Rationality results for the exterior and the symmetric square L-function (with an appendix by Nadir Matringe)

Mathematische Annalen, Sep 2017

Let \(G=\mathrm{GL}_{2n}\) over a totally real number field F and \(n\ge 2\). Let \(\Pi \) be a cuspidal automorphic representation of \(G(\mathbb {A})\), which is cohomological and a functorial lift from SO\((2n+1)\). The latter condition can be equivalently reformulated that the exterior square L-function of \(\Pi \) has a pole at \(s=1\). In this paper, we prove a rationality result for the residue of the exterior square L-function at \(s=1\) and also for the holomorphic value of the symmetric square L-function at \(s=1\) attached to \(\Pi \). As an application of the latter, we also obtain a period-free relation between certain quotients of twisted symmetric square L-functions and a product of Gauß sums of Hecke characters.

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Rationality results for the exterior and the symmetric square L-function (with an appendix by Nadir Matringe)

Rationality results for the exterior and the symmetric square L-function (with an appendix by Nadir Matringe) Harald Grobner 0 0 Fakultät für Mathematik, University of Vienna , Oskar-Morgenstern-Platz 1, 1090 Vienna , Austria Let G = GL2n over a totally real number field F and n ≥ 2. Let be a cuspidal automorphic representation of G(A), which is cohomological and a functorial lift from SO(2n + 1). The latter condition can be equivalently reformulated that the exterior square L -function of has a pole at s = 1. In this paper, we prove a rationality result for the residue of the exterior square L -function at s = 1 and also for the holomorphic value of the symmetric square L -function at s = 1 attached to . As an application of the latter, we also obtain a period-free relation between certain quotients of twisted symmetric square L -functions and a product of Gauß sums of Hecke characters. Mathematics Subject Classification Primary 11F67; Secondary 11F41 · 11F70 · 11F75 · 22E55 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Notation and conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Facts and conventions for cuspidal automorphic representations . . . . . . . . . . . . . . . . . . . . 4 Top-degree periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 An Aut(C)-rational assignment for Whittaker functions . . . . . . . . . . . . . . . . . . . . . . . . 6 An integral representation of the residue of the exterior square L-function . . . . . . . . . . . . . . . - Contents H.G.’s research has been supported by the Austrian Science Fund (FWF), stand-alone research project P 25974 and 2016’s START-prize, grant number Y 966. 7 A rationality result for the exterior square L-function . . . . . . . . . . . . . . . . . . . . . . . . . . 8 A rationality result for the Rankin–Selberg L-function . . . . . . . . . . . . . . . . . . . . . . . . . 9 A rationality result for the symmetric square L-function . . . . . . . . . . . . . . . . . . . . . . . . 10 Applications for quotients of symmetric square L-functions . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction 1.1 General background Let F be an algebraic number field and let be a cuspidal automorphic representation of GL2(AF ). Rationality results for special values of the associate automorphic Lfunction L(s, ) have been studied by several authors over the last decades. For the scope of this paper, we would like to mention Manin and Shimura, who were the first to study special values of L(s, ) in the particular case, when F is totally real, i.e., when comes from a Hilbert modular form, cf. [ 27 ] and [33], and Kurchanov, who treated the case of a CM-field F in a series of papers, cf. [ 24,25 ]. Shortly later, Harder published some articles, see [ 15,16 ], in which he described a general approach to such rationality results. In [ 15 ], Harder considered the case of an arbitrary number field F , while in [ 16 ], he extended the methods of the above authors to some automorphic representations, which do not necessarily come from cusp forms (for F imaginary quadratic). The case of GL2 over a general number field F has also been considered independently by Hida, cf. [ 18 ] and later on also by Shimura, see [34]. It took some time until extensions of these results to general linear groups GLn of higher rank n were available. Important achievements include Ash–Ginzburg, [1], Kazhdan–Mazur–Schmidt [ 22 ], and Mahnkopf [ 26 ]. Guided by the above methods, meanwhile, there is a growing number of results that have been proved about the rationality of special values of certain automorphic Lfunctions attached to GLn. As a selection of examples, relevant to the present paper, we refer to Raghuram [ 28,29 ], Harder–Raghuram [17], Grobner–Harris [ 11 ]; Grobner– Raghuram [ 14 ], Grobner–Harris–Lapid [ 12 ] and Balasubramanyam–Raghuram [2]. In all of these references, the corresponding rationality result is obtained by writing the special L-value at hand as an algebraic multiple of a certain period invariant.1 This period is defined by comparison of a rational structure on a cohomology space, attached to the given automorphic representation , with a rational structure on a model-space of (the finite part of) , such as a Whittaker model or a Shalika model. (The word “rational structure” here refers to a subspace of the vector space, carrying the action of , which is essentially defined over the field of rationality of and at the same time stable under the group action.) While the first rational structure on the cohomology space is purely of geometric nature and has its origin in the cohomology of arithmetic groups (or better: the cohomology of arithmetically defined locally symmetric spaces), the latter rational structure is defined by reference to the uniqueness of the given modelspace. 1 The approach taken in [ 12 ], however, is a certain, basis-free variation of the latter. In this paper, we continue the above considerations. But while most of the aforementioned papers deal with special values of the Rankin–Selberg L-function (by some variation or the other), the principal L-function, or the Asai L-functions, here we would like to study the algebraicity of the exterior square L-function and the symmetric square L-function, attached to a cuspidal automorphic representation of the general linear group. 1.2 The main results of this paper To put ourselves in medias res, let F be a totally real number field and let G = GLN /F , N = 2n with n ≥ 2. The restrictions on F and the index of the general linear groups under consideration are owed to the inevitable, as it will become clear below. Indeed, let be a cuspidal automorphic representation of G(A) and let Wψ ( ) be its ψ Whittaker model. As we want to exploit the results of Bump–Friedberg [ 6 ], we shall assume that the partial exterior square L-function L S(s, , 2) of has a pole at s = 1. (Here, S is a finite set of places of F , containing all archimedean ones, such that for a place v ∈/ S, the local components v and ψv are unramified.) In particular, this forces N = 2n to be even, see [20, Theorem 2], and furthermore to be self-dual, ∼= ∨, and to have trivial central character. Our first main result gives a rationality statement for the residue Ress=1(L S(s, , 2)) of the exterior square L-function. More precisely, we obtain the following result: Theorem 1.1 Let F be a totally real number field and G = GL2n/F , n ≥ 2. Let be a unitary cuspidal automorphic representation of G(A), which is cohomological with respect to an irreducible, self-contragredient, algebraic, finite-dimensional representation Eμ of G∞. Assume that satisfies the equivalent conditions of Proposition 3.5, i.e., the partial exterior square L-function L S(s, , 2) has a pole at s = 1. Then, for every σ ∈ Aut(C), there is a non-trivial period pt (σ ), defined by a comparison of a given rational structure on the Whittaker model of σ f and a rational structure on a realization of σ f in cohomology in top degree t , and a non-trivial archimedean period pt (σ ∞), such that σ where “∼Q( f )” means up to multiplication of the right hand side by an element in the number field Q( f ). This is proved in details in Sect. 7.4, see Theorem 7.4. For a precise definition of the periods pt (σ ) and pt (σ ∞), as well as for a complete list of choices which enter their involved definitions, we refer to Proposition 4.3 and Remark 4.4, respectively (7.2) and Remark 7.3. The non-vanishing of the archimedean period pt (σ ∞) is shown—building on a result of Sun—in our Theorem 7.1. The number field Q( f ) in the theorem is (by Strong Multiplicity One) the aforementioned field of rationality of the cuspidal automorphic representation . See Sects. 2.5 and 3.3. The key result, which we use, in order to derive the above theorem, is a certain integral-representation, obtained by Bump–Friedberg [ 6 ], of the residue Ress=1(L S(s, , 2)) of the exterior square L-function in terms of integrating over a cycle Z (A)H (F )\H (A). Here, Z is the centre of G and H = GLn × GLn, suitably embedded into G, cf. 2.2. More precisely, if one combines the three main results of [ 6 ], then, under the assumptions made in the theorem, one obtains the following equality, shown in our Theorem 6.1: Here, is a certain global Schwartz–Bruhat function on An, chosen with care in Sect. 6.1, and cn · ˆ (0) is the (non-zero) residue at s = 1 of an Eisenstein series attached to a section fs = ⊗v fv,s , which is defined by . See Sects. 6.1 and 6.4 for the precise definitions of the terms appearing in (1.2). What one should observe is that the value of the partial L-function L S(n, 1) of the trivial character of A at n appears in the formula. In order for the pole of L S(s, , 2) at s = 1 not to cancel with the pole of L S(n, 1) at n = 1, we assumed n ≥ 2, which explains the corresponding assumption in Theorem 1.1 (resp. Theorem 7.4). (As for the case of n = 1, L S(s, , 2) = L S(s, 1), the analogue of Theorem 1.1 would boil down to a rationality result for the central critical value of the L-function of unitary cusp forms of GL2(A), which is known, e.g., by Harder [15]. Therefore, considering only n ≥ 2 is not a serious restriction.) Observe that the top degree t , mentioned in Theorem 1.1, where has non-trivial cohomology, equals the dimension of the locally symmetric spaces, which are associated to the cycle Z (A)H (F )\H (A), cf. Sect. 5.2. (Here, we necessarily have to use that F is totally real, which explains the last obstruction, set in the beginning.) As a consequence, we may use the de Rham isomorphism. Together with (1.2) and Matringe’s equivariance-result (Theorem A) in the Appendix, this finally gives Theorem 1.1. We point out that, if satisfies the assumptions made in the theorem, then automatically satisfies the assumptions made in Grobner–Raghuram [ 14 ]. Hence, the non-zero periods ω 0 ( f ) and ω( ∞) constructed in loc. cit. are well-defined. See our Sect. 7.5 below for details. If we define the non-zero, top-degree Whittaker–Shalika periods, then we may get rid of the L-factor L( 21 , f ) in Theorem 1.1, as long as it does not vanish. The following result is Corollary 7.6. Corollary 1.3 Let be as in the statement of Theorem 1.1 (resp. Theorem 7.4). If L( 21 , f ) is non-zero, then where “∼Q( f )” means up to multiplication of the right hand side by an element in the number field Q( f ). In order to obtain our second main theorem on the symmetric square L-function, we need a version of one of the main results of Grobner–Harris–Lapid [ 12 ] and Balasubramanyam–Raghuram [2], which is tailored to our present situation at hand. This is achieved in Sect. 8, applying [2] to our particular case. The aforementioned result reads as follows: Theorem 1.3 Let be a self-dual, unitary, cuspidal automorphic representation of G(A) (with trivial central character), which is cohomological with respect to an irreducible, self-contragredient, algebraic, finite-dimensional representation Eμ of G∞. Then, for every σ ∈ Aut(C), . ∞) where “∼Q( f )× ” means up to multiplication by a non-trivial element in the number field Q( f ). In the statement of the latter theorem, pt ( ) is the top-degree period defined above, while pb( ) is defined analogously, but using the lowest degree b, where has nontrivial cohomology. The non-vanishing archimedean period p( ∞) is defined in (7.1). We refer to Sects. 8.2 and 8.3 for precise assertions and definitions concerning these periods, in particular Remarks 8.1 and 8.4. The second main theorem of this paper finally deals with the value of the symmetric square L-function at s = 1. Recall that we have L S(s, × ) = L S(s, , Sym2) · L S(s, , 2). As by assumption L S(s, , 2) carries the (simple) pole of L S(s, × ) at s = 1, the symmetric square L-function L S(s, , Sym2) is holomorphic and non-vanishing at s = 1. Our second main theorem hence follows by combining Theorem 1.1 (resp. Theorem 7.4) with Theorem 1.3 (resp. Theorem 8.5). We obtain, see Theorem 9.2, Theorem 1.4 Let be a unitary cuspidal automorphic representation of G(A), as in the statement of Theorem 1.1. Then, for every σ ∈ Aut(C), σ L S(1, , Sym2) ∼Q( f ) L( 21 , f ) pb( ) pb( ∞) where “∼Q( f )” means up to multiplication of L S(1, , Sym2) by an element in the number field Q( f ). Similar to before, we may define bottom-degree Whittaker–Shalika periods. Set Pb( ) := pb( ) · ω 0 ( f ) and Pb( Then, we have the following corollary, see Corollary 9.4, in which we may get once more rid of the L-factor L( 21 , f ), if it is non-zero. Corollary 1.4 Let be as in the statement of Theorem 1.4 (resp. Theorem 9.2). If L( 21 , f ) is non-zero, then L S(1, , Sym2) ∼Q( f )× Pb( ) Pb( ∞), where “∼Q( f )× ” means up to multiplication of L S(1, , Sym2) by a non-zero element in the number field Q( f ). On a final note, we may also derive a theorem for quotients of symmetric square Lfunctions, which is independent of all periods appearing in this paper. We hope that this application of Theorem 1.4—our third main result—serves as an interesting example of the strength of the relation provided by Theorem 1.4 between the symmetric square L-function and our a priori only abstract Whittaker period pb( ). More precisely, we obtain (cf. Theorem 10.1) Theorem 1.5 Let be a cuspidal automorphic representation of G(A) and let χ1 and χ2 be two Hecke characters of finite order, such that ⊗ χi , i = 1, 2, both satisfy the conditions of Corollary 1.4. If χ1 and χ2 have moreover the same infinity-type, i.e., χ1,∞ = χ2,∞, then, where “∼Q( f ,χ1, f ,χ2, f )× ” means up to multiplication by a non-zero element in the composition of number fields Q( f ), Q(χ1, f ) and Q(χ2, f ). It shall be noted that, whereas the quantities on the left hand side of the above equation all depend crucially on , the right hand side is not only independent of the all periods considered in this paper, but completely independent of . 2 Notation and conventions 2.1 Number fields In this paper, F denotes a totally real number field of degree d = [F : Q] with ring of integers O. For a place v, let Fv be the topological completion of F at v. Let S∞ be the set of archimedean places of F . If v ∈/ S , we let Ov be the local ring of integers of Fv ∞ with unique maximal ideal ℘v. Moreover, A denotes the ring of adèles of F and A f its finite part. We use the local and global normalized absolute values, the first being denoted by | · |v, the latter by · . The fact that F has no complex place is crucial, see Sect. 5.2. Once and for all, we fix a non-trivial additive character ψ : F \A → C× as in [14, §2.7]. 2.2 Algebraic groups and real Lie groups Throughout this paper G denotes GL2n/F , n ≥ 2, the general linear group over F . Although much of the paper works also for GLN with N arbitrary (e.g., the Diagram 5.2), it will be crucial for the main result that N = 2n is even (because only then, the exterior square L-function may have a pole, [20, Theorem 2, p. 224]) and that n ≥ 2 (because the ζ -function attached to F has a pole at n, if n = 1). Let H be GLn × GLn over F . We identify H with a subgroup of G, defined as the image of the homomorphism J : GLn × GLn → GL2n, where ⎧ gi, j J (g, g )k,l := ⎨ gi, j The center of G/F is denoted Z /F . We write G∞ := RF/Q(G)(R) (resp., H∞ := RF/Q(H )(R) or Z∞ := RF/Q(Z )(R)), where RF/Q stands for Weil’s restriction of scalars. Lie algebras of real Lie groups are denoted by the same letter, but in lower case gothics. At an archimedean place v ∈ S∞ we let Kv be a maximal compact subgroup of the real Lie group G(Fv) = GL2n(R). It is isomorphic to O(2n). We write K ◦ v for the connected component of the identity of Kv, which is isomorphic to S O(2n). We set K∞ := Kv and K ∞◦ := v∈S∞ Kv◦. Moreover, we denote by K H,v the intersection Kv ∩ Hv∈(SF∞v), which is a maximal compact subgroup of H (Fv), isomorphic to O(n)× O(n). As before, we write K H◦,v for the connected component of the identity and we let K H,∞ := v∈S∞ K H,v and K H◦,∞ := v∈S∞ K H◦,v. Let AG be the multiplicative group of positive real numbers R+, being diagonally embedded into the center Z∞ of G∞. It is a direct complement of the group G(A)(1) := {g ∈ G(A)| det(g) = 1} in G(A). According to our conventions, the Lie algebra of the real Lie group AG is denoted aG . Furthermore, we let mG := g∞/aG , mH := h∞/aG and s := z∞/aG . Observe that these spaces are Lie subalgebras of g . ∞ 2.3 Coefficient modules In this paper, Eμ denotes an irreducible, algebraic representation of G∞ on a finite-dimensional complex vector space. It is determined by its highest weight μ = (μv)v∈S∞ , whose local components at an archimedean place v may be identified with μv = (μ1,v, . . . , μ2n,v) ∈ Z2n, μ1,v ≥ μ2,v ≥ · · · ≥ μ2n,v. We assume that Eμ is self-dual, i.e., it is isomorphic to its contragredient, Eμ =∼ Eμ∨, or, in other words, that μ j,v + μ2n− j+1,v = 0, 1 ≤ j ≤ n at all places v ∈ S . Clearly, this condition implies that μn,v ≥ 0 ≥ μn+1,v for all v ∈ ∞ S . The self-duality hypothesis, hence incorporates that dimC Hom H(C)(Eμv , C) = 1 ∞ for all v ∈ S∞. (See [14, Proposition 6.3.1].) 2.4 Cohomology of locally symmetric spaces Define the orbifolds SG := G(F )\G(A)/ AG K ∞◦ = G(F )\G(A)1/K ◦ ∞ and S˜H := H (F )\H (A)/ AG K H◦,∞. A representation Eμ as in Sect. 2.3 defines a locally constant sheaf Eμ on SG , whose espace étalé is G(A)1/K ∞◦ ×G(F) Eμ (with the discrete topology on Eμ). Along the sphroeapferonmSa˜Hp ,Jwh:icSh˜Hwe→willSaGg,aiwnhdiecnhoitse binydEuμce.dLebtyHcJq,(SSGec,tE. μ2).2(r,ewspe. Halcsqo(S˜oHbt,aEinμ)a) be the corresponding space of sheaf cohomology with compact support. This is an admissible G(A f )-module (resp. H (A f )-module), cf. [31, Corollary 2.13]. Observe that the proper map J from above gives rise to a non-trivial H (A f )-equivariant map q q q Jμ : Hc (SG , Eμ) → Hc (S˜H , Eμ). 2.5 Complex automorphisms and rational structures For σ ∈ Aut(C), let us define the σ -twist σν of an (abstract) representation ν of G(A f ) (resp., G(Fv), v ∈/ S∞) on a complex vector space W , following Waldspurger [36], I.1: If W is a C-vector space with a σ -linear isomorphism φ : W → W then we set σν := φ ◦ ν ◦ φ−1. This definition is independent of φ and W up to equivalence of representations, whence we may always take W := W ⊗σ C, i.e., the abelian group W endowed with the scalar multiplication λ ·σ w := σ −1(λ)w. Furthermore, if ν∞ = v∈S∞ νv is an (abstract) representation of the real Lie group G∞, we let σν∞ := νσ −1v, v∈S∞ σν := σν∞ ⊗ σν f . interpreting v ∈ S∞ as an embedding of fields v : F → R. Combining these two definitions, we may define the σ -twist on a global representation ν = ν∞ ⊗ ν f of G(A) as We recall also the definition of the rationality field of a representation from [36], I.1. If ν is any of the representations considered above, then let S(ν) be the group of all automorphisms σ ∈ Aut(C) such that σν ∼= ν. Then the rationality field Q(ν) of ν is defined as the fixed-field of S(ν) within C, i.e., Q(ν) := {z ∈ C|σ (z) = z for all σ ∈ S(ν)}. We say that a representation ν on a C-vector space W is defined over a subfield F ⊂ C, if there is a F-vector subspace WF ⊂ W , stable under the given action, such that the canonical map WF ⊗F C → W is an isomorphism. The following lemma is due to Clozel, [7, p. 122 and p. 128]. (See also [13, Lemma 7.1].) Lemma 2.6 Let Eμ be an irreducible, algebraic representation as in Sect. 2.3. As a representation of the diagonally embedded group G(F ) → G∞, σEμ is isomorphic to the abstract representation Eμ ⊗σ C. Moreover, as a representation of G(F ), Eμ is defined over Q(Eμ). We fix once and for all a Q(Eμ)-structure on Eμ as a representation of G(F ). Clearly, this also fixes a Q(Eμ)-structure on Eμ as a representation of H (F ). As a consequence, the G(A f )-module Hcq (SG , Eμ) and the H (A f )-module Hcq (S˜H , Eμ) carry a fixed, natural Q(Eμ)-structure, cf. [7, p. 123]. Moreover, this also pins down natural σ -linear, equivariant isomorphisms Hμσ,q : Hcq (SG , Eμ)−∼→Hcq (SG , σEμ) and H˜ μσ,q : Hcq (S˜H , Eμ)−∼→Hcq (S˜H , σEμ) for all σ ∈ Aut(C), cf. [7, p. 128]. The following lemma is obvious. (2.1) Lemma 2.7 For all σ ∈ Aut(C) the following diagram commutes, q Hc (SG , Eμ) σ,q Hμ Hcq (SG , σEμ) q Jμ q Jσμ q Hc (S˜H , Eμ) H˜ σμ,q Hcq (S˜H , σEμ) 3 Facts and conventions for cuspidal automorphic representations 3.1 Cohomological cusp forms In this paper, we let be an irreducible unitary cuspidal automorphic representation of G(A) with trivial central character. Furthermore, we assume that is self-dual, i.e., ∼= ∨. This is no loss of generality, as the main result will only hold for such cuspidal representations. (Compare this to Proposition 3.5 below.) Recall that has a (unique) Whittaker model (with respect to ψ ). We write W ψ : → W ψ ( ) for the realization of in its Whittaker model Wψ ( ) by the ψ -Fourier coefficient. Recall that there is a canonical decomposition Wψ ( ) = ⊗vWψv ( v), in the sense that each space Wψv ( v) is canonically determined by the uniqueness of local Whittaker models. We will furthermore assume that is cohomological: By this we understand that there is an irreducible, algebraic representation Eμ of G∞, as in Sect. 2.3, such that the archimedean component ∞ of has non-vanishing (mG , K ◦ )-cohomology with respect to Eμ, i.e., ∞ H q (mG , K ◦ , ∞ ∞ ⊗ Eμ) = 0, for some degree q. Lemma 3.2 Let ρ∞ be an irreducible unitary (g∞, K ◦ )-module with trivial AG action. Then the following assertions are equivalent: ∞ 1. H ∗(mG , K ◦ , ρ∞ ⊗ Eμ) = 0, ∞ 2. H ∗(g∞, K ∞◦, ρ∞ ⊗ Eμ) = 0, 3. H ∗(g∞, (Z∞ K∞)◦, ρ∞ ⊗ Eμ) = 0. Proof This follows combining the following well-known results on relative Lie algebra cohomology :[ 5 ], I. 1.3 (the Künneth rule), I. 5.1, I. Theorem 5.3 (Wigner’s lemma) and II. Proposition 3.1 (all cochains are closed and harmonic). As a consequence, the archimedean component ∞ of a cuspidal automorphic representation , as above, is cohomological in our sense, if and only if ∞ has nonvanishing (g∞, K ∞◦)-cohomology or equivalently, non-vanishing (g∞, (Z∞ K∞)◦)cohomology with respect to the same algebraic, self-dual coefficient module Eμ (although the degrees and dimensions of non-trivial cohomology spaces may change). H qT(hmeG c,oKm∞◦p,one∞nt ⊗grEouμp) iπn0e(Gac∞h)de∼=greeK. ∞Fo/rKa∞◦chaacratsctoern th∈e πc0o(hGom∞o)l∗o,gwyhgicrhouwpes identify with = ( 1, . . . , d ) ∈ (Z/2Z)d ∼= π0(G∞)∗, one obtains a corresponding π0(G∞)-isotypic component H q (mG , K ◦ , ∞ Eμ)[ ]. Put t := dn(n + 1) − 1. ∞ ⊗ (3.1) Then, dimC H t (mG , K ◦ , ∞ for all ∈ π0(G∞)∗. This is a direct consequence of the formula in Clozel [7, Lemma 3.14] (see also [26, 3.1.2] or [13, 5.5]), the Künneth rule ([5, I. 1.3]) and the fact that s is a d − 1-dimensional abelian real Lie algebra, whence H q (s, C) ∼= q Cd−1. Observe furthermore, that (for all degrees q and characters ∈ π0(G∞)∗) there is a natural G(A f )-equivariant inclusion q : H q (mG , K ◦ , ∞ q ⊗ Eμ)[ ] → Hc (SG , Eμ). (3.2) This is well-known and follows from [4, §5]. 3.3 Rational structures We have the following result: Proposition 3.3 Let be a cuspidal automorphic representation of G(A), which is cohomological with respect to Eμ. Then, the σ -twisted representation σ is also cuspidal automorphic and it is cohomological with respect to σEμ. For every ∈ π0(G∞)∗, the irreducible unitary G(A f )-module H t (mG , K ∞◦, ⊗ Eμ)[ ] is defined over the rationality field Q( f ). This field is a number field, containing Q(Eμ). Proof This is essentially due to Clozel [ 7 ]. In order to derive the above result from [ 7 ], observe that ∞ is “regular algebraic” in Clozel’s sense, if and only if it is cohomological in our sense: This follows using Lemma 3.2 and [13, Theorem 6.3]. Hence, σ f is the non-archimedean part of a cuspidal automorphic representation, which is cohomological with respect to σEμ by [7, Theorem 3.13]. By uniqueness, see e.g. [13, 5.5], the archimedean part of this cuspidal automorphic representation Eμ)[ ] is defined ove∞r Qas( deffi)n(eSdeeabaolsvoe.[1B3y, C[7o,rPorlloaproys8it.i7o]n.).3F.1i]n,aHllyt,(imtiGs,aKn ◦im,plicit is isomorphic to σ ∞ ⊗ consequence of [7, Theorem 3.13] and its proof that Q( f ) is a number field containing Q(Eμ). For a detailed exposition of the latter assertion, we refer to [13, Theorem 8. 1] and the proof of [13, Corollary 8.7]. Definition 3.4 The Q( f )-structure on H t (mG , K ∞◦, ⊗ Eμ)[ ] is unique up to homotheties, i.e., up to multiplication by non-zero complex numbers, cf. [7, Proposition 3.1]. As Q(Eμ) ⊆ Q( f ), we may fix the Q( f )-structure on H t (mG , K ∞◦, ⊗ Eμ)[ ] which is induced by t , cf. (3.2), and our choice of a Q(Eμ)-structure on Hct (SG , Eμ), cf Sect. 2.5. 3.4 Lifts from SO(2n + 1) We resume the assumptions made on from Sect. 3.1. As a last part of notation for , let us introduce S = S( , ψ ), which is a (sufficiently large) finite set of places of F , containing S∞ and such that outside S, both and ψ are unramified. Proposition 3.5 Let be a cuspidal automorphic representation of G(A) as in Sect. 3.1 above. Then the following assertions are equivalent: 1. The partial exterior square L-function, L S(s, , 2), has a pole at s = 1, 2. is the lift of an irreducible unitary generic cuspidal automorphic representation of the split special orthogonal group SO(2n + 1) in the sense of [8, §1]. Proof With our assumptions on this is [8, Theorem 7.1]. This result is recalled for convenience, as it provides an alternative description of what it means that the exterior square L-function of has a pole at s = 1. We will have to make this assumption later, in order to obtain our main theorems. See, Theorems 6.1, 7.4, 9.2, and 10.1. It is not referred to until Sect. 6.4. In any case, the above result is accompanied by Proposition 3.6 Let be a cuspidal automorphic representation of G(A) as in Sect. 3.1. Assume that satisfies the equivalent conditions of Proposition 3.5, i.e., the partial exterior square L-function, L S(s, , 2), has a pole at s = 1. Then, for all σ ∈ Aut(C), σ is a cuspidal automorphic representation of G(A) as in Sect. 3.1, which satisfies the equivalent conditions of Proposition 3.5, i.e., the partial exterior square L-function, L S(s, σ , 2), has a pole at s = 1. Proof The first assertion has already been proved in Proposition 3.3. For the second assertion, observe that the set S does not change under the action of Aut(C) and then combine [14, Theorem 3.6.2] and [20, Theorem 1, p. 213]. 4 Top-degree periods 4.1 The map W σ Recall the unique Whittaker model Wψ f ( f ) = ⊗v∈/S∞ Wψv ( v) of f , its decomposition being canonical. Given a Whittaker function ξv ∈ Wψv ( v) on G(Fv), v ∈/ S∞, and σ ∈ Aut(C), we may define a Whittaker function σ ξv ∈ Wψv (σ v) by σ ξv(gv) := σ (ξv(tσ,v · gv)), (4.1) where tσ,v is the (uniquely determined) diagonal matrix in G(Ov), having 1 as its last entry, which conjugates ψv to σ ◦ ψv. (Observe that tσ,v does not depend on ψv). See [26, 3.3] and [30, 3.2]. This provides us a σ -linear intertwining operator W σ : Wψv ( v) → Wψv (σ ξv → σ ξv, v) for all σ ∈ Aut(C). In particular, we get a Q( v) structure on Wψv ( v) by taking the subspace of Aut(C/Q( v))-invariant vectors. By the same procedure, we obtain a canonical Q( f ) structure on Wψ f ( f ). (Cf. [15, p. 80], [26, 3.3] or [30, Lemma 3.2].) 4.2 The map Ft Let 0 := ((−1)n−1, . . . , (−1)n−1) ∈ π0(G∞)∗. This choice of a character of the component group is forced upon us by the proof of Theorem 7.1 and so we restrict our attention from now on to it. Recall the Q( f )-rational structure on H t (mG , K ∞◦, ⊗ Eμ)[ 0], chosen in Definition 3.4 above and recall the canonical Q( f )-rational structure on the Whittaker model Wψ f ( f ) of f just fixed in Sect. 4.1 above. As it has been mentioned briefly in the introduction, our top-degree Whittaker period—in abbreviated symbol pt ( )— will be determined by the comparison of these two rational structures. Hence, in order to actually compare them, we have to specify a concrete comparison isomorphism F t : Wψ f ( f )−∼→H t (mG , K ∞◦, It is the purpose of this section to explain this choice carefully, as it is all crucial for the definition of our periods. Our choices will be guided by the ideas in [15, p. 79], [26, 3.3 & 5.1.4], [29, 3.2.5], and [14, 4.1]. The first data we will fix once and for all consists of Choice 4.2 1. A basis {X j } of mG /k∞, which fixes the dual-basis {X ∗j } of (mG /k∞)∗; given a multi-index i = (i1, . . . , it ), we abbreviate Xi∗ := Xi∗1 ∧ · · · ∧ Xi∗t . 2. Vectors eα := ⊗v∈S∞ eα,v ∈ Eμ = ⊗v∈S∞ Eμv , such that {eα,v}α defines a basis of Eμv for all v ∈ S . ∞ 3. For each v ∈ S , mulit-index i = (i1, . . . , it ) and α as above a Whittaker function ∞ ξv,0i,α ∈ Wψv ( v), such that (putting ξ∞0,i,α := ⊗v∈S∞ ξv,0i,α ∈ Wψ∞ ( ∞)) the vector [Wψ∞ ( t ∞)] := dim Eμ is a generator of the one-dimensional space C-vector space H t (mG , K ◦ , Wψ∞ ∞ ( ∞) ⊗ Eμ)[ 0]. (We may and will also assume that {X j } is the extension of a given ordered basis {Y j } of mH /kH,∞ along our embedding J : H → G. This assumption, however, will only be important later on. See, e.g., Sects. 5.2 and 7.1.) By [5, II. Proposition 3.1] and the uniqueness of the archimedean Whittaker model and its canonical decomposition into local factors Wψ∞ ( ∞) = ⊗v∈S∞ Wψv ( v) this generator [Wψ∞ ( ∞)]t is well-defined. For the sake of readability we suppress its various dependencies, listed in Choice 4.2 above, in its notation. Next recall (e.g. from Sect. 2.5) that σ ∈ Aut(C) acts on objects at infinity, which are parameterized by S , by permuting the archimedean places. Having given a gEeμn)e[ra0t]orhe[Wnceψ∞pr(ov∞id)e]st∞uosf wthiethonaen-datiumreanlscihooniaclespoafcae gHe nt(emraGto,rK[W∞◦,ψW∞ (ψσ∞ (∞∞)]t) o⊗f H t (mG , K ∞◦, Wψ∞ (σ ∞) ⊗ σEμ)[ 0]: Wψ∞ (σ t ∞) := where σ ξ∞0,i,α = ⊗v∈S∞ ξσ0−1v,i,α (observe that 0 does not change, when its local components are permuted) and σeα = ⊗v∈S∞ eα,σ −1v. Finally, this entails the description of the desired “comparison isomorphism” mentioned at the beginning of this subsection, i.e., of a fixed choice of isomorphism of G(A f )-modules defined by F t : Wψ f ( f )−∼→H t (mG , K ∞◦, dim Eμ ξ f → F t (ξ f ) := where ϕi,α := (W ψ )−1(ξ∞0,i,α ⊗ ξ f ) ∈ . It is important to observe that we did not have to decompose the global map W ψ computing the ψ -Fourier coefficient, hence there are no hidden ambiguities in this definition: A complete set of dependencies of our comparison isomorphism F t is hence given by Choice 4.2. In light of Proposition 3.3 and our discussion above, we obtain isomorphisms Fσ for all σ ∈ Aut(C) with t the same precise set of dependencies. 4.3 The map Hμσ,t As a last ingredient in this section, we define a σ -linear, G(A f )-equivariant isomor phism H σ,t μ : H t (mG , K ◦ , ∞ ⊗ Eμ)[ 0]−∼→H t (mG , K ◦ , σ ∞ ⊗ σEμ)[ 0]. To that end, recall the embedding q from (3.2) and the σ -linear isomorphism Hμσ,q from (2.1). Observe that Im(Hμσ,t ◦ t ) =Im( tσ ). Indeed, by multiplicity one and strong multiplicity one for the discrete automorphic spectrum of G(A), the σ f -isotypic component of the G(A f )-module Hct (SG , σEμ) is precisely the image oHfct (HStG(m,EGμ,)Kc∞o◦m, mσute⊗,thσiEsμs)h.oAwss tthhaet nImat(uHraμσl,tac◦tiont )of=πIm0((G ∞tσ))aansdcolafimGe(dA. Sf)inocne t σ is injective, the map μ := ( tσ )−1 ◦ Hμ ◦ H σ,t σ,t t is hence a well-defined σ -linear, G(A f )-equivariant isomorphism mapping H t (mG , K ∞◦, ⊗ Eμ)[ 0] onto H t (mG , K ∞◦, σ ⊗μσEμ)[ 0]σ,tastodetshieresdu.b(mShoodrutllye speaking, this amounts to say that the restriction H σ,t of Hμ H t (mG , K ∞◦, ⊗ Eμ)[ 0] of Hct (SG , Eμ) has image H t (mG , K ∞◦, σ ⊗ σEμ)[ 0].) 4.4 Top-degree Whittaker periods Recall the maps W σ (Sect. 4.1), F t (Sect. 4.2) and H σ,t (Sect. 4.3). There is the μ following result: Proposition 4.3 For every σ ∈ Aut(C), there is a non-zero complex number pt (σ ) (a “period”), uniquely determined up to multiplication by elements in Q(σ f )×, such that the normalized maps Fσt := pt (σ )−1 Fσt make the following diagram commutative: Wψ f ( f ) W σ Wψ f (σ f ) Ft t Fσ H t (mG , K ◦ , ∞ Proof This is essentially due to the uniqueness of essential vectors for v, v ∈/ S∞: Otherwise put, the proof of Proposition/Definition 3.3 in Raghuram–Shahidi [ 30 ] goes through word for word in our (slightly different) situation at hand. Remark 4.4 A lot of choices have been made in order to give the definition of our topdegree Whittaker periods, while (almost) none of them is reflected explicitly in our choice of notation “ pt ( )”. So, for the sake of precision, we would like to summarize comprehensively at one place on which data, i.e., fixed chosen ingredients, pt ( ) actually depends: 1. , ψ and the cohomological degree t . 2. The fixed concrete choices of a Q( f )-rational structure on the canonical Whit t3a.k4e).r model Wψ f ( f ) (Sect. 4.1) and on H t (mG , K ∞◦, ⊗ Eμ)[ 0] (Definition 3. The concrete choice of a comparison isomorphism F t (Sect. 4.2), which depends itself precisely on the data fixed in Choice 4.2. 4. The σ -linear intertwining operator W σ : Wψv ( v) → Wψv (σ biguously in Sect. 4.1 and 5. The σ -linear intertwining operator H σ,t defined unambiguously in Sect. 4.3. μ v) defined unam The (Whittaker) periods pt ( ) defined by Proposition 4.3 are the analogues of the (Shalika) periods ω ( f ) defined in Grobner–Raghuram DefinitionProposition 4.2.1. The idea behind the construction of pt ( ) (as of ω ( f )), however, goes back to [ 15,26,30 ]. 5 An Aut(C)-rational assignment for Whittaker functions 5.1 The map Tμ Let Eμ = ⊗v∈S∞ Eμv be an irreducible, algebraic representation as in Sect. 2.3. We have dimC Hom H(C)(Eμv , C) = 1 for all v ∈ S . Let us fix Tμv ∈ Hom H(C)(Eμv , C) ∞ and set Tμ := ⊗v∈S∞ Tμv ∈ Hom RF/Q(H)(C)(Eμ, C). For σ ∈Aut(C), we obtain Tσμ = ⊗v∈S∞ Tμσ−1v ∈ Hom RF/Q(H)(C) (σEμ, C). The map induced on cohomology, Hct (S˜H , σEμ) → Hct (S˜H , C) will be denoted by the same letter Tσμ. 5.2 The de-Rham-isomorphism R So far, we have not made any choice of a Haar measure on H (A f ). From this section on, we will restrict our possible choices on Q-valued Haar measures on H (A f ). In Sect. 6.3 we will specify our concrete choice of a measure in details. (So far, this is not necessary.) Let dh f be any Q-valued Haar measure of H (A f ). It is important to notice that we have dimR S˜H = dn(n + 1) − 1 = t , cf. Sect. 3.1, because we assumed that F is totally real. Knowing this, a short moment of thought shows that we obtain a surjective map R : Hct (S˜H , C) → C, induced by the de Rham-isomorphism: Indeed, let K f be a compact open subgroup of H (A f ) and set Then it is easy to see that S˜H is homeomorphic to the projective limit S˜HK f := H (F )\H (A)/ AG K H◦,∞ K f . S˜H =∼ lim S˜HK f ←− K f running over the compact open subgroups K f of H (A f ), partially ordered by opposite inclusion, [31, Proposition 1.9]. As dimR S˜HK f = t for all K f , we may use the deRham-isomorphism to define a surjective map Hct (S˜HK f , C) → C. More precisely, each of the (finitely many, cf. [3, Theorem 5.1]) connected components of S˜HK f is homeomorphic to a quotient of H ∞◦/ AG K H◦,∞ by a discrete subgroup of H (F ). Recall the ordered basis {Y j } of mH /kH,∞, from Sect. 4.2. It determines a choice of an orientation on H ◦ / AG K H◦,∞, whence on each connected component of S˜HK f and so ∞ finally also on S˜HK f . Hence, the de-Rham-isomorphism provides us a surjection R K f : Hct (S˜HK f , C) → Hct (S˜HK f , C) → R : Hct (S˜H , C) → C, as mentioned above. (Compare this also to the considerations in [14, 6.4], [11, 3.8], [12, 5.1] and [29, 3.2.3] as well as to the corresponding references therein.) 5.3 In summary: a rational diagram In the following proposition, we abbreviate H t (mG , K ∞◦, ⊗ Eμ)[ 0] by H t ( ⊗ Eμ)[ 0] (with analogous notation for the cohomology of the σ -twisted representa tions). Recollecting what we observed in Sects. 4.1–5.2, we find Proposition 5.1 The following diagram is commutative: Its horizontal arrows are linear, whereas its vertical arrows are σ -linear. Proof The first square from the left is commutative by Proposition 4.3, while the second square is commutative by the definition of H σ,t in Sect. 4.3. Commutativity μ of the third square is the assertion of Lemma 2.7. The fourth square commutes by the very definition of Tσμ in Sect. 5.1, while commutativity of the last square is due to the Q-rationality of the measure on H (A f ), Sect. 5.2. (5.2) 6 An integral representation of the residue of the exterior square L-function In this section, we will recapitulate some results from Jacquet–Shalika [ 21 ] and Bump– Friedberg [ 6 ]. 6.1 Eisenstein series and a result of Jacquet–Shalika We resume the notation and assumptions made in the previous sections. In addition, for any integer m ≥ 2, we will now fix once and for all a Schwartz–Bruhat function = ⊗v v ∈ S (Am ): We assume that v is the characteristic function of Ovm at all v ∈/ S , while at the archimedean places v ∈ S , we assume to have chosen (O(m)∞ ∞ finite) local components v, such the global Schwartz–Bruhat function satisfies ˆ (0) = 0. Here, we wrote ˆ (x ) := (y)ψ (t y · x )d y for the Fourier transform of (at x ) with respect to the self-dual Haar measure d y on Am , i.e., the unique Haar measure on Am which satisfies ˆˆ (x ) = x ∈ Am . Let (−x ) for all s fv,s (gv) := | det(gv)|v v(t · (0, . . . , 0, 1)gv)|t |vms d×t and for e(s) tion) where fs (g) := ⊗v fv,s (gv) = det(g) s (t · (0, . . . , 0, 1)g) t ms d×t A× 0. Then fs ∈ IndGGLLmm −(A1()A)×GL1(A)[δsP ], (unnormalised parabolic inducδP h 0 0 a = det(h) · a −(m−1) is the modulus character of the standard parabolic subgroup P of GLm , with Levi subgroup L = GLm−1 × GL1. Clearly, the analogous assertion holds for the local components fv,s . There is the following result due to Jacquet–Shalika [21, Lemma 4.2]. Lemma 6.2 The Eisenstein series associated with fs , formally defined as E ( fs , )(g) := fs (γ g), γ ∈P(F)\GLm (F) extends to a meromorphic function on e(s) > 0. It has a simple pole at s = 1 with constant residue Ress=1(E ( fs , ))(g) = cm · ˆ (0). Here, cm is a certain non-zero complex number. 6.3 Measures When dealing with rationality results of special values of L-functions, the choice of measures is all-important. In this section, we specify our choices of measures, which will be guided by the explicit choices made in Bump–Friedberg [ 6 ]. Let m ≥ 2 be again any integer and consider the group GLm /F . A measure dg of GLm (A) will be a product dg = v dgv of local Haar measures of GLm (Fv). We write dgvB F for the local Haar measure of GLm (Fv) chosen in Bump–Friedberg [ 6 ]. See loc. cit. (3.2), p. 61. At v ∈/ S∞, these measures assign rational volumes to compact open subgroups of GLm (Fv). Furthermore, the product measure dg Bf F := v∈/S∞ dgvB F gives rational volumes to compact open subgroups of GLm (A f ). At v ∈/ S , we define our choice of a measure to be the one of Bump–Friedberg, ∞ dgv := dgvB F whereas at an archimedean place v ∈ S , we let dgv be the local Haar measures of ∞ GLm (R) such that S O(m) has volume 1. If we let m = 2n as in Sect. 2.2, we hence have chosen a measure dg on G(A) = GL2n(A). Recall the group H = GLn × GLn, Sect. 2.2. We will use the notation (g, g ), to specify an element of H (A) (and use analogous notation locally). A measure of H (A) will be the product of a measures dg and dg as chosen above for m = n of the two isomorphic copies of GLn(A) inside H (A). As Z ⊂ H , also the volume voldg×dg (Z (F )\Z (A)/ AG ) is well-defined and finite. 6.4 A result of Bump–Friedberg Let Un be the group of upper triangular matrices in GLn, having 1 on the diagonal and let Zn be the centre of GLn. Recall the finite set of places S = S( , ψ ) from Sect. 3.4. By assumption, outside S, both and ψ are unramified (and ψ normalised). Let ξ = ⊗vξv ∈ Wψ ( ) ∼= ⊗vWψv ( v) be a Whittaker function, such that for v ∈/ S, ξv is invariant under G(Ov) and normalized such that ξv(i dv) = 1. Recall the section fs = ⊗v fv,s from Sect. 6.1, defined by the choice of a Schwartz–Bruhat function = ⊗v v, where we now let m = n. Following Bump–Friedberg [6, p. 53], we define the integral Z (ξ, fs ) := Un(A)\GLn(A) Zn(A)Un(A)\GLn(A) ξ( J (g, g )) fs (g) dg dg . It factors over all places of F as Z (ξ, fs ) = v Zv(ξv, fv,s ), where Zv(ξv, fv,s ) := Un(Fv)\GLn(Fv) Zn(Fv)Un(Fv)\GLn(Fv) ξv( J (gv, gv)) fv,s (gv) dgv dgv. Recall the value L S(n, 1) of the partial L-function of the trivial character 1 of A× at n. Since we assumed that n ≥ 2, this number is well-defined and non-zero. The following result is crucial for us: Theorem 6.1 Let be a cuspidal automorphic representation of G(A) as in Sect. 3.1. Let ϕ := (W ψ )−1(ξ ) ∈ be the inverse image of a Whittaker function ξ as in Sect. 6.4 above and assume that satisfies the equivalent conditions of Proposition 3.5, i.e., the partial exterior square L-function, L S(s, , 2), has a pole at s = 1. Then, Proof With our preparatory work, this is a direct consequence of our choice of measures in Sect. 6.3 and the three main results of Bump–Friedberg [6, Theorem 1, Theorem 2 and Theorem 3]. Indeed, our Lemma 6.2 together with [6, Theorem 1 and Theorem 2], identify the left hand side with , where L(n, 1) = v L(n, 1v) is the global L-function of the trivial character 1 of A× at n. As by assumption n ≥ 2, L(n, 1) is well-defined and non-zero. Factorizing Z (ξ, fs ) as in Sect. 6.4, and using the description of Zv (ξv, fv,s ), v ∈/ S, in [6, Theorem 3], Zv(ξv, fv,s ) = L( 21 , v) · L(s, v, 2) L(n, 1v) we obtain Ress=1 Z (ξ, fs ) L(n, 1) = since by assumption L S(s, , 2) carries the (simple) pole of the above expression. 6.5 Consequences for the σ -twisted case Let be a cuspidal automorphic representation of G(A) as in Sect. 3.1 and assume that the partial exterior square L-function, L S(s, , 2), has a pole at s = 1. Then by Proposition 3.6, σ satisfies the same conditions. Hence, we see that once satisfies the assumptions made in Theorem 6.1, then automatically also σ satisfies them, i.e., Theorem 6.1 holds for the whole Aut(C)-orbit of . As we are going to use this in the proof of the main results, let us render this more precise. Let ξ = ⊗vξv ∈ Wψ ( ) ∼= ⊗vWψv ( v) be a Whittaker function, such that for v ∈/ S, ξv is invariant under G(Ov) and normalized such that ξv(i dv) = 1. Given σ ∈Aut(C), let σ ξ ∈ Wψ (σ ) be the σ -twisted Whittaker function, cf. Sect. 4.1 (the action of σ on the archimedean part of ξ being by permutations as in Sect. 4.2), and let σϕ := (W ψ )−1(σ ξ ) ∈ σ be the corresponding cuspidal automorphic form. Recall our Schwartz–Bruhat function ∈ S (An) from Sect. 6.1, with m = n now. We define the constant cn( , σ ) := σ voldg×dg (Z (F )\Z (A)/ AG ) cn ˆ (0) cn ˆ (0) · voldg×dg (Z (F )\Z (A)/ AG ) . This is done purely for cosmetic reasons, as it will become clear below (see the proof of Theorem 7.4). By Sect. 6.1, cn( , σ ) is non-zero. Let σ ∈ S (An) be the Schwartz– Bruhat function which is defined as follows: At v ∈/ S∞, (σ )v := v, whereas (σ )∞ := cn( , σ )−1 · ∞. As in Sect. 6.1, we obtain a function σ fs = ⊗vσ fv,s ∈ IndGGLLnn−(A1()A)×GL1(A)[δsP ] and an associated Eisenstein series E (σ fs , σ ). Clearly, E (σ fs , σ ) satisfies the assertions of Lemma 6.2, with being replaced by σ . In summary, with this notation, saying that Theorem 6.1 holds for the whole Aut(C) orbit of , amounts to the equation cnσ ˆ (0) = σϕ( J (g, g )) dg dg Z(A)H(F)\H(A) L S( 1 , σ ) · Ress=1(L S(s, σ , 2)) 2 L S(n, 1)2 v∈S Zv(σ ξv, σ fv,s ) . L(n, 1v) (6.2) 7 A rationality result for the exterior square L-function 7.1 Archimedean considerations The integral representation of the exterior square L-function in Theorem 6.1 allows us to combine the results of Sects. 5 and 6. Before we derive out first main result, we need a non-vanishing theorem, which is an application of Sun’s main result in [35]. Recall the generator [Wψ∞ ( ∞)]t = i=(i1,...,it ) dαi=m1Eμ Xi∗ ⊗ ξ∞0,i,α ⊗ eα of the cohomology space H t (mG , K ∞◦, Wψ∞ ( ∞) ⊗ Eμ)[ 0] from Sect. 4.2. Recall furthermore, that the basis {X ∗j } of mG /k∞ was the extension of a given ordered basis {Y j∗}tj=1 of mH /kH,∞, whence, for each multi-index i there is a well-defined complex number s(i ), such that the restriction of Xi∗ to t (mH /kH,∞)∗ along the injection t (mH /kH,∞)∗ → t (mG /k∞)∗, induced by J , equals s(i ) · Y1 ∧ .. ∧ Yt . As a last ingredient, before we can state the aforementioned non-vanishing theorem, we need the following lemma: Lemma 7.2 For all v ∈ S∞, and Kv◦-finite ξv ∈ Wψv ( v), the integrals Zv(s , ξv, fv,1) := ξv( J (gv, gv)) × fv,1(gv) Un(Fv)\GLn(Fv) Zn(Fv)Un(Fv)\GLn(Fv) det(gv) s −1/2 dgvdgv det(gv) v are a holomorphic multiple (in s ) of the local archimedean L-function L(s , v). Proof This follows combining Theorem 6.1 with [ 9 ], Proposition 2.3 and Proposition 3.1 loc. cit. . It follows that the factor Z∞(ξ∞0,i,α, f∞,1) := v∈S∞ Zv(ξv,0i,α, fv,1) of the product v∈S Zv(ξv, fv,1) is well-defined. Indeed, using [ 23 ], Theorem 2 and Theorem 3 loc. cit., it is easy to see that n k=1 L(s , v) = h(s ) · s + μv,k + n − k + 21 , where h(s ) is holomorphic and non-vanishing for all s ∈ C. Since μv,k ≥ 0 for all 1 ≤ k ≤ n, by the self-duality hypotheses, cf. Sect. 2.3, L(s , v) is holomorphic at s = 21 , whence so is Zv(ξv,0i,α, fv,1) = Zv( 2 , ξv,0i,α, fv,1) by Lemma 7.2. Finally, we 1 let ct ( ∞) := (L S(n, 1)2)−1 · i=(i1,...,it ) α=1 dim Eμ s(i ) · Tμ(eα) · . Here, both numbers L(n, 1∞) = L(n, 1v) = π −dn/2 ( n2 )d and L S(n, 1) are non-zero. We claim that Sun’s aforemv∈eSn∞tioned result now implies the following Theorem 7.1 The number ct ( Proof As a first step and in order to be able to apply Sun’s result ([35], Theorem C), we reduce the problem of showing that ct ( ∞) is non-zero to showing that a similarly defined number, dt ( v) is non-zero. This latter number will only depend on one archimedean place v ∈ S , whence we find ourselves back in the setting of [35]. ∞ To this end, observe that there is a projection Lt : t (mG /k∞)∗−∼→ a (g∞/c∞)∗ ⊗ bs∗ r (g∞/c∞)∗ ⊗ a+b=t where c∞ := z∞ ⊕ k∞ and r = t − d + 1. By reasons of degree, Lt induces an isomorphism of (one-dimensional) vector spaces Lt : H t (mG , K ∞◦, Wψ∞ ( H r (g∞, (Z∞ K∞)◦, Wψ∞ ( ∈ r (g∞/c∞)∗ ⊗ Ld−1(Xi ), where Lr (Xi ) ∈ r (g∞/c∞)∗ and Ld−1(Xi ) ∈ t factor L = Lr ⊗ Ld−1. As z∞ ⊂ h∞, and as moreover r = dimR h∞/cH,∞, where cH,∞ := z∞ ⊕ kH,∞, we also have a canonical isomorphism t (mH /kH,∞)∗−∼→ r (h∞/cH,∞)∗ ⊗ Hence, Lt and Lt factor over the injection t (mH /kH,∞)∗ → by J . As a consequence, ct ( ∞) is a non-trivial multiple of t (mG /k∞)∗ induced dt ( ∞) := i=(i1,...,it ) α=1 dim Eμ u(i ) · Tμ(eα) · , where u(i ) is the uniquely defined complex number, such that the restriction of Lr (Xi∗) tdot ( r (h)∞=/cH,∞v∈)S∗∞edqtu(alsv)u, (wi)h·erLere(aYc1h∧lo.c.a∧l fYatc)t.orTdhet(nuvm) biserdedfi(tned∞a)nfaaloctgoorussalys (using∞ r (h∞/cH,∞)∗ ∼= rv=r v∈S∞ rv (hv/cH,v)∗). Therefore, we may finish the proof by showing that dt ( v) is non-zero for all v ∈ S∞ and we are in the situation considered by Sun [35]. Let v ∈ S∞ be an arbitrary archimedean place. For sake of simplicity, we drop the subscript “v” now everywhere, so, e.g., = v, H = G Ln(R) × G Ln(R), μ = μv, g = gl2n(R) and analogously for all other local archimedean objects. The local integrals Z (ξ, f1) define a non-zero homomorphism Z (., f1) ∈ Hom H (Wψ ( ), C). This follows from [ 6 ], Theorem 2 and Lemma 7.2. Hence, if we let χ := 1 × 1 be the trivial character of H , then Z (., f1) can be taken as the map ϕχ in Sun’s Theorem C [35]. Next, recall Tμ ∈ Hom H(C)(Eμ ⊗ C) from Sect. 5.1. If we set w1 := 0 =: w2, then we may take Tμ to be the non-zero homomorphism ϕw1,w2 from [35, Theorem C]. Hence, loc. cit. , Theorem C, asserts that the map D : Hom( r g/c, Wψ ( ) ⊗ Eμ) −→ Hom( r h/cH , χ ⊗ C) r h → D(h) := (Z (., f1) ⊗ Tμ) ◦ h ◦ ∧ j2n is non-zero on the one-dimensional sub-space H r (g, (Z K )◦, Wψ ( ) ⊗ Eμ)[ 0]. (Here, j2n is Sun’s notation for the embedding h/cH → g/c.) By the onedimensionality of the latter cohomology space, it is hence non-zero on Lr ([Wψ ( )]t ). But, then, D computes D(Lr ([Wψ ( )]t )) = (Z (., f1) ⊗ Tμ) ◦ Lr ([Wψ ( )]t ) ◦ ∧r j2n Hence, reintroducing the subscript “v”, and recalling that L(n, 1v) = π −n/2 (n/2) = 0, the number dt ( v) is non-zero for all archimedean places, whence so is ct ( ∞). 7.3 Definition of the archimedean top-degree period As a consequence of Proposition 3.6, we may hence define the archimedean periods pt (σ ∞) := ct (σ (7.2) for all σ ∈Aut(C). Remark 7.3 Analogously to the case of the top-degree Whittaker period pt ( ), which we dealt with in Remark 4.4, (almost) none of the various choices entering the definition of our archimedean top-degree period pt ( ∞) may be found in its notation. For the sake of precision, we would like to summarize at this place on which data, i.e., fixed chosen ingredients, pt ( ∞) actually depends: 1. ∞, ψ∞, the cohomological degree t , as well as the archimedean measures dgv and dgv chosen for all v ∈ S∞ in Sect. 6.3 2. The fixed generator [Wψ∞ ( ∞)]t (cf. Sect. 4.2) of the one-dimensional cohomology space H t (mG , K ∞◦, Wψ∞ ( ∞) ⊗ Eμ)[ 0]. We remark further that this generator depends itself precisely on the data fixed in Choice 4.2 3. The concrete choice of an intertwining operator Tμv ∈ HomH(C)(Eμv , C) for all v ∈ S∞ (Sect. 5.1) 4. The sections fv,1 ∈ IndGGLLnn−(F1v(F)v)×GL1(Fv)[δP(Fv)] chosen unambiguously in Sect. 6.1 for all v ∈ S . ∞ It is hence clear that the archimedean top-degree period pt ( ∞) depends exclusively on data, which is associated with objects at archimedean places (which explains its name); and that its definition and existence is independent of the definition and proof of existence of our global Whittaker periods pt ( ) from Sect. 4.4. 7.4 Rationality of the residue of the exterior square L-function at s = 1 This is our first main theorem. For the precise definitions of pt ( ) and pt ( ∞), a comprehensive list of their individual dependencies as well as for their mutual independence, we refer to Sects. 4.4, 7.3, Remarks 4.4 and 7.3 Theorem 7.4 Let F be a totally real number field and G = GL2n/F , n ≥ 2. Let be a unitary cuspidal automorphic representation of G(A) (self-dual and with trivial central character), which is cohomological with respect to an irreducible, selfcontragredient, algebraic, finite-dimensional representation Eμ of G∞. Assume that satisfies the equivalent conditions of Proposition 3.5, i.e., the partial exterior square L-function L S(s, , 2) has a pole at s = 1. Then, for every σ ∈ Aut(C), σ = ∞) . where “∼Q( f )” means up to multiplication of the right hand side by an element in the number field Q( f ). Proof Let be as in the statement of the theorem. We consider the commutative diagram (5.2) in Proposition 5.1: Let t := R ◦ Tμ ◦ Jμ ◦ t ◦ F t be the composition of the upper horizontal arrows, and analogously, let σ be the composition of the lower horizontal arrows. Let ξ f = ⊗v∈/S∞ ξv ∈ Wψ f ( f ) ∼= ⊗v∈/S∞ Wψv ( v) be a Whittaker function, such that for v ∈/ S, ξv is invariant under G(Ov) and normalized such that ξv(i dv) = 1. Given σ ∈Aut(C), let W σ (ξ f ) = σ ξ f ∈ Wψ f (σ f ) be the σ -twisted Whittaker function, cf. Sect. 4.1. Then, Proposition 5.1 says that σ ( (ξ f )) = σ (σ ξ f ). (7.6) In order to prove the theorem, we make both sides of this equation explicit. To that end, let [Wψ∞ ( ∞)]t (resp. [Wψ∞ (σ ∞)]t ) the generators of the respective cohomology spaces, Sect. 4.2. For each i and α, let ϕi,α := (W ψ )−1(ξ∞0,i,α ⊗ ξ f ) ∈ (resp. σϕi,α := (W ψ )−1(σ ξ∞0,i,α ⊗ σ ξ f ) ∈ σ ) be the corresponding cuspidal automorphic form. Recall our Schwartz-Bruhat function ∈ S (An) (resp. σ ∈ S (An)) from Sect. 6.1 (resp. Sect. 6.5), with m = n now. Inserting these functions into Theorem 6.1 [likewise, also into (6.2)] and recalling the definition of our archimedean periods ∞) from (7.2) shows that Eq. (7.6), induced by our Diagram (5.2), ⎛ voldg×dg (Z (F)\Z(A)/ AG ) L S( 1 , ) · Ress=1(L S(s, , 2)) σ ⎝ cn · ˆ (0) · 2 pt ( ) pt ( ∞) (Recall that was assumed to have trivial central character.) Invoking our cosmetically tuned choice for σ ∈ S (An) from Sect. 6.5, and observing that L(n, 1v) = (1 − |Ov/℘v|−n)−1 ∈ Q× for v ∈ S\S∞, this simplifies to = L S( 1 , σ ) · Ress=1(L S(s, σ , 2)) 2 pt (σ ) pt (σ ∞) v∈S\S∞ v∈S\S∞ ⎞ Zv(ξv, fv,s )⎠ Since σ (L( 21 , v)) = L( 21 , σ v) = 0 for all v ∈ S\S∞, cf. [29, Proposition 3.17], and recalling once more that S = S( , ψ ) = S(σ , ψ ), we may rewrite this by ⎛ L( 21 , f ) · Ress=1(L S(s, , 2)) σ ⎝ pt ( ) pt ( ∞) = ∞) IOnbdseeerdv,eiftthhaetfinitve∈pS\roS∞duZctv(ξvv,∈Sf\vS,s∞) Z(avn(ξdv, fvv,∈s)S\wS∞ereZzve(rσoξ,vt,hσenfvb,sy))thaerheolnoomno-zreprhoy. of L S(s , ) at s = 21 and the holomorphy of the integral Z∞(s , ξ∞0,i,α, f∞,1) at s = 21 (Lemma 7.2 and the paragraph below), L S( 21 , ) · L S(s, , 2) L S(n, 1)2 Zv(ξv, fv,1) L(n, 1v) would have no pole at s = 1 (Here we let ξv = ξv,0i,α at an archimedean place.). However, reading the proof of Theorem 6.1 backwards, respectively, by [6, Theorem 1 and Theorem 3], the latter expression equals as meromorphic functions in s. By [6, Theorem 1] and our assumption that is a functorial lift from SO(2n + 1), cf. Proposition 3.5, the integral Z (ξ, fs ) has a pole at s = 1, whereas L(n, 1) does not by the assumption that n ≥ 2. Hence, we arrived at a contradiction. We may therefore finish the proof of the first assertion of Theorem 7.4 by showing that σ ⎝ Zv(ξv, fv,s )⎠ = v∈S\S∞ Observing that by a simple change of variable and by our specific choice of fv,s = σ fv,s , Zv(σ ξv, σ fv,s ) = Zv(σ ◦ ξv, fv,s ), this is achieved by Matringe in Theorem A of the Appendix. The last assertion of the theorem follows by strong multiplicity one for cuspidal automorphic representations of G(A). 7.5 Whittaker–Shalika periods and the exterior square L-function Theorem 7.4 above is accompanied by the following corollary. Recall the non-zero Shalika periods ω ( f ) from Grobner–Raghuram [ 14 ]: These were defined by comparing a Q( f )-rational structure on a Shalika model of f and a Q( f )-rational structure on H r (g∞, (Z∞ K∞)◦, ⊗ Eμ)[ ]. For details, we refer to [14, Definition/Proposition 4.2.1]. Observe that ω 0 ( f ) is well-defined, if we assume that satisfies the assumptions made in the statement of Theorem 7.4: Indeed, as these assumptions include that the partial exterior square L-function L S(s, , 2) has a pole at s = 1, has a (1, ψ )-Shalika model by [14, Theorem 3.1.1]. (The extremely careful reader may also recall Lemma 3.2 at this place.) Moreover, by the same reasoning, also the archimedean Shalika period ω( ∞) = ω( ∞, 0) from [14, Theorem 6.6.2] is well-defined (and non-zero). A complete list of all choices, which enter the definition of these Shalika periods ω 0 ( f ) and ω( ∞), can be extracted (similar to our considerations leading to Remarks 4.4 and 7.3 above) from [14, Definition/Proposition 4.2.1 and Theorem 6.6.2], where they have been constructed in details. We do not provide such a list here, for the reason that neither ω 0 ( f ) nor ω( ∞) appear in the statement of the main theorems (but only in some corollaries). Define the Whittaker–Shalika periods Obviously the left hand side of (7.3) is uninteresting, if L( 21 , f ) = 0. Hence, we allow ourselves to make the strong assumption that L( 21 , f ) is non-zero in order to derive the following result: Corollary 7.6 Let then be as in the statement of Theorem 7.4. If L( 21 , f ) is non-zero, where “∼Q( f )” means up to multiplication of the right hand side by an element in the number field Q( f ). Proof This is obvious invoking Theorem 7.4 and [14, Theorem 7.1.2]. 8 A rationality result for the Rankin–Selberg L-function 8.1 The content of this section is very closely related to Grobner–Harris–Lapid [12, §4– §5] and a special case of Balasubramanyam–Raghuram [2, §2–§3]. Indeed, the main result, Theorem 8.5, of this section is Theorem 5.3 from [ 12 ] (but with the totally imaginary field E from [ 12 ] being replaced by the totally real field F as a groundfield), respectively Theorem 3.3.11 from [2] (but with the L-value L(1, Ad0, π ) from [2] being replaced by the residue of L S(s, × ∨) at s = 1). For the reason of these close analogies we allow ourselves to be rather brief, when it comes to details. Nevertheless, we think it is worthwhile writing down the following, already for reasons of notation, and in order to give precise statements of results in what follows. 8.2 Bottom-degree Whittaker periods Let be as in Sect. 3.1 and let b := dn2. Then, dimC H b(mG , K ◦ , ∞ for all ∈ π0(G∞)∗. As in Sect. 3.1, this is a direct consequence of the formula in Clozel [ 7 ], Lemma 3.14 and the Künneth rule. It is hence clear that the entire discussion of Sects. 3.3 and 4.1–4.4 carries over to (mG , K ◦ )-cohomology in degree ∞ q = b. In particular, let 1 := ((−1)n, . . . , (−1)n ) ∈ π0(G∞)∗, i.e., the inverse of the character 0. We obtain a Q( f )-structure on H b(mG , K ∞◦, ⊗ Eμ)[ 1] imposed by the Q(Eμ)-structure on Hcb(ψS∞−G1 (, Eμ) (exactly as in Definition 3.4) and we may fix once and for all a generator [W ∞)]b of the one-dimensional cohomology space H b(mG , K ∞◦, Wψ∞−1 ( ∞) ⊗ Eμ)[ 1], [W in complete analogy to Sect. 4.2, replacing the degree of cohomology t by b in Choice 4.2. Observe that here we exchanged the non-trivial additive character ψ by its inverse ψ −1 and (for notational clearness only), also the index α by β. Moreover, in light of Proposition 3.6, for all σ ∈Aut(C), we obtain non-trivial Whittaker periods pb(σ ), unique up to multiplication by elements in Q(σ f )×, such that Wψ −f1 ( f ) W W σ ψ −f1 (σ f ) Fb b Fσ H b(mG , K ◦ , ∞ ⊗ Eμ)[ 1] Hμσ,b H b(mG , K ◦ , σ ∞ ⊗ σEμ)[ 1] commutes. This is the analogue of Proposition 4.3, whose proof goes through word for word in the current situation, i.e., for cohomology in degree b instead of t . See [30, Proposition/Definition 3.3]. In the above diagram, Hμσ,b := ( σb )−1 ◦ Hμσ,b ◦ b is the restriction of Hμσ,b to H b(mG , K ∞◦, ⊗ Eμ)[ 1], this map being well-defined following by the same argument as in Sect. 4.3. We leave it to the reader to fill in the remaining details. Remark 8.1 A comprehensive list of all ingredients on which our bottom-degree period pb( ) depends is now easily accomplished reading through Remark 4.4, mutatis mutandis, i.e., replacing t by b, ψ by ψ −1 and 0 by 1. 8.3 Another archimedean period Recall the Schwartz–Bruhat function = ⊗v v ∈ S (A2n) from Sect. 6.1 with m = 2n in this case. Let U2n be the subgroup of upper triangular matrices in G = GL2n, whose diagonal entries are all equal to 1. For each v ∈ S∞ we let ξv ∈ Wψv ( v) (resp. ξv ∈ Wψv−1 ( v)) be a local Whittaker function, which is S O(2n)-finite from the right. For such Whittaker functions, the local zeta-integrals v(s, ξv, ξv, v) := U2n(Fv)\GL2n(Fv) × v((0, . . . , 0, 1)gv)| det(gv)|sv dgv ξv(gv)ξv(gv) converge for e(s) ≥ 1, cf. [21, Proposition (3.17)]. If ξ∞ ∈ Wψ∞ ( ∞) ∼= ⊗v∈S∞ Wψv ( v) (resp. ξ∞ = ⊗v∈S∞ ξv ∈ ⊗v∈S∞ Wψv−1 ( v)) is K ◦ -finite, we abbreviate ∞ = ⊗v∈S∞ ξv Wψ∞−1 ( ∞) ∼= ∞(s, ξ∞, ξ∞, ∞) := v(s, ξv, ξv, v). v∈S∞ Furthermore, by assumption Eμ =∼ Eμ∨. So, the canonical pairing Eμ × Eμ∨ → C induces a pairing Eμ × Eμ → C, which we will denote by eα, eβ := eβ∨(eα). As a last ingredient, recall our generators [Wψ∞ ( ∞)]t and [Wψ∞−1 ( ∞)]b from Sects. 4.2 and 8.2. Similar to Sect. 7.1, we let s(i , j ) be the unique complex number, such that Xi∗ ∧ X ∗j = s(i , j ) · X1 ∧ · · · ∧ Xt+b. Putting things together, consider ∞) := i=(i1,...,it ) j=( j1,..., jb) α=1 β=1 ∞(1, ξ∞0,i,α, ξ∞1, j,β , ∞). Then there is the following theorem, which follows from Proposition 5.0.3 in [2]. Theorem 8.2 The number c( Proof We may adapt the argument given at the beginning of the proof of Theorem 7.1, to see that the non-vanishing of c( ∞) may be reduced to showing the non-vanishing of a similarly defined number d( v), which only depends on one given archimedean place v ∈ S∞. Indeed, there is a projection u+v=b Mb : b(mG /k∞)∗−∼→ u (g∞/c∞)∗ ⊗ b(g∞/c∞)∗ ⊗ 0s∗, where we wrote again c∞ := z∞ ⊕ k∞. By reasons of degrees of cohomology, Mb induces an isomorphism of (one-dimensional) vector spaces W ∞) ⊗ Eμ)[ 1]−∼→H b(g∞, (Z∞ K∞)◦, whose effect on the generator [Wψ∞−1 ( ∞)]b is by mapping X ∗j to Mb(X j ) ∈ b(g∞/c∞)∗ ⊗ 0s∗. Whence, at the cost of re-scaling Mb(X j ) by the non-trivial factor in 0s∗ = R, we may and will assume that Mb(X j ) ∈ b(g∞/c∞)∗. Recall the projection Lt = Lr ⊗ Ld−1 and the isomorphism Lt = Lr ⊗ Ld−1 from the proof of 7.1.2 Moreover, observe that there is an isomorphism N t+b : t+b(mG /k∞)∗−∼→ r+b(g∞/c∞)∗ ⊗ which we factor similarly to Lt as N t+b(X1 ∧ · · · ∧ Xt+b) = Nr+b(X1 ∧ · · · ∧ Xt+b) ⊗ Nd−1(X1 ∧ · · · ∧ Xt+b), where Nr+b(X1 ∧ · · · ∧ Xt+b) ∈ r+b(g∞/c∞)∗ and Nd−1(X1 ∧ · · · ∧ Xt+b) ∈ d−1s∗. It hence follows that the number c( ∞) is a non-trivial multiple of 2 Observe the difference between the last factors in Lt and Mb: While 0s∗ = R by convention, the isomorphism d−1s∗ ∼= R is not canonical, for which we introduced the notational factor Md−1. ∞) := i=(i1,...,it ) j=( j1,..., jb) α=1 β=1 ∞), where u(i , j ) is the uniquely defined complex number, such that Lr (Xi∗) ∧ Mb(X ∗j ) = u(i , j ) · Nr+b(X1 ∧ · · · ∧ Xt+b). The number d( ∞) factors as v∈S∞ d( v), where each local factor d( v) is defined analogously d( ∞) = (using r+b(g∞/c∞)∗ ∼= 2n2+n−1(gv/cv)∗). Therefore, we may finish the proof by showing that d( v) is non-zero for all v ∈ S . This is the reduction to a single ∞ archimedean place v ∈ S , mentioned at the beginning of the proof. The result hence ∞ follows by [2, Proposition 5.0.3]. In view of the latter non-vanishing result and Proposition 3.6, we may define p(σ ∞) := c(σ for all σ ∈ Aut(C). Remark 8.4 Analogously to Remark 7.3, let us recollect at one place the various choices which enter the definition of our archimedean period p( ∞), since (almost) none of them appear in its notation: 1. ∞, ψ∞, the cohomological degrees b and t , as well as the archimedean measures dgv chosen for all v ∈ S∞ in Sect. 6.3. 2. The fixed generators [Wψ∞ ( ∞)]t and [Wψ∞−1 ( ∞)]b from Sects. 4.2 and 8.2. We remark further that these generators depend themselves on the data fixed in Choice 4.2. 3. The concrete choice of an archimedean Schwartz–Bruhat function ∞ = ⊗v∈S∞ v from Sect. 6.1 with m = 2n. Ipteirsiohdesnccoencsliedaerretdhastoefxairstienntcheisapnadpdeer,fipntit(ion),opfbp(( )∞an)dispitn(dependent of the other ∞). 8.4 Rationality of the residue of the Rankin–Selberg L-function at s = 1 Having set up our additional notation above, we obtain the main result of this section. Theorem 8.5 Let F be a totally real number field and G = GL2n/F , n ≥ 2. Let be a self-dual, unitary, cuspidal automorphic representation of G(A) (with trivial central character), which is cohomological with respect to an irreducible, self-contragredient, algebraic, finite-dimensional representation Eμ of G∞. Then, for every σ ∈ Aut(C), Ress=1(L S(s, × )) pt ( ) pb( ) p( ∞) . ∞) Ress=1(L S(s, ∞), where “∼Q( f )× ” means up to multiplication by a non-trivial element in the number field Q( f ). Proof The first assertion follows from Theorem 3.3.11 of [2]. The second assertion of Theorem 8.5 follows from the first one, applying strong multiplicity one for the cuspidal automorphic spectrum of G(A) and recalling that Ress=1(L S(s, × )) is non-zero. In fact, Ress=1(L S(s, × )) = 0 is well-known and is a consequence of Theorem 8.2 together with [ 21 ] (5), p. 550 and Proposition (2.3) in loc. cit.. 9 A rationality result for the symmetric square L-function 9.1 Definition of the archimedean bottom-degree period Let be a cuspidal automorphic representation of G(A) as in Sect. 3.1 and σ ∈ Aut(C). Recall the archimedean periods pt (σ ∞) from (7.2) and p(σ ∞) from (7.1). We define our bottom-degree, archimedean period by pb(σ ∞) . ∞) Remark 8.1 Of course, by Theorems 7.1 and 8.2, pb(σ ∞) is well-defined and nonzero. For the sake of the reader, we remark that a complete list of all concrete choices on which pb( ∞) actually depends, is hence given by merging the precise lists provided by Remarks 7.3 and 8.4. 9.2 Rationality of the symmetric square L-function at s = 1 Recall our bottom-degree periods pb( ) and pb( ∞), defined in Sects. 8.2 and 9.1, respectively. For a complete list of all data, entering the respective definition, we refer to Remarks 8.1 and 8.1 above. The following result is our second main theorem. Theorem 9.2 Let F be a totally real number field and G = GL2n/F , n ≥ 2. Let be a unitary cuspidal automorphic representation of G(A) (self-dual and with trivial central character), which is cohomological with respect to an irreducible, selfcontragredient, algebraic, finite-dimensional representation Eμ of G∞. Assume that satisfies the equivalent conditions of Proposition 3.5, i.e., the partial exterior square L-function L S(s, , 2) has a pole at s = 1. Then, for every σ ∈ Aut(C), L( 21 , f ) pb( ) pb( ∞) L S(1, , Sym2) L( 21 , σ L S(1, , Sym2) ∼Q( f ) L( 21 , f ) pb( ) pb( ∞) (9.3) where “∼Q( f )” means up to multiplication of L S(1, , Sym2) by an element in the number field Q( f ). Proof Recall that L S(s, × ) = L S(s, , Sym2) · L S(s, , 2) as meromorphic functions in s, whence, by the assumptions on , we obtain )) = L S(1, , Sym2) · Ress=1(L S(s, , 2)). Since L S(1, , Sym2) is non-zero (cf. [32, Theorem 5.1]), the first assertion of the theorem follows from Theorems 7.4 and 8.5. The second assertion is now again a consequence of strong multiplicity one for the cuspidal automorphic spectrum of G(A). 9.3 Whittaker–Shalika periods and the symmetric square L-function Corollary 9.4 Let then As in the case of the exterior square L-function, we obtain a corollary of our second main theorem, Theorem 9.2, using the main results of our paper [ 14 ]. Recall the non-zero Shalika periods ω ( f ) and ω( ∞) = ω( ∞, 0) from Sect. 7.5 above, respectively from [14, Definition/Proposition 4.2.1 and Theorem 6.6.2], therein, their existence being guaranteed as in Sect. 7.5. Define the Whittaker–Shalika periods Pb( ) := pb( ) · ω 0 ( f ) and Pb( Analogously to the situation considered in Sect. 7.5 above, the right hand side of (9.3) is uninteresting if L( 21 , f ) = 0. Hence, we allow ourselves to make the strong assumption that L( 21 , f ) is non-zero in order to obtain the following result. be as in the statement of Theorem 9.2. If L( 21 , f ) is non-zero, L S(1, , Sym2) ∼Q( f )× Pb( ) Pb( ∞), where “∼Q( f )× ” means up to multiplication of L S(1, , Sym2) by a non-zero element in the number field Q( f ). Proof This follows directly from Theorem 9.2 and [14, Theorem 7.1.2]. 10 Applications for quotients of symmetric square L-functions 10.1 Gauß sums of algebraic Hecke characters It is the purpose of this section to provide a result, independent of the all the periods mentioned above for certain quotients of symmetric square L-functions. ordv(DF ). Here, DF stands for the absolute different of F , that is, D−F1 = {x ∈ F : T rF/Q(x O) ⊂ Z}. Recall our fixed non-trivial additive character ψ : F \A → C× from Sect. 2.1. The Gauß sum of χ f with respect to y and ψ is now defined as G (χ f , ψ f , y) = v∈/S∞ G (χv, ψv, yv), where the local Gauß sum G (χv, ψv, yv) is defined as G (χv, ψv, yv) = × Ov χv(uv)−1ψv(yvuv) duv. For almost all v, we have G (χv, ψv, yv) = 1, and for all v we have G (χv, ψv, yv) = 0. (See, for example, Godement [10, Eq. 1.22].) Note that, unlike in [37], we do not normalize the Gauß sum to make it have absolute value one. For the sake of easing notation and readability we suppress its dependence on ψ and y, and denote G (χ f , ψ f , y) simply by G (χ f ). 10.2 An application of Theorem 9.2 Theorem 10.1 Let F be a totally real number field and G = GL2n/F , n ≥ 2. Let be any cuspidal automorphic representation of G(A) and let χ1 and χ2 be two Hecke characters of finite order, such that ⊗ χi , i = 1, 2, both satisfy the conditions of Corollary 9.4. If χ1 and χ2 have moreover the same infinity-type, i.e., χ1,∞ = χ2,∞, then, where “∼Q( f ,χ1, f ,χ2, f )× ” means up to multiplication by a non-zero element in the composition of number fields Q( f ), Q(χ1, f ) and Q(χ2, f ). It shall be noted that, whereas the quantities on the left hand side of the above equation all depend crucially on , the right hand side is not only independent of the all periods considered in this paper, but completely independent of . Proof Recall the Whittaker–Shalika periods Pb( ⊗ χi ) = pb( ⊗ χi ) · ω 0 ( f ⊗ χi, f ) and Pb( ∞ ⊗ χi,∞) = pb( ∞ ⊗ χi,∞) · ω( ∞ ⊗ χi,∞), i = 1, 2, from Sect. 9.3. We remind the reader that since both ⊗ χi , i = 1, 2, satisfy the assumptions of Theorem 9.2, all periods appearing in their definition are well-defined and non-zero, cf. Sect. 7.5. Since it follows directly from the definition of rationality fields that Q( f ⊗ χ1, f )Q( f ⊗ χ2, f ) = Q( f , χ1, f , χ2, f ), our Corollary 9.4 (or, alternatively, Theorem 9.2 together with [14, Theorem 7.1.2]) implies that L S (1, L S (1, ⊗ χ1) · ω 0 ( ⊗ χ2) · ω 0 ( ∼Q( f ,χ1, f ,χ2, f )× f ⊗ χ1, f ) pb( f ⊗ χ2, f ) pb( P b( P b( ⊗ χ1) P b( ⊗ χ2) P b( ∞ ⊗ χ1,∞) · ω ( ∞ ⊗ χ2,∞) · ω ( ∞ ⊗ χ1,∞) ∞ ⊗ χ2,∞) ∞ ⊗ χ1,∞) . ∞ ⊗ χ2,∞) Moreover, the infinity-types of χ1 and χ2 are equal by assumption, which implies that ∞ ⊗ χ1,∞ and ∞ ⊗ χ2,∞ are not only isomorphic, but literally identical. As a consequence, the contribution of all archimedean periods above cancels out, and we are left with L S (1, L S (1, ⊗ χ1) · ω 0 ( ⊗ χ2) · ω 0 ( f ⊗ χ1, f ) f ⊗ χ2, f ) . (We remark aside that in order to see this cancellation it would also have been enough to know that the χi are of finite order, since then ∞ ⊗ χi,∞ ∼= ∞, for i = 1, 2, see [14, 5.3]. However, the equality ∞ ⊗ χ1,∞ = ∞ ⊗ χ2,∞ makes the cancellation even more obvious.) It is exactly the main result of [30, Theorem 4.1.], that—if χ1 and χ2 have the same infinity-type, which we assume—one has the relation Furthermore, it is the main result of section 5 of [14, Theorem 5.2.1], that we obtain ω 0 ( f ⊗ χi, f ) ∼Q( f ,χi, f )× ω ( f ) · G (χi, f )n . where ∈ K∞/K ∞◦ is the same for both i = 1, 2, because we assumed that χ1 and χ2 have the same infinity-type. Inserting the relations (10.3) and (10.4) into (10.2), we obtain L S (1, L S (1, ⊗ χ1, Sym2) ⊗ χ2, Sym2) ∼Q( f ,χ1, f ,χ2, f )× pb( ) · G (χ2, f )n(2n−1) · ω ( pb( ) · G (χ1, f )n(2n−1) · ω ( f )· G (χ1, f )n f ) · G (χ2, f )n (10.3) (10.4) = G (χ1, f )2n2 G (χ2, f )−2n2 . This shows the claim. Acknowledgements Open access funding provided by University of Vienna. I am grateful to Erez Lapid for suggesting me to work in this direction. Moreover, I would like to thank the anonymous referee for suggesting (and motivating) me to think of further applications of Theorem 1.4, which resulted in Theorem 1.5 above. Thanks are also due to her/him for insisting to clarify all the dependencies of our periods, which certainly improved the readability and clearness of the present paper. Appendix In this appendix,3 F denotes a non-archimedean local field with valuation v and absolute value |.| (normalised as usual). We will write |g| for | det(g)| when g is a square matrix, O for the ring of integers of F and let ℘ = O be the maximal ideal of O. Proposition A Let φ ∈ Cc∞(F), χ a character of F×, and m ≥ 0 and integer. Then, T (q−s , χ , m, φ) := φ(x) χ (x) v(x)m |x|s d×x (with vold×x (O×) = 1) converges for |q−s | < |χ ( )|−1 and can be extended to an element of L(s, χ )m · C[q±s ]. Moreover, if σ ∈ Aut(C), then σ (T (q−s , χ , m, φ)) = T (σ (q−s ), σ (χ ), m, σ (φ)). Proof For k ∈ Z, we set and ck (χ , φ) := φ( k x) χ (x) d×x, c(χ ) := χ (x) d×x, O× so that we have c(χ ) = 0, if χ is ramified, and c(χ ) = 1, if χ is unramified. As O× is compact, take U an open compact subgroiu=p1o1fφO(× fikxxii)nχg(xxi →), hφen(ce,k xas) and χ , and let O× = i=1 xi U , then ck (χ , φ) = σ (1/ ) = 1/ , we have σ (ck (χ , φ)) = ck (σ (χ ), σ (φ)). Let a be a positive integer such that the support of φ is contained in ℘a−1. Let b ≥ a be such that φ is constant on ℘b. We have T (q−s , χ , m, φ) = ck (χ , φ)km χ ( )kq−ks + c(χ )kmχ ( )kq−ks . a≤k≤b k≥b F× O× We set ck (χ , φ)km χ ( )kq−ks 3 Nadir Matringe: Université de Poitiers, Laboratoire de Mathématiques et Applications, Téléport 2 - BP 30179, Boulevard Marie et Pierre Curie, 86962, Futuroscope Chasseneuil Cedex. Email: E-mail address: . and Suppose that χ is ramified, i.e., non trivial on O×. Then B(q−s , χ , m, φ) := c(χ )km χ ( )k q−ks . T (q−s , χ , m, φ) = A(q−s , χ , m, φ). σ (T (q−s , χ , m, φ)) = σ (ck (χ , φ))km σ (χ ( ))k σ (q−ks ) = A(σ (q−s ), σ (χ ), m, σ (φ)) = T (σ (q−s ), σ (χ ), m, σ (φ)), ck (σ (χ ), σ (φ))km σ (χ ( ))k σ (q−ks ) which shows the claim in this case. Suppose now that χ is unramified. Then there is P ∈ Q[X, X −1] (which can be determined explicitly, notice that the coefficients of P are in Q, hence σ -invariant) such that B(q−s , χ , m, φ) = km χ ( )k q−ks = P(χ ( )q−s )/(1 − χ ( )q−s )m , which implies that σ (B(q−s , χ , m, φ)) = P(σ (χ ( ))σ (q−s ))/(1 − σ (χ ( ))σ (q−s ))m = B(σ (q−s ), σ (χ ), m, σ (φ)). This implies again σ (T (q−s , χ , φ)) = T (σ (q−s ), σ (χ ), σ (φ)). We denote by Pn the mirabolic subgroup of Gn = G L(n, F ), and by An the diagonal torus of Gn, which is contained in the standard Borel Bn with unipotent radical Nn. For k ∈ {1, . . . , n − 1}, the group Gk embeds naturally in Gn, so the center Zk of Gk embeds in An, and An = Z1 · · · Zn (direct product). The following result follows from Proposition 2.2 of [ 19 ]. We fix a non-trivial additive character ψ of F . If zi belongs to Zi ⊂ An, we set t (zi ) to be the element of F ∗ such that zi = di ag(t (zi ), In−i ) of F ∗, non-negative integers (miξk )ik ∈Ik , and functions (φiξk )ik ∈Ik such that Proposition B Let π be an irreducible generic representation of Gn , and ξ ∈ Wψ (π ). For each k ∈ {1, . . . , n − 1}, there exists a finite set Ik , a string of characters (cik )ik ∈Ik ξ(z1 · · · zn−1) = n−1 k=1 ik ∈Ik k=1 cik (t (zk )) v(t (zk ))miξk φiξk (t (zk )). (The characters cik , which we allow to be equal, depend only on π .) We denote by wn the element of the symmetric group Sn naturally embedded in Gn, defined by 1 2 · · · m − 1 m 1 3 · · · 2m − 3 2m − 1 when n = 2m is even, and by 1 2 · · · m − 1 m m + 1 m + 2 · · · 2m 2m + 1 1 3 · · · 2m − 3 2m − 1 2m + 1 2 · · · 2m − 2 2m when n = 2m + 1 is odd. We denote by Ln the standard Levi subgroup of Gn which is G (n+1)/2 × G n/2 embedded by the map (g1, g2) → diag(g1, g2). We denote by Hn the group Lnwn = wn−1 Lnwn, by J (g1, g2) the matrix wn−1diag(g1, g2)wn of Hn (with diag(g1, g2) ∈ Ln). Let r be a positive integer. Thanks to the Iwasawa decomposition Gr = Nr · Ar · Gr (O), if χ is an unramified character of Ar , then the map is well defined on Gr . For example, if δr is the modulus character of the maximal parabolic subgroup of type (r − 1, 1) restricted to Ar , we have a map δ˜r on Gr . Similarly, if λ : z1 · · · zr ∈ Ar → |t (z1) · · · t (zr−1)|, the map λ˜ is also defined on Gr , and left invariant under Zr . Theorem A Let π be an irreducible generic representation of Gn with trivial central character and ξ ∈ Wψ (π ). Set m = (n + 1)/2 and m = n/2 . The integral Nm\Gm Zm Nm \Gm Z (ξ, q−s ) := ξ(J (h, g)) δ˜m (g) |g|s λ˜(h)s dg dh (with the normalisations dg = d×a dk with d×a( Am(O)) = 1 and dk(Gm(O)) = 1, dh = d×b dk with d×b(Zm (O)\ Am (O)) = 1 and dk(Gm (O)) = 1) converges absolutely for |q−s | small enough. It extends to an element of C(q−s ), which satisfies that for all σ ∈ Aut(C), one has σ (Z (ξ, q−s )) = Z (σ ◦ ξ, σ (q−s )). Proof Let δ be the modulus character of Bm and δ that of Bm . Let U × U be a compact open subgroup of Gm (O) × Gm(O) such that J (U ×U ) fixes ξ on the right, and write Gm (O) × Gn (O) = j=1 x j U × y j U . We have Z (ξ , q−s ) = Am −1 ξ j ( J (b, a)) δm (a) δ (a)−1 δ (b)−1|a|s |b|s d×a d×b, where ξ j (g) = ξ (g J (x j , y j )). We identify Am × Am −1 (a, b) → J (b, a), and set χ the character of An−1 J (b, a) → δm (a)δ (a)−1δ (b)−1. The previous integral becomes with An−1 by defined by Z (ξ , q−s ) = ξ j (z) χ (z) |z|s d×z, We set χk to be the restriction of χ to Zk . It takes values in qZ ⊂ Q. 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Harald Grobner. Rationality results for the exterior and the symmetric square L-function (with an appendix by Nadir Matringe), Mathematische Annalen, 2017, 1-41, DOI: 10.1007/s00208-017-1590-7