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On the Conductor of Certain Local L-Functions

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On the Conductor of Certain Local L-Functions J. Ramanujan Math. Soc. 31, No.2 (2016) 137–156 On the conductor of certain local L-functions U. K. Anandavardhanan1 and Amiya Kumar Mondal2 1Department of Mathematics, Indian Institute of Technology Bombay, Mumbai 400 076, India e-mail: [email protected] 2School of Mathematics, Tata Institute of Fundamental Research, Mumbai 400 005, India e-mail: [email protected] Communicated by: R. Dipendra Prasad Received: August 14, 2015 Abstract. The conductor formula of Bushnell, Henniart and Kutzko [BHK98] computes the conductor of a pair of supercuspidal representations of general linear groups over a p-adic field. This is the conductor of the tensor product lift. In this paper, we give an explicit formula for the conductors of the symmetric and exterior square lifts, under the assumption that p = 2. 2000 Mathematics Subject Classification: 11F33, 11F70, 11F85. 1. Introduction Let F be a p-adic field. Let oF be the ring of integers of F and let pF denote the unique maximal ideal of oF .Letψ be a continuous nontrivial additive character of F.Letn(ψ) be the largest integer n such that ψ is trivial on p−n. π ( ) = , If i ’s are irreducible admissible representations of GLni F for i 1 2 respectively, then Jacquet, Piatetski-Shapiro and Shalika [JPSS83] attach an L-function L(s,π1 × π2) and an epsilon factor (s,π1 × π2,ψ)to the given data, where s ∈ C. The epsilon factor (s,π1 × π2,ψ)satisfies the following relation − f (π1×π2,ψ)s (s,π1 × π2,ψ)= q (0,π1 × π2,ψ). The integer f (π1 × π2,ψ) for an unramified character ψ of F, i.e., when n(ψ) = 0, is called the conductor for the pair (π1,π2) and is denoted by f (π1 × π2). The conductor formula of Bushnell, Henniart and Kutzko [BHK98] gives an explicit formula for f (π1 × π2), when π1 and π2 are 137 138 U. K. Anandavardhanan and Amiya Kumar Mondal supercuspidal representations of general linear groups. The ingredients in this explicit conductor formula are in terms of the data attached to π1 and π2 via the theory of compact induction of Bushnell and Kutzko [BK93]. It follows from the explicit conductor formula that f (π1 × π2) is a non-negative integer. We refer to Section 5 for more details regarding the formula of Bushnell, Henniart and Kutzko. ρ → (C) Let WF be the Weil-Deligne group of F and let πi : WF GLni be the Langlands parameters of πi ,wherei = 1, 2. Then, under the local Langlands correspondence, the conductor for pairs f (π1 × π2) equals the ρ ⊗ ρ (π × π ) Artin conductor of π1 π2 . Thus, f 1 2 is nothing but the conductor of the tensor product lift π1 π2, which is an irreducible admissible ( ) representation of GLn1n2 F . A natural question is to ask for explicit conductor formulas in the spirit of the formula of Bushnell, Henniart and Kutzko [BHK98] for other functorial lifts. In particular, we ask this for the following three functorial lifts: (i) the symmetric square lift from GLn(F) to GLn(n+1) (F), (ii) the exterior square 2 lift from GLn(F) to GLn(n−1) (F), and (iii) the Asai lift from GLn(F) to ( ) / 2 GLn2 F0 when F F0 is a quadratic extension. It is conceivable that the conductor formulas in these three cases should follow from the formula of Bushnell, Henniart and Kutzko. This is because these functorial lifts are intimately connected to the tensor product lift. Indeed, we have: ∼ 2 2 ρπ ⊗ ρπ = Sym ρπ ⊕∧ ρπ and σ ∼ ρπ ⊗ ρπ = r(ρπ )| , WF where σ is the nontrivial automorphism of F/F0 and r denotes tensor induction from W to W . Since the Artin conductor is additive, the explicit F F0 conductor formula for the tensor product gives rise to conductor formulas for the sum of conductors of the symmetric and exterior square lifts and also the sum of conductors of the Asai lift and the twisted Asai lift. Here, the twisted Asai lift is the Asai lift tensored with the quadratic character associated to F/F0. In other words, we will have explicit conductor formulas for these three functorial lifts the moment we can calculate the difference of the relevant conductors. It turns out that this is not very hard if we assume that the residue characteristic is odd. Under our assumption, the necessary calculations can be carried out on the Galois side of the local Langlands correspondence. The Asai case has been done in [AM15]. The following is [AM15, Theorem 1.4]. On the conductor of certain local L-functions 139 Theorem 1.1. Let F/F0 be a quadratic extension of p-adic fields. If it is ramified, assume also that p = 2.Letκ be a character of F× which restricts × ω / / π to the quadratic character F F0 of F0 associated to F F0.Let be an irreducible supercuspidal representation of GLn(F) and let r(π) be its Asai ( ) lift to GLn2 F0 . Then, ( (π)) + ( ,π, ) = ( (π) ⊗ ω ) + ( ,π ⊗ κ, ). f r deg L s r f r F/F0 deg L s r Here, L(s,π,r) is the Asai L-function of π. 2 For an irreducible admissible representation π of GLn(F),letSym(π) (resp. ∧2(π)) be its symmetric square (resp. exterior square) lift to 2 2 GLn(n+1) (F) (resp. GLn(n−1) (F)). Let L(s,π,Sym ) (resp. L(s,π,∧ )) 2 2 denote the symmetric square (resp. exterior square) L-function. In this paper, we prove: Theorem 1.2. Let π be an irreducible supercuspidal representation of GLn(F), where F is a p-adic field with p = 2. Then, f (Sym2(π)) + deg L(s,π,Sym2) = f (∧2(π)) + deg L(s,π,∧2) + f (π) + deg L(s,π). Theorem 1.2, together with the conductor formula of Bushnell, Henniart and Kutzko (cf. Theorem 5.1), gives explicit conductor formulas for these functorial lifts, thanks to the identity f (π × π) = f (Sym2(π)) + f (∧2(π)). (1) and ⎧ ⎨ f (r(π)) if F/F0 is unramified f (π × π σ ) = ⎩ ( (π)) + ( (π) ⊗ ω ) − 2 / f r f r F/F0 n if F F0 is ramified, (2) where in the second case, we have assumed that p = 2. We leave the statement of these explicit conductor formulas to Section 6 (cf. Theorem 6.1). We hope that these explicit conductor formulas will have applications in number theory. In this regard, we refer to the recent work of White [Whi15] where an easy upper bound for the conductor of the symmetric square L-function plays a crucial role in explicating an error term that comes in a beyond endoscopic problem [Whi15, Proposition 4.4]. Remark 1.3. ThedegreeoftheL-functions appearing in Theorem 1.1 and Theorem 1.2 can also be made explicit in terms of the inducing data of the 140 U. K. Anandavardhanan and Amiya Kumar Mondal representation π. This is either n/e or n/2e,wheree is the oF -period of the principal oF -order in Mn(F) in the simple stratum defining a maximal simple type occurring in π [AM15, Theorems 1.1,1.2 and 1.3]. Remark 1.4. Theorem 1.2, interpreted on the Galois side of the local Langlands correspondence, is a particular case of a slightly more general statement about Adams operators, and the statement is neater when presented in terms of the Swan conductor instead of the Artin conductor. We do this in Section 4 (cf. Theorem 4.1) on a suggestion by Guy Henniart. 2. Preliminaries 2.1 Artin conductor We first recall the definition of the Artin conductor [Ser79, Chapter VI]. Let F be a p-adic field with ring of integers oF with its unique maximal ideal pF and residue field kF = oF /pF .LetE be a finite Galois extension of F with Galois group G = Gal(E/F).LetoE and kE denote the ring of integers and the residue field of E respectively. The integer e = e(E/F) (resp. f = f (E/F)) denotes the ramification index of E/F (resp. the degree of kE /kF ). Now oE is generated by some x ∈ oE as an oF -algebra; oE = oF [x]. Define the order function on G by iG (s) = v E (s(x) − x), where v E is some normalized valuation defined on E. It is easy to see that −1 iG (tst ) = iG (s) for all s, t ∈ G. Now define an integer valued function on G by ⎧ ⎨⎪− f · iG (s), s = 1 ( ) = aG s ( ), = . (3) ⎩⎪ f iG s s 1 s=1 Clearly, aG is a class function on G. We have [Ser79, Chapter VI, Theorem 1]: Theorem 2.1. The function aG is the character of a representation of G. If ρ is a (virtual) representation of G, its Artin conductor is defined as the multiplicity of ρ in the Artin representation; 1 f (ρ) = (χρ, a ) = χρ(s)a (s), G G |G| G s∈G where χρ is the (virtual) character of ρ. Note that the Artin conductor is non-negative if ρ is a representation. On the conductor of certain local L-functions 141 The ramification groups Gi ¡ G are defined by Gi ={g ∈ G | iG (g) ≥ i + 1}. We have [Ser79, Chapter VI, Propositions 1 and 2]: Proposition 2.2. Let Fnr ⊂ E be the maximal unramified extension of F in E. · ( ) ∈ f aG0 g if g G0 aG (g) = 0 if g ∈ G0. Moreover, ∞ 1 = G0 , aG0 IndG ui (G0 : Gi ) i i=0 where ui is the augmentation representation of Gi .
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