For more informationabout local factors, we refer [26], [14], [15]. N 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 iso-morphism. 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 representa-tion π up to L -packet. On the other hand, if the irreducible admissible representations underconsideration have additional structures, then one may still expect a certain subcollection oflocal γ -factors classifies the irreducible representation π up to equivalence.For GL n ( F ), we have the following theorem. Theorem 6.5 ( Jacquet-Liu, 2016, [24] ). Let π , π be irreducible generic representationsof GL n ( F ) . Suppose that they have the same central character. If γ ( s, π × τ, ψ ) = γ ( s, π × τ, ψ ) as functions of the complex variable s , for all irreducible generic representations τ of GL r ( F ) with (cid:54) r (cid:54) [ n ] , then π ∼ = π .Remark . 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 GL ( F ) in terms of L - and (cid:15) -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 GL n side except VolkerHeiermann’s [21] work, because we do not have general twisting formula for local root numbersfor Galois representations. Therefore, to translate local converse theorems to Galois side, weneed 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 ρ m = ρ ⊗ (cid:102) χ F be a U -isotopic Heisenberg representation of G F of dimension prime to p . Let ψ be a nontrivialadditive character of F . Let ρ , ρ be two finite dimensional complex representations of G F with det( ρ ) ≡ det( ρ ) , and j ( ρ m ) > · max { β ( ρ ) , β ( ρ ) } . If W ( ρ ⊗ ρ m , ψ ) = W ( ρ ⊗ ρ m , ψ ) , then ρ ≡ ρ or ρ ≡ ρ ⊗ µ , where µ : F × → C × is an unramified character whose orderdivides dim( ρ i ) , i = 1 , .Proof. By the given condition j ( ρ m ) > · max { β ( ρ ) , β ( ρ ) } , then by using Theorem 1.3, wecan write W ( ρ ⊗ ρ m , ψ ) = W ( ρ ⊗ ρ m , ψ ) = ⇒ (6.6) det( ρ )( γ ) W ( ρ m , ψ ) dim( ρ ) = det( ρ )( γ ) W ( ρ m , ψ ) dim( ρ ) . Again by using Theorem 1.2, from equation (6.6), we have(6.7)det( ρ )( γ ) · W ( χ F , ψ ) dim( ρ ⊗ ρ m ) · det( ρ )( c dim( ρ ) ) = det( ρ )( γ ) · W ( χ F , ψ ) dim( ρ ⊗ ρ m ) · det( ρ )( c dim( ρ ) ) . Since det( ρ ) ≡ det( ρ ) on F × , and χ F is arbitrary character of a ( χ F ) (cid:62)
2, then from aboveequation (6.7), we can conclude that dim( ρ ) = dim( ρ ).This gives: Case-1: ρ ∼ = ρ ⊗ µ , where µ : G × F → C × is a character of G F with , hence µ can beconsidered a character of F × (via class field theory). Case-2: ρ ≡ ρ , and it is one of our assertions.When we are in Case-1, by the given assumption det( ρ ) ≡ det( ρ ) = det( ρ ⊗ µ ) =det( ρ ) · µ dim( ρ ) , then the order of µ must be a divider of dim( ρ ) = dim( ρ ) . Now we are left to prove that µ is unramified, and which follows from the given condition W ( ρ ⊗ ρ m , ψ ) = W ( ρ ⊗ ρ m , ψ ).This completes the proof. (cid:3) AppendixExample 7.1 ( Explicit description of Heisenberg representations of dimensionprime to p ). Let F/ Q p be a local field, and G F be the absolute Galois group of F . Let ρ = ρ ( X, χ K ) be a Heisenberg representation of G F of dimension m prime to p . Then fromCorollary 3.9 the alternating character X = X η is U -isotropic for a character η : U F /U F → C × .Here from Lemma 3.6 we can say m = (cid:112) [ F × : Rad(X η )] = η divides q F − U F is a pro-p-group and gcd ( m, p ) = 1, we have ( U F ) m = U F ⊂ F × m , and therefore F × /F × m ∼ = Z m × Z m , is a bicyclic group of order m . So by class field theory there is precisely one extension K/F such that Gal(K / F) ∼ = Z m × Z m and the norm group N K/F := N K/F ( K × ) = F × m .We know that U F /U F is a cyclic group of order q F −
1, hence (cid:92) U F /U F ∼ = U F /U F . By thegiven condition m | ( q F − U F /U F has exactly one subgroup of order m . Then numberof elements of order m in U F /U F is ϕ ( m ), the Euler’s ϕ -function of m . In this setting, wehave η ∈ (cid:92) U F /U F ∼ = (cid:92) F F × /U F ∧ U F with η = m . This implies that up to 1-dimensionalcharacter twist there are ϕ ( m ) representations corresponding to X η where η : U F /U F → C × is of order m . According to Corollary 1.2 of [38], all dimension-m-Heisenberg representationsof G F = Gal(F / F) are given as(1H) ρ = ρ ( X η , χ K ) , where χ K : K × /I F K × → C × is a character such that the restriction of χ K to the subgroup K × F corresponds to X η under the map (3.3), and(2H) F × /F × m ∧ F × /F × m ∼ = K × F /I F K × , N EXTENSION OF DELIGNE-HENNIART’S TWISTING FORMULA AND ITS APPLICATIONS 27 which is given via the commutator in the relative Weil-group W K/F (for details arithmeticdescription of Heisenberg representations of a Galois group, see [38], pp. 301-304). Thecondition (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 χ K = χ has been fixed, then all other χ K are given as(7.1) χ K = ( χ F ◦ N K/F ) · χ , for arbitrary characters of F × . For an optimal choice χ K = χ , and order of χ we need thefollowing lemma. Lemma 7.2.
Let
K/F be the extension of F/ Q p for which Gal(K / F) = Z m × Z m . The K × F and I F K × are as above. Then the sequence (7.2) 1 → U K K × F /U K I F K × → U K /U K I F K × N K/F −−−→ U F /U F → U F /U F ∩ F × m → is exact, and the outer terms are both of order m , hence inner terms are both cyclic of order q F − .Proof. The sequence is exact because F × m = N K/F ( K × ) is the group of norms, and F × /F × m ∼ = Z m × Z m implies that the right hand term is of order m . By our assumption the order of K × F /I F K × is m . Now we consider the exact sequence(7.3) 1 → U K ∩ K × F /U K ∩ I F K × → K × F /I F K × → U K K × F /U K I F K × → . Since the middle term has order m , the left term must have order 1, because U K is a pro-p-group and gcd ( m, p ) = 1. Hence the right term is also of order m . So the outer termsof the sequence (7.2) have both order m , hence the inner terms must have the same order q F − U F : U F ], and they are cyclic, because the groups U F /U F and U K /U K are bothcyclic. (cid:3) We now are in a position to choose χ K = χ as follows :(1) we take χ as a character of K × /U K I F K × ,(2) we take it on U K K × F /U K I F K × as it is prescribed by the above Explicit Lemma ?? , inparticular, χ restricted to that subgroup (which is cyclic of order m ) will be faithful.(3) we take it trivial on all primary components of the cyclic group U K /U K I F K × whichare not p i -primary, where m = (cid:81) ni =1 p a i i .(4) we take it trivial for a fixed prime element π K .Under the above optimal choice of χ , we have Since gcd ( m, p ) = 1, we have U F · F × m = ( < ζ > × U F )( < π mF > × < ζ m > × U F ) = < π mF > × < ζ > × U F ,where ζ is a ( q F − U F /U F ∩ F × m = U F · F × m /F × m = < π mF > × < ζ > × U F / < π mF > × < ζ m > × U F ∼ = Z m .Hence | U F /U F ∩ F × m | = m . Lemma 7.3.
Denote ν p ( n ) := as the highest power of p for which p ν p ( n ) | n . The character χ must be a character of order m q F − := (cid:89) l | m l ν l ( q F − , which we will call the m -primary part of q F − , so it determines a cyclic extension L/K of degree m q F − which is totally tamely ramified, and we can consider the Heisenberg repre-sentation ρ = ( X, χ ) of G F = Gal(F / F) is a representation of Gal(L / F) , which is of order m · m q F − .Proof. By the given conditions, m | q F −
1. Therefore we can write q F − (cid:89) l | m l ν l ( q F − · (cid:89) p | q F − , p (cid:45) m p ν p ( q F − = m q F − · (cid:89) p | q F − , p (cid:45) m p ν p ( q F − , where l, p are prime, and m q F − = (cid:81) l | m l ν l ( q F − .From the construction of χ , π K ∈ Ker( χ ), hence the order of χ comes from the restrictionto U K . Then the order of χ is m q F − , because from Lemma 7.2, the order of U K /U K I F K is q F −
1. Since order of χ is m q F − , by class field theory χ determines a cyclic extension L/K of degree m q F − , hence N L/K ( L × ) = Ker( χ ) = Ker( ρ ) . This means G L is the kernel of ρ ( X, χ ), hence ρ ( X, χ ) is actually a representation of G F /G L ∼ = Gal(L / F).Since G L is normal subgroup of G F , hence L/F is a normal extension of degree [ L : F ] =[ L : K ] · [ K : F ] = m q F − · m . Thus Gal(L / F) is of order m · m q F − .Moreover, since [ L : K ] = m q F − and gcd ( m, p ) = 1, L/K is tame. By construction wehave a prime π K ∈ Ker( χ ) = N L / K (L × ), hence L/K is totally ramified extension. (cid:3)
Lemma 7.4. (Here L , K , and F are the same as in Lemma 7.3) Let F ab /F be the maximalabelian extension. Then we have L ⊃ L ∩ F ab ⊃ K ⊃ F, { } ⊂ G (cid:48) ⊂ Z ( G ) ⊂ G = Gal(L / F) , where [ L : L ∩ F ab ] = | G (cid:48) | = m and [ L : K ] = | Z ( G ) | = m q F − .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 concludeGal(L / L ∩ F ab ) = [Gal(L / F) , Gal(L / F)] =: G (cid:48) . Since Gal(L / F) = G F / Ker( ρ ), and [[ G F , G F ] , G F ] ⊆ Ker( ρ ), from relation (3.3) we have G (cid:48) = [ G F , G F ] / Ker( ρ ) ∩ [G F , G F ] = [G F , G F ] / [[G F , G F ] , G F ] ∼ = K × F / I F K × . Again from sequence 7.3 we have | U K K × F /U K I F K × | = | K × F /I F K × | = m . Hence | G (cid:48) | = m . N EXTENSION OF DELIGNE-HENNIART’S TWISTING FORMULA AND ITS APPLICATIONS 29
From the Heisenberg property of ρ , we have [[ G F , G F ] , G F ] ⊆ Ker( ρ ), hence Gal(L / F) =G F / Ker( ρ ) is a two-step nilpotent group. This gives [ G (cid:48) , G ] = 1, hence G (cid:48) ⊆ 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 ) = (cid:112) [G : Z] = mTherefore the order of Z is m q F − and Z = Gal(L / K). (cid:3)
Remark
Special case: m = 2 , hence p (cid:54) = 2 ) . Now if we take m = 2, hence p (cid:54) = 2, andchoose χ as the above optimal choice, then we will have m q F − = 2 q F − = 2-primary factorof the number q F −
1, and Gal(L / F) is a 2-group of order 4 · q F − .When q F ≡ − q F is of the form q F = 4 l −
1, where l (cid:62)
1. So we can write q F − l − l − q F ≡ − χ is 2 q F − = 2. Then Gal(L / F) will be of order 8 if and only if q F ≡ − i (cid:54)∈ F . And if q F ≡ q F − m for someinteger m (cid:62)
1, hence 2 q F − (cid:62)
4. Therefore when q F ≡ / F) willbe at least 16.
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