ON THE SHARPNESS OF THE BOUND FOR THE LOCAL CONVERSE THEOREM OF p-ADIC GLprime

We introduce a novel ultrametric on the set of equivalence classes of cuspidal irreducible representations of a general linear group GLN over a nonarchimedean local field, based on distinguishability by twisted gamma factors. In the case that N is prime and the residual characteristic is greater than or equal to ⌊ N 2 ⌋ , we prove that, for any natural number i ≤ ⌊ N 2 ⌋ , there are pairs of cuspidal irreducible representations whose logarithmic distance in this ultrametric is precisely −i. This implies that, under the same conditions on N , the bound ⌊ N 2 ⌋ in the Local Converse Theorem for GLN is sharp.


Introduction
Let F be a non-archimedean local field and fix a non-trivial additive character ψ of F .Given irreducible generic representations π and τ of GL N (F ) and GL r (F ), respectively, the twisted-gamma factor γ(s, π × τ, ψ) is a function of a complex variable s, defined either by using Rankin-Selberg convolution [JPSS83] or by using the Langlands-Shahidi method [Sha84].
If we now fix π, the γ(s, π ×τ, ψ), as τ runs through the irreducible generic representations of GL r (F ), with r ≥ 1, give a set of important invariants of π.A natural question to ask is, how large do we need to allow r to be in order to completely determine π using these invariants?This is usually called the Local Converse Problem for GL N .It is an easy consequence of the work of Jacquet, Piatetskii-Shapiro, and Shalika [JPSS83] that r ≤ N .By the work of Henniart [Hen93], r ≤ N − 1 for N ≥ 2, and by the work of Chen [Che06] and the work of Cogdell and Piatetskii-Shapiro [CPS99], r ≤ N − 2 for N ≥ 3. The following recently proved theorem was known as Jacquet's conjecture; it was originally formulated for generic representations but was reduced to the case of cuspidal representations in [JNS15].
The bound N 2 is suggested by an analogous conjecture for automorphic representations (see [CPS99, Conjecture 1] and [ALSX16,Introduction]), and it is expected to be sharp.It is easy to construct examples of generic (non-cuspidal) representations showing that the bound N 2 is sharp for the generic version of Theorem A, but so far no sharpness results exist for cuspidal representations of GL N (F ) when N > 4. Indeed previous results have given families of representations for which only twisting by characters is required (for example, simple cuspidal representations in [AL16] and [BH14]).
In this paper, we show that the bound N 2 is indeed sharp in Theorem A when N is a prime distinct from the residual characteristic p of F and p ≥ N 2 .In fact, we prove much more.
For π 1 , π 2 inequivalent cuspidal irreducible representations of GL N (F ), we define v γ (π 1 , π 2 ) to be the minimal integer i ≥ 0 for which there exists a cuspidal irreducible representation ρ of GL i (F ) with as functions of the complex variable s. (When i = 0, so that ρ is the trivial representation of the trivial group, we interpret this as meaning that γ(s, π 1 , ψ F ) = γ(s, π 2 , ψ F ).) Thus the sharpness result would say that there are cuspidal irreducible representations π 1 , π 2 such that v γ (π 1 , π 2 ) = N 2 .We prove the following finer result.

Theorem B.
Let N be a prime distinct from p and let i be an integer with The pairing v γ induces an ultrametric d γ on the set of equivalence classes of cuspidal irreducible representations of GL N (F ) by where q is the cardinality of the residue field of F and where we understand v γ (π, π) = +∞, so that d γ (π, π) = 0.This ultrametric takes values in q −i | 0 ≤ i ≤ N 2 ∪ {0}, and Theorem B implies that when N is a prime distinct from p and p ≥ N 2 , the ultrametric takes all these values.
It is not entirely clear to the authors how the ultrametric d γ should be interpreted.We note only that it is quite different from the distance function induced by the ultrametric on endo-classes recently defined by Bushnell-Henniart in [BH17].
When N = 1, Theorem B is empty.When N = 2, 3, it is easy to find two cuspidal representations of GL N (F ) with the same standard gamma factors and with different central characters (hence their GL 1 (F )-twisted gamma factors are different by [JNS15, Corollary 2.7]).Thus Theorem B is new for N ≥ 5, though in fact the proof works for any odd prime N .
For composite N , analogous examples for the case N = 4 have already been constructed in [Che96, Section 5].It seems likely that similar ideas to those in this paper should allow one to prove the same result for arbitrary N .
On the other hand, the tameness assumption (that N = p and that i < p in Theorem B) is central to our proof.Indeed, we prove instead the analogous result for γ-factors of representations of the Weil group of F (see Theorem 3.2) and transfer it via the local Langlands correspondence.The main idea is to find two totally ramified admissible pairs (E, θ 1 ), (E, θ 2 ) of a particular depth (depending on i), which agree on U 2 E , on the group μ F of roots of unity in F of order coprime to p, and on a fixed uniformizer E of E. It is the choice of depth which allows one to verify that the corresponding cuspidal representations π 1 , π 2 satisfy d γ (π 1 , π 2 ) = i.
We end the introduction with a brief summary of the contents of each section.In Section 2, we recall Howe's construction of irreducible representations of the Weil group via admissible pairs and the formula for computing -factors, as well as some further results on representations of Weil groups.In Section 3, we construct the admissible pairs which will allow us to prove Theorem B and complete the proof in Section 4.

Local Langlands parameters
In this section, we recall basic properties of irreducible local Langlands parameters for GL N when the residual characteristic is prime to N , as constructed by Howe, and formulas for computing epsilon factors of these representations.We mainly refer to [How77,Moy86,BH06].
We begin by fixing notation.Let F be a non-archimedean local field, with ring of integers O F , maximal ideal P F in O F , and residue field of cardinality q and characteristic p.We write We fix once and for all a non-trivial additive character ψ F of F of level 1; that is, ψ F is trivial on P F but not on O F .For any finite extension E/F , we put ψ E = ψ F •tr E/F and additive character of E; if E/F is tamely ramified, then ψ E is also of level 1. Write W F for the Weil group of F .
Throughout the paper, we assume that N ≥ 1 is an integer which is prime to p.There is then a nice parametrization of irreducible representations of W F of dimension N using admissible (quasi-)characters, introduced by Howe.

Definition 2.1 ([How77]
).An admissible pair of degree N with respect to F is a pair (E, θ) consisting of an extension E/F of degree N and a character θ of E × such that: Two admissible pairs (E 1 , θ 1 ) and (E 2 , θ 2 ) with respect to F are said to be conjugate over By local class field theory, we identify characters of F × with those of W F .The following theorem says that irreducible N -dimensional representations of W F are parametrized by conjugacy classes of admissible pairs of degree N with respect to F (see [Moy86, Theorem 2.2.2] when F has characteristic zero or [BH05, A.2 Proposition, A.3 Theorem] in general).
Theorem 2.2.If (E, θ) is an admissible pair of degree N with respect to F , then the representation Ind W F W E θ is irreducible.Moreover, two such admissible pairs induce to equivalent representations of W F if and only if they are conjugate, and each irreducible N -dimensional representation of W F is induced from an admissible pair of degree N .
. There is and c θ is well-defined modulo P . We say that θ is represented by c θ .We observe, for later use, that if θ 1 , θ 2 are ramified characters of F × with unequal conductoral exponents f 1 , f 2 respectively and represented by c 1 , c 2 respectively, then the character θ 1 θ 2 has conductoral exponent max{f 1 , f 2 } and is represented by For θ a ramified character of F × represented by c θ as above, we define the Gauss sum Note that if f is even, then this sum is just trivial, so G(θ, ψ F ) = 1.
We do not recall the definition of the L-functions or -factors of finite-dimensional semisimple representations of W F , instead referring to [BH06,§29].For now, we recall only that L(s, θ) = 1 for any ramified character θ, and that, for ϕ any finitedimensional semisimple representation of W F , the γ-factor is given by In particular, if θ is a ramified character, then γ(s, θ, ψ F ) = (s, θ, ψ F ).The following theorem is [Moy86, (2.3.17)] when F has characteristic zero and can be extracted from the results in [BH06, § §23.4-.6] in general (see especially 23.5 Theorem and 23.6 Proposition).
Theorem 2.3.Let θ be a ramified character of F × represented by c θ .Then For E/F an extension of F , we write λ E/F (ψ F ) for the Langlands constant associated to E/F and ψ F (see [BH06,§30.4]).The following lemma, together with Theorem 2.3, will allow us to compute the -and γ-factors of the representations of W F which will interest us.

Lemma 2.4. For any finite extension E/F and any ramified character
The formula is also true when γ is replaced by .
Proof.The exercise in [BH06, §29.3], together with the hypothesis that θ is ramified, gives L(s, , so the same identity holds if we replace by γ. As well as inducing, we will need to restrict characters.Let θ be a ramified character of F × of conductor f θ and represented by c θ .For E/F a tamely ramified extension, we write θ E = θ • N E/F , a ramified character of conductor e(f θ − 1) + 1, where e = e(E/F ) is the ramification index of E/F .Then, recalling that we have the character ψ E = ψ F • tr E/F , the character θ E is also represented by c θ (see for example [BH06, 18.1 Proposition]).Note that, viewed as a character of the Weil group W E , the character θ E is simply the restriction of θ.
We end this section with two lemmas which we will need for computing -factors of pairs.Lemma 2.5.Let E and L be field extensions of F , and let φ and λ be characters of E × and L × respectively.Then Proof.This is just the Mackey induction formula, recalling that local class field theory tells us that the restriction to W E of φ, viewed as a character of W F , is given by φ The following lemma shows that, in the situation of Lemma 2.5 which we will be considering in this paper, the double coset space W L \W F /W E is actually a singleton.
Lemma 2.6.Let E/F be a totally (tamely) ramified extension of degree N and let L/F be a finite extension whose ramification index is prime to pN .Then the double coset space W L \W F /W E is a singleton.
Proof.Put e = e(L/F ) and f = f (L/F ) and let K/F be the Galois closure of the compositum EL, which is a tamely ramified extension of ramification index eN and some residue degree fd.We have which we prove is a singleton.
Let F be a uniformizer of F and let ζ ∈ K be a root of unity of order q fd − 1.By [Has80, p. 251], we can write K = F [ζ, K ], where eN K = F ζ c for a certain c, depending on K, such that eN divides both q fd − 1 and c(q − 1).Moreover, by [Has80, p. 252], the Galois group Gal(K/F ) has presentation Then Gal(K/E) = ρτ j , τ N and Gal(K/L) = ρ f τ k , τ e , for some integer j, k.
The coset space Gal(K/F )/Gal(K/E) has {τ i | 0 ≤ i ≤ N − 1} as a set of representatives.The action of τ e ∈ Gal(K/L) on this coset space is given by τ i Gal(K/E) → τ i+e Gal(K/E).Since e, N are coprime, this action is transitive, and the result follows.

Construction
From now until the end of the paper, we assume that N ≥ 3 is a prime different from p. Let i ≤ N 2 be a natural number.In this section, we explicitly construct a family of irreducible N -dimensional representations of W F (depending on i) with the property that their twisted γ-functions by irreducible r-dimensional representations of W F are equal, for r < i, but for which there exists an irreducible idimensional representation of W F whose twisted γ-function distinguishes them.In particular, the local Langlands correspondence then implies that the corresponding cuspidal representations of GL N (F ) show that the bound N 2 in Theorem A is sharp.
The idea is to pick two totally ramified admissible pairs (E, θ 1 ), (E, θ 2 ) of a particular depth (depending on i) which agree on U 2 E , on the group μ F of roots of unity in F of order coprime to p, and on a fixed uniformizer E of E. To identify the correct depth, we begin with a simple numerical lemma.Lemma 3.1.Let i ≤ N 2 be a natural number.Then there exist unique integers k, v with 2 ≤ k < N such that Nv − ik = 1.Moreover, if r < i is a natural number, then there is no integer u such that |Nu − rk| = 1.
As an example, we note that when i = N 2 , we have k = 2. Proof.The first assertion is just Bézout's Lemma, since i, N are coprime (since N is prime).The integers k < N are then coprime, and the solutions to |Nu − rk| = 1 are given by r = ±i + Nt, u = ±v + kt (for the same choice of sign).In particular, since 0 < i < N 2 , the only solution with 0 < r < N 2 is given by r = i, and the result follows.
We now fix a natural number i ≤ N 2 and let k be the integer given by Lemma 3.1.We also fix a uniformizer F in F and the totally tamely ramified extension and extend it to a character φ 0 of U 2 E .We consider the set Φ of characters of E × such that (i) φ| U 2 E = φ 0 ; (ii) φ(ζ) = 1, for all ζ ∈ μ F , the group of roots of unity of order prime to p in F ; (iii) φ( E ) = 1.Note that the final condition implies that φ(β) = 1 also.Moreover, the restriction of φ to The set Φ has cardinality q, since we may construct such a character by extending φ 0 arbitrarily to U 1 E , and then conditions (ii), (iii) determine it uniquely.Each φ ∈ Φ has conductoral exponent k +1 and is represented by β, by construction.Since the conductoral exponent is not congruent to 1 (mod N ), the restriction of φ to U 1 E does not factor through N E/F , so that (E, φ) is an admissible pair over F of degree N .Moreover, the admissible pairs (E, φ), for φ ∈ Φ, are inequivalent: The following theorem is our main result.
Theorem 3.2.Let (L, λ) be an admissible pair of degree r < i, with p r.Then, for φ ∈ Φ, the γ-function γ(s, Moreover, if p i, then, for any distinct φ 1 , φ 2 ∈ Φ, there is an admissible pair (L, λ) of degree i such that γ(s, ).We will prove Theorem 3.2 in the following section and first derive its consequence (via the local Langlands correspondence) for cuspidal representations of GL N (F ), Theorem B of the introduction.Recall that, as in the introduction, for π 1 , π 2 cuspidal irreducible representations of GL N (F ), we define v γ (π 1 , π 2 ) to be the minimal non-negative number i for which there exists a cuspidal irreducible representation ρ of GL i (F ) with Corollary 3.3.Let N ≥ 3 be a prime distinct from p and let 0 ≤ i ≤ min p − 1, N 2 be an integer.Then there exist cuspidal irreducible representations In particular, we see that the bound N 2 for r in Theorem A is indeed sharp, for N a prime different from p and p ≥ N 2 .Proof of Corollary 3.3.Let φ 1 , φ 2 be distinct characters in Φ; then, by the remarks before the statement of Theorem 3.2, the admissible pairs (E, φ j ) are non-conjugate.For j = 1, 2, write π j for the cuspidal irreducible representation of GL N (F ) corresponding to Ind W F W E φ j via the local Langlands correspondence.Since i < p, for each r ≤ i the local Langlands correspondence gives a bijection between the cuspidal irreducible representations of GL r (F ) and the representations Ind W F W L λ for equivalence classes of admissible pairs (L, λ).Since the local Langlands correspondence preserves γ-factors of pairs, the result follows immediately from Theorem 3.2.
Note that in the proof of Corollary 3.3, we have not explicitly identified cuspidal irreducible representations π 1 , π 2 of GL N (F ) such that v γ (π 1 , π 2 ) = i in terms of the inducing data (i.e., the types) which give them, but only in terms of their Langlands parameters.In order to identify the inducing data one would need to use Bushnell-Henniart's (essentially) tame local Langlands correspondence [BH10] and, in particular, compute the so-called rectifier for the admissible pairs concerned.
n contains no rational number with denominator e or (b) n is prime to e and jb ≡ −1 (mod n), for j = 1, . . ., In fact, we will only use the conditions (b) in Lemma 4.1; the conditions (a), which are more straightforward, are included with a view to the possibility of extending the results here to composite N .
Proof.The proof of (ii) is exactly analogous to that of (i), so we prove only the latter.We put a = val L (α) and write Gal(K/E) for the set of embeddings of K in the separable closure of F which fix E.
(i)(a) We have where E j (α −1 ) denotes the j th symmetric polynomial in {σ(α −1 ) | σ ∈ Gal(K/E)}.Since val F (β) > val F (α) we have b n > a e and, by the hypothesis, also b−1 n > a e .Since Gal(K/E) preserves the valuation val F , we get, for each j = 1, . . ., [K : E], Then the natural restriction map Gal(K/E) → Gal(L/M ) is injective, since K = LE.By the uniqueness of unramified extensions of F of given degree, E/M and L/M have coprime residue class degrees.Since, by hypothesis, they also have coprime ramification index, they have coprime degrees.This implies that [K : E] = [L : M ] so the injective map Gal(K/E) → Gal(L/M ) is in fact a bijection.
Now we specialize to the situation of Theorem 3.2, so that is a natural number, and k is the integer given by Lemma 3.1; in particular, if r < i, then rk ≡ ±1 (mod N ).A simple application of Lemma 4.1 yields the following.
Corollary 4.2.With notation as in Lemma 4.1, suppose that E/F is totally ramified of degree N , that b = −k, and that Again, the proofs are similar so we treat only (i).Since i, N are coprime, we have E ∩ L = F and, as in the proof of Lemma 4.1, we also have Finally, we are ready to prove Theorem 3.2.
Proof of Theorem 3.2.Let (L, λ) be an admissible pair of degree r ≤ i. Writing f λ = u + 1 for its conductoral exponent, the character λ is represented by some (choice of) element α ∈ P −u L , so that Let e = e(L/F ) be the ramification index of L/F and put K = EL, an extension of F of degree rN and of ramification index eN .We put φ K = φ • N K/E and λ K = λ • N K/L , ramified characters of K × of conductoral exponent ke + 1 and uN + 1 respectively, represented by β and α respectively.Note that ke + 1 = uN + 1, since N is prime and 1 ≤ k, e < N , so N is prime to ke.
We set θ = φ K ⊗ λ K , a character of K × .Then Lemmas 2.5 and 2.6 imply that is a ramified character of K × of conductoral exponent max{ke, uN }+1 and is represented by α+β.In particular, Lemma 2.4 implies that γ(s, where ψ K = ψ F • tr K/F .Putting this together, Theorem 2.3 implies that the γfactor γ(s, Only the final two terms on the right hand side may depend on the choice of φ ∈ Φ.In fact, the Gauss sum G(θ, ψ K ), which is defined as in (2.1), does not.Indeed, the conductoral exponent of θ is f = max{ke, uN } + 1, and θ = φ K λ K is represented by α + β, both of which are independent of φ ∈ Φ.Thus it is sufficient to observe that φ Thus the only term which may depend on the choice of φ ∈ Φ is the term θ(α + β) −1 .We prove that if r < i, then it does not.Since θ = φ K λ K , we need only show that φ(N K/E (α + β)) is independent of φ.We split into two cases: • If val K (β) < val K (α), then we write β +α = β(1+β −1 α).By Corollary 4.2 we have N K/E (1 + β −1 α) ∈ U 2 E , and the independence follows since all characters in Φ agree on β and on U 2 E .• If val K (β) > val K (α), then we write β + α = α(1 + βα −1 ).Since α ∈ L while [K : E] = [L : F ], we have N K/E (α) = N L/F (α) ∈ F × .By Corollary 4.2 again, N K/E (1 + βα −1 ) ∈ U 2 E , and the independence follows since all characters in Φ agree on F × and on U 2 E .This completes the proof of the first assertion of Theorem 3.2.Now let φ 1 , φ 2 ∈ Φ be distinct characters.We must show that there is an admissible pair (L, λ) of degree i such that γ(s, Ind W F W E φ 1 ⊗ Ind W F W L λ, ψ F ) = γ(s, Ind W F W E φ 2 ⊗ Ind W F W L λ, ψ F ).We put v = (ik +1)/N , which is a positive integer, by the choice of k in Lemma 3.1.Then, from the arguments above, it is sufficient to find a totally ramified extension L/F of degree i and an element α ∈ L × of valuation val L (α) = −v such that, writing K = EL as above, Moreover, since then val K (β) = −ik > −(ik + 1) = val K (α), and all φ ∈ Φ agree on N L/F (α) ∈ F × , it is sufficient to check that Since φ 1 , φ 2 agree on U 2 E but differ on U 1 E , there is a root of unity ζ ∈ μ F such that φ 1 (1 , where i L = (−1) i+1 ζ N F , and set α = ζ k −v L .Since, by construction, i, v are coprime, the minimum polynomial of α −1 over F is X i − N L/F (α −1 ) = X i − ζ −ki+vN v F so that the j th symmetric polynomial in the conjugates of α −1 is zero for 1 ≤ j < i.In particular, repeating the analysis in the proof of Lemma 4.1(i)(b), we get Thus, if λ is a character of L × represented by α, then the admissible pair (L, λ) is as required.

4.
Proof of Theorem 3.2 Before starting to prove Theorem 3.2, we need the following numerical lemma.Lemma 4.1.Suppose E = F [β]/F is a finite extension of ramification index n and val E (β) = b is prime to n.Let L/F be a finite extension of ramification index e, let α ∈ L, and set K = LE.(i) Suppose val F (β) > val F (α) and either (a) the interval b−1 n , b n contains no rational number with denominator e or (b) n is prime to e and jb ≡ 1 [K : E] = [L : F ] < i.Then, for 1 ≤ j ≤ [K : E], we have −kj ≡ 1 (mod N ) by the choice of k, and the result follows from Lemma 4.1(i)(b).