Local Galois representations associated to additive polynomials

For an additive polynomial and a positive integer, we define an irreducible smooth representation of a Weil group of a non-archimedean local field. We study several invariants of this representation. We deduce a necessary and sufficient condition for it to be primitive.


Introduction
Let p be a prime number and q a power of it.An additive polynomial R(x) over F q is a one-variable polynomial with coefficients in F q such that R(x + y) = R(x) + R(y).It is known that R(x) has the form e i=0 a i x p i (a e = 0) with an integer e ≥ 0. Let F be a non-archimedean local field with residue field F q .We take a separable closure F of F .Let W F be the Weil group of F /F .Let v F (•) denote the normalized valuation on F .We take a prime number ℓ = p.For a non-trivial character ψ : F p → Q × ℓ , a non-zero additive polynomial R(x) over F q and a positive integer m which is prime to p, we define an irreducible smooth W F -representation τ ψ,R,m over Q ℓ of degree p e if v F (p) is sufficiently large.This is unconditional if F has positive characteristic.The integer m is related to the Swan conductor exponent of τ ψ,R,m .As m varies, the isomorphism class of τ ψ,R,m varies.
Let C R denote the algebraic affine curve defined by a p − a = xR(x) in A 2 Fq = Spec F q [a, x].This curve is studied in [6] and [1] in detail.For example, the smooth compactification of C R is proved to be supersingular if (p, e) = (2, 0).The automorphism group of C R contains a semidirect product Q R of a cyclic group and an extra-special p-group ((2.5)).Let F be an algebraic closure of F q .Then a semidirect group Q R ⋊ Z acts on the base change C R,F := C R × Fq F as endomorphisms, where 1 ∈ Z acts on C R,F as the Frobenius endomorphism over F q .The center Z(Q R ) of Q R is identified with F p , which acts on C R as a → a+ζ for ζ ∈ F p .Let H 1 c (C R,F , Q ℓ ) be the first étale cohomology group of C R,F with compact support.Each element of Z(Q R ) is fixed by the action of Z on Q R .Thus its ψ-isotypic part H 1 c (C R,F , Q ℓ )[ψ] is regarded as a Q R ⋊ Z-representation.We construct a concrete Galois extension over F whose Weil group is isomorphic to a subgroup of Q R ⋊ Z associated to the integer m (Definition 3.1 and (3.8)).Namely we will define a homomorphism Θ R,m : W F → Q R ⋊ Z in (3.12).As a result, we define τ ψ,R,m to be the composite This is a smooth irreducible representation of W F of degree p e .
We state our motivation and reason why we introduce and study τ ψ,R,m .It is known that the reductions of concentric affinoids in the Lubin-Tate curve fit into this type of curves C R with special R.For example, see [16] and [17].When R is a monomial and m = 1, the representation τ ψ,R,m is studied in [9] and [10] in detail.In these papers, the reduction of a certain affinoid in the Lubin-Tate space is related to C R in some sense and the supercuspidal representation π of GL p e (F ) which corresponds to τ ψ,R,m under the local Langlands correspondence explicitly.The homomorphism Θ R,1 with R(x) = x p e (e ∈ Z ≥1 ) does appear in the work [9].An irreducible representation of a group is said to be primitive if it is not isomorphic to an induction of any representation of a proper subgroup.The representation τ ψ,R,m in [9] and [10] is primitive and this property makes it complicated to describe π in a view point of type theory.It is an interesting problem to do the same thing for general τ ψ,R,m .In this direction, it would be valuable to know when τ ψ,R,m is primitive.We expect that another C R will be related to concentric affinoids in the Lubin-Tate spaces as in [9].
We briefly explain the content of each section.In §2, we state several things on the curves C R and the extra-special p-subgroups contained in the automorphism groups of the curves.
In §3.1 and §3.2, we construct the Galois extension mentioned above and define τ ψ,R,m .Let d R := gcd{p i + 1 | a i = 0}.We show that the Swan conductor exponent of τ ψ,R,m equals m(p e + 1)/d R (Corollary 3.15).In §3.3, we study primitivity of τ ψ,R,m .In particular, we write down a necessary and sufficient condition for τ ψ,R,m to be primitive.By this, we give examples such that τ ψ,R,m is primitive (Example 3.29).The necessary and sufficient condition is that a symplectic module (V R , ω R ) associated to τ ψ,R,m is completely anisotropic (Corollary 3.28).If R is a monomial, (V R , ω R ) is studied in §3.4 in more detail.In Proposition 3.44, a primary module in the sense of [11, §9] is constructed geometrically by using the Künneth formula.
Our aim in §4 is to show the following theorem.
If τ ψ,R,m is imprimitive, it is written as a form of an induced representation of a certain explicit W F ′ -representation τ ′ ψ,R 1 ,m associated to a finite extension F ′ /F .The proof of the above theorem depends on several geometric properties of C R developed in [6] and [1].See the beginning of §4 for more details.

Notation
Let k be a field.Let µ(k) denote the set of all roots of unity in k.For a positive integer r, let µ r (k For a positive integer i, let A i k and P i k be an i-dimensional affine space and a projective space over k, respectively.For a scheme X over k and a field extension l/k, let X l denote the base change of X to l.For a closed subset Z of a variety X, we regard Z as a closed subscheme with reduced scheme structure.
Throughout this paper, we set q := p f with a positive integer f .For a positive integer i, we simply write Nr q i /q and Tr q i /q for the norm map and the trace map from F q i to F q , respectively.
Let X be a scheme over F q , let F q : X → X be the q-th power Frobenius endomorphism.Let F be an algebraic closure of F q .Let Fr q : X F → X F be the base change of F q .This endomorphism Fr q is called the Frobenius endomorphism of X over F q .
For a Galois extension l/k, let Gal(l/k) denote the Galois group of the extension.
2 Extra-special p-groups and affine curves . Let A k be the set of all additive polynomials with coefficients in k.
Let p be a prime number.We simply A q for A Fq .Let R(x) := e i=0 a i x p i ∈ A q with e ∈ Z ≥0 and a e = 0. Let (2.1) We always assume (p, e) = (2, 0). ( We simply write µ r for µ r (F) for a positive integer r.Let If a i = 0, we have α p i = α −1 and α p e−i = α for α ∈ µ d R .Hence we have This is linear with respect to x and y.By (2.3), we have Proof.The former equality follows from be the group whose group law is given by This is well-defined according to (2.3) and Lemma 2.2. ( If e = 0, we have p = 2 by (2.2).We have For a group G and elements g, Proof.This is directly checked.We omit the details.
For a group G, let Z(G) denote its center and [G, G] the commutator subgroup of G.
(1) The group H R is non-abelian.We have ( The group H R is an extra-special p-group.The pairing ω R : ) is a non-degenerate symplectic form.
Proof.We show (1).Let This implies that H R is non-abelian according to Lemma 2.4.
Clearly we have Lemma 2.4.This implies β = 0. Hence we obtain Z(H R ) ⊂ Z.The last claim is easily verified.
The curve C R is studied in [6] and [1].We take a prime number ℓ = p.For a finite abelian group A, let A ∨ denote the character group Hom According to Lemma 2.6(1), we identify a character ψ ∈ F ∨ p with a character of Z(H R ).
(1) Let W ⊂ V R be an F p -subspace of dimension e, which is totally isotropic with respect to ω R .Let W ′ ⊂ H R be the inverse image of W by the natural map

Local Galois representation
In this section, we define an irreducible smooth W F -representation τ ψ,R,m and determine several invariants associated to it.In §3.2.2, we determine the Swan conductor exponent of τ ψ,R,m .In §3.3, we determine the symplectic module associated to τ ψ,R,m , and give a necessary and sufficient condition for τ ψ,R,m to be primitive.As a result, we obtain several examples such that τ ψ,R,m is primitive.If R is a monomial, we calculate invariants of the root system corresponding to (V R , ω R ) defined in [11] ( Lemma 3.36).

Galois extension
For a valued field K, let O K denote the valuation ring of K.
Let F be a non-archimedean local field.Let F be a separable closure of F .Let F denote the completion of F .Let v(•) denote the unique valuation on F such that v(̟) = 1 for a uniformizer ̟ of F , which we now fix.We simply write O for O F .Let p be the maximal ideal of O.
Let q be the cardinality of the residue field of O F .We take R(x) = e i=0 a i x p i ∈ A q .For an element a ∈ F q , let a ∈ µ(F ) ∪ {0} be its Teichmüller lift.Let Similarly as (2.3), we have (3.1) Definition 3.1.Let m be a positive integer prime to p. Let α R,̟ , β R,m,̟ , γ R,m,̟ ∈ F be elements such that The integer m controls the depth of ramification of the resulting field extension F (α R , β R,m , γ R,m )/F .We will understand this later in §3.2.
In the following, we simply write a σ , b σ , c σ for a R,σ , b R,σ , c R,σ , respectively.
For an element x ∈ O, let x denote the image of x by the map O → O/p.In the following proofs, for simplicity, we often write α, β and γ for α R , β R,m and γ R,m , respectively.
Using v(β) < 0 in Remark 3.2 and (3.2), we have ∆(x . By letting x = b σ and applying the previous relationship, we deduce that where we have used (3.3) for the last congruence.Hence we obtain We easily check that Hence the first equalities in (3.7) follow.We compute where we use the second condition in (3.5) for the second congruence.We have where we substitute σ(β) = a −m σ (β + b σ ), (3.5) and (3.1) for the second congruence.The last equality in (3.7)  Proof.There exists s ∈ Z ≥1 such that the coefficients of all polynomials in (3.2) and (3.5) have the form: In the sequel, we assume that the conditions (3.2) and (3.5) are satisfied.Let F ur denote the maximal unramified extension of F in F .

Galois representations associated to additive polynomials 3.2.1 Construction of Galois representation
We assume that (3.2) and (3.5) are satisfied.If the characteristic of F is positive, these are unconditional.If the characteristic of F is zero, these conditions are satisfied if the absolute ramification index of F is large enough as in Lemma 3.6.
) .Hence the representation τ ψ,R,m is a smooth irreducible representation of W F by Lemma 2.8 (1).
Let G F := Gal(F /F ).We consider a general setting in the following lemma.
Lemma 3.11.Let τ be a continuous representation of G F over Q ℓ such that there exists an unramified continuous character Hence so is τ .Since τ is irreducible and smooth, we have dim τ < ∞.We will show that τ ′ is imprimitive if and only if τ is imprimitive.
First, assume an isomorphism τ ′ ≃ Ind G F H η ′ with a proper subgroup H.We can check Ker τ ′ ⊂ H. Hence H is open.Hence we can write H = G F ′ with a finite extension F ′ /F .Hence we obtain an isomorphism τ ≃ Ind To the contrary, assume τ ≃ Ind W F H σ. In the same manner as above with replacing G F by W F , the subgroup H is an open subgroup of W F of finite index by dim τ < ∞.Hence we can write H = W F ′ with a finite extension . By the assumption, the image σ ′ (W F ′ ) is finite.Hence the smooth W F ′ -representation σ ′ extends to a smooth representation of G F ′ , for which we write σ ([2, Proposition 28.6]).

The restriction of Ind
Lemma 3.12.The eigenvalues of have the forms ζ √ q with roots of unity ζ.The automorphism Fr * q is semi-simple.Proof.The claims follow from Proposition 2.10.

The cohomology group
, we obtain a continuous representation of G F .We denote this representation by τ ψ,R,m .Let φ : G F → Q × ℓ be the unramified character sending a geometric Frobenius to √ q −1 .The image of G F by the twist τ ′ := τ ψ,R,m ⊗ φ is finite by Lemma 3.12.By fixing an isomorphism C is primitive if and only if τ ′ is primitive.We obtain the claim by applying Lemma 3.11 with τ = τ ψ,R,m and τ = τ ψ,R,m .

Swan conductor exponent
In the sequel, we compute the Swan conductor exponent Sw(τ ψ,R,m ).
We simply write α, β, γ for α R , β R,m , γ R,m in Definition 3.1, respectively.We consider the unramified field extension F r /F of degree r such that N := F r (α, β, γ) is Galois over F .Let T := F r (α) and M := T (β).Then we have otherwise.
Proof.We easily have  Hence the former claim follows from We can check

Symplectic module associated to Galois representation
Let ρ : W F → PGL p e (Q ℓ ) denote the composite of τ ψ,R,m : W F → GL p e (Q ℓ ) with the natural map GL p e (Q ℓ ) → PGL p e (Q ℓ ).Let ρ ′ be the projective representation associated to τ ′ = τ ψ,R,m ⊗ φ.
, the proof of Lemma 2 in 28.6]).This implies the claim.
Let F ρ denote the kernel field of ρ and T ρ the maximal tamely ramified extension of The homomorphism ρ induces an injection ρ : G → PGL p e (Q ℓ ).Let V R be as in Lemma 2.6.Let τ denote the composite Lemma 3.17.We have an isomorphism ρ(H) ) and K := F ur (α m R ).By (3.8), we have The subfield K is the maximal tamely ramified extension of F in L. We have F ρ ⊂ L and . By Lemma 2.9, we have H R ∩Ker τ = Z(H R ).
Hence the claim follows from the isomorphism V R Let σ ∈ H 0 .We take a lifting σ ∈ G ։ H 0 of σ.Let H 0 act on H by σ •σ ′ := σσ ′ σ −1 for σ ′ ∈ H.This is well-defined because H is abelian by Lemma 3.17.We regard H ≃ V R as an F p [H 0 ]-module.By Lemma 3.12, we can take a positive integer r such that rZ ⊂ Ker τ and where the isomorphism is given by (α, 0) → τ and (1, −1) → σ.The groups H 0 and H are supersolvable.We consider the commutative diagram where every map is canonical and surjective.
Proof.These are directly checked.
We can regard V R as an F p [H ]-module via ϕ.Let ω R be as in Lemma 2.6(2).
Proof.The claim for h = α follows from (2.4).For h = (1, −1), the claim follows from Definition 3.20.Let G be a finite group.Let V be an Let ω : V × V → F p be a symplectic form.We say that the pair (V, ω) is symplectic if ω is non-degenerate and satisfies ω(gv, gv Proof.The claim follows from Lemma 2.6(2) and Lemma 3.19.
Let k be a field.We say that a polynomial f (1) Assume that E(x) is monic and V is stable under σ.Then we have E(x) ∈ F q [x].
(2) Let r be a positive integer.Assume that V is stable under µ r -multiplication.Then we have E(αx) = αE(x) for α ∈ µ r .
(2) We say that f (x) ∈ A q is prime if it does not admit a non-trivial decomposition Definition 3.26.Let (V, ω) be a symplectic F p [H ]-module.Then (V, ω) is said to be completely anisotropic if V does not admit a non-zero totally isotropic F p [H ]submodule.
For an Assume that V R is not completely anisotropic.We take a non-zero totally isotropic [13, 4 in Chapter 1], there exists a monic reduced polynomial ] by Lemma 3.24 (1).Since V ′ is stable by τ , we have f (αx) = αf (x) for α ∈ µ d R,m by Lemma 3.18 and Lemma 3.24 (2).There exist Hence the converse is shown.
(2) The W F -representation τ ψ,R,m is primitive if and only if there does not exist a non- Proof.The claim (1) follows from Corollary 3.13, Lemma 3.17 We show (4).We assume that there exists a non-zero which is regarded as a ring with multiplication ϕ 1 • ϕ 2 := ϕ 1 (ϕ 2 (x)) for ϕ 1 , ϕ 2 ∈ A q,s .
In the following, we give examples such that E R (x) is prime.We write d R = p t + 1 with t ≥ 0. Then we have E R ∈ A q,t .We write q = p f .Assume f | t.We have By f | t, we have the ring isomorphism Φ : ) is a reciprocal polynomial.The number of the monic irreducible reciprocal polynomials is calculated in [3, Theorems 2 and 3].
In general, we do not know a necessary and sufficient condition on R(x) for E R (x) to be prime.The number of prime elements in A q,s is calculated in [4] and [12].Proposition 3.30.Assume d R,m ∈ {1, 2}.There exists an unramified finite extension Proof.For a positive integer r, let F r be the unramified extension of F of degree r.We take a non-zero element β ∈ V R .Let t be the positive integer such that F q t = F q (β).Let H t ⊂ H be the subgroup generated by σ t , τ .The subspace Proof.We take a non-zero element β ∈ V R .Then F p β is a totally isotropic symplectic submodule of the symplectic module V R associated to τ ψ,R,m | W Tρ .Hence τ ψ,R,m | W Tρ is imprimitive by Corollary 3.28(1).

Root system associated to irreducible F p [H ]-module
A root system associated to an irreducible F p [H ]-module is defined in [11].We determine the root system associated to V R in the situation of Corollary 3.28 (3).
We recall the definition of a root system.
(2) Let W = Φ(α, β) be a root system.Let a = a(W ) be the minimal positive integer with α q a = α, b = b(W ) the minimal positive integer with α p b = α q x , β p b = β with x ∈ Z, and c = c(W ) the minimal non-negative integer with α p b = α q c .Let e ′ = e ′ (W ) and f ′ = f ′ (W ) be the orders of α and β, respectively.These integers are independent of (α, β) in W .
(4) We say that a root system W belongs to H d,r if e ′ | d and af ′ | r.
(5) Let W = Φ(α, β) be a root system which belongs to H d,r .Let M(W ) be the F-module with the basis

2])
(1) There exists an irreducible (2) The map W → M(W ) defines a one-to-one correspondence between the set of root systems belonging to H d,r and the set of isomorphism classes of irreducible We go back to the original situation.Assume that R(x) = a e x p e and F p (µ d R,m ) = F p 2e .Let H be as in (3.9).In the above notation, we have Proposition 3.34.We write q = p f .Let e 1 := gcd(f, 2e) and β := Nr q/p e 1 (−a −(p e −1) e ).Let α ∈ µ d R,m be a primitive d R,m -th root of unity.We consider the root system W := Φ(α, β).
(2) The root system W belongs to H . ( Proof.We show (1).We simply write a, b, c for a(W ), b(W ), c(W ), respectively.By definition, a is the minimal natural integer such that α q a = α.By F p (α) = F p 2e , a is the minimal positive integer satisfying f a ≡ 0 (mod 2e).Thus we obtain a = 2e/e 1 .By definition, b is the minimal natural integer such that α p b = α q x with some integer x and β p b = β.The first condition implies that f x ≡ b (mod 2e).Hence b is divisible by e 1 .By β ∈ F × p e 1 , we have By definition, c is the minimal non-negative integer such that α p b = α q c .This is equivalent to e 1 = b ≡ f c (mod 2e).

Künneth formula and primary module
Classification results in [11] We recall classification results on completely anisotropic symplectic modules given in [11] restricted to the case p = 2.
Theorem 3.37.([11, Theorem 9.1]) Let (V, ω) = n i=1 (V i , ω i ) be a direct sum of irreducible symplectic F p [H ]-modules.Assume that p = 2. Then (V, ω) is completely anisotropic if and only if, for each isomorphism class, the modules of type B or C occur at most once and of type A at most twice among V 1 , . . ., V n .
Assume that p = 2. Let (M(W ), 0) denote the unique symplectic module on M(W ) which is of type B or C by Theorem 3.35.Let (M(W ), 0), (M(W ), 1) denote the two symplectic modules on M(W ) in the case where p = 2 and M(W ) is of type A. We denote by (M(W ), 2) the completely anisotropic symplectic module on M(W ) ⊕ M(W ), where M(W ) is of type A. Let k be a positive integer.Let R := {R i } 1≤i≤k with R i ∈ A q .We consider the k-dimensional affine smooth variety X R defined by Let the notation be as in (3.3).Let m = {m i } 1≤i≤k , where m i is a positive integer.We have the homomorphism .12) Definition 3.39.We define a smooth W F -representation τ ψ,R,m to be the inflation of the where δ is the diagonal map.Hence the claim follows.
Remark 3.41.Let + :  Let T R be the maximal tamely ramified extension of F in F R .We have the restriction map (2).We give a recipe to make an example of (M(W ), 2) below.
This is well-defined by Lemma 2.2 and (4.2).By Lemma 4.1, we have E R (β) = 0. Since E R 1 (r(x)) is separable, the claim follows.We have The space V ′ R is stable by the q-th power map.Hence we can consider the semidirect product Q ′ R,m ⋊ Z.The map π induces π ′ : Proposition 4.7.We have an isomorphism τ ψ,R,m ≃ τ ′′ ψ,R 1 ,m as W F -representations.
Proof.By Lemma 4.6, we have the injection as Q ′ R,m ⋊ Z-representations.Hence we have a non-zero homomorphism as Q R,m ⋊ Z-representations by Frobenius reciprocity.Since the target is irreducible by Lemma 2.8(2), the map (4.5) is surjective.By comparing the dimensions, (4.5) is an isomorphism.By inflating it by Θ R,m , we obtain the claim.

(
a j x p i y) p j + (x R(y)) p i .Let p[x] := pO[x] and p[x, y] := pO[x, y].We assume that
Thus the first claim follows.The second claim follows from the first one and [11, Theorem 4.1].If gcd(p e + 1, m) = 1, we have d R,m = d R = p e + 1. Hence the third claim follows from F p (µ p e +1 ) = F p 2e .Example 3.29.For a positive integer s, we consider the set and with the action of H by τ m = αm, σ a m = βm, θ b m = σ −c m.Theorem 3.33.([11, Theorems 7.1 and 7.

Theorem 3 .
38. ([11, Theorem 8.2]) Each completely anisotropic symplectic F p [H ]module is isomorphic to one and only one symplectic module of the form n i=1(M(W i ), ν i ),where W 1 , . . ., W n are mutually different root systems belonging to H .
is irreducible by[8, 16.14(2)Satz].The claim follows from this.Let ρ ψ,R i ,m i denote the projective representation associated to τ ψ,R i ,m i .Let F i denote the kernel field of ρ ψ,R i ,m i and T i the maximal tamely ramified extension of F in F i .The field T i is called the tame kernel field of ρ