Rationality results for the exterior and the symmetric square Lfunction (with an appendix by Nadir Matringe)
Rationality results for the exterior and the symmetric square Lfunction (with an appendix by Nadir Matringe)
Harald Grobner 0
0 Fakultät für Mathematik, University of Vienna , OskarMorgensternPlatz 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 periodfree 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 Topdegree periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 An Aut(C)rational assignment for Whittaker functions . . . . . . . . . . . . . . . . . . . . . . . . 6 An integral representation of the residue of the exterior square Lfunction . . . . . . . . . . . . . . .

Contents
H.G.’s research has been supported by the Austrian Science Fund (FWF), standalone research project
P 25974 and 2016’s STARTprize, grant number Y 966.
7 A rationality result for the exterior square Lfunction . . . . . . . . . . . . . . . . . . . . . . . . . .
8 A rationality result for the Rankin–Selberg Lfunction . . . . . . . . . . . . . . . . . . . . . . . . .
9 A rationality result for the symmetric square Lfunction . . . . . . . . . . . . . . . . . . . . . . . .
10 Applications for quotients of symmetric square Lfunctions . . . . . . . . . . . . . . . . . . . . . .
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 CMfield 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 Lvalue 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 modelspace
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, basisfree 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 Lfunction (by some
variation or the other), the principal Lfunction, or the Asai Lfunctions, here we
would like to study the algebraicity of the exterior square Lfunction and the
symmetric square Lfunction, 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 Lfunction 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 selfdual,
∼= ∨, 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 Lfunction. 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, selfcontragredient, algebraic, finitedimensional
representation Eμ of G∞. Assume that satisfies the equivalent conditions of Proposition 3.5,
i.e., the partial exterior square Lfunction L S(s, , 2) has a pole at s = 1. Then,
for every σ ∈ Aut(C), there is a nontrivial 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 nontrivial 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 nonvanishing 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 integralrepresentation, obtained by Bump–Friedberg [
6
], of the residue
Ress=1(L S(s, , 2)) of the exterior square Lfunction 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 (nonzero) 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 Lfunction 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 Lfunction 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 nontrivial
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 equivarianceresult (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
nonzero periods ω 0 ( f ) and ω( ∞) constructed in loc. cit. are welldefined. See our
Sect. 7.5 below for details. If we define the nonzero, topdegree Whittaker–Shalika
periods,
then we may get rid of the Lfactor 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 nonzero, 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 Lfunction,
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 selfdual, unitary, cuspidal automorphic representation of
G(A) (with trivial central character), which is cohomological with respect to an
irreducible, selfcontragredient, algebraic, finitedimensional representation Eμ of
G∞. Then, for every σ ∈ Aut(C),
.
∞)
where “∼Q( f )× ” means up to multiplication by a nontrivial element in the number
field Q( f ).
In the statement of the latter theorem, pt ( ) is the topdegree period defined above,
while pb( ) is defined analogously, but using the lowest degree b, where has
nontrivial cohomology. The nonvanishing 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 Lfunction 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 Lfunction L S(s, , Sym2) is holomorphic and nonvanishing
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 bottomdegree 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 Lfactor L( 21 , f ), if it is nonzero.
Corollary 1.4 Let be as in the statement of Theorem 1.4 (resp. Theorem 9.2). If
L( 21 , f ) is nonzero, then
L S(1, , Sym2) ∼Q( f )× Pb( ) Pb(
∞),
where “∼Q( f )× ” means up to multiplication of L S(1, , Sym2) by a nonzero
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
Lfunction 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 infinitytype,
i.e., χ1,∞ = χ2,∞, then,
where “∼Q( f ,χ1, f ,χ2, f )× ” means up to multiplication by a nonzero 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 nontrivial 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 Lfunction 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
finitedimensional 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 selfdual, 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 selfduality 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 nontrivial 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 Cvector 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 fixedfield of S(ν) within C, i.e.,
Q(ν) := {z ∈ Cσ (z) = z for all σ ∈ S(ν)}.
We say that a representation ν on a Cvector space W is defined over a subfield F ⊂ C,
if there is a Fvector 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 selfdual,
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 nonvanishing
(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 wellknown 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, nonvanishing (g∞, (Z∞
K∞)◦)cohomology with respect to the same algebraic, selfdual coefficient module Eμ
(although the degrees and dimensions of nontrivial 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 − 1dimensional 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 wellknown 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 nonarchimedean 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 nonzero 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 Lfunction, 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 Lfunction 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 Lfunction, 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 Lfunction, 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 Topdegree 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 topdegree 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 dualbasis {X ∗j } of (mG /k∞)∗;
given a multiindex 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 , mulitindex 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 onedimensional space Cvector 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 welldefined. 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 wthiethonaendatiumreanlscihooniaclespoafcae 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 welldefined 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 Topdegree 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 nonzero 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 deRhamisomorphism 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 Qvalued 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 Qvalued 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 Rhamisomorphism: 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
deRhamisomorphism 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 deRhamisomorphism 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
Qrationality of the measure on H (A f ), Sect. 5.2.
(5.2)
6 An integral representation of the residue of the exterior square
Lfunction
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 selfdual 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 nonzero complex number.
6.3 Measures
When dealing with rationality results of special values of Lfunctions, the choice of
measures is allimportant. 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 welldefined 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 Lfunction of the trivial character 1 of A×
at n. Since we assumed that n ≥ 2, this number is welldefined and nonzero. 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 Lfunction, 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 Lfunction of the trivial character 1 of
A× at n. As by assumption n ≥ 2, L(n, 1) is welldefined and nonzero. 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 Lfunction, 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 nonzero. 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 Lfunction
7.1 Archimedean considerations
The integral representation of the exterior square Lfunction 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 nonvanishing 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 multiindex i there is a welldefined 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 nonvanishing 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 Lfunction 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 welldefined. 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 nonvanishing for all s ∈ C. Since μv,k ≥ 0 for all
1 ≤ k ≤ n, by the selfduality 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
nonzero. 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 nonzero to showing that a
similarly defined number, dt ( v) is nonzero. 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 (onedimensional) 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 nontrivial 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 nonzero 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 nonzero 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 nonzero 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 nonzero on the onedimensional subspace 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 nonzero 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 nonzero for all archimedean places, whence so is ct ( ∞).
7.3 Definition of the archimedean topdegree 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 topdegree Whittaker period pt ( ), which
we dealt with in Remark 4.4, (almost) none of the various choices entering the
definition of our archimedean topdegree 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 onedimensional
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 topdegree 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 Lfunction 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) (selfdual and with
trivial central character), which is cohomological with respect to an irreducible,
selfcontragredient, algebraic, finitedimensional representation Eμ of G∞. Assume that
satisfies the equivalent conditions of Proposition 3.5, i.e., the partial exterior square
Lfunction 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 SchwartzBruhat 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))thaerheolnoomnozreprhoy.
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 Lfunction
Theorem 7.4 above is accompanied by the following corollary. Recall the nonzero
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 welldefined, 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 Lfunction 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 welldefined (and nonzero). 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 nonzero 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 nonzero,
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 Lfunction
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 Lvalue 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 Bottomdegree 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 onedimensional 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 nontrivial 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 nontrivial
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 welldefined
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 bottomdegree
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 zetaintegrals
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 nonvanishing of c( ∞) may be reduced to showing the nonvanishing
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 (onedimensional) 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 rescaling Mb(X j ) by the nontrivial
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
nontrivial 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 nonzero 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 nonvanishing 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 Lfunction 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 selfdual, unitary, cuspidal automorphic representation of G(A) (with trivial central
character), which is cohomological with respect to an irreducible, selfcontragredient,
algebraic, finitedimensional 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 nontrivial 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
nonzero. In fact, Ress=1(L S(s, × )) = 0 is wellknown 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 Lfunction
9.1 Definition of the archimedean bottomdegree 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 bottomdegree, archimedean period by
pb(σ
∞) .
∞)
Remark 8.1 Of course, by Theorems 7.1 and 8.2, pb(σ ∞) is welldefined 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 Lfunction at s = 1
Recall our bottomdegree 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) (selfdual and with
trivial central character), which is cohomological with respect to an irreducible,
selfcontragredient, algebraic, finitedimensional representation Eμ of G∞. Assume that
satisfies the equivalent conditions of Proposition 3.5, i.e., the partial exterior square
Lfunction 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 nonzero (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 Lfunction
Corollary 9.4 Let
then
As in the case of the exterior square Lfunction, we obtain a corollary of our second
main theorem, Theorem 9.2, using the main results of our paper [
14
]. Recall the
nonzero 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 nonzero in order to obtain the following result.
be as in the statement of Theorem 9.2. If L( 21 , f ) is nonzero,
L S(1, , Sym2) ∼Q( f )× Pb( ) Pb(
∞),
where “∼Q( f )× ” means up to multiplication of L S(1, , Sym2) by a nonzero
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 Lfunctions
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 Lfunctions.
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 nontrivial 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 infinitytype, i.e., χ1,∞ = χ2,∞,
then,
where “∼Q( f ,χ1, f ,χ2, f )× ” means up to multiplication by a nonzero 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 welldefined and nonzero,
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 infinitytypes 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
infinitytype, 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 infinitytype. 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 nonarchimedean 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 xs 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: Email 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 nontrivial 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 ∗, nonnegative 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) gs 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)−1as bs 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) zs d×z,
We set χk to be the restriction of χ to Zk . It takes values in qZ ⊂ Q. If we now apply
the second proposition of this appendix, we obtain that
is the sum for k between 1 and n − 1, i ∈ Ik , and j ∈ {1, . . . , } of
Z (ξ , q−s )
1 n−1
k=1
T
(q−s )k , cik χk , miξ j , φiξ j .
k k
This implies, according to the first proposition of this appendix, that
is the sum for k between 1 and n − 1, i ∈ Ik , and j ∈ {1, . . . , } of
1 n−1
k=1
T
(σ (q−s ))k , σ (cik )χk , mξ j
ik , σ
φξ j
ik
.
This means that σ (Z (ξ , q−s )) is equal to Z (σ ◦ ξ , σ (q−s )).
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