Periods of cusp forms on GSp4

Periods of cusp forms on GSp

₄

Shih-Yu Chen

Academia Sinica

22nd Autumn Workshop on Number Theory November 3, 2019

We present our recent progress on the following topics:

(1) Automorphic analogue of Yoshida’s period relation forGSp₄. (2) Algebraicity of the symmetric sixth powerL-functions forGL2. (3) Algebraicity of critical values of the Rankin-SelbergL-functions for

GSp₄×GL2.

Deligne’s conjecture for the spinor L-functions for GSp

₄

π: an irr. cusp. auto. rep. ofGSp₄(A) with trivial central character.

We assumeπisglobally genericandπ∞ is adiscrete series representation.

π∞|Sp₄(R)=D(λ₁,λ₂)⊕D(−λ₂,−λ₁),

whereD(λ₁,λ₂)is the discrete series representation ofSp₄(R) with Blattner parameter (λ1, λ2)∈Z² such that 1−λ1≤λ2≤ −1.

Q(π): the rationality field ofπ.

L(s, π): the spinorL-function ofπ.

We assumeπisstable, that is, the functorial lift ofπtoGL4(A) is cuspidal.

Deligne’s conjecture for the spinor L-functions for GSp

₄

M: the hypothetical motive attached to the spinorL-functionL(s, π).

M is a motive overQ with coefficients inQ(π) of rank 4 and of pure weightw =λ1−λ2−1.

c^±(M)∈(Q(π)⊗_QC)^×: Deligne’s periods attached toM.

MotivicL-function ofM: L(M,s) =

L^(∞) s−w

2, π^σ

σ

, whereσruns over a complete set of coset representatives of Aut(C)/Aut(C/Q(π)).

Conjecture (Deligne (1977))

Let m∈Zbe a critical point for M. For any finite order characterχof Q^×\A^×, we have

L(M⊗χ,m) (2π√

−1)^2m·G(χ)²·c^{(−1)msgn(χ)}(M) ∈Q(π).

Here G(χ)is the Gauss sum ofχ.

Deligne’s conjecture for the spinor L-functions for GSp

₄

We have the following automorphic analogue of Deligne’s conjecture.

Theorem (Grobner−Raghuram (2014), Januszewski (2016), Jiang−Sun−Tian (2019))

There exist c^±(π)∈C^×such that L(¹₂+m, π×χ) G(χ)²·c^{(−1)msgn(χ)}(π)

!σ

= L(¹₂+m, π^σ×χ^σ) G(χ^σ)²·c^{(−1)msgn(χ)}(π^σ)

for allσ∈Aut(C), all critical points ¹₂+m∈ ¹₂+Zof L(s, π), and any finite order characterχofQ^×\A^×.

Remark

The theorem is a special case of the results on the algebraicity of the critical values of the twisted standardL-functions of irr. regular algebraic,symplectic cusp. auto. rep. ofGL2n.

Yoshida’s period relation

π^hol: the unique irr. holo. cusp. auto. rep. of GSp₄(A) such that πf^hol'πf.

fhol: a non-zerovector-valued holomorphic cusp form associated toπ^hol normalized so that its Fourier coefficients belong toQ(π).

c^±(Sym²(M)): Deligne’s periods attached toSym²(M).

Theorem (Yoshida (2001)) We have

c⁺(Sym²(M)) (2π√

−1)^{6−3λ1+3λ2}·c⁺(M)·c⁻(M)·(kf_hol^σ k)σ

∈Q(π).

Deligne’s conjecture for the adjoint L-functions for GSp

₄

L(s, π,Ad): the adjointL-function ofπ.

MotivicL-function ofSym²(M):

L(Sym²(M),s) =

L^(∞)(s−w, π^σ,Ad)

σ.

Conjecture (Deligne (1977))

Let m∈Zbe a critical point for M. We have L(Sym²(M),m) (2π√

−1)^d(−1)mm·c^(−1)m(Sym²(M)) ∈Q(π).

Here d⁺= 6and d⁻= 4.

In particular, when m is evem, we have L(Sym²(M),m) (2π√

−1)^{6+6m−3λ1+3λ2}·c⁺(M)·c⁻(M)·(kf_hol^σ k)σ

∈Q(π)

Main result

Following is our main result, which can be regarded as an automorphic analogue of Yoshida’s period relation.

Theorem (C.-)

Assume thatλ1+λ2≥4andλ2≤ −5. Forσ∈Aut(C), we have L(1, π,Ad)

π³·c⁺(π)·c⁻(π)· kfholk σ

= L(1, π^σ,Ad)

π³·c⁺(π^σ)·c⁻(π^σ)· kf_hol^σ k.

Remark

(1) The result holds for any irr. (tempered) stablecusp. auto. rep. πof GSp₄(A) with trivial central character such thatπ∞ is adiscrete series representation.

(e.g. π=πF with Hecke eigenformF∈S2−λ₂,λ₁+λ₂−2(Sp₄(Z)).) (2) In caseλ1+λ2= 2,πp is unramified for all primesp, and there exists a

quadratic characterχofQ^×\A^×withχ∞(−1) =−1 such that L ¹₂, π

L ¹₂, π×χ

6= 0. Then the theorem follows from the explicit refinement of B¨ocherer’s conjecture proved by Furusawa−Morimoto.

Symmetric sixth power L-functions for GL

₂

τ ⊂ A0(PGL2(A)):non-dihedralwithτ∞'D(κ)for someκ≥2.

π⊂ A0(GSp₄(A)): the automorphic descent ofSym³τ. Corollary

Assumeκ≥6. Forσ∈Aut(C), we have L(1, τ,Sym⁶)

π⁶· kfτk³· kFτk ^σ

= L(1, τ^σ,Sym⁶) π⁶· kfτ^σk³· kFτ^σk. Here

fτ is the normalized newform ofτ,

Fτ is a non-zero vector-valued holomorphic cusp form associated toπ^hol normalized so that its Fourier coefficients belong toQ(τ).

Theorem (Morimoto) Forσ∈Aut(C), we have

kFτk kfτk³

σ

=±kFτ^σk kfτ^σk³.

Symmetric sixth power L-functions for GL

₂

Combining the corollary with Morimoto’s result, we obtain the following theorem.

Theorem

Forσ∈Aut(C), we have

L(1, τ,Sym⁶) π⁶· kfτk⁶

σ

=±L(1, τ^σ,Sym⁶) π⁶· kf_τ^σk⁶ .

Remark

(1) Morimoto proved in a different way that the ratio is inQ,

elliptic modular form→Hilbert modular form, all critical values,

twisted symmetric fourth powerL-function.

(2) Whenτ has level 1 andFτ is a Hecke eigenform, we callFτ the

Kim−Ramakrishnan−Shahidi lift offτ. Katsurada−Takemori conjectured that, after suitably normalized, a prime ideal dividing the ratio but not dividing Γ(2κ) gives a congruence betweenFτ and non K-R-S lift.

Proof of the main result

We show that there existΩ^W±(π)∈C^×, call theWhittaker periodsofπ, satisfying the following assertions:

(1) Forσ∈Aut(C), we have L ¹₂+m, π

L ^λ1+λ₂²⁻¹, π×χ π^−2λ1·G(χ)²·Ω^W₊(π)

!σ

=L ¹₂+m, π^σ

L ^λ1+λ₂²⁻¹, π^σ×χ^σ π^−2λ1·G(χ^σ)²·Ω^W₊(π^σ) for any finite order characterχofQ^×\A^×and critical points ¹₂+msuch that (−1)^{m+λ1 +2λ2}χ∞(−1) = 1.

(2) Forσ∈Aut(C), we have L(m, π×τ)

π^−λ1+λ2·G(χτ)²·Ω^W₋(π)· kfτk σ

= L(m, π^σ×τ^σ)

π^−λ1+λ2·G(χ^στ)²·Ω^W₋(π^σ)· kfτ^σk for any irr. cusp. auto. rep. τ ofGL2(A) satisfying:

(i) ωτ=χτ| |^r

Afor some finite order characterχτ ofQ^×\A^×andr∈Z, (ii) τ∞⊗ | |^−r/2

R 'D(`) for someλ1+λ2+ 1≤`≤λ1with`≡r( mod 2), and any critical pointsm∈Zwithm>−^r₂.

Proof of the main result

(3) Forσ∈Aut(C), we have kfπk

Ω^W₊(π)·Ω^W₋(π) σ

= kfπ^σk Ω^W₊(π^σ)·Ω^W₋(π^σ). Herefπis thenormalized newform ofπ(to be explained later).

(1)+ results ofJanuszewski, Jiang−Sun−Tian: ifλ1+λ2≥4, then Ω^W₊(π)

π^2λ1·c⁺(π)·c⁻(π) ^σ

= Ω^W₊(π^σ) π^2λ1·c⁺(π^σ)·c⁻(π^σ).

(2)+ results ofFurusawa, B¨ocherer−Heim, Pitale−Schmidt, Saha, Morimoto:

ifλ2≤ −5, then

Ω^W−(π)

π^4+λ1−λ2· kfholk ^σ

= Ω^W−(π^σ) π^4+λ1−λ2· kf_hol^σ k.

Therefore, by (3) we have kfπk

π^4+3λ1−λ2·c⁺(π)·c⁻(π)· kfholk σ

= kfπ^σk

π^4+3λ1−λ2·c⁺(π^σ)·c⁻(π^σ)· kf_hol^σ k.

Proof of the main result

Our main theorem then follows from the following result:

Forσ∈Aut(C), we have L(1, π,Ad)

π^{−1−3λ1+λ2}· kfπk σ

= L(1, π^σ,Ad) π^{−1−3λ1+λ2}· kfπ^σk.

C.−Ichino (2019): We computed the explicit value for the ratio whenπ has square-free paramodular conductor.

In the rest of this talk, we

sketch the construction of the Whittaker periods Ω^W±(π),

sketch the proof of the corresponding algebraicity results for critical L-values.

Notation

For (k1,k2)∈Z²withk1≥k2, let (ρ_(k1,k2),V_(k1,k2)) be the irr. alg. rep. of U(2) defined by

ρ(k₁,k₂)=Sym^k1−k2⊗det^k2,

V(k₁,k₂)=hXⁱY^k1−k2−i|0≤i≤k1−k2i_Q⊗_QC. Maximal unipotent subgroup ofGSp₄:

U=





















1 ∗ ∗ ∗

0 1 ∗ ∗

0 0 1 0

0 0 ∗ 1











∈GSp₄









 .

ψU:U(Q)\U(A)−→C^×defined by

ψU





















1 x ∗ ∗

0 1 ∗ y

0 0 1 0

0 0 −x 1





















=ψ(−x−y),

whereψ:Q\A−→C^×so thatψ∞(x) =e^2π

√−1x.

Rational structures via the Whittaker models

Forϕ∈π, define the global Whittaker function Wϕ,ψ_U(g) =

Z

U(Q)\U(A)

ϕ(ug)ψU(u)du.

W(πv, ψU,v): the space of Whittaker functions ofπv with respect toψU,v. W(πf, ψU,f) =N0

pW(πp, ψU,p).

Forσ∈Aut(C), define theσ-linear isomorphism ofGSp₄(A^f)-modules:

tσ:W(πf, ψU,f)−→ W(π^σ_f, ψU,f), tσW(g) =σ

W

diag(u⁻²,u⁻¹,u,1)g

. Hereσ|_Qab=rec(a·u) witha·u∈R^×>0·bZ^×and

rec:Q^×\A^×−→Gal(Q^ab/Q) is the geometrically normalized reciprocity map.

Rational structures via the Whittaker models

Recall

π∞|Sp₄(R)=D(λ₁,λ₂)⊕D(−λ₂,−λ₁), for some (λ1, λ2)∈Z²such that 1−λ1≤λ2≤ −1.

Write

π⁺= (π⊗_CV(λ₁,λ₂))^U(2), π⁻= (π⊗_CV(−λ₂,−λ₁))^U(2). Note thatπ⁺'π⁻'πf asGSp₄(A^f)-modules.

Forf ∈π⁺,h∈π⁻, we have

f =

λ1−λ₂

X

i=0

(−1)ⁱ λ1−λ2

i

!

·P_i⁺(f)⊗X^{λ1−λ2−i}Yⁱ,

h=

λ₁−λ₂

X

i=0

λ1−λ2

i

!

·P_i⁻(h)⊗XⁱY^{λ1−λ2−i}

for some uniquely determinedP_i⁺(f),P_i⁻(h)∈πfor 0≤i≤λ1−λ2.

Rational structures via the Whittaker models

Let 0≤i ≤λ1−λ2.

Wi ∈ W(π∞, ψU,∞): in the minimalU(2)-type ofD(−λ₂,−λ₁)with weight (−λ1+i,−λ2−i)normalized so that (following T. Moriyama)

Wi(1)

= (2√

−1)^{3λ1 +2λ2−i}π⁻¹²e^−2π

× Z c₁+√

−1∞

c₁−√

−1∞

ds1

2π√

−1 Z c₂+√

−1∞

c₂−√

−1∞

ds2

2π√

−1(4π³)

−s1 +λ1 +1−i

2 (4π)

−s2 +λ2 +i 2

×Γ

s1+s2−2λ2+ 1 2

Γ

s1+s2+ 1 2

Γs1

2

Γ−s2

2

Γ(s1+i) Γ(s1) , wherec1,c2∈Rsatisfyc1+c2+ 1>0 andc1>0>c2.

Wi ∈ W(π∞, ψ_U,∞): in the minimalU(2)-type ofD(λ₁,λ₂)with weight (λ1−i, λ2+i).

Rational structures via the Whittaker models

DefineGSp₄(Af)-module isomorphisms

π⁺−→ W(πf, ψU,f), f 7−→W_f⁺ π⁻−→ W(πf, ψ_U,f), h7−→W_h⁻ by

W_P+

i (f),ψ_U =Wi·W_f⁺, W_P−

i (h),ψ_U =Wi·W_h⁻ for 0≤i≤λ1−λ2.

Define theσ-linear isomorphisms ofGSp₄(Af)-modules π⁺−→(π^σ)⁺, f 7−→f^σ π⁻−→(π^σ)⁻, h7−→h^σ by

W_f⁺σ =tσW_f⁺, W_h⁻σ =tσW_h⁻. fπ∈π⁺: thenormalized newformofπdefined so thatW_f⁺

π ∈ W(πf, ψU,f) is the paramodular newform withW_f⁺_π(1) = 1. It is clear thatfπ^σ=fπ^σ for σ∈Aut(C).

Rational structures via the cohomology

PutK∞=R^×·U(2)⊂GSp₄(R) and (cf. Oshima’s and Horinaga’s talk)

p⁻=

( √−1A A

A √

−1A

!

A=^tA∈M2(Q) )

⊗_QC⊂Lie(GSp₄(R))_C.

Let (ρ,Vρ) be an irr. alg. rep. ofK∞. Consider the complexes with respect to the Lie algebra differential operator:

C_sia,ρ^q = Csia^∞(GSp₄(Q)\GSp₄(A))⊗_C

q

^(p⁻)^∗⊗_CVρ

!K∞

,

C_rda,ρ^q = Crda^∞(GSp₄(Q)\GSp₄(A))⊗_C

q

^(p⁻)^∗⊗_CVρ

!K∞

forq≥0.

H^q(Vρ^can)andH^q(Vρ^sub): theq-th cohomology groups with respect to the complexesC_sia,ρ^∗ andC_rda,ρ^∗ , respectively.

H_!^q(Vρ): the image of the morphismH^q(V_ρ^sub)−→H^q(V_ρ^can) induced by the inclusionC_rda,ρ^∗ −→C_sia,ρ^∗ .

Rational structures via the cohomology

Theorem (Harris, Milne)

(1) H^q(Vρ^can)and H^q(Vρ^sub)are admissibleGSp₄(Af)-modules and have canonical rational structures overQ.

(2) H_!^q(Vρ)is semisimple.

(3) Forσ∈Aut(C), conjugation byσinduces natural σ-linear GSp₄(A^f)-module isomorphism:

Tσ:H^q(V_ρ^can)−→H^q(V_ρ^can).

Similar assertion holds for H^q(Vρ^sub)and H_!^q(Vρ).

(4) We have a natural injective homomorphism ofGSp₄(A^f)-modules

cl: A0(GSp₄(A), ρ)⊗_C

q

^(p⁻)^∗⊗_CVρ

!K∞

−→H_!^q(Vρ)

for each q∈Z^≥0. HereA0(GSp₄(A), ρ)is the space of cusp forms on GSp₄(A)which are eigenfunctions of the Casimir operator ofGSp₄(R) with certain eigenvalue depending onρ.

Rational structures via the cohomology

We apply the results of Harris to (ρ,Vρ) equal to

(ρ⁺,V⁺) := (ρ(λ₁−3,λ₂−1),V(λ₁−3,λ₂−1)), (ρ⁻,V⁻) := (ρ(−λ₂−2,−λ₁),V(−λ₂−2,−λ₁)).

In these two cases,πoccurs inA0(GSp₄(A), ρ).

Proposition

The homorphismsclinduceisomorphismsofGSp₄(Af)-modules:

cl:

"

π⊗_C

2

^(p⁻)^∗⊗_CV⁺

#K∞

−→H!²(Vρ⁺)[πf]'πf,

cl:

"

π⊗_C

1

^(p⁻)^∗⊗_CV⁻

#K∞

−→H_!¹(V_ρ−)[πf]'πf.

Remark

We use Arthur’s multiplicity formula to deduce that A0(GSp₄(A))[πf] =π⊕π^hol.

Whittaker periods

FixQ-rationalinjectiveK∞-equivariant homomorphisms:

V(λ₁,λ₂)−→

2

^(p⁻)^∗⊗_CV⁺=⇒cl:π⁺−→H_!²(Vρ⁺)[πf],

V(−λ₂,−λ₁)−→

1

^(p⁻)^∗⊗_CV⁻=⇒cl:π⁻−→H!¹(V_ρ−)[πf].

Note that theQ-rational embeddings are unique up to homotheties overQ^×. Lemma

There existΩ^W+(π),Ω^W−(π)∈C^×, unique up toQ(π)^×, such that cl

π^+,Aut(C/Q(π))

Ω^W₊(π) =H_!²(Vρ⁺)[πf]^Aut(C/Q(π)), cl

π^{−,Aut(C/Q(π))}

Ω^W₋(π) =H_!¹(Vρ⁻)[πf]^Aut(C/Q(π)).

We call Ω^W±(π) theWhittaker periodsofπ. By the uniqueness up toQ(π)^×, we can normalize the Whittaker periods so that

Tσ

cl(f) Ω^W_ε (π)

= cl(f^σ)

Ω^W_ε (π^σ), f ∈π^ε.

Algebraicity of critical L-values for GSp

₄

× GL

₂

Theorem (C.-, in progress) (1) Forσ∈Aut(C), we have





L ¹₂+m, π L_λ

1 +λ2−1

2 , π×χ

π^−2λ1·G(χ)²·Ω^W₊(π)





σ

=

L ¹₂+m, π^σ L_λ

1 +λ2−1

2 , π^σ×χ^σ π^−2λ1·G(χ^σ)²·Ω^W₊(π^σ)

for any finite order characterχofQ^×\A^×and critical points ¹₂+m such that(−1)^{m+λ1 +2λ2}χ∞(−1) = 1.

(2) Forσ∈Aut(C), we have

L(m, π×τ)

π^−λ1 +λ2·G(χ_τ)²·Ω^W₋(π)· kfτk

!σ

= L(m, π^σ×τ^σ)

π^−λ1 +λ2·G(χ^σ_τ)²·Ω^W₋(π^σ)· kfτ σk

for any irr. cusp. auto. rep. τ ofGL2(A)satisfying:

(i) ωτ=χτ| |^r

Afor some finite order characterχτ ofQ^×\A^×and r∈Z, (ii) τ∞⊗ | |^−r/2

R 'D(`)for someλ1+λ2+ 1≤`≤λ1with`≡r(mod2), and any critical points m∈Zwith m>−^r₂.

Algebraicity of critical L-values for GSp

₄

× GL

₂

We sketch the proof of (2). Writeω=ωτ andχ=χτ.

LetG⁰={(g1,g2)∈GL2×GL2|det(g1) = det(g2)}and we regard it as a subgroup ofGSp₄by the embedding

G⁰−→GSp₄,

a1 b1

c1 d1

!

, a2 b2

c2 d2

!!

7−→











a1 0 b1 0 0 a2 0 b2

c1 0 d1 0 0 c2 0 d2









 .

Ingredients:

(1) Integral representation ofL(s, π×τ).

(2) Cohomological interpretation of the global integral.

(3) Local zeta integrals:

Explicit calculation of the archimedean local zeta integral.

Galois equivariance property of thep-adic local zeta integral.

Algebraicity of critical L-values for GSp

₄

× GL

₂

(1) Integral representation ofL(s, π×τ) (Piatetski-Shapiro−Soudry):

Forϕ1∈π,ϕ2∈τ, andfs∈Ind^GL_B(^2(A)

A) (| |^s−

1 2

A | |^−s+

1 2

A ω⁻¹) be a good section, we have the basic identity

Z(ϕ1, ϕ2,E(fs)) :=

Z

A^×G⁰(Q)\G⁰(A)

ϕ1(g)E(g1;fs)ϕ2(g2)dg

= Z

A^×N⁰(A)\G⁰(A)

Wϕ₁,ψ_U(g)fs(g1)Wϕ₂,ψ(g2)dg.

Heredg is the Tamagawa measure onA^×\G⁰(A).

Forκ≥1 and Φ∈ S(A²f), letEs^[κ](ω,Φ) =E(f_ω,Φ[κ]⊗Φ,s), where

f_ω,Φ[κ]⊗Φ,s(g) =|det(g)|^s_A Z

A^×

(Φ^[κ]⊗Φ)((0,t)g)ω(t)|t|^2s_A d^×t and

Φ^[κ](x,y) = 2^−κ(x+√

−1y)^κe^−π(x2+y2). ThenEs^[κ](ω,Φ)|_s=κ−r

2 is a holo. auto. form of weightκ.

Algebraicity of critical L-values for GSp

₄

× GL

₂

(2) Cohomological interpretation of the global integral.

Letm∈Zwith−^r₂ <m≤ ^`−λ1−λ₂ ^2−r (right-half critical points).

H!¹(V_ρ−)⊗_CH!¹(V(`−2;−r))⊗<sub>C</sub>H<sup>0</sup>(V<sub>(λ</sub><sup>can</sup><sub>1+λ2</sub>−`;r))−→C, [f1]⊗[f2]⊗[f3]7−→Z

P_`−λ⁻ ₂(f1),f2,X

`−λ1−λ2−r

2 −m

+ ·f3

is a well-definedQ-rationaltrilinear form. Here

V_(κ;r): the automorphic line bundle onM_GL2associated to the character a·e

√−1θ7−→a^r·e^κ

√−1θ

ofR^×·U(1).

X+=−¹₄

√−1 −1

−1 −√

−1

!

∈Lie(GL2(R))_Cis the weight raising differential operator.

Algebraicity of critical L-values for GSp

₄

× GL

₂

Forϕ1∈π⁻,ϕ2∈τ with weight−`, and Φ∈ S(A²f), we have

Tσ

[ϕ1] Ω^W₋(π)

= [ϕ^σ₁]

Ω^W₋(π^σ) ∈H!¹(V_ρ−)[πf^σ], Tσ

[ϕ2] kfτk(2π√

−1)^`−r2

!

= [ϕ^σ2] kfτ^σk(2π√

−1)<sup>`−r2</sup> ∈H!<sup>1</sup>(V(`−2;−r))[τf^σ], Tσ

[Es^[2m+r](ω,Φ)|s=m] G(χ)(2π√

−1)^−m

!

= [Es^[2m+r](ω^σ,Φ^σ)|s=m] G(χ^σ)(2π√

−1)^−m ∈H⁰(V_(λ^can_1+λ2−`;r)).

We conclude that







 Z

P_`−λ⁻ ₂(ϕ1), ϕ2,X

`−λ1−λ2−r

2 −m

+ ·Es^[2m+r](ω,Φ)|s=m

Ω^W₋(π)· kfτk(2π√

−1)^`−r2 ·G(χ)(2π√

−1)^−m









σ

= Z

P_`−λ⁻

2(ϕ^σ₁), ϕ^σ₂,X

`−λ1−λ2−r

2 −m

+ ·Es^[2m+r](ω^σ,Φ^σ)|s=m

Ω^W−(π^σ)· kfτ^σk(2π√

−1)^`−r2 ·G(χ^σ)(2π√

−1)^−m

Algebraicity of critical L-values for GSp

₄

× GL

₂

(3) Local zeta integrals:

Let

(i) W`−λ<sub>2</sub>∈ W(π∞, ψU,∞): normalized with weight (λ1+λ2−`, `), (ii) W<sub>−`</sub>⁰ ∈ W(τ∞, ψ∞): weight−`withW<sub>−`</sub>⁰ (1) =e^−2π,

(iii) f_ω

∞,Φ<sup>[`−λ</sup>1−λ2 ],s: normalized with weight`−λ1−λ2. We have

Z∞(W`−λ<sub>2</sub>,W−`⁰ ,f_ω∞,Φ[`−λ1−λ2 ],s)

∈(√

−1)^{λ1 +2λ2} ·π

3λ1−λ2

2 ·L(s, π∞×τ∞)·Q^×. Letp be a prime. LetW1∈ W(πp, ψ_U,p),W2∈ W(τp, ψ_p), and Φ∈ S(Q²p). Forσ∈Aut(C), we have

Zp(W1,W2,fω_p,Φ,s) ε(0, χp, ψp)

σ

= Zp(tσW1,tσW2,fω_p^σ,Φ^σ,s) ε(0, χ^σp, ψp) , L(s, πp×τp)^σ=L(s, πp^σ×τp^σ).

The assertion then follows from (1)-(3) with good choice of datum (W1,W2,Φ)∈ W(πf, ψ_U,f)× W(τf, ψ_f)× S(A²f).

The end.

Thank you for your attention.