AN EXPLICIT CONDUCTOR FORMULA FOR GL(n)×GL(1) AND FUNCTORIAL DEPTH PRESERVATION

We prove an explicit formula for the conductor of an irreducible, admissible representation of GL(n, F ) twisted by a character of F× where the field F is local and non-archimedean. The components of our formula are analysed, both explicitly and on average, for application to the analytic study of automorphic forms on GL(n). We also demonstrate that “functoriality preserves depth” for such Rankin–Selberg convolutions and give a precise conjecture in the GL(n)×GL(m) case. 1. THE TWISTED CONDUCTOR PROBLEM Let F denote a non-archimedean local field of characteristic zero and let n ≥ 2. For an irreducible, admissible representation π of GL(n, F ) and a quasi-character χ of F×, form the twist χπ = (χ ◦ det) ⊗ π. Our main result (Theorem 1.6) is an explicit formula for the conductor a(χπ), c.f. the Artin conductor, as defined in §2.1. This formula is given by a(χπ) = a(π) + ∆χ(π)− δχ(π) (1) where ∆χ(π) and δχ(π) are non-negative integers as defined in Theorem 1.6; they denote a dominant and a non-twist-minimal interference term, respectively. We give detailed analysis of these terms in §2.3, answering questions such as “for how many χ is there interference?” Our primary motivation was originally to formulate a tactile formula for a(χπ), sensitive to fluctuations in the analytic behaviour of automorphic forms on GL(n). A second objective soon became to prove an explicit formula for the depth of any irreducible, admissible representation of GL(n, F ). The depth ρ(π) is a rational number defined analogously the integer a(π), when considered from the point of view of newform theory (see §2.1). In fact in Theorem 3.3 we prove a formula explicitly relating the two. However, unlike the conductor, the depth is widely expected to be preserved under Langland’s functoriality conjecture and in particular the transfers of representations therein (see [15–17, 19]). In the case at hand, the Rankin–Selberg transfer from GL(n) × GL(1) to GL(n), we provide an affirmative answer to this question: the depth ρ(π χ) = ρ(χπ) is preserved (see Theorem 3.6). Moreover, our results encourage us to predict an explicit formula asserting depth preservation under the GL(n) × GL(m) to GL(mn) Rankin–Selberg transfer for all m ≥ 1 (see Conjecture 3.7). Date: 14 June 2017. 1 AN EXPLICIT CONDUCTOR FORMULA FOR GL(n)×GL(1) 2 As an example, computing a(χπ) in the limit a(χ) → ∞ is straightforward: from Proposition 1.2 and Equation (5) we deduce a(χπ) = na(χ) (2) whenever a(χ) > a(π). In this case ∆χ(π) = na(χ) − a(π) and δχ(π) = 0. Bushnell–Henniart extend (2) by proving the upper bound a(χπ) ≤ max{a(π), a(χ)}+ (n− 1)a(χ), (3) permitting extra summands in the presence of smaller values of a(χ). This is a special case of [4, Theorem 1], and indeed our own Theorem 1.6. In fact this bound is sharp in that it is attained for some π and χ; as in (2) for example. However, in general such examples become sparse, rendering (3) as rather coarse as one averages over χ for large a(π) → ∞. In such cases, understanding the integers ∆χ(π) and δχ(π) exactly is of crucial importance for numerous problems in analytic number theory. For instance, when investigating the analytic behaviour of automorphic forms and their L-functions, such a formula is the unavoidable consequence of two techniques: taking harmonic GL(1)-averages combined with the use of the functional equation for GL(n)×GL(1)-L-functions. Such character twists arise in the work of Nelson–Pitale–Saha [18], who address the quantum unique ergodicity conjecture for holomorphic cusp forms with “powerful” level (see [18, Remarks 1.9 & 3.16]). The current record for upper and lower bounds for the sup-norm of a Maaßnewform on GL(2) in the level aspect [24–26] depends crucially on the n = 2 case of Theorem 1.6. An advantage of working locally there is that such conductor formulae automatically hold in the number field setting, where the strongest bounds for the sup-norm are proved in a forthcoming work of Edgar Assing. Of a more constructive flavour, in [1], Brunault computed the value of ramification indeces of modular parameterisation maps of various elliptic curves (of conductor N ) over Q. Whenever a newform attached to E is “twist minimal”, Brunault could prove that this index was trivial (equal to 1), holding in particular whenever N is square-free. This problem was recently solved by Saha and the present author [6] in full generality. In our solution, the subtleties behind evaluating conductors of twists explicitly gives rise to the few examples of non-trivial ramification indices. These results all concern the case n = 2, where the conductor formula for twists of supercuspidal representations was given by Tunnell [30, Proposition 3.4] in his thesis; see [6, Lemma 2.7] for the general case. Tunnell himself applied his formula to count isomorphism classes of supercuspidal representations of fixed (odd) conductor (see [30, Theorem 3.9]). He used this observation to prove the local Langlands correspondence for GL(2, F ) in the majority of cases. The present result is suggestive of similar applications: a bound for local Whittaker newforms (and a corresponding global sup-norm bound) in the level aspect; 1The purpose of [4] is to establish a variant of (3) for GL(n) × GL(m)-conductors where, more generally, n,m ≥ 1. AN EXPLICIT CONDUCTOR FORMULA FOR GL(n)×GL(1) 3 bounds for matrix coefficients of local representations, and estimates relating to the Voronoı̆ summation problem for GL(n), to name a few. Here, in §3, we give an application of our formula to the study of depth of irreducible, admissible representations of GL(n, F )×GL(1, F ). In §2 we provide a detailed analysis of the terms ∆χ(π) and δχ(π) in (1). Lastly, in §4, we give a uniform proof of (1) for all quasi-square-integrable representations (see Proposition 1.2); we use such representations as building blocks in the general case, which we now establish in §1.1. 1.1. A minimal classification and the explicit conductor formula. 1.1.1. The Langlands classification for GL(n, F ). Let AF (n) denote the set of (equivalence classes of) the irreducible, admissible representations of GL(n, F ). The natural building blocks that describe AF (n) are the quasi-square-integrable representations; that is, the π ∈ AF (n) for which there exists an α ∈ R such that the matrix coefficients of | · |π are square-integrable on GL(n, F ) modulo its centre. The so-called ‘Langlands classification’ (due to Berstein–Zelevinsky) describes the structure of each representation in the graded ring AF = ⊕n≥1AF (n) in terms of the subset S G F of quasi-square-integrable representations. By [33, Theorems 9.3 & 9.7], one deduces an addition law on S G F , by which S G F generates a free commutative monoid Λ. The classification is then the assertion that there is a bijection between AF and the semi-group of non-identity elements in Λ, thus endowing AF with the addition law . Crucially, the maps (AF , ) → (C, · ) given by applying Lor ε-factors are homomorphisms of semi-groups (see [31, §2.5] for their definitions). Both expositions [20, 31] provide excellent background on this topic. The upshot of this classification being that for any π ∈ AF (n) there exists a unique partition n1 + · · · + nr = n alongside a collection of quasi-squareintegrable representations πi ∈ S G F ∩AF (ni) for 1 ≤ i ≤ r such that π = π1 · · · πr (4) and, for any quasi-character χ of F×, a(χπ) = a(χπ1) + · · ·+ a(χπr). (5) Equation (5) follows from the definition of the ε-factor and formula (11) in §2.1. 1.1.2. Minimality and the formula for quasi-square-integrable representations. Definition 1.1. An irreducible, admissible representation π of GL(n, F ) is called twist minimal if a(π) is the least integer amongst the conductors a(χπ), ranging over all quasi-characters χ of F×. In particular, if a quasi-square-integrable representation π ∈ S G F is not twist minimal then n | a(π). For these representations, the notion of twist-minimality is sufficient to give an exact formula. AN EXPLICIT CONDUCTOR FORMULA FOR GL(n)×GL(1) 4 Proposition 1.2. Let π be an irreducible, admissible, square-integrable representation of GL(n, F ) and let χ be a quasi-character of F×. Then a(χπ) ≤ max{a(π), na(χ)} (6) with equality in (6) whenever π is twist minimal or a(π) 6= na(χ). Remark 1.3. In practice, one handles those π which are not twist minimal as follows: tautologically, write π = μπ where μ is a quasi-character of F× and π is twist minimal. Then Proposition 1.2 implies a(χπ) = max{a(π), na(χμ)}. It is the collusion of the characters χ and μ that give rise to any degeneracies. Let us draw attention to the conductor formula of Bushnell–Henniart–Kutzko [5, Theorem 6.5] for the (more general) GL(n) × GL(m)-pairs of supercuspidal representations. There they deploy the full structure theory of supercuspidal representations to proving an outstanding identity, relating the conductor to the respective inducing data. Proposition 1.2 may indeed be derived from their work. However, in our case with m = 1, the formula is more simple and holds uniformly for the larger set S G F , which contains the supercuspidals (c.f. the known formula for the Steinberg representation [23, p. 18]). Indeed, we are able to give an elementary proof of Proposition 1.2. This promotes our philosophy that, as far as the conductor is concerned, the set S G F (and in particular the subset of twist minimal elements) contains sufficient and necessary information to explicitly determine the conductor via the decomposition given in (4). We defer our proof of Proposition 1.2 until §4.4. The arguments made there are also used determine a result on the central character. Proposition 1.4. Let π be an irreducible, admissible, square-integrable representation of GL(n, F ) with central character π|F× = ωπ. Then a(ωπ) ≤ a(π) n . (7) Remark 1.5. As a rule of thumb, problems arising in the “highly ramified central character aspect” come from interactions between the components in π1 · · · πr for r ≥ 2. We mean this in the sense that otherwise a(ωπ) does not influence a(π). As such it should be handled in a separate fashion, as we do in the present work. 1.1.3. The general formula. We arrive at our main result, having defined a sufficient and necessary set of properties of representations π ∈ AF in order to give a fully explicit formula for the conductor. Theorem 1.6. Let π be an irreducible, admissible representation of GL(n, F ) given in terms of quasi–square–integrable representations πi of GL(ni, F ), as described in (4), where n = n1 + · · · + nr; write π = π1 · · · πr. Let χ be a quasi-character of F×. Then a(χπ) = a(π) + ∆χ(π)− δχ(π) AN EXPLICIT CONDUCTOR FORMULA FOR GL(n)×GL(1) 5 where ∆χ and δχ are defined by the semi-group homomorphisms (AF , ) → (Z≥0,+) given by their values on the representations πi = μiπ i , the twist of a minimal representation π i and a quasi-character μ of F ×, as follows: ∆χ(πi) = { max{nia(χ)− a(πi), 0} if a(χ) 6= a(μi) 0 if a(χ) = a(μi) and δχ(πi) = { a(πi)−max{a(π i ), nia(χμi)} if a(χ) = a(μi) 0 if a(χ) 6= a(μi). Both terms are non-negative for any π and χ. Proof. By applying Proposition 1.2 to Equation (4) we have