Period Relations for Automorphic Forms on

Unitary Groups and Critical Values of L-Functions

Lucio Guerberoff

Received: March 16, 2016 Revised: October 12, 2016

Communicated by Otmar Venjakob

Abstract. In this paper we explore some properties of periods attached to automorphic representations of unitary groups over CM fields and the critical values of their L-functions. We prove a formula expressing the critical values in the range of absolute convergence in terms of Petersson norms of holomorphic automorphic forms. On the other hand, we express the Deligne period of a related motive as a product of quadratic periods and compare the two expressions by means of Deligne’s conjecture.

2010 Mathematics Subject Classification: 11F67 (Primary) 11F70, 11G18, 11R39, 22E55 (Secondary).

1. Introduction

The goal of the present paper is to study critical values of the standard L-functions of cohomological automorphic representations of unitary groups, and relate them to the motivic expression predicted by Deligne’s conjecture ([Del79a]). This extends previous results by Harris ([Har97]) from quadratic imaginary fields to arbitrary CM fields L. Let K be the maximal totally real subfield of L, and let G be a similitude unitary group attached to an n-dimensional hermitian space over L. Fix a CM type Φ for L/K, and suppose that G has sigature (r_{τ} , s_{τ}) at each τ ∈ Φ. Let π be a cuspidal cohomological automorphic representation of G(𝔸). We can parametrize the weight of π by a tuple of integers ((a_{τ,1}, …, a_{τ,n})_{τ∈Φ}; a_{0}). See Section 3 for details. We assume that the corresponding algebraic representation of G_{ℂ} is defined over ℚ, and that π^{∨} ≅ π ⊗ ∥ν∥^{2a_{0}}, where ν is the similitude factor. Let ψ be an algebraic Hecke character of L of infinity type (m_{τ})_{τ:L↪ℂ}. Our main theorem is the following.

Theorem 1. Suppose that π satisfies Hypothesis 4.5.1 and contributes to antiholomorphic cohomology. If m > n is an integer satisfying

m ≤ min\{a_{τ,r_{τ}} + s_{τ} + m_{τ} − m\overline{τ} , a_{τ,s_{τ}} + r_{τ} + m\overline{τ} − m_{τ} \}_{τ∈Φ},

then

L^{S}(m − \frac{n − 1}{2}, π ⊗ ψ, St) ∼ (2πi)^{[K:ℚ](mn−n(n−1)/2)−2a_{0}}D_{K}^{⌊ \frac{n+1}{2} ⌋/2}P (ψ)Q^{hol}(π).

In this expression, the members belong to E(π) ⊗ E(ψ) ⊗ ℂ, where E(π) and E(ψ) are number fields over which π_{f} and ψ are defined, and ∼ means up to multiplication by an element of E(π) ⊗ E(ψ) ⊗ L′, with L′ being the Galois closure of L in ℂ. We refer the reader to Section 4, in particular to Theorem 4.5.1, for a precise and detailed explanation of the notation. Let us mention that the factor Q^{hol}(π) is an automorphic quadratic period attached to π. This is basically defined as a Petersson norm of an arithmetic holomorphic vector in π. The element P (ψ) is an explicit expression involving CM periods attached to ψ. The method of proof of Theorem 1 follows the lines of [Har97], and is based on earlier work by Shimura ([Shi76]). It is based on the doubling method, and allows us to write the L-function as an integral of a holomorphic automorphic form against a certain Eisenstein series. Roughly speaking, Li proved in [Li92] that the L-function can be written in terms of global and local Piatetski-Shapiro-Rallis zeta integrals and an inner product between automorphic forms. The inequality that m needs to satisfy in the hypotheses of the theorem governs the existence of a differential operator for automorphic vector bundles, constructed in [Har86]. By carefully choosing the sections defining the Eisenstein series and using these differential operators, we can see that the zeta integrals are rational over L′, and we can interpret the inner product as the automorphic quadratic period. See Section 4 for more details. In the final sections of the paper, we interpret the formula in Theorem 1 motivically to obtain period relations.

1.1. Background and motivation. The first results concerning the expression of special values of automorphic L-functions as Petersson norms were due to Shimura, especially in the case of Hilbert modular forms (see [Shi76], [Shi78], [Shi83a], [Shi88]). Petersson norms are to be interpreted as quadratic periods, as in Shimura’s conjectures concerning the factorization of periods. In [Har97], Harris generalized the quadratic periods to the setting of coherent cohomology of Shimura varieties, in this case attached to unitary groups of hermitian spaces over quadratic imaginary fields. In this paper, we treat the case of arbitrary CM fields. The motivic interpretation, given below, relates the expression of Theorem 1 to Deligne’s conjecture on critical values. It should be noted that quadratic periods have their own importance independently of any reference to motives. In particular, a proper understanding of them is key to the construction of p-adic L-functions.

Period relations have been studied by several authors such as Shimura, Yoshida, Oda, Schappacher, Panchishkin, Blasius, Hida and Harris, among others. One of the basic principles in their prediction is Tate’s conjecture. For instance, Tate’s conjecture is used crucially in Blasius’s proof of Deligne’s conjecture for Hecke L-series ([Bla86]). See also [Bla97]. It has long been known by specialists, dating back to Shimura, Deligne and Langlands, that there should be motivic relations, and hence period relations, predicted by relations between automorphic forms on different groups. In this paper, this manifestation takes the form of period relations for unitary groups of different signatures. For related recent results, we remark that Jie Lin in her recent Paris thesis ([Lin15]) conjectures a similar formula as that of Theorem 1, but without the discriminant factor, which is assumed to belong to the coefficient field. This is used to prove results generalizing those of [GH16].

1.2. Motivic interpretation. We can interpret motivically the formula in Theorem 1 as follows. Suppose that Π is a cuspidal, cohomological, self-dual automorphic representation of GL_{n}(𝔸_{K}). There is a conjectural motive M over K, with coefficients in a number field E ⊂ ℂ, attached to Π. The ℓ-adic realizations of M have already been constructed (see [CHL], [Shi11], [CH13], [Sor]), and the existence of M will mostly play a heuristic role. We refer to Section 2 for details about motives. For simplicity in what follows, we fix the embedding E ↪ ℂ, and all the L-values and periods will be considered to be complex numbers via this embedding. Under certain assumptions, we can descend Π_{L} to an automorphic representation π of the unitary group G ([Lab11]; see also [Mok15], [KMSW14]). Write ψ|_{𝔸_{K}} = ψ_{0}∥ · ∥^{−w}, with w being the weight of ψ and ψ_{0} a finite order character, and let χ = ψ^{2}(ψ_{0} ∘ N_{L/K})^{−1}. The relation between the standard L-function of π ⊗ ψ and M is encompassed in the following formula:

L (s − \frac{n − 1}{2}, π ⊗ ψ, St) = L(M ⊗ RM(χ), s + w).

Here RM(χ) is the restriction of scalars from L to K of the motive M(χ) attached to χ. In Section 2, we determine explicitly the set of critical integers of L(M ⊗ RM(χ), s). The motive M is regular of weight n − 1, in the sense that the Hodge components M_{σ}^{pq} have dimension 0 or 1 for each σ. We let the Hodge numbers be (p_{i}(σ), q_{i}(σ)), where p_{1}(σ) > ⋯ > p_{n}(σ). We assume that m_{τ} ≠ m\overline{τ} for any τ, and take the CM type Φ in such a way that m_{τ} > m\overline{τ} for τ ∈ Φ. We show that if M ⊗ RM(χ) has critical values, then for each σ there exists r_{σ} = 0, …, n such that

n − 1 − 2p_{r_{σ}}(σ) < 2m_{τ} − 2m\overline{τ} < n − 1 − 2p_{r_{σ}+1}(σ).

Moreover, suppose that the signatures of G are given by r_{τ} = r_{σ} for τ ∈ Φ extending σ (starting from Π and ψ, we can always find G with these signatures). The set of critical integers of the form m + w for M ⊗ RM(χ) is governed by two inequalities (see (2.5.2)), one of which is precisely the inequality on Theorem 1. Thus, we can see the corresponding values of the L-function of π ⊗ ψ as critical values of M ⊗ RM(χ). Deligne’s conjecture predicts that these are, up to multiplication by an element in the coefficient field, equal to the Deligne periods c^{+}(M ⊗ RM(χ))(m + w). The motive M is also equipped with a polarization M ≅ M^{∨}(1 − n). The methods of Section 2 allow us to write the σ-periods c_{σ}^{+}(M ⊗ RM(χ)) in terms of another set of periods Q_{j,σ} of M, called quadratic periods. We refer to the main text for a precise definition. We prove the following result, first proved in [Har97] when K = ℚ.

Theorem 2. If m + w is critical for M ⊗ RM(χ), then

c^{+} (M ⊗ RM(χ)(m + w)) ∼ (2πi)^{[K:ℚ]mn+w ∑_{σ} r_{σ}−s_{σ}}δ(M)Q(χ) ∏_{σ} ∏_{j=1}^{s_{σ}} Q_{j,σ}.

We can actually obtain a more precise formula involving only the c_{σ}^{+}(M ⊗ RM(χ)) for a single σ (see Theorem 2.5.1). Here, Q(χ) is an explicit expression involving CM periods attached to χ. We stress that the conjectural existence of the motive M attached to Π plays a heuristic role, but Theorem 2 is proved for any family of realizations M (such as a motive for absolute Hodge cycles) with the properties of being regular and polarized over a totally real field. Comparing this expression with that of Theorem 1, we match the CM periods and get that the prediction of Deligne’s conjecture for M ⊗RM(χ) is translated in the following statement:

(1.2.1)  ∏_{σ} ∏_{j=1}^{s_{σ}} Q_{j,σ} ∼ (2πi)^{−[K:ℚ]n(n−1)/2}δ(M)^{−1}D_{K}^{n/2}Q^{hol}(π).

We can state this relation without making reference to the quadratic periods Q_{j,σ} by interpreting them as automorphic periods obtained from automorphic representations of different unitary groups, whose signatures are (n, 0) at all places except at one place, where the signature is (n−1, 1). These automorphic representations contribute in coherent cohomology to the different stages of the Hodge filtration (as opposed to the single holomorphic stage which gives rise to Q^{hol}(π)). We show that the relation (1.2.1) is predicted by Tate’s conjecture as well, as we explained above. If one is willing to assume this conjecture, then this implies Deligne’s conjecture for the motives M ⊗RM(χ). We should stress here that what it actually implies is a slightly weaker version of Deligne’s conjecture. Namely, the automorphic methods only allow us to relate the expressions up to multiples by elements in the Galois closure L′ of L in ℂ. Also, we only obtain the version of Deligne’s conjecture obtained by fixing an embedding of the coefficient field. This last issue does not arise when K = ℚ, since the Hodge components M_{σ}^{pq} of the motives in question are free over E⊗ℂ, something which is almost never true when K is bigger than ℚ (for example, this is already false for motives M(χ) attached to algebraic Hecke characters).

1.3. Organization of the paper. In Section 2, we recall the basic facts and main properties about motives, realizations and their periods. The reader who is interested solely in the main theorem on critical values of automorphic L-functions on unitary groups can skip Section 2 and go directly to Sections 3 and 4. The reason we include Section 2 before is that we use some of the terminology regarding Hodge-de Rham structures and polarizations in Section 3. In Section 2, we make emphasis in regular polarized motives over totally real fields, and in Theorem 2.5.1, we prove the factorization of c_{σ}^{+}(M ⊗ RM(χ)) in term of quadratic periods. In Section 3, we introduce unitary groups and their associated Shimura varieties. We set up the notation for the parameters of representations giving rise to automorphic vector bundles, and give a brief overview of the main properties of the Hodge-de Rham structures attached to cohomological automorphic representations, constructed in [Har94]. We write down the action of complex conjugation and the polarizations in terms of automorphic forms, and we give an automorphic definition of quadratic periods in the setting of coherent cohomology. Section 4 contains our main theorem on critical values of cohomological automorphic L-functions. In the first subsections we set up the doubling method and we recall the relation, proved by Li ([Li92]), between the standard Lfunctions and Piatetski-Shapiro-Rallis zeta integrals, and in Subsection 4.5 we prove the main theorem. Finally, in Section 5, we postulate period relations obtained by comparing the results of Sections 2 and 4 by means of Deligne’s conjecture. This section is hypothetical in nature. We start by recalling some basic facts about transfer and descent for automorphic representations of unitary groups and GL_{n}, along with several motivic expectations, including the existence of the motive M described above, as well different relations between the Hodge-de Rham structures attached to nearly equivalent automorphic representations of unitary group, which are consequences of Tate’s conjecture. We show how they imply the period relations embodied in Deligne’s conjecture. The arguments are heuristic, depending on these motivic expectations, but we can write down the predicted period relations concretely in terms of automorphic forms on unitary groups.

Acknowledgements. The author wants to thank Michael Harris for many helpful answers, his advice and his support. The author also thanks Daniel Barrera, Don Blasius and Jie Lin for several useful discussions, and for comments on earlier versions of this paper. Finally, the author thanks the referee for helpful comments and suggestions. Part of this work was carried out while the author was a guest at the Max Planck Institute for Mathematics in Bonn, Germany. It is a pleasure to thank the Institute for its hospitality and the excellent working conditions.

Notation and conventions. We fix an algebraic closure ℂ of ℝ, a choice of i = \sqrt{−1}, and we let \overline{ℚ} denote the algebraic closure of ℚ in ℂ. We let c ∈ Gal(ℂ/ℝ) denote complex conjugation on ℂ, and we use the same letter to denote its restriction to \overline{ℚ}. Sometimes we also write c(z) = \overline{z} for z ∈ ℂ. We let Γ_{ℚ} = Gal(\overline{ℚ}/ℚ). For a number field K, we let 𝔸_{K} and 𝔸_{K,f} denote the rings of adèles and finite adèles of K respectively. When K = ℚ, we write 𝔸 = 𝔸_{ℚ} and 𝔸_{f} = 𝔸_{ℚ,f}. If K is a number field (resp. a finite extension of ℚ_{p} for some prime number p), and \overline{K} is a fixed algebraic closure, let K^{ab} be the maximal abelian extension of K inside \overline{K}. We let art_{L} : 𝔸_{L}^{×} → Gal(K^{ab}/K) (resp. art_{L} : L^{×} → Gal(K^{ab}/K)) be the Artin reciprocity map of class field theory. We normalize these maps so that the global reciprocity map is compatible with the local ones, and the local map takes a uniformizer to a geometric Frobenius element. A CM field L is a totally imaginary quadratic extension of a totally real field K. A CM type Φ for L/K is a choice of one of the two possible extensions to L of each embedding of K. All vector spaces will be finite-dimensional except otherwise stated. By a variety over a field K we will mean a geometrically reduced scheme of finite type over K. For a field K, we let 𝔾_{m,K} denote the usual multiplicative group over K. For any algebraic group G over K, we let Lie(G) denote its Lie algebra and Ad : G → GL_{Lie(G)} the adjoint representation. A reductive algebraic group will always be assumed to be connected. We let 𝕊 = R_{ℂ/ℝ}𝔾_{m,ℂ}. We denote by c the complex conjugation map on 𝕊, so for any ℝ-algebra A, this is c ⊗_{ℝ} 1_{A} : (ℂ ⊗_{ℝ} A)^{×} → (ℂ ⊗_{ℝ} A)^{×}. We usually also denote it by z ↦ \overline{z}, and on complex points it should not be confused with the other complex conjugation on 𝕊(ℂ) = (ℂ ⊗_{ℝ} ℂ)^{×} on the second factor. A tensor product without a subscript between ℚ-vector spaces will always mean tensor product over ℚ. For any number field K, we denote by J_{K} = Hom(K, ℂ). For σ ∈ J_{K}, we let \overline{σ} = cσ. Let E and K be number fields, and σ ∈ J_{K}. If α, β ∈ E ⊗ ℂ, we write α ∼_{E⊗K,σ} β if either β = 0 or if β ∈ (E ⊗ ℂ)^{×} and α/β ∈ (E ⊗ K)^{×}, viewed as a subset of (E ⊗ ℂ)^{×} via σ. There is a natural isomorphism E ⊗ℂ ≃ ∏_{φ∈J_{E}} ℂ given by e⊗z ↦ (φ(e)z)_{φ} for e ∈ E and z ∈ ℂ. Under this identification, we denote an element α ∈ E ⊗ ℂ by (α_{φ})_{φ∈J_{E}}. We write α ∼_{E;K;σ} β if either β = 0 or β ∈ (E ⊗ ℂ)^{×} and α_{φ}/β_{φ} ∈ φ(E)σ(K) ⊂ ℂ for all φ ∈ J_{E}. The relation α ∼_{E⊗K,σ} β implies α ∼_{E;K;σ} β, but the converse is not necessarily true. When K is given from the context as a subfield of ℂ, we write ∼_{E⊗K} (resp. ∼_{E;K}) for ∼_{E⊗K,1} (resp. ∼_{E;K;1}), where 1 : K ↪ ℂ is the given embedding. Suppose that r = (r_{φ})_{φ∈J_{E}} is a tuple of nonnegative integers. Given Q_{1}, …, Q_{n} in E ⊗ ℂ (with n ≥ r_{φ} for all φ), we denote by

∏_{j=1}^{r} Q_{j} ∈ E ⊗ ℂ

the element whose φ-th coordinate is ∏_{j=1}^{r_{φ}} Q_{j,φ}. In particular, this defines x^{r} for x ∈ E ⊗ ℂ.

We choose Haar measures on local and adelic points of unitary groups as in the Introduction of [Har97].

2. Factorization of Deligne’s periods

In this section, we start by introducing the notation we will use for motives and realizations. We work with the category of realizations (as in [Jan90], §2) instead of the category of motives for absolute Hodge cycles. All of the unexplained notions can be found in [Del79a] (see also [Sch88], [Jan90], [Pan94] and [Yos94]). We then recall the basic facts about periods, and introduce polarized regular realizations and their quadratic periods, extending the results of [Har97] from ℚ to a totally real field. The main result of the section is Theorem 2.5.1, expressing the σ-periods of motives of the form M ⊗ RM(χ) in terms of quadratic periods.

2.1. Motives and realizations. Let K be a number field. We fix an algebraic closure \overline{K} of K and we let Γ_{K} = Gal(\overline{K}/K). For a number field E, by a pure realization over K with coefficients in E, of weight w ∈ ℤ, we will mean the following data.

• For each σ ∈ J_{K}, an E-vector space M_{σ} of dimension d (independent of σ), together with E-linear isomorphisms F_{σ} : M_{σ} → M\overline{σ} which satisfy F_{σ}^{−1} = F\overline{σ}. Each M_{σ} is endowed with a ℚ-Hodge structure of weight w

M_{σ} ⊗ ℂ = ⨁_{pq} M_{σ}^{pq},

where each M_{σ}^{pq} is an E ⊗ ℂ-submodule, such that F_{σ,c} = F_{σ} ⊗ c sends M_{σ}^{pq} to M pq ; \overline{σ}. We let F_{σ,ℂ} = F_{σ} ⊗1_{ℂ}. The filtration on M_{σ} ⊗ℂ induced by the Hodge decomposition is called the Hodge filtration.

• A free E ⊗ K-module M_{dR} of rank d, together with a decreasing filtration F^{•}(M_{dR}) by (not necessarily free) E ⊗ K-submodules; this is called the de Rham filtration.

• For each finite place λ of E, an E_{λ}-vector space M_{λ} of dimension d, endowed with a continuous action of Γ_{K}.

• For each σ ∈ J_{K}, an E ⊗ ℂ-linear isomorphisms I_{∞,σ} : M_{σ} ⊗ ℂ → M_{dR} ⊗_{K,σ} ℂ compatible with the Hodge and de Rham filtrations. We also require that c_{dR,σ}I_{∞,σ} = I_{∞,\overline{σ}}F_{σ,c}, where c_{dR,σ} = 1_{M_{dR}} ⊗_{K,σ} c.

• For each σ ∈ J_{K}, for each extension σ̃ : \overline{K} ↪ ℂ of σ to \overline{K}, and for each λ, E_{λ}-linear isomorphisms I_{σ̃,λ} : M_{σ} ⊗_{E} E_{λ} → M_{λ}. Moreover, if σ is real, then the automorphism F_{σ} ⊗_{E} 1_{E_{λ}} of M_{σ} ⊗_{E} E_{λ} corresponds, via I_{σ̃,λ} to the action of the element of Γ_{K} given by the complex conjugation deduced from σ̃.

For an object M as above, the integer d is called the rank of M. A morphisms between a pure realization M and a pure realization N is defined to be a family of maps ((f_{σ})_{σ∈J_{K}} , f_{dR}, (f_{λ})_{λ}), where f_{σ} : M_{σ} → N_{σ} is an E-linear morphism of Hodge structures for each σ, f_{dR} : M_{dR} → N_{dR} are E ⊗ K-linear maps preserving the de Rham filtrations, and f_{λ} : M_{λ} → N_{λ} are E_{λ}-linear Γ_{K}-equivariant maps. We require all these maps to correspond under the comparison isomorphisms. The category ℛ(K)_{E} is defined to be the category whose objects are direct sums of pure realizations. It is a semi-simple Tannakian category over E, whose objects are simply called realizations. A Hodge-de Rham structure consists of the same data, without the ℓ-adic realizations (see [Har94]). As in Deligne’s article [Del79a], a motive will mean a pure motive for absolute Hodge cycles (see [DMOS82], Section 6, for details). We denote by ℳ(K)_{E} the category of motives over K with coefficients in E. There is a fully faithful functor from the category of motives ℳ(K)_{E} to ℛ(K)_{E}, which identifies ℳ(K)_{E} with the full Tannakian subcategory of ℛ(K)_{E} generated by the cohomologies of smooth projective varieties over K ([Jan90]). If M is a motive, we also denote by M the realization it defines in ℛ(K)_{E}. Since E ⊗ ℂ ≃ ℂ^{J_{E}}, we can write M_{σ} ⊗ ℂ ≃ ⨁_{φ∈J_{E}} M_{σ}(φ), where M_{σ}(φ) = (M_{σ} ⊗ ℂ) ⊗_{E⊗ℂ,φ} ℂ ≃ M_{σ} ⊗_{E,φ} ℂ. Since the Hodge decomposition M_{σ}^{pq} is E ⊗ ℂ-stable, this gives each factor M_{σ}(φ) a decomposition

M_{σ}(φ) = ⨁_{p,q} M_{σ}^{pq}(φ),

where M_{σ}^{pq}(φ) = M_{σ}^{pq} ⊗_{E⊗ℂ,φ} ℂ. This has the property that complex conjugation sends M_{σ}^{pq}(φ) to M_{σ}^{qp}(\overline{φ}). We put

h_{σ}^{pq}(φ) = dim_{ℂ} M_{σ}^{pq}(φ) = dim_{ℂ} gr^{p}(M_{dR}) ⊗_{E⊗K,φ⊗σ} ℂ.

Thus, h_{σ}^{pq}(φ) = h_{σ}^{qp}(\overline{φ}) = hpq ; \overline{σ}(\overline{φ}). We say that M ∈ ℛ(K)_{E} is regular if h_{σ}^{pq}(φ) ≤ 1 for every pair of integers p, q and every σ ∈ J_{K}, φ ∈ J_{E}. If M is regular of rank d and pure of weight w, then given σ and φ, there are d numbers p_{1}(σ, φ), …, p_{d}(σ, φ) with the property that M_{σ}^{pq}(φ) ≠ 0 (and has complex dimension 1) if and only if p = p_{i}(σ, φ) for some i. For fixed σ and φ, we order these numbers in such a way that p_{1}(σ, φ) > ⋯ > p_{d}(σ, φ). We let q_{i}(σ, φ) = w − p_{i}(σ, φ). Note that q_{i}(σ, φ) = p_{d+1−i}(σ, \overline{φ}) = p_{d+1−i}(\overline{σ}, φ). For a realization M, we denote its dual by M^{∨}, and for realizations M, N, we denote their tensor product (over E) by M ⊗_{E} N. We similarly adopt the standard linear algebra notation for exterior products. If M is a realization with coefficients in ℚ, and N is a realization with coefficients in E, then we can naturally see M ⊗ (R_{E/ℚ}N), denoted by M ⊗ N, as a realization with coefficients in E, where R_{E/ℚ}N is restriction of coefficients from E to ℚ. Let E and E′ be number fields, and write E ⊗ E′ ≅ ∏_{j=1}^{m} F_{j}, where F_{j} are number fields. Let M ∈ ℛ(K)_{E} and N ∈ ℛ(K)_{E′}, of rank d and e respectively. We define realizations (M ⊗N)^{(j)} ∈ ℛ(K)_{F_{j}} of rank de by taking (M ⊗N)^{(j)} = (M ×_{E} F_{j})⊗_{F_{j}} (N ×_{E′} F_{j}). By M ⊗N we mean the collection {(M ⊗N)^{(j)}}_{j=1}^{m}, and we often say that it is a realization with coefficients in E ⊗ E′. We denote by ℳ^{0}(K)_{E} the category of Artin motives over K with coefficients in E. This is equivalent to the category of continuous, finite-dimensional representations of Γ_{K} on E-vector spaces. In this setting, we can describe the realizations of an Artin motive M, viewed as a representation V of Γ_{K}, as follows. For every σ ∈ J_{K}, M_{σ} ≅ V, the isomorphism depending on an extension of σ to \overline{K}. The Hodge structures are purely of type (0, 0). Also, M_{λ} ≅ V ⊗_{E} E_{λ} and M_{dR} ≅ (V ⊗ \overline{K})^{Γ_{K}}. If ϵ : 𝔸_{K}^{×}/K^{×} → E^{×} is a finite order character, we denote by [ϵ] the Artin motive in ℳ^{0}(K)_{E} given by the character Γ_{K} → E^{×} obtained from ϵ by class field theory. Let χ : 𝔸_{K}^{×}/K^{×} → ℂ^{×} be an algebraic Hecke character. Recall that this means that χ is continuous, and that for every embedding σ ∈ J_{K}, there exist an integer n_{σ}, such that if v is the infinite place of K induced by σ and x ∈ (K_{v}^{×})^{+}, then

χ(x) = σ(x)^{−n_{σ}} if v is real;
χ(x) = σ(x)^{−n_{σ}} \overline{σ}(x)^{−n\overline{σ}} if v is complex.

The integer n_{σ} + n\overline{σ} = w(χ) is independent of σ, and is called the weight of χ. The tuple (n_{σ})_{σ∈J_{K}} is called the infinity type of χ. Let T^{K} = Res_{K/ℚ} 𝔾_{m,K}. Consider the group of characters X^{∗}(T^{K}), which is naturally identified with ℤ^{J_{K}}. For η ∈ X^{∗}(T^{K}), we denote by X(η) the set of algebraic Hecke characters χ of K of infinity type η. Let ℚ(χ) ⊂ ℂ denote the field generated by the values of χ on 𝔸_{K,f}^{×} (the finite idèles). Then ℚ(χ) is either ℚ or a CM field. Let E ⊂ ℂ be a number field containing ℚ(χ). We denote by M(χ) ∈ ℳ(K)_{E} the motive of weight w(χ) attached to χ, as in [Sch88]. For n ∈ ℤ, the motive attached to the character χ(x) = ∥x∥^{n}, where ∥ · ∥ is the idèlic norm, is the Tate motive ℚ(n) ∈ ℳ(K)_{ℚ}. For any M ∈ ℛ(K)_{E}, we let M(n) = M ⊗ ℚ(n). For later use, we record the Hodge decomposition of M(χ). Let ℚ(η) be the field of definition of the character η, that is, the fixed field in \overline{ℚ} of the stabilizer of η ∈ X^{∗}(T^{K}) under the natural action of Γ_{ℚ}. Then ℚ(η) ⊂ ℚ(χ). For any γ ∈ Γ_{ℚ} (or γ ∈ Aut(ℂ)), we let χ^{γ} ∈ X(η^{γ}) be the algebraic Hecke character of infinity type η^{γ} whose values on 𝔸_{K,f}^{×} are obtained by applying γ to the values of χ. The tuple of integers parametrizing η^{γ} is given by (n(σ, γ))_{σ∈J_{K}}, where n(σ, γ) = n_{γ^{−1}σ}. Note that η^{γ}, and hence n(σ, γ), only depend on the restriction of γ to ℚ(η). In particular, we can define n(σ, φ) for any φ ∈ J_{ℚ(χ)}. Then M(χ)_{σ}^{pq}(φ) ≠ 0 if and only if p = n(σ, φ) and q = n(σ, \overline{φ}). Note however that different embeddings φ could give rise to the same numbers, so M(χ)_{σ}^{n(σ,φ),n(σ,\overline{φ})} ≠ M(χ)_{σ}^{n(σ,φ),n(σ,\overline{φ})}(φ) in general. Since M(χ) is of rank 1, it is regular and p_{1}(σ, φ) = n(σ, φ). In order to properly define the L-function of a realization M, we need to impose that assumption that (M_{λ})_{λ} is a strictly compatible system of λadic representations over E (see [Del79a], 1.1 for details). We will assume from now on without further mention that this holds. If φ ∈ J_{E}, then L(φ, M, s) = ∏_{v} L_{v}(φ, M, s), where L_{v}(φ, M, s) is the corresponding Euler factor at v of the system (M_{λ})_{λ}. We also assume that L(φ, M, s) converges absolutely for Re(s) large enough and has a meromorphic continuation to ℂ. We define L^{∗}(M, s) = (L(φ, M, s))_{φ∈J_{E}} ∈ ℂ^{J_{E}} ≃ E ⊗ ℂ. We can complete the Lfunction with the factors at infinity L_{σ}(φ, M, s) for each σ ∈ J_{K}. This is done following Serre’s recipe ([Ser70]) as in 5.2 of [Del79a]. We define L_{∞}(φ, M, s) = ∏_{σ∈J_{K}} L_{σ}(φ, M, s) and L_{∞}^{∗}(M, s) = (L_{∞}(φ, M, s))_{φ∈J_{E}} ∈ ℂ^{J_{E}} ≃ E ⊗ ℂ. It is not hard to see that L_{∞}(φ, M, s) does not depend on φ (see 2.9 of [Del79a]). Note that L^{∗}(M(n), s) = L^{∗}(M, s + n), and similarly for the factors at infinity. Let M ∈ ℛ(K)_{E}. We say that an integer n is critical for M if for every φ ∈ J_{E} (or equivalently, for one φ), neither L_{∞}(φ, M, s) nor L_{∞}(φ, M^{∨}, 1 − s) have a pole at s = n. We say that M has critical values if there exists an integer critical for M, and we say that M is critical if 0 is critical for M. Thus, M has critical values if and only if M(n) is critical for some n ∈ ℤ.

2.2. Periods. Let M ∈ ℛ(K)_{E}. From now on, unless otherwise stated, K is a totally real number field. In this subsection we recall the definition of periods. We refer to [Del79a], [Pan94] and [Yos94] for details. For each σ ∈ J_{K}, we define δ_{σ}(M) ∈ (E ⊗ ℂ)^{×} to be the determinant of I_{∞,σ}, calculated with respect to an E-basis of M_{σ} and an E ⊗ K-basis of M_{dR}. This is well defined modulo (E ⊗ K)^{×} (contained in (E ⊗ ℂ)^{×} via σ). We define δ(M) = δ(Res_{K/ℚ} M), with respect to the unique element of J_{ℚ}. A realization M ∈ ℛ(K)_{E} is said to be special if it is pure of some weight w, and if for every σ ∈ J_{K}, F_{σ,ℂ} acts on M_{σ}^{w/2,w/2} by a scalar ε = ±1, independent of σ. It is easily checked that a pure realization with critical values is special (see for instance (1.3.1) of [Del79a]). Suppose that M is special. For each σ ∈ J_{K}, let M_{σ}^{±} ⊂ M_{σ} denote the ±-eigenspace for F_{σ}. It’s easy to see that the dimension of M_{σ}^{±} is independent of σ, and we denote this common dimension by d^{±} = dim_{E} M_{σ}^{±}. We can also choose appropriate terms F^{±}(M_{dR}) ⊂ M_{dR} such that M_{dR}^{±} = M_{dR}/F^{∓}(M_{dR}) is a free E ⊗ K-module of rank d^{±} and the map

I_{∞,σ}^{±} : M_{σ}^{±} ⊗ ℂ → M_{dR}^{±} ⊗_{K,σ} ℂ

given by the composition of the projection M_{dR}⊗_{K,σ}ℂ → M_{dR}^{±}⊗_{K,σ}ℂ with I_{∞,σ} and with the inclusion M_{σ}^{±} ⊗ℂ ↪ M_{σ} ⊗ℂ is an E ⊗ℂ-linear isomorphism. We define c_{σ}^{±}(M) = det(I_{∞,σ}^{±}) ∈ (E ⊗ ℂ)^{×}, where the determinants are computed in terms of an E-basis of M_{σ}^{±} and an E ⊗ K-basis of M_{dR}^{±}. Note that these quantities are defined modulo (E ⊗ K)^{×} ⊂ (E ⊗ ℂ)^{×} (via σ). The relation between these periods and the usual Deligne periods c^{±}(Res_{K/ℚ} M), which we denote by c^{±}(M), is given by the following factorization formula, proved in [Yos94] or [Pan94] (we also include a similar formula for the δ’s):

c^{±}(M) ∼_{E⊗K′} D_{K}^{d^{±}/2} ∏_{σ} c_{σ}^{±}(M). (2.2.1)

δ(M) ∼_{E⊗K′} D_{K}^{d/2} ∏_{σ} δ_{σ}(M). (2.2.2)

Here D_{K} is the discriminant of K, and K′ ⊂ \overline{ℚ} is the Galois closure of K in \overline{ℚ}. The following is the main conjecture of [Del79a].

Conjecture 2.2.1 (Deligne). If M is critical and L(φ, M, 0) ≠ 0 for some φ, then

L^{∗}(M, 0) ∼_{E} c^{+}(M).

The conjecture is aimed at motives rather than general realizations. We formulate a weaker version that we will need later. The hypotheses are as in Deligne’s conjecture.

Conjecture 2.2.2. Let F ⊂ ℂ be a number field. By the weak Deligne conjecture up to F -factors, we mean the statement

L(M, s) ∼_{E;F} c^{+}(M).

Regarding the notation in this conjecture, recall that this means that the φcomponents differ by a multiple in φ(E)F.

Remark 2.2.1. If M is of rank d and t ∈ ℤ, then δ_{σ}(M(t)) ∼ (2πi)^{td}δ_{σ}(M). If M is special, then so is M(t). If t is even, then c_{σ}^{±}(M(t)) ∼ (2πi)^{td^{±}} · c_{σ}^{±}(M), while if t is odd, then c_{σ}^{±}(M(t)) ∼ (2πi)^{td^{∓}} · c_{σ}^{∓}(M). In all these formulas, ∼ means ∼_{E⊗K,σ}. Let ϵ : Γ_{K} → E^{×} be a continuous character, and let [ϵ] ∈ ℳ^{0}(K)_{E} be the corresponding Artin motive. For each σ ∈ J_{K}, choose σ̃ : \overline{K} ↪ ℂ an extension of σ to \overline{K}. Then [ϵ]_{σ} ≅ E and [ϵ]_{dR} ≅ (E ⊗ \overline{K})^{Γ_{K}}. The map

I_{∞,σ}^{−1} : (E ⊗ \overline{K})^{Γ_{K}} ⊗_{K,σ} ℂ ≅ E ⊗ ℂ

is given by ((e ⊗ λ) ⊗ z) ↦ (e ⊗ σ̃(λ)z). If {ζ} is an E ⊗ K-basis of [ϵ]_{dR}, and we choose the natural E-basis {1} of E, then det(I_{∞,σ}) = σ̃(ζ)^{−1}, so that δ_{σ}([ϵ]) ∼_{E⊗K,σ} σ̃(ζ)^{−1} (see also p. 104, [Sch88]). Now, note that the Frobenius automorphism F_{σ} acts on the one-dimensional E-vector space [ϵ]_{σ} by the sign ϵ(c_{σ}), where c_{σ} ∈ Γ_{K} is a complex conjugation attached to the place σ. Suppose that this scalar ε does not depend on σ. This means that [ϵ] is special. If ε = −1, then c_{σ}^{+}([ϵ]) ∼ 1 and c_{σ}^{−}([ϵ]) ∼ δ_{σ}([ϵ]). If ε = 1, then c_{σ}^{+}([ϵ]) ∼ δ_{σ}([ϵ]) and c_{σ}^{−}([ϵ]) ∼ 1. Finally, note that if M is a special realization, then so is M ⊗_{E} [ϵ] and c_{σ}^{±}(M ⊗_{E} [ϵ]) ∼ c_{σ}^{±ε}(M)δ_{σ}([ϵ])^{d^{±ε}}.

Remark 2.2.2. It’s easy to see that F^{∓}(M_{dR}^{∨}) ⊂ M_{dR}^{∨} is the annihilator of F^{±}(M_{dR}). It follows that there are natural isomorphisms (F^{±}(M_{dR}))^{∨} ≃ (M^{∨})_{dR}^{±}. In particular, F^{±}(M_{dR}) is E ⊗ K-free of rank d^{±}.

Remark 2.2.3. Let M ∈ ℛ(K)_{E} be a realization of rank d and weight w, assumed to be special. Since E ⊗ K is a product of fields, we can extend any E ⊗ K-basis of the free rank d^{±} module F^{±}(M_{dR}) to an E ⊗ K-basis of the free rank d module M_{dR}. Replacing M_{dR} by the appropriate F^{+}(M_{dR}) or F^{−}(M_{dR}) as an intermediate step, we can choose the bases consistently, so it follows that we can find a basis {ω_{1}, …, ω_{d}} of M_{dR} such that {ω_{1}, …, ω_{d^{±}}} is a basis of F^{±}(M_{dR}). For simplicity of notation, ω_{i} will also denote the element ω_{i} ⊗_{K,σ} 1 ∈ M_{dR} ⊗_{K,σ} ℂ, since σ will be understood throughout. Let {e_{1}, …, e_{d^{+}}} (resp. {f_{1}, …, f_{d^{−}}}) be an E-basis of M_{σ}^{+} (resp. of M_{σ}^{−}), and write

I_{∞,σ}^{−1}(ω_{j}) = ∑_{i=1}^{d^{+}} a_{ij,σ}^{+}e_{i} + ∑_{i=1}^{d^{−}} a_{ij,σ}^{−}f_{i}, j = 1, …, d

in M_{σ} ⊗ ℂ, with a_{ij,σ}^{±} ∈ E ⊗ ℂ. Then

(2.2.3)  c_{σ}^{±}(M^{∨}) ∼_{E⊗K,σ} det(P_{σ}^{±}),

where P_{σ}^{±} = ((a_{ij,σ}^{±})_{i,j=1,...,d^{±}}). Indeed, this follows from Remark 2.2.2: letting I_{∞,σ}^{∨} be the comparison maps for M^{∨}, the equations above mean precisely that

I_{∞,σ}^{∨}(e_{j}^{∨}) = ∑_{i=1}^{d^{+}} a_{ji,σ}^{+}ω_{i}^{∨},

where {e_{1}^{∨}, …, e_{d^{+}}^{∨}} is the dual basis of M_{σ}^{∨,+} and {ω_{1}^{∨}, …, ω_{d^{+}}^{∨}} is the dual basis of (F^{+}(M_{dR}))^{∨}, thus proving (2.2.3) for c_{σ}^{+}. The case of c_{σ}^{−} is completely similar. Furthermore, suppose that {Ω_{1}, …, Ω_{d}} is an E ⊗ ℂ-basis of M_{σ} ⊗ ℂ with the property that the change of basis matrix with respect to {ω_{1}, …, ω_{d}} is unipotent. More precisely, suppose that

I_{∞,σ}(Ω_{i}) = ω_{i} + ∑_{j=1}^{i−1} r_{ji,σ}ω_{j}

with r_{ji,σ} ∈ E ⊗ ℂ. We let r_{ii,σ} = 1 and r_{ji,σ} = 0 if j > i. Write

(2.2.4)  Ω_{j} = ∑_{i=1}^{d+} ã_{ij,σ}^{+}e_{i} + ∑_{i=1}^{d−} ã_{ij,σ}^{−}f_{i}, j = 1, …, d

and let P̃_{σ}^{±} = ((ã_{ij,σ}^{±})_{i,j=1,...,d^{±}}). If R_{σ}^{±} denotes the matrix constructed from the first d^{±} rows and columns of R_{σ} = (r_{ij,σ})_{i,j=1,...,d}, then R_{σ}^{±} is upper triangular with diagonal entries 1, and the formula P̃_{σ}^{±} = P_{σ}^{±}R_{σ}^{±} implies that

(2.2.5)  c^{±}(M^{∨}) ∼_{E⊗K,σ} det P̃_{σ}^{±}.

Similarly, if we let

P_{σ} = ( \begin{matrix}(a_{ij,σ}^{+}) & \begin{matrix}1≤i≤d^{+} \\ 1≤j≤d \end{matrix} \\ (a_{ij,σ}^{−}) & \begin{matrix}1≤i≤d^{−} \\ 1≤j≤d \end{matrix} \end{matrix} ) ∈ GL_{d}(E ⊗ ℂ)

and

P̃_{σ} = ( \begin{matrix}(ã_{ij,σ}^{+}) & \begin{matrix}1≤i≤d^{+} \\ 1≤j≤d \end{matrix} \\ (ã_{ij,σ}^{−}) & \begin{matrix}1≤i≤d^{−} \\ 1≤j≤d \end{matrix} \end{matrix} ) ∈ GL_{d}(E ⊗ ℂ),

then δ_{σ}(M) ∼_{E⊗K,σ} det(P_{σ})^{−1}, and the formula P̃_{σ} = P_{σ}R_{σ} implies that

δ_{σ}(M) ∼_{E⊗K,σ} det(P̃_{σ})^{−1}.

2.3. Polarizations and quadratic periods. Let M ∈ ℛ(K)_{E} be pure of weight w, and let ϵ : Γ_{K} → E^{×} be a continuous character. Let A = [ϵ] denote the corresponding rank 1 Artin motive. An A-polarization of M is a morphism of realizations

⟨, ⟩ : M ⊗_{E} M → A(−w)

which is non-degenerate in the sense that the induced map M → M^{∨}(−w)⊗_{E} A is an isomorphism of realizations. If ϵ = 1, so A(−w) = E(−w), we speak of polarizations. We will mostly be interested in polarizations, but the added versatility of an A-polarization will be useful in one particular, yet very important case, and it gives the advantage that many arguments do not need to be repeated. We will not make use of the λ-adic polarizations, so in fact we only require the existence of the Betti and de Rham polarizations, compatible as they must be (for this we could simply work in the category of Hodge-de Rham structures instead of realizations). The fact that ⟨, ⟩ is a morphism of realizations immediately implies the Hodge-Riemann bilinear relations:

• ⟨F^{p}(M_{dR}), F^{q}(M_{dR})⟩ = 0 if p + q > w;

• ⟨, ⟩ defines non-degenerate pairings M_{σ}^{pq} ⊗_{E⊗ℂ}M_{σ}^{qp} → A_{σ} ⊗ℂ whenever p + q = w. Moreover, these pairings are rational, that is, they descend to non-degenerate pairings gr^{p}(M_{dR}) ⊗_{E⊗K} gr^{q}(M_{dR}) → A_{dR}.

Let M ∈ ℛ(K)_{E} be a regular, special realization endowed with a Apolarization. Fix once and for all an E ⊗K-basis \{ζ\} of A_{dR} = (E ⊗\overline{K})^{Γ_{K}}. For each σ ∈ J_{K} and φ ∈ J_{E}, let p_{1}(σ, φ) > ⋯ > p_{d}(σ, φ) be the Hodge numbers defined in Subsection 2.1. As in Remark 2.2.3, we can choose an E ⊗ Kbasis {ω_{1,σ}, …, ω_{d,σ}} of M_{dR} such that {ω_{1,σ}, …, ω_{d^{±},σ}} is an E ⊗ K-basis of F^{±}(M_{dR}). Moreover, letting ω_{i,σ}(φ) = ω_{i,σ} ⊗_{E⊗K,φ⊗σ} 1 ∈ M_{dR} ⊗_{E⊗K,φ⊗σ} ℂ, we claim that we can choose the basis in such a way that {ω_{1,σ}(φ), …, ω_{i,σ}(φ)} is a ℂ-basis of F^{p_{i}(σ,φ)}(M_{dR}) ⊗_{E⊗K,φ⊗σ} ℂ for each i = 1, …, d and each φ ∈ J_{E}. To see this, we need some extra notation that will also be useful later. Write E ⊗ K ≅ ∏_{μ=1}^{m} K_{μ}, where the K_{μ}/K are finite extensions. Fixing the embedding σ, this decomposition induces a bijection between J_{E} and ∐_{μ=1}^{m} Hom_{σ}(K_{μ}, ℂ), where we define the latter as the sets of embeddings of K_{μ} extending σ. It’s easy to see that two embeddings φ and φ′ give rise to the same index μ if and only if φ′ = h ∘ φ for some h ∈ Aut(ℂ) such that h ∘ σ = σ. Now, notice that M_{σ}^{pq}(φ) ≠ 0 if and only if gr^{p}(M_{dR}) ⊗_{E⊗K,φ⊗σ} ℂ ≠ 0. The isomorphism

gr^{p}(M_{dR}) ⊗_{E⊗K,φ′⊗σ} ℂ ≅ gr^{p}(M_{dR}) ⊗_{E⊗K,φ⊗σ} ℂ ⊗_{ℂ,h} ℂ

implies that p_{i}(σ, φ′) = p_{i}(σ, φ) for all i = 1, …, d. This shows that we can unambiguously define p_{i}(σ, μ) by declaring them to be p_{i}(σ, φ), where φ ∈ J_{E} is an embedding corresponding to the index μ. Finally, there is a natural isomorphism M_{dR} ≃ ⨁_{μ} M_{dR} ⊗_{E⊗K} K_{μ}, so we can construct a basis of M_{dR} from a family of bases of the spaces M_{dR} ⊗_{E⊗K} K_{μ}. Once we fix μ, we can choose the corresponding basis by taking the first i elements in F^{p_{i}(σ,μ)}, which proves our claim.

For each i, σ and φ, let

I_{∞,σ,i}(φ) : M_{σ}^{p_{i}(σ,φ),q_{i}(σ,φ)}(φ) → gr^{p_{i}(σ,φ)}(M_{dR}) ⊗_{E⊗K,φ⊗σ} ℂ

be the induced isomorphism obtained from I_{∞,σ}. The image \overline{ω_{i,σ}(φ)} = \overline{ω_{i,σ}}(φ) of ω_{i,σ}(φ) in the right hand side is a ℂ-basis, and we let

Ω_{i,σ}(φ) = I_{∞,σ,i}(φ)^{−1}(\overline{ω_{i,σ}(φ)})

be the corresponding ℂ-basis of M_{σ}^{p_{i}(σ,φ),q_{i}(σ,φ)}(φ). Since I_{∞,σ}(φ)(Ω_{i})−ω_{i,σ}(φ) belongs to F^{p_{i}(σ,φ)+1}, we can write

I_{∞,σ}(φ)(Ω_{i,σ}(φ)) = ω_{i,σ}(φ) + ∑_{j=1}^{i−1} r_{ji,σ}(φ)ω_{j,σ}(φ)

for some r_{ji,σ}(φ) ∈ ℂ. We let r_{ji,σ}(φ) = 0 if j > i and r_{ii,σ}(φ) = 1. Let R_{σ}(φ) = (r_{ij,σ}(φ))_{i,j=1,...,d}, so that R_{σ}(φ) is the change of basis matrix from {I_{∞,σ}(φ)(Ω_{i,σ}(φ))}_{i=1,...,d} to {ω_{i,σ}(φ)}_{i=1,...,d}. Note that R_{σ}(φ) is an upper triangular matrix with diagonal entries 1. We let r_{ij,σ} ∈ E ⊗ ℂ be the elements whose φ-components are r_{ij,σ}(φ) for every φ, and R_{σ} be the corresponding upper triangular matrix in GL_{d}(E ⊗ ℂ) with diagonal entries 1. Under the isomorphism M_{σ} ⊗ ℂ ≃ ⨁_{φ} M_{σ}(φ), let Ω_{i,σ} ∈ M_{σ} ⊗ ℂ be the element whose φ-component is Ω_{i,σ}(φ), so that we can write

I_{∞,σ}(Ω_{i,σ}) = ω_{i,σ} + ∑_{j=1}^{i−1} r_{ji,σ}ω_{j,σ}

and R_{σ} is the change of basis matrix from {I_{∞,σ}(Ω_{i,σ})}_{i=1,...,d} to {ω_{i,σ} ⊗_{K,σ} 1}_{i=1,...,d} in M_{dR} ⊗_{K,σ} ℂ, as in Remark 2.2.3. For each i = 1, …, d, define the elements μ_{i,σ} ∈ E ⊗ K by

⟨ω_{i,σ}, ω_{d+1−i,σ}⟩_{dR} = μ_{i,σ}ζ.

If φ ∈ J_{E}, then

⟨ω_{i,σ}(φ), ω_{d+1−i,σ}(φ)⟩_{dR,σ,φ} = μ_{i,σ}(φ)ζ,

where μ_{i,σ}(φ) = (φ ⊗ σ)(μ_{i,σ}) ∈ ℂ. From the second Hodge-Riemann bilinear relation, it follows that (φ ⊗ σ)(μ_{i,σ}) ∈ ℂ^{×}. The following lemma implies that μ_{i,σ} ∈ (E ⊗ K)^{×}.

Lemma 2.3.1. Let E and K be number fields, and fix σ ∈ J_{K}. Suppose that x ∈ E ⊗ K is an element such that (φ ⊗ σ)(x) ∈ ℂ^{×} for every φ ∈ J_{E}. Then x ∈ (E ⊗ K)^{×}.

Proof. Write E ⊗ K ≅ ∏_{μ} K_{μ} as before, and let x_{μ} ∈ K_{μ} be the μ-component of x. We need to show that x_{μ} ≠ 0 for any μ. For this, it’s enough to see that σ_{μ}(x_{μ}) ∈ ℂ^{×} for some embedding σ_{μ} of K_{μ}. Since we can choose an arbitrary embedding, we can take one extending σ, so that the pair (μ, σ_{μ}) determines an embedding φ ∈ J_{E} with the property that (φ⊗σ)(x) = σ_{μ}(x_{μ}). The lemma follows. □

The first Hodge-Riemann bilinear equation implies that

⟨ω_{i,σ}(φ), ω_{d+1−i,σ}(φ)⟩_{dR,σ,φ} = ⟨I_{∞,σ}(φ)(Ω_{i,σ}(φ)), I_{∞,σ}(φ)(Ω_{d+1−i,σ}(φ))⟩_{dR,σ,φ}

for every φ, and thus

(2.3.1)  ⟨Ω_{i,σ}, Ω_{d+1−i,σ}⟩_{σ} = σ(μ_{i,σ})δ_{σ}(A)^{−1} ∈ E ⊗ ℂ.

(see Remark 2.2.1). Since the realization is regular, there exists λ_{i,σ}(φ) ∈ ℂ^{×} for every i, σ and φ, such that

F_{σ,ℂ}(φ)(Ω_{i,σ}(φ)) = λ_{i,σ}(φ)Ω_{d+1−i,σ}(φ).

Applying F_{σ,ℂ}(φ) a second time, we get that λ_{i,σ}(φ)λ_{d+1−i,σ}(φ) = 1. In particular, if d is odd, λ_{(d+1)/2,σ}(φ) = ±1, and the sign is independent of φ, because in fact it is the scalar ε by which F_{σ,ℂ} acts on M_{σ}^{w/2,w/2}. Thus, if we let λ_{i,σ} ∈ (E ⊗ ℂ)^{×} be the elements whose φ-components are λ_{i,σ}(φ), then λ_{i,σ}λ_{d+1−i,σ} = 1, and λ_{(d+1)/2,σ} = ε = ±1 if d is odd. For i = 1, …, d, let Q_{i,σ} ∈ E ⊗ ℂ be defined by the formula

Q_{i,σ} = ⟨Ω_{i,σ}, F_{σ,ℂ}(Ω_{i,σ})⟩_{σ}.

From the definition of the scalars λ_{i,σ} and (2.3.1), it follows that

Q_{i,σ} = λ_{i,σ}σ(μ_{i,σ})δ_{σ}(A)^{−1}

and it’s an element of ∈ (E ⊗ ℂ)^{×}. Moreover,

Q_{i,σ}^{−1}F_{σ,ℂ}(Ω_{i,σ}) = σ(μ_{i,σ}^{−1})δ_{σ}(A)Ω_{d+1−i,σ}.

Lemma 2.3.2. Suppose that A = E(0) is trivial. Let 0 ≤ r < s ≤ d be integers such that d = r + s. Then

∏_{i=1}^{r} Q_{i,σ} ∼_{E⊗K,σ} ∏_{i=1}^{s} Q_{i,σ}.

Proof. We write ∼ for ∼_{E⊗K,σ} throughout. Since Q_{i,σ} ∼ λ_{i,σ} and λ_{i,σ}λ_{d+1−i,σ} = 1, it follows that Q_{i,σ} ∼ Q_{d+1−i,σ}^{−1}. If r + 1 ≤ i ≤ s, then r + 1 ≤ d + 1 − i ≤ s as well. Finally, Q_{(d+1)/2,σ} ∼ 1 if d is odd, because in this case λ_{(d+1)/2,σ} = ±1. This is enough to prove the lemma. □

Proposition 2.3.1. Let M ∈ ℛ(K)_{E} be a regular, special realization of weight w and rank d, endowed with an A-polarization, and let σ ∈ J_{K}. Then

c_{σ}^{+}(M)c_{σ}^{−}(M) ∼_{E⊗K,σ} (2πi)^{−dw}δ_{σ}(M)^{−1}δ_{σ}(A)^{d+⌊d/2⌋} ∏_{j=1}^{⌊d/2⌋} Q_{j,σ}.

Proof. We pick up the notation from Remark 2.2.3, so that δ_{σ}(M) ∼ det(P̃_{σ})^{−1}. Now, recalling that Q_{j,σ} = λ_{j,σ}σ(μ_{j,σ})δ_{σ}(A)^{−1}, by applying F_{σ,ℂ} to (2.2.4) we obtain that

(2.3.2) ã_{i,d+1−j,σ}^{±} = ±δ_{σ}(A)^{−1}σ(μ_{j,σ})Q_{j,σ}^{−1}ã_{ij,σ}^{±} i = 1, …, d^{±}, j = 1 …, d.

It follows that δ_{σ}(M)^{−1} ∼ δ_{σ}(A)^{−⌊d/2⌋} ∏_{j=1}^{⌊d/2⌋} Q_{j,σ}^{−1} det(H_{σ}), where

H_{σ} = ( \begin{matrix}(ã_{ij,σ}^{+})\begin{matrix}1≤i≤d^{+} \\ 1≤j≤d^{+} \end{matrix} & (ã_{i,d+1−j,σ}^{+})\begin{matrix}1≤i≤d^{+} \\ d^{+}+1≤j≤d \end{matrix} \\ (ã_{ij,σ}^{−})\begin{matrix}1≤i≤d^{−} \\ 1≤j≤d^{+} \end{matrix} & (−ã_{i,d+1−j,σ}^{−})\begin{matrix}1≤i≤d^{−} \\ d^{+}+1≤j≤d \end{matrix} \end{matrix} )

if either d is even or d = 2m − 1 is odd and d^{−} = m − 1, and

H_{σ} = ( \begin{matrix}(ã_{ij,σ}^{+})\begin{matrix}1≤i≤m−1 \\ 1≤j≤m \end{matrix} & (ã_{i,d+1−j,σ}^{+})\begin{matrix}1≤i≤m−1 \\ m+1≤j≤d \end{matrix} \\ (ã_{ij,σ}^{−})\begin{matrix}1≤i≤m \\ 1≤j≤m \end{matrix} & (−ã_{i,d+1−j,σ}^{−})\begin{matrix}1≤i≤m \\ m+1≤j≤d \end{matrix} \end{matrix} )

if d = 2m − 1 is odd and d^{−} = m. Suppose first that d = 2m is even. Then, applying elementary column operations, we can take H_{σ} to the matrix

( \begin{matrix}(2ã_{ij,σ}^{+})\begin{matrix}1≤i≤d^{+} \\ 1≤j≤d^{+} \end{matrix} & (ã_{i,d+1−j,σ}^{+})\begin{matrix}1≤i≤d^{+} \\ d^{+}+1≤j≤d \end{matrix} \\ (0)\begin{matrix}1≤i≤d^{−} \\ 1≤j≤d^{+} \end{matrix} & (−ã_{i,d+1−j,σ}^{−})\begin{matrix}1≤i≤d^{−} \\ d^{+}+1≤j≤d \end{matrix} \end{matrix} ) .

It follows that (2.3.3)

δ_{σ}(M)^{−1} ∼ δ_{σ}(A)^{−⌊d/2⌋} ∏_{j=1}^{⌊d/2⌋} Q_{j,σ}^{−1} · det ((ã_{ij,σ}^{+})_{i,j=1}^{d^{+}}) · det ((ã_{ij,σ}^{−})_{i,j=1}^{d^{−}}) .

Suppose now that d = 2m − 1 is odd and that d^{−} = m − 1. Then we can take H_{σ} to

( \begin{matrix}(2ã_{ij,σ}^{+})\begin{matrix}1≤i≤m \\ 1≤j≤m−1 \end{matrix} & (ã_{i,m,σ}^{+})_{1≤i≤m} & (ã_{i,m−j,σ}^{+})\begin{matrix}1≤i≤m \\ 1≤j≤m−1 \end{matrix} \\ (0)\begin{matrix}1≤i≤m−1 \\ 1≤j≤m−1 \end{matrix} & (ã_{i,m,σ}^{−})_{1≤i≤m−1} & (−ã_{i,m−j,σ}^{−})\begin{matrix}1≤i≤m−1 \\ 1≤j≤m−1 \end{matrix} \end{matrix} ) .

But we also have that F_{σ,ℂ}(Ω_{m,σ}) = λ_{m,σ}Ω_{m,σ}, with λ_{m,σ} = ±1. Since we are assuming that d^{+} = m, it follows that λ_{m,σ} = ε = 1. This implies that a_{i,m,σ}^{−} = 0 for every i = 1, …, m − 1, and thus we obtain (2.3.3). The case where d^{−} = m is analyzed in a similar way, and in all cases we obtain the formula (2.3.3). Now, it follows from (2.2.5) that c_{σ}^{±}(M^{∨}) ∼ det ((ã_{ij,σ}^{±})_{i,j,=1}^{d^{±}}). Finally, the A-polarization defines an isomorphism M^{∨} ⊗_{E} A ≅ M(w), and since

c_{σ}^{+}(M^{∨} ⊗_{E} A)c_{σ}^{−}(M^{∨} ⊗_{E} A) ∼ c_{σ}^{+}(M^{∨})c_{σ}^{−}(M^{∨})δ_{σ}(A)^{d}

and

c_{σ}^{+}(M(w))c_{σ}^{−}(M(w)) ∼ c_{σ}^{+}(M)c_{σ}^{−}(M)(2πi)^{wd},

it follows that

δ_{σ}^{−1}(M) ∼ ∏_{j=1}^{[d/2]} Q_{j,σ}^{−1}c_{σ}^{+}(M)c_{σ}^{−}(M)(2πi)^{wd}δ_{σ}(A)^{−d−⌊d/2⌋},

which is the expression we wanted to obtain. □

Remark 2.3.1. The fact that M has an A-polarization implies that

(2.3.4)  δ_{σ}(M)^{2} ∼_{E⊗K,σ} δ_{σ}(A)^{d}(2πi)^{−wd}.

Assume, moreover, that M is regular and special. Then this expression can also be written in terms of discriminants as in 1.4.12, [Har97]. One can even take the square root on both sides when the pairing ⟨, ⟩ is alternated (and hence d is even) to get δ_{σ}(M) ∼ δ_{σ}(A)^{d/2}(2πi)^{−wd/2}.

2.4. The motives RM(χ). Let L/K be a totally imaginary quadratic extension of the totally real field K. Let χ be an algebraic Hecke character of L with infinity type (n_{τ})_{τ∈J_{L}}. Let RM(χ) = R_{L/K}M(χ) ∈ ℛ(K)_{ℚ(χ)}. Then RM(χ) is of rank 2 over ℚ(χ), pure of weight w(χ). Note that given σ ∈ J_{K} and ρ ∈ J_{ℚ(χ)}, RM(χ)_{σ}^{pq}(ρ) ≠ 0 if and only if (p, q) = (n(τ, ρ), n(τ, \overline{ρ})) or (p, q) = (n(τ, \overline{ρ}), n(τ, ρ)). Moreover, RM(χ) is regular if and only if n_{τ} ≠ n\overline{τ} for every τ ∈ J_{L}, in which case we call χ critical. The numbers p_{i}^{χ}(σ, ρ) (i = 1, 2) for the motive RM(χ) are given by \{p_{1}^{χ}(σ, ρ), p_{2}^{χ}(σ, ρ)\} = \{n(τ, ρ), n(\overline{τ}, ρ)\}. For each σ ∈ J_{K} and ρ ∈ J_{ℚ(χ)}, define

t_{σ,ρ}(χ) = p_{1}^{χ}(σ, ρ) − p_{2}^{χ}(σ, ρ).

Let ε_{L} : 𝔸_{K}^{×}/K^{×} → {±1} denote the quadratic character attached to L/K. Since K is totally real, χ|_{𝔸_{K}^{×}/K^{×}} = χ_{0}∥ · ∥^{−w(χ)}, where χ_{0} is of finite order. It’s easy to see, as in (1.6.2) of [Har97], that Λ_{ℚ(χ)}^{2}RM(χ) ≅ [χ_{0}ε_{L}](−w(χ)). This map defines a morphism

(2.4.1)  ⟨, ⟩ : RM(χ) ⊗_{ℚ(χ)} RM(χ) → [χ_{0}ε_{L}](−w(χ))

which is alternated and non-degenerate; in particular, it is an A-polarization of RM(χ), with A = [χ_{0}ε_{L}]. For every σ ∈ J_{K}, we let δ_{σ}(χ) = δ_{σ}(RM(χ)). By Remark 2.3.1, δ_{σ}(χ) ∼_{ℚ(χ)⊗K,σ} (2πi)^{−w(χ)}δ_{σ}[χ_{0}ε_{L}]. Assume from now on that χ is critical, and let c_{σ}^{±}(χ) = c_{σ}^{±}(RM(χ)). Thus, by Proposition 2.3.1,

(2.4.2)  c_{σ}^{+}(χ)c_{σ}^{−}(χ) ∼_{ℚ(χ)⊗K,σ} (2πi)^{−2w(χ)}δ_{σ}(χ)^{−1}δ_{σ}[χ_{0}ε_{L}]^{3}Q_{σ}(χ) ∼
(2πi)^{−w(χ)}δ_{σ}[χ_{0}ε_{L}]^{2}Q_{σ}(χ).

The element Q_{σ}(χ) is defined as in Subsection 2.3. To be more precise and set up some further notation, we recall its definition. We take a ℚ(χ) ⊗ Kbasis {ω_{σ}(χ), ω′_{σ}(χ)} of RM(χ)_{dR} with the property that {ω_{σ}(χ)} is a basis of F^{±}(RM(χ)_{dR}). For every ρ ∈ J_{ℚ(χ)}, we let Ω_{σ}(χ)(ρ) = I_{∞,σ,1}(ρ)^{−1}(\overline{ω_{σ}(χ)(ρ)}) and Ω′_{σ}(χ)(ρ) = I_{∞,σ,2}(ρ)^{−1}(\overline{ω′_{σ}(χ)(ρ)}), and we let Ω_{σ}(χ), Ω′_{σ}(χ) ∈ RM(χ)_{σ}⊗ℂ be the elements whose ρ-components are the Ω_{σ}(χ)(ρ) and Ω′_{σ}(χ)(ρ) respectively. Then Q_{σ}(χ) = ⟨Ω_{σ}(χ), F_{σ,ℂ}(Ω_{σ}(χ))⟩_{σ}. We also write

I_{∞,σ}(Ω′_{σ}(χ)) = x_{σ}(χ)ω_{σ}(χ) + ω′_{σ}(χ)

with x_{σ}(χ) ∈ E ⊗ ℂ. Let χ be critical. For τ ∈ J_{L} and ρ ∈ J_{ℚ(χ)}, let e_{τ,ρ} = 1 if n(τ, ρ) > n(\overline{τ}, ρ) and e_{τ,ρ} = −1 if n(τ, ρ) < n(\overline{τ}, ρ). Let e_{τ} = (e_{τ,ρ})_{ρ∈J_{ℚ(χ)}} ∈ (ℚ(χ) ⊗ ℂ)^{×}. Note that e_{τ} = −e\overline{τ}, and in particular, e_{τ} ∼_{ℚ(χ)} e\overline{τ}. See also H.4, [Bla97].

Lemma 2.4.1. Let τ ∈ J_{L} and σ = τ|_{K}. Then

c_{σ}^{−}(χ) ∼_{ℚ(χ)⊗K,σ} e_{τ} c_{σ}^{+}(χ).

Proof. Let \{γ_{τ} \} be a ℚ(χ)-basis of M(χ)_{τ}, and let γ\overline{τ} = F_{τ} (γ_{τ}), a basis of M(χ)\overline{τ}. We can then form a basis \{(γ_{τ} , 0), (0, γ\overline{τ})\} of RM(χ)_{σ} = M(χ)_{τ} ⊕ M(χ)\overline{τ}. Then \{(γ_{τ} , γ\overline{τ})\} (resp. \{(γ_{τ} , −γ\overline{τ})\}) is a ℚ(χ)-basis of RM(χ)_{σ}^{+} (resp. RM(χ)_{σ}^{−}). Write

I_{∞,σ}((γ_{τ} , 0)_{ℂ}) = a_{σ}ω_{σ}(χ) + b_{σ}ω′_{σ}(χ),
I_{∞,σ}((0, γ\overline{τ})_{ℂ}) = c_{σ}ω_{σ}(χ) + d_{σ}ω′_{σ}(χ),

with a_{σ}, b_{σ}, c_{σ} and d_{σ} in ℚ(χ) ⊗ ℂ. It follows that c_{σ}^{+}(χ) ∼_{ℚ(χ)⊗K,σ} b_{σ} + d_{σ} and c_{σ}^{−}(χ) ∼_{ℚ(χ)⊗K,σ} b_{σ} − d_{σ}. Now fix ρ ∈ J_{ℚ(χ)}. Suppose that e_{τ,ρ} = −1, so that n(τ, ρ) < n(\overline{τ}, ρ), p_{1}^{χ}(σ, ρ) = n(\overline{τ}, ρ) and F^{±}(RM(χ)_{dR}⊗_{ℚ(χ)⊗K,ρ⊗σ}ℂ = F^{n(\overline{τ},ρ)}⊗_{ℚ(χ)⊗K,ρ⊗σ}ℂ. Then γ_{\overline{τ},ρ} ∈ M(χ)\overline{τ} (ρ) contributes to the (n(\overline{τ}, ρ), n(τ, ρ)) Hodge component, so it is in F^{p_{1}}, and thus d_{σ,ρ} = 0. Similarly, if e_{τ,ρ} = 1, then b_{σ,ρ} = 0. This proves that b_{σ} + d_{σ} = e_{τ} (b_{σ} − d_{σ}), which is what we wanted. □

For future reference, let a_{σ}^{±}(χ) ∼ c_{σ}^{±}(RM(χ)^{∨}). We deduce the following formulas (where τ is any element of J_{L} extending σ):

(2.4.3)  Q_{σ}(χ) ∼_{ℚ(χ)⊗K,σ} (2πi)^{w(χ)}δ_{σ}[χ_{0}ε_{L}]^{−2}e_{τ} c_{σ}^{+}(χ)^{2}.
(2.4.4)  a_{σ}^{−}(χ) ∼_{ℚ(χ)⊗K,σ} e_{τ} a_{σ}^{+}(χ).

The following lemma will be useful to ignore integer powers of δ_{σ}[ε_{L}] if one is willing to work over the Galois closure L′ of L.

Lemma 2.4.2. For any σ ∈ J_{K}, δ_{σ}[ε_{L}] ∈ L′ ⊂ ℂ.

Proof. By the calculations in Remark 2.2.1, we can take δ_{σ}[ε_{L}] = σ̃(ζ)^{−1}. Here σ̃ is an extension of σ to an embedding of \overline{K} in ℂ, and ζ is a K-basis of (ℚ⊗\overline{K})^{Γ_{K}}, where γ ∈ Γ_{K} act by sending q ⊗z (q ∈ ℚ, z ∈ \overline{K}) to ε_{L}(γ)q ⊗γ(z). Thus, we can take ζ = 1 ⊗ α, where α ∈ L^{×} satisfies ι(α) = −α, which shows that σ̃(ζ)^{−1} is in L′. □

2.5. Periods of Hecke twists. Let L/K be a CM extension, E′ any number field, M ∈ ℛ(K)_{E} and N ∈ ℛ(L)_{E′}. Denote by RN = R_{L/K}N ∈ ℛ(K)_{E′} (restriction of scalars).

Lemma 2.5.1. Suppose that M and N are pure. Then M ⊗ RN has critical values if and only if for every σ ∈ J_{K}, (M ⊗ RN)_{σ}^{pp} = 0 for every p ∈ ℤ.

Proof. For simplicity, we will assume that E ⊗ E′ is a field. Let w and w_{N} denote the weights of M and N respectively. Note that if σ ∈ J_{K}, then

(M ⊗ RN)_{σ}^{pq} = ⨁_{a+b=p} M_{σ}^{a,w−a} ⊗_{ℂ} RN_{σ}^{b,w_{N} −b}.

Suppose that, for some p ∈ ℤ and σ ∈ J_{K}, (M ⊗ RN)_{σ}^{pp} ≠ 0. Then there exists a pair (a, b) with a + b = p = w − a + w_{N} − b and M_{σ}^{a,w−a} ⊗_{ℂ} RN_{σ}^{b,w_{N} −b} ≠ 0.

Two possibilities arise. First, suppose that w − a ≠ a. Then w_{N} − b ≠ b, and there are two different summands

(M ⊗ RN)_{σ}^{pp} ⊃ (M_{σ}^{a,w−a} ⊗_{ℂ} RN_{σ}^{b,w_{N} −b}) ⊕ (M_{σ}^{w−a,a} ⊗_{ℂ} RN_{σ}^{w_{N} −b,b}) .

These two nonzero summands are interchanged by F_{σ} acting on (M ⊗ RN)_{σ}^{pp}, and thus F_{σ} does not act on (M⊗RN)_{σ}^{pp} as a scalar. In particular, M⊗RN does not have critical values, because it’s not special. Now suppose that w−a = a, so that w_{N} −b = b as well. There is a nonzero F_{σ}-stable summand M_{σ}^{a,a}⊗_{ℂ}RN_{σ}^{b,b}, but by definition of RN, this splits as

(M_{σ}^{a,a} ⊗_{ℂ} N_{τ}^{b,b}) ⊕ (M_{σ}^{a,a} ⊗_{ℂ} N b,b ; \overline{τ}) ,

where τ, \overline{τ} are the two embeddings of L over σ; also here, Frobenius interchanges these two nonzero summands, and again this implies that it does not act on (M ⊗ RN)_{σ}^{pp} as scalars. For the converse, it’s easily seen (using, for instance, (1.3.1) of [Del79a]) that (M ⊗ RN)(t) is critical, with t = ⌊\frac{w+w(χ)+1}{2}⌋ (the symbol ⌊, ⌋ denotes the floor function). □

Let χ be an algebraic Hecke character χ of L of infinity type (n_{τ})_{τ∈J_{L}}. From now on, unless otherwise stated, we will assume for simplicity that E ⊗ ℚ(χ) is a field.

Proposition 2.5.1. Suppose that M ∈ ℛ(K)_{E} is regular. Then M ⊗ RM(χ) has critical values if and only if, for every σ ∈ J_{K} and every ρ ∈ J_{ℚ(χ)}, t_{σ,ρ}(χ) is not equal to any of the values w − 2p_{i}(σ, φ) (for any i = 1, …, d and any φ ∈ J_{E}).

Proof. For σ ∈ J_{K}, ρ ∈ J_{ℚ(χ)} and φ ∈ J_{E}, (2.5.1)

(M ⊗RM(χ))_{σ}^{pq}(φ⊗ρ) = ⨁_{i=1}^{d} M_{σ}^{p_{i}(σ,φ),q_{i}(σ,φ)}(φ)⊗_{ℂ}RM(χ)_{σ}^{p−p_{i}(σ,φ),q−q_{i}(σ,φ)}(ρ).

The proposition follows from this and Lemma 2.5.1. □

If M is regular, σ ∈ J_{K}, φ ∈ J_{E}, and i = 0, …, d, consider the interval I_{i}(σ, φ) = (w − 2p_{i}(σ, φ), w − 2p_{i+1}(σ, φ)), where for consistency of notation we take p_{0}(σ, φ) = +∞ and p_{d}(σ, φ) = −∞. By the last proposition, if M is regular and M ⊗ RM(χ) has critical values, then for each σ ∈ J_{K}, φ ∈ J_{E} and ρ ∈ J_{ℚ(χ)}, there is a number r = r_{σ,φ,ρ}(χ) ∈ {0, …, d} such that t_{σ,ρ}(χ) ∈ I_{r}(σ, φ). Write E ⊗ ℚ(χ) ⊗ K ≅ ∏_{μ=1}^{m} K_{μ}, where the K_{μ}/K are finite extensions. By the same reasoning as in Subsection 2.3, we can unambiguously define p_{i}(σ, μ), p_{i}^{χ}(σ, μ) and r_{σ,μ}(χ), by declaring them to be p_{i}(σ, φ), p_{i}^{χ}(σ, ρ) and r_{σ,φ,ρ}(χ) respectively, where (φ, ρ) is any pair of embeddings corresponding to the index μ. This will be useful later to construct a suitable rational basis of (M ⊗ RM(χ))_{dR}.

Retaining the hypothesis that M and χ are regular, with M ⊗ RM(χ) having critical values, we can determine the set of critical integers of M ⊗ RM(χ) as follows. Fix σ ∈ J_{K}, φ ∈ J_{E} and ρ ∈ J_{ℚ(χ)}. Then (2.5.1) shows that

{(p, q) : h_{σ}^{pq}(φ ⊗ ρ) ≠ 0, p < q} =
{(p_{i}(σ, φ) + p_{1}^{χ}(σ, ρ), q_{i}(σ, φ) + p_{2}^{χ}(σ, ρ))}_{i=r+1}^{d} ∪
{(p_{i}(σ, φ) + p_{2}^{χ}(σ, ρ), q_{i}(σ, φ) + p_{1}^{χ}(σ, ρ))}_{i=d+1−r}^{d} ,

where r = r_{σ,φ,ρ}(χ). Note that the two sets in the union may have non-trivial intersection. However, the elements in the intersection contribute with h_{σ}^{pq}(φ⊗ ρ) = 2, while the elements that are not in the intersection contribute with h_{σ}^{pq}(φ ⊗ ρ) = 1. It follows from the construction of L_{∞}(φ ⊗ ρ, M ⊗ RM(χ), s) that this equals

∏_{σ∈J_{K}} ( ∏_{i=r_{σ,φ,ρ}(χ)+1}^{d} Γ_{ℂ}(s − p_{i}(σ, φ) − p_{1}^{χ}(σ, ρ))

∏_{i=d+1−r_{σ,φ,ρ}(χ)}^{d} Γ_{ℂ}(s − p_{i}(σ, φ) − p_{2}^{χ}(σ, ρ))) .

Thus, letting

υ_{φ,ρ}^{(1)}(χ) = max{p_{r_{σ,φ,ρ}(χ)+1}(σ, φ)+p_{1}^{χ}(σ, ρ), p_{d+1−r_{σ,φ,ρ}(χ)}(σ, φ)+p_{2}^{χ}(σ, ρ)}_{σ∈J_{K}} ,

the set of poles of L_{∞}(φ ⊗ ρ, M ⊗ RM(χ), s) consists of the set of integers m such that m ≤ υ_{φ,ρ}^{(1)}(χ) Similarly, letting

υ_{φ,ρ}^{(2)}(χ) = min{p_{r_{σ,φ,ρ}(χ)}(σ, φ) + p_{1}^{χ}(σ, ρ), p_{d−r_{σ,φ,ρ}(χ)}(σ, φ) + p_{2}^{χ}(σ, ρ)}_{σ∈J_{K}} ,

the set of poles of L_{∞}(φ⊗ρ, (M ⊗RM(χ))^{∨}, 1−s) consists of the set of integers m such that m ≥ υ_{φ,ρ}^{(2)}(χ) + 1. We conclude that the set of critical integers of M ⊗ RM(χ) consists of the set of integers m such that

(2.5.2)  υ_{φ,ρ}^{(1)}(χ) < m ≤ υ_{φ,ρ}^{(2)}(χ).

The following theorem is the main result of this section, and it generalizes Proposition 1.7.6 of [Har97]. The notation ⌈d/2⌉ refers to the ceiling function.

Theorem 2.5.1. Let M be a regular, special realization over K with coefficients in E, of rank d and weight w, endowed with a polarization. Suppose that χ is a critical algebraic Hecke character of L of weight w(χ). Assume that M⊗RM(χ) has critical values. Let r_{σ} = (r_{σ,φ,ρ}(χ))_{φ⊗ρ} and s_{σ} = (d − r_{σ,φ,ρ}(χ))_{φ⊗ρ}. Then

c_{σ}^{+}(M ⊗ RM(χ)) ∼

(2πi)^{−⌈d/2⌉w(χ)}δ_{σ}([χ_{0}ε_{L}])^{r_{σ}}δ_{σ}(M)a_{σ}^{∗}(χ)Q_{σ}(χ)^{r_{σ}−⌈d/2⌉} ∏_{j=1}^{s_{σ}} Q_{j,σ},

where a_{σ}^{∗}(χ) = 1 if d is even, and a_{σ}^{∗}(χ) = a_{σ}^{±}(χ) if d is odd, with ± = − if d^{+} > d^{−} and ± = + if d^{−} > d^{+}. In the formula, ∼ means ∼_{(E⊗ℚ(χ))⊗K,σ}.

Remark 2.5.1. If K = ℚ, then r_{1} = (r_{1,φ,ρ}), and the integers r_{1,φ,ρ} are actually independent of the embeddings. This extra simplicity comes from the fact that the nonzero Hodge components of M and RM(χ) are free of rank 1 over E ⊗ ℂ and ℚ(χ) ⊗ ℂ respectively in this case (see Proposition 2.5 of [Del79a]), which is not true in general if [K : ℚ] > 1.

Proof. Recall that we are assuming that E⊗ℚ(χ) is a field; the general case basically follows from this case. The polarization of M and the [χ_{0}ε_{L}]-polarization of RM(χ) (2.4.1), define an A-polarization

M ⊗ RM(χ) ≅ (M ⊗ RM(χ))^{∨} ⊗_{E⊗ℚ(χ)} [χ_{0}ε_{L}](−w − w(χ))

in ℛ(K)_{E⊗ℚ(χ)}, with A = [χ_{0}ε_{L}]. Note that (χ_{0}ε_{L})(c_{σ}) = (−1)^{w(χ)+1} for any complex conjugation c_{σ} ∈ Γ_{K} attached to any σ ∈ J_{K}. Hence, the motive [χ_{0}ε_{L}] is special, and by Remark 2.2.1, we can write (2.5.3)

c_{σ}^{+}(M ⊗ RM(χ)) ∼ (2πi)^{(−w−w(χ))d}δ_{σ}[χ_{0}ε_{L}]^{d}c_{σ}^{(−1)^{w+1}}((M ⊗ RM(χ))^{∨}).

Here, and in the rest of the proof, we write ∼ for ∼_{(E⊗ℚ(χ))⊗K,σ}. For σ ∈ J_{K}, φ ∈ J_{E} and ρ ∈ J_{ℚ(χ)}, denote by I_{∞,σ}(φ ⊗ ρ) the comparison isomorphism between (M ⊗ RM(χ))_{σ}(φ ⊗ ρ) and (M ⊗ RM(χ))_{dR} ⊗_{(E⊗ℚ(χ))⊗K,φ⊗ρ⊗σ} ℂ. Note that d^{±}(M ⊗ RM(χ)) = d, F^{+}((M ⊗ RM(χ))_{dR}) = F^{−}((M ⊗ RM(χ)_{dR}), and

I_{∞,σ}(φ, ρ)^{−1}(F^{±}((M ⊗ RM(χ))_{dR}) ⊗_{(E⊗ℚ(χ))⊗K,φ⊗ρ⊗σ} ℂ) =

( ⨁_{i=1}^{r_{σ,φ,ρ}(χ)} M_{σ}^{p_{i}(σ,φ),q_{i}(σ,φ)}(φ) ⊗_{ℂ} RM(χ)_{σ}^{p_{1}^{χ}(σ,ρ),p_{2}^{χ}(σ,ρ)}(ρ)) ⊕

( ⨁_{i=1}^{d−r_{σ,φ,ρ}(χ)} M_{σ}^{p_{i}(σ,φ),q_{i}(σ,φ)}(φ) ⊗_{ℂ} RM(χ)_{σ}^{p_{2}^{χ}(σ,ρ),p_{1}^{χ}(σ,ρ)}(ρ)) . (2.5.4)

Also,

(M ⊗ RM(χ))_{σ}^{+} = (M_{σ}^{+} ⊗ RM(χ)_{σ}^{+}) ⊕ (M_{σ}^{−} ⊗ RM(χ)_{σ}^{−}) ,
(M ⊗ RM(χ))_{σ}^{−} = (M_{σ}^{−} ⊗ RM(χ)_{σ}^{+}) ⊕ (M_{σ}^{+} ⊗ RM(χ)_{σ}^{−}) .

Choose E-bases {e_{i,σ}}_{i=1,...,d^{+}}, {f_{i,σ}}_{i=1,...,d^{−}} of M_{σ}^{+} and M_{σ}^{−} respectively, and choose ℚ(χ)-bases {e_{σ}(χ)} and {f_{σ}(χ)} of RM(χ)_{σ}^{+} and RM(χ)_{σ}^{−} respectively. Also, choose an E ⊗ K-bases {ω_{i,σ}}_{i=1,...,d} of M_{dR} as in Subsection 2.3. Take as well a ℚ(χ) ⊗ K-basis {ω_{σ}(χ), ω′_{σ}(χ)} as in Subsection 2.4. In particular, {ω_{σ}(χ)(ρ)} is a ℂ-basis of RM(χ)_{σ}^{p_{1}^{χ}(σ,ρ),p_{2}^{χ}(σ,ρ)}(ρ). For each i = 1, …, d, we let

ω_{i,σ}(χ) = ω_{i,σ} ⊗ ω_{σ}(χ), ω′_{i,σ}(χ) = ω_{i,σ} ⊗ ω′_{σ}(χ).

Let {e_{i,σ} ⊗ e_{σ}(χ)}_{i=1,...,d^{+}} ∪ {f_{i,σ} ⊗ f_{σ}(χ)}_{i=1,...,d^{−}} and

{e_{i,σ} ⊗ f_{σ}(χ)}_{i=1,...,d^{+}} ∪ {f_{i,σ} ⊗ e_{σ}(χ)}_{i=1,...,d^{−}}

be the E ⊗ ℚ(χ)-bases of (M ⊗ RM(χ))_{σ}^{+} and (M ⊗ RM(χ))_{σ}^{−} respectively, obtained from the chosen bases for M and RM(χ). Their union is a basis of (M ⊗ RM(χ))_{σ}. Consider also the E ⊗ ℚ(χ) ⊗ K-basis of (M ⊗ RM(χ))_{dR} given by {ω_{i,σ}(χ), ω′_{i,σ}(χ)}_{i=1,...,d}. We want to use Remark 2.2.3 to compute the Deligne periods of the dual of M ⊗RM(χ). For that we need an appropriate basis of its de Rham realization which contains a basis of the corresponding ±-step. Using (2.5.4), we see that the basis {ω_{i,σ}(χ), ω′_{i,σ}(χ)}_{i=1,...,d} does not exactly meet this in the first place, since the indexing for each embedding φ⊗ρ will depend on r_{σ,φ,ρ}(χ), which may change for different choices. However, we can form another basis by reordering this basis for each embedding, in the following way. Recall the notation E ⊗ ℚ(χ) ⊗ K ≅ ∏_{μ} K_{μ} introduced before, so that we can speak of the numbers r_{μ} = r_{σ,μ}(χ). We can write (M ⊗ RM(χ))_{dR} = ⨁_{μ}(M ⊗ RM(χ))_{dR,μ}. For each μ, if x ∈ (M ⊗ RM(χ))_{dR}, x_{μ} will denote the element x ⊗_{E⊗ℚ(χ)⊗K} 1_{K_{μ}}. Consider the ordered basis

{ω_{i,σ}(χ)_{μ}}_{i=1}^{r_{μ}} ∪ {ω′_{i,σ}(χ)_{μ}}_{i=1}^{d−r_{μ}} ∪ {ω_{i,σ}(χ)_{μ}}_{i=r_{μ}+1}^{d} ∪ {ω′_{i,σ}(χ)_{μ}}_{i=d+1−r_{μ}}^{d} .

These bases define an E ⊗ ℚ(χ) ⊗ K-basis {γ_{1,σ}, …, γ_{2d,σ}} of M ⊗ RM(χ)_{dR} with the property that the first d elements form a basis of F^{±}, because the decomposition (2.5.4) actually takes place over K_{μ} (where μ corresponds to (φ, ρ)). Let Ω_{i,σ}, Ω_{σ}(χ) and Ω′_{σ}(χ) be the elements constructed in Subsection 2.3. Let Ω_{i,σ}(χ) = Ω_{i,σ} ⊗ Ω_{σ}(χ) and Ω′_{i,σ}(χ) = Ω_{i,σ} ⊗ Ω′_{σ}(χ). Consider the ordered ℂ-basis of (M ⊗ RM(χ))_{σ}(φ ⊗ ρ) given by

{Ω_{i,σ}(χ)(φ ⊗ ρ)}_{i=1}^{r_{φ,ρ}} ∪ {Ω′_{i,σ}(χ)(φ ⊗ ρ)}_{i=1}^{d−r_{φ,ρ}} ∪
{Ω_{i,σ}(χ)(φ ⊗ ρ)}_{i=r_{φ,ρ}+1}^{d} ∪ {Ω′_{i,σ}(χ)(φ ⊗ ρ)}_{i=d+1−r_{φ,ρ}}^{d} ,

where r_{φ,ρ} = r_{σ,φ,ρ}(χ). Name the elements of this ordered basis as

{Γ_{1,σ}(φ ⊗ ρ), …, Γ_{2d,σ}(φ ⊗ ρ)}.

Putting these bases together, we get an E ⊗ ℚ(χ) ⊗ ℂ-basis {Γ_{1,σ}, …, Γ_{2d,σ}} of (M ⊗ RM(χ))_{σ} ⊗ ℂ. It’s easy to see, given our choices, that {I_{∞,σ}(Γ_{1,σ}), …, I_{∞,σ}(Γ_{2d,σ})} is obtained from {γ_{1,σ}, …, γ_{2d,σ}} by means of an upper triangular matrix with diagonal entries 1, as in Remark 2.2.3. Thus, to compute c_{σ}^{±}((M ⊗ RM(χ))^{∨}), we can use the basis {Γ_{1,σ}, …, Γ_{2d,σ}}. We write

Ω_{j,σ} = ∑_{i=1}^{d^{+}} ã_{ij,σ}^{+}e_{i,σ} + ∑_{i=1}^{d−} ã_{ij,σ}^{−}f_{i,σ}, j = 1, …, d, (2.5.5)

Ω_{σ}(χ) = ã_{σ}^{+}(χ)e_{σ}(χ) + ã_{σ}^{−}(χ)f_{σ}(χ), (2.5.6)

Ω′_{σ}(χ) = b̃_{σ}^{+}(χ)e_{σ}(χ) + b̃_{σ}^{−}(χ)f_{σ}(χ),

(2.5.7)  Γ_{j,σ} = ∑_{i=1}^{d^{+}} b_{ij,σ}^{+}e_{i,σ} ⊗ e_{σ}(χ) + ∑_{i=1}^{d−} c_{ij,σ}^{+}f_{i,σ} ⊗ f_{σ}(χ)+

∑_{i=1}^{d^{+}} b_{ij,σ}^{−}e_{i,σ} ⊗ f_{σ}(χ) + ∑_{i=1}^{d^{−}} c_{ij,σ}^{−}f_{i,σ} ⊗ e_{σ}(χ).

Then

c_{σ}^{±}((M ⊗ RM(χ))^{∨}) ∼ det T_{σ}^{±},

where

T_{σ}^{±} = ( \begin{matrix}(b_{ij,σ}^{±}) & \begin{matrix}i=1,...,d^{+} \\ j=1,...,d \end{matrix} \\ (c_{ij,σ}^{±}) & \begin{matrix}i=1,...,d^{−} \\ j=1,...,d \end{matrix} \end{matrix} ) .

We now compute each φ ⊗ ρ-component det(T_{σ}^{±})_{φ,ρ} = det(T_{σ,φ,ρ}^{±}). By the shape of the basis {Γ_{1,σ}(φ ⊗ ρ), …, Γ_{2d,σ}(φ ⊗ ρ)}, equations (2.5.5), (2.5.6) and (2.5.7) imply that

T_{σ,φ,ρ}^{±} = ( \begin{matrix}(ã_{σ,ρ}^{±}(χ)ã_{ij,σ,φ}^{+})\begin{matrix}1≤i≤d^{+} \\ 1≤j≤r \end{matrix} & (b̃_{σ,ρ}^{±}(χ)ã_{ij,σ,φ}^{+})\begin{matrix}1≤i≤d^{+} \\ 1≤j≤d−r \end{matrix} \\ (ã_{σ,ρ}^{∓}(χ)ã_{ij,σ,φ}^{−})\begin{matrix}1≤i≤d^{−} \\ 1≤j≤r \end{matrix} & (b̃_{σ,ρ}^{∓}(χ)ã_{ij,σ,φ}^{−})\begin{matrix}1≤i≤d^{−} \\ 1≤j≤d−r \end{matrix} \end{matrix} ) ,

where r = r_{σ,φ,ρ}(χ). Note that d − r < r because p_{1}^{χ}(σ, ρ) > p_{2}^{χ}(σ, ρ). Now, as in Subsection 2.3, there are elements λ_{i,σ} ∈ (E ⊗ℂ)^{×}, λ_{σ}(χ) ∈ (ℚ(χ)⊗ ℂ)^{×}, μ_{i,σ} ∈ (E ⊗ K)^{×} and μ_{σ}(χ) ∈ (ℚ(χ) ⊗ K)^{×} such that

F_{σ,ℂ}(Ω_{i,σ}) = λ_{i,σ}Ω_{d+1−i,σ}, F_{σ,ℂ}(Ω_{σ}(χ)) = λ_{σ}(χ)Ω′_{σ}(χ),
Q_{i,σ} = λ_{i,σ}σ(μ_{i,σ}), Q_{σ}(χ) = λ_{σ}(χ)σ(μ_{σ}(χ))δ_{σ}[χ_{0}ε_{L}]^{−1}.

This, together with (2.5.5) and (2.5.6), implies that

ã_{ij,σ}^{±} = ±λ_{j,σ}ã_{i,d+1−j,σ}^{±},
ã_{σ}^{±}(χ) = ±λ_{σ}(χ)b̃_{σ}^{±}(χ).

Thus,

det(T_{σ}^{±}) = σ(μ_{σ}(χ))^{d−r_{σ}}δ_{σ}[χ_{0}ε_{L}]^{r_{σ}−d}Q_{σ}(χ)^{r_{σ}−d}(ã_{σ}^{±}(χ))^{d^{+}}(ã_{σ}^{∓}(χ))^{d−} det(V_{σ}^{±}),

where V_{σ}^{±} ∈ GL_{d}(E ⊗ ℚ(χ) ⊗ ℂ) is the matrix such that its φ, ρ-components are

V_{σ,φ,ρ}^{±} = ( \begin{matrix}(ã_{ij,σ,φ}^{+})\begin{matrix}1≤i≤d^{+} \\ 1≤j≤r \end{matrix} & (±ã_{ij,σ,φ}^{+})\begin{matrix}1≤i≤d^{+} \\ 1≤j≤d−r \end{matrix} \\ (ã_{ij,σ,φ}^{−})\begin{matrix}1≤i≤d^{−} \\ 1≤j≤r \end{matrix} & (∓ã_{ij,σ,φ}^{−})\begin{matrix}1≤i≤d^{−} \\ 1≤j≤d−r \end{matrix} \end{matrix} ) .

Note that σ(μ_{σ}(χ))^{d−r_{σ}}, as an element of E⊗ℚ(χ)⊗ℂ, belongs to E⊗ℚ(χ)⊗K via σ. This follows from the decomposition E ⊗ ℚ(χ) ⊗ K ≅ ∏_{μ} K_{μ} and the fact that the r_{σ,φ,ρ}(χ) only depend on the index μ. Using the same reasoning with (−1)^{d−r_{σ}}, we get

c_{σ}^{±}((M ⊗ RM(χ))^{∨}) ∼ δ_{σ}[χ_{0}ε_{L}]^{r_{σ}−d}Q_{σ}(χ)^{r_{σ}−d}(ã_{σ}^{±}(χ))^{d^{+}}(ã_{σ}^{∓}(χ))^{d−} det(V_{σ}^{+}).

Applying elementary column operations and using the relations (2.3.2), together with the fact that the powers of 2 and −1 that appear in the process belong to E ⊗ ℚ(χ) ⊗ K by the reasoning above, we conclude that

det(V_{σ}^{+}) ∼ det((ã_{ij,σ,φ}^{+})_{i,j=1,...,d^{+}}) det((ã_{ij,σ,φ}^{−})_{i,j=1,...,d^{−}}) ∏_{j=d^{+}+1}^{r_{σ}} Q_{j,σ}

(for this, one needs to use at some point that when d = 2k − 1 is odd, Q_{k,σ} ∈ (E ⊗ K)^{×}). Using (2.3.3) (note that A is trivial in this case), we get

det(V_{σ}^{+}) ∼ δ_{σ}(M)^{−1} ∏_{j=1}^{⌊d/2⌋} Q_{j,σ} ∏_{j=d^{+}+1}^{r_{σ}} Q_{j,σ}.

Then it’s easy to see, again using that Q_{k,σ} ∼ 1, that

det(V_{σ}^{+}) ∼ δ_{σ}(M)^{−1} ∏_{j=1}^{r_{σ}} Q_{j,σ}.

We can use Lemma 2.3.2 (rather, it’s generalization to powers indexed by embeddings of E ⊗ ℚ(χ); the formula in the lemma still holds because the relevant scalars in E ⊗ ℚ(χ) ⊗ K are powers of the σ(μ_{j,σ}), which we can index by the fields K_{μ} as above) to obtain

det(V_{σ}^{+}) ∼ δ_{σ}(M)^{−1} ∏_{j=1}^{s_{σ}} Q_{j,σ}.

Note that, by Remark 2.2.3, ã_{σ}^{±}(χ) ∼_{ℚ(χ)⊗K,σ} a_{σ}^{±}(χ) ∼_{ℚ(χ)⊗K,σ} c_{σ}^{±}(RM(χ)^{∨}). Also, by the fact that RM(χ)^{∨} ≅ RM(χ)(w(χ)) ⊗_{ℚ(χ)} [χ_{0}ε_{L}] and by (2.4.2), we can write

c_{σ}^{+}(RM(χ)^{∨})c_{σ}^{−}(RM(χ)^{∨}) ∼
c_{σ}^{+}(χ)c_{σ}^{−}(χ)(2πi)^{2w(χ)}δ_{σ}[χ_{0}ε_{L}]^{−2} ∼ (2πi)^{w(χ)}Q_{σ}(χ).

We put together all these formulas with (2.5.3) to arrive at the main formulas in the statement of the theorem (we also use (2.3.4)). □

Remark 2.5.2. There is an apparent difference between the formulas of the theorem and those of Proposition 1.7.6 of [Har97]. The main point is that we are also leaving the factor δ_{σ}(M) instead of replacing it with powers of (2πi) and discriminant factors. We don’t need to do that in this paper.

The following proposition follows from Theorem 2.5.1 and its proof when M = ℚ(0). Alternatively, it’s a simple consequence of the isomorphism RM(χ) ≅ RM(χ)^{∨} ⊗_{ℚ(χ)} [χ_{0}ε_{L}](−w(χ)).

Proposition 2.5.2. Let χ be a critical algebraic Hecke character of L. Then

c_{σ}^{±}(χ) ∼_{ℚ(χ)⊗K} (2πi)^{−w(χ)}δ[χ_{0}ε_{L}]a_{σ}^{∓}(χ).

Remark 2.5.3. Suppose that M and χ are as in Theorem 2.5.1. Then we can also express the periods c_{σ}^{−}(M ⊗ RM(χ)) in a similar fashion. Looking at the proof of the theorem, along with (2.4.4), the following is clear:

c_{σ}^{−}(M ⊗ RM(χ)) ∼_{(E⊗ℚ(χ))⊗K,σ} e′_{σ}c_{σ}^{+}(M ⊗ RM(χ)),

where e′_{σ} = 1 if d is even, and e′_{σ} = e_{τ} if d is odd, for any τ ∈ J_{L} extending σ.

2.6. CM periods. In this subsection we recall the relationship between Deligne periods and CM periods for algebraic Hecke characters. The theory of CM periods as we will use it is explained in [Har93], to which we refer for details. Let L/K be a CM extension, and let T^{L} = Res_{L/ℚ} 𝔾_{m,L}. Then (T^{L})_{ℂ} ≃ ∏_{τ∈J_{L}} 𝔾_{m,ℂ}. Suppose that η ∈ X^{∗}(T^{L}) and χ ∈ X(η), and assume that χ is critical, so that n_{τ} ≠ n\overline{τ} for all τ ∈ J_{L}. Given any morphism h : 𝕊 → (T^{L})_{ℝ}, the pair (T^{L}, h) is a Shimura datum. In op. cit., as in the Appendix of [HK91], a CM period p(χ; h) ∈ ℂ^{×} is constructed, well defined modulo (ℚ(χ)E_{h})^{×}, where E_{h} is the reflex field of (T^{L}, h). For example, for any Ψ ⊂ J_{L} such that Ψ ∩ \overline{Ψ} = ∅ (for instance Ψ = \{τ\} for a single τ), we can naturally construct a map h_{Ψ} : 𝕊 → T_{ℝ}^{L}; we denote the corresponding periods by p(χ; Ψ), and the reflex field by E_{Ψ}. Concretely, E_{Ψ} is the subfield of \overline{ℚ} fixed by the elements γ ∈ Γ_{ℚ} such that γΨ = Ψ. Attached to η is a CM type Φ_{η}, defined by the fact that n_{τ} > n\overline{τ} if and only if τ ∈ Φ_{η} (note that what we call n_{τ} here is −λ(τ) in [Har93]). We also write Φ_{χ} = Φ_{η}. The reflex field E_{Φ_{η}} of Φ_{η} is contained in ℚ(χ) (in fact in ℚ(η) ⊂ ℚ(χ), where ℚ(η) is the field of definition of η). If γ ∈ Aut(ℂ), the characters η^{γ} and χ^{γ} ∈ X(η^{γ}) only depend on the restriction of γ to ℚ(χ), and hence we can look at the family (χ^{ρ}, η^{ρ})_{ρ∈J_{ℚ(χ)}}. If we look at embeddings ρ of ℚ(χ)E_{Ψ}, then we can also define Ψ^{ρ} = ρ̃Ψ, for any extension ρ̃ of ρ to ℂ. We write

p(χ; Ψ) = (p(χ^{ρ}; Ψ^{ρ}))_{ρ∈J_{ℚ(χ)E_{Ψ}}} ∈ ℚ(χ)E_{Ψ} ⊗ ℂ.

If Ψ = Φ_{η}, then p(χ; Φ_{η}) = (p(χ^{ρ}; Φ_{η^{ρ}}))_{ρ∈J_{ℚ(χ)}} ∈ ℚ(χ) ⊗ ℂ (note that Φ_{η^{ρ}} = ρΦ_{η}). The following formula is due to Blasius. We use the statement given as Proposition 1.8.1 of [Har93] (corrected as in the Introduction to [Har97], that changes χ for χ̌), combined with Deligne’s conjecture for the motive M(χ) (proved by Blasius in [Bla86]), to get

c^{+}(χ) ∼_{ℚ(χ)} D_{K}^{1/2}p(χ̌; Φ_{η}).

Here χ̌ = χ^{ι,−1} (not to be confused with the dual χ^{∨} = χ^{−1}). In fact, Blasius’s constructions should provide the following more precise statement, which we will assume: for every σ ∈ J_{K},

(2.6.1)  c_{σ}^{+}(χ) ∼_{ℚ(χ)⊗K,σ} p(χ̌; τ),

where τ ∈ Φ_{χ}. Let M and χ be as in Theorem 2.5.1. For each ρ ∈ J_{ℚ(χ)} and φ ∈ J_{E}, let c^{±}(M ⊗RM(χ))_{φ,ρ} ∈ ℂ be the ρ⊗φ-component of c^{±}(Res_{K/ℚ}(M ⊗RM(χ))) ∈ E ⊗ ℚ(χ) ⊗ ℂ. We let s_{σ,φ,ρ} = d − r_{σ,φ,ρ} and s_{σ} as in Theorem 2.5.1. Define Q^{s}(M) ∈ E ⊗ ℚ(χ) ⊗ ℂ by

Q^{s}(M) = ∏_{σ∈J_{K}} ∏_{j=1}^{s_{σ}} Q_{j,σ},

and let Q_{φ,ρ}^{s}(M) ∈ E ⊗ ℂ be its φ ⊗ ρ-component. Suppose that m ∈ ℤ is a critical integer for M ⊗ RM(χ). For simplicity of notation, we let r_{σ} = r_{σ,φ,ρ} and s_{σ} = s_{σ,φ,ρ} in the following expressions. By (2.2.1), (2.2.2), (2.4.3), (2.4.4), Remark 2.5.3, Proposition 2.5.2, (2.6.1) and Theorem 2.5.1, we can write

c^{+}(M ⊗ RM(χ)(m))_{φ,ρ} ∼_{φ(E)ρ(ℚ(χ))K′}

(2πi)^{emd−w(χ) ∑_{σ} s_{σ}}δ(M)_{φ} ( ∏_{σ∈J_{K}} δ_{σ}[χ_{0}ε_{L}]_{ρ}^{s_{σ}} (p((χ^{ρ})ˇ ; ρτ))^{r_{σ}−s_{σ}}) Q_{φ,ρ}^{s}(M),

where K′ is the Galois closure of K in \overline{ℚ}, we choose the embeddings τ in Φ_{χ}, and we write ρτ for ρ̃τ, with ρ̃ an extension of ρ. If we moreover assume that χ_{0} is trivial, then by Lemma 2.4.2 we can write

(2.6.2)  c^{+}(M ⊗ RM(χ)(m))_{φ,ρ} ∼_{φ(E)ρ(ℚ(χ))L′}

(2πi)^{emd−w(χ) ∑_{σ} s_{σ}}δ(M)_{φ} ( ∏_{σ∈J_{K}} (p((χ^{ρ})ˇ ; ρτ))^{r_{σ}−s_{σ}}) Q_{φ,ρ}^{s}(M).

3. Hodge-de Rham structures for unitary groups and automorphic periods

In this section we introduce the Hodge-de Rham structures attached to automorphic representations of unitary groups, which come from the cohomology of automorphic vector bundles and local systems on the corresponding Shimura varieties. On the first subsections we introduce the varieties in question, and setup the notation that we will use throughout the rest of the paper regarding weights and automorphic vector bundles. Many of the things that we say here are valid for more general Shimura varieties, but we restrict ourselves to the unitary group case to keep a reasonable length, and because it’s ultimately the case that we will use in the rest of the paper. In the later subsections, we define automorphic quadratic periods for cohomological automorphic representations. For generalities regarding Shimura varieties, automorphic vector bundles, and conjugation, our main references are [DMOS82] and [Mil90].

3.1. The groups. In this paper we will work with the following unitary groups. Let K be a totally real field of degree e = [K : ℚ] and let L/K be a totally imaginary quadratic extension. Let V be a finite-dimensional Lvector space and h : V × V → L a non-degenerate hermitian form relative to the non-trivial automorphism ι ∈ Gal(L/K). Let n = dim_{L} V. Let GU_{∗} denote the similitude unitary group of the pair (V, h) over K. Thus, for a K-algebra R,

GU_{∗}(R) = {g ∈ Aut_{F ⊗_{K}R}(V ⊗_{K}R) | h_{R}(gu, gv) = ν(g)h_{R}(u, v), u, v ∈ V ⊗_{k}R},

where ν(g) ∈ R^{×} and h_{R} : V ⊗_{K} R × V ⊗_{K} R → F ⊗_{K} R is defined by h_{R}(u ⊗ a, v ⊗ b) = h(u, v) ⊗ ab. The map g ↦ ν(g) defines a morphism ν : GU_{∗} → 𝔾_{m,K}, and its kernel is the unitary group U_{∗} over K. The center of GU_{∗} is Res_{L/K} 𝔾_{m,L}. We let GU_{0} = Res_{K/ℚ} GU_{∗} and U = Res_{K/ℚ} U_{∗}. Finally, we let GU = GU(V) be the subgroup of GU_{0} consisting of automorphisms g ∈ GU_{0}(R) for which ν(g) ∈ R^{×} ⊂ (K ⊗R)^{×}. All of these groups are reductive algebraic groups. Throughout the rest of this subsection, we let G = GU. The center Z of G, which is connected, is the subtorus of T^{L} = Res_{L/ℚ} 𝔾_{m,L} given by

Z(R) = {x ∈ (L ⊗_{ℚ} R)^{×} | N_{L⊗_{ℚ}R/K⊗_{ℚ}R}(x) ∈ R^{×}}.

Note that this is actually the center of the (abstract) group G(R). We will fix an L-basis β = {v_{1}, …, v_{n}} of V, orthogonal for h. For each τ ∈ J_{L}, let V_{τ} = V ⊗_{L,τ} ℂ, and let h_{τ} : V_{τ} × V_{τ} → ℂ be the non-degenerate hermitian form (relative to ℂ/ℝ) defined by h_{τ} (u ⊗ z, v ⊗ w) = τ(h(u, v))z\overline{w}. We let (r_{τ} , s_{τ}) denote the signature of (V_{τ} , h_{τ}). Choose once and for all a CM type Φ for the extension L/K. Then there are isomorphisms L ⊗_{ℚ} ℝ ≅ ∏_{τ∈Φ} ℂ and V ⊗_{ℚ} ℝ ≅ ∏_{τ∈Φ} V_{τ} which induce isomorphisms

(3.1.1)  G_{ℝ} ≅ ( ∏_{τ∈Φ} GU(r_{τ} , s_{τ}))′ , G_{ℂ} ≅ ∏_{τ∈Φ} GL_{n,ℂ} ×𝔾_{m,ℂ},

where the symbol ′ means that we are looking at tuples where all the elements have the same multiplier ν. The second isomorphism is defined over the Galois closure L′ of L in \overline{ℚ} ⊂ ℂ. Let T ⊂ G be the subgroup of elements of G which, considered as L-linear automorphisms of V, are diagonal with respect to β. Then T is a maximal torus of G, and it maps to the subgroup of diagonal matrices in (3.1.1). Let B ⊂ G_{ℂ} (or G_{L′}) be the Borel subgroup of G_{ℂ} containing T_{ℂ} that maps to ∏_{τ∈Φ} B_{n,ℂ} × 𝔾_{m,ℂ} under the second isomorphism of (3.1.1), where B_{n,ℂ} ⊂ GL_{n,ℂ} is the group of upper triangular matrices.

Remark 3.1.1. The group Aut(ℂ) acts on the complex points G(ℂ) by functoriality. Let g = ((X_{τ})_{τ∈Φ}, ν) ∈ G(ℂ). We can explicitly describe the action of an element γ ∈ Aut(ℂ), but we will only need the formula when γ = c. If g ∈ G(ℂ), then c(g) = ((\overline{ν}I_{r_{τ} ,s_{τ}} X_{τ}^{∗,−1}I_{r_{τ} ,s_{τ}})_{τ∈Φ}, \overline{ν}).

3.2. The Shimura varieties. Fix (V, h, Φ) as in the previous subsection. Given an orthogonal basis β and the corresponding isomorphism (3.1.1), we define x : 𝕊 → G_{ℝ} as x = (x_{τ})_{τ∈Φ}, where

(3.2.1)  x_{τ} (z) = ( \begin{matrix}zI_{r_{τ}} & 0 \\ 0 & \overline{z}I_{s_{τ}} \end{matrix} )

for an ℝ-algebra R and z ∈ 𝕊(R). We denote by X the G(ℝ)-conjugacy class of x. The pair (G, X) satisfies Deligne’s axioms (2.1.1.1-3, [Del79b]) for a Shimura datum, unless (V, h) is totally definite (i.e., unless r_{τ} s_{τ} = 0 for all τ ∈ J_{L}; in this case, we can still attach to G a Shimura variety of dimension 0, but we will omit this case from our discussion and assume henceforward that (V, h) is not totally definite). The point x of (3.2.1) factors through T_{ℝ}, where T is the maximal torus constructed in the previous subsection from the same orthogonal basis, and hence (T, x) is a CM pair. The reflex field E = E(G, X), the field of definition of the G(ℂ)-conjugacy class of the Hodge cocharacter μ_{x}, is the field generated over ℚ by the set \{∑_{τ∈Φ} τ(b)r_{τ} + \overline{τ}(b)s_{τ} : b ∈ L\}. In particular, it is contained in L′ and hence it’s CM or totally real. The Shimura varieties over E defined by the datum (G, X) will be denoted by S_{E}. For any F ⊃ E, we let S_{F} = S_{E} ×_{E} F. For any compact open subgroup U ⊂ G(𝔸_{f}), which will always be assumed to be sufficiently small, we denote by S_{U,E} the Shimura varieties with level U. Let γ ∈ Aut(ℂ). We denote by (^{γ,x}G,^{γ,x}X) the conjugate Shimura datum, and by ^{γ,x}S the corresponding Shimura variety, so that γS_{E} ≅ ^{γ,x}S_{γ(E)} (Theorem II.4.2, [Mil90]). In the particular case of γ = c complex conjugation, we can naturally identify the pair (^{c,x}G,^{c,x}X) with (G, \overline{X}), where \overline{X} = \{h ∘ c : h ∈ X\} (see [BG16]). We denote by \overline{S}_{E} the Shimura variety attached to (G, \overline{X}).

Remark 3.2.1. We can also identify the pair (G, \overline{X}) with the Shimura datum defined by the hermitian space (V, −h) and the CM type Φ, or by the hermitian space (V, h) and the CM type \overline{Φ} = \{\overline{τ} : τ ∈ Φ\}.

3.3. Roots, weights and representations. Let (G, X) be the pair attached to (V, h, Φ) as in the last subsections, and let (T, x) be the CM pair defined in (3.2.1). Let K_{x} denote the centralizer of x in G_{ℝ}, i.e., the schemetheoretic centralizer of the (scheme-theoretic) image of x in G_{ℝ}. Then K_{x} ≅ (∏_{τ∈Φ} Z_{τ} K_{τ} )′, where Z_{τ} = 𝕊 is the center of GU(r_{τ} , s_{τ}) and K_{τ} = U(r_{τ}) × U(s_{τ}) is embedded diagonally. Also, K_{x,ℂ} ≅ (∏_{τ∈Φ} GL_{r_{τ} ,ℂ} × GL_{s_{τ} ,ℂ}) × 𝔾_{m,ℂ}. Note that T_{ℝ} ⊂ K_{x}. Let R denote the set of roots of the pair (G_{ℂ}, T_{ℂ}), and write R = R_{c} ∐ R_{n}, where R_{c} denotes the roots which are also roots of (K_{x,ℂ}, T_{ℂ}) (these are called compact roots). We let Λ = X^{∗}(T). Using (3.1.1), we make once and for all the identification Λ ≅ (∏_{τ∈Φ} ℤ^{n}) × ℤ, and we write elements μ ∈ Λ as tuples of integers μ = ((a_{τ,1}, …, a_{τ,n})_{τ∈Φ}; a_{0}). Thus, μ corresponds to the character of T_{ℂ} given by ((diag(t_{τ,1}, …, t_{τ,n}))_{τ∈Φ}; t_{0}) ↦ t_{0}^{a_{0}} ∏_{τ∈Φ} ∏_{i=1}^{n} t_{τ,i}^{a_{τ,i}}. For each a ∈ ℤ, we let μ_{a} = ((0, …, 0)_{τ∈Φ}; a) ∈ Λ. For 1 ≤ i ≤ n and τ ∈ Φ, we let e_{i,τ} denote the element of Λ which hast the i-th standard vector of ℤ^{n} in coordinate τ, and 0 everywhere else, including the multiplier coordinate. We can write the set of roots R as R ≅ ∐_{τ∈Φ} R_{τ}, where R_{τ} = {e_{i,τ} − e_{j,τ} |1 ≤ i ≠ j ≤ n}. We can write the set of compact roots R_{c} as R_{c} ≅ ∐_{τ∈Φ} R_{c,τ}, where R_{c,τ} = {e_{i,τ} − e_{j,τ} |1 ≤ i ≠ j ≤ r_{τ} or r_{τ} + 1 ≤ i ≠ j ≤ n}. Let 𝔤 = Lie(G) and 𝔨_{x} = Lie(K_{x}). The map Ad ∘x : 𝕊 → GL_{𝔤_{ℝ}} induces a Hodge decomposition on 𝔤_{ℝ}, which has Hodge type {(0, 0), (−1, 1), (1, −1)}, and 𝔤_{ℝ}^{0,0} = 𝔨_{x,ℂ}. Let 𝔭_{x}^{±} = 𝔤_{ℝ}^{∓1,±1}. Identify X^{∗}(𝕊_{ℂ}) ≅ ℤ⊕ℤ. Note that if α ∈ R, then αx_{ℂ} ∈ X^{∗}(𝕊_{ℂ}) equals either (0, 0), (−1, 1) or (1, −1). We denote by R_{x}^{0,0}, R_{x}^{−1,1} and R_{x}^{1,−1} the corresponding subsets, so that R = R_{x}^{0,0} ∐ R_{x}^{−1,1} ∐ R_{x}^{1,−1}. Note that R_{x}^{0,0} = R_{c}, and if α = e_{i,τ} − e_{j,τ} ∈ R, then α ∈ R_{x}^{1,−1} (resp. R_{x}^{−1,1}) if and only if 1 ≤ i ≤ r_{τ} and r_{τ} + 1 ≤ j ≤ n (resp. r_{τ} + 1 ≤ i ≤ n and 1 ≤ j ≤ r_{τ}). Let R_{c,τ,x}^{+} ⊂ R_{c,τ} consist of the elements e_{i,τ} − e_{j,τ} for which i < j, and let R_{c,x}^{+} = ∐_{τ∈Φ} R_{c,τ,x}^{+}. This is a set of positive roots in R_{c}. Let B_{c,x} denote the corresponding Borel subgroup of K_{x,ℂ} containing T_{ℂ}; then B_{c,x} ≅ ∏_{τ∈Φ} (B_{r_{τ} ,ℂ} × B_{s_{τ} ,ℂ}) × 𝔾_{m,ℂ}. Let R_{x}^{+} = R_{c,x}^{+} ∐ R_{x}^{1,−1}. This is a set of positive roots in R. Concretely, R_{x}^{+} = ∐_{τ∈Φ} R_{τ,x}^{+} where R_{τ,x}^{+} consists of the elements e_{i,τ} − e_{j,τ} in R_{τ} such that i < j. Let B_{x} ⊂ G_{ℂ} be the corresponding Borel subgroup containing T_{ℂ}; then B_{x} ≅ ∏_{τ∈Φ} B_{n,ℂ}×𝔾_{m,ℂ}, i.e., it is the group that we called B before. We use the notation B_{x} because we will soon work with B\overline{x} which is a different group. Note that B_{c,x} = B_{x} ∩ K_{x,ℂ}. Let P_{x} ⊂ G_{ℂ} be the subgroup fixing the filtration on the category of representations of G_{ℂ} defined by the cocharacter μ_{x} : 𝔾_{m,ℂ} → G_{ℂ} (see Proposition I.1.7 of [Mil90]). This is a parabolic subgroup with contains K_{x,ℂ} as a Levi component. The Lie algebra of P_{x} is 𝔓_{x} = 𝔨_{x,ℂ}⊕𝔭_{x}^{−}, and Lie(R_{u}P_{x}) = 𝔭_{x}^{−}. Note that we can identify 𝔭_{x}^{+} (resp. 𝔭_{x}^{−}) with the holomorphic (resp. antiholomorphic) tangent space of X at the point x, and P_{x} with the subgroup of tuples in ∏_{τ} GL_{n,ℂ} ×𝔾_{m,ℂ} for which each τ-component is bock lower triangular with respect to the partition n = r_{τ} +s_{τ}. The groups P_{x} and K_{x,ℂ} are defined over L′ ⊂ ℂ, and the maximal torus T_{L′} of K_{x,L′} is split. We denote by Λ_{x}^{+} and Λ_{c,x}^{+} the set of dominant weights for R_{x}^{+} and R_{c,x}^{+} respectively. If μ ∈ Λ_{x}^{+} (resp. λ ∈ Λ_{c,x}^{+}), we denote by (W_{μ}, ρ_{μ}) (resp. (V_{λ}, r_{λ})) the irreducible representation of G_{ℂ} (resp. K_{x,ℂ}) with highest weight μ (resp. λ). We say that a representation (W, ρ) of G_{ℂ} is defined over a subfield F ⊂ ℂ if there exists a representation (W_{F} , ρ_{F}) of G_{F} such that the extension of scalars (W_{F} , ρ_{F}) ⊗_{F} ℂ is isomorphic to (W, ρ) as representations of G_{ℂ}. Note that all the representations of G_{ℂ} are defined over L′. The set Λ_{x}^{+} (resp. Λ_{c,x}^{+}) consists of tuples μ = ((a_{τ,1}, …, a_{τ,n})_{τ∈Φ}; a_{0}) as above with a_{τ,1} ≥ ⋯ ≥ a_{τ,n} (resp. a_{τ,1} ≥ ⋯ ≥ a_{τ,r_{τ}} and a_{τ,r_{τ+1}} ≥ ⋯ ≥ a_{τ,n}) for every τ ∈ Φ. For example, the adjoint action of G_{ℂ} on 𝔤_{ℂ} restricts to an action of K_{x,ℂ} on 𝔓_{x} and on 𝔭_{x}^{±}; note that we can identify 𝔭_{x}^{+} with the dual of 𝔭_{x}^{−} via the Killing form. The space Λ^{d}𝔭_{x}^{+} has dimension 1, and its highest weight in Λ_{c,x}^{+} is ((s_{τ} , …, s_{τ} , −r_{τ} , …, −r_{τ})_{τ∈Φ}; 0), with s_{τ} appearing r_{τ} times. As another example, what we called μ_{a} = ((0, …, 0_{τ∈Φ}; a) is the highest weight of the character of G_{ℂ} given by the a-th power ν^{a} of the multiplier ν : G → 𝔾_{m,ℚ}, which is obviously defined over ℚ. For any μ ∈ Λ, we let ξ(μ) = 2a_{0} +∑_{τ,i} a_{τ,i}.

Remark 3.3.1. Let w_{X} denote the weight morphism of (G, X). If μ ∈ Λ_{x}^{+} then ρ_{μ} ∘ w_{X,ℂ} : 𝔾_{m,ℂ} → GL_{W_{μ}} takes t ∈ ℂ^{×} to t^{−ξ(μ)} id_{W_{μ}}, because the central character of W_{μ} is the restriction of μ to Z_{ℂ}.

Remark 3.3.2. Our parametrization differs slightly from that of [Har97]. Namely, a highest weight parametrized in the form ((a_{τ,1}, …, a_{τ,n})_{τ∈Φ}; a_{0}) ∈ Λ_{x}^{+} corresponds to the representation that in the parametrization of op. cit. has the same a_{τ,i}’s and c = ξ(μ).

We denote by 𝒲 (resp. 𝒲_{c}) the Weyl group of (G_{ℂ}, T_{ℂ}) (resp. (K_{x,ℂ}, T_{ℂ})), and by ℓ the length function on 𝒲 with respect to R_{x}^{+} (note that if w ∈ 𝒲_{c}, then ℓ(w) is also the length with respect to R_{c,x}^{+} because w preserves R_{x}^{±1,∓1}). Let w_{0} (resp. w_{0,c}) be the longest element of 𝒲 (resp. 𝒲_{c}), so that ℓ(w_{0}) = |R_{x}^{+}| = \frac{n(n−1)e}{2} and ℓ(w_{0,c}) = |R_{c,x}^{+}| = |R_{x}^{+}| − d. For any w ∈ 𝒲, we let w^{♭} = w_{0,c}ww_{0}, and we let w_{0}^{1} = 1^{♭} = w_{0,c}w_{0}. Note that (w^{♭})^{♭} = w for any w. For any integer n, we let 𝔖_{n} denote the symmetric group on {1, …, n}, acting on (a_{1}, …, a_{n}) ∈ ℤ^{n} by σ(a_{1}, …, a_{n}) = (a_{σ^{−1}(1)}, …, a_{σ^{−1}(n)}). We let u_{n} ∈ 𝔖_{n} denote the order reversing permutation i ↦ n+1−i. We can identify 𝒲 (resp. 𝒲_{c}) with ∏_{τ∈Φ} 𝔖_{n} (resp. ∏_{τ∈Φ} 𝔖_{r_{τ}} × 𝔖_{s_{τ}}), with the natural inclusion of the latter into the former. Then we can see w_{0} (resp. w_{0,c}) as the tuple (u_{n})_{τ∈Φ} (resp. (u_{r_{τ}} , u_{s_{τ}})_{τ∈Φ}), and w_{0}^{1} = (w_{0,τ}^{1} )_{τ∈Φ}, where

w_{0,τ}^{1} (i) = \{ \begin{matrix}i + r_{τ} & if & 1 ≤ i ≤ s_{τ} \\ i − s_{τ} & if & s_{τ} + 1 ≤ i ≤ n. \end{matrix}

If μ = ((a_{τ,1}, …, a_{τ,n})_{τ∈Φ}; a_{0}) ∈ Λ_{x}^{+} (resp. λ = ((a_{τ,1}, …, a_{τ,n})_{τ∈Φ}; a_{0}) ∈ Λ_{c,x}^{+}), then the dual representation W_{μ}^{∨} of G_{ℂ} (resp. V_{λ}^{∨} of K_{x,ℂ}) has highest weight

μ^{∨} = −w_{0}(μ) = ((−a_{τ,n}, …, −a_{τ,1})_{τ∈Φ}; −a_{0})

(resp.

λ^{∨} = −w_{0,c}(λ) = ((−a_{τ,r_{τ}} , …, −a_{τ,1}, −a_{τ,n}, …, −a_{τ,r_{τ} +1})_{τ∈Φ}; −a_{0}) ).

We define 𝒲^{1} = {w ∈ 𝒲 : w(R_{x}^{+}) ⊃ R_{c,x}^{+}}; this is a set of coset representatives of shortest length for 𝒲_{c}\𝒲. Concretely, 𝒲^{1} is the set of tuples (w_{τ})_{τ∈Φ} with w_{τ} ∈ 𝔖_{n} a permutation such that w_{τ}^{−1}(i) < w_{τ}^{−1}(j) whenever 1 ≤ i < j ≤ r_{τ} or r_{τ} + 1 ≤ i < j ≤ n. The element w_{0}^{1} belongs to 𝒲^{1}, has length d, and is the longest element of 𝒲^{1}. More generally, if w ∈ 𝒲^{1} then w^{♭} ∈ 𝒲^{1} and ℓ(w^{♭}) = d − ℓ(w). If μ ∈ Λ_{x}^{+} and w ∈ 𝒲^{1}, then w ∗ μ = w(μ + ρ_{x}) − ρ_{x} ∈ Λ_{c,x}^{+}, where

ρ_{x} = \frac{1}{2} ∑_{α∈R_{x}^{+}} α = ((\frac{1}{2}(n − 1), \frac{1}{2}(n − 3), …, \frac{1}{2}(1 − n))_{τ∈Φ} ; 0) ∈ Λ ⊗ ℝ.

Remark 3.3.3. Suppose that V has signature (n − 1, 1) at some τ_{0} ∈ Φ, and signatures (n, 0) at all other places. Then we can write 𝒲^{1} = {w_{1}, …, w_{n}}, where w_{i} = (w_{i,τ})_{τ∈Φ}, w_{i,τ} = 1 if τ ≠ τ_{0}, and w_{i,τ_{0}} is the permutation that sends i to n and is order perserving on the other n−1 elements, so w_{i,τ_{0}}(x) = x for 1 ≤ x < i, w_{i,τ_{0}}(i) = n and w_{i,τ_{0}}(x) = x − 1 for i + 1 ≤ x ≤ n. Note that ℓ(w_{j}) = n − j.

There is a natural action of Aut(ℂ) on Λ. For μ ∈ Λ, γ ∈ Aut(ℂ) and t ∈ T (ℂ), μ^{γ}(t) = γμ(γ^{−1}(t)). By choosing the Borel B_{x} ⊂ G_{ℂ}, there is a second action μ ↦ γ(μ) of γ ∈ Aut(ℂ) on Λ, the ∗-action, defined as γ(μ)(t) = μ^{γ}(n^{−1}tn), where conjugation by n ∈ G(ℂ) takes the Borel pair (γ(B_{x}), γ(T_{ℂ})) to (B_{x}, T_{ℂ}). This action preserves R, R_{x}^{+} and Λ_{x}^{+}, and is trivial on Aut(ℂ/F) for any F ⊂ ℂ such that G_{F} is split (in particular, on Aut(ℂ/L′)). Concretely, let μ = ((a_{τ,1}, …, a_{τ,n})_{τ∈Φ}; a_{0}) ∈ Λ. We can explicitly compute the parameters μ^{γ} and γ(μ). We only write here the action for γ = c. By Remark 3.1.1,

μ^{c} = ((−a_{τ,1}, …, −a_{τ,n})_{τ∈Φ}; a_{0} + ∑_{i,τ} a_{τ,i})

and

c(μ) = ((−a_{τ,n}, …, −a_{τ,1})_{τ∈Φ}; a_{0} + ∑_{i,τ} a_{τ,i}) .

If μ ∈ Λ_{x}^{+}, then c(μ) = μ^{∨} + μ_{ξ(μ)}.

Remark 3.3.4. The equation c(μ) = μ means that a_{τ,i} = −a_{τ,n+1−i} for every τ ∈ Φ and i = 1, …, n. In this case, ξ(μ) = 2a_{0} and W_{μ} ≅ W_{μ}^{∨} ⊗_{ℂ} W_{μ_{2a_{0}}} .

Remark 3.3.5. Suppose that W_{μ} is defined over ℚ. Then γ(μ) = μ for any γ ∈ Aut(ℂ). To show this, we can replace ℂ by \overline{ℚ}. Let ρ : G → GL_{W_{μ,ℚ}} denote the descent of the representation to ℚ. Since it’s irreducible, it follows from Théoreme 7.2 of [Tit71] that ρ is isomorphic to ^{ℚ}ρ_{λ} for some λ ∈ Λ_{x}^{+}, where the notation is as in op. cit. Moreover, in Lemme 7.4 of op. cit., we must have d = 1 and r = 1 because ρ is absolutely irreducible, so λ ∈ (Λ_{x}^{+})^{Gal(\overline{ℚ}/ℚ)}. Furthermore, ρ\overline{ℚ} must be the irreducible representation with highest weight λ, and thus μ = λ, which is invariant under Gal(\overline{ℚ}/ℚ). In particular, if W_{μ} is defined over ℚ, then W_{μ} ≅ W_{μ}^{∨} ⊗ W_{μ_{2a_{0}}} as representations over ℚ. Conversely, if γ(μ) = μ for every γ ∈ Aut(ℂ), then c(μ) = μ. It’s easy to see that μ ∈ Q ⊕ Λ_{0}, where Q is the subgroup of Λ generated by R and Λ_{0} is the subgroup orthogonal to the coroots. It follows from Théorème 3.3 of [Tit71] that W_{μ} is defined over ℚ.

Consider now the pair (G, \overline{X}) and its special point \overline{x}, and consider the same context and notation in this case. Then K\overline{x} = K_{x}. The Hodge decomposition of 𝔤_{ℂ} for \overline{x} is the complex conjugate of that of x, so that 𝔭± ; \overline{x} = 𝔭_{x}^{∓}, and

R±1,∓1 ; \overline{x} = R_{x}^{∓1,±1}. We choose R_{c,\overline{x}}^{+} ⊂ R_{c} to be R_{c,\overline{x}}^{+} = −(R_{c,x}^{+}) = ∐_{τ∈Φ} R_{c,τ,\overline{x}}^{+},

where R_{c,τ,\overline{x}}^{+} consists of the elements e_{i,τ} − e_{j,τ} such that 1 ≤ j < i ≤ r_{τ} or

r_{τ} + 1 ≤ j < i ≤ n. Then let R+ ; \overline{x} = R_{c,\overline{x}}^{+} ∐ R1,−1 ; \overline{x} = −(R_{x}^{+}) = ∐_{τ∈Φ} R_{τ,\overline{x}}^{+},

where R_{τ,\overline{x}}^{+} consists of the elements e_{i,τ} − e_{j,τ} with i > j. This is a set of positive roots for R. By making reference to R+ ; \overline{x} or R_{c,\overline{x}}^{+} instead of R_{x}^{+} or R_{c,x}^{+}, we can similarly define, using \overline{x}, all the objects defined so far with a subindex x. Note that under the natural identification cG_{ℂ} ≃ G_{ℂ}, the groups c(B_{x}), c(B_{c,x}) and c(P_{x}) correspond respectively to B\overline{x}, B_{c,\overline{x}} and P\overline{x}. Concretely, B_{c,\overline{x}} ≅ ∏_{τ∈Φ}(B_{r_{τ} ,ℂ}^{−} × B_{s_{τ} ,ℂ}^{−}) × 𝔾_{m,ℂ}, B\overline{x} ≅ (∏_{τ∈Φ} B_{n,ℂ}^{−}) × 𝔾_{m,ℂ}, and P\overline{x} has block upper triangular matrices. Also, Λ+ ; \overline{x} = −(Λ_{x}^{+}) consists of elements μ = ((a_{τ,1}, …, a_{τ,n})_{τ∈Φ}; μ_{0}) such that a_{τ,1} ≤ ⋯ ≤ a_{τ,n} for every τ ∈ Φ. Similarly, Λ_{c,\overline{x}}^{+} = −(Λ_{c,x}^{+}) consists of those μ satisfying a_{τ,1} ≤ ⋯ ≤ a_{τ,r_{τ}} and a_{τ,r_{τ} +1} ≤ ⋯ ≤ a_{τ,n}. Finally, ρ\overline{x} = ρ_{x}^{c} = −ρ_{x}. This gives an action of 𝒲 on Λ ⊗ ℝ taking Λ+ ; \overline{x} to Λ_{c,\overline{x}}^{+}; if needed, we will distinguish it from the other one by writing w\overline{∗}μ = w(μ + ρ\overline{x}) − ρ\overline{x}. This is equal to −(w ∗ (−μ)). If c(μ) = μ, then (w ∗ μ)^{c} = w\overline{∗}μ^{c}. Let (W, ρ) be a representation of G_{ℂ}. We define the conjugate representation (W^{c}, ρ^{c}) of G_{ℂ} by taking W^{c} = W as a real vector space, with complex conjugate ℂ-action, and by taking ρ^{c}(g) = ρ(\overline{g}) for g ∈ G(ℂ). If (W, μ) is irreducible and μ ∈ Λ_{x}^{+} is its R_{x}^{+}-highest weight, then (W^{c}, ρ^{c}) is irreducible, and its R+ ; \overline{x}-highest weight is μ^{c}. Equivalently, (W^{c}, ρ^{c}) has R_{x}^{+}-highest weight w_{0}(μ^{c}) = −(μ^{c})^{∨} = (−μ^{c})^{∨}. If W_{μ} is defined over ℚ, then c(μ) = μ and this implies (−μ^{c})^{∨} = μ, so there exists an isomorphism, unique up to scalars, between W^{c} and W. If λ ∈ Λ_{c,x}^{+}, we define in a similar way the conjugate representation (V_{λ}^{c}, r_{λ}^{c}), which has R_{c,\overline{x}}^{+}-highest weight λ^{c}. We can do a similar construction starting with a representation (W, ρ) of P_{x} and obtain a representation (W^{c}, ρ^{c}) of P\overline{x}. For future reference, we introduce the following operation on compact weights. For λ ∈ Λ_{c,x}^{+}, define λ^{♭} to be the R_{c,x}^{+}-highest weight of the representation V_{λ}^{♭} = V_{λ}^{∨} ⊗_{ℂ} Λ^{d}(𝔭_{x}^{+})^{∨} ⊗ V_{μ_{ξ(λ)}}. Concretely, if λ = ((a_{τ,1}, …, a_{τ,n})_{τ∈Φ}; a_{0}), then λ^{♭} is given by

((−a_{τ,r_{τ}} − s_{τ} , …, −a_{τ,1} − s_{τ} , −a_{τ,n} + r_{τ} , …, −a_{τ,r_{τ} +1} + r_{τ})_{τ∈Φ}; ξ(λ) − a_{0}) .

Note that V_{λ}^{♭} ≅ V_{λ}^{c} ⊗_{ℂ} Λ^{d}(𝔭_{x}^{+})^{∨}. It’s easily checked that if c(μ) = μ and w ∈ 𝒲^{1}, then (w ∗ μ)^{♭} = w^{♭} ∗ μ. Suppose that (W, ρ) is any representation of P_{x}. The map ρ∘μ_{x} : 𝔾_{m,ℂ} → GL_{W} defines a grading W = ⨁_{p} W^{p} with W^{p} = {w ∈ W : ρ(μ_{x}(z))(w) = z^{−p}w ∀z ∈ ℂ^{×}}. For each p ∈ ℤ, we define F^{p}(W) = ⨁_{p′≥p} W^{p′}. Then F^{p}(W) is P_{x}-stable, and it’s called the Hodge filtration of the representation (W, ρ), even though it’s not necessarily attached to a Hodge structure. Suppose in particular that (W, ρ) is an irreducible representation of K_{x,ℂ}, extended to P_{x} in such a way that the unipotent radical R_{u}P_{x} acts trivially on W. Then W = W^{p} for a unique p ∈ ℤ. If λ ∈ Λ_{c,x}^{+} is the R_{c,x}^{+}-dominant weight of (W, ρ), we usually write (p_{λ}, q_{λ}) = (p, −ξ(λ) − p). Concretely, if λ = ((a_{τ,1}, …, a_{τ,n})_{τ∈Φ}; a_{0}), then

\begin{matrix}p_{λ} = −a_{0} − ∑_{τ∈Φ} ∑_{i=1}^{r_{τ}} a_{τ,i}, \\ q_{λ} = −a_{0} − ∑_{τ∈Φ} ∑_{i=r_{τ} +1}^{n} a_{τ,i}. \end{matrix}

Remark 3.3.6. If λ ∈ Λ_{c,\overline{x}}^{+}, we can also write (p_{λ}, q_{λ}) for the pair associated to the filtration obtained from \overline{x} instead of x. With this convention, if λ ∈ Λ_{c,x}^{+}, then p_{λ^{c}} = p_{λ}.

3.4. Automorphic vector bundles and their conjugates. We refer to [Mil90] for the definition and main properties of automorphic vector bundles, and just set up the relevant notation here. Automorphic vector bundles on S_{ℂ} or S_{U,ℂ} are attached to homogeneous bundles on the compact dual Hermitian symmetric space of (G, X). Once we fix x ∈ X, these are given by representations of P_{x}. An automorphic vector bundle associated with a representation (W, ρ) of P_{x} which extends to G_{ℂ} is called flat. It is endowed with a natural regular flat connection. The associated local system on ℂ-vector spaces is denoted by W̃, and it has a structure of a local system on F -vector spaces for any F ⊂ ℂ over which (W, ρ) is defined as a representation of G (see Proposition II.3.3 of [Mil90]). On the other hand, automorphic vector bundles attached to representations of P_{x} which are trivial on the unipotent radical of P_{x} (that is, representations of K_{x,ℂ}) are called fully decomposed. Automorphic vector bundles have canonical models over finite extensions of E. For instance, a flat automorphic vector bundle has a canonical model over EF, where F is as above. Any fully decomposed automorphic vector bundle ℰ has a canonical model over L′. This follows from the fact that K_{x,L′} is a split group. We stress that L′ is not necessarily optimal. Usually, the letter ℰ will denote an automorphic vector bundle over S_{ℂ}, and ℰ_{U} will denote its descent to S_{U,ℂ}. If it has a canonical model over F ⊃ E, we write ℰ_{U,F} for the corresponding bundle over F. The Hodge filtration on representations of P_{x} induces a filtration on the corresponding automorphic vector bundles, which is independent of x. Let γ ∈ Aut(ℂ), and fix a special point x ∈ X. We can then conjugate automorphic vector bundles, in the sense that if ℰ is an automorphic vector bundle over S_{ℂ}, then we can construct another automorphic vector bundle γ,xℰ over ^{γ,x}S_{ℂ} (see [Mil90], III.5). If ℰ is associated with the representation (W, ρ) of P_{x}, then ^{γ,x}ℰ is associated with a representation (W^{γ,x}, ρ^{γ,x}) of P_{^{γ}x}, described in the following way. There is a γ-semilinear isomorphism W → W^{γ,x} that takes the ρ-action of p ∈ P_{x}(ℂ) to the ρ^{γ,x}-action of ^{γ,x}γ(p) ∈ P_{^{γ}x}(ℂ). In particular, if γ = c and we forget about x in the notation, then we obtain the representation (W^{c}, ρ^{c}) that we defined in Subsection 3.3. If λ ∈ Λ_{c,x}^{+}, then we denote by ℰ_{λ} the automorphic vector bundle over S_{ℂ} obtained from V_{λ}. If λ ∈ Λ_{c,\overline{x}}^{+}, then ℰ_{λ} will be the automorphic vector bundle over \overline{S}_{ℂ} obtained from V_{λ}. With these conventions, we can identify ^{c}ℰ_{λ} with ℰ_{λ^{c}} (see Subsection 3.3). Fix U ⊂ G(𝔸_{f}). There is a family of compactifications S_{U,ℂ,Σ} of S_{U,ℂ}, depending on auxiliary data Σ, which are smooth, projective, and with the property that Z_{U,Σ} = S_{U,ℂ,Σ} − S_{U,ℂ} is a divisor with normal crossings. For the main properties of these compactifications and their conjugates, see [Har90] and [BHR94]. They have canonical models S_{U,E,Σ} over E. If ℰ is an automorphic vector bundle over S_{ℂ}, there is a canonical extension ℰ_{U,Σ}^{can} of ℰ_{U} to a vector bundle over S_{U,ℂ,Σ}. Let ℐ_{U,Σ} denote the sheaf of ideals defining Z_{U,Σ}, and define the subcanonical extension as ℰ_{U,Σ}^{sub} = ℰ_{U,Σ}^{can} ⊗_{𝒪_{S_{U,ℂ,Σ}}} ℐ_{U,Σ}. If ℰ has a canonical model over F ⊃ E, then the canonical and subcanonical extensions ℰ_{U,Σ}^{?} also have canonical models ℰ_{U,F,Σ}^{?} over F. Conjugation of automorphic vector bundles extends to these extensions (Proposition 1.4.3 of [BHR94]). Given U and two toroidal data Σ and Σ′, there are canonical isomorphisms H^{i}(S_{U,F,Σ}, ℰ_{U,F,Σ}^{?}) ≃ H^{i}(S_{U,F,Σ′}, ℰ_{U,F,Σ′}^{?}), where ? = can or sub (Proposition 2.4 of [Har90]). We define H^{i}(S_{U,F} , ℰ^{?}) = \underrightarrow{lim}_{Σ} H^{i}(S_{U,F,Σ}, ℰ_{U,F,Σ}^{?}). Let H_{!}^{i}(S_{U,F} , ℰ) be the image of H^{i}(S_{U,F} , ℰ^{sub}) in H^{i}(S_{U,F} , ℰ^{can}) under the natural map induced from ℰ_{F}^{sub} → ℰ_{F}^{can}. This space is called the interior cohomology of S_{U,F} with coefficients in ℰ. All these cohomologies spaces are finite-dimensional over F. We also define

H^{i}(S_{F} , ℰ^{?}) = \underrightarrow{lim}_{U} H^{i}(S_{U,F} , ℰ^{?})

and H_{!}^{i}(S_{F} , ℰ) to be the image of H^{i}(S_{F} , ℰ^{sub}) in H^{i}(S_{F} , ℰ^{can}). There is an action G(𝔸_{f}) on H_{!}^{i}(S_{F} , ℰ), which makes it into a smooth admissible representation of G(𝔸_{f}) ([Har90], Proposition 2.6). For any automorphic vector bundle ℰ over S_{U,ℂ}, let ℰ′ = Ω_{S_{U,ℂ}}^{d} ⊗ ℰ^{∨} be its Serre dual. Then there is a non-degenerate pairing (Serre duality)

H_{!}^{d−i}(S_{U,ℂ}, ℰ′) ⊗_{ℂ} H_{!}^{i}(S_{U,ℂ}, ℰ) → ℂ,

ratioanl over any F over which ℰ has a canonical model (Corollary 3.8.5, [Har90]). We can combine the isomorphism cS_{U,F} → \overline{S}_{U,c(F)} and its extension to toroidal compactifications with base change for cohomology, and we get a c-semilinear, G(𝔸_{f})-equivariant isomorphism

c_{coh} : H_{!}^{i}(S_{F} , ℰ) → H_{!}^{i}(\overline{S}_{c(F)},^{c}ℰ).

3.5. The Hodge-de Rham structures M(W). Let (W, ρ) be an absolutely irreducible representation of G, with R_{x}^{+}-highest weight μ = ((a_{τ,1}, …, a_{τ,n})_{τ∈Φ}; a_{0}). Then ρ ∘ w_{X,ℚ}(t) = t^{−ξ} for each t ∈ ℚ^{×}, where ξ = ξ(μ) = 2a_{0}. In Theorems 2.2.7 and 2.3.1 of [Har94], a construction is given of a Hodge-de Rham structure (rather, a G(𝔸_{f})-admissible Hodge-de Rham structure, with the obvious definition) M(W)^{i} associated to W, using the i-th degree cohomology of the local system W̃ and its conjugates for the Betti realizations and the hypercohomology of a certain complex of automorphic vector bundles, called the Faltings (dual) BGG complex, for the de Rham realization. We refer to [Har94] for the details of the construction.

Remark 3.5.1. We are assuming that (W, ρ) is a representation of G as an algebraic group over ℚ. This assumption is made for simplicity of notation. Otherwise, the field of definition of W_{μ} must be incorporated into the field of coefficients of the Hodge-de Rham structures. Also, following Subsection 2.2 of [Har97], one can start with any irreducible representation of G_{ℂ}, which will be defined over some number field, and take the sum of its Galois conjugates to obtain a representation over ℚ. To simplify notation, we will assume from now on that (W, ρ) is defined over ℚ. We can identify which representations are defined over ℚ using Remark 3.3.5.

Remark 3.5.2. The additional data of the ℓ-adic sheaves associated to W (Proposition II.3.3, [Mil90]) makes M(W)^{i} a realization (or a G(𝔸_{f})-admissible realization). We will not use them in this part of the paper. Ideally, M(W)^{i} is a motive for absolute Hodge cycles, but this is not relevant for our calculations.

In any case, M(W)^{i} is a pure Hodge-de Rham structure of weight i − ξ, over L′ (or over a field over which the Faltings complex has a canonical model; this complex is formed of direct sums of fully decomposable bundles, and so it always has a canonical model over L′), and it has coefficients in ℚ. We write M(W) = M(W)^{d}. Letting 1 : L′ ↪ ℂ denote the given inclusion, the Hodge component of M(W)_{1}^{i} of type (p, i − ξ − p) is given by

(3.5.1)  (M(W)_{1}^{i})^{p,i−ξ−p} = ⨁_{\begin{matrix}w∈𝒲^{1} \\ p_{w∗μ}=p \end{matrix}} H_{!}^{i−ℓ(w)}(S_{ℂ}, ℰ_{w∗μ}).

The summands with ℓ(w) = i form what is called the holomorphic part of M^{i}(W)_{1} ⊗ ℂ. Those with ℓ(w) = 0 form the anti-holomorphic part. We usually refer to the summands in (3.5.1) as Weyl components. Any w ∈ 𝒲^{1} defines a Weyl component, contributing to the Hodge component of type (p, q) with p = p_{w∗μ}. Note that q = i + q_{w∗μ} ≠ q_{w∗μ}. On the case of interest in this paper, i will be equal to d, in which case q = d + q_{w∗μ} can also be written as q = p_{w^{♭}∗μ}.

3.6. Automorphic forms as cohomology classes. From now on, we will write K_{x} for K_{x}(ℝ) when it appears as an argument for relative Lie algebra cohomology. Thus, for example, we write H^{q}(𝔓_{x}, K_{x}; V) for a (𝔓_{x}, K_{x}(ℝ))- module V when we mean H^{q}(𝔓_{x}, K_{x}(ℝ); V). Let (W, ρ) (resp. (V, r)) be a representation of G over ℚ (resp. of K_{x,ℂ} over ℂ). Let 𝒜 (resp. 𝒜_{0}, resp. 𝒜_{(2)}) be the space of automorphic forms (resp. cuspidal, resp. square-integrable automorphic forms) on G(ℚ)\G(𝔸) in the sense of [BJ79], with respect to some maximal compact subgroup K_{∞} ⊂ G(ℝ). There are canonical inclusions ([Sch90] 4.2)

(3.6.1) H^{i}(𝔤_{ℂ}, K_{x}; 𝒜_{0} ⊗_{ℂ} W_{ℂ}) ⊂ M(W)_{1}^{i} ⊗ ℂ ⊂ H^{i}(𝔤_{ℂ}, K_{x}; 𝒜_{(2)} ⊗_{ℂ} W_{ℂ}).

Let c_{B} denote complex conjugation on the second factor of H_{!}^{i}(W) ⊗ ℂ. There is a similar picture for the fully decomposed automorphic vector bundle ℰ attached to (V, r). This case is contained in [Har90]. There are canonical inclusions

(3.6.2)  H^{i}(𝔓_{x}, K_{x}; 𝒜_{0} ⊗_{ℂ} V) ⊂ H_{!}^{i}(S_{ℂ}, ℰ) ⊂ H^{i}(𝔓_{x}, K_{x}; 𝒜_{(2)} ⊗_{ℂ} V).

In some cases, these inclusions are isomorphisms. See for instance [Har90], 5.3.2. The first inclusion in (3.6.2) is an isomorphism whenever i = 0 or i = d ([Har90] 5.4.2). Moreover, for i = 0 we can also write

(3.6.3)  H^{0}(𝔓_{x}, K_{x}; 𝒜(G) ⊗_{ℂ} V) = H^{0}(S_{ℂ}, ℰ^{can}).

3.7. Complex conjugation. Keep the assumptions of Subsection 3.5, and assume moreover that i = d. We are interested in studying how c_{B} acts on a coherent cohomology class represented in terms of automorphic forms. It’s illustrating to start with a general λ ∈ Λ_{c,x}^{+}, so we look at the spaces H_{!}^{q}(S_{ℂ}, ℰ_{λ}). By (3.6.2), we can view it sitting between H^{q}(𝔓_{x}, K_{x}; 𝒜_{0} ⊗_{ℂ} V_{λ}) and H^{q}(𝔓_{x}, K_{x}; 𝒜_{(2)} ⊗_{ℂ} V_{λ}). We can decompose 𝒜_{0} and 𝒜_{(2)} into a countable direct sum of irreducible unitary (𝔤_{ℂ}, K_{x})-modules π. By Proposition 4.5 of [Har90] (see also op. cit., Lemma 5.2.3), each such π contributes to cohomology if and only if χ_{π}(C) = ⟨λ + ρ_{x}, λ + ρ_{x}⟩ − ⟨ρ_{x}, ρ_{x}⟩, where in our case ⟨, ⟩ is the standard inner product on vectors, C is the Casimir element of 𝔤_{ℂ} and χ_{π} is the infinitesimal character of π. In this case, all the corresponding cochains will be closed. If we let 𝒜_{?,λ} denote the sum of the π that contribute, we obtain a natural identification (where ? = 0 or (2))

H^{q}(𝔓_{x}, K_{x}; 𝒜_{?} ⊗_{ℂ} V_{λ}) = (𝒜_{?,λ} ⊗_{ℂ} Λ^{q}(𝔭_{x}^{+}) ⊗_{ℂ} V_{λ})^{K_{x}(ℝ)}.

Now, let c_{𝒜} denote complex conjugation of functions on 𝒜_{?}. It’s easy to see that c_{𝒜} preserves 𝒜_{?,λ} in a K_{x}(ℂ)-equivariant way. Similarly, we can conjugate elements of Λ^{q}(𝔭_{x}^{+})⊗_{ℂ}V_{λ} and land in Λ^{q}(𝔭_{x}^{−})⊗_{ℂ}V_{λ}^{c} in a K_{x}(ℝ)-equivariant way. Moreover, we can identify V_{λ}^{c} with V_{λ^{♭}} ⊗_{ℂ} Λ^{d}(𝔭_{x}^{+}), and using the Killing form, this becomes V_{λ^{♭}} ⊗_{ℂ} Λ^{d}(𝔭_{x}^{−})^{∨}. Thus, there is a K_{x}(ℝ)-equivariant, c-semilinear isomorphism from 𝒜_{?,λ} ⊗_{ℂ} Λ^{q}(𝔭_{x}^{+}) ⊗_{ℂ} V_{λ} to 𝒜_{?,λ} ⊗_{ℂ} Λ^{q}(𝔭_{x}^{−}) ⊗_{ℂ} Λ^{d}(𝔭_{x}^{−})^{∨} ⊗_{ℂ} V_{λ^{♭}}. We can see the latter space, after contracting, as Hom_{ℂ}(Λ^{d−q}(𝔭_{x}^{−}), 𝒜_{?,λ}⊗_{ℂ}V_{λ^{♭}}).

Lemma 3.7.1. Suppose that ∑_{τ,i} λ_{τ,i} = 0. Then 𝒜_{?,λ} = 𝒜_{?,λ^{♭}}.

Proof. An elementary computation shows that ⟨λ + ρ_{x}, λ + ρ_{x}⟩ = ⟨λ^{♭} + ρ_{x}, λ^{♭} + ρ_{x}⟩ if the multiplier factors of λ and λ^{♭} coincide, which is precisely the condition that ∑_{τ,i} λ_{τ,i} = 0. □

Remark 3.7.1. For any w ∈ 𝒲^{1}, λ = w ∗μ satisfies the condition of the lemma because W is defined over ℚ.

Thus, under the conditions of the lemma, we get, for any q = 0, …, d, csemilinear isomorphisms H^{q}(𝔓_{x}, K_{x}; 𝒜_{?} ⊗_{ℂ} V_{λ}) → H^{d−q}(𝔓_{x}, K_{x}; 𝒜_{?} ⊗ V_{λ^{♭}}). The element x(i) ∈ K_{x}(ℝ) acts on 𝒜_{?} by right translation, preserving 𝒜_{?,λ}, and on V_{λ} by r_{λ}(x(i)). Composing the previous map with x(i) gives c-semilinear isomorphisms

(3.7.1)  c_{aut} : H^{q}(𝔓_{x}, K_{x}; 𝒜_{?} ⊗_{ℂ} V_{λ}) → H^{d−q}(𝔓_{x}, K_{x}; 𝒜_{?} ⊗ V_{λ^{♭}}).

The following proposition is proved in [Har97] (Corollary 2.5.9), and allows to express the conjugation c_{B} in terms of automorphic forms. The statement in op. cit. only covers the case w = w_{0}^{1}, or cohomology in degrees 0 and d, but the proof extends to any q = 0, …, d (see [Har97], Remark 2.5.13, 2.).

Proposition 3.7.1. Let W be a representation of G, absolutely irreducible with highest weight μ ∈ Λ_{x}^{+}. Then c_{B} : H_{!}^{d}(W) ⊗ ℂ → H_{!}^{d}(W) ⊗ ℂ takes the Weyl component corresponding to w to the Weyl component corresponding to w^{♭}, and the following diagram commutes up to F^{×}-multiples, where F ⊃ E is a field over which ℰ_{w∗μ} and ℰ_{w^{♭}∗μ} have a canonical model:

\begin{matrix}H^{d−ℓ(w)}(𝔓_{x}, K_{x}; 𝒜_{0} ⊗_{ℂ} V_{w∗μ}) & H^{d−ℓ(w^{♭})}(𝔓_{x}, K_{x}; 𝒜_{0} ⊗_{ℂ} V_{w^{♭}∗μ}) \\ H_{!}^{d−ℓ(w)}(S_{ℂ}, ℰ_{w∗μ}) & H_{!}^{d−ℓ(w^{♭})}(S_{ℂ}, ℰ_{w^{♭}∗μ}) \\ H^{d−ℓ(w)}(𝔓_{x}, K_{x}; 𝒜_{(2)} ⊗_{ℂ} V_{w∗μ}) & H^{d−ℓ(w^{♭})}(𝔓_{x}, K_{x}; 𝒜_{(2)} ⊗_{ℂ} V_{w^{♭}∗μ}). \end{matrix} c_{aut} c_{B} c_{aut}

In this diagram, the vertical arrows are the inclusions from (3.6.2).

We introduce now another operation Fr in terms of automorphic forms, as follows. As in (3.6.2), the space H_{!}^{d−ℓ(w)}(\overline{S}_{ℂ}, ℰ_{(w∗μ)^{c}}) sits between the spaces

H^{d−ℓ(w)}(𝔓\overline{x}, K_{x}; 𝒜_{?} ⊗_{ℂ} V_{(w∗μ)^{c}}) = (𝒜_{?,(w∗μ)^{c}} ⊗_{ℂ} Λ^{d−ℓ(w)}(𝔭_{x}^{−}) ⊗_{ℂ} V_{(w∗μ)^{c}})^{K_{x}(ℝ)}

for ? = 0 and ? = (2). An easy computation shows that 𝒜_{?,(w∗μ)^{c}} = 𝒜_{?,(w∗μ)}, which by Lemma 3.7.1 equals 𝒜_{?,(w∗μ)^{♭}}. On the other hand, there is a natural identification of K_{x}(ℝ)-modules between V_{(w∗μ)^{c}} and Λ^{d}(𝔭_{x}^{+}) ⊗_{ℂ} V_{(w∗μ)^{♭}} (see Subsection 3.3). Finally, contracting via the Killing form, we obtain ℂ-linear isomorphisms

H^{d−ℓ(w)}(𝔓\overline{x}, K_{x}; 𝒜_{?} ⊗_{ℂ} V_{(w∗μ)^{c}}) → H^{d−ℓ(w^{♭})}(𝔓_{x}, K_{x}; 𝒜_{?} ⊗_{ℂ} V_{(w∗μ)^{♭}}),

and an isomorphism in coherent cohomology sitting in between them, which we denote by

(3.7.2)  Fr : H_{!}^{d−ℓ(w)}(\overline{S}_{ℂ}, ℰ_{(w∗μ)^{c}}) → H_{!}^{d−ℓ(w^{♭})}(S_{ℂ}, ℰ_{w^{♭}∗μ}).

Recall that we also have a c-semilinear isomorphism

c_{coh} = H_{!}^{q}(S_{ℂ}, ℰ_{w∗μ}) → H_{!}^{q}(\overline{S}_{ℂ}, ℰ_{(w∗μ)^{c}})

In the next Proposition, c_{B} really denotes the inverse of the map in Proposition 3.7.1, although in terms of Betti conjugation it’s obviously equal to its own inverse, so we use the same notation. Alternatively, we use Proposition 3.7.1 with w and w^{♭} interchanged.

Proposition 3.7.2. Under the assumptions of Proposition 3.7.1, the following ℂ-linear automorphism

c_{B} ∘ Fr ∘c_{coh} : H_{!}^{d−ℓ(w)}(S_{ℂ}, ℰ_{w∗μ}) → H_{!}^{d−ℓ(w)}(S_{ℂ}, ℰ_{w∗μ})

is rational over F, where F ⊃ E is any field over which ℰ_{w∗μ} has a canonical model.

Proof. This is proved exactly as in [Har97], Lemma 2.5.12. □

3.8. Cohomological automorphic representations. Let Aut_{G} denote the set of automorphic representations of G, Aut_{G}^{0} the set of cuspidal automorphic representations, and Temp_{G} ⊂ Aut_{G}^{0} the set of essentially tempered automorphic representations (an automorphic representation π is called essentially tempered if its twist by a character of G(ℚ)\G(𝔸) is tempered at all places). From now on we will only work with essentially tempered representations, although many of the things we say are true in more general cases. We say that π = π_{∞} ⊗ π_{f} ∈ Temp_{G} is cohomological of type μ ∈ R_{x}^{+} if H^{∗}(𝔤_{ℂ}, K_{x}; π_{∞} ⊗_{ℂ} W_{μ}) ≠ 0. We assume almost all the time that W_{μ} is defined over ℚ, so as to place ourselves in the setting of the last subsections. We write W = W_{μ} if μ is understood. The condition of π ∈ Temp_{G} being cohomological is equivalent to the statement that π_{∞} is a discrete series representation that belongs to the L-packet whose infinitesimal character is that of W^{∨}. Moreover, it follows from Theorem II.5.4 of [BW80] (see also VI, 1.4, (1) therein) that π actually contributes to middle-dimensional cohomology only, that is, H^{∗}(𝔤_{ℂ}, K_{x}; π_{∞} ⊗_{ℂ} W_{ℂ}) = H^{d}(𝔤_{ℂ}, K_{x}; π_{∞} ⊗_{ℂ} W_{ℂ}) ≠ 0 (we thank the referee for the precise reference). We let Coh_{G,μ} the set of π ∈ Temp_{G} which are cohomological of type μ. Assume from now on that π ∈ Coh_{G,μ}. As in [Har97], 2.7, we can define the standard L-function of π as L^{S}(s, π, St) = L^{S}(s, BC(π_{0}), St). Here St refers to the standard representation of the L-group of GL_{n} over L, π_{0} is an irreducible constituent of the restriction of π to U(𝔸), and BC(π_{0}) is the base change of π_{0} to an irreducible admissible representation of GL_{n}(𝔸_{L}^{S}), for a big enough finite set of places S of L. The base change is defined locally at archimedean places, at split places, and at places of K where the local unitary group U_{v} and π_{0,v} are unramified. Under our assumptions, it is known that BC(π_{0}) is the restriction to GL_{n}(𝔸_{L}^{S}) of an automorphic representation Π of GL_{n}(𝔸_{L}), so we can actually define L(s, π, St) at all places as L(s, Π, St). We define the motivic normalization by

L^{mot,S}(s, π, St) = L^{S} (s − \frac{n − 1}{2}, π, St) .

The representation π_{f} is defined over a number field E(π). We denote the corresponding model by π_{f,0} (note that E(π) may not always be taken to be the fixed field under the stabilizer of π_{f}). We can take E(π) to be a CM field (see [BHR94], Theorem 4.4.1, and [Har97], 2.6). To make things simpler, we take E(π) to contain L′; this is still a CM field. We let J_{π} = J_{E(π)}, and for each σ ∈ J_{π}, we let π_{f}^{σ} = π_{f,0} ⊗_{E(π),σ} ℂ. Throughout, we make the assumption that π_{f}^{σ} is essentially tempered for any σ. For ? = τ : L′ ↪ ℂ or ? = dR, define

M(π, W)_{?} = Hom_{ℚ[G(𝔸_{f})]}(π_{f,0}, M(W)_{?}).

These data define a Hodge-de Rham structure M(π, W) over L′ with coefficients in E(π), which is pure of weight d−ξ. We let M^{+}(π, W) = R_{L′/(L′)^{+}}M(π, W), a Hodge-de Rham structure over (L′)^{+}, the maximal totally real subfield of L′, with coefficients in E(π). Just as in the case of M(W), we can endow M(π, W) and M^{+}(π, W) with the additional structure of a realization using ℓ-adic cohomology. It is well known that M(π, W) is non-trivial if π ∈ Coh_{G,μ}. When W is understood from the context, we will denote these realizations by M(π) and M^{+}(π) respectively.

3.9. Polarizations. Recall that there is an isomorphism W ≅ W^{∨} ⊗ W_{μ_{ξ}} as representations of G (see Subsection 3.3), with ξ = 2a_{0}. As in [Har97], 2.6.8, this induces a perfect G(𝔸_{f})-equivariant pairing of Hodge-de Rham structures

(3.9.1)  ⟨, ⟩ : M(W) ⊗ M(W) → ℚ(ξ − d).

Remark 3.9.1. The G(𝔸_{f})-action on ℚ(π)(ξ − d) is given by the character ∥ν∥^{−ξ} : G(𝔸) → ℚ^{×}, where ∥ · ∥ is the adelic norm on 𝔸^{×}.

Let w ∈ 𝒲^{1}. It’s easy to see that the Serre dual (ℰ_{w∗μ})′ is isomorphic to ℰ_{w^{♭}∗μ^{∨}}, and we can write Serre duality

(3.9.2)  H_{!}^{2d−i−ℓ(w^{♭})}(S_{ℂ}, (ℰ_{w∗μ})′) × H_{!}^{i−ℓ(w)}(S_{ℂ}, ℰ_{w∗μ}) → ℂ

(note that 2d − i − ℓ(w^{♭}) = d − (i − ℓ(w))). Consider the de Rham pairing ⟨, ⟩_{dR} : M(W)_{dR} ⊗_{L′} M(W)_{dR} → L′. For τ ∈ J_{L′}, there is a complex pairing ⟨, ⟩_{dR,τ} : M(W)_{dR,τ} ⊗_{ℂ} M(W)_{dR,τ} → ℂ, where M(W)_{dR,τ} = M(W)_{dR} ⊗_{L′,τ} ℂ. We will only be explicit for τ = 1, and usually ignore the subscript 1. Then, in terms of Weyl components, the pairing ⟨, ⟩_{dR,1} of (3.9.1) is, up to the ν^{ξ}-twist, Serre duality (3.9.2) with i = d. Thus, we can write

(3.9.3)  ⟨, ⟩_{dR} : H_{!}^{d−ℓ(w^{♭})}(S_{ℂ}, ℰ_{w^{♭}∗μ}) ⊗_{ℂ} H_{!}^{d−ℓ(w)}(S_{ℂ}, ℰ_{w∗μ}) → ℂ.

This pairing is actually L′-rational, in the sense that it descends to a pairing

⟨, ⟩_{dR} : H_{!}^{d−ℓ(w^{♭})}(S_{L′}, ℰ_{w^{♭}∗μ}) ⊗_{L′} H_{!}^{d−ℓ(w)}(S_{L′}, ℰ_{w∗μ}) → L′.

In particular, taking w = 1, we obtain a pairing

⟨, ⟩_{dR} : H_{!}^{0}(S_{L′}, ℰ_{w_{0}^{1}∗μ}) ⊗_{L′} H_{!}^{d}(S_{L′}, ℰ_{μ}) → L′.

We can write these pairings in terms of vector-valued functions. Namely, the pairings (3.9.3) sit (with respect to the inclusions (3.6.2)) between pairings (3.9.4) ⟨, ⟩_{dR} : H^{d−ℓ(w^{♭})}(𝔓_{x}, K_{x}; 𝒜_{?} ⊗_{ℂ} V_{w^{♭}∗μ}) × H^{d−ℓ(w)}(𝔓_{x}, K_{x}; 𝒜_{?} ⊗_{ℂ} V_{w∗μ}) → ℂ

for ? = 0 or (2). Now, as K_{x,ℂ}-representations, V_{w^{♭}∗μ} ≅ V_{w∗μ}^{∨}⊗_{ℂ}Λ^{d}(𝔭_{x}^{+})⊗_{ℂ}V_{μ_{ξ}}. If f (resp. f′) is an element in the first (resp. second) factor of (3.9.4), then we can see f (resp. f′) as an element in (Λ^{d−ℓ(w^{♭})}(𝔭_{x}^{+}) ⊗_{ℂ} 𝒜_{?,w^{♭}∗μ} ⊗_{ℂ} V_{w^{♭}∗μ})^{K_{x}(ℝ)} (resp. in (Λ^{d−ℓ(w)}(𝔭_{x}^{+})⊗_{ℂ} 𝒜_{?,w∗μ} ⊗_{ℂ} V_{w∗μ})^{K_{x}(ℝ)}). After choosing an L′-rational isomorphism Λ^{2d}(𝔭_{x}) ≅ L′, contraction of coefficients and multiplication of automorphic forms, an operation we denote by g ↦ [f(g), f′(g)], defines an element in 𝒜_{?} ⊗_{ℂ} V_{μ_{ξ}}. Keeping in mind our choices for Haar measures, we can write (3.9.4) as

⟨f, f′⟩_{dR} = ∫_{G(ℚ)Z_{G}(𝔸)\G(𝔸)}[f(g), f′(g)]∥ν(g)∥^{ξ}dg.

Remark 3.9.2. We can also write these pairings in terms of scalarvalued automorphic forms, as in [Har97], 2.6.11. To be more precise, let f, f′ be as above. We can now see them as elements f ∈ Hom_{K_{x}(ℝ)}(V_{w^{♭}∗μ}^{∨} ⊗_{ℂ} Λ^{d−ℓ(w^{♭})}(𝔭_{x}^{−}), C^{∞}(G(ℚ)\G(𝔸))) and f′ ∈ Hom_{K_{x}(ℝ)}(V_{w∗μ}^{∨} ⊗_{ℂ} Λ^{d−ℓ(w)}(𝔭_{x}^{−}), C^{∞}(G(ℚ)\G(𝔸))). We assume for simplicity that V_{w∗μ} ⊗ Λ^{d−ℓ(w)}(𝔭_{x}^{+}) is irreducible. This is true, for example, if w = 1 or w = w_{0}^{1}, which are the cases that will concern us in this paper. The general case follows by decomposing this space into irreducible components. It’s easy to see that

V_{w^{♭}∗μ}^{∨} ⊗_{ℂ} Λ^{d−ℓ(w^{♭})}(𝔭_{x}^{−}) ≃ V_{w∗μ} ⊗_{ℂ} Λ^{d−ℓ(w)}(𝔭_{x}^{+}) ⊗_{ℂ} V_{μ_{ξ}}.

Let L ⊂ V_{w∗μ}^{∨} ⊗_{ℂ} Λ^{d−ℓ(w)}(𝔭_{x}^{−}) (resp. L′ ⊂ V_{w^{♭}∗μ}^{∨} ⊗_{ℂ} Λ^{d−ℓ(w^{♭})}(𝔭_{x}^{−})) be the lowest (resp. highest) weight spaces. Then f and f′ define smooth functions on G(𝔸) by restricting to L and L′ and evaluating at a basis element. Then (see [Har97], Proposition 2.6.12)

(3.9.5)  ⟨f, f′⟩_{dR} = ∫_{G(ℚ)Z_{G}(𝔸)\G(𝔸)} f(g)f′(g)∥ν(g)∥^{ξ}dg.

Assume from now on that π satisfies π^{∨} ≅ π ⊗ ∥ν∥^{ξ}. Then ⟨, ⟩ defines polarizations with the same notation

⟨, ⟩ : M(π, W) ⊗_{E(π)} M(π, W) → E(π)(ξ − d)

and

⟨, ⟩ : M^{+}(π, W) ⊗_{E(π)} M^{+}(π, W) → E(π)(ξ − d).

over L′ and L^{′+} respectively.

Remark 3.9.3. Suppose that β : π_{f,0,ℂ} → H_{!}^{0}(S_{ℂ}, ℰ_{w_{0}^{1}∗μ}) is a ℚ[G(𝔸_{f})]- equivariant map. We can see the target as Hom_{K_{x}(ℝ)}(V_{w_{0}^{1}∗μ}^{∨}, 𝒜_{0,w_{0}^{1}∗μ}). For any v ∈ π_{f,0}, the image via β(v) of a lowest weight vector of V_{w_{0}^{1}∗μ}^{∨} in 𝒜_{0} corresponds to a holomorphic form f in π. Similarly, if β : π_{f,0,ℂ} → H_{!}^{d}(S_{ℂ}, ℰ_{μ}), then β gives rise to an antiholomorphic form in π.

3.10. Automorphic quadratic periods. Let w ∈ 𝒲^{1}, and let β be an element of Hom_{ℚ[G(𝔸_{f})]}(π_{f,0}, H_{!}^{d−ℓ(w)}(S_{L′}, ℰ_{w∗μ})). We define Rβ to be the pair (β, c_{coh}(β)). Define Fr Rβ = (Fr c_{coh}(β), Fr β), where Fr is the map defined in (3.7.2). We define the automorphic quadratic period Q(π; β) by

Q(π; β) = ⟨Rβ, Fr Rβ⟩ ∈ E(π) ⊗ ℂ,

where the pairing ⟨, ⟩ is defined in Subsection 3.9. As in [Har97], (2.8.7.4), we see that

Q(π; β) ∼_{E(π)⊗L′} ⟨β, Fr c_{coh}(β)⟩.

Using Proposition 3.7.2, we can replace this by ⟨β, c_{B}(β)⟩.

Remark 3.10.1. We can express Q(π; β) in terms of integrals of the form (3.9.5). Namely, suppose that β gives rise to an automorphic form f on G(𝔸), as in Remark 3.9.2. Using Remark 3.7.1, it’s not hard to see λ(x(i)) = ±1, where λ is the highest weight of V_{w∗μ} ⊗_{ℂ} Λ^{d−ℓ(w)}(𝔭_{x}^{+}). Determining the sign should not be complicated but we don’t care about it in this paper. In any case, the map c_{aut} of (3.7.1) takes f to ±\overline{f}. Moreover, by Proposition 3.7.1, this differs from c_{B} by a rational multiple in L′, and thus

(3.10.1)  Q(π; β) ∼_{E(π)⊗L′} ∫_{G(ℚ)Z(𝔸)\G(𝔸)} f(g)\overline{f}(g)∥ν(g)∥^{ξ}dg.

Remark 3.10.2. When studying L-functions in the next section, we will need to pair forms in π with forms in π^{∨}, rather than with π ≅ π^{∨} ⊗∥ν∥^{−ξ}. To remedy this, we introduce a new operation on automorphic forms and cohomology classes. Fix λ ∈ Λ_{c,x}^{+}, with ξ = ξ(λ). Let λ̂ = λ−μ_{ξ}. Define a map 𝒜_{?} ⊗_{ℂ} V_{λ} → 𝒜_{?} ⊗_{ℂ} V_{λ} ⊗_{ℂ} V_{μ_{−ξ}} by taking f ⊗ v ↦ f̂⊗ v ⊗ 1, where f̂(g) = (2πi)^{ξ}∥ν(g)∥^{ξ}f(g) for f ∈ 𝒜. It’s easy to see that this map is (𝔓_{x}, K_{x}(ℝ))-linear, and hence it thus induces a map in (𝔓_{x}, K_{x})-cohomology. There is a map

H_{!}^{q}(S_{ℂ}, ℰ_{λ}) → H_{!}^{q}(S_{ℂ}, ℰλ̂)
α ↦ α̂

sitting in between as in (3.6.2), which is rational over L′ (see [Har97], 2.8.9.2). Now let β : Hom_{ℚ[G(𝔸_{f})]}(π_{f,0}, H_{!}^{d−ℓ(w)}(S_{L′}, ℰ_{w∗μ})). By Proposition 3.7.1,

c_{B}(β) ∈ Hom_{ℚ[G(𝔸_{f})]}(π_{f,0}, H_{!}^{d−ℓ(w^{♭})}(S_{L′}, ℰ_{w^{♭}∗μ})). Note that w^{♭} ∗ μ − μ_{ξ} =

w^{♭}∗μ^{∨}. Define c_{B}(β)^{∨} to be the composition of α ↦ α̂ : H_{!}^{d−ℓ(w^{♭})}(S_{ℂ}, ℰ_{w^{♭}∗μ}) →

H_{!}^{d−ℓ(w^{♭})}(S_{ℂ}, ℰ_{w∗μ^{∨}}) with

π_{f,0}^{∨} → π_{f,0}^{∨} ⊗ ∥ν∥^{−ξ} ≅ π_{f,0} ⟶^{c_{B}(β)} H_{!}^{d−ℓ(w^{♭})}(S_{ℂ}, ℰ_{w^{♭}∗μ}).

Then we define Q(π, π^{∨}; β) = ⟨β, c_{B}(β)^{∨}⟩, where this pairing comes from Serre duality between H_{!}^{d−ℓ(w)}(S_{ℂ}, ℰ_{w∗μ}) and H_{!}^{d−ℓ(w^{♭})}(S_{ℂ}, ℰ_{w^{♭}∗μ^{∨}}) (see (3.9.2)). As in [Har97], 2.8.9.5, it follows from (3.10.1) that

Q(π, π^{∨}; β) ∼_{E(π)⊗L′} ∫_{G(ℚ)Z(𝔸)\G(𝔸)} f(g)ˆ\overline{f}(g)dg,

where f lifts β as above. Thus

(3.10.2)  Q(π, π^{∨}; β) ∼_{E(π)⊗L′} (2πi)^{ξ}Q(π; β).

As in [Har97], 2.9, we can extend the definition of automorphic quadratic periods to the case where we twist by an algebraic Hecke character ψ. The main reason that we switched from χ to ψ in the notation for algebraic Hecke characters is that in Section 5 we will combine the results of Sections 2 and 4, but the algebraic Hecke characters will be slightly different (although closely related; see Section 5 for details). Recall that we have fixed an orthogonal basis of V, giving rise to a particular map x : 𝕊 → G_{ℝ} (see (3.2.1)). This depends on Φ, but it will be fixed throughout. Note that G ⊂ Res_{L/ℚ} GL_{V}, and thus there is a map det : G → T^{L} = Res_{L/ℚ} 𝔾_{m,L}, and we can consider the composition det ∘x : 𝕊 → (T^{L})_{ℝ}. Using Φ to identify (T^{L})_{ℝ} with ∏_{τ∈Φ} 𝕊, the map det ∘x sends an element z to the tuple (z^{r_{τ}} \overline{z}^{s_{τ}})_{τ∈Φ}, and thus det ∘x = ∏_{τ∈Φ} h_{\{τ\}}^{r_{τ}}h_{\{\overline{τ}\}}^{s_{τ}} with the notation of Subsection 2.6 (using sets Ψ consisting of a single element). Just as in §1 of [Har93], we can normalize the CM periods p(ψ; Ψ) in such a way that

(\frac{p(ψ; det ∘x)}{∏_{τ∈Φ} p(ψ; \{τ\})^{r_{τ}} p(ψ; \{\overline{τ}\})^{s_{τ}}} )^{γ} = \frac{p(ψ^{γ}; det ∘x)}{∏_{τ∈Φ} p(ψ^{γ}; \{τ\})^{r_{τ}} p(ψ^{γ}; \{\overline{τ}\})^{s_{τ}}}

for all γ ∈ Aut(ℂ). In particular,

\frac{p(ψ; det ∘x)}{∏_{τ∈Φ} p(ψ; \{τ\})^{r_{τ}} (ψ; \{\overline{τ}\})^{s_{τ}}} ∈ ℚ(ψ).

Moreover, Lemma 1.6 of op. cit. implies that p(ψ; \{\overline{τ}\}) ∼ p(ψ^{ι}; \{τ\}), and using 1.4 (c) of op. cit., it follows that

(3.10.3)  p(ψ; det ∘x) ∼_{E(ψ)} ∏_{τ∈Φ} p(ψ^{r_{τ}} (ψ^{ι})^{s_{τ}}; {τ}).

Here we are taking E(ψ) to be a sufficiently big field containing ℚ(ψ) and the relevant reflex fields (in practice, we will take E(ψ) = L′ℚ(ψ)).

Remark 3.10.3. After conjugating ψ and η, as explained in Subsection 2.6, as well as the Shimura datum (T^{L}, det ∘x)), we can consider the tuple p(ψ; det ∘x) = (p(ψ^{ρ}; (det ∘x))^{ρ})_{ρ∈J_{E(ψ)}} as an element in (E(ψ) ⊗ ℂ)^{×}, and (3.10.3) remains valid.

Let S(det ∘x) be the Shimura variety attached to the pair (T^{L}, det ∘x). The map det : G → T^{L} sends X to {det ∘x}, and thus defines a map det : S_{ℂ} → S(det ∘x)_{ℂ}, which is rational over E(T^{L}, det ∘x) ⊃ E. Let W_{η} be the 1dimensional ℚ(η)-vector space on which η acts. This defines the local system W̃_{η} on ℚ(η)-vector spaces over S(det ∘x)(ℂ) that is used in the construction of the Hodge-de Rham structure M(W_{η}). We can pull it back to a local system det^{∗}(W̃_{η}) on ℚ(η)-vector spaces over S(ℂ). In terms of parameters, note that η ∘ det : T_{ℂ} → 𝔾_{m,ℂ}, in our parametrization of X^{∗}(T), equals

μ(η) = ((n_{τ} − n\overline{τ} , …, n_{τ} − n\overline{τ})_{τ∈Φ}; n ∑_{τ∈Φ} n\overline{τ} ) .

Now, let the notation and assumptions be as in Subsection 3.8. The local system W̃ ⊗ det^{∗}(W̃_{η}) on ℚ(η)-vector spaces is attached to the absolutely irreducible representation of G_{ℚ(η)} whose highest weight is μ + μ(η). Suppose that ψ ∈ X(η^{−1}), and let π ⊗ ψ = π ⊗ ψ ∘ det. We can consider the Hodge-de Rham structure M(π ⊗ ψ, W_{μ+μ(η)}), which is defined over L′ and has coefficients in E(π, ψ) = E(π) ⊗ E(ψ). Note that W_{μ+μ(η)} is only defined over ℚ(η), and rarely over ℚ, but we can still construct the Hodge-de Rham structures as in Theorems 2.2.7 and 2.3.1 of [Har94].

Let β be an element as in Subsection 3.10 for M(π, W). Suppose that

s_{ψ} : E(ψ) → H_{!}^{0}(S(det ∘x)_{ℂ}, ℰ_{η})

is a nonzero ℚ[T^{L}(𝔸_{f})]-linear map. Here ℰ_{η} refers to the automorphic vector bundle attached to the character η. Assume that s_{ψ} is rational for the de Rham structure, that is, rational with respect to the canonical model of ℰ_{η} over L′. Such nonzero s_{ψ} is unique up to an (E(ψ) ⊗ L′)^{×}-multiple (using the given embedding of L′ to view E(ψ) ⊗ L′ ⊂ E(ψ) ⊗ ℂ). Pulling s_{ψ} back to S_{ℂ} via det, and combining it with β, we form

(3.10.4)  β(ψ) = β ⊗ s_{ψ} : π_{f,0} ⊗ E(ψ) → H_{!}^{d−ℓ(w)}(S_{L′}, ℰ_{w∗(μ+μ(η))}).

We can similarly take a nonzero map s_{ψ^{−1}} to define β(ψ^{−1}). As in Subsection 3.10, we define

Q(π; ψ; β) = ⟨R(β(ψ)), Fr R(β(ψ^{−1}))⟩ ∈ E(π, ψ) ⊗ ℂ,

where the pairing comes from the polarization on M(π ⊗ ψ, W_{μ+μ(η)}). The calculations in [Har97], 2.9, show that

(3.10.5) Q(π; ψ; β) ∼_{E(π,ψ)⊗L′} Q(π; β)p(ψ; det ∘x)^{−1}p(ψ^{−1}; det ∘\overline{x})^{−1}.

Remark 3.10.4. As in Remark 3.10.2, we modify these quadratic periods, and define Q(π, π^{∨}; ψ; β) analogously.

4. Critical values of L-functions

In this section we prove our main result (Theorem 4.5.1) concerning the critical values of L^{S,mot}(s, π ⊗ ψ, St) for a cohomological automorphic representation π of a unitary group G and an algebraic Hecke character ψ of L. In the first subsection, we introduce the double hermitian space in order to set up the doubling method. In the following subsections, we define a family of differential operators, Eisenstein series and Piatetski-Shapiro-Rallis zeta integrals, invoking Li’s formula relating them to the standard L-function. In Subsection 4.5, we analyze the rationality properties of these zeta integrals by carefully choosing the sections defining the Eisenstein series, and relate the remaining factors of Li’s formula to automorphic quadratic periods and CM periods. Then we put everything together to prove Theorem 4.5.1.

4.1. The double hermitian space. Let L/K be a CM extension, and let (V, h_{V}) be any hermitian vector space over L/K of dimension n, as in Subsection 3.1. Define −V to be the space V with hermitian form −h_{V}, and let 2V = V ⊕ (−V), equipped with its natural hermitian form h_{2V} = h_{V} ⊕ h_{−V}. We fix an orthogonal basis β = {v_{1}, …, v_{n}} for (V, h_{V}) (and a CM type Φ), after which we can choose x : 𝕊 → G_{ℝ} and x^{−} : 𝕊 → G_{ℝ} as before, where G = GU(V) = GU(−V). We let G^{(2)} = GU(2V). This is a unitary group with signatures (n, n) at all places. Choosing the orthogonal basis β^{(2)} = {(v_{1}, 0), …, (v_{n}, 0), (0, v_{1}), …, (0, v_{n})} of 2V, we construct x^{(2)} : 𝕊 → G_{ℝ}^{(2)} in a similar fashion as x. Let G^{♯} ⊂ G × G be subgroup of pairs with the same similitude factor, and let x^{♯} : 𝕊 → G_{ℝ}^{♯} be the map (x, x^{−}). We denote by (G, X), (G^{−}, X^{−}) = (G, X^{−}) = (G, \overline{X}), (G^{(2)}, X^{(2)}) and (G^{♯}, X^{♯}) the corresponding Shimura data, and by S, S^{−}, S^{(2)} and S^{♯} the corresponding Shimura varieties. All the reflex fields are contained in the Galois closure L′ ⊂ \overline{ℚ} of L in \overline{ℚ}, and the reflex field of (G^{(2)}, X^{(2)}) is ℚ. Notice that there is a natural embedding

i : (G^{♯}, X^{♯}) ⊂ (G^{(2)}, X^{(2)}).

We denote by the same letter the embedding of Shimura varieties i : S^{♯} → S^{(2)}. Our choices give the following identifications:

K_{x,ℂ} ≅ ( ∏_{τ∈Φ} GL_{r_{τ} ,ℂ} × GL_{s_{τ} ,ℂ}) × 𝔾_{m,ℂ},

K_{x^{−},ℂ} ≅ ( ∏_{τ∈Φ} GL_{s_{τ} ,ℂ} × GL_{r_{τ} ,ℂ}) × 𝔾_{m,ℂ},

K_{x^{♯},ℂ} ≅ ( ∏_{τ∈Φ} GL_{r_{τ} ,ℂ} × GL_{s_{τ} ,ℂ} × GL_{s_{τ} ,ℂ} × GL_{r_{τ} ,ℂ}) × 𝔾_{m,ℂ},

K_{x^{(2)},ℂ} ≅ ( ∏_{τ∈Φ} GL_{n,ℂ} × GL_{n,ℂ}) × 𝔾_{m,ℂ}.

The inclusion K_{x^{♯},ℂ} ⊂ K_{x^{(2)},ℂ} coming from the embedding of Shimura data is explicitly given, at each place τ, by taking a tuple of matrices (A, D, A^{−}, D^{−}) (with A, D^{−} ∈ GL_{r_{τ}} and D, A^{−} ∈ GL_{s_{τ}}) to the pair

(( \begin{matrix}A & 0 \\ 0 & A^{−} \end{matrix} ) , ( \begin{matrix}D & 0 \\ 0 & D^{−} \end{matrix} )) ∈ GL_{n} × GL_{n} .

For future reference, we write the highest weight for the action of K_{x^{♯},ℂ} on the 1-dimensional space Λ^{2d}(𝔭_{x^{♯}}^{+}). In the usual parametrization, where we split 2n = r_{τ} + s_{τ} + s_{τ} + r_{τ}, it is given by

((s_{τ} , …, s_{τ}; −r_{τ} , …, −r_{τ}; r_{τ} , …, r_{τ}; −s_{τ} , …, −s_{τ})_{τ∈Φ}; 0) .

Suppose that (V, r) (resp. (V^{−}, r^{−})) is an irreducible representation of K_{x,ℂ} (resp. K_{x^{−},ℂ}). Thus, V is given as a tuple (V_{τ})_{τ∈Φ}, with V_{τ} an irreducible representation of GL_{r_{τ} ,ℂ} × GL_{s_{τ} ,ℂ}, and a characer χ_{V} of 𝔾_{m,ℂ}. We similarly define V_{τ}^{−} and χ_{V^{−}}. We define an irreducible representation (V, V^{−})^{♯} of K_{x^{♯},ℂ} by taking the family (V_{τ} , V_{τ}^{−})_{τ∈Φ} and the character χ_{V} χ_{V^{−}}. If V (resp. V^{−}) has highest weight λ = ((a_{τ,1}, …, a_{τ,n})_{τ∈Φ}; a_{0}) ∈ Λ_{c,x}^{+} (resp. λ^{−} = ((a_{τ,1}^{−}, …, a_{τ,n}^{−})_{τ∈Φ}; a_{0}^{−}) ∈ Λ_{c,x^{−}}^{+}), then, in the usual parametrization, the highest weight of (V, V^{−})^{♯} is

(λ, λ^{−})^{♯} = ((a_{τ,1}, …, a_{τ,n}, a_{τ,1}^{−}, …, a_{τ,n}^{−})_{τ∈Φ}; a_{0} + a_{0}^{−}) .

We denote by ℰ_{(λ,λ^{−})^{♯}} the corresponding automorphic vector bundle over S_{ℂ}^{♯}. We can also see it as the pullback via the natural map S_{ℂ}^{♯} ↪ S_{ℂ} × \overline{S}_{ℂ} of the external tensor product of ℰ_{λ} and ℰ_{λ^{−}}. We let (V^{∗}, r^{∗}) be the representation of K_{x^{−},ℂ} obtained by taking the dual (V^{∨}, r^{∨}) of (V, r), seen as a representation of K_{x^{−},ℂ} via swapping the GL-factors at each place in the identifications above. For any ℓ ∈ ℤ, define λ^{♯}(ℓ) = (λ, λ^{∗})^{♯} ⊗ ν^{ℓ}, so that

λ^{♯}(ℓ) = ((a_{τ,1}, …, a_{τ,n}, −a_{τ,n}, …, −a_{τ,1})_{τ∈Φ}; ℓ) .

4.2. A family of differential operators. Fix an integer m, and define

Λ_{m} = ((−m, …, −m; m, …, m)_{τ∈Φ}; 0) ∈ Λ_{c,x^{(2)}}^{+},

where there are n entries of the form −m and n of the form m. This is the highest weight of the character K_{x^{(2)},ℂ} → 𝔾_{m,ℂ} defined by

((g_{τ} , g′_{τ})_{τ∈Φ}; z) → ∏_{τ∈Φ} det(g_{τ})^{−m} det(g′_{τ})^{m}

(for g_{τ} , g′_{τ} ∈ GL_{n}). We write 𝕍_{m} for ℂ, endowed with this action. In particular, the corresponding automorphic vector bundle over S_{ℂ}^{(2)} is a line bundle which we denote by ℰ_{m} = ℰ_{Λ_{m}}. It has a canonical model over L′. The pullback i^{∗}ℰ_{m} is the automorphic vector bundle over S_{ℂ}^{♯}, whose associated K_{x^{♯},ℂ}-representation is the 1-dimensional representation with highest weight

i^{∗}Λ_{m} = ((−m, …, −m; m, …, m; −m, …, −m; m, …, m)_{τ∈Φ}; 0) .

Suppose that (V, r) is an irreducible representation of K_{x,ℂ}, and look at the representation V_{i^{∗}Λ_{−m}} ⊗_{ℂ}V_{λ^{♯}(ℓ)}, for another integer ℓ ∈ ℤ. In the terminology of [Har86], taking into account the different sign conventions, this representation is positive if and only if a_{τ,1} + m ≤ 0 and −a_{τ,n} + m ≤ 0 for every τ ∈ Φ. The following Proposition follows from 7.11.11 of op. cit..

Proposition 4.2.1. Let the notation and assumptions be as above, and let η be an algebraic character of T^{L} = Res_{L/ℚ} 𝔾_{m,L}, identified with the tuple of integers (m_{τ})_{τ∈J_{L}}. Let μ = ((a_{τ,1}, …, a_{τ,n})_{τ∈Φ}; a_{0}) ∈ Λ_{x}^{+}, and let Λ = Λ(μ; η^{−1}) = w_{0}^{1} ∗ (μ − μ(η)) ∈ Λ_{c,x}^{+}. Then, for each m such that

(4.2.1) \frac{n}{2} ≤ m ≤ min\{−a_{τ,s_{τ} +1} + s_{τ} + m_{τ} − m\overline{τ} , a_{τ,s_{τ}} + r_{τ} + m\overline{τ} − m_{τ} \}_{τ∈Φ},

there exists a nonzero, L′-rational, differential operator

Δ_{m}(Λ; ℓ) : ℰ_{m}|_{S_{ℂ}^{♯}} → ℰ_{Λ^{♯}(ℓ)}.

Remark 4.2.1. These differential operators extend to toroidal compactifications and canonical extensions, and the functorial properties of the latter ([Har90]) give rise to L′-rational maps

Δ_{m}(Λ; ℓ) : H^{0}(S_{L′}^{(2)}, ℰ_{m}^{can}) → H^{0}(S_{L′}^{♯}, ℰ_{Λ^{♯}(ℓ)}^{can}).

Using (3.6.3), we get a differential operator

Δ_{m}(Λ; ℓ) : H^{0}(𝔓_{x^{(2)}}, K_{x^{(2)}}; 𝒜(G^{(2)}) ⊗_{ℂ} 𝕍_{m}) → H^{0}(𝔓_{x^{♯}}, K_{x^{♯}}; 𝒜(G^{♯}) ⊗_{ℂ} V_{Λ^{♯}(ℓ)}).

As in Section 7 of [Har86], Δ_{m}(Λ; ℓ) is induced by an element ∂_{m}(Λ; ℓ) ∈ U(𝔭_{x^{(2)}}^{+}) ⊗_{ℂ} Hom(𝕍_{m}, V_{Λ^{♯}(ℓ)}). In this description, the universal enveloping algebra U(𝔭_{x^{(2)}}^{+}) acts by differentiating an element of 𝒜(G^{(2)}) and restricting to G^{♯}(𝔸).

4.3. Eisenstein series. For this subsection, we follow [Har07] (see also [Har97], [Har93] and [Har08]). We will use Siegel-type Eisenstein series, and for this it is more convenient to identify the group G^{(2)} with the similitude unitary group over ℚ attached to L/K and the skew-hermitian matrix S_{n} ∈ GL_{2n}(ℚ) ⊂ GL_{2n}(L) given by

S_{n} = ( \begin{matrix}0 & I_{n} \\ −I_{n} & 0 \end{matrix} ) .

(In the definition of hermitian spaces and their associated groups, we might equally work with skew-hermitian forms instead of hermitian forms). Let GU(S_{n}) denote the similitude unitary group attached to the matrix S_{n} over L/K. Thus, for a ℚ-algebra R, GU(S_{n})(R) consists of matrices X ∈ GL_{2n}(L⊗ R) such that X^{∗}S_{n}X = ν(X)S_{n} for some scalar ν(X) ∈ R^{×}. To identify G^{(2)} with GU(S_{n}), let α ∈ L be a nonzero totally imaginary element, that is, such that ι(α) = −α. Fix an L-basis β = {v_{1}, …, v_{n}} of V, and let Q ∈ GL_{n}(L) be the matrix of h_{V} in this basis, so that Q_{ij} = h_{V} (v_{i}, v_{j}). Consider the L-basis β^{(2)} = {(v_{1}, 0), …, (v_{n}, 0), (0, v_{1}), …, (0, v_{n})} of 2V, and let Q^{(2)} ∈ GL_{2n}(L) be the matrix of h_{2V} with respect to β^{(2)}. Thus,

Q^{(2)} = ( \begin{matrix}Q & 0 \\ 0 & −Q \end{matrix} ) .

Let η ∈ GL_{2n}(L) be the matrix

η = ( \begin{matrix}I_{n} & I_{n} \\ \frac{α}{2}Q & −\frac{α}{2}Q \end{matrix} ) .

Then the map sending g ∈ G^{(2)}(R) to η[g]_{β^{(2)}}η^{−1} (where [g]_{β^{(2)}} is the matrix of g with respect to β^{(2)}) gives an isomorphism G^{(2)} ≅ GU(S_{n}). We can also write this isomorphism in terms of another convenient basis of 2V. Namely, let γ = {v′_{1}, …, v′_{2n}}, where v′_{i} = (v_{i}, v_{i}) for i = 1, …, n and v′_{i} = (v_{i}, −v_{i}) for i = n + 1, …, 2n. Then we can write the isomorphism g ↦ η[g]_{β^{(2)}}η^{−1} as g ↦ η′[g]_{γ}η^{′−1}, where

η′ = ( \begin{matrix}I_{n} & 0 \\ 0 & \frac{α}{2}I_{n} \end{matrix} ) .

We will identify G^{(2)} with GU(S_{n}) using these choices. Given g ∈ G^{(2)}(R) (for some ℚ-algebra R), we write

g = ( \begin{matrix}A & B \\ C & D \end{matrix} )

for the description of g as a 2n × 2n-matrix in GU(S_{n})(R), and call the entries A(g) = A, B(g) = B, C(g) = C, D(g) = D, the Siegel coordinates of g. We let GP ⊂ G^{(2)} be the subgroup described by the condition C(g) = 0. This is a rational, maximal parabolic subgroup of G^{(2)}. For an algebraic Hecke character α of L, consider the induced representation I_{n}(s, α) for s ∈ ℂ, as in [Har07], (1.2). We will only treat the case where α is trivial. The case of more general α (or more precisely, α with infinity type (−κ, 0) at each τ ∈ Φ) should follow exactly like our case, but for simplicity of notation we will only concentrate on trivial α. We thus denote I(s) = I_{n}(s, 1). Concretely,

I(s) = {f : G^{(2)}(𝔸) → ℂ : f(pg) = δ_{GP,𝔸}(p, s)f(g), g ∈ G^{(2)}(𝔸), p ∈ GP (𝔸)},

where δ_{GP,𝔸}(p, s) = ∥N_{L/K} det A(p)∥_{𝔸_{K}}^{\frac{n}{2} +s}∥ν(p)∥_{𝔸_{K}}^{− \frac{n^{2}}{2} −ns}. We similarly define the local inductions I(s)_{v} and finite and archimedean inductions I(s)_{f} and I(s)_{∞}, and we require the functions to be K_{x^{(2)}}(ℝ)-finite. A section of I(s) is a function φ(·, ·) that to each s ∈ ℂ assigns an element φ(·, s) ∈ I(s), with a certain continuity property (see [Har97], (3.1.8)). We define local sections similarly. The first variable is usually denoted by g, meaning that φ(g, s) ∈ I(s) is a function of g ∈ G^{(2)}(𝔸). For Re(s) ≫ 0, the Eisenstein series

E_{φ,s}(g) = ∑_{γ∈GP (ℚ)\G^{(2)}(ℚ)} φ(γg, s)

converges absolutely to an automorphic form on G^{(2)}(𝔸), and this extends meromorphically to a function of s ∈ ℂ. For normalization purposes, we also include a possible shift in the construction. Namely, suppose that φ(g, s) is a section but with φ(g, s) ∈ I(s+a) for some fixed a ∈ ℤ. Then we can still form the Eisenstein series E_{φ,s}(g) using the same formula. Of course, if we define φ′(g, s) = φ(g, s − a), this is a true section and the corresponding Eisenstein series satisfy E_{φ,s}(g) = E_{φ′,s+a}(g). For an integer m such that 2m ≥ n, consider the elements

𝕁_{m} (g, s + m − \frac{n}{2}) ∈ I (s + m − \frac{n}{2})_{∞}

defined in [Har07], (1.2.7) (taking κ = 0; note that there is a misprint: there should be an extra factor ν(g)^{ns} multiplying the expression, so that it transforms correctly under δ). These are constructed from certain elements 𝕁_{m} ∈ I(m − \frac{n}{2})_{∞} which generate irreducible, unitarizable (𝔤_{ℂ}^{(2)}, K_{x^{(2)}}(ℝ))- modules 𝔻_{m} ⊂ I(m − \frac{n}{2})_{∞} (denoted by 𝔻(m, 0) in op. cit.). This module has the property that its generated by its 𝔭_{x^{(2)}}^{−}-torsion, and its lowest K_{x^{(2)}}- type is given by the character Λ_{−m}. Then

H^{0}(𝔓_{x^{(2)}}, K_{x^{(2)}}; 𝒜(G^{(2)}) ⊗_{ℂ} 𝕍_{m}) ≅ Hom_{𝔤_{ℂ}^{(2)},K_{x^{(2)}} (ℝ)}(𝔻_{m}, 𝒜(G^{(2)})).

Combining this with (3.6.3), we obtain an isomorphism

(4.3.1)  Hom_{𝔤_{ℂ}^{(2)},K_{x^{(2)}} (ℝ)}(𝔻_{m}, 𝒜(G^{(2)})) ≅ H^{0}(S_{ℂ}^{(2)}, ℰ_{m}^{can}).

Suppose that φ_{f}(·, s) ∈ I(s)_{f} is a section at the finite idèles. Define

(4.3.2) φ(g, s) = 𝕁_{m} (g, s + m − \frac{n}{2}) ⊗ φ_{f} (g, s + m − \frac{n}{2}) ∈ I (s + m − \frac{n}{2}) .

If m > n, then E_{φ,s}(g) has no pole at s = 0 ([Shi83b]), so we can speak of the automorphic form E_{φ} = E_{φ,0} on G^{(2)}(𝔸). Using (4.3.1), we can define an element also denoted by E_{φ} ∈ H^{0}(S_{ℂ}^{(2)}, ℰ_{m}^{can}), which is the map 𝔻_{m} → 𝒜(G^{(2)}) sending 𝕁_{m} to E_{φ}. The rationality properties of E_{φ} are dictated by the rationality properties of its constant term φ, which we can identify with a holomorphic automorphic form on the point boundary stratum attached to (G^{(2)}, X^{(2)}) and the parabolic GP. We summarize these rationality properties in the next proposition, which follows from the results and techniques of [Har84], [Har86] and [GH93].

Proposition 4.3.1. Let m > n, and let F ⊂ \overline{ℚ} be a number field containing L′. Then E_{φ} ∈ H^{0}(S_{ℂ}^{(2)}, ℰ_{m}^{can}) is rational over F if and only if φ_{f} (·, m − \frac{n}{2}) takes values in (2πi)^{enm}F ℚ^{ab} and satisfies the following reciprocity law: if g ∈ G^{(2)}(𝔸_{f}) and a ∈ 𝔸_{f}^{×} is an element such that art_{ℚ}(a) ∈ Gal(ℚ^{ab}/F ∩ℚ^{ab}), then

((2πi)^{−enm}φ_{f} (g, m − \frac{n}{2}))^{art_{ℚ}(a)} = (2πi)^{−enm}φ_{f} (g, m − \frac{n}{2}) .

Proof. First of all, we need to identify the Shimura datum (G_{P} , F_{P}), in the notation of [Har86], Section 5. After going through the definitions, we find that, in Siegel coordinates, G_{P} = G_{h}A_{P} ⊂ G^{(2)}, where for a ℚ-algebra R

G_{h}(R) = \{aI_{2n} ∈ G(R), a ∈ (L ⊗ R)^{×}, a\overline{a} ∈ R^{×}\}

and

A_{P} (R) = \{( \begin{matrix}aI_{n} & 0 \\ 0 & dI_{n} \end{matrix} ) ∈ G(R), a, d ∈ R^{×}\} .

The boundary component F_{P} in our case is a single point, corresponding to the map h_{P} : 𝕊 → G_{P,ℝ} described by

h_{P} (z) = ( \begin{matrix}z\overline{z}I_{n} & 0 \\ 0 & I_{n} \end{matrix} ) .

The constant term of E_{φ}, identified with a holomorphic automorphic form on Sh(G_{P} , h_{P}), is a section of a certain automorphic line bundle ℰ_{m,P} that we can obtain as follows. There is a character λ̃ of P, whose restriction λ to G_{P} gives rise to this line bundle, and whose restriction to P (ℝ)^{+} (the elements with positive multiplier) is the character δ_{GP,ℝ}(·, m− \frac{n}{2})^{−1}. Putting all together, we obtain that the character λ̃ : P → 𝔾_{m,ℚ} is given in Siegel coordinates by

λ̃(p) = (N_{L/ℚ} det(A(p)))^{−m}ν(p)^{enm}

(recall that e = [K : ℚ]), and ℰ_{m,P} is given by its restriction λ : G_{P} → 𝔾_{m,ℚ}. Note that the restriction of ℰ_{m,P} to Sh(𝔾_{m,ℚ}, N) (where N is the norm) is the Tate automorphic vector bundle ℚ(−enm), and that Sh(G_{P} , h_{P}) is covered by the translates of Sh(𝔾_{m,ℚ}, N). Here we are including 𝔾_{m,ℚ} in G_{P} by taking a to the diagonal matrix with A(g) = aI_{n}, D(g) = I_{n}. The rest of the proof follows from the arguments in §3, (A.2.4) and (A.2.5) of [GH93]. □

Now, let m > n be an integer satisfying (4.2.1), and φ defined by (4.3.2) with φ_{f} satisfying the hypotheses of Proposition 4.3.1. Also, let ℓ be an arbitrary integer. Using the usual action of U(𝔭_{x^{(2)}}^{+}) on smooth functions on G^{(2)}(ℝ), we can define a section Δ_{m}(Λ; ℓ)φ by means of the element ∂_{m}(Λ; ℓ). Then

Δ_{m}(Λ; ℓ)E_{φ} ∈ H^{0}(S_{ℂ}^{♯}, ℰ_{Λ^{♯}(ℓ)}^{can}) ≅ H^{0}(𝔓_{x^{♯}}, K_{x^{♯}}; 𝒜^{♯} ⊗_{ℂ} V_{Λ^{♯}(ℓ)})

is represented as a scalar automorphic form, as in Remark 3.9.2, by the restriction to G^{♯}(𝔸) of E_{Δ_{m}(Λ;ℓ)φ}. We can similarly define Δ_{m}(Λ; ℓ)𝕁_{m}.

4.4. Piatetski-Shapiro-Rallis zeta integrals. Let π be a cuspidal automorphic representation of G(𝔸). We always consider π (rather, the space on which π acts) to be a specific irreducible subspace of 𝒜_{0}(G). Similarly for its dual and their twists. Let

(·, ·)_{G} : π ⊗ π^{∨} → ℂ

be the pairing given as

(f, f′)_{G} = ∫_{Z(A)G(ℚ)\G(𝔸)} f(g)f′(g)dg.

Suppose that φ(g, s) is a section, f ∈ π and f′ ∈ π^{∨}. The (modified) Piatetski-Shapiro-Rallis zeta integral is defined to be

Z(s, f, f′, φ) = ∫_{Z^{♯}(𝔸)G^{♯}(ℚ)\G^{♯}(𝔸)} E_{φ,s}(i(g, g′))f(g)f′(g′)dgdg′,

where Z^{♯} is the center of G^{♯}. Decompose π = ⊗′_{v}π_{v}, π^{∨} = ⊗′_{v}π_{v}^{∨}, where v runs through places of ℚ, and suppose that f and f′ are factorizable relative to these decompositions. Thus, f = ⊗′_{v}f_{v} and f′ = ⊗′_{v}f′_{v}. At almost all places, π_{v} is unramified, and f′_{v} and f′_{v} are normalized spherical vectors with (f_{v}, f′_{v}) = 1. Suppose as well that the section φ(g, s) is factorizable as ∏′_{v} φ_{v}. Define the local integrals

Z_{v}(s, f, f′, φ) = ∫_{U_{v}} φ_{v}(i(h_{v}, 1), s)c_{f,f′,v}(h_{v})dh_{v},

where U_{v} is the local unitary group for V. Here

c_{f,f′,v}(h_{v}) = (f_{v}, f′_{v})^{−1}(π_{v}(h_{v})f_{v}, f′_{v})

is a normalized matrix coefficient for π_{v}. Let S be a big enough set of places (namely, we can take S to contain the set consisting of the archimedean places, the set of places at which G_{v} is not quasi-split, and the set of places where π_{v} is ramified or f_{v} or f′_{v} is not a standard spherical vector). Define

d^{S}(s) = ∏_{j=0}^{n−1} L^{S}(2s + n − j, ε_{L}^{j}).

Then Li proved in [Li92] that

(4.4.1)  d^{S} (s − \frac{n}{2}) Z (s − \frac{n}{2}, f, f′, φ) =

(f, f′)_{G} ∏_{v∈S} Z_{v} (s − \frac{n}{2}, f, f′, φ) L^{mot,S} (s, π, St) .

Writing S = S_{∞} ∪S_{f}, we let Z_{f}(s, f, f′, φ) be the product of the local integrals for v ∈ S_{f}, and Z_{∞}(s, f, f′, φ) be the local archimedean factor.

Lemma 4.4.1. With the previous definitions,

d^{S} (m − \frac{n}{2}) ∼_{ℚ} (D_{K}^{1/2})^{⌊ \frac{n+1}{2} ⌋}δ([ε_{L}])^{⌊ \frac{n}{2} ⌋}(2πi)^{e(2mn−n(n−1)/2)}.

Proof. Each factor L^{S}(2m−j, ε_{L}^{j}) differs from the full L-function L(2m−j, ε_{L}^{j}) by a multiple in ℚ^{×}. Consider first the case when j is even, so that L(2m − j, ε_{L}^{j}) = ζ_{K}(2m − j). Since 2m − j is a positive even integer, we can write

ζ_{K}(2m − j) ∈ (D_{K}^{1/2}(2πi)^{e(2m−j)})ℚ^{×}

using the Klingen-Siegel Theorem. The factors with j odd correspond to special values of L(s, ε_{L}) at odd positive integers. These integers are critical for Res_{K/ℚ}[ε_{L}]. Deligne’s conjecture (a theorem in this case) says then that

L(2m − j, ε_{L}) ∼_{ℚ} c^{−}([ε_{L}])(2πi)^{e(2m−j)}

for 0 ≤ j ≤ n − 1 odd. Note that c^{−}([ε_{L}]) ∼_{ℚ} δ([ε_{L}]) by (2.2.1), (2.2.2) and Remark 2.2.1. The final formula follows by combining all these factors. □

4.5. The main theorem. In this subsection we derive our main theorem for critical values of the standard L-function of an automorphic representation π, in the range of absolute convergence. We recollect the notation that we used in the previous subsections. From now on, suppose that π ∈ Coh_{G,μ}, with W_{μ} defined over ℚ, and suppose that π^{∨} ≃ π ⊗ ∥ν∥^{ξ}, where ξ = ξ(μ). Recall that we are fixing a CM type Φ for L/K. We let M(π) ∈ ℛ(L′)_{E(π)} be the Hodgede Rham structure M(π, W). We fix an algebraic Hecke character ψ of L, of infinity type (m_{τ})_{τ∈J_{L}}. We denote by η the corresponding algebraic character of T^{L} = Res_{L/ℚ} 𝔾_{m,L}. We let Λ = Λ(μ; η^{−1}) and ℓ = n ∑_{τ∈Φ}(m_{τ} −m\overline{τ}). We fix an integer m > n satisfying the inequalities (4.2.1), and we let Δ_{m} = Δ_{m}(Λ; ℓ). Recall that we are taking E(ψ) = L′ℚ(ψ). We let E(π, ψ) = E(π) ⊗ E(ψ). We assume that π contains antiholomorphic automorphic forms. As in Remark 3.9.3, we can find a nonzero, L′-rational, G(𝔸_{f})-equivariant map β : π_{f,0} → H_{!}^{d}(S_{ℂ}, ℰ_{μ}). Note that μ^{c} = μ^{∨} + μ_{ξ}. Let β′ : π_{f,0}^{∨} → H_{!}^{d}(\overline{S}_{ℂ}, ℰ_{μ^{∨}}) be the G(𝔸_{f})-map defined as the composition

π_{f,0}^{∨} → π_{f,0}^{∨} ⊗ ∥ν∥^{−ξ} ≅ π_{f,0} ⟶^{c_{coh}(β)} H_{!}^{d}(\overline{S}_{ℂ}, ℰ_{μ^{c}}) ⟶^{α↦α̂} H_{!}^{d}(\overline{S}_{ℂ}, ℰ_{μ^{∨}}),

where the map α ↦ α̂ is the one we defined in Subsection 3.10.2. Let γ = β(ψ) and γ′ = β′(ψ^{−1}) be as in (3.10.4). Cup product and pullback via S_{ℂ}^{♯} ↪ S_{ℂ}×\overline{S}_{ℂ} give rise to an element which we denote by (γ, γ′)^{♯}. This element contributes to H_{!}^{2d}(S_{ℂ}^{♯}, ℰ_{((μ+μ(η),μ^{∨}−μ(η))}_{♯}). Note that we can identify

ℰ_{((μ+μ(η),μ^{∨}−μ(η))}_{♯} = ℰ′_{Λ^{♯}(ℓ)} = ℰ_{Λ^{♯}(ℓ)}^{∨} ⊗ 𝕂^{♯}

as automorphic vector bundles, where 𝕂^{♯} is the canonical bundle on S_{ℂ}^{♯}. As in Subsection 3.10, the elements γ and γ′ give rise to automorphic forms f in π ⊗ ψ and f′ ∈ π^{∨} ⊗ ψ^{−1}.

Let

φ̃_{m,∞}(·, s) = (Δ_{m}𝕁_{m}) (·, s + m − \frac{n}{2}) ∈ I (s + m − \frac{n}{2})_{∞} .

The following result is proved by Garrett in [Gar08]; see [Har08], (4.4) for details.

Lemma 4.5.1. The archimedean local factor Z_{∞} (m − \frac{n}{2}, f, f′, φ̃_{m,∞}) belongs to (L′)^{×}.

Lemma 4.5.2. There exists a finite section φ_{f}(·, s) ∈ I(s)_{f} with φ_{f}(·, m − \frac{n}{2}) taking values in ℚ such that

Z_{f} (m − \frac{n}{2}, f, f′, φ_{f}) ∈ (L′)^{×}.

Proof. The existence of φ_{f} with φ_{f}(·, m − \frac{n}{2}) ∈ ℚ and Z_{f} (m − \frac{n}{2}, f, f′, φ_{f}) ≠ 0 follows from results of Li ([Li92]), as in Lemma 3.5.7 of [Har97] and the discussion after it. The fact that it belongs to L′ follows from our choices and Lemma 6.2.7 of [Har93]. □

From now on, take φ_{f} as in Lemma 4.5.2, and define

φ_{m}(·, s) = 𝕁_{m} (·, s + m − \frac{n}{2}) ⊗ (2πi)^{enm}φ_{f} (·, s + m − \frac{n}{2}) ∈ I (s + m − \frac{n}{2})

and

φ̃_{m}(·, s) = φ̃_{m,∞}(·, s) ⊗ (2πi)^{enm}φ_{f} (·, s + m − \frac{n}{2}) ∈ I (s + m − \frac{n}{2}) ,

so that Δ_{m}φ_{m} = φ̃_{m}. We let E_{m} = E_{φ_{m}} and Ẽ_{m} = Eφ̃_{m}.

Lemma 4.5.3. Let the notation and assumptions be as above. Then

Z (m − \frac{n}{2}, f, f′, φ̃_{m}) ∈ L′.

Proof. It follows from Proposition 4.3.1 that E_{m} and Ẽ_{m} are rational over L′. The Eisenstein series Ẽ_{m}, when restricted to G^{♯}(𝔸), equals Δ_{m}E_{m} (as scalars automorphic forms). Let

ℒ_{m} : H^{2d}(S_{L′}^{♯}, ℰΛ^{♯}(ℓ)^{′}) → L′

be the L′-linear map defined by pairing with Δ_{m}E_{m} via Serre duality. Then, as in Remark 3.9.2, we can write

ℒ_{m} ((γ, γ′)^{♯}) = Z (m − \frac{n}{2}, f, f′, φ̃_{m}) .

Since γ and γ′ are rational over L′, this shows that the zeta function belongs to L′. □

From now on, we assume the following hypothesis.

Hypothesis 4.5.1. If σ ∈ J_{π} then dim_{ℂ[G(𝔸_{f})]}(π_{f}^{σ}, H_{!}^{d}(S_{ℂ}, ℰ_{μ})) ≤ 1.

Using the map c_{B}, we can see that dim_{ℂ[G(𝔸_{f})]}(π_{f}^{σ}, H_{!}^{0}(S_{ℂ}, ℰ_{w_{0}^{1}∗μ})) ≤ 1 as well. The statement of Hypothesis 4.5.1 is part of Arthur’s multiplicity conjectures for unitary groups (in which case the analogous statement should hold for the Weyl components corresponding to any w ∈ 𝒲^{1}, but we are only considering the cases w = 1 and w = w_{0}^{1} here). It is reasonable to expect a proof of these multiplicity conjectures to appear soon, and we refer the reader to [KMSW14] and their forthcoming sequels for more details. Under this condition, we can unambiguously define

Q^{hol}(π) = Q(π; α)

and

Q^{ahol}(π) = Q(π; β),

where β is as above and α is a holomorphic vector (rational over L′ for the de Rham structure, as usual).

Lemma 4.5.4. Suppose that π satisfies Hypothesis 4.5.1. Then

Q^{hol}(π)Q^{ahol}(π) ∼_{E(π)⊗L′} 1.

Proof. This follows from the reasoning in Subsection 2.3. □

We are now ready to state our main theorem, which follows from all the results so far. We include all the relevant hypothesis.

Theorem 4.5.1. Let π ∈ Coh_{G,μ}, with W_{μ} defined over ℚ. Assume that π^{∨} ≅ π ⊗ ∥ν∥^{ξ}, that it satisfies Hypothesis 4.5.1 and that it contributes to antiholomorphic cohomology. Let ψ be an algebraic Hecke character of L, with infinity type (m_{τ})_{τ∈J_{L}}, and let m > n be an integer satisfying (4.2.1). Then

L^{S,mot}(m, π ⊗ ψ, St) ∼_{E(π,ψ)⊗L′}

(2πi)^{e(mn−n(n−1)/2)−ξ}(D_{K}^{1/2})^{⌊ \frac{n+1}{2} ⌋}δ([ε_{L}])^{⌊ \frac{n}{2} ⌋}Q^{hol}(π)p(ψ; det ∘x)p(ψ^{−1}; det ∘\overline{x}).

Proof. Recall that (4.4.1) for φ̃_{m} says that

(4.5.1)  d^{S} (m − \frac{n}{2}) Z (m − \frac{n}{2}, f, f′, φ̃_{m}) =

(f, f′)_{G}Z_{f} (m − \frac{n}{2}, f, f′, φ̃_{m}) Z_{∞}(m − \frac{n}{2}, f, f′, φ̃_{m})L^{mot,S} (m, π, St) .

Note that Z_{f} (m − \frac{n}{2}, f, f′, φ̃_{m}) = (2πi)^{emn}Z_{f} (m − \frac{n}{2}, f, f′, φ_{f}) ∈ (2πi)^{enm}L′ by Lemma 4.5.2. Also, the zeta integrals Z_{∞}(m − \frac{n}{2}, f, f′, φ̃_{m}) and Z (m − \frac{n}{2}, f, f′, φ̃_{m}) belong to L′ by Lemmas 4.5.1 and 4.5.3. The factor (f, f′)_{G} can be written, by (3.10.2) (rather, its twisted version), as

(f, f′)_{G} ∼_{E(π,ψ)⊗L′} (2πi)^{ξ}Q(π; ψ; β).

It follows from this and (4.5.1) that

L^{S,mot}(m, π ⊗ ψ, St) ∼_{E(π,ψ)⊗L′} (2πi)^{−enm−ξ}d^{S} (m − \frac{n}{2}) Q(π; ψ; β)^{−1}.

This formula, together with (3.10.5) and Lemma 4.4.1, implies that

L^{mot,S} (m, π ⊗ ψ, St) ∼_{E(π,ψ)⊗L′} Q(π; β)^{−1}(2πi)^{e(mn−n(n−1)/2)−ξ}

(D_{K}^{1/2})^{⌊ \frac{n+1}{2} ⌋}δ([ε_{L}])^{⌊ \frac{n}{2} ⌋}p(ψ; det ∘x)p(ψ^{−1}; det ∘\overline{x}).

Finally, the theorem follows by combining this with Lemma 4.5.4. □

Remark 4.5.1. The right hand side of this formula contains Q^{hol}(π), which ultimately depends on the choice of CM type Φ. This choice is also reflected on the CM periods, because the map x implicitly depends on Φ.

Remark 4.5.2. Since we are working over L′, we can use Lemma 2.4.2 and (2.2.2) and rewrite the statement of Theorem 4.5.1 as

(4.5.2)  L^{S,mot}(m, π ⊗ ψ, St) ∼_{E(π,ψ)⊗L′}
(2πi)^{e(mn−n(n−1)/2)−ξ}D_{K}^{n/2}Q^{hol}(π)p(ψ; det ∘x)p(ψ^{−1}; det ∘\overline{x}).

Thus, if n is even we can ignore the discriminant factor.

Remark 4.5.3. The formula in Theorem 4.5.1 proves a version of Conjecture 5.1.1 of the recent Jie Lin’s thesis at Paris ([Lin15]). The discriminant factor does not appear in her formula, as she assumes that it belongs to the coefficient field. Galois equivariance of the formula will appear in a forthcoming joint work with Lin.

5. Period relations

In this section, we analyze Deligne’s conjecture for certain motives of the form M ⊗ RM(χ) coming from automorphic forms, comparing the results of Sections 2 and 4. From this comparison, we deduce period relations, and we explain how they are also reflected by Tate’s conjecture. We start by recalling some results regarding base change and descent for automorphic representations of unitary groups and GL_{n}. This will be the link in translating the results of Section 4 into motivic terms. Some of the objects in this section are conjectural and used solely for heuristic purposes, so this section is hypothetical in nature. Nevertheless, we do write down concretely the period relations that we obtain in the end in terms of automorphic forms on unitary groups.

5.1. Transfer and descent. We fix as before a CM extension L/K. We suppose from now on that L = KE for some quadratic imaginary field E/ℚ (with a fixed inclusion E ⊂ \overline{ℚ}). Suppose that G is a unitary group attached to an n-dimensional hermitian space over L/K, and Π is a cuspidal, cohomological, conjugate self-dual representation of GL_{n}(𝔸_{L}). Then we expect the existence of a descent π to G (actually Π should descend to an L-packet, but for our purposes we just choose one member of the corresponding L-packet). This has been proved in a significant number of cases ([Lab11]; see also [Mok15] and [KMSW14]). We will furthermore assume that π ∈ Coh_{G,μ}, contributes to antiholomorphic cohomology, and satisfies π^{∨} ≅ π ⊗ ∥ν∥^{ξ}. The parameter μ = ((a_{τ,1}, …, a_{τ,n})_{τ∈Φ}; a_{0}) is obtained from the weight of Π. We will give more details later in a special case. Suppose that G′ is another unitary group attached to an n-dimensional hermitian spaces over L/K. We say that two representations π and π′ of G and G′ respectively are nearly equivalent if π_{v} ≃ π′_{v} for almost all places v (this only makes sense for the places where the local groups are isomorphic, which is the case for all v except a finite number of them). If this is the case, the partial L-functions L^{S}(s, π, St) and L^{S}(s, π′, St) agree. The general Langlands philosophy predicts that, given π, there exists a represetantion π′ (more precisely, an L-packet) which is nearly equivalent to π. This usually involves a two step process: first, transfer π to a cohomological automorphic representation Π of GL_{n}(𝔸_{L}) (this involves also an algebraic Hecke character ϕ_{π} of E; see the appendix to [Gol14] by Shin), and then descend Π to G′ as in the previous paragraph.

5.2. Motivic interpretation. Suppose that π ∈ Coh_{G,μ}. Let ϕ_{π} be the Hecke character of E mentioned above, obtained by base change from π. Concretely, ϕ_{π} is given by the restriction to 𝔸_{E}^{×} ⊂ Z(𝔸) of the conjugate χ_{π}^{c} of the central character of π. Its relation to the base change Π is that the central character of Π satisfies χ_{Π}|_{𝔸_{E}^{×}} = ϕ_{π}^{c}/ϕ_{π}. The character ϕ_{π} is an algebraic Hecke character of infinity type (a_{0}, a_{0}). Then, there is a motive M(ϕ_{π})_{L′} = M(ϕ_{π}) ×_{E} L′ ∈ ℳ(L′)_{ℚ(ϕ_{π})} of rank 1 and weight ξ attached to ϕ_{π}. Assume that Π is cuspidal. We expect the existence of a motive M(Π) over L, of rank n and weight n − 1, with coefficients in a number field E(Π), such that its λ-adic representations give rise to the Galois representations attached to Π (whose construction is due to many people, see [CHL], [Shi11], [CH13], [Sor]). The normalizations are such that, outside a finite set of places S,

L_{v} (s − \frac{n − 1}{2}, Π^{σ}) = L_{v}(σ, M(Π), s).

We do not need to know that M(Π) actually exists. Its existence will be used for heuristic purposes, and we will just deduce some consequences of it in terms of period relations. Take E(Π) = E(π) big enough, containing ℚ(ϕ). Fixing the CM type Φ, we let (r_{τ} , s_{τ}) be the signature of G at each place τ, and we let t_{s} = \frac{1}{2} ∑_{τ∈Φ} s_{τ} (s_{τ} − 1). We expect the existence of an isomorphism of motives over L′

M(π) ⊗_{E(π)} M(ϕ_{π})_{L′} ≅ (⨂_{τ∈Φ} Λ^{s_{τ}} (M(Π) ×_{L,τ} L′)) (t_{s}).

(see [HLS11], 3.2.2-3.2.4). The corresponding Galois representations have been shown to be equal in many cases, and Tate’s conjecture would imply this expected isomorphism. In particular, if G has signature (n, 0) at all places τ ≠ τ_{0} and signature (n − 1, 1) at τ_{0}, then

(5.2.1)  M(π) ⊗_{E(π)} M(ϕ_{π})_{L′} ≅ M(Π) ×_{L,τ_{0}} L′.

Suppose that for each τ ∈ Φ, there is a group G_{τ}, whose signature at τ is (n − 1, 1) and whose signature at all other places is (n, 0), and suppose that there is an automorphic representation π_{τ} of G_{τ} (𝔸) which is nearly equivalent to π. Then, by the formulas above, we expect the existence of an isomorphism

(5.2.2)  M(π) ⊗_{E} M(ϕ)_{L′} ≅ (⨂_{τ∈Φ} Λ^{s_{τ}} (M(π_{τ}) ⊗_{E} M(ϕ)_{L′})) (t_{s}),

where ϕ = ϕ_{π} = ϕ_{π_{τ}} and E = E(π) = E(π_{τ}) for all τ. In this paper we will mostly be concerned with the case of Π self-dual. In this case, the motive M(Π) should be the base change from K to L of a polarized regular motive M(Π_{0}) over K, where Π_{0} is a cuspidal, self-dual, cohomological automorphic representation of GL_{n}(𝔸_{K}) whose base change to GL_{n}(𝔸_{L}) is Π. Under these conditions, M(Π)^{∨} = M(Π)(n − 1).

Remark 5.2.1. If Π is the base change of π with π^{∨} ≅ π ⊗∥ν∥^{ξ}, as we assume, then Π is self-dual. Indeed, this follows from the uniqueness of the weak base change from G(𝔸) to GL_{1}(𝔸_{E}) × GL_{n}(𝔸_{L}), since the GL_{n}(𝔸_{L})-component is the weak base change of an irreducible constituent of the restriction to the unitary group, and π and π ⊗ ∥ν∥^{ξ} have the same such restriction.

Remark 5.2.2. Suppose that (r_{τ})_{τ∈Φ} is a tuple of integers with r_{τ} ∈ {0, …, n} for each τ. Then, by the classification of hermitian spaces, there exists a hermitian space V of dimension n such that the associated unitary group has signature (r_{τ} , n − r_{τ}) at each τ ∈ Φ. Moreover, if n is odd, or if n is even and ne − 2 ∑_{τ∈Φ} r_{τ} ≡ 0(4), we can impose the condition that the local unitary group at finite places v of K is quasi-split for all v. In the other cases, we can impose the same condition for all finite places except a single v, inert over ℚ and split in L.

5.3. Deligne’s conjecture. Let Π_{0} be a cuspidal automorphic representation of GL_{n}(𝔸_{K}), self-dual, and cohomological of type (a_{σ,1}, …, a_{σ,n})_{σ∈J_{K}} in the usual parametrization. For any τ ∈ J_{L}, we let a_{τ,i} = a_{σ,i} if τ restricts to σ. We let E = E(Π_{0}), and we take the freedom to make E bigger if necessary. We let M = M(Π_{0}) be the conjectural motive attached to Π_{0} as in the previous subsection. We can recover the Hodge types p_{i}(σ, 1) (with 1 : E ↪ ℂ the given embedding) from the infinity type of Π_{0,∞}. The formula is

(5.3.1)  p_{i}(σ, 1) = a_{σ,i} + n − i

for any σ ∈ J_{K}. We can similarly compute the Hodge types p_{i}(σ, φ) for different embeddings φ ∈ J_{E} by looking at the conjugates of Π_{0}. We let Π = Π_{0,L} be the base change of Π_{0} to L. Thus, Π is an automorphic representation of GL_{n}(𝔸_{L}), conjugate self-dual (and self-dual), cohomological of type (a_{τ,1}, …, a_{τ,n})_{τ∈J_{L}}. We assume moreover that Π_{0} ≅ Π̸_{0} ⊗ ε_{L} (this is always the case when n is odd), which implies that Π is cuspidal. Accordingly we expect a motive M(Π) = M ×_{K} L attached to Π with the same Hodge types, as in the previous subsection. We note that, starting with K and Π_{0}, there always exists a CM extension L/K such that Π_{0} ≅ Π̸_{0} ⊗ ε_{L} (see Section 1 of [Clo13]).

We fix an algebraic Hecke character ψ of L, of infinity type (m_{τ})_{τ∈J_{L}} and weight w = m_{τ} + m\overline{τ}. We assume ψ to be critical in the sense of Subsection 2.4, so that m_{τ} ≠ m\overline{τ} for any τ ∈ J_{L}. Write ψ|_{𝔸_{K}^{×}} = ψ_{0}∥ · ∥^{−w}, with ψ_{0} of finite order, and let

χ = ψ^{2}(ψ_{0} ∘ N_{L/K})^{−1}.

Recall that we previously defined χ̌ = χ^{ι,−1}. If we let ψ̃ = ψ/ψ^{ι}, then

ψ̃/χ̌ = ∥ · ∥^{−w}.

We let (n_{τ})_{τ∈J_{L}} be the infinity type of χ, so that n_{τ} = 2m_{τ} and χ is critical of weight 2w. Note that χ|_{𝔸_{K}^{×}} = ∥ · ∥^{−2w}, so that the finite order character that we denoted by χ_{0} in Subsection 2.4 is trivial. In everything that we say, we need to fix a CM field Φ, and we will take it to be Φ = Φ_{ψ} = Φ_{χ}. Thus, an embedding τ ∈ J_{L} belongs to Φ if and only if m_{τ} > m\overline{τ}. In the notation of Section 2, for σ ∈ J_{K},

p_{1}^{χ}(σ, 1) = n_{τ} (5.3.2)

p_{2}^{χ}(σ, 1) = n\overline{τ}

for τ ∈ Φ extending σ. The reason we introduce the character χ is the following. If G is a unitary group and π is a descent of Π from GL_{n} to G, then

L^{mot,S}(s, π ⊗ ψ, St) = L^{S}(M ⊗ RM(χ), s + w)

(see (3.5.2) of [Har97]). More precisely, the right hand side should be replaced by L^{S}(1, M ⊗ RM(χ), s + w), where 1 is the given embedding of E ⊗ ℚ(χ). For simplicity of notation, we will only deal with this component, but everything that follows also works after fixing a different embedding. Suppose that M ⊗ RM(χ) has critical values. Then, as in Subsection 2.5, we can find integers r_{σ} = r_{σ,1,1}(χ) ∈ {0, …, n}, for each embedding σ ∈ J_{K}, such that n_{τ} − n\overline{τ} belongs to the interval (n − 1 − 2p_{r_{σ}}(σ, 1), n − 1 − 2p_{r_{σ}+1}(σ, 1)), where τ ∈ Φ extends σ. Here we take p_{0}(σ, 1) = +∞ and p_{n+1}(σ, 1) = −∞. For τ ∈ J_{L}, we let r_{τ} = r_{σ} if τ restricts to σ. We let s_{σ} = s_{τ} = n − r_{τ}. Assuming that there are critical values, we can express the set of integers m which are critical for L^{S}(M ⊗ RM(χ), s + w) using (2.5.2). Namely, it consists of the m ∈ ℤ satisfying

υ^{(1)} < m + w ≤ υ^{(2)}.

Thus, m is a critical integer for L^{S}(M ⊗ RM(χ), s + w) if and only if

υ^{(1)} − w < m ≤ υ^{(2)} − w.

Combining the definitions of the υ^{(i)} with (5.3.1) and (5.3.2), we can write

υ^{(1)} − w = max\{a_{τ,r_{τ} +1} + s_{τ} − 1 + m_{τ} − m\overline{τ} , a_{τ,s_{τ} +1} + r_{τ} − 1 + m\overline{τ} − m_{τ} \}_{τ∈Φ}

and

υ^{(2)} − w = min\{a_{τ,r_{τ}} + s_{τ} + m_{τ} − m\overline{τ} , a_{τ,s_{τ}} + r_{τ} + m\overline{τ} − m_{τ} \}_{τ∈Φ}.

Consider the tuple (r_{τ})_{τ∈Φ} and let G be a unitary group with signatures (r_{τ} , n − s_{τ})_{τ∈Φ}, as in Remark 5.2.2. In accordance with the discussion of Subsection 5.1, we assume the existence of a descent π ∈ Coh_{G} of Π_{L}. Here μ = ((a_{τ,1}, …, a_{τ,n})_{τ∈Φ}; 0). We assume furthermore that π is self-dual, contributes to antiholomorphic cohomology and satisfies Hypothesis 4.5.1. The assumption that a_{0} = 0 is reflected in the self-duality of π, and it’s made for simplifying purposes. In turn, we are assuming that the character ϕ of Subsection 5.2 is trivial. We will use the results of Section 4 with the group G, the representation π and the Hecke character ψ. The crucial fact is that the second inequality m ≤ υ^{(2)} − w is exactly the second inequality in (4.2.1). It follows, under all these assumptions, using version (4.5.2) of Theorem 4.5.1, and assuming m > n, that

L^{S}(1, M ⊗ RM(χ), m + w) ∼_{E(ψ)EL′}
(2πi)^{e(mn−n(n−1)/2)}D_{K}^{n/2}Q^{hol}(π)p(ψ; det ∘x)p(ψ^{−1}; det ∘\overline{x}).

Note that we are taking E(π) = E. Also, we took E(ψ) to contain L′, so we could write ∼_{E(ψ)E}, but we choose to leave the L′ in the notation for possible future refinements. Deligne’s conjecture would then predict the following results.

Proposition 5.3.1. With the previous hypotheses, Deligne’s conjecture for the motive M ⊗ RM(χ) implies that

c^{+}(M ⊗ RM(χ)(m + w))_{1} ∼_{E(ψ)EL′}
(2πi)^{e(mn−n(n−1)/2)}D_{K}^{n/2}Q^{hol}(π)p(ψ; det ∘x)p(ψ^{−1}; det ∘\overline{x}).

Here c^{+}(M ⊗ RM(χ))_{1} means we are taking the restriction of scalars from K to ℚ of M ⊗ RM(χ); see Subsection 2.2.

Remark 5.3.1. The prediction is actually a consequence of the weaker Conjecture 2.2.2 over L′ for M ⊗ RM(χ).

In what follows, we will rewrite the prediction of Proposition 5.3.1 and get an expression only depending on automorphic quadratic periods and quadratic periods of the motive M. First, we use (3.10.3), together with Lemma 1.6 of [Har93], to get

p(ψ; det ∘x)p(ψ^{−1}; det ∘\overline{x}) ∼_{E(ψ)} ∏_{τ∈Φ} p(ψ̃^{r_{τ} −s_{τ}}; τ).

Since ψ̃ = χ̌∥ · ∥^{−w}, using Proposition 1.4 and Lemma 1.8.3 of [Har93], we can write this as

(5.3.3)  (2πi)^{w ∑_{τ} r_{τ} −s_{τ}} ( ∏_{τ∈Φ} p (χ̌; τ)^{r_{τ} −s_{τ}} ) .

Now, note that χ_{0} is trivial, and hence by (2.6.2) we can write

c^{+}(M ⊗ RM(χ)(m + w))_{1} ∼_{E(ψ)EL′}

(2πi)^{emn+w ∑_{τ} r_{τ} −s_{τ}} δ(M)_{1} ( ∏_{τ∈Φ} p (χ̌; τ)^{r_{τ} −s_{τ}} ) Q_{1}^{s}(M)

(recall that the weight of χ is 2w and rank of M is n). Let

∂(M) = (2πi)^{−en(n−1)/2}δ(M)^{−1}D_{K}^{n/2}.

Then, by (5.3.3), Deligne’s prediction (Proposition 5.3.1) is equivalent to the statement

(5.3.4)  Q_{1}^{s}(M) ∼_{E(ψ)EL′} ∂(M)_{1}Q^{hol}(π).

For example, if n is even, then δ(M) ∼_{E⊗K′} D_{K}^{n/2} ∏_{σ∈J_{K}} δ_{σ}(M) by (2.2.2), and δ_{σ}(M) ∼_{E⊗K,σ} (2πi)^{−n(n−1)/2}, so in this case ∂(M) ∼_{E⊗K′} 1 and (5.3.4) is equivalent to the statement

Q_{1}^{s}(M) ∼_{E(ψ)EL′} Q^{hol}(π).

Remark 5.3.2. As we said before, we can work with arbitrary embeddings of the coefficient field. In this case, Conjecture 2.2.2 would mean

Q^{s}(M) ∼_{E(ψ)⊗E;L′} ∂(M)Q̃^{hol}(π).

The notation Q̃^{hol}(π) refers to the fact that for each embedding of the coefficient field, the corresponding signatures, and hence the transfers, are different, and we take Q̃^{hol}(π) to consist of the corresponding holomorphic automorphic periods in each component.

5.4. Applications of Tate’s conjecture. Keep the assumptions and notation of the last subsection. Moreover, assume that we have isomorphisms as in (5.2.1) and (5.2.2). As mentioned in Subsection 5.2, this would follow from Tate’s conjecture. We can interpret the quadratic periods Q_{j,σ}(M) in terms of automorphic quadratic periods. For this, fix σ ∈ J_{K}, and τ ∈ Φ extending σ. Let G_{τ} and π_{τ} be as in (5.2.2). Then

M(π_{τ}) ≅ M ×_{K,σ} L′.

Suppose that {ω_{1,σ}, …, ω_{n,σ}} is an E ⊗ K-basis of M_{dR}, as in Subsection 2.3, and denote by the same letters the E ⊗ L′-basis of M_{dR} ×_{K,σ} L′ ≅ M(π_{τ})_{dR} obtained by extension of scalars to L′ via σ. Let {Ω_{1,σ}, …, Ω_{d,σ}} be the basis of M_{σ} ⊗ ℂ constructed in Subsection 2.3. Now, by Remark 3.3.3, the Hodge decomposition of M_{σ} ⊗ ℂ is given by

M_{σ} ⊗ ℂ ≅ M(π_{τ})_{1} ⊗ ℂ = ⨁_{j=1}^{n} M(π_{τ})^{p_{j},q_{j}}.

Here p_{j} = a_{σ,j} + n − j, q_{j} = n − 1 − p_{j} (more precisely, the decomposition is obtained after tensoring with ℂ over E with the given embedding) and M(π_{τ})^{p_{j},q_{j}} is the Weyl component corresponding to the element w_{j} defined in Remark 3.3.3. Moreover, the basis is chosen in a such a way that {ω_{1,σ}, …, ω_{j,σ}} is a basis of the filtration stage F^{p_{j}}(M(π_{τ})_{dR}) and Ω_{j,σ} defines a nonzero rational element of the Hodge component corresponding to the element w_{j} ∈ 𝒲^{1}. It follows at once from the definitions that

(5.4.1)  Q_{j,σ} ∼ Q(π_{τ}; β_{j}),

where β_{j} contributes to coherent cohomology with respect to the Weyl component of type w_{j}. From (5.2.2), we obtain the period relation

(5.4.2)  Q^{hol}(π) ∼_{E(ψ)EL′} ∏_{τ∈Φ} ∏_{j=1}^{s_{τ}} Q_{j,σ}.

Once again, this relation would follow from Tate’s conjecture, so if we assume its veracity, we can prove the weak version of Deligne’s conjecture predicted in (5.3.4) when n is even. When n is odd, we invoke another consequence of Tate’s conjecture. Assuming that the central character of Π_{0} is trivial, as we do, Tate’s conjecture predicts the existence of an isomorphism

Λ^{n}(M) ≅ E(−n(n − 1)/2)

of motives over K and coefficients in E. Then

δ_{σ}(M) ∼_{E⊗K,σ} δ_{σ}(Λ^{n}(M)) ∼_{E⊗K,σ}
δ_{σ}(E(−n(n − 1)/2)) ∼_{E⊗K,σ} (2πi)^{−n(n−1)/2}.

It follows, using (2.2.2), that

δ(M) ∼_{E⊗K′} D_{K}^{n/2}(2πi)^{−en(n−1)/2}.

Thus, ∂(M) ∼_{E⊗K′} 1 in this case as well, and hence the weak version of Deligne’s conjecture predicted in (5.3.4) follows from (5.4.2).

Remark 5.4.1. We don’t need the full strength of Tate’s conjecture for this. We only need to know that an isomorphism at the level of λ-adic realizations implies an isomorphism at the level of Hodge-de Rham structures.

Remark 5.4.2. When K = ℚ, Harris proved in a series of works ([Har99], [Har07], [Har08]) that the relations are true up to an unknown factor depending on the infinity type of the automorphic representation. It remains as a project for the coming future to extend these results to the situation of a general totally real field K.

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1458 Lucio Guerberoff

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Lucio Guerberoff

Department of Mathematics University College London 25 Gordon Street London WC1H 0AY, UK