Algebraicity of the central critical values of twisted triple product L -functions
aa r X i v : . [ m a t h . N T ] S e p ALGEBRAICITY OF THE CENTRAL CRITICAL VALUES OF TWISTED TRIPLEPRODUCT L -FUNCTIONS SHIH-YU CHEN
Abstract.
We study the algebraicity of the central critical values of twisted triple product L -functionsassociated to motivic Hilbert cusp forms over a totally real ´etale cubic algebra in the totally unbalancedcase. The algebraicity is expressed in terms of the cohomological period constructed via the theory ofcoherent cohomology on quaternionic Shimura varieties developed by Harris [Har90b]. As an application,we generalize our previous result [CC19] on Deligne’s conjecture for certain automorphic L -functions forGL × GL . We also establish a relation for the cohomological periods under twisting by algebraic Heckecharacters. Introduction
Let E be a totally real cubic extension over Q and Σ E = {∞ , ∞ , ∞ } be the set of embeddings of E into R . Let Π = N v π v be an irreducible unitary cuspidal automorphic representation of (R E / Q GL / E )( A Q ) =GL ( A E ) with central character ω Π , where v runs through the places of Q and R E / Q denotes Weil’s restrictionof scalars. We assume Π is motivic of weight ( κ , κ , κ ) ∈ Z ≥ , that is, π ∞ is a discrete series representationof weight ( κ , κ , κ ) with κ i corresponding to ∞ i and κ i ≡ κ j (mod 2) for 1 ≤ i, j ≤
3. We assume furtherthat κ + κ + κ ≡ κ ≥ κ ≥ κ . Consider the (finite) Asai cube L -function L ( ∞ ) ( s, Π , As) = Y p L ( s, π p , As)of Π . A critical point for L ( s, Π , As) is a half-integer m + which is neither a pole nor a zero of thearchimedean local factors L ( s, π ∞ , As) and L (1 − s, π ∞ , As). We have the following conjecture on the alge-braicity of the critical values of L ( s, Π , As) in terms of Shimura’s Q -invariant [Shi88, §
7] which we shall nowrecall. Let I ⊆ Σ E . Suppose there exists a quaternion algebra B over E such that • B is unramified at places in I and ramified at places in Σ E r I ; • there exists an irreducible cuspidal automorphic representation Π B of B × ( A E ) associated with Π bythe Jacquet–Langlands correspondence.Then the period invariant Q ( Π , I ) ∈ C × / Q × is define to be the class represented by the Petersson normof a non-zero Q -rational vector-valued cusp form on B × ( A E ) associated to Π B (we refer to [Shi81, §
2] forthe notion of Q -rational automorphic forms). Note that Q ( Π , I ) dose not depend on the choice of B and Q -rational automorphic forms (cf. [Yos94, Theorem 6.6]) and is always defined when ♯ I = 1 , Conjecture 1.1.
Let m + be critical for L ( s, Π , As) . We have L ( ∞ ) ( m + , Π , As)(2 π √− m · q ( Π , As) ∈ Q , where q ( Π , As) = ( π κ + κ + κ +2 Q ( Π , Σ E ) if κ < κ + κ ,π κ +2 Q ( Π , {∞ } ) if κ ≥ κ + κ . Similar conjecture was proposed by Blasius [Bla87] if we replace E by Q × Q × Q . One can also propose aconjecture if we replace E by K × Q for some real quadratic extension K over Q . We remark that the conjectureis compatible with Deligne’s conjecture [Del79] by Yoshida’s calculation of the motivic Asai periods in [Yos94,(5.11)] (see also Remark 1.7). When κ < κ + κ , the conjecture was proved by Garrett and Harris in [GH93,Theorem 4.6] for | m | > m = 0. When κ ≥ κ + κ ,we have the following result which is a special case of our main result Theorem 1.4 (see Remark 1.5). heorem 1.2. Assume κ ≥ κ + κ , ω Π | A × Q is trivial, and Hom GL ( Q p ) ( π p , C ) = 0 for all rational primes p . Then Conjecture 1.1 holds for m = 0 . Remark 1.3.
By the results of Prasad [Pra90] and [Pra92], Hom GL ( Q p ) ( π p , C ) = 0 whenever π p is a principalseries representation.1.1. Main results.
Let E be a totally real ´etale cubic algebra over a totally real number field F . Let Π = N v π v be an irreducible cuspidal automorphic representation of (R E / F GL / E )( A F ) = GL ( A E ) withcentral character ω Π , where v runs through the places of F . We have the Asai cube representationAs : L (R E / F GL / E ) −→ GL( C ⊗ C ⊗ C )of the L -group L (R E / F GL / E ) of R E / F GL / E . The associated automorphic L -function L ( s, Π , As) = Y v L ( s, π v , As)is called the twisted triple product L -function of Π . We denote by L ( ∞ ) ( s, Π , As) the L -function obtainedby excluding the archimedean L -factors. Let Σ E (resp. Σ F ) be the set of non-zero algebra homomorphismsfrom E (resp. F ) into R . We assume that ω Π | A × F is trivial and Π is motivic (cf. § κ = X w ∈ Σ E κ w w ∈ Z ≥ [Σ E ] . We say Π is totally unbalanced (resp. totally balanced) if for all v ∈ Σ F we have2 max w | v { κ w } − X w | v κ w ≥ . < . In the totally balanced case, the algebraicity of L ( s, Π , As) at the critical points (except for s = − , ) wereproved by Garrett–Harris [GH93] and C.–Cheng [CC19] in terms of the Petersson norm of the normalizednewform of Π and the result is compatible with Deligne’s conjecture [Del79]. The aim of this paper is toprove, in the totally unbalanced case, the algebraicity of the central critical value L ( , Π , As) in terms ofHarris’ cohomological period which we shall now describe. Our result is compatible with Deligne’s conjecturewhich predict that the algebraicity can be expressed in terms of the (conjectural) motivic periods in [Yos94](cf. Remark 1.7). Suppose the global root number ε ( Π , As) of Π with respect to the Asai cube representationis equal to 1. By the results of Prasad [Pra90] and [Pra92] and Loke [Lok01], there exists a unique quaternionalgebra D over F so that there exists an irreducible cuspidal automorphic representation Π D = N v π Dv of D × ( A E ) associated to Π by the Jacquet–Langlands correspondence and such thatHom D × ( F v ) ( π Dv , C ) = 0for all places v of F . Note that D is totally indefinite if and only if Π is totally unbalanced. In this case, foreach subset I of Σ E , we denote by Ω I ( Π D ) ∈ C × (resp. Ω I (( Π D ) ∨ ) ∈ C × ) the period obtained by comparingthe rational structures on π Df = N v ∤ ∞ π Dv (resp. ( π Df ) ∨ = N v ∤ ∞ ( π Dv ) ∨ ) via the zeroth and ♯ I -th coherentcohomology of certain automorphic line bundles on toroidal compactification of the quaternionic Shimuravariety associated to D × (see § σ ∈ Aut( C ), there exists a unique motivic irreducible cuspidal automorphic representation σ Π ofGL ( A E ) such that its finite part is isomorphic to the σ -conjugate of the finite part of Π . We denote by σ κ the weight of σ Π . The rationality field Q ( Π ) of Π is define to be the fixed field of { σ ∈ Aut( C ) | σ Π = Π } and is a number field. For each v ∈ Σ F , let ˜ v ( κ ) ∈ Σ E be the homomorphism such that max w | v { κ w } = κ ˜ v ( κ ) .Put I κ = { ˜ v ( κ ) | v ∈ Σ F } ⊂ Σ E . Following is our main result for totally unbalanced Π . When E = F × F × F and D is the matrix algebra,the theorem was proposed and proved by Harris [Har89]. Following the ideas in [Har89], we generalize theresult of Harris to arbitrary totally real ´etale cubic algebra E over F . Theorem 1.4.
Let Π be a motivic irreducible cuspidal automorphic representation of GL ( A E ) . Assumethat ω Π | A × F is trivial and Π is totally unbalanced. If ε ( Π , As) = − , then L ( , σ Π , As) = 0 for all σ ∈ Aut( C ) . (2) Suppose ε ( Π , As) = 1 . For σ ∈ Aut( C ) , we have σ L ( ∞ ) ( , Π , As) D / E (2 π √− F : Q ] · Ω I κ ( Π D ) · Ω I κ (( Π D ) ∨ ) ! = L ( ∞ ) ( , σ Π , As) D / E (2 π √− F : Q ] · Ω I σκ ( σ Π D ) · Ω I σκ ( σ ( Π D ) ∨ ) . Here D E is the absolute discriminant of E / Q and D is the unique quaternion algebra over F suchthat Hom D × ( F v ) ( π Dv , C ) = 0 for all places v of F . In particular, we have L ( ∞ ) ( , Π , As) D / E (2 π √− F : Q ] · Ω I κ ( Π D ) · Ω I κ (( Π D ) ∨ ) ∈ Q ( Π ) . Moreover, when D is the matrix algebra, we can replace Ω I κ (( Π D ) ∨ ) by Ω I κ ( Π ) . Remark 1.5.
Assume κ ∈ Z ≥ [Σ E ]. Let I ⊆ Σ E . By the result of Harris [Har94, Theorem 1] and the periodrelation in Lemma 2.8 below, suppose that Shimura’s period invariant Q ( Π , I ) ∈ C × / Q × is defined, then wehave Q ( Π , I ) = (2 π √− − P w ∈ I κ w · Ω I ( Π ) (mod Q × ) . Therefore, if D is the matrix algebra and Q ( Π , I κ ) is defined, then we can express the algebraicity of L ( ∞ ) ( , Π , As) in terms of Q ( Π , I κ ). In particular, Theorem 1.2 holds. Remark 1.6.
Let σ ∈ Aut( C ) and I be a subset of Σ E which is admissible with respect to κ (see § σ (cid:18) Ω I ( Π D )Ω I ( Π ) (cid:19) = Ω σ I ( σ Π D )Ω σ I ( σ Π ) . Similarly for Π ∨ . This conjecture holds trivially for I = ∅ and is known for I = Σ E (cf. Corollary 2.10). Bythe period relation in Theorem 1.11 below, we also expect that σ (cid:18) Ω I ( Π ∨ )Ω I ( Π ) (cid:19) = Ω σ I ( σ Π ∨ )Ω σ I ( σ Π ) . Remark 1.7.
We remark that the theorem is compatible with Deligne’s conjecture. Assume E is a field.The other cases can be verified in a similar way. Suppose κ w ≥ w ∈ Σ E . Let M ( Π ) be the(conjectural) motive attached to Π of rank 2 over E with coefficients in Q ( Π ) satisfying conditions in [Yos94,Conjecture 4.1] with k therein replaced by 0. Let Σ Q ( Π ) be the set of embeddings from Q ( Π ) to C andidentify Q ( Π ) ⊗ Q C with C Σ Q ( Π ) in a natural way. We also identify Q ( Π ) as a subfield of Q ( Π ) ⊗ Q C by thediagonal embedding. For w ∈ Σ E , let c ± w ( M ( Π )) = ( c ± w ( σ, M ( Π ))) σ ∈ ( Q ( Π ) ⊗ Q C ) × , δ w (Art ω − Π ) = ( δ w ( σ, Art ω − Π )) σ ∈ ( Q ( Π ) ⊗ Q C ) × be the covariantly defined w -periods in [Yos94, § ω − Π is the Artin motive attached to ω − Π defined as in [Del79, §
6] and σ ∈ Σ Q ( Π ) . Comparing [Yos94, (4.14)] with [Har89, Theorem 3.5.1] (see alsoTheorem 4.7 below) on the algebraicity of Rankin–Selberg L -functions for GL × GL , it is natural to expectthat (Ω I σκ ( σ Π )) σ ≡ (2 π √− − [ F : Q ] Y w ∈ I σκ c + w ( σ, M ( Π )) c − w ( σ, M ( Π )) δ w ( σ, Art ω − Π ) − σ (mod Q ( Π ) × ) . On the other hand, let As( M ( Π )) be the (conjectural) Asai motive associated to M ( Π ), that is, As( M ( Π )) = N Ω Res E / F M ( Π ) in the notation of [Yos94, Conjecture 1.8] with Ω = Gal( Q / F ) / Gal( Q / E ). Then we have L (R F / Q As( M ( Π )) , s ) = ( L ( ∞ ) ( s + , σ Π , As)) σ and Deligne’s conjecture predicts that L (R F / Q As( M ( Π )) , m )(2 π √− F : Q ] m · c ( − m (R F / Q As( M ( Π ))) ∈ Q ( Π ) or all critical points m ∈ Z for R F / Q As( M ( Π )). Here c ± (R F / Q As( M ( Π ))) are Deligne’s periods attachedto R F / Q As( M ( Π )). Now we explicate Yoshida’s calculation [Yos94, (5.9)] in our case, we then deduce fromthe totally unbalanced condition that c ± (R F / Q As( M ( Π ))) ∈ (2 π √− F : Q ] G ( σ ω Π ) Y w ∈ I κ c + w ( σ, M ( Π )) c − w ( σ, M ( Π )) δ w ( σ, Art ω − Π ) − σ · ( e E × ) Σ Q ( Π ) , where G ( σ ω Π ) is the Gauss sum of σ ω Π and e E is the Galois closure of E over Q . Note that the assumption ω Π | A × F is trivial and (1.2) imply that ( G ( σ ω Π )) σ ∈ Q ( Π ) × . Therefore, our main result is compatible withDeligne’s conjecture for m = −
1, at least modulo e E × .As an application to Theorem 1.4, we generalize our previous result [CC19], which is compatible withDeligne’s conjecture, on the algebraicity of the central critical value of certain automorphic L -functions forGL × GL . Let Π = N v π v , Π ′ = N v π ′ v be motivic irreducible cuspidal automorphic representations ofGL ( A F ) with central characters ω Π , ω Π ′ and of weights ℓ, ℓ ′ ∈ Z ≥ [Σ F ], respectively. For each subset I of Σ F which is admissible with respect to ℓ ′ , let Ω I ( Π ′ ) ∈ C × be the Harris’ period of Π recalled in § ( Π ) be the Gelbart–Jacquet lift [GJ78] of Π , which is an isobaric automorphic representation ofGL ( A F ) and is the functorial lift of the symmetric square representation of GL associated to Π . Let L ( s, Sym ( Π ) × Π ′ ) = Y v L ( s, Sym ( π v ) × π ′ v )the Rankin–Selberg automorphic L -function for GL ( A F ) × GL ( A F ) associated to Sym ( Π ) × Π ′ . We denoteby L ( ∞ ) ( s, Sym ( Π ) × Π ′ ) the L -function obtained by excluding the archimedean L -factors. Theorem 1.8.
Suppose the following conditions hold: (i) ω Π ω Π ′ is trivial; (ii) ℓ ′ − ℓ ∈ Z ≥ [Σ F ] ; (iii) there exists a totally real quadratic extension K over F such that ε ( Π ′ ⊗ ω Π ω K / F ) = ε (Sym ( Π ) × Π ′ ) , where ε ( ⋆ ) is the global root number of ⋆ and ω K / F is the quadratic Hecke character of A × F associatedto K / F by class field theory.Then we have σ L ( ∞ ) ( , Sym ( Π ) × Π ′ ) D / F (2 π √− [ F : Q ](1+ r ) · G ( ω Π ) · Ω Σ F ( Π ′ ) · p (( − r sgn( ω Π ) , Π ′ ) ! = L ( ∞ ) ( , Sym ( σ Π ) × σ Π ′ ) D / F (2 π √− [ F : Q ](1+ r ) · G ( σ ω Π ) · Ω Σ F ( σ Π ′ ) · p (( − r sgn( σ ω Π ) , σ Π ′ ) for all σ ∈ Aut( C ) . Here r ∈ Z is defined so that | ω Π | = | | r A F , G ( ω Π ) is the Gauss sum of ω Π , sgn( ω Π ) ∈{± } Σ F is the signature of ω Π , and p ( ε, Π ′ ) are the periods for Π ′ defined for ε ∈ {± } Σ F in [Shi78] . Remark 1.9.
Condition (iii) is satisfied when either π ′ v is not a discrete series representation for any finiteplace v or the conductors of Π and Π ′ are square-free. We also refer to Lemma 2.9 for the period relationbetween Petersson norm and Ω Σ F ( Π ′ ). Remark 1.10.
When ℓ ′ − ℓ ∈ Z < [Σ F ], the algebraicity of L ( ∞ ) ( , Sym ( σ Π ) × σ Π ′ ) was proved by severalauthors. Suppose that F = Q and the conductors of Π and Π ′ are square-free, it is proved in [Ich05,Corollary 2.6], [Xue19, Theorem 1.1], [PdVP19, Corollary 1.4], [CC19, Theorem A], and [Che20b, Corollary1.2] in terms of Petersson norm and Shimura’s period [Shi77] and the result is compatible with Deligne’sconjecture. The algebraicity is also proved in [Rag16] in terms of certain cohomological period.Another result we would like to present in this paper is on the behavior of the cohomological periods undertwisting by algebraic Hecke characters. More precisely, let Π be a motivic irreducible cuspidal automorphicrepresentation of GL ( A F ) such that its weight belongs to Z ≥ [Σ F ]. Let M ( Π ) be the (conjectural) motive ttached to Π of rank 2 over F with coefficients in Q ( Π ) satisfying conditions in [Yos94, Conjecture 4.1]with k therein replaced by − r , where | ω Π | = | | r A F for some r ∈ Z . Let I be a subset of Σ F . Denote byΩ I ( Π ) ∈ C × the Harris’ period of Π . As mentioned in Remark 1.7, we expect that(Ω σ I ( σ Π )) σ ≡ (2 π √− − ♯ I ( √− r ♯ I Y v ∈ σ I c + v ( σ, M ( Π )) c − v ( σ, M ( Π )) δ v ( σ, Art ω − Π ) − ! σ (mod Q ( Π , I ) × ) . Here we enlarge the coefficients to Q ( Π , I ) = Q ( Π ) · Q ( I ), Art ω − Π is the Artin motive attached to ω − Π , and c ± v ( M ( Π )) = ( c ± v ( σ, M ( Π ))) σ ∈ ( Q ( Π , I ) ⊗ Q C ) × , δ v (Art ω − Π ) = ( δ v ( σ, Art ω − Π )) σ ∈ ( Q ( Π , I ) ⊗ Q C ) × are the covariantly defined v -periods in [Yos94, § v ∈ Σ F . On the other hand, let χ be analgebraic Hecke character of A × F . Enlarging the coefficients to Q ( Π , I, χ ) = Q ( Π ) · Q ( I ) · Q ( χ ) and applying[Yos94, Proposition 3.1] to M ⊗ N = M ( Π ) ⊗ Art χ − = M ( Π ⊗ χ ), we have c + v ( M ( Π ⊗ χ )) c − v ( M ( Π ⊗ χ )) δ v (Art ω − Π χ − ) − = c + v ( M ( Π )) c − v ( M ( Π )) δ v (Art ω − Π ) − for each v ∈ Σ F . Therefore, it is natural to expect that(Ω σ I ( σ Π ⊗ σ χ )) σ ≡ (Ω σ I ( σ Π )) σ (mod Q ( Π , I, χ ) × ) . Indeed, we have the following result.
Theorem 1.11.
Let Π be a motivic irreducible cuspidal automorphic representation of GL ( A F ) and χ analgebraic Hecke character of A × F . Suppose the weight of Π belongs to Z ≥ [Σ F ] . For I ⊆ Σ F and σ ∈ Aut( C ) ,we have σ (cid:18) Ω I ( Π )Ω I ( Π ⊗ χ ) (cid:19) = Ω σ I ( σ Π )Ω σ I ( σ Π ⊗ σ χ ) . An outline of the proof.
There are two main ingredients for the proof of Theorem 1.4: • Ichino’s central value formula for L ( , Π , As); • cohomological interpretation of the global trilinear period integral.We have the global trilinear period integral I D ∈ Hom C ( Π D ⊗ ( Π D ) ∨ , C ) in (5.1) defined by integration on D × ( A F ) × D × ( A F ) of cusp forms in Π D ⊗ ( Π D ) ∨ . In [Ich08], Ichino established a formula which decomposesthe global trilinear period integral I D into a product of L ( , Π , As) and the local trilinear period integrals I Dv defined in (5.2). The formula is a special case of the refined Gan–Gross–Prasad conjecture proposed byIchino–Ikeda [II10]. Note that our choice of the quaternion algebra D guarantees the non-vanishing of I Dv (cf. Lemma 5.5). The Galois equivariant property of these local trilinear period integrals at non-archimedeanplaces is proved in Lemma 5.3, and the calculation for the archimedean local trilinear periods integral wassettled in [CC19] and is recalled in Lemma 5.4. On the other hand, consider the automorphic line bundles L ( κ,r ) and L ( κ ( I κ ) ,r ) associated to the algebraic characters ρ ( κ,r ) and ρ ( κ ( I κ ) ,r ) of ( R × · SO(2)) Σ E in (2.4)on the quaternion Shimura variety for the Shimura datum (R E / Q ( D ⊗ F E ) × , ( H ± ) Σ E ). Let H ( L sub( κ,r ) ) and H [ F : Q ] ( L sub( κ ( I κ ) ,r ) ) be the zeroth and [ F : Q ]-th coherent cohomology groups of the subcanonical extensions L sub( κ,r ) and L sub( κ ( I κ ) ,r ) , respectively, on a toroidal compactification of the quaternion Shimura variety. These co-homology groups have canonical Q ( κ )-rational structures and admit natural action of D × ( A E ,f ). Inside thesecoherent cohomology groups, we have the cuspidal cohomology groups H ( L ( κ,r ) ) and H [ F : Q ]cusp ( L ( κ ( I κ ) ,r ) )which are admissible semisimple D × ( A E ,f )-submodules and consisting of cohomology classes represented bycusp forms on D × ( A E ). The representation π Df occurs with multiplicity one in the π Df -isotypic components H ( L ( κ,r ) )[ π Df ] and H [ F : Q ]cusp ( L ( κ ( I κ ) ,r ) )[ π Df ] and these isotypic components have canonical Q ( Π )-rationalstructures inherit from that of L sub( κ,r ) and L sub( κ ( I κ ) ,r ) (see Lemmas 2.3 and 2.5). Note that each class in H ( L ( κ,r ) )[ π Df ] (resp. H [ F : Q ]cusp ( L ( κ ( I κ ) ,r ) )[ π Df ]) is uniquely represented by a holomorphic cusp form in Π D (resp. by a cusp form in Π D which is anti-holomorphic at w ∈ I κ and holomorphic elsewhere). The periodΩ I κ ( Π D ) ∈ C × is the non-zero complex number, unique up to Q ( Π ) × , such that ϕ I κ Ω I κ ( Π D ) epresents a Q ( Π )-rational cusp form in H [ F : Q ]cusp ( L ( κ ( I κ ) ,r ) )[ π Df ] for any Q ( Π )-rational holomorphic cusp form ϕ ∈ Π D . Here ϕ I κ is the right translation of ϕ by τ I κ ∈ GL ( R ) Σ E = D × ( E ∞ ) with( τ I κ ) w = − ! if w ∈ I κ , . Under the totally unbalanced condition, we constructed in § δ ( κ )]rational over Q ( κ ) from L ( κ ( I κ ) ,r ) to L ′ (2 , , the automorphic line bundle associated to the algebraic character ρ ′ (2 , of ( R × · SO(2)) Σ F on the quaternion Shimura variety for the Shimura datum (R F / Q D × , ( H ± ) Σ F ). Itinduces a D × ( A F ,f )-module homomorphism[ δ ( κ )] : H [ F : Q ] ( L sub( κ ( I κ ) ,r ) ) −→ H [ F : Q ] (( L ′ (2 , ) sub )which is rational over Q ( κ ). Moreover, if a class in H [ F : Q ]cusp ( L ( κ ( I κ ) ,r ) )[ π Df ] is represented by a cusp form ϕ ,then its image under [ δ ( κ )] is represented by ( X ( κ ) · ϕ ) | D × ( A F ) for some differential operator X ( κ ) ∈ U ( gl Σ E , C )defined in (3.4). Composing the trilinear differential operator with the Q -rational trace map H [ F : Q ] (( L ′ (2 , ) sub ) −→ C in Lemma 2.2, we deduce that Z A × F D × ( F ) \ D × ( A F ) X ( κ ) · ϕ I κ ( g )Ω I κ ( Π D ) dg Tam ∈ Q ( Π )for any Q ( Π )-rational holomorphic cusp form ϕ ∈ Π D . Here dg Tam is the Tamagawa measure. Similarassertions hold for ( Π D ) ∨ if we replace r by − r and π Df by ( π Df ) ∨ . Therefore, we have I D ( X ( κ ) · ϕ I κ ⊗ X ( κ ) · ϕ I κ )Ω I κ ( Π D ) · Ω I κ (( Π D ) ∨ ) ∈ Q ( Π )for Q ( Π )-rational holomorphic cusp forms ϕ ∈ Π D and ϕ ∈ ( Π D ) ∨ . Combining with Ichino’s formula,we obtain the algebraicity of L ( , Π , As). We mention one subtlety in the proof. In order to apply Ichino’sformula, it is necessary to compare L (1 , Π , Ad), which is the special value of the adjoint L -function of Π at s = 1, with the Petersson bilinear pairing (2.18) on Π D × ( Π D ) ∨ . It is known that L (1 , Π , Ad) is essentiallyequal to the Petersson pairing of Q ( Π )-rational holomorphic cusp form in Π D × ( Π D ) ∨ (see Lemma 2.9 andCorollary 2.10). On the other hand, the rationality of the global trilinear period integral I D is related to therational structure of H [ F : Q ]cusp ( L ( κ ( I κ ) ,r ) )[ π Df ] as we have explained above. This is the key reason why we needto compare the rational structures on H ( L ( κ,r ) )[ π Df ] and H [ F : Q ]cusp ( L ( κ ( I κ ) ,r ) )[ π Df ].Theorem 1.11 is actually a direct consequence of the algebraicity of the rightmost critical value of thetwisted Rankin–Selberg L -function L ( s, Π × Π ′ × χ ) for suitable motivic irreducible cuspidal automor-phic representation Π ′ of GL ( A F ). The algebraicity is expressed in terms of the cohomological periodsΩ I ( Π ) , Ω Σ E r I ( Π ′ ), the Gauss sum G ( χ ω Π ω Π ′ ), and some other elementary factors. The algebraicity wasproved by Shimura in [Shi78, Theorem 4.2] when I = Σ F and by Harris in [Har89, Theorem 3.5.1] for general Π , Π ′ , and χ = . We generalize the result to arbitrary twist by algebraic Hecke character χ in Theorem4.7. By applying Theorem 4.7 to the triplets ( Π , Π ′ , χ ) and ( Π ⊗ χ, Π ′ , ), the period relation in Theorem1.11 follows immediately.This paper is organized as follows. In §
2, we recall the theory of coherent cohomology on quaternionShimura varieties based on the general results of Harris [Har90b]. The cohomological periods Ω I ( Π ) foradmissible I ⊆ Σ E are defined in Proposition 2.6. In §
3, we construct the trilinear differential operator underthe totally unbalanced condition. We specialize the results of Harris in [Har85, §
3] and [Har86, §
7] to thenatural inclusion (R F / Q D × , ( H ± ) Σ F ) ⊂ (R F / Q ( D ⊗ F E ) × , ( H ± ) Σ E )of Shimura data and the automorphic line bundles L ( κ ( I κ ) ,r ) and L ′ (2 , . In §
4, we prove the algebraicityof critical values of twisted Rankin–Selberg L -functions for GL × GL over F . This section is logicallyindependent of the proof of Theorem 1.4 except for § §
4. In §
5, we prove our main results Theorems 1.4, 1.8, and 1.11. .3. Notation.
Fix a totally real number field F . Let A F (resp. A ) be the ring of adeles of F (resp. Q ) and A F ,f (resp. A f ) be its finite part. Let o F be the ring of integers of F and D F the absolute discriminant of F / Q . We denote by ˆ o F the closure of o F in A F ,f . Let ψ Q = N v ψ Q v be the standard additive character of Q \ A defined so that ψ Q p ( x ) = e − π √− x for x ∈ Z [ p − ] ,ψ R ( x ) = e π √− x for x ∈ R . The standard additive character ψ F of F \ A F is defined by ψ F = ψ Q ◦ tr F / Q . For α ∈ F , let ψ α F be the additivecharacter defined by ψ α F ( x ) = ψ F ( αx ). Similarly we define ψ α F v for α ∈ F v .Let E ba a totally real ´etale algebra over F . Let Σ E be the set of non-zero algebra homomorphismsfrom E to R . We identify E ∞ = E ⊗ Q R with R Σ E so that the w -th coordinate of R Σ E corresponds to thecompletion of E at w . Let κ = Σ w ∈ Σ E κ w w ∈ Z [Σ E ]. For σ ∈ Aut( C ), define σ κ = Σ w ∈ Σ E σ κ w w ∈ Z [Σ E ]with σ κ w = κ σ − ◦ w . Let Q ( κ ) be the fixed field of { σ ∈ Aut( C ) | σ κ = κ } . For each subset I of Σ E , let κ ( I ) = Σ w ∈ Σ E κ ( I ) w w ∈ Z [Σ E ] defined by κ ( I ) w = ( − κ w if w ∈ I,κ w if w / ∈ I, and let Q ( I ) be the fixed field of { σ ∈ Aut( C ) | σ I = I } . Note that Q ( κ ( I )) ⊆ Q ( κ ) · Q ( I ). Consider the mapthe power set of Σ E −→ { , , · · · , [ E : Q ] } × Z [Σ E ] ,I ( ♯ I, κ ( I )) . We say a subset I of Σ E is admissible with respect to κ if the fiber of ( ♯ I, κ ( I )) under the above map containsonly I . In this case, we have Q ( κ ( I )) = Q ( κ ) · Q ( I ). It is clear that the empty set and Σ E are admissible.We will use the notion of admissibility only when κ ∈ Z ≥ [Σ E ]. In this case, any subset I such that κ w ≥ w ∈ I is admissible. We assume E = F × · · · × F n for some totally real number fields F , · · · , F n over F . Let e E be the composite of the Galois closure of F i over Q for 1 ≤ i ≤ n . We identify Σ E with the disjoint union of Σ F i for 1 ≤ i ≤ n in a natural way. Let( κ, r ) ∈ Z [Σ E ] × Z [Σ E ]. We say ( κ, r ) is motivic if κ w ≡ r w (mod 2) for w ∈ Σ E and r w = r w ′ whenever w, w ′ ∈ Σ F i for some 1 ≤ i ≤ n .In GL , let B be the Borel subgroup consisting of upper triangular matrices and N be its unipotent radical,and put a ( ν ) = (cid:18) ν
00 1 (cid:19) , d ( ν ) = (cid:18) ν (cid:19) , m ( t ) = (cid:18) t t − (cid:19) , n ( x ) = (cid:18) x (cid:19) for ν, t ∈ GL and x ∈ G a . Let gl be the Lie algebra of GL ( R ) and gl , C be its complexification. We have gl , C = C · Z ⊕ C · H ⊕ C · X + ⊕ C · X − , where Z = (cid:18) (cid:19) , H = (cid:18) −√− √− (cid:19) , X + = (cid:18) √− − − −√− (cid:19) , X − = (cid:18) √− −√− (cid:19) . We denote by U ( gl , C ) the universal enveloping algebra of gl , C . LetSO(2) = (cid:26) k θ = (cid:18) cos θ sin θ − sin θ cos θ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) θ ∈ R / π Z (cid:27) . For κ ∈ Z ≥ and ε ∈ {± } , let D ( κ ) ε be the irreducible admissible ( gl , SO(2))-module characterized so thatthere exists a non-zero v ∈ D ( κ ) ε such that Z · v = 0 , H · v = εκ · v , X − ε · v = 0 . Let D ( κ ) = D ( κ ) + ⊕ D ( κ ) − , which is an irreducible admissible ( gl , O(2))-module. Note that D ( κ ) is theO(2)-finite part of the (limit of) discrete series representation of GL ( R ) with weight κ . For r ∈ Z such that ≡ r (mod 2) and α ∈ R × , let W ± ( κ,r ) ,ψ α R be the Whittaker function for D ( κ ) ⊗ | | r/ with respect to ψ α R ofweight ± κ normalized so that W ± ( κ,r ) ,ψ α R (1) = e − πα . The explicit formula is given by W ± ( κ,r ) ,ψ α R ( z n ( x ) a ( y ) k θ ) = z r ( ± αy ) ( κ + r ) / e π √− α ( x ±√− y ) ±√− κθ · I R > ( ± αy )(1.1)for x ∈ R , y, z ∈ R × , and k θ ∈ SO(2).Let σ ∈ Aut( C ). Define the σ -linear action on C ( X ), which is the field of formal Laurent series in variable X over C , as follows: σ P ( X ) = ∞ X n ≫−∞ σ ( a n ) X n for P ( X ) = P ∞ n ≫−∞ a n X n ∈ C ( X ). For a complex representation π of a group G on the space V π of π , let σ π be the representation of G defined σ π ( g ) = t ◦ π ( g ) ◦ t − , where t : V π → V π is a σ -linear isomorphism. Note that the isomorphism class of σ π is independent of thechoice of t . We call σ π the σ -conjugate of π . When v is a finite place of F and f is a complex-valued functionon F mv or ( F × v ) m for some m ∈ Z ≥ , we define σ f by σ f ( x ) = σ ( f ( x )) for x ∈ F mv or x ∈ ( F × v ) m .For an algebraic Hecke character χ of A × F , the Gauss sum G ( χ ) of χ is defined by G ( χ ) = D − / F Y v ∤ ∞ ε (0 , χ v , ψ F v ) , where ε ( s, χ v , ψ F v ) is the ε -factor of χ v with respect to ψ F v defined in [Tat79]. For σ ∈ Aut( C ), define Heckecharacter σ χ of A × F by σ χ ( x ) = σ ( χ ( x )). It is easy to verify that σ ( G ( χ )) = σ χ ( u ) G ( σ χ ) ,σ (cid:18) G ( χχ ′ ) G ( χ ) G ( χ ′ ) (cid:19) = G ( σ χ σ χ ′ ) G ( σ χ ) G ( σ χ ′ )(1.2)for algebraic Hecke characters χ, χ ′ of A × F , where u ∈ b Z × is the unique element such that σ ( ψ F ( x )) = ψ F ( ux )for x ∈ A F ,f . Let sgn( χ ) ∈ {± } Σ F be the signature of χ defined by sgn( χ ) v = χ v ( −
1) for v ∈ Σ F .2. Periods of motivic quaternionic modular forms
Coherent cohomology groups on the quaternionic Shimura variety.
Let E = F × · · · × F n be a totally real ´etale algebra over F , where F , · · · , F n are totally real number fields over F . For 1 ≤ i ≤ n ,let D i be a totally indefinite quaternion algebra over F i . Put D = D × · · · × D n . Let G = R E / Q D × = R F / Q D × × · · · R F n / Q D × n be a connected reductive linear algebraic group over Q . We identify G ( R ) with GL ( R ) Σ E via the identificationof E ∞ with R Σ E . Let h : R C / R G m → G R be the homomorphism defined by h ( x + √− y ) = (cid:18)(cid:18) x y − y x (cid:19) , · · · , (cid:18) x y − y x (cid:19)(cid:19) (2.1)on R -points. Let X be the G ( R )-conjugacy class containing h . Then ( G, X ) is a Shimura datum and theassociated Shimura variety Sh(
G, X ) = lim ←− K Sh K ( G, X ) = lim ←− K G ( Q ) \ X × G ( A f ) / K is called the quaternionic Shimura variety associated to ( G, X ), where K runs through neat open compactsubgroups of G ( A f ). It is a pro-algebraic variety over C with continuous G ( A f )-action and admits canonicalmodel over Q . Let K ∞ be the stabilizer of h in G ( R ). Note that K ∞ = Z G ( R ) · SO(2) Σ E and we have isomorphisms G ( R ) /K ∞ −→ X −→ ( H ± ) Σ E , gK ∞ ghg − g · ( √− , · · · , √− . (2.2) ere H ± = C r R is the union of the upper and lower half-planes and G ( R ) acts on ( H ± ) Σ E by the linearfractional transformation. Let g and k be the Lie algebras of G ( R ) and K ∞ , respectively. The Hodgedecomposition on g C induced by Ad ◦ h is given by g C = p + ⊕ k C ⊕ p − (2.3)with p + = g ( − , C , p − = g (1 , − C , and k C = g (0 , C . Here g ( p,q ) C = { X ∈ g C | h ( z ) − Xh ( z ) = z − p z − q X for z ∈ C } . We identify g with gl Σ E via the identification of E ∞ with R Σ E .Let ( κ, r ) ∈ Z [Σ E ] × Z [Σ E ] such that κ w ≡ r w (mod 2) for w ∈ Σ E . We denote by C ( κ,r ) the complex field C equipped with the action ρ ( κ,r ) of K ∞ given by ρ ( κ,r ) ( a · k θ ) z = Y w ∈ Σ E a − r w w e −√− κ w θ w · z (2.4)for a = ( a w ) w ∈ Σ E ∈ ( R × ) Σ E and k θ = ( k θ w ) w ∈ Σ E ∈ SO(2) Σ E , and z ∈ C . Conversely, any one-dimensionalalgebraic representation of K ∞ over C is of this form. We say ρ ( κ,r ) is motivic if ρ ( κ,r ) | Z G ( R ) is the basechange of a Q -rational character of Z G . It is easy to see that ρ ( κ,r ) is motivic if and only if ( κ, r ) is motivic(cf. § A (2) ( G ( A )) (resp. A ( G ( A ))) be the space of essentially square-integrable automorphic forms(resp. cusp forms) on G ( A ). Let C ∞ sia ( G ( A )) = { ϕ ∈ C ∞ ( G ( Q ) \ G ( A )) | X · ϕ is slowly increasing for all X ∈ U ( g C ) } ,C ∞ rda ( G ( A )) = { ϕ ∈ C ∞ ( G ( Q ) \ G ( A )) | X · ϕ is rapidly decreasing for all X ∈ U ( g C ) } . Let P = p − ⊕ k C be a parabolic subalgebra of g C . We have the ( P , K ∞ )-modules A ⋆ ( G ( A )) ⊗ C C ( κ,r ) , C ∞ ⋆ ′ ( G ( A )) ⊗ C C ( κ,r ) for ⋆ ∈ { (2) , } and ⋆ ′ ∈ { sia , rda } , where the action of P on C ( κ,r ) factors through k C . Consider the complexeswith respect to the Lie algebra differential operator (cf. [BW00, Chapter I]): C q (2) , ( κ,r ) = A (2) ( G ( A )) ⊗ C q ^ p + ⊗ C C ( κ,r ) ! K ∞ ,C q cusp , ( κ,r ) = A ( G ( A )) ⊗ C q ^ p + ⊗ C C ( κ,r ) ! K ∞ ,C q sia , ( κ,r ) = C ∞ sia ( G ( A )) ⊗ C q ^ p + ⊗ C C ( κ,r ) ! K ∞ ,C q rda , ( κ,r ) = C ∞ rda ( G ( A )) ⊗ C q ^ p + ⊗ C C ( κ,r ) ! K ∞ (2.5)for q ∈ Z ≥ . The corresponding q -th cohomology groups of the above complexes are denoted respectively by H q (2) ( L ( κ,r ) ) , H q cusp ( L ( κ,r ) ) , H q ( L can( κ,r ) ) , H q ( L sub( κ,r ) ) . Note that G ( A f ) acts on the above complexes by right translation. This in turn defines G ( A f )-module struc-tures on the cohomology groups. It is clear that H q cusp ( L ( κ,r ) ) and H q (2) ( L ( κ,r ) ) are semisimple G ( A f )-modules.By the results of Harris [Har90a] and [Har90b], the relative Lie algebra cohomology groups H q ( L can( κ,r ) ) and H q ( L sub( κ,r ) ) are isomorphic to the q -th coherent cohomology groups of the canonical and subcanonical ex-tension, respectively, of certain automorphic line bundle L ( κ,r ) on Sh( G, X ) to its toroidal compactification.Thus the terminology is justified. The natural inclusions A ( G ( A )) A (2) ( G ( A )) C ∞ rda ( G ( A )) C ∞ sia ( G ( A )) nduce the following commutative diagram for G ( A f )-module homomorphisms:(2.6) H q cusp ( L ( κ,r ) ) H q (2) ( L ( κ,r ) ) H q ( L sub( κ,r ) ) H q ( L can( κ,r ) ) . Let H q ! ( L ( κ,r ) ) be the image of the homomorphism H q ( L sub( κ,r ) ) → H q ( L can( κ,r ) ). By Theorem 2.1-(3) below, thehomomorphism H q cusp ( L ( κ,r ) ) → H q ( L sub( κ,r ) ) is injective. We identify H q cusp ( L ( κ,r ) ) with a G ( A f )-submoduleof H q ( L sub( κ,r ) ) by this injection.We recall in the following theorem some results of Harris [Har85], [Har90b] and Milne [Mil83] specialized tothe Shimura datum ( G, X ). Let A (2) ( G ( A ) , ( κ, r )) be the space of essentially square-integrable automorphicforms ϕ on G ( A ) such that • ϕ ( ag ) = Q w ∈ Σ E a r w w · ϕ ( g ) for a = ( a w ) w ∈ Σ E ∈ Z G ( R ) = ( R × ) Σ E and g ∈ G ( A ); • ϕ is an eigenfunction of the Casimir operator of g C with eigenvalue Q w ∈ Σ E ( κ w − κ w ).Let A ( G ( A ) , ( κ, r )) be the subspace of A (2) ( G ( A ) , ( κ, r )) consisting of cusp forms on G ( A ). We fix a Q ( κ )-rational structure on C ( κ,r ) spanned by a non-zero vector v ( κ,r ) ∈ C ( κ,r ) . We also fix a e E -rational structureon p ± with e E -basis given by { X ± ,w | w ∈ Σ E } , where X ± ,w is defined so that its w -component is equal to X ± and zero otherwise. For σ ∈ Aut( C ), we havethe σ -linear isomorphisms C ( κ,r ) −→ C ( σ κ,r ) , z · v ( κ,r ) σ ( z ) · v ( σ κ,r ) ; p ± −→ p ± , X w ∈ Σ E z w · X ± ,w X w ∈ Σ E σ ( z w ) · X ± ,σ ◦ w . (2.7)Note that we obtain a Q -rational structure on p ± by taking the Aut( C )-invariants with respect to the above σ -linear isomorphism. Theorem 2.1.
Assume ( κ, r ) ∈ Z [Σ E ] × Z [Σ E ] is motivic. (1) For σ ∈ Aut( C ) , with respect to the σ -linear isomorphisms in (2.7), conjugation by σ induces natural σ -linear G ( A f ) -module isomorphisms T σ : H q ( L sub( κ,r ) ) −→ H q ( L sub( σ κ,r ) ) , T σ : H q ( L can( κ,r ) ) −→ H q ( L can( σ κ,r ) ) , and such that the diagram H q ( L sub( κ,r ) ) H q ( L sub( σ κ,r ) ) H q ( L can( κ,r ) ) H q ( L can( σ κ,r ) ) T σ T σ is commute. Moreover, H q ( L sub( κ,r ) ) and H q ( L can( κ,r ) ) are admissible G ( A f ) -modules and have canon-ical rational structures over Q ( κ ) given by taking the Galois invariants with respect to T σ for σ ∈ Aut( C / Q ( κ )) . (2) We have H q cusp ( L ( κ,r ) ) = A ( G ( A ) , ( κ, r )) ⊗ C q ^ p + ⊗ C C ( κ,r ) ! K ∞ ,H q (2) ( L ( κ,r ) ) = A (2) ( G ( A ) , ( κ, r )) ⊗ C q ^ p + ⊗ C C ( κ,r ) ! K ∞ . (3) The composite of the left vertical and lower horizontal homomorphisms in (2.6) is an injective G ( A f ) -module homomorphism H q cusp ( L ( κ,r ) ) → H q ! ( L ( κ,r ) ) . The image of the homomorphism H q (2) ( L ( κ,r ) ) → H q ( L can( κ,r ) ) in (2.6) contains H q ! ( L ( κ,r ) ) . In partic-ular, H q ! ( L ( κ,r ) ) is a semisimple G ( A f ) -module. Let K be a neat open compact subgroup of G ( A f ). For ⋆ = sia or rda, we have the complexe K C ∗ ⋆, ( κ,r ) analogous to (2.5) with C ∞ ⋆ ( G ( Q ) \ G ( A )) replacing by C ∞ ⋆ ( G ( Q ) \ G ( A ) / K ). We denote by H q K ( L can( κ,r ) )(resp. H q K ( L sub( κ,r ) )) the corresponding q -th cohomology group if ⋆ = sia (resp. ⋆ = rda). Note that H q K ( L can( κ,r ) )and H q K ( L sub( κ,r ) ) are finite dimensional vector spaces over C and isomorphic to the q -th coherent cohomol-ogy groups of the canonical and subcanonical extension, respectively, of certain automorphic line bundle K L ( κ,r ) on Sh K ( G, X ) to its toroidal compactification. For each g ∈ G ( A f ), let K g = g − K g . The naturalisomorphism Sh K g ( G, X ) → Sh K ( G, X ) induces isomorphisms H q K ( L can( κ,r ) ) −→ H q K g ( L can( κ,r ) ) , H q K ( L sub( κ,r ) ) −→ H q K g ( L sub( κ,r ) ) . (2.8)Then similar assertions as in Theorem 2.1-(1) hold for H q K ( L can( κ,r ) ) and H q K ( L sub( κ,r ) ) so that the corresponding σ -linear isomorphisms T σ in (1) are compatible with (2.8). Moreover, the natural morphisms of complexes K C ∗ ⋆, ( κ,r ) → C ∗ ⋆, ( κ,r ) induce G ( A f )-module isomorphismslim −→ K H q K ( L can( κ,r ) ) −→ H q ( L can( κ,r ) ) , lim −→ K H q K ( L sub( κ,r ) ) −→ H q ( L sub( κ,r ) )which are compatible with T σ for all σ ∈ Aut( C ). Here the G ( A f )-module structure on the direct limits aredefined by the isomorphisms (2.8).Let 2 = (2 , · · · ,
2) and 0 = (0 , · · · , K L sub(2 , is isomorphic to the dualizingsheaf for toroidal compactification of Sh K ( G, X ) (cf. [Har90b, Proposition 2.2.6]) with trace map H [ E : Q ] K ( L sub(2 , ) −→ C , ω Z Sh K ( G,X ) ω, where Z Sh K ( G,X ) ω = Z G ( Q ) \ X × G ( A f ) / K ϕ ◦ ι K ( x ) dµ K if ω is represented by ϕ ⊗ V w ∈ Σ E X + ,w ⊗ v (2 , ∈ K C [ E : Q ]rda , (2 , . Here ι K : G ( Q ) \ X × G ( A f ) / K −→ G ( Q ) \ G ( A ) /K ∞ K is the natural isomorphism, and the measure on G ( Q ) \ X × G ( A f ) / K is defined as follows: • on G ( A f ) / K , we take the counting measure; • on X , we take the measure (cid:18) dz ∧ dz π √− (cid:19) [ E : Q ] on ( H ± ) Σ E via the isomorphism (2.2).Under this normalization of measure, we have the following Galois equivariant property: σ Z Sh K ( G,X ) ω ! = Z Sh K ( G,X ) T σ ω (2.9)for ω ∈ H [ E : Q ] K ( L sub(2 , ) and σ ∈ Aut( C ) (cf. [Har90b, (3.8.4)]). In the following lemma, we show that the familyof trace maps as K varies can be normalized to define a trace map H [ E : Q ] ( L sub(2 , ) → C . Lemma 2.2.
We have the G ( A f ) -equivariant trace map H [ E : Q ] ( L sub(2 , ) −→ C , ω Z Sh(
G,X ) ω, where Z Sh(
G,X ) ω = [ˆ o × E : o × E · U ] − X a ∈ E × \ A × E / E ×∞ U Z Z G ( A ) G ( Q ) \ G ( A ) ϕ ( ag ) dg Tam11 f ω is represented by ϕ ⊗ V w ∈ Σ E X + ,w ⊗ v (2 , ∈ C [ E : Q ]rda , (2 , . Here dg Tam is the Tamagawa measure on Z G ( A ) \ G ( A ) and U is any open compact subgroup of A × E ,f such that ϕ is right U -invariant. Moreover, thetrace map satisfies the Galois equivariant property: σ Z Sh(
G,X ) ω ! = Z Sh(
G,X ) T σ ω for ω ∈ H [ E : Q ] ( L sub(2 , ) and σ ∈ Aut( C ) .Proof. We identify Z G ( A ) with A × E . Let ω ∈ H [ E : Q ] ( L sub(2 , ) be a class represented by ϕ ⊗ V w ∈ Σ E X + ,w ⊗ v (2 , ∈ C [ E : Q ]rda , (2 , . For any neat open compact subgroup K of G ( A f ) such that ϕ is right K -invariant, we let U K = Z G ( A f ) ∩ K and ω K ∈ H [ E : Q ] K ( L sub(2 , ) be the class represented by ϕ ⊗ V w ∈ Σ E X + ,w ⊗ v (2 , consideredas an element in K C [ E : Q ]rda , (2 , . By [IP18, Lemmas 6.1 and 6.3], there exists a non-zero rational number C depending only on G such that Z Sh K ( G,X ) ω K = Z G ( Q ) \ X × G ( A f ) / K ϕ ◦ ι K ( x ) dµ K = C · [ K : K ] · [ˆ o × E : U K ] − · Z G ( Q ) \ G ( A ) /Z G ( R ) U K ϕ ( g ) dg Tam . Here K is any maximal open compact subgroup of G ( A f ) containing K . We remark that the above formulawas proved in [IP18, Lemmas 6.3] for U K = ˆ o × E . The general case can be proved in a similar way. We rewritethe formula as C − · [ K : K ] − · [ o × E · U K : U K ] · Z Sh K ( G,X ) ω K = [ˆ o × E : o × E · U K ] − · Z G ( Q ) \ G ( A ) /Z G ( R ) U K ϕ ( g ) dg Tam = [ˆ o × E : o × E · U K ] − · X a ∈ E × \ A × E / E ×∞ U K Z Z G ( A ) G ( Q ) \ G ( A ) ϕ ( ag ) dg Tam
Note that the constant C − · [ K : K ] − · [ o × E · U K : U K ] depends only on G and K . Let U be any open compactsubgroup of A × E ,f such that ϕ is right U -invariant. Then we have[ˆ o × E : o × E · ( U ∩ U K )] − X a ∈ E × \ A × E / E ×∞ ( U∩U K ) Z Z G ( A ) G ( Q ) \ G ( A ) ϕ ( ag ) dg Tam = [ˆ o × E : o × E · ( U ∩ U K )] − · [ E × E ×∞ U : E × E ×∞ ( U ∩ U K )] X a ∈ E × \ A × E / E ×∞ U Z Z G ( A ) G ( Q ) \ G ( A ) ϕ ( ag ) dg Tam = [ˆ o × E : o × E · U ] − X a ∈ E × \ A × E / E ×∞ U Z Z G ( A ) G ( Q ) \ G ( A ) ϕ ( ag ) dg Tam . We conclude that the trace map ω R Sh(
G,X ) ω is well-defined. Finally, the Galois equivariance propertyfollows from C ∈ Q × and (2.9). This completes the proof. (cid:3) Rational structures via the coherent cohomology.
Let Π = N v π v be an irreducible cuspidalautomorphic representation of G ( A ). Let π f = N p π p be the finite part of Π . We assume that Π is motivic,that is, there exists motivic ( κ, r ) ∈ Z ≥ [Σ E ] × Z [Σ E ] such that π ∞ = ⊠ w ∈ Σ E ( D ( κ w ) ⊗ | | − r w / ) . We call κ (resp. ( κ, r )) the weight (resp. motivic weight) of Π . When E is a field, we necessary have r =( r, · · · , r ) for some r ∈ Z and also call ( κ, r ) ∈ Z ≥ [Σ E ] × Z the motivic weight of Π . Note that Π occurs in A ( G ( A ) , ( κ, r )). For each motivic ( κ ′ , r ′ ) ∈ Z [Σ E ] × Z [Σ E ] and ⋆ ∈ { cusp , (2) , ! } , we denote by H q⋆ ( L ( κ ′ ,r ′ ) )[ π f ]the π f -isotypic component of π f in H q⋆ ( L ( κ ′ ,r ′ ) ). emma 2.3. Let ( κ ′ , r ′ ) ∈ Z [Σ E ] × Z [Σ E ] be motivic. (1) If r = r ′ or ( κ w − κ ′ w )( κ w + κ ′ w − = 0 for some w ∈ Σ E , then H q⋆ ( L ( κ ′ ,r ′ ) )[ π f ] = 0 for all q and ⋆ ∈ { cusp , (2) , ! } . (2) If r = r ′ and ( κ w − κ ′ w )( κ w + κ ′ w −
2) = 0 for all w ∈ Σ E , then the homomorphism in Theorem2.1-(3) induces an isomorphism of G ( A f ) -modules H q cusp ( L ( κ ′ ,r ) )[ π f ] → H q ! ( L ( κ ′ ,r ) )[ π f ] . Moreover,the multiplicity of π f in H q cusp ( L ( κ ′ ,r ) )[ π f ] is equal to the number of subsets I of Σ E such that κ ′ = κ ( I ) , q = ♯ I. Proof.
For each subset I of Σ E , let ε ( I ) w = ( − if w ∈ I, + if w / ∈ I for w ∈ Σ E and π ∞ ,I = ⊠ w ∈ Σ E ( D ( κ w ) ε ( I ) w ⊗ | | − r w / )be an irreducible admissible ( g , K ∞ )-module. Note that we have a ( g , K ∞ )-module isomorphism π ∞ ≃ M I ⊆ Σ E π ∞ ,I . Specializing [Har90b, Theorem 4.6.2] to the (limit of) discrete series representation π ∞ ,I , we havedim π ∞ ,I ⊗ C q ^ p + ⊗ C C ( κ ′ ,r ′ ) ! K ∞ = ( κ ′ , r ′ ) = ( κ ( I ) , r ) or q = ♯ I, κ ′ , r ′ ) = ( κ ( I ) , r ) and q = ♯ I. (2.10)On the other hand, by the strong multiplicity one theorem, we have A ( G ( A ) , ( κ, r ))[ π f ] = A (2) ( G ( A ) , ( κ, r ))[ π f ] = Π . The assertions then follow from Theorem 2.1-(2)-(4) and (2.10). This completes the proof. (cid:3)
For σ ∈ Aut( C ), let σ Π be the irreducible admissible representation of G ( A ) defined by σ Π = σ π ∞ ⊗ σ π f , where σ π f is the σ -conjugate of π f and σ π ∞ is the representation of G ( R ) = GL ( R ) Σ E so that its w -componentis equal to the σ − ◦ w -component of π ∞ . The following lemma is well-known and can be deduced from theresult of Shimura [Shi78, Proposition 1.6]. When κ w ≥ w ∈ Σ E , the lemma was also proved byWaldspurger [Wal85] and Harder [Har87] and is based on the study of ( g , K ∞ )-cohomology. We provideanother proof based on the results of Harris [Har90b], which is ( P , K ∞ )-cohomological in natural. Lemma 2.4.
For σ ∈ Aut( C ) , the representation σ Π is a motivic irreducible cuspidal automorphic represen-tation of G ( A ) of motivic weight ( σ κ, r ) . Moreover, the rationality field Q ( Π ) of Π is equal to the fixed fieldof { σ ∈ Aut( C ) | σ Π = Π } .Proof. Fix σ ∈ Aut( C ). Since H ( L ( κ,r ) )[ π f ] ≃ π f , we have H ( L ( σ κ,r ) )[ σ π f ] = T σ ( H ( L ( κ,r ) )[ π f ]) ≃ σ π f . On the other hand, the homomorphism in Theorem 2.1-(3) is an isomorphism by [Har90b, Proposition 5.4.2].It follows that H ( L ( σ κ,r )) )[ σ π f ] = (cid:0) A ( G ( A ) , ( σ κ, r ))[ σ π f ] ⊗ C C ( σ κ,r ) (cid:1) K ∞ ≃ σ π f . We conclude that A ( G ( A ) , ( σ κ, r ))[ σ π f ] = Π ′ for some irreducible cuspidal automorphic representation Π ′ = π ′∞ ⊗ σ π f of G ( A ) such that (cid:0) π ′∞ ⊗ C C ( σ κ,r ) (cid:1) K ∞ = 0 . This implies that there exists a non-zero vector v ∈ π ′∞ such that π ′∞ ( a · k θ ) v = Y w ∈ Σ E a r w w e √− σ κ w θ w · v (2.11)for a = ( a w ) w ∈ Σ E ∈ ( R × ) Σ E and k θ = ( k θ w ) w ∈ Σ E ∈ SO(2) Σ E . In particular, we have( H, · · · , H ) · v = Y w ∈ Σ E σ κ w · v = Y w ∈ Σ E κ w · v. ote that the Casimir operator of gl , C is given by H − H − X + X − . We deduce from the second condition defining the space A ( G ( A ) , ( σ κ, r )) that( X + X − , · · · , X + X − ) · v = 0 . (2.12)By [JL70, Lemma 5.6], we see that (2.11) and (2.12) imply that π ′∞ = ⊠ w ∈ Σ E ( D ( σ κ w ) ⊗ | | − r w / ) . Therefore Π ′ is isomorphic to σ Π . For the second assertion, assume σ π f = π f . Then it follows from thestrong multiplicity one theorem that σ Π = Π . This completes the proof. (cid:3) Lemma 2.5.
Let I ⊆ Σ E . For any field extension Q ( Π , I ) ⊆ F ⊆ C , we have the F -rational structure on H ♯ I cusp ( L ( κ ( I ) ,r ) )[ π f ] by taking the Aut( C /F ) -invariants H ♯ I cusp ( L ( κ ( I ) ,r ) )[ π f ] Aut( C /F ) = n c ∈ H ♯ I cusp ( L ( κ ( I ) ,r ) )[ π f ] (cid:12)(cid:12)(cid:12) T σ c = c for all σ ∈ Aut( C /F ) o . Here Q ( Π , I ) = Q ( Π ) · Q ( I ) . Moreover, suppose that I is admissible with respect to κ , then the G ( A f ) -module H ♯ I cusp ( L ( κ ( I ) ,r ) )[ π f ] is isomorphic to π f and the F -rational structures are unique up to homotheties.Proof. By the commutativity of the diagram in Theorem 2.1-(1) and Lemma 2.3-(2), we have H ♯ I cusp ( L ( σ κ ( σ I ) ,r ) )[ σ π f ] = T σ ( H ♯ I cusp ( L ( κ ( I ) ,r ) )[ π f ])(2.13)for all σ ∈ Aut( C ). Let Q ( Π ) · Q ( I ) ⊆ F ⊆ C be a field extension. By Theorem 2.1-(1), H ♯ I ( L sub( κ ( I ) ,r ) ) admitsa F -rational structure given by H ♯ I ( L sub( κ ( I ) ,r ) ) Aut( C /F ) = n c ∈ H ♯ I ( L sub( κ ( I ) ,r ) ) (cid:12)(cid:12)(cid:12) T σ c = c for all σ ∈ Aut( C /F )) o . By (2.13), H ♯ I cusp ( L ( κ ( I ) ,r ) )[ π f ] is invariant by T σ for σ ∈ Aut( C /F ). Therefore, by [Clo90, Lemme 3.2.1], wehave a F -rational structure on H ♯ I cusp ( L ( κ ( I ) ,r ) )[ π f ] given by H ♯ I cusp ( L ( κ ( I ) ,r ) )[ π f ] ∩ H ♯ I ( L sub( κ ( I ) ,r ) ) Aut( C /F ) = H ♯ I cusp ( L ( κ ( I ) ,r ) )[ π f ] Aut( C /F ) . Finally, suppose that I is admissible with respect to κ . Then H ♯ I cusp ( L ( κ ( I ) ,r ) )[ π f ] ≃ π f is irreducible byLemma 2.3-(2) and the uniqueness up to homotheties then follows from Schur’s lemma. This completes theproof. (cid:3) Harris’ periods.
Let Π be a motivic irreducible cuspidal automorphic representation of G ( A ) of motivicweight ( κ, r ) ∈ Z ≥ [Σ E ] × Z [Σ E ]. Let Π hol be the subspace of Π consisting of holomorphic cusp forms ϕ ∈ Π ,that is, ϕ ( gk θ ) = e √− P w ∈ Σ E κ w θ w ϕ ( g )for k θ = ( k θ w ) w ∈ Σ E ∈ SO(2) Σ E and g ∈ G ( A ). Let I be a subset of Σ E . Define τ I ∈ G ( R ) = GL ( R ) Σ E by τ I = ( a ( ε ( I ) w )) w ∈ Σ E , (2.14)where ε ( I ) w = ( − w ∈ I, w / ∈ I. For ϕ ∈ Π hol , let ϕ I ∈ Π defined by ϕ I ( g ) = ϕ ( g · τ I )(2.15)for g ∈ G ( A ). We then have the homomorphism of G ( A f )-modules: ξ I : Π hol −→ H ♯ I cusp ( L ( κ ( I ) ,r ) )[ π f ] , ϕ ϕ I ⊗ ^ w ∈ I X + ,w ⊗ v ( κ ( I ) ,r ) . (2.16)Here we fix an ordering of the wedge V w ∈ I X + ,w once and for all such that V w ∈ I X + ,w V w ∈ σ I X + ,w underthe σ -linear isomorphism in (2.7) for all σ ∈ Aut( C ). Note that ξ I is an isomorphism if and only if I is dmissible with respect to κ by Lemma 2.3-(2). By taking I = ∅ to be the empty set, we identify Π hol with H ( L ( κ,r ) )[ π f ] by the isomorphism ξ ∅ . Moreover, for σ ∈ Aut( C ) we have the σ -linear isomorphism Π hol −→ σ Π hol , ϕ σ ϕ (2.17)defined so that the diagram Π hol σ Π hol H ( L ( κ,r ) )[ π f ] H ( L ( σ κ,r ) )[ σ π f ] ξ ∅ ξ ∅ T σ is commute. Comparing the Q ( Π , I )-rational structures on H ( L ( κ,r ) )[ π f ] and H ♯ I cusp ( L ( κ ( I ) ,r ) )[ π f ] definedin Lemma 2.5, we obtain the following result. Proposition 2.6.
Let I ⊆ Σ E be admissible with respect to κ . There exists Ω I ( Π ) ∈ C × , unique up to Q ( Π , I ) × , such that ξ I (cid:16) Π Aut( C / Q ( Π ,I ))hol (cid:17) Ω I ( Π ) = H ♯ I cusp ( L ( κ ( I ) ,r ) )[ π f ] Aut( C / Q ( Π , I)) . Moreover, we can normalize the periods so that T σ (cid:18) ξ I ( ϕ )Ω I ( Π ) (cid:19) = ξ σ I ( σ ϕ )Ω σ I ( σ Π ) for ϕ ∈ Π hol and σ ∈ Aut( C ) . Remark 2.7.
For non-admissible I ⊆ Σ E , the multiplicity of π f in H ♯ I cusp ( L ( κ ( I ) ,r ) )[ π f ] is greater than oneand it is not known whether the equality T σ ◦ ξ I ( Π hol ) = ξ σ I ( σ Π hol ) holds for any σ ∈ Aut( C ). Therefore,we do not know whether ξ I ( Π hol ) is defined over Q ( Π , I ). However, since H ♯ I cusp ( L ( κ ( I ) ,r ) )[ π f ] is defined over Q ( Π ) · Q ( κ ( I )), it follows that ξ I ( Π hol ) is define over some finite extension F over Q ( Π ) · Q ( κ ( I )). Hecnethere exists Ω I ( Π ) ∈ C × , unique up to F × , such that ξ I (cid:16) Π Aut( C /F )hol (cid:17) Ω I ( Π ) = ξ I ( Π hol ) ∩ H ♯ I cusp ( L ( κ ( I ) ,r ) )[ π f ] Aut( C /F ) . We have assume that E = F × · · · × F n and D = D × · · · × D n for some totally real number fields F i andsome totally indefinite quaternion algebra D i over F i . Then Π = Π × · · · × Π n for some motivic irreducible cuspidal automorphic representation Π i of D × i ( A F i ) for 1 ≤ i ≤ n . We identifyΣ E with the disjoint union of Σ F i for 1 ≤ i ≤ n in a natural way. Then I = n G i =1 I ∩ Σ F i . We have the following period relation.
Lemma 2.8.
Let I ⊆ Σ E be admissible with respect to κ . For σ ∈ Aut( C ) , we have σ (cid:18) Ω I ( Π ) Q ni =1 Ω I ∩ Σ F i ( Π i ) (cid:19) = Ω σ I ( σ Π ) Q ni =1 Ω σ I ∩ Σ F i ( σ Π i ) . Proof.
Indeed, we have Sh(
G, X ) = Sh( G , X ) × · · · × Sh( G n , X n ) , where ( G i , X i ) = (R F i / Q D × i , ( H ± ) Σ F i ). Write( κ, r ) = ( κ , r ) × · · · × ( κ n , r n )under the identification Z ≥ [Σ E ] × Z [Σ E ] = ( Z ≥ [Σ F ] × Z [Σ F ]) × · · · × ( Z ≥ [Σ F n ] × Z [Σ F n ]) . hen we have L ( κ ( I ) ,r ) = L ( κ i ( I ∩ Σ F ) ,r ) × · · · × L ( κ i ( I ∩ Σ F n ) ,r n ) . Then it follows from the K¨unneth formula that we have a canonical G ( A f )-module isomorphism H ♯ I ( L ⋆ ( κ ( I ) ,r ) ) ≃ M q + ··· + q n = ♯ I n O i =1 H q i ( L ⋆ ( κ i ( I ∩ Σ F i ) ,r i ) )for ⋆ ∈ { sub , can } . Taking the π f -isotypic parts and note that H q i ! ( L ( κ i ( I ∩ Σ F i ) ,r i ) )[ π i,f ] is zero unless q i = ♯ I ∩ Σ F i by Lemma 2.3 and the admissibility of I , we thus obtain an isomorphism H ♯ I ! ( L ( κ ( I ) ,r ) )[ π f ] ≃ n O i =1 H ♯ I ∩ Σ F i ! ( L ( κ i ( I ∩ Σ F i ) ,r i ) )[ π i,f ] . We deduce from Lemma 2.3-(2) that H ♯ I cusp ( L ( κ ( I ) ,r ) )[ π f ] ≃ n O i =1 H ♯ I ∩ Σ F i cusp ( L ( κ i ( I ∩ Σ F i ) ,r i ) )[ π i,f ] . In the above isomorphism, we normalize the Q ( κ i )-rational structure on C ( κ i ,r i ) for 1 ≤ i ≤ n such that v ( κ,r ) = n O i =1 v ( κ i ,r i ) under the isomorphism ( ρ ( κ,r ) , C ( κ,r ) ) ≃ n O i =1 ( ρ ( κ i ,r i ) , C ( κ i ,r i ) ) . Then for σ ∈ Aut( C ), we have T σ = T (1) σ ⊗ · · · ⊗ T ( n ) σ . Here T ( i ) σ : H ♯ I ∩ Σ F i ( L sub( κ i ( I ∩ Σ F i ) ,r i ) ) −→ H ♯ I ∩ Σ F i ( L sub( σ κ i ( σ I ∩ Σ F i ) ,r i ) )is the σ -linear isomorphism in Theorem 2.1 -(1). The assertion then follows at once. (cid:3) For I = Σ E , the period Ω Σ E ( Π ) can be expressed in terms of the Petersson pairing of holomorphic cuspforms. Let h , i : Π hol × Π ∨ hol → C be the G ( A f )-equivariant Petersson bilinear pairing defined by h ϕ , ϕ i = Z Z G ( A ) G ( Q ) \ G ( A ) ϕ Σ E ( g ) ϕ ( g ) dg Tam . (2.18)Here dg Tam is the Tamagawa measure on Z G ( A ) \ G ( A ). Lemma 2.9.
We have σ (cid:18) h ϕ , ϕ i Ω Σ E ( Π ) (cid:19) = h σ ϕ , σ ϕ i Ω Σ E ( σ Π ) for ϕ ∈ Π hol , ϕ ∈ Π ∨ hol , and σ ∈ Aut( C ) .Proof. We have the morphism for complexes C [ E : Q ]rda , (2 − κ,r ) × C , ( κ, − r ) −→ C [ E : Q ]sia , (2 , ϕ ⊗ ^ w ∈ Σ E X + ,w ⊗ v (2 − κ,r ) , ϕ ⊗ v ( κ, − r ) ! ϕ ϕ ⊗ ^ w ∈ Σ E X + ,w ⊗ v (2 , . This induces G ( A f )-module homomorphism of cohomology groups H [ E : Q ] ( L sub(2 − κ,r ) ) × H ( L can( κ, − r ) ) −→ H [ E : Q ] ( L sub(2 , ) , ( c , c ) c ∧ c . The homomorphism satisfies the Galois equivariant property T σ ( c ∧ c ) = T σ c ∧ T σ c or all σ ∈ Aut( C ). Composing with the trace map in Lemma 2.2, we obtain the G ( A f )-equivariant homo-morphism Π hol × Π ∨ hol −→ C , ( ϕ , ϕ ) Z Sh(
G,X ) ξ Σ E ( ϕ ) ∧ ξ ∅ ( ϕ )which satisfies σ Z Sh(
G,X ) ξ Σ E ( ϕ )Ω Σ E ( Π ) ∧ ξ ∅ ( ϕ )Ω ∅ ( Π ∨ ) ! = Z Sh(
G,X ) ξ Σ E ( σ ϕ )Ω Σ E ( σ Π ) ∧ ξ ∅ ( σ ϕ )Ω ∅ ( σ Π ∨ )for all σ ∈ Aut( C ). Note that we may take Ω ∅ ( Π ∨ ) = 1 by definition. Since the class ξ Σ E ( ϕ ) ∧ ξ ∅ ( ϕ ) in H [ E : Q ] ( L sub(2 , ) is represented by ϕ Σ E · ϕ ⊗ V w ∈ Σ E X + ,w ⊗ v (2 , , by Lemma 2.2 we have Z Sh(
G,X ) ξ Σ E ( ϕ ) ∧ ξ ∅ ( ϕ ) = [ A × E : E × E ×∞ ˆ o × E ] · h ϕ , ϕ i . This completes the proof. (cid:3)
Let L ( s, Π , Ad) = Y v L ( s, π v , Ad)be the adjoint L -function of Π , where Ad is the adjoint representation of L G on pgl ( C ) [ E : Q ] . Note that L ( s, Π , Ad) is holomorphic and non-zero at s = 1. Combining with the results of Shimura and Takase, weobtain the following corollary. Corollary 2.10.
We have σ L (1 , Π , Ad)(2 π √− − P w ∈ Σ E κ w · π [ E : Q ] · Ω Σ E ( Π ) ! = L (1 , σ Π , Ad)(2 π √− − P w ∈ Σ E κ w · π [ E : Q ] · Ω Σ E ( σ Π ) for all σ ∈ Aut( C ) .Proof. By Lemma 2.8, it suffices to consider the case when E is a field. Let Π ′ be the Jacquet-Langlandstransfer of Π to GL ( A E ). By a variant of the result [HK91, Theorem 12.3] (see also [Shi81, Theorem 3.8]),we have σ (cid:18) h ϕ , ϕ ih ϕ , ϕ i (cid:19) = h σ ϕ , σ ϕ ih σ ϕ , σ ϕ i (2.19)for ϕ ∈ Π hol , ϕ ∈ Π ∨ hol , ϕ ∈ Π ′ hol , ϕ ∈ ( Π ′ ) ∨ hol with h ϕ , ϕ i 6 = 0 and σ ∈ Aut( C ). We remark thatalthough the assertion in [HK91, Theorem 12.3] is stated for E = Q , one can prove (2.19) for general totallyreal number fields following the same argument in [HK91, § σ L (1 , Π , Ad)(2 π √− − P w ∈ Σ E κ w · π [ E : Q ] · h ϕ , ϕ i ! = L (1 , σ Π , Ad)(2 π √− − P w ∈ Σ E κ w · π [ E : Q ] · h σ ϕ , σ ϕ i for ϕ ∈ Π ′ hol , ϕ ∈ ( Π ′ ) ∨ hol with h ϕ , ϕ i 6 = 0, and σ ∈ Aut( C ). We remark that the factor (2 π √− − P w ∈ Σ E κ w is obtained from the comparison between rational structures on Π ′ hol given by the zeroth coherent cohomol-ogy and by the Whittaker model (cf. [GH93, (A.4.6)] or Lemma 4.2 below). The assertion then followsimmediately from Lemma 2.9. (cid:3) Trilinear differential operators
Let E be a totally real ´etale cubic algebra over a totally real number field F . Let D be a totally indefinitequaternion algebra over F . Let G ′ = R F / Q D × , G = R E / Q ( D ⊗ F E ) × be connected reductive linear algebraic groups over Q . We identify G ′ ( R ) and G ( R ) with GL ( R ) Σ F andGL ( R ) Σ E via the identifications F ∞ = R Σ F and E ∞ = R Σ E , respectively. Let X ′ (resp. X ) be the G ′ ( R )-conjugacy class (resp. G ( R )-conjugacy class) containing h ′ (resp. h ) defined as in (2.1). The natural diagonal mbedding F → E induces the natural injective morphism G ′ → G , which defines the inclusion of Shimuradata ( G ′ , X ′ ) ⊂ ( G, X ) . We begin with a well-known lemma.
Lemma 3.1.
Let ( ℓ , ℓ , ℓ ) ∈ Z ≥ such that ℓ ≥ ℓ + ℓ and ℓ + ℓ + ℓ ≡ . Let v ℓ and v ℓ be non-zero vectors in D ( ℓ ) + and D ( ℓ ) + of weights ℓ and ℓ , respectively. Then there exists a non-zeroelement in U ( gl , C ) of the form X m +2 m = ℓ − ℓ − ℓ c m ,m ( ℓ , ℓ , ℓ )( X m + ⊗ X m + ) , unique up to scalars, such that H · X m +2 m = ℓ − ℓ − ℓ c m ,m ( ℓ , ℓ , ℓ )( X m + v ℓ ⊗ X m + v ℓ )= ℓ · X m +2 m = ℓ − ℓ − ℓ c m ,m ( ℓ , ℓ , ℓ )( X m + v ℓ ⊗ X m + v ℓ ) , (3.1) and X − · X m +2 m = ℓ − ℓ − ℓ c m ,m ( ℓ , ℓ , ℓ )( X m + v ℓ ⊗ X m + v ℓ ) = 0 . (3.2) Here the action of gl , C on D ( ℓ ) + ⊗ D ( ℓ ) + is given by X · ( v ⊗ v ′ ) = X · v ⊗ v ′ + v ⊗ X · v ′ . Moreover, the coefficients can be normalized so that c m ,m ( ℓ , ℓ , ℓ ) ∈ Q .Proof. The existence and uniqueness follow directly from the decomposition (cf. [Rep76]) D ( ℓ ) + ⊗ D ( ℓ ) + | gl , C = ∞ M j =0 D ( ℓ + ℓ + 2 j ) + . For the rationality of the differential operator, note that H · v ℓ = ℓ · v ℓ , H · v ℓ = ℓ · v ℓ , X + X − − X − X + = − H. Therefore, by a simple induction argument, we see that the linear equations defined by (3.2) have coefficientsin Q . In particular, the solutions can be chosen to be rational numbers. (cid:3) For a triplet ( ℓ , ℓ , ℓ ) ∈ Z ≥ such that ℓ + ℓ + ℓ ≡ { ℓ , ℓ , ℓ } ≥ ℓ + ℓ + ℓ , we fix a choice of c m ,m ( ℓ , ℓ , ℓ ) ∈ Q for each pair of non-negative integers m , m with 2 m + 2 m =2 max { ℓ , ℓ , ℓ }− ℓ + ℓ + ℓ such that the similar assertion in Lemma 3.1 holds. Define X ( ℓ , ℓ , ℓ ) ∈ U ( gl , C )by X ( ℓ , ℓ , ℓ ) = P m +2 m = ℓ − ℓ − ℓ c m ,m ( ℓ , ℓ , ℓ )(1 ⊗ X m + ⊗ X m + ) if ℓ ≥ ℓ + ℓ , P m +2 m = ℓ − ℓ − ℓ c m ,m ( ℓ , ℓ , ℓ )( X m + ⊗ ⊗ X m + ) if ℓ ≥ ℓ + ℓ , P m +2 m = ℓ − ℓ − ℓ c m ,m ( ℓ , ℓ , ℓ )( X m + ⊗ X m + ⊗
1) if ℓ ≥ ℓ + ℓ . Let ( κ, r ) ∈ Z ≥ [Σ E ] × Z [Σ E ] be motivic. We assume κ satisfies the totally unbalanced condition2 max w | v { κ w } − X w | v κ w ≥ v ∈ Σ F and X w ∈ Σ E r w = 0 . For each v ∈ Σ F , let v (1) , v (2) , v (3) ∈ Σ E be the extensions of v and ˜ v ( κ ) ∈ Σ E the homomorphism such thatmax w | v { κ w } = κ ˜ v ( κ ) . Put I κ = { ˜ v ( κ ) | v ∈ Σ F } ⊂ Σ E . et L ′ (2 , and L ( κ ( I κ ) ,r ) be the automorphic line bundles on Sh( G ′ , X ′ ) and Sh( G, X ) defined by the motivicalgebraic representations ρ ′ (2 , of K ′∞ and ρ ( κ ( I κ ) ,r ) of K ∞ in (2.4), respectively. Here(2 ,
0) = ((2 , · · · , , (0 , · · · , ∈ Z [Σ F ] × Z [Σ F ] . Proposition 3.2.
There exists a homogeneous Q ( κ ) -rational differential operator [ δ ( κ )] from L ( κ ( I κ ) ,r ) to L ′ (2 , satisfying the following conditions: (1) Let [ δ ( κ )] : H [ F : Q ] ( L sub( κ ( I κ ) ,r ) ) −→ H [ F : Q ] (( L ′ (2 , ) sub ) be the induced G ′ ( A f ) -module homomorphism. Then we have T σ ◦ [ δ ( κ )] = [ δ ( σ κ )] ◦ T σ for all σ ∈ Aut( C ) . (2) If a class in H [ F : Q ] ( L sub( κ ( I κ ) ,r ) ) is represented by ϕ ⊗ V w ∈ I κ X + ,w ⊗ v ( κ ( I κ ) ,r ) , then its image under [ δ ( κ )] in H [ F : Q ] (( L ′ (2 , ) sub ) is represented by ( X ( κ ) · ϕ ) | G ′ ( A ) ⊗ ^ v ∈ Σ F X + ,v ⊗ v (2 , . Here X ( κ ) ∈ U ( gl Σ E , C ) is defined by X ( κ ) = O v ∈ Σ F X ( κ v (1) , κ v (2) , κ v (3) ) . (3.4) Proof.
Recall g and k (resp. g ′ and k ′ ) are the Lie algebras of G ( R ) and K ∞ = Z G ( R ) · SO(2) Σ E (resp. G ′ ( R )and K ′∞ = Z G ′ ( R ) · SO(2) Σ F ), respectively, and we have the Hodge decompositions for g C and g ′ C as in (2.3).We identify g and g ′ with gl Σ E and gl Σ F , respectively. For each w ∈ Σ E (resp. v ∈ Σ F ), let g w (resp. g ′ v ) bethe w -component of g (resp. v -component of g ′ ). Let P = p − ⊕ k C , P ′ = ( p ′ ) − ⊕ k ′ C . By [Har85, Theorem 4.8] and [Har86, Lemma 7.2], each element δ ∗ ∈ Hom U ( P ′ ) ( C ∨ (2 , , U ( g C ) ⊗ U ( P ) C ∨ ( κ ( I κ ) ,r ) ) = Hom U ( P ′ ) ( C ( − , , U ( g C ) ⊗ U ( P ) C ( − κ ( I κ ) , − r ) )gives rise to a homogeneous differential operator [ δ ] from L ( κ ( I κ ) ,r ) to L ′ (2 , . Here the action of P and P ′ on C ( − κ ( I κ ) , − r ) and C ( − , factor through k C and k ′ C , respectively. We define [ δ ( κ )] be the homogeneousdifferential operator corresponds to the element δ ( κ ) ∗ ∈ Hom U ( P ′ ) ( C ( − , , U ( g C ) ⊗ U ( P ) C ( − κ ( I κ ) , − r ) )defined by δ ( κ ) ∗ ( v ( − , ) = X ( κ ) · (1 ⊗ v ( − κ ( I κ ) , − r ) ) , where X ( κ ) ∈ U ( g C ) = U ( gl Σ E , C ) is in (3.4). We shall show that δ ( κ ) ∗ is indeed U ( P ′ )-equivariant. For( ℓ, t ) ∈ Z × Z with ℓ ≡ t (mod 2), let C ( ℓ,t ) be the complex field equipped with the action of R × · SO(2) givenby ak θ · z = a − t e −√− ℓθ · z for a ∈ R × and k θ ∈ SO(2). Thus we have C ( − κ ( I κ ) , − r ) = O w ∈ Σ E C ( − κ ( I κ ) w , − r w ) as algebraic characters of K ∞ = ( R × · SO(2)) Σ E . In the above isomorphism, we fix v w ∈ C ( − κ ( I κ ) w , − r w ) foreach w ∈ Σ E such that v ( − κ ( I κ ) , − r ) = O w ∈ Σ E v w . Note that we also have U ( g C ) ⊗ U ( P ) C ( − κ ( I κ ) , − r ) = O w ∈ Σ E U ( g C ,w ) ⊗ U ( P w ) C ( − κ ( I κ ) w , − r w ) . e write D w = U ( g C ,w ) ⊗ U ( P w ) C ( − κ ( I κ ) w , − r w ) for each w ∈ Σ E . Let v ∈ Σ F . The action of g ′ v on D v (1) ⊗ D v (2) ⊗ D v (3) is given by X · ( v ⊗ v ⊗ v ) = X · v ⊗ v ⊗ v + v ⊗ X · v ⊗ v + v ⊗ v ⊗ X · v for X ∈ g ′ v and v i ∈ D v ( i ) for i = 1 , ,
3. Then we have Z · X ( κ v (1) , κ v (2) , κ v (3) ) · O i =1 (1 ⊗ v v ( i ) ) ! = 0 ,H · X ( κ v (1) , κ v (2) , κ v (3) ) · O i =1 (1 ⊗ v v ( i ) ) ! = 2 · X ( κ v (1) , κ v (2) , κ v (3) ) · O i =1 (1 ⊗ v v ( i ) ) ! ,X − · X ( κ v (1) , κ v (2) , κ v (3) ) · O i =1 (1 ⊗ v v ( i ) ) ! = 0 . (3.5)Indeed, suppose v (1) = ˜ v ( κ ), then κ ( I κ ) v (1) = 2 − κ v (1) , κ ( I κ ) v (2) = κ v (2) , and κ ( I κ ) v (3) = κ v (3) . Thus we have D v (2) = D ( κ v (2) ) + ⊗ | | r v (2) / , D v (3) = D ( κ v (3) ) + ⊗ | | r v (3) / . Also note that H · (1 ⊗ v v (1) ) = (2 − κ v (1) ) · (1 ⊗ v v (1) ) , X − · (1 ⊗ v v (1) ) = 0by definition. Thus (3.5) follows from the condition r v (1) + r v (2) + r v (3) = 0, (3.1), and (3.2). Since X ( κ ) · (1 ⊗ v ( − κ ( I κ ) , − r ) ) = O v ∈ Σ F X ( κ v (1) , κ v (2) , κ v (3) ) · O i =1 (1 ⊗ v v ( i ) ) ! , we deduce from (3.5) that δ ( κ ) ∗ is U ( P ′ )-equivariant. Moreover, it is clear that the diagram C ( − , U ( g C ) ⊗ U ( P ) C ( − κ ( I κ ) , − r ) C ( − , U ( g C ) ⊗ U ( P ) C ( − σ κ ( I σκ ) , − r ) δ ( κ ) ∗ δ ( σ κ ) ∗ is commutative for all σ ∈ Aut( C ), where the vertical homomorphisms are induced by the σ -linear isomor-phisms in (2.7). The assertions (1) and (2) then follow from the construction of δ ( κ ) ∗ . This completes theproof. (cid:3) Algebraicity of Rankin–Selberg L -functions for GL × GL The aim of this section is to prove Theorem 4.7 on the algebraicity of critical values of Rankin–Selberg L -functions, from which Theorem 1.11 follows immediately as we will show in § q -expansion principle. For an automorphic form ϕ on GL ( A F ) and a non-trivial additive character ψ of F \ A F , let W ϕ,ψ be the Whittaker function of ϕ with respect to ψ defined by W ϕ,ψ ( g ) = Z F \ A F ϕ ( n ( x ) g ) ψ ( x ) dx Tam . Here dx Tam is the Tamagawa measure on A F . Let ( κ, r ) ∈ Z ≥ [Σ F ] × Z [Σ F ] be motivic, that is, r = ( r, · · · , r )for some r ∈ Z and κ v ≡ r (mod 2) for v ∈ Σ F . Consider the automorphic line bundle L ( κ,r ) on the Shimuravariety Sh(R F / Q GL / F , ( H ± ) Σ F ) and the zeroth coherent cohomology group H ( L can( κ,r ) ). Let ϕ ⊗ v ( κ,r ) ∈ H ( L can( κ,r ) ). For σ ∈ Aut( C ), let σ ϕ be the automorphic form on GL ( A F ) defined so that T σ ( ϕ ⊗ v ( κ,r ) ) = σ ϕ ⊗ v ( σ κ,r ) . (4.1)Note that the notation is compatible with (2.17). Let f ϕ : H Σ F × GL ( A F ,f ) → C defined by f ϕ ( τ, g f ) = Y v ∈ Σ F y − ( κ v + r ) / v · ϕ (( n ( x v ) a ( y v )) v ∈ Σ F · g f ) or τ = ( x v + √− y v ) v ∈ Σ F ∈ H Σ F and g f ∈ GL ( A F ,f ). Since ϕ is a holomorphic, we have the Fourierexpansion f ϕ ( τ, g f ) = X α ∈ F W α ( ϕ, g f ) e π √− P v ∈ Σ F v ( α ) τ v . By the results of Harris [Har86, Theorem 6.4] and [BHR94, (1.1.13)] (see also [GH93, (A.4.6) and (A.4.7)]),we have the following theorem on the algebraicity of the Fourier coefficients.
Theorem 4.1 ( q -expansion principle) . Let ϕ ⊗ v ( κ,r ) ∈ H ( L can( κ,r ) ) . For σ ∈ Aut( C ) , we have σ (cid:16) (2 π √− − P v ∈ Σ F ( κ v + r ) / · W α ( ϕ, a ( u ) − g f ) (cid:17) = (2 π √− − P v ∈ Σ F ( κ v + r ) / · W α ( σ ϕ, g f ) for α ∈ F and σ ∈ Aut( C ) . Here u ∈ b Z × is the unique element such that σ ( ψ F ( x )) = ψ F ( ux ) for x ∈ A F ,f . Let ψ = N v ψ v be a non-trivial additive character of F \ A F . Suppose ϕ is cuspidal. There exists a uniqueWhittaker function W ( ∞ ) ϕ,ψ on GL ( A F ,f ) with respect to ψ f = N v ∤ ∞ ψ v such that W ϕ,ψ = Y v ∈ Σ F W +( κ v ,r ) ,ψ v · W ( ∞ ) ϕ,ψ . Here the archimedean Whittaker function W +( κ v ,r ) ,ψ v is defined in § Lemma 4.2.
Let ϕ ⊗ v ( κ,r ) ∈ H ( L can( κ,r ) ) . Suppose ϕ is cuspidal. For σ ∈ Aut( C ) , we have σ (cid:16) (2 π √− − P v ∈ Σ F ( κ v + r ) / · W ( ∞ ) ϕ,ψ ( a ( u ) − g f ) (cid:17) = (2 π √− − P v ∈ Σ F ( κ v + r ) / · W ( ∞ ) σ ϕ,ψ ( g f ) for g f ∈ GL ( A F ,f ) . Here u ∈ b Z × is the unique element such that σ ( ψ F ( x )) = ψ F ( ux ) for x ∈ A F ,f .Proof. We have the Fourier expansion ϕ ( g ) = X α ∈ F × W ϕ,ψ α ( g ) = X α ∈ F × W ϕ,ψ ( a ( α ) g )for g ∈ GL ( A F ). Comparing the Fourier expansions of f ϕ and ϕ , by (1.1), we see that W α ( ϕ, g f ) = (Q v ∈ Σ F v ( α ) ( κ v + r ) / · W ( ∞ ) ϕ,ψ ( a ( αβ − ) g f ) if α ≫ , . Here β ∈ F × is the element such that ψ = ψ β F . Note that σ Y v ∈ Σ F v ( α ) ( κ v + r ) / ! = Y v ∈ Σ F v ( α ) ( κ σ − ◦ v + r ) / = Y v ∈ Σ F v ( α ) ( σ κ v + r ) / . The assertion then follows from Theorem 4.1. (cid:3)
Eisenstein series.
Let µ and µ be Hecke characters of A × F . Consider the space I ( µ , µ , s ) consistingof smooth and right (O(2) Σ F × GL (ˆ o F ))-finite functions f s : GL ( A F ) → C such that f s ( n ( x ) a ( a ) d ( d ) g ) = µ ( a ) µ ( d ) | ad − | s A F · f s ( g )for x ∈ A F , a, d ∈ A × F , and g ∈ GL ( A F ). It is a (( gl Σ F , O(2) Σ F ) × GL ( A F ,f ))-module in a natural way. For aplace v of F , we can define the space I ( µ ,v , µ ,v , s ) in a similar way. When v ∤ ∞ and σ ∈ Aut( C ), we havethe σ -linear isomorphism I ( µ ,v , µ ,v , s ) −→ I ( σ µ ,v , σ µ ,v , s ) , f s,v σ f s,v defined by σ f s,v ( n ( x ) a ( a ) d ( d ) k ) = σ µ ,v ( a ) σ µ ,v ( d ) | ad − | s F v · σ ( f s,v ( k ))for x ∈ F v , a, d ∈ F × v , and k ∈ GL ( o F v ). For f s ∈ I ( µ , µ , s ), we define the Eisenstein series E ( g ; f s ) = X γ ∈ B ( F ) \ GL ( F ) f s ( γg ) . he above series converges absolutely for Re( s ) ≫ s ∈ C . Let S ( A F ) = N v S ( F v ) be the space of Bruhat–Schwartz functions on A F . For Φ ∈ S ( A F ), we define theGodement section f µ ,µ , Φ ,s ∈ I ( µ , µ , s ) by f µ ,µ , Φ ,s ( g ) = µ (det( g )) | det( g ) | s A F Z A × F Φ((0 , t ) g ) µ µ − ( t ) | t | s A F d × t std . Here d × t std = Q v d × t std v is the standard measure on A × F defined so that d × t std v = dt v | t v | if v ∈ Σ F and dt v is the Lebesgue measure on R = F v , and vol( o × F v , d × t std v ) = 1 if v ∤ ∞ . For a place v of F , we can define f µ ,v ,µ ,v , Φ v ,s for Φ v ∈ S ( F v ) in a similar way. When v ∤ ∞ and σ ∈ Aut( C ), it is easy to verify that σ f µ ,v ,µ ,v , Φ v ,s = f σ µ ,v , σ µ ,v , σ Φ v ,s . (4.2)Now we assume that µ and µ are algebriac Hecke characters of A × F with | µ | = | | n A F and | µ | = | | n A F forsome n , n ∈ Z , and µ µ has parallel signature withsgn( µ µ ) = ( − n + n . For κ ∈ Z ≥ , let Φ [ κ ] ∈ S ( R ) defined byΦ [ κ ] ( x, y ) = 2 − κ ( x + √− y ) κ e − π ( x + y ) . For Φ ∈ S ( A F ,f ) and κ ∈ Z ≥ with κ ≡ n + n (mod 2), we define the Eisenstein series E [ κ ] ( µ , µ , Φ) by E [ κ ] ( µ , µ , Φ) = E ( f µ ,µ , ⊗ v ∈ Σ F Φ [ κ ] ⊗ v ∤ ∞ Φ ,s ) | s =( κ − n + n ) / . We have the following result on the algebraicity of Eisenstein series. When κ ≥
3, the series defining E [ κ ] ( µ , µ , Φ) is absolutely convergent and the lemma below then follows from (4.2) and the result [Har84,Theorem 3.2.1] of Harris, which is proved by geometric method. Here we prove the algebraicity for any κ ≥ Lemma 4.3.
Let Φ ∈ S ( A F ,f ) and κ ∈ Z ≥ with κ ≡ n + n (mod 2) . Then E [ κ ] ( µ , µ , Φ) is holomorphicof motivic weight ( κ, n + n ) = (( κ, · · · , κ ) , n + n ) ∈ Z ≥ [Σ F ] × Z . Moreover, for σ ∈ Aut( C ) , we have σ E [ κ ] ( µ , µ , Φ) σ (cid:16) D / F (2 π √− − [ F : Q ]( κ + n + n ) / · G ( µ − ) (cid:17) = E [ κ ] ( σ µ , σ µ , σ Φ) D / F (2 π √− − [ F : Q ]( κ + n + n ) / · G ( σ µ − ) . Here σ E [ κ ] ( µ , µ , Φ) is defined in (4.1).Proof. The first assertion is clear. Indeed, for each v ∈ Σ F we have D ( κ ) ⊗ | | ( n + n ) / ⊂ I ( µ ,v , µ ,v , s ) | s =( κ − n + n ) / and f µ ,v ,µ ,v , Φ [ κ ] ,s ( gk θ ) = e √− κθ f µ ,v ,µ ,v , Φ [ κ ] ,s ( g )for k θ ∈ SO(2) and g ∈ GL ( F v ) = GL ( R ). To prove the second assertion, we may assume that Φ = N v ∤ ∞ Φ v is a pure tensor. We write Φ v = Φ [ κ ] for v ∈ Σ F . For any f s ∈ I ( µ , µ , s ), we have the Fourier expansion E ( g ; f s ) = X α ∈ F W E ( f s ) ,ψ α F ( g )= f s ( g ) + M f s ( g ) + X α ∈ F × W E ( f s ) ,ψ F ( a ( a ) g ) , where M f s ∈ I ( µ , µ , − s ) is defined by the intertwining integral M f s ( g ) = Z A F f s (cid:18)(cid:18) −
11 0 (cid:19) n ( x ) g (cid:19) dx Tam . For f s = f µ ,µ , ⊗ v Φ v ,s , we have (cf. [Ike89, (5.1.8)]) M f µ ,µ , ⊗ v Φ v ,s ( g )= L (2 s − , µ µ − ) Y v ε (2 s − , µ ,v µ − ,v , ψ F v ) − L (2 − s, µ − ,v µ ,v ) − f µ ,v ,µ ,v , b Φ v , − s ( g v ) . ere b Φ v is the symplectic Fourier transform of Φ v with respect to ψ F defined by b Φ v ( x, y ) = Z F v Φ v ( u, w ) ψ F v ( uy − wx ) dudw and the Haar measures du and dw are the self-dual Haar measures on F v with respect to ψ F v . Note that thelocal factors appearing in the above infinite product are holomorphic in s and equal to 1 for almost all v .Since κ ≡ n + n (mod 2), we have L ( κ − n + n − , µ µ − ) = 0 by the functional equation for L ( s, µ µ − ).In particular, we have M f µ ,µ , ⊗ v Φ v ,s | s =( κ − n + n ) / = 0. For the non-zero Fourier coefficient, we have thefollowing factorization into local Whittaker functions (cf. [Ike89, (5.1.6)]): W E ( f s ) ,ψ F ( g ) = D − / F µ (det( g )) | det( g ) | s A F Z A × F ^ ρ ( g )Φ( t, − t − ) µ µ − ( t ) | t | s − A F d × t std = D − / F Y v µ ,v (det( g v )) | det( g v ) | s F v Z F × v ^ ρ ( g v )Φ v ( t v , − t − v ) µ ,v µ − ,v ( t v ) | t v | s − F v d × t std v = D − / F Y v W ψ F v ( g v ; f µ ,v ,µ ,v , Φ v ,s ) . Here ^ ρ ( g v )Φ v is the partial Fourier transform of ρ ( g v )Φ v with respect to ψ F v defined by ^ ρ ( g v )Φ v ( x, y ) = Z F v Φ v (( x, w ) g ) ψ F v ( wy ) dw std and dw std is the standard measure on F v defined so that dw std is the Lebesgue measure if v ∈ Σ F andvol( o F v , dw std ) = 1 if v ∤ ∞ . Note that the factor D − / F is the ratio between the standard and Tamagawameasures on A F . For v ∈ Σ F , it is easy to show that (cf. [CH20, Lemma 4.5]) f µ ,v ,µ ,v , Φ v ,s (1) = 2 − κ ( √− κ π − ( s +( κ + n − n ) / Γ( s + κ + n − n )and W ψ F v ( f µ ,v ,µ ,v , Φ v ,s ) | s =( κ − n + n ) / = W +( κ,n + n ) ,ψ F v . We conclude that for g f = ( g v ) v ∤ ∞ ∈ GL ( A F ,f ) and α ∈ F , W α ( E [ κ ] ( µ , µ , Φ) , g f )= ( − π √− − [ F : Q ] κ Γ( κ ) [ F : Q ] Q v ∤ ∞ f µ ,v ,µ ,v , Φ v ,s ( g v ) | s =( κ − n + n ) / if α = 0 ,D − / F N F / Q ( α ) ( κ + n + n ) / Q v ∤ ∞ W ψ F v ( g v ; f µ ,v ,µ ,v , Φ v ,s ) | s =( κ − n + n ) / if α ≫ , . (4.3)Let σ ∈ Aut and u = ( u p ) p ∈ b Z × the unique element such that σ ( ψ F ( x )) = ψ F ( ux ) for x ∈ A F ,f . For v ∤ ∞ and v | p for some rational prime p , we have σ W ψ F v ( a ( u p ) − g v ; f µ ,v ,µ ,v , Φ v ,s ) = σ µ ,v ( u p ) − W ψ F v ( g v ; f σ µ ,v , σ µ ,v , σ Φ v ,s )(4.4)as rational functions in q − sv , where q v is the cardinality of the residue field of F v . Indeed, ^ ρ ( g v )Φ v ( t v , − t − v )vanishes when | t v | F v is either sufficiently small or sufficiently large, thus W ψ F v ( g v ; f µ ,v ,µ ,v , Φ v ,s ) is a finitesum of polynomials in C [ q sv , q − sv ]. In particular, we have σ W ψ F v ( a ( u p ) − g v ; f µ ,v ,µ ,v , Φ v ,s )= σ µ ,v ( u − p det( g v )) | det( g v ) | s F v Z F × v σ (cid:16) ^ ρ ( a ( u p ) − g v )Φ v (cid:17) ( t v , − t − v ) σ µ ,vσ µ − ,v ( t v ) | t v | s − F v d × t std v = σ µ ,v ( u p ) − σ µ ,v (det( g v )) | det( g v ) | s F v Z F × v σ (cid:16) ^ ρ ( a ( u p ) − g v )Φ v (cid:17) ( u p t v , − u − p t − v ) σ µ ,vσ µ − ,v ( t v ) | t v | s − F v d × t std v . or the partial Fourier transform, we have σ (cid:16) ^ ρ ( a ( u p ) − g v )Φ v (cid:17) ( x, y ) = Z F v σ Φ v (( x, w ) a ( u p ) − g v ) σ ψ F v ( wy ) dw std = Z F v σ Φ v (( u − p x, w ) g v ) ψ F v ( wu p y ) dw std = ^ ρ ( g v ) σ Φ v ( u − p x, u p y ) . Thus (4.4) holds. Also note that when µ ,v , µ ,v are unramified, F v / Q p is unramified, Φ v = I o F v , and g v ∈ GL ( o F v ), we have W ψ F v ( g v ; f µ ,v ,µ ,v , Φ v ,s ) = 1 . Therefore, by (1.2), (4.3), and (4.4) we have σ W α ( E [ κ ] ( µ , µ , Φ) , g f ) D / F G ( µ − ) ! = W α ( E [ κ ] ( σ µ , σ µ , σ Φ) , g f ) D / F G ( σ µ − )for α ∈ F × . This together with Theorem 4.1 imply that the holomorphic automorphic forms σ E [ κ ] ( µ , µ , Φ) σ (cid:16) D / F (2 π √− − [ F : Q ]( κ + n + n ) / · G ( µ − ) (cid:17) and E [ κ ] ( σ µ , σ µ , σ Φ) D / F (2 π √− − [ F : Q ]( κ + n + n ) / · G ( σ µ − )have the same Fourier coefficients for α = 0 and same motivic weight ( κ, n + n ). Thus they must be equal.This completes the proof. (cid:3) L -functions for GL × GL . Let Π = N v π v and Π ′ = N v π ′ v be motivic irreducible cuspidal auto-morphic representations of GL ( A F ) with motivic weights ( ℓ, r ) ∈ Z ≥ [Σ F ] × Z and ( ℓ ′ , r ′ ) ∈ Z ≥ [Σ F ] × Z ,respectively, and χ an algebraic Hecke character of A × F with | χ | = | | r A F for some r ∈ Z . Let L ( s, Π × Π ′ × χ ) = Y v L ( s, π v × π ′ v × χ v )be the twisted Rankin–Selberg L -function of Π × Π ′ × χ defined by the tensor representation L (GL / F × GL / F × GL / F ) −→ GL( C ⊗ C ⊗ C ) . We denote by L ( ∞ ) ( s, Π × Π ′ × χ ) the L -function obtained by excluding the archimedean L -factors. Astandard unfolding argument (cf. [Jac72, § Z A × F GL ( F ) \ GL ( A F ) ϕ ( g ) ϕ ′ ( g ) E ( g ; f s ) dg Tam = D − F ζ F (2) − · L ( s, Π × Π ′ × χ ) · Y v Z ∗ ( W v , W ′ v , f s,v )(4.5)as meromorphic functions in s ∈ C , for ϕ ∈ Π , ϕ ′ ∈ Π ′ , and f s ∈ I ( χ, χ − ω − Π ω − Π ′ , s ) such that W ϕ,ψ F = Y v W v , W ϕ ′ ,ψ F = Y v W ′ v , f s = Y v f s,v . Here Z ∗ ( W v , W ′ v , f s,v ) = Z ( W v , W ′ v , f s,v ) L ( s, π v × π ′ v × χ v )and Z ( W v , W ′ v , f s,v ) is the local zeta integral defined by Z ( W v , W ′ v , f s,v ) = Z F × v N ( F v ) \ GL ( F v ) W v ( g v ) W ′ v ( a ( − g v ) f s,v ( g v ) dg std v and dg std v is the standard measure defined so that dg std v is the quotient measure of the measure on PGL ( F v )defined in § F v if v ∈ Σ F , and vol( o × F v N ( o F v ) \ GL ( o F v )) = 1 if v ∤ ∞ .Note that the factor D − / F ζ F (2) − is the ratio between the Tamagawa and standard measures. We have the ollowing lemmas on basic properties of the non-archimedean local zeta integrals and explicit calculation ofcertain archimedean local zeta integrals. Lemma 4.4.
Assume v is a finite place of F and v | p for some rational prime p . Let W v and W ′ v beWhittaker functions of π v and π ′ v with respect to ψ F v , respectively, and f s,v ∈ I ( χ v , χ − v ω − Π ,v ω − Π ′ ,v , s ) . (1) The integral Z ( W v , W ′ v , f s,v ) is absolutely convergent for Re( s ) sufficiently large and admits mero-morphic continuation to s ∈ C . Moreover, if f s,v = f χ v ,χ − v ω − Π ,v ω − Π ′ ,v , Φ v ,s for some Φ v ∈ S ( F v ) ,then Z ∗ ( W v , W ′ v , f χ v ,χ − v ω − Π ,v ω − Π ′ ,v , Φ v ,s ) is a polynomial in C [ q sv , q − sv ] . Here q v is the cardinality of theresidue field of F v . (2) Suppose π v , π ′ v , χ v , ψ F v are unramified. Let W ◦ v and ( W ′ v ) ◦ be the right GL ( o F v ) -invariant Whittakerfunctions of π v and π ′ v with respect to ψ F v , respectively, normalized so that W ◦ v (1) = ( W ′ v ) ◦ (1) = 1 and Φ ◦ v = I o F v . Then we have Z ∗ ( W ◦ v , ( W ′ v ) ◦ , f χ v ,χ − v ω − Π ,v ω − Π ′ ,v , Φ ◦ v ,s ) = 1 . (3) For σ ∈ Aut( C ) , we have σ Z ∗ ( W v , W ′ v , f χ v ,χ − v ω − Π ,v ω − Π ′ ,v , Φ v ,s ) = σ χ v ( u p ) − Z ∗ ( t σ W v , t σ W ′ v , f σ χ v , σ χ − v σ ω − Π ,vσ ω − Π ′ ,v , σ Φ v ,s ) as polynomials in C [ q sv , q − sv ] . Here u p ∈ Z × p is the unique element such that σ ψ F v = ψ u F v and t σ W v isthe Whittaker function of σ π v with respect to ψ F v defined by t σ W v ( g ) = σ W v ( a ( u p ) − g ) . Proof.
The assertions (1) and (2) were proved in [Jac72, Theorem 14.7]. Let σ ∈ Aut( C ). Analogous to theproof of Lemma 5.2 below, we can show that σ L ( s, π v × π ′ v × χ v ) = L ( s, σ π v × σ π ′ v × σ χ v )as rational functions in q − sv . To prove assertion (3), it suffices to show that σ Z ( W v , W ′ v , f χ v ,χ − v ω − Π ,v ω − Π ′ ,v , Φ v ,s ) = σ χ v ( u p ) − Z ( t σ W v , t σ W ′ v , f σ χ v , σ χ − v σ ω − Π ,vσ ω − Π ′ ,v , σ Φ v ,s ) . We recall a type of integral Z F × v f ( a ) µ ( a ) | a | s F v (log q v | a | F v ) n d × a, where f ∈ S ( F v ), µ is a character of F × v , and n ∈ Z ≥ . The integral converges absolutely for Re( s ) sufficientlylarge and defines a rational function in q − sv . Moreover, we have σ (cid:18)Z F × v f ( a ) µ ( a ) | a | s F v (log q v | a | F v ) n d × a (cid:19) = Z F × v σ f ( a ) σ µ ( a ) | a | s F v (log q v | a | F v ) n d × a (4.6)as rational functions in q − sv (cf. [Gro18, Proposition A]). Here vol( o × F v , d × a ) = 1. It is well-known that(cf. [Jac72, Lemma 14.3]) there exist locally constant functions f , f , f ′ , f ′ on F v × GL ( o F v ) with compactsupport, characters µ , µ , µ ′ , µ ′ of F × v , and integers n , n , n ′ , n ′ such that W v ( a ( a ) k ) = f ( a, k ) µ ( a )(log q v | a | F v ) n + f ( a, k ) µ ( a )(log q v | a | F v ) n ,W ′ v ( a ( − a ) k ) = f ′ ( a, k ) µ ′ ( a )(log q v | a | F v ) n ′ + f ′ ( a, k ) µ ′ ( a )(log q v | a | F v ) n ′ for a ∈ F × v and k ∈ GL ( o F v ). Therefore, we have Z ( W v , W ′ v , f χ v ,χ − v ω − Π ,v ω − Π ′ ,v , Φ v ,s )= Z GL ( o F v ) Z F × v W v ( a ( a ) k ) W ′ v ( a ( − a ) k ) f χ v ,χ − v ω − Π ,v ω − Π ′ ,v , Φ v ,s ( a ( a ) k ) d × a | a | F v dk = X ≤ i,j ≤ Z GL ( o F v ) f χ v ,χ − v ω − Π ,v ω − Π ′ ,v , Φ v ,s ( k ) Z F × v f i ( a, k ) f ′ j ( a, k ) µ i µ ′ j χ v ( a ) | a | s − F v (log q v | a | F v ) n i + n ′ j d × adk. ere vol(GL ( o F v ) , dk ) = 1. It then follows from (4.2) and (4.6) that σ Z ( W v , W ′ v , f χ v ,χ − v ω − Π ,v ω − Π ′ ,v , Φ v ,s )= X ≤ i,j ≤ Z GL ( o F v ) f σ χ v , σ χ − v σ ω − Π ,vσ ω − Π ′ ,v , σ Φ v ,s ( k ) Z F × v σ f i ( a, k ) σ f ′ j ( a, k ) σ µ iσ µ ′ jσ χ v ( a ) | a | s − F v (log q v | a | F v ) n i + n ′ j d × adk = Z GL ( o F v ) Z F × v σ W v ( a ( a ) k ) σ W ′ v ( a ( a ) k ) f σ χ v , σ χ − v σ ω − Π ,vσ ω − Π ′ ,v , σ Φ v ,s ( a ( a ) k ) d × a | a | F v dk = σ χ v ( u p ) − Z GL ( o F v ) Z F × v t σ W v ( a ( a ) k ) t σ W ′ v ( a ( a ) k ) f σ χ v , σ χ − v σ ω − Π ,vσ ω − Π ′ ,v , σ Φ v ,s ( a ( a ) k ) d × a | a | F v dk = σ χ v ( u p ) − Z ( t σ W v , t σ W ′ v , f σ χ v , σ χ − v σ ω − Π ,vσ ω − Π ′ ,v , σ Φ v ,s ) . This completes the proof. (cid:3)
Lemma 4.5.
Assume v ∈ Σ F and ℓ v > ℓ ′ v . Let m be an integer such that ≤ m + r + r + r ′ ≤ ℓ v − ℓ ′ v . Put m ′ = 2 m + 2 r + r + r ′ . For integers m , m ∈ Z ≥ such that m ′ + 2 m + 2 m = ℓ v − ℓ ′ v , we have Z ( W − ( ℓ v ,r ) ,ψ F v , X m + · W +( ℓ ′ v ,r ′ ) ,ψ F v , X m + · f χ v ,χ − v ω − Π ,v ω − Π ′ ,v , Φ [ m ′ ] ,s ) | s = m = χ v ( − − m ′ + m − ℓ ′ v − m ′ ( √− ℓ v π · (2 π √− − ( ℓ v + ℓ ′ v +3 m ′ ) / · Γ (cid:16) ℓ v + ℓ ′ v + m ′ − (cid:17) Γ (cid:16) ℓ v − ℓ ′ v + m ′ (cid:17) . Proof.
Note that we have W − ( ℓ v ,r ) ,ψ F v ( a ( − a )) = a ( ℓ v + r ) / e − πa · I R > ( a ) ,X m + · W +( ℓ ′ v ,r ′ ) ,ψ F v ( a ( a )) = (2 √− m m X j =0 ( − π ) j (cid:18) m j (cid:19) Γ( ℓ ′ v + m )Γ( ℓ ′ v + j ) a ( ℓ ′ v + r ′ ) / j e − πa · I R > ( a ) ,X m + · f χ v ,χ − v ω − Π ,v ω − Π ′ ,v , Φ [ m ′ ] ,s | s = m ( a ( a )) = ( − π √− m f χ v ,χ − v ω − Π ,v ω − Π ′ ,v , Φ [ m ′ +2 m ,s | s = m ( a ( a ))= 2 − m ′ + m ( √− m ′ + m π − m ′ Γ( m ′ + m ) χ v ( a ) | a | m for a ∈ F × v = R × . Here we refer to [CC19, Lemma 3.3] for the second equation. Hence Z ( W − ( ℓ v ,r ) ,ψ F v , X m + · W +( ℓ ′ v ,r ′ ) ,ψ F v , X m + · f χ v ,χ − v ω − Π ,v ω − Π ′ ,v , Φ [ m ′ ] ,s ) | s = m = Z R × W − ( ℓ v ,r ) ,ψ F v ( a ( a ))( X m + · W +( ℓ ′ v ,r ′ ) ,ψ F v )( a ( − a ))( X m + · f χ v ,χ − v ω − Π ,v ω − Π ′ ,v , Φ [ m ′ ] ,s | s = m )( a ( a )) d × a | a | = χ v ( − − m ′ + m + m ( √− m ′ + m + m π − m ′ Γ( ℓ ′ v + m )Γ( m ′ + m ) × m X j =0 ( − π ) j (cid:18) m j (cid:19) Γ( ℓ ′ v + j ) − Z ∞ a ( ℓ v + ℓ ′ v + m ′ ) / j − e − πa d × a = χ v ( − − m ′ + m + m ( √− m ′ + m + m π − m ′ Γ( ℓ ′ v + m )Γ( m ′ + m ) × (4 π ) − ( ℓ v + ℓ ′ v + m ′ ) / m X j =0 ( − j (cid:18) m j (cid:19) Γ (cid:16) ℓ v + ℓ ′ v + m ′ + j − (cid:17) Γ( ℓ ′ v + j ) . By [Ike98, Lemma 2.1], we have m X j =0 ( − j (cid:18) m j (cid:19) Γ (cid:16) ℓ v + ℓ ′ v + m ′ + j − (cid:17) Γ( ℓ ′ v + j ) = ( − m Γ (cid:16) ℓ v + ℓ ′ v + m ′ − (cid:17) Γ (cid:16) ℓ v − ℓ ′ v + m ′ (cid:17) Γ( ℓ ′ v + m )Γ( m ′ + m ) . The assertion thus follows. (cid:3)
Remark 4.6.
When ℓ v < ℓ v , we have similar formula by exchanging ℓ v and ℓ ′ v and excluding χ v ( − .4. Algebraicity of the critical values.
We keep the notation of § I = { v ∈ Σ F | ℓ v > ℓ ′ v } , J = { v ∈ Σ F | ℓ v < ℓ ′ v } . It is clear that I and J are admissible with respect to ℓ and ℓ ′ , respectively.The following result is on the algebraicity of the critical values of L ( s, Π × Π ′ × χ ). We generalize theresult [Har89, Theorem 3.5.1] of Harris where χ is trivial. Theorem 4.7.
Assume that ℓ v = ℓ ′ v for all v ∈ Σ F . Let m ∈ Z such that ≤ m + r + r + r ′ ≤ | ℓ v − ℓ ′ v | for all v ∈ Σ F . We have σ (cid:18) L ( ∞ ) ( m, Π × Π ′ × χ )(2 π √− [ F : Q ](2 m +2 r + r + r ′ ) ( √− P v ∈ I ℓ v + P v ∈ J ℓ ′ v · G ( χ ω Π ω Π ′ ) · Ω I ( Π ) · Ω J ( Π ′ ) (cid:19) = L ( ∞ ) ( m, σ Π × σ Π ′ × σ χ )(2 π √− [ F : Q ](2 m +2 r + r + r ′ ) ( √− P v ∈ I ℓ v + P v ∈ J ℓ ′ v · G ( σ χ σ ω Π σ ω Π ′ ) · Ω σ I ( σ Π ) · Ω σ J ( σ Π ′ ) for all σ ∈ Aut( C ) .Proof. Consider the ´etale cubic algebra E = F × F × F over F . We have Σ E = Σ (1) F ⊔ Σ (2) F ⊔ Σ (3) F , where Σ ( i ) F is the set of algebra homomorphisms from E into R which are non-zero on its i -th component. For each v ∈ Σ F , there are three extensions v (1) , v (2) , v (3) ∈ Σ E of v . We arrange the index so that v ( i ) ∈ Σ ( i ) F . Fix m ∈ Z such that12 ≤ m + r + r + r ′ ≤ | ℓ v − ℓ ′ v | v ∈ Σ F and put m ′ = 2 m + 2 r + r + r ′ . Let ( κ, r ) ∈ Z ≥ [Σ E ] × Z [Σ E ] be motivic defined by κ w = ℓ v if w = v (1) ∈ Σ (1) F ,ℓ ′ v if w = v (2) ∈ Σ (2) F ,m ′ if w = v (3) ∈ Σ (3) F , r w = r if w = v (1) ∈ Σ (1) F ,r ′ if w = v (2) ∈ Σ (2) F , − r − r ′ if w = v (3) ∈ Σ (3) F . Note that κ satisfied the totally unbalanced condition (3.3). Recall for each v ∈ Σ F , we define ˜ v ( κ ) ∈ Σ E bethe homomorphism corresponding to max w | v { κ w } , that is, κ ˜ v ( κ ) = max w | v { κ w } . Therefore,˜ v ( κ ) = ( v (1) if v ∈ I,v (2) if v ∈ J, and I κ = { ˜ v ( κ ) | v ∈ Σ F } = I ⊔ J ⊔ ∅ with respect to the disjoint union Σ E = Σ (1) F ⊔ Σ (2) F ⊔ Σ (3) F . Consider the automorphic line bundle L ( κ ( I κ ) , r ) on the Shimura varietySh(R E / Q GL / E , ( H ± ) Σ E ) = Sh(R F / Q GL / F , ( H ± ) Σ F ) × Sh(R F / Q GL / F , ( H ± ) Σ F ) × Sh(R F / Q GL / F , ( H ± ) Σ F )and the autmorphic line bundles L ′ ( ℓ ( I ) ,r ) , L ′ ( ℓ ′ ( J ) ,r ′ ) , and L ′ ( m ′ , − r − r ′ ) on Sh(R F / Q GL / F , ( H ± ) Σ F ). We have L ( κ ( I κ ) , r ) = L ′ ( ℓ ( I ) ,r ) × L ′ ( ℓ ′ ( J ) ,r ′ ) × L ′ ( m ′ , − r − r ′ ) . Let [ δ ( κ )] : L ( κ ( I κ ) , r ) → L ′ (2 , be the trilinear differential operator constructed in Proposition 3.2. In thiscase, it induces a homomorphism[ δ ( κ )] : H ♯ I (( L ′ ( ℓ ( I ) ,r ) ) sub ) ⊗ H ♯ J (( L ′ ( ℓ ′ ( J ) ,r ′ ) ) sub ) ⊗ H (( L ′ ( m ′ , − r − r ′ ) ) can ) −→ H [ F : Q ] (( L ′ (2 , ) sub )which satisfies the Galois equivariant property: T σ ([ δ ( κ )]( c ⊗ c ⊗ c )) = [ δ ( σ κ )]( T σ c ⊗ T σ c ⊗ T σ c )(4.7) or all σ ∈ Aut( C ). Moreover, if c , c , c are represented by ϕ ⊗ ^ v ∈ I X + ,v ⊗ v ( ℓ ( I ) ,r ) , ϕ ⊗ ^ v ∈ J X + ,v ⊗ v ( ℓ ′ ( J ) ,r ′ ) , ϕ ⊗ v ( m ′ , − r − r ′ ) , respectively, then [ δ ( κ )]( c ⊗ c ⊗ c ) is represented by( X ( κ ) · ( ϕ , ϕ , ϕ )) | GL ( A F ) ⊗ ^ v ∈ Σ F X + ,v ⊗ v (2 , . Here ( ϕ , ϕ , ϕ ) is the automorphic form on GL ( A E ) = GL ( A F ) × GL ( A F ) × GL ( A F ) defined by( ϕ , ϕ , ϕ )( g , g , g ) = ϕ ( g ) · ϕ ( g ) · ϕ ( g ) , and X ( κ ) ∈ U ( gl Σ E , C ) is the differential operator defined in (3.4). For ϕ ∈ Π hol , ϕ ′ ∈ Π ′ hol , and Φ ∈ S ( A F ,f ),let c ( ϕ, ϕ ′ , χ, Φ) ∈ H ♯ I (( L ′ ( ℓ ( I ) ,r ) ) sub ) ⊗ H ♯ J (( L ′ ( ℓ ′ ( J ) ,r ′ ) ) sub ) ⊗ H (( L ′ ( m ′ , − r − r ′ ) ) can ) be the class defined by c ( ϕ, ϕ ′ , χ, Φ) = ξ I ( ϕ ) ⊗ ξ J ( ϕ ′ ) ⊗ (cid:16) E [ m ′ ] ( χ, χ − ω − Π ω − Π ′ , Φ) ⊗ v ( m ′ , − r − r ′ ) (cid:17) . Here ξ I and ξ J are defined in (2.16). By Proposition 2.6 and Lemma 4.3, this cohomology class satisfies thefollowing Galois equivariant property: T σ c ( ϕ, ϕ ′ , χ, Φ) D / F (2 π √− − [ F : Q ]( m + r ) · G ( χω Π ω Π ′ ) · Ω I ( Π ) · Ω J ( Π ′ ) ! = c ( σ ϕ, σ ϕ ′ , σ χ, σ Φ) D / F (2 π √− − [ F : Q ]( m + r ) · G ( σ χ σ ω Π σ ω Π ′ ) · Ω σ I ( σ Π ) · Ω σ J ( σ Π ′ )for all σ ∈ Aut( C ). It then follows from Lemma 2.2 and (4.7) that σ R Sh(R F / Q GL / F , ( H ± ) Σ F ) [ δ ( κ )] c ( ϕ, ϕ ′ , χ, Φ) D / F (2 π √− − [ F : Q ]( m + r ) · G ( χω Π ω Π ′ ) · Ω I ( Π ) · Ω J ( Π ′ ) ! = R Sh(R F / Q GL / F , ( H ± ) Σ F ) [ δ ( σ κ )] c ( σ ϕ, σ ϕ ′ , σ χ, σ Φ) D / F (2 π √− − [ F : Q ]( m + r ) · G ( σ χ σ ω Π σ ω Π ′ ) · Ω σ I ( σ Π ) · Ω σ J ( σ Π ′ ) . By the explicit realization of [ δ ( κ )] described above and Lemma 2.2 again, we conclude that σ R A × F GL ( F ) \ GL ( A F ) X ( κ ) · ( ϕ I , ( ϕ ′ ) J , E [ m ′ ] ( χ, χ − ω − Π ω − Π ′ , Φ))( g, g, g ) dg Tam D / F (2 π √− − [ F : Q ]( m + r ) · G ( χω Π ω Π ′ ) · Ω I ( Π ) · Ω J ( Π ′ ) = R A × F GL ( F ) \ GL ( A F ) X ( σ κ ) · ( σ ϕ σ I , ( σ ϕ ′ ) σ J , E [ m ′ ] ( σ χ, σ χ − σ ω − Π σ ω − Π ′ , σ Φ))( g, g, g ) dg Tam D / F (2 π √− − [ F : Q ]( m + r ) · G ( σ χ σ ω Π σ ω Π ′ ) · Ω σ I ( σ Π ) · Ω σ J ( σ Π ′ )(4.8)for all σ ∈ Aut( C ). On the other hand, assume W ϕ,ψ F = Y v ∈ Σ F W +( ℓ v ,r ) ,ψ F v Y v ∤ ∞ W v , W ϕ ′ ,ψ F = Y v ∈ Σ F W +( ℓ ′ v ,r ′ ) ,ψ F v Y v ∤ ∞ W ′ v , and Φ = N v ∤ ∞ Φ v , we have the integral representation of L ( ∞ ) ( m, Π × Π ′ × χ ) recalled in (4.5): Z A × F GL ( F ) \ GL ( A F ) X ( κ ) · ( ϕ I , ( ϕ ′ ) J , E [ m ′ ] ( χ, χ − ω − Π ω − Π ′ , Φ))( g, g, g ) dg Tam = D − F ζ F (2) − · L ( ∞ ) ( m, Π × Π ′ × χ ) · Y v ∤ ∞ Z ∗ ( W v , W ′ v , f χ v ,χ − v ω − Π ,v ω − Π ′ ,v , Φ v ,s ) | s = m × Y v ∈ I X m ′ +2 m +2 m = ℓ v − ℓ ′ v c m ,m ( ℓ v , ℓ ′ v , m ′ ) Z ( W − ( ℓ v ,r ) ,ψ F v , X m + · W +( ℓ ′ v ,r ′ ) ,ψ F v , X m + · f χ v ,χ − v ω − Π ,v ω − Π ′ ,v , Φ [ m ′ ] ,s ) | s = m × Y v ∈ J X m ′ +2 m +2 m = ℓ ′ v − ℓ v c m ,m ( ℓ v , ℓ ′ v , m ′ ) Z ( X m + · W +( ℓ v ,r ) ,ψ F v , W − ( ℓ ′ v ,r ′ ) ,ψ F v , X m + · f χ v ,χ − v ω − Π ,v ω − Π ′ ,v , Φ [ m ′ ] ,s ) | s = m . ote that ζ F (2) ∈ D / F · π [ F : Q ] · Q × . By (1.2) and Lemmas 4.2 and 4.4, we have W σ ϕ,ψ F = σ (2 π √− − Σ v ∈ Σ F ( ℓ v + r ) / (2 π √− − Σ v ∈ Σ F ( ℓ v + r ) / Y v ∈ Σ F W +( σ ℓ v ,r ) ,ψ F v Y v ∤ ∞ t σ W v ,W σ ϕ ′ ,ψ F = σ (2 π √− − Σ v ∈ Σ F ( ℓ ′ v + r ′ ) / (2 π √− − Σ v ∈ Σ F ( ℓ ′ v + r ′ ) / Y v ∈ Σ F W +( σ ℓ ′ v ,r ′ ) ,ψ F v Y v ∤ ∞ t σ W ′ v , (4.9)and σ G ( χ ) Y v ∤ ∞ Z ∗ ( W v , W ′ v , f χ v ,χ − v ω − Π ,v ω − Π ′ ,v , Φ v ,s ) | s = m = G ( σ χ ) Y v ∤ ∞ Z ∗ ( t σ W v , t σ W ′ v , f σ χ v , σ χ − v σ ω − Π ,vσ ω − Π ′ ,v , σ Φ v ,s ) | s = m . (4.10)By Lemma 4.5, we have Y v ∈ I X m ′ +2 m +2 m = ℓ v − ℓ ′ v c m ,m ( ℓ v , ℓ ′ v , m ′ ) Z ( W − ( ℓ v ,r ) ,ψ F v , X m + · W +( ℓ ′ v ,r ′ ) ,ψ F v , X m + · f χ v ,χ − v ω − Π ,v ω − Π ′ ,v , Φ [ m ′ ] ,s ) | s = m × Y v ∈ J X m ′ +2 m +2 m = ℓ ′ v − ℓ v c m ,m ( ℓ v , ℓ ′ v , m ′ ) Z ( X m + · W +( ℓ v ,r ) ,ψ F v , W − ( ℓ ′ v ,r ′ ) ,ψ F v , X m + · f χ v ,χ − v ω − Π ,v ω − Π ′ ,v , Φ [ m ′ ] ,s ) | s = m = C ( κ, m ′ ) · ( √− P v ∈ I ℓ v + P v ∈ J ℓ ′ v · π [ F : Q ] · (2 π √− − P v ∈ Σ F ( ℓ v + ℓ ′ v +3 m ′ ) / . (4.11)Here C ( κ, m ′ ) ∈ Q is the rational number C ( κ, m ′ ) = Y v ∈ I χ v ( − − [ F : Q ] m ′ F : Q ] − [ F : Q ] m ′ − P v ∈ I ℓ v − P v ∈ J ℓ ′ v Y v ∈ Σ F Γ (cid:16) ℓ v + ℓ ′ v + m ′ − (cid:17) Γ (cid:16) | ℓ v − ℓ ′ v | + m ′ (cid:17) × Y v ∈ Σ F X m ′ +2 m +2 m = | ℓ v − ℓ ′ v | ( − m c m ,m ( ℓ v , ℓ ′ v , m ′ )Note that C ( κ, m ′ ) is non-zero as we will show in the proof of Lemma 5.4 below. Finally, for each v ∤ ∞ , welet the triplet ( W v , W ′ v , Φ v ) be chosen so that (cf. [CH20, § Z ∗ ( W v , W ′ v , f χ v ,χ − v ω − Π ,v ω − Π ′ ,v , Φ v ,s ) = 1 . The theorem then follows from (4.8)-(4.11). This completes the proof. (cid:3) Proof of main results
In this section, we prove the main results Theorems 1.4, 1.8, and 1.11 of this paper. In § § § Π = N v π v be an irreducible cuspidal automorphic representation of GL ( A E )with central character ω Π , where v runs through the places of F . We assume the following conditions hold: • ω Π | A × F is trivial; • Π is motivic of weight ( κ, r ) ∈ Z ≥ [Σ E ] × Z [Σ E ]; • κ satisfies the totally unbalanced condition2 max w | v { κ w } − X w | v κ w ≥ v ∈ Σ F . .1. Ichino’s formula.
In this subsection, we recall Ichino’s central value formula [Ich08] for L ( , Π , As) interms of global trilinear period integral, which is a special case of the refined Gan–Gross–Prasad conjectureproposed by Ichino–Ikeda [II10]. Let D be a totally indefinite quaternion algebra over F such that thereexists an irreducible cuspidal automorphic representation Π D = N v π Dv of D × ( A F ) associated to Π by theJacquet–Langlands correspondence. Define the functional I D ∈ Hom D × ( A F ) × D × ( A F ) ( Π D ⊗ ( Π D ) ∨ , C ) by theglobal trilinear period integral I D ( ϕ ⊗ ϕ ) = Z A × F D × ( F ) \ D × ( A F ) Z A × F D × ( F ) \ D × ( A F ) ϕ ( g ) ϕ ( g ) dg Tam1 dg Tam2 . (5.1)Here dg Tam1 and dg Tam2 are the Tamagawa measures on A × F \ D × ( A F ). For each place v of F , we fix a non-zero D × ( E v )-equivariant bilinear pairing h , i v : π Dv × ( π Dv ) ∨ −→ C . Let dg v be the Haar measure on F v \ D × ( F v ) defined as follows: • If v is a finite place, let dg v be the Haar measure normalized so that vol( o × F v \ o × D v , dg v ) = 1. Here o D v is a maximal order of D ( F v ). • If v is a real place, let dg v = dx v dy v | y v | dk v for g v = n ( x v ) a ( y v ) k v with x v ∈ R , y v ∈ R × , and k v ∈ SO(2). Here dx v and dy v are the Lebesguemeasures and dk v is the Haar measure on SO(2) such that vol(SO(2) , dk v ) = 2.Let I Dv ∈ Hom D × ( F v ) × D × ( F v ) ( π Dv ⊗ ( π Dv ) ∨ , C ) be the functional define by the local trilinear period integrals I Dv ( ϕ ,v ⊗ ϕ ,v ) = ζ F v (2) ζ E v (2) · L (1 , π v , Ad) L ( , π v , As) · Z F × v \ D × ( F v ) h π Dv ( g v ) ϕ ,v , ϕ ,v i v dg v . (5.2)Here ζ E v ( s ) and ζ F v ( s ) are the local zeta functions of E v and F v , respectively. Note that L ( , π v , As) = 0(cf. [Che20a, Lemma 3.1]) and the integral is absolutely convergent by [Ich08, Lemma 2.1]. When D isunramified at v , we also write I v = I Dv . We normalize the pairings h , i v so that if ϕ = N v ϕ ,v ∈ Π D and ϕ = N v ϕ ,v ∈ ( Π D ) ∨ , then h ϕ ,v , ϕ ,v i v = 1 for almost all v and Z A × E D × ( E ) \ D × ( A E ) ϕ ( g ) ϕ ( g ) dg Tam = Y v h ϕ ,v , ϕ ,v i v . Here dg Tam is the Tamagawa measure on A × E \ D × ( A E ). Let C D be the constant such that C D · Y v dg v is the Tamagawa measure on A × F \ D × ( A F ). Then we have (cf. [IP18, Lemma 6.1]) C D = Y v ∈ Σ D ( q v − − · D − / F · ζ F (2) − , (5.3)where Σ D is the set of places of F at which D is ramified, q v is the cardinality of the residue field of F v ,and ζ F is the completed Dedekind zeta function of F . We have the following central value formula of Ichino[Ich08]. Theorem 5.1 (Ichino) . As functionals in
Hom D × ( A F ) × D × ( A F ) ( Π D ⊗ ( Π D ) ∨ , C ) , we have I D = C D c · ζ E (2) ζ F (2) · L ( , Π , As) L (1 , Π , Ad) · Y v I Dv . Here ζ E ( s ) and ζ F ( s ) are the completed Dedekind zeta functions of E and F , respectively, and c = if E = F × F × F , if E = K × F for some totally real quadratic extension K of F , if E is a field . .2. Local trilinear period integrals.
For each finite place v of F and σ ∈ Aut( C ), we fix σ -linearisomorphisms t σ,v : π Dv −→ σ π Dv , t ∨ σ,v : ( π Dv ) ∨ −→ σ ( π Dv ) ∨ . By abuse of notation, we denote by the same notation I Dv ∈ Hom D × ( F v ) × D × ( F v ) ( σ π Dv ⊗ σ ( π Dv ) ∨ , C ) the func-tional defined as in (5.2) with respect to the D × ( F v )-equivariant bilinear pairing h , i v : σ π Dv × σ ( π Dv ) ∨ −→ C defined by h t σ,v ϕ ,v , t ∨ σ,v ϕ ,v i v = σ h ϕ ,v , ϕ ,v i v (5.4)for ϕ ,v ∈ π Dv and ϕ ,v ∈ ( π Dv ) ∨ . In the following lemmas, we show that the local L -factors and the localtrilinear period integrals satisfy the Galois equivariant property. Lemma 5.2.
Let v be a finite place of F . For σ ∈ Aut( C ) , we have σL (1 , π v , Ad) = L (1 , σ π v , Ad) , σL ( , π v , As) = L ( , σ π v , As) . Proof.
We prove the assertion for L ( , π v , As) in the case when E v is a field. The assertion for L (1 , π v , Ad)or arbitrary E v can be proved in a similar way and we omit it. Let W ′ E v and W ′ F v be the Weil–Deligne groupsof E v and F v , respectively. Fix σ ∈ Aut( C ). Let χ E v ,σ and χ F v ,σ be the quadratic characters of E × v and F × v ,respectively, defined by χ E v ,σ = σ ( | | / E v ) · | | − / E v , χ F v ,σ = σ ( | | / F v ) · | | − / F v . For n ≥
1, we identify the Langlands dual group L (R E v / F v GL n ) of R E v / F v GL n with GL n ( C ) ⋊ Gal( F v / F v )(cf. [Bor79, § F v / F v ) on GL n ( C ) is the permutation of components induced by thenatural homomorphism Gal( F v / F v ) → Gal( E ′ v / F v ) with E ′ v equals to the Galois closure of E v / F v . We have anatural one to one correspondence described in [Bor79, Lemma 4.5] between the set of L -parameters W ′ F v → L (R E v / F v GL n ) and the set of n -dimensional admissible representations W ′ E v → GL n ( C ). We will identify thetwo sets via this correspondence and note that the correspondence is compatible with σ -conjugation. Wealso identify characters of E × v with 1-dimensional admissible representations of W ′ E v via the local class fieldtheory. Let As be the Asai cube representation of L (R E v / F v GL ) on C ⊗ C ⊗ C so that the restriction ofAs to GL ( C ) is defined byAs( g , g , g ) · ( v ⊗ v ⊗ v ) = ( g · v , g v · v , g · v )and the action of Gal( F v / F v ) on C ⊗ C ⊗ C is the permutation of components induced by the naturalhomomorphism Gal( F v / F v ) → Gal( E ′ v / F v ). It is easy to verify that σ (As ◦ φ ) = As ◦ σ φ, As ◦ ( φ ⊗ χ ) = (As ◦ φ ) ⊗ χ | F × v for 2-dimensional admissible representation φ of W ′ E v and character χ of E × v . On the other hand, for anyadmissible representation Φ : W ′ F v → GL ( C ), by [Clo90, Lemme 4.6] and [Hen01, Propri´et´e 3, §
7] we have σ L ( s + , Φ ) = L ( s + , σ Φ ⊗ χ F v ,σ )as rational functions in q − sv . Let φ π v : W ′ E v → GL ( C ) be the admissible representation associated to π v by the local Langlands correspondence established in [Hen99] and [HT01]. Note that we have (cf. [Hen01,Propri´et´e 3, § σ φ π v = φ σ π v ⊗ χ E v ,σ . We conclude that σ L ( s + , π v , As) = σ L ( s + , As ◦ φ π v )= L ( s + , σ (As ◦ φ π v ) ⊗ χ F v ,σ )= L ( s + , (As ◦ σ φ π v ) ⊗ χ F v ,σ )= L ( s + , (As ◦ φ σ π v ) ⊗ χ E v ,σ | F × v · χ F v ,σ )= L ( s + , As ◦ φ σ π v )= L ( s + , σ π v , As) . e obtain the assertion by evaluating at s = 0. This completes the proof. (cid:3) Lemma 5.3.
Let v be a finite place of F . For σ ∈ Aut( C ) , we have σI Dv ( ϕ ,v ⊗ ϕ ,v ) = I Dv ( t σ,v ϕ ,v ⊗ t ∨ σ,v ϕ ,v ) for ϕ ,v ∈ π Dv and ϕ ,v ∈ ( π Dv ) ∨ .Proof. Let ϕ ,v ∈ π Dv and ϕ ,v ∈ ( π Dv ) ∨ . Note that by definition (5.4) we have σ h π Dv ( g v ) ϕ ,v , ϕ ,v i v = h σ π Dv ( g v ) t σ,v ϕ ,v , t ∨ σ,v ϕ ,v i v for g v ∈ D × ( E v ) and σ ∈ Aut( C ). Together with Lemma 5.2, it suffices to show that σ Z F × v \ D × ( F v ) h π Dv ( g v ) ϕ ,v , ϕ ,v i v dg v ! = Z F × v \ D × ( F v ) σ h π Dv ( g v ) ϕ ,v , ϕ ,v i v dg v (5.5)for all σ ∈ Aut( C ). If D is ramified at v , then F v \ D × ( F v ) is compact. Therefore the local period integral isa finite sum and equality (5.5) holds trivially. Suppose D is unramified at v and identify D × with GL . Let K v = o × F v \ GL ( o F v ). By the Cartan decomposition, we have Z F × v \ GL ( F v ) h π v ( g v ) ϕ ,v , ϕ ,v i v dg v = Z K v Z K v Z F × v h π v ( k ,v a ( t v ) k ,v ) ϕ ,v , ϕ ,v i v I o F v ( t v )vol( K v a ( t v ) K v , dg v ) d × t v dk ,v dk ,v , where dk ,v , dk ,v , and d × t v are Haar measures normalized so thatvol( K v , dk ,v ) = vol( K v , dk ,v ) = vol( o × F v , d × t v ) = 1 . Note that vol( K v a ( t v ) K v , dg v ) = ( | t v | F v = 1 , | t v | − F v (1 + q − v ) if | t v | F v < t v ∈ o F v r { } . It is well-known that there exist characters χ ,v and χ ,v of E × v depending only on π v andlocally constant functions f ,v and f ,v on E v × GL ( o E v ) × GL ( o E v ) such that h π v ( k ,v a ( t v ) k ,v ) ϕ ,v , ϕ ,v i v = χ ,v ( t v ) f ,v ( t v , k ,v , k ,v ) + χ ,v ( t v ) f ,v ( t v , k ,v , k ,v )for ( t v , k ,v , k ,v ) ∈ ( o E v r { } ) × GL ( o E v ) × GL ( o E v ). For a character χ of F × v and a locally constantfunction f : F v → C with compact support, let Z ( χ, f ) be the Tate integral defined by Z ( χ, f ) = Z F × v χ ( t v ) f ( t v ) d × t v . It is easy to show that the integral converges absolutely when | χ | = | | λ F v for some λ > σZ ( χ, f ) = Z ( σ χ, σ f )(5.6)for all σ ∈ Aut( C ) when both sides are absolutely convergnet. We have Z F × v \ GL ( F v ) h π v ( g v ) ϕ ,v , ϕ ,v i v dg v = [ K v : U v ] − X k ,v ∈ K v /U v X k ,v ∈ K v /U v h Z ( χ ,v | F × v · | | − F v , f (1) k ,v ,k ,v ) + Z ( χ ,v | F × v · | | − F v , f (2) k ,v ,k ,v ) i . (5.7)Here U v is an open compact normal subgroup of K v such that both ϕ ,v and ϕ ,v are U v -invariant and f ( i ) k ,v ,k ,v ( t v ) = t v / ∈ o F v ,f i,v ( t v , k ,v , k ,v ) if | t v | F v = 1 , (1 + q − v ) f i,v ( t v , k ,v , k ,v ) if | t v | F v < i = 1 , t v ∈ F v . We remark that | χ i,v | F × v | = | | λ i F v for some λ i > (cid:3) ow we consider the archimedean local trilinear period integrals. The totally unbalanced condition impliesthat D is unramified at v and π Dv = π v for each v ∈ Σ F . Let π ∞ = N v ∈ Σ F π v be a representation of D × ( E ∞ ) = GL ( R ) Σ E . Define I ∞ ∈ Hom GL ( F ∞ ) × GL ( F ∞ ) ( π ∞ ⊗ π ∨∞ , C ) and h , i ∞ : π ∞ × π ∨∞ → C by I ∞ = O v ∈ Σ F I v , h , i ∞ = Y v ∈ Σ F h , i v . We recall in the following lemma our previous calculation of local trilinear period integral.
Lemma 5.4.
Let ϕ ( κ,r ) ∈ π ∞ , ϕ ( κ, − r ) ∈ π ∨∞ be non-zero vectors of weight κ and X ( κ ) ∈ U ( gl Σ E , C ) be thedifferential operator defined in (3.4). We have I ∞ ( X ( κ ) · π ∞ ( τ I κ ) ϕ ( κ,r ) ⊗ X ( κ ) · π ∨∞ ( τ I κ ) ϕ ( κ, − r ) ) h ϕ ( κ,r ) , π ∨∞ ( τ Σ E ) ϕ ( κ, − r ) i ∞ ∈ (2 π √− P v ∈ Σ F (2 max w | v { κ w }− P w | v κ w ) · Q × . Here τ I κ , τ Σ E ∈ GL ( R ) Σ E are defined in (2.14).Proof. Let v ∈ Σ F and v (1) , v (2) , v (3) ∈ Σ E be the extensions of v . Note that π v = ⊠ i =1 ( D ( κ v ( i ) ) ⊗ | | r v ( i ) / ) , π ∨ v = ⊠ i =1 ( D ( κ v ( i ) ) ⊗ | | − r v ( i ) / ) . Let v v ( i ) ∈ ( D ( κ v ( i ) ) ⊗ | | r v ( i ) / ) and v ∨ v ( i ) ∈ ( D ( κ v ( i ) ) ⊗ | | − r v ( i ) / ) be non-zero vectors of weight κ v ( i ) for i = 1 , ,
3. Put v v = v v (1) ⊗ v v (2) ⊗ v v (3) and v ∨ v = v ∨ v (1) ⊗ v ∨ v (2) ⊗ v ∨ v (3) . We may assume v (1) = ˜ v ( κ ). Then I v ( X ( κ v (1) , κ v (2) , κ v (3) ) · π v (( τ I κ ) v ) v v ⊗ X ( κ v (1) , κ v (2) , κ v (3) ) · π ∨ v (( τ I κ ) v ) v ∨ v ) h v v , π v ( a ( − v ∨ v i v = ζ R (2) − · L (1 , π v , Ad) L ( , π v , As) · X m ,m ,m ′ ,m ′ c m ,m ( κ v (1) , κ v (2) , κ v (3) ) c m ′ ,m ′ ( κ v (1) , κ v (2) , κ v (3) ) × Z R × \ GL ( R ) h π v ( g ( τ I κ ) v )( v v (1) ⊗ X m + v v (2) ⊗ X m + v v (3) ) , π ∨ v (( τ I κ ) v )( v ∨ v (1) ⊗ X m + v ∨ v (2) ⊗ X m + v ∨ v (3) ) i v h v v , π v ( a ( − v ∨ v i v . Here m , m , m ′ , m ′ runs through non-negative integers such that2 m + 2 m = 2 m ′ + 2 m ′ = κ v (1) − κ v (2) − κ v (3) . By Lemma 3.1, the vector X m +2 m = κ v (1) − κ v (2) − κ v (3) c m ,m ( κ v (1) , κ v (2) , κ v (3) )( X m + v v (2) ⊗ X m + v v (3) )has weight κ v (1) and the ( gl , O(2))-module generated by it under the diagonal action is isomorphic to D ( κ v (1) ) ⊗ | | r v (1) / . It then follows from the Schur orthogonality relations that the above integral is non-zero.On the other hand, by [CC19, Proposition 4.1 and Corollary 4.4], the above integral is equal to(2 π √− κ v (1) − κ v (2) − κ v (3) · κ v (1) − κ v (2) − κ v (3) +1 · X m +2 m = κ v (1) − κ v (2) − κ v (3) ( − m c m ,m ( κ v (1) , κ v (2) , κ v (3) ) . We conclude that I ∞ ( X ( κ ) · π ∞ ( τ I κ ) ϕ ( κ,r ) ⊗ X ( κ ) · π ∨∞ ( τ I κ ) ϕ ( κ, − r ) ) h ϕ ( κ,r ) , π ∨∞ ( τ Σ E ) ϕ ( κ, − r ) i ∞ = Y v ∈ Σ F I v ( X ( κ v (1) , κ v (2) , κ v (3) ) · π v (( τ I κ ) v ) v v ⊗ X ( κ v (1) , κ v (2) , κ v (3) ) · π ∨ v (( τ I κ ) v ) v ∨ v ) h v v , π v ( a ( − v ∨ v i v ∈ (2 π √− P v ∈ Σ F (2 max w | v { κ w }− P w | v κ w ) · Q × . This completes the proof. (cid:3)
Let v be a place of F . By the results of Prasad [Pra90], [Pra92] and Loke [Lok01]. We havedim C Hom D × ( F v ) ( π Dv , C ) ≤ . Lemma 5.5.
Let v be a place of F . Then Hom D × ( F v ) ( π Dv , C ) = 0 if and only if I Dv = 0 . roof. When v ∈ Σ F , the assertion follows from Lemma 5.4. Assume v is a finite place. The assertion can beproved word by word following the proof of [Che20b, Corollary 9.6] except we replace R and [LZ97, Theorem4.12] therein by F v and [KR92, Corollary 3.7], respectively. (cid:3) Proof of Theorem 1.4.
Let ε ( s, Π , As) = Y v ε ( s, π v , As , ψ v )be the global Asai ε -factor of Π , where N v ψ v is a non-trivial additive character of F \ A F and ε ( s, π v , As , ψ v )is the local Asai ε -factor of π v with respect to ψ v defined via the Weil–Deligne representation. Note that ε ( s, π v , As , ψ v ) = 1 for all finite places v of F such that E v is unramified over F v , π v is unramified, and ψ v is ofconductor o F v . By our assumption that ω Π | A × F is trivial, we have ε ( , π v , As , ψ v ) ∈ {± } and is independentof the choice of ψ v . We write ε ( , π v , As) = ε ( , π v , As , ψ v ) . Recall the global root number ε ( Π , As) is defined by ε ( Π , As) = ε ( , Π , As) ∈ {± } . On the other hand, analogous to the proof of Lemma 5.2, we can show that σε ( , π v , As) = ε ( , σ π v , As)for all finite places v of F and σ ∈ Aut( C ). In particular, we have ε ( Π , As) = ε ( σ Π , As)for all σ ∈ Aut( C ). We see that assertion (1) of Theorem 1.4 follows immediately from the expected functionalequation L ( s, Π , As) = ε ( s, Π , As) L (1 − s, Π , As) . (5.8)Indeed, let L PSR ( s, Π , As) = Y v L PSR ( s, π v , As) , ε
PSR ( s, Π , As) = Y v ε PSR ( s, π v , As , ψ v )be the global Asai L -function and ε -factor of Π defined by the Rankin–Selberg method as well as the local zetaintegrals developed by Piatetski-Shapiro–Rallis [PSR87] and Ikeda [Ike89]. In the Rankin–Selberg context,the functional equation L PSR ( s, Π , As) = ε PSR ( s, Π , As) L PSR (1 − s, Π , As)holds and is a direct consequence of the functional equation of Siegel Eisenstein seires. On the other hand,by [KS02] and [Che20a, Corollary 1.4], we have L ( s, π v , As) = L PSR ( s, π v , As) , ε ( s, π v , As , ψ v ) = ε PSR ( s, π v , As , ψ v )for all places v of F . Hence the functional equation (5.8) holds.Now we assume ε ( Π , As) = 1. Let K be the quadratic discriminant algebra of E / F and ω K / F = Q v ω K v / F v the quadratic character of F × \ A × F associated to K / F by class field theory. By the assumption ε ( Π , As) = 1,there exists a unique quaternion algebra D over F such that D is ramified at v if and only if ε ( , π v , As) · ω K v / F v ( −
1) = − . In particular, it follows from the totally unbalanced condition that D is totally indefinite. By the results ofPrasad [Pra90], [Pra92] and Loke [Lok01], the above sign condition implies that there exists an irreduciblecuspidal automorphic representation Π D = N v π Dv of D × ( A E ) associated to Π by the Jacquet–Langlandscorrespondence such that Hom D × ( F v ) ( π Dv , C ) = 0for all places v of F . Fix non-zero vectors ϕ ( κ,r ) ∈ π ∞ = N v ∈ Σ F π v and ϕ ( κ, − r ) ∈ π ∨∞ = N v ∈ Σ F π ∨ v of weight κ , that is, π ∞ ( k θ ) ϕ ( κ,r ) = Y w ∈ Σ E e √− κ w θ w · ϕ ( κ,r ) ,π ∨∞ ( k θ ) ϕ ( κ, − r ) = Y w ∈ Σ E e √− κ w θ w · ϕ ( κ, − r )34 or k θ = ( k θ w ) w ∈ Σ E ∈ SO(2) Σ E . Fix σ ∈ Aut( C ). Let ϕ ( σ κ,r ) ∈ σ π ∞ and ϕ ( σ κ, − r ) ∈ σ π ∨∞ be the vectors ofweight σ κ such that if ϕ = ϕ ( κ,r ) ⊗ (cid:16)N v ∤ ∞ ϕ ,v (cid:17) ∈ Π D hol and ϕ = ϕ ( κ, − r ) ⊗ (cid:16)N v ∤ ∞ ϕ ,v (cid:17) ∈ ( Π D hol ) ∨ , then σ ϕ = ϕ ( σ κ,r ) ⊗ O v ∤ ∞ t σ,v ϕ ,v ∈ σ Π D hol , σ ϕ = ϕ ( σ κ, − r ) ⊗ O v ∤ ∞ t ∨ σ,v ϕ ,v ∈ σ ( Π D hol ) ∨ Here the σ -linear isomorphisms t σ,v , t ∨ σ,v for finite places v of F are fixed in § Π D hol −→ σ Π D hol , ϕ σ ϕ, ( Π D hol ) ∨ −→ σ ( Π D hol ) ∨ , ϕ ∨ σ ϕ ∨ are defined in (2.17). By Lemma 5.5, for each finite place v of F , there exist ϕ ,v ∈ π Dv and ϕ ,v ∈ ( π Dv ) ∨ such that I Dv ( ϕ ,v ⊗ ϕ ,v ) = 0 . Moreover, when v is a finite place such that E v is unramified over F v and π v is unramified, then D isunramified at v and we choose ϕ ,v and ϕ ,v be non-zero GL ( o E v )-invariant vectors. In this case, by [Ich08,Lemma 2.2], we have I Dv ( ϕ ,v ⊗ ϕ ,v ) = h ϕ ,v , ϕ ,v i v . With this local choice, we put ϕ = ϕ ( κ,r ) ⊗ (cid:16)N v ∤ ∞ ϕ ,v (cid:17) ∈ Π D hol and ϕ = ϕ ( κ, − r ) ⊗ (cid:16)N v ∤ ∞ ϕ ,v (cid:17) ∈ ( Π D hol ) ∨ .We also fix ϕ = ϕ ( κ,r ) ⊗ (cid:16)N v ∤ ∞ ϕ ,v (cid:17) ∈ Π D hol and ϕ = ϕ ( κ, − r ) ⊗ (cid:16)N v ∤ ∞ ϕ ,v (cid:17) ∈ ( Π D hol ) ∨ such that h ϕ , ϕ i = Z Z G ( A ) G ( Q ) \ G ( A ) ϕ ( g ) ϕ ( g · t Σ E ) dg Tam = 0 . Let X ( κ ) ∈ U ( gl Σ E , C ) be the differential operator defined in (3.4). By Ichino’s formula Theorem 5.1, we have I D ( X ( κ ) · ϕ I κ ⊗ X ( κ ) · ϕ I κ ) h ϕ , ϕ i = C D c · ζ E (2) ζ F (2) · L ( , Π , As) L (1 , Π , Ad) · Y v ∤ ∞ I Dv ( ϕ ,v ⊗ ϕ ,v ) h ϕ ,v , ϕ ,v i v · I ∞ ( X ( κ ) · π ∞ ( τ I κ ) ϕ ( κ,r ) ⊗ X ( κ ) · π ∨∞ ( τ I κ ) ϕ ( κ, − r ) ) h ϕ ( κ,r ) , π ∨∞ ( τ Σ E ) ϕ ( κ, − r ) i ∞ ,I D ( X ( σ κ ) · ( σ ϕ ) I σκ ⊗ X ( σ κ ) · ( σ ϕ ) I σκ ) h σ ϕ , σ ϕ i = C D c · ζ E (2) ζ F (2) · L ( , σ Π , As) L (1 , σ Π , Ad) · Y v ∤ ∞ I Dv ( t σ,v ϕ ,v ⊗ t ∨ σ,v ϕ ,v ) h t σ,v ϕ ,v , t ∨ σ,v ϕ ,v i v × I ∞ ( X ( σ κ ) · σ π ∞ ( τ I σκ ) ϕ ( σ κ,r ) ⊗ X ( σ κ ) · σ π ∨∞ ( τ I σκ ) ϕ ( σ κ, − r ) ) h ϕ ( σ κ,r ) , σ π ∨∞ ( τ Σ E ) ϕ ( σ κ, − r ) i ∞ . Here τ I κ , τ Σ E and ϕ I κ , ϕ I κ are defined in (2.14) and (2.15), respectively. By Lemma 2.2 and Proposition 3.2,we have the Galois equivariant property of the global trilinear period integral that σ I D ( X ( κ ) · ϕ I κ ⊗ X ( κ ) · ϕ I κ )Ω I κ ( Π D ) · Ω I κ (( Π D ) ∨ ) ! = I D ( X ( σ κ ) · ( σ ϕ ) I σκ ⊗ X ( σ κ ) · ( σ ϕ ) I σκ )Ω I σκ ( σ Π D ) · Ω I σκ ( σ ( Π D ) ∨ ) . (5.9)On the other hand, by Lemma 2.9 and Corollary 2.10, we have σ L (1 , Π , Ad)(2 π √− − P w ∈ Σ E κ w · π F : Q ] · h ϕ , ϕ i ! = L (1 , σ Π , Ad)(2 π √− − P w ∈ Σ E κ w · π F : Q ] · h σ ϕ , σ ϕ i . (5.10)By Lemmas 5.3 and 5.4, we have σ Y v ∤ ∞ I Dv ( ϕ ,v ⊗ ϕ ,v ) h ϕ ,v , ϕ ,v i v = Y v ∤ ∞ I Dv ( t σ,v ϕ ,v ⊗ t ∨ σ,v ϕ ,v ) h t σ,v ϕ ,v , t ∨ σ,v ϕ ,v i v , (5.11) nd I ∞ ( X ( κ ) · π ∞ ( τ I κ ) ϕ ( κ,r ) ⊗ X ( κ ) · π ∨∞ ( τ I κ ) ϕ ( κ, − r ) ) h ϕ ( κ,r ) , π ∨∞ ( τ Σ E ) ϕ ( κ, − r ) i ∞ = I ∞ ( X ( σ κ ) · σ π ∞ ( τ I σκ ) ϕ ( σ κ,r ) ⊗ X ( σ κ ) · σ π ∨∞ ( τ I σκ ) ϕ ( σ κ, − r ) ) h ϕ ( σ κ,r ) , σ π ∨∞ ( τ Σ E ) ϕ ( σ κ, − r ) i ∞ ∈ (2 π √− P v ∈ Σ F (2 max w | v { κ w }− P w | v κ w ) · Q × . (5.12)By (5.3) and the result of Siegel [Sie69], we have C D ∈ D / F · ζ F (2) − · Q × , ζ F (2) ∈ D / F · π [ F : Q ] · Q × , ζ E (2) ∈ D / E · π F : Q ] · Q × . Also note that Y v ∈ Σ F L ( , π v , As) ∈ π − P v ∈ Σ F max w | v { κ w } · Q × . The algebraicity for L ( , Π , As) then follows from (5.9)-(5.12). Finally, assume D is the matrix algebra, weshow that Ω I κ (( Π D ) ∨ ) = Ω I κ ( Π ∨ ) can be replaced by Ω I κ ( Π ). When L ( , Π , As) = 0, the assertion holdsby assertion (1). Thus we may assume L ( , Π , As) = 0. In this case, by Ichino’s formula Theorem 5.1, thereexists ϕ ∈ Π hol such that Z A × F GL ( F ) \ GL ( A F ) X ( κ ) · ϕ I κ ( g ) dg Tam = 0 . By Lemma 2.2 and Proposition 3.2, we have σ Z A × F GL ( F ) \ GL ( A F ) X ( κ ) · ϕ I κ ( g )Ω I κ ( Π ) dg Tam ! = Z A × F GL ( F ) \ GL ( A F ) X ( σ κ ) · σ ϕ I σκ ( g )Ω I σκ ( σ Π ) dg Tam . On the other hand, since ω Π | A × F is trivial, we have ϕ | GL ( A F ) = ( ϕ ⊗ ω − Π ) | GL ( A F ) . By Lemma 2.2 andProposition 3.2 again, we have σ Z A × F GL ( F ) \ GL ( A F ) X ( κ ) · ϕ I κ ( g )Ω I κ ( Π ) dg Tam ! = ω Π (det( τ I κ )) · σ (cid:18) Ω I κ ( Π ∨ )Ω I κ ( Π ) (cid:19) · σ Z A × F GL ( F ) \ GL ( A F ) X ( κ ) · ( ϕ ⊗ ω − Π ) I κ ( g )Ω I κ ( Π ∨ ) dg Tam ! = ω Π (det( τ I κ )) · σ (cid:18) Ω I κ ( Π ∨ )Ω I κ ( Π ) (cid:19) · Z A × F GL ( F ) \ GL ( A F ) X ( σ κ ) · σ ( ϕ ⊗ ω − Π ) I σκ ( g )Ω I σκ ( σ Π ∨ ) dg Tam . Here σ ( ϕ ⊗ ω − Π ) is defined in (2.17) with Π replaced by Π ∨ . By the q -expansion principle, we have σ ( ϕ ⊗ ω − Π ) = σ ϕ ⊗ σ ω − Π . Indeed, by Lemma 4.2, we have W ( ∞ ) σ ( ϕ ⊗ ω − Π ) ,ψ E ( g f ) = σ (2 π √− − P w ∈ Σ E ( κ w − r w ) / (2 π √− − P w ∈ Σ E ( κ w − r w ) / · σ (cid:16) W ( ∞ ) ϕ ⊗ ω − Π ,ψ F ( a ( u ) − g f ) (cid:17) = σ ω Π ( u ) · σ (2 π √− − P w ∈ Σ E ( κ w − r w ) / (2 π √− − P w ∈ Σ E ( κ w − r w ) / · σ (cid:16) W ( ∞ ) ϕ,ψ F ( a ( u ) − g f ) (cid:17) · σ ω − Π (det( g f ))= σ ω Π ( u ) · σ (2 π √− P w ∈ Σ E r w (2 π √− P w ∈ Σ E r w · W ( ∞ ) σ ϕ,ψ F ( g f ) · σ ω − Π (det( g f ))= W ( ∞ ) σ ϕ ⊗ σ ω − Π ,ψ E ( g f )for all g f ∈ GL ( A E ,f ). Here u ∈ b Z × is the unique element such that σ ( ψ E ( x )) = ψ E ( ux ) for x ∈ A E ,f . Notethat the last equality follows from the conditions that ω Π | A × F is trivial and P w ∈ Σ E r w = 0. Therefore, we ave ω Π (det( τ I κ )) · Z A × F GL ( F ) \ GL ( A F ) X ( σ κ ) · σ ( ϕ ⊗ ω − Π ) I σκ ( g ) dg Tam = Z A × F GL ( F ) \ GL ( A F ) X ( σ κ ) · σ ϕ I σκ ( g ) · ω − Π (det( g )) dg Tam = Z A × F GL ( F ) \ GL ( A F ) X ( σ κ ) · σ ϕ I σκ ( g ) dg Tam . We conclude that σ (cid:18) Ω I κ ( Π ∨ )Ω I κ ( Π ) (cid:19) = Ω I σκ ( σ Π ∨ )Ω I σκ ( σ Π ) . This completes the proof.5.4.
Proof of Theorem 1.8.
First we recall the result of Shimura [Shi78] on the algebraicity of specialvalues of the twisted standard L -functions for motivic irreducible cuspidal automorphic representations ofGL ( A F ). We also refer to [RT11] for a different proof. Theorem 5.6 (Shimura) . Let Π be a motivic irreducible cuspidal automorphic representation of GL ( A F ) of motivic weight ( ℓ, r ) ∈ Z ≥ [Σ F ] × Z and central character ω Π . There exist complex numbers p ( ε, Π ) ∈ C × defined for ε ∈ {± } Σ F satisfying the following assertions: (1) We have σ L ( ∞ ) ( m + , Π ⊗ χ )(2 π √− [ F : Q ] m · G ( χ ) · p (( − m sgn( χ ) , Π ) ! = L ( ∞ ) ( m + , σ Π ⊗ σ χ )(2 π √− [ F : Q ] m · G ( σ χ ) · p (( − m sgn( σ χ ) , σ Π ) for any finite order Hecke character χ of A × F , σ ∈ Aut( C ) , and m ∈ Z such that − min v ∈ Σ F { ℓ v } − r < m < min v ∈ Σ F { ℓ v } − r . (2) We have σ p ( ε, Π ) · p ( − ε, Π )(2 π √− [ F : Q ](1+ r ) ( √− P v ∈ Σ F ℓ v · G ( ω Π ) · Ω Σ F ( Π ) ! = p ( σ ε, σ Π ) · p ( − σ ε, σ Π )(2 π √− [ F : Q ](1+ r ) ( √− P v ∈ Σ F ℓ v · G ( σ ω Π ) · Ω Σ F ( σ Π ) for any ε ∈ {± } Σ F and σ ∈ Aut( C ) . Remark 5.7.
In [Shi78, Theorem 4.3], the theorem was stated for motivic ( ℓ, r ) ∈ Z ≥ [Σ F ] × Z . It isstraightforward to extend the results to ( ℓ, r ) ∈ Z ≥ [Σ F ] × Z by [Shi78, (4.16)] and the non-vanishing theoremof Friedberg–Hoffstein [FH95]. We also refer to Lemma 2.9 for the period relation between Petersson normand Ω Σ F ( Π ).Now we begin the proof of Theorem 1.8. Let Π = N v π v , Π ′ = N v π ′ v be motivic irreducible cuspidalautomorphic representations of GL ( A F ) with central characters ω Π , ω Π ′ and of weights ℓ, ℓ ′ ∈ Z ≥ [Σ F ],respectively. By assumption (i), the automorphic representation Sym ( Π ) × Π ′ of GL ( A F ) × GL ( A F ) isself-dual. Since Sym ( Π ) × Π ′ is isobaric, we have the global functional equation L ( s, Sym ( Π ) × Π ′ ) = ε ( s, Sym ( Π ) × Π ′ ) L (1 − s, Sym ( Π ) × Π ′ ) . Recall the global root number ε (Sym ( Π ) × Π ′ ) is defined by ε (Sym ( Π ) × Π ′ ) = ε ( , Sym ( Π ) × Π ′ ) ∈ {± } . Analogous to the proof of Lemma 5.2, we can show that ε (Sym ( Π ) × Π ′ ) = ε (Sym ( σ Π ) × σ Π ′ )for all σ ∈ Aut( C ). In particular, if ε (Sym ( Π ) × Π ′ ) = −
1, then it follows from the global functionalequation that L ( , Sym ( σ Π ) × σ Π ′ ) = 0 or all σ ∈ Aut( C ). Therefore, we may assume that ε (Sym ( Π ) × Π ′ ) = 1. By assumption (iii) and thenon-vanishing theorem of Friedberg–Hoffstein [FH95], there exists a totally real quadratic extension K over F such that the base change lift Π K = N v π K ,v of Π to GL ( A K ) is cuspidal and L ( , Π ′ ⊗ ω Π ω K / F ) = 0 . Consider the totally real ´etale cubic algebra E = K × F over F and the motivic irreducible cuspidal automorphicrepresentation Π K × Π ′ of GL ( A E ). We identify Σ E with Σ K ⊔ Σ F in a natural way. Note that the weight κ ∈ Z ≥ [Σ E ] of Π K × Π ′ is given as follows: for w ∈ Σ E lying over v ∈ Σ F , we have κ w = ( ℓ v if w ∈ Σ K ,ℓ ′ v if w ∈ Σ F . In particular, Π K × Π ′ is totally unbalanced and I κ = Σ F . Let D be the unique totally indefinite quaternionalgebra over F so that there exists an irreducible cuspidal automorphic representation Π D K × ( Π ′ ) D = O v ( π D K ,v × ( π ′ v ) D )of D × ( A E ) associated to Π K × Π ′ by the Jacquet–Langlands correspondence such thatHom D × ( F v ) ( π D K ,v × ( π ′ v ) D , C ) = 0for all places v of F . By Lemma 2.8, we have the period relation σ (cid:18) Ω Σ F ( Π D K × ( Π ′ ) D )Ω ∅ ( Π D K ) · Ω Σ F (( Π ′ ) D ) (cid:19) = Ω Σ F ( σ Π D K × σ ( Π ′ ) D )Ω ∅ ( σ Π D K ) · Ω Σ F ( σ ( Π ′ ) D )(5.13)for all σ ∈ Aut( C ). Note that by definition we may take Ω ∅ ( Π D K ) = 1 and by Corollary 2.10 we have σ (cid:18) Ω Σ F ( Π ′ ) Ω Σ F ( Π ′ D ) · Ω Σ F ( Π ′∨ ) D ) (cid:19) = Ω Σ F ( σ Π ′ ) Ω Σ F ( σ Π ′ D ) · Ω Σ F ( σ ( Π ′∨ ) D )(5.14)for all σ ∈ Aut( C ). Also we have the well-known fact for the rationality of the quadratic Gauss sum that D / K G ( ω K / F ) ∈ Q × . (5.15)Therefore, by (5.13)-(5.15) and Theorem 1.4, we have σ L ( ∞ ) ( , Π K × Π ′ , As) D / F (2 π √− F : Q ] · G ( ω K / F ) · Ω Σ F ( Π ′ ) ! = L ( ∞ ) ( , σ Π K × σ Π ′ , As) D / F (2 π √− F : Q ] · G ( ω K / F ) · Ω Σ F ( σ Π ′ ) for all σ ∈ Aut( C ). Finally, we have the factorization of L -functions: L ( s, Π K × Π ′ , As) = L ( s, Sym ( Π ) × Π ′ ) L ( s, Π ′ ⊗ ω Π ω K / F ) . Theorem 1.8 then follows immediately from Theorem 5.6 of Shimura for the central critical value L ( ∞ ) ( , Π ′ ⊗ ω Π ω K / F ) = L ( ∞ ) ( r + , Π ′ ⊗ | | − r A F ω Π ω K / F )and the condition that L ( , Π ′ ⊗ ω Π ω K / F ) = 0. This completes the proof.5.5. Proof of Theorem 1.11.
Let Π = N v π v be a motivic irreducible cuspidal automorphic representationof GL ( A F ) and χ an algebraic Hecke character of A × F . Assume that Π has motivic weight ( ℓ, r ) ∈ Z ≥ [Σ F ] × Z and | χ | = | | r A F for some r ∈ Z . Let I be a subset of Σ F . We fix a motivic irreducible cuspidal automorphicrepresentation Π ′ = N v π ′ v of GL ( A F ) with motivic weight ( ℓ ′ , r ′ ) ∈ Z ≥ [Σ F ] × Z satisfying the followingconditions: • min v ∈ Σ F {| ℓ v − ℓ ′ v |} ≥ • I = { v ∈ Σ F | ℓ v > ℓ ′ v } .The existence of Π ′ is guaranteed by the assumption that ℓ ∈ Z ≥ [Σ F ] and the result of Weinstein [Wei09].Consider the Rankin–Selberg L -function L ( s, Π × Π ′ × χ ) = L ( s, ( Π ⊗ χ ) × Π ′ × )for the triplets ( Π , Π ′ , χ ) and ( Π ⊗ χ, Π ′ , ). Note that at the rightmost critical point m = − r − r + r ′ v ∈ Σ F {| ℓ v − ℓ ′ v |} , e have L ( m, Π × Π ′ × χ ) = 0 by the condition that min v ∈ Σ F {| ℓ v − ℓ ′ v |} ≥
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Institute of Mathematics, Academia Sinica, 6F, Astronomy-Mathematics Building, No. 1, Sec. 4, Roosevelt Road,Taipei 10617, Taiwan, ROC
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