1. Formulation of the conjecture

CONJECTURE

ATSUSHI ICHINO AND TAMOTSU IKEDA

1. Formulation of the conjecture

We would like to formulate a relation between a certain period of automorphic forms on orthogonal group and some L-value. Our con- jecture can be considered as a refinement of global Gross-Prasad con- jecture [9].

Letk be a global field with char.k = 2. Let (V₁, Q₁) and (V₀, Q₀) be quadratic forms with rank n+ 1 and n, respectively. In the following, the subscriptiis either 0 or 1. We denote the special orthogonal group of (V_i, Q_i) by G_i. We assume there is an embeddingι:V₀ →V₁. Then we have an embedding of corresponding orthogonal groups ι : G₀ → G₁. When n= 2, we assume V₀ is not split.

Let π_i ⊗_vπ_i,v be an irreducible square integrable automorphic representation of G_i. There is a canonical inner product ∗,∗ on π_i defined by

ϕ_i, ϕ_i=

Gi(k)\Gi( )

ϕ_i(g_i)ϕ_i(g_i)dg_i,

where dg_i is the Tamagawa measure on G_i(A). We choose a Haar measure dg_i,v for each v. There exist positive number C_i such that dg_i = C_i

vdg_i,v. The character χ_Qi is the quadratic character of A^×_k/k^× corresponding to k(

(−1)^mdetQ_i)/k.

Put Δ_G1,v =

ζ_v(2)ζ_v(4)· · ·ζ_v(2m), dim_kV₁ = 2m+ 1, ζ_v(2)ζ_v(4)· · ·ζ_v(2m−2)·L_v(m, χ_Q1), dim_kV₁ = 2m, Δ_G1 =

v

Δ_G1,v

Choose decomposable vectors ϕ_i = ⊗vϕ_i,v ∈ ⊗vπ_i,v. Since π_i,v is a unirary representation, there is a norm ∗ on π_i,v for any place v of k.

We are interested in the periodϕ₁|G0, ϕ₀whereϕ₁|G0 is the restric- tion of ϕ₁ toG₀(A).

1

The L-group ^LG_i of G_i is a semi-direct product ˆG_iW_k. Here, W_k is the Weil group of k and

Gˆ_i =

SO(m,C) if m = dimV_i is even, Sp_(m−1)/2(C) if m = dimV_i is odd.

We denote by st the standard representation of ^LG_i. The (complete) standardL-function is denoted byL(s, π_i,st). For simplicity, we some- times denote L(s, π_i,st) by L(s, π_i). For v /∈ S, the Euler factor for L(s, π_i,st) is given by det(1−st(A_πi,v)·q_v^−s)⁻¹. Here, A_πi,v is the Sa- take parameter of π_i,v. We consider the tensor product L-function L(s, π₁ ⊗π₀). The Euler factor of L(s, π₁⊗π₀) for v /∈ S is given by det(1−st₁(A_π1,v)⊗st₀(A_π0,v)·q_v^−s)⁻¹, where st₁ and st₀ are the standard representations of ^LG₁ and ^LG₀, respectively.

Consider the adjoint representation Ad : ^LG_i →GL(Lie(^LG_i)). The associated L-functionL(s, π_i,Ad) is called the adjointL-function. We put

Pπ1,π0(s) = L(s, π₁π₀)

L(s+ ¹₂, π₁,Ad)L(s+ ¹₂, π₀,Ad).

LetS be a set of bad primes containing all archimedean places. We may and do assume the following conditions:

(UR1) G₁ andG₀ are unramified over k_v outside S.

(UR2) π₁ and π₀ are unramified outsideS.

(UR3) Ifv /∈S, thenϕ_1,v andϕ_0,v are unramified vectors andϕ_1,v= ϕ_0,v= 1.

(UR4) For v /∈ S, the Haar measures dg_1,v and dg_0,v are the standard measures. (By the standard measure, we mean the Haar mea- sure for which the volume of the hyperspecial maximal compact subgroup is 1).

Rallis and Bernstein announced (independently) that they proved that

dim Hom_G0,v(π_1,v⊗π˜_0,v,C)≤1

for non-archimedean place v of k. Although their proofs are not pub- lished yet, we admit this as a hypothesis. For archimedean prime, this hypothesis is verified in many cases (but not in general).

We consider the matrix coefficient Φ_ϕ1,v,ϕ

1,v(g₁) =ϕ_1,v, π_1,v(g₁)ϕ_1,vv, g₁ ∈G_1,v, Φ_ϕ0,v,ϕ

0,v(g₀) =ϕ_0,v, π_0,v(g₀)ϕ_0,vv, g₀ ∈G_0,v.

Put

I(ϕ_1,v, ϕ_0,v) =

G0,v

Φ_ϕ1,v,ϕ1,v(ι(g₀))Φ_ϕ0,v,ϕ0,v(g₀)dg₀, α_v(ϕ_1,v, ϕ_0,v) =Δ⁻¹_G

1,vPπ1,v,π0,v(¹₂)⁻¹I(ϕ_1,v, ϕ_0,v).

Proposition 1. If both π_1,v and π_0,v are tempered, then the integral I(ϕ_1,v, ϕ_0,v) is absolutely convergent.

In fact, it is enough to assume that π_1,v and π_0,v are sufficiently

“close” to tempered. We expect that the quantity α_v(ϕ_1,v, ϕ_0,v) is meaningful whenever the irreducible admissible representations π_1,v and π_0,v are unitary and dim Hom_G0,v(π_1,v⊗π˜_0,v,C) = 1.

Proposition 2. Let v be a non-archimedean place. Assume that the conditions (UR1), (UR2) (UR3) and (UR4) hold. Assume, moreover, v 2. Then we have α_v(ϕ_1,v, ϕ_0,v) = 1 in the range of absolute conver- gence of I(ϕ_1,v, ϕ_0,v).

The proof uses the result of [17], which treats only the split case. We can prove an analogue of [17] for unramified quasi-split case.

Conjecture 1.1. Assume that both π_1,v and π_0,v are tempered. Then (1) The integral α_v(ϕ_1,v, ϕ_0,v) is non-negative for any ϕ_1,v ∈ π_1,v

and ϕ_0,v ∈π_0,v.

(2) There existϕ_1,v ∈π_1,v andϕ_0,v ∈π_0,v such thatα_v(ϕ_1,v, ϕ_0,v)>

0 if and only if dim Hom_G0,v(π_1,v, π_0,v) = 1.

Now our global conjecture can be formulated as follows.

Conjecture 1.2. Assume that π₁ ⊗_vπ_1,v and π₀ ⊗_vπ_0,v are tem- pered cuspidal automorphic representation. Then there should be an integer β such that

|ϕ₁|G0, ϕ₀|²

ϕ₁, ϕ₁ϕ₀, ϕ₀ = 2^βΔ_G1C₀P_π1,π0(¹₂)

v∈S

α_v(ϕ_1,v, ϕ_0,v) ϕ_1,v²· ϕ_0,v² holds.

We expect that ifπ₁ andπ₀ are generic cuspidal automorphic repre- sentations, then the same relation holds even when π₁ and π₀ are not (known to be) tempered.

Remark 1. When π₁ and π₀ are tempered, the L-factors L(s, π_1,v,Ad), L(s, π_0,v,Ad), andL(s, π_1,vπ_0,v) should be holomorphic for Re(s)>0.

Therefore our conjecture is equivalent to

|ϕ₁|G0, ϕ₀|²

ϕ₁, ϕ₁ϕ₀, ϕ₀ = 2^βΔ^S_G1C₀P_π^S_1,π0(¹₂)

v∈S

I(ϕ_1,v, ϕ_0,v) ϕ_1,v²· ϕ_0,v².

Here, Δ^S_G1 and P_π^S_1,π0(s) are the partial Euler products. In particular, the definition of theL-factors for bad primes plays no role in this case.

Now we consider the case whenπ₁orπ₀are not necessarily tempered.

We assume thatα_v(ϕ_1,v, ϕ_0,v) is meaningful for anyv. Note that in this case the definition of theL-factors for bad primes are crucial. Note also that in this case theL-factors depends not only onπ_i,v, but also on the Arthur packets whichπ_i,v belong to. By Proposition 2, we may assume α_v(ϕ_1,v, ϕ_0,v) = 1 for v /∈S. In this case, we conjecture as follows.

Conjecture 1.3. Assume that ϕ₁ ∈ π₁ and ϕ₀ ∈ π₀ are square- integrable automorphic forms. Then

(1) The integral ϕ₁|G0, ϕ₀ should be finite.

(2) Assume that dim Hom_G0,v(π_1,v⊗π˜_0,v,C) = 1 for any v. Then exists an integer β such that

|ϕ₁|G0, ϕ₀|²

ϕ₁, ϕ₁ϕ₀, ϕ₀ = 2^βΔ_G1C₀P_π1,π0(¹₂)

v∈S

α_v(ϕ_1,v, ϕ_0,v) ϕ_1,v²· ϕ_0,v². Remark 2. At the conference, we claimed that the integralϕ₁|G0, ϕ₀ is finite, but there was a gap in our proof. Therefore we include this in the conjecture.

2. Relation to the Arthur conjecture

We recall the conjecture of Langlands and Arthur [1] for automorphic representation of reductive algebraic groups. We assume, for simplicity, G is a semi-simple algebraic groups defined over k. We put

Lv =

W_kv ×SU(2) if v is non-archimedean, W_kv if v is archimedean.

A Langlands parameter is a homomorphism ϕ_v :L_v → ^LGwhich satis- fies certain additional conditions. Two Langlands parameters are said to be equivalent if they are conjugate by an element of ˆG. Langlands conjectured that for each Langlands parameter, there exists a finite set Π_ϕv(G) of irreducible admissible representations of G(k_v). The finite set Π_ϕv(G) is called the L-packet of ϕ_v. The set Π(G_v) of equiva- lence classes of irreducible admissible representations of G_v should be decomposed into a disjoint union

Π(G_v) =

ϕv

Π_ϕv(G).

The L-packet Π_ϕv(G) should contain a tempered representation if and only if the Langlands parameterϕ_v has a bounded image, in which case ϕ_v is called tempered. If ϕ_v is tempered, then all members of Π_ϕv(G) should be tempered.

A homomorphism ψ_v : Lv ×SL₂(C) → ^LG is called an Arthur pa- rameter if ψ_v|Lv is a tempered Langlands parameter and if ψ_v|SL2()

is holomorphic. Arthur conjectured that for each Arthur parameter ψ_v, there exists a finite set of unitary representations Π_ψv(G). The set Π_ψv(G) is called the A-packet of ψ_v. A-packets are not necessarily disjoint.

Langlands conjectured that there exists a locally compact group L_k such that the equivalence classes of irreduciblen-dimensional represen- tation of Lk is in one-to-one correspondence with the set of irreducible cuspidal automorphic representations of GL_n(A_k). There should be a homomorphism ι_v :Lv → Lk for eachv.

Arthur conjectured that for each irreducible discrete series automor- phic representations π ⊗vπ_v of G(A) is associated to a homomor- phism

ψ :Lk×SL₂(C)→ ^LG

in such a way that π_v belongs to Π_ψ◦ιv(G) for each v. In this case, ψ is called the Arthur parameter of π. The set Π_ψ(G) of irreducible discrete series automorphic representations with Arthur parameter ψ is called the A-packet of ψ.

It is known that the Arthur parameter ψ : Lk × SL₂(C) → ^LG associated with a discrete series automorphic representation should be elliptic in the sense that Im(ψ) is not contained in any proper Levi subgroup of ^LG. This is the case if and only if Cent_Gˆ(Im(ψ)) is finite.

We put

Sψ = Cent_Gˆ(Im(ψ)).

Now we go back to the situation that G₁ = SO(n + 1) and G₀ = SO(n). Assume thatπ₁ (resp. π₀) is a tempered cuspidal automorphic representation of G₁ (resp. G₀) with Arthur parameter ψ₁ (resp. ψ₀).

Note that in this case the groups S_ψ1 andS_ψ0 are elementary 2-abelian groups.

Conjecture 2.1. Assume that both π₁ and π₀ are tempered cuspidal automorphic representations. Then the constant 2^β in Conjecture 1.2 is equal to 1/(|Sψ1| · |Sψ0|), i.e., we have

|ϕ₁|_G0, ϕ₀|²

ϕ₁, ϕ₁ϕ₀, ϕ₀ = Δ_G1C₀

|Sψ1|·|Sψ0|Pπ1,π0(¹₂)

v∈S

α_v(ϕ_1,v, ϕ_0,v) ϕ_1,v²· ϕ_0,v².

Remark 3. At the conference, we conjectured the the constant 2^β de- pends only n, but it seems the constant 2^β depends on the full Arthur parameter ψ.

3. Examples

• The case when π₁ and π₀ are trivial.

Assume, for simplicity, k is a function filed with Discriminant D_k. Assume that both G₁ and G₀ are quasi-split and that both π₁ and π₀ are trivial. Obviously, we have

|ϕ₁|_G0, ϕ₀|²

ϕ₁, ϕ₁ϕ₀, ϕ₀ = 1.

Consider first the case n = dimV₀ = 2m. Note that we excluded the case when V₀ is the split hyperbolic plane. Let K be the extension of k which corresponds to χ_Q0. Then we have

Δ_G1 = m j=1

ζ(2j),

C₀ =D^{−(2m−1)/2}_K/k D^{−m(2m−1)/2}_k L(m, χ_Q0)⁻¹

m−1

j=1

ζ(2j)⁻¹,

Pπ1,π0(¹₂) =L(1−m, χ) _m−1

j=1 ζ(−2j+ 1) _m

j=1ζ(2j) .

From the functional equationζ(s) =D^(1−2s)/2_k ζ(1−s), and L(s, χ_Q0) = (D_K/kD_k)^(1−2s)/2L(1−s, χ_Q0), the conjecture holds with β= 0 in this case.

Now consider the case n= 2m+ 1. In this case, Δ_G1 =L(m+ 1, χ_Q1)

m j=1

ζ(2j),

C₀ =D^−m(2m+1)/2_k m j=1

ζ(2j),

P_π1,π0(¹₂) =L(m+ 1, χ_Q1)⁻¹ _m

j=1ζ(−2j+ 1) _m

j=1ζ(2j) . In this case, we see the conjecture also holds with β = 0.

• The case n= 2.

This example is due to Waldspurger [25]. PutG₁ = PGL₂. Then an irreducible cuspidal automorphic representationπ ofG₁ can be consid- ered as a representation of GL₂(A) with trivial central character. Let T be an anisotropic torus ofG₁ corresponding to a quadratic extension K/k. Then a character ω of T(A)/T(k) can be regarded as a charac- ter of A^×_K whose restriction to A^×_k is trivial. The base change of π to GL₂(AK) is denoted by Π. Choose a cusp formϕ=⊗_vϕ_v ∈π ⊗_vπ_v. Then among other things, Waldspurger [25] proved that

|

T(k)\T( )ϕ(ι(t))ω⁻¹(t)dt|²

ϕ, ϕ = ζ(2)L(¹₂,Π⊗ω⁻¹) 2L(1, π,Ad)

v∈S

α(ϕ_v, T_v, ω_v) ϕ_v² . (cf. [25], p. 222, Proposition 7). Here,

α(ϕ_v, T_v, ω_v) = L(1, χ_Tv)L(1, π_v,Ad) ζ_v(2)L(¹₂,Π_v⊗ω_v⁻¹)

Tv

Φ_ϕv,ϕv(ι(t_v))ω⁻¹(t_v)dt_v. Note that in this equation, the Haar measure of T(A) is the prod- uct measure

vdt_v. It is the measure such that the total volume of T(k)\T(A) is equal to 2L(1, χ_K/k). Now take the Tamagawa measure for T(A). Then we have C₀ = L(1, χ_K/k)⁻¹. Note that Δ_G1 = ζ(2).

Therefore Waldspurger’s theorem implies

|ϕ|T( ), ω|² ϕ, ϕω, ω = 1

4Δ_G1C₀ L(¹₂,Π⊗ω⁻¹) L(1, π,Ad)L(1, χ_K/k)

v∈S

α_v(ϕ_v, ω_v) ϕ_v² . Note that in this case, we have|Sψ1|=|Sψ0|= 2.

• The Jacquet conjecture.

We consider the case n = 3 andk =Q. We assume that G₁ and G₀ are split overQ. In this case, the Jacquet conjecture is compatible with our conjecture. For example, we consider the case every finite place is unramified. Put G₁ = SO(2,2) and G₀ = SO(2,1) = PGL₂. The local measures dg_0,v is defined byd^×a dn dk, whereg₀ =ank is the standard Iwasawa decomposition d^×a andn is the standard measures onk_v^× and k_v, respectively, and dk is the Haar measure of the standard maximal compact subgroup of PGL₂(k_v) such that the total measure is 1. We denote by Λ(s, π_i) the complete L-function for L(s, π_i) and so on. We put ξ(s) = π^−s/2Γ(2/s)ζ(s). Then Δ_G1 = ξ(2)² = π²/36, and C₀ = 2Vol(SL₂(Z)\H1)⁻¹ = 6/π. Take Hecke eigenforms f_i ∈ S_κi(SL₂(Z)) (i = 1,2,3). We assume κ₁ +κ₂ = κ₃. We denote the lift of f_i to GL₂(A) byfi. Let τ_i be the irreducible automorphic representation of PGL₂(A) generated by fi. Note that f1 ×f2 induces a cusp form on

SO(2,2)(A) and its restriction to SO(2,1) isf1f2. Putπ₁=τ₁τ₂ and π₀ =τ₃. Then

Pπ1,π0(¹₂) = Λ(¹₂, τ₁×τ₂×τ₃) ₃

i=1Λ(1, τ_i,Ad). Then the main theorem of Harris-Kudla [13] implies

Λ(¹₂, τ₁ ×τ₂×τ₃) = 2^2κ3+2|f₁f₂, f₃|². It is well known that

Λ(1, τ_i,Ad) = 2^κif_i, f_i.

Here the left hand side is the completeL-function, and the right hand side is the usual Petersson inner product.

As both the Tamagawa numbers of SO(2,2) and SO(2,1) are equal to 2, we have

|f₁f₂,f₃|²

f1 ×f2,f1×f2f3,f3 = π 3

|f₁f₂, f₃|² ₃

i=1f_i, f_i. After a little calculation, we have

α_∞(ϕ_1,∞×ϕ_2,∞, ϕ_3,∞) = 2.

Putting altogether, we have

|f1f2,f3|² f1×f2,f1 ×f2f3,f3

= 1

4Δ_G1C₀Λ(¹₂, τ₁×τ₂×τ₃) ₃

i=1Λ(1, τ_i,Ad) ·α_∞(ϕ_1,∞×ϕ_2,∞, ϕ_3,∞).

Therefore in this case it seems our conjecture also holds for β = −2.

Note that in this case we have |Sψ1| = |Sψ0| = 2. We can carry out a similar calculation for holomorphic modular forms with square-free levels by using the result of Watson [26].

• Restriction of the Saito-Kurokawa lifting to the diagonal subset.

In this example, n = 4 and k = Q. Let κ be an odd number. Let f ∈ S_2κ(SL₂(Z)) and g ∈ S_κ+1(SL₂(Z)) be normalized Hecke eigen- forms. We denote the Kohnen plus subspace by S_κ+(1/2)⁺ (Γ₀(4)) ⊂ S_κ+(1/2)(Γ₀(4)). Let h ∈ S_κ+(1/2)⁺ (Γ₀(4)) a Hecke eigenform associated tof. LetF ∈S_κ+1(Sp₂(Z)) Saito-Kurokawa lift of h. Then it is shown in Ichino [14] that

Λ(2κ,Sym²(g)⊗f) = 2^κ+1f, f h, h

|F| ₁× 1, g×g|² g, g² .

We interpret this result in terms of automorphic representations. Let ϕ₁ be the automorphic form onG₁(A) = SO(3,2)(A) corresponding to F. Similarly, let ϕ₀ be the automorphic form on G₀(A) = SO(2,2)(A) corresponding to g ×g. As in the last case, let dg_0,v be the Haar measure of G₀(Qv) defined in terms of the Iwasawa decomposition.

Then we have , and

Δ_G1 =ξ(2)ξ(4), C₀ =ξ(2)⁻², ϕ₁, ϕ₁= F,F

ξ(2)ξ(4), ϕ₀, ϕ₀=g, g²

2ξ(2)²,

|ϕ₁|G0, ϕ₀|²

ϕ₁, ϕ₁ϕ₀, ϕ₀ = ξ(4) 2ξ(2)

|F| ₁× 1, g×g|² g, g .

Let τ and σ be the automorphic representations of GL₂(A) generated by f and g, respectively. Then we have

Λ(2κ,Sym²(g)⊗f) =Λ(¹₂,Ad(σ)τ), f, f=2^−2κΛ(1, τ,Ad) F,F

h, h =2^κ−2

π ξ(2)Λ(³₂, τ) It follows that

|ϕ₁|_G0, ϕ₀|²

ϕ₁, ϕ₁ϕ₀, ϕ₀ =π· ξ(4)

ξ(2) · Λ(¹₂,Ad(σ)τ) ξ(2)Λ(³₂, τ)Λ(1, τ,Ad). By easy calculation, we have

P_π1,π0(s) = Λ(s− ¹₂, σ,Ad)Λ(s,Ad(σ)τ)

ξ(s+ ³₂)Λ(s+ 1, τ)Λ(s+¹₂, σ,Ad)Λ(s+ ¹₂, τ,Ad). It follows that

P_π1,π0(¹₂) = Λ(0, σ,Ad)Λ(¹₂,Ad(σ)τ) ξ(2)Λ(³₂, τ)Λ(1, σ,Ad)Λ(1, τ,Ad)

= Λ(¹₂,Ad(σ)τ) ξ(2)Λ(³₂, τ)Λ(1, τ,Ad)

by the functional equation. After a little calculation, we can show that α_∞(ϕ_1,∞, ϕ_0,∞) = 2π. Putting together, we have

|ϕ₁|G0, ϕ₀|² ϕ₁, ϕ₁ϕ₀, ϕ₀ = 1

2Δ_G1C₀Pπ1,π0(¹₂)· α_∞(ϕ_1,∞, ϕ_0,∞) ϕ_1,∞²· ϕ_0,∞².

Therefore in this case our conjecture seems to hold withβ =−1.

• Restriction of the hermitian Maass lift.

Now we discuss the case n = 5 and k = Q. We assume that G₁ is quasi-split, but not split. We denote the splitting field ofG₁ byK. Let

−D and χ be the discriminant and the associated Dirichlet character for K/Q. Note that G₁ = SO(4,2)SU(2,2)/{±1}.

We putG₀ = SO(3,2). Note that G₀ is isogenous to Sp₂. Letκ >0 be an even integer andf ∈S_2κ−2(SL₂(Z)) be a normalized Hecke eigen- form. We denote the irreducible cuspidal automorphic representation of SL₂(A) generated byf byσ. Then we haveL(s, σ) =L(s+κ−³₂, f).

Let

h=

N>0 N≡0,1(4)

c(N)q^N ∈S_κ−(1/2)⁺ (Γ₀(4))

is a Hecke eigenform corresponding to f, and F the Saito-Kurokawa lift of h.

It is well-known (cf. [18]) that F,F

h, h = 2^κ−3π⁻¹ξ(2)Λ(³₂, σ).

For simplicity, we assume that c(D)= 0.

Now let Γ_K = SU(2,2)(Q)∩GL₄(O_K) be the special hermitian modu- lar group. Let g ∈S_κ−1(Γ₀(D), χ) be a primitive form andG ∈S_κ(Γ_K) be the hermitian Maass lift of g. Thus G is a modular form on the hermitian upper half space

H₂ ={Z ∈M₂(C)|₂^√1₋₁(Z− ^tZ)¯ >0} of degree 2 with respect to Γ_K. We may assume that G = 0.

We denote the irreducible cuspidal automorphic representation of GL₂(A) generated byg (resp.f) byτ (resp.σ). Letπ₁ (resp.π₀) be the irreducible cuspidal automorphic representation ofG₁(A) (resp.G₀(A)) generated by G (resp. F). Then both π₁ and π₀ are non-tempered. It is easy to check that

L(s, π₁) = Λ(s,Sym²(τ))ξ(s+ 1)ξ(s)ξ(s−1), L(s, π₀) = Λ(s, σ)ξ(s+ ¹₂)ξ(s− ¹₂),

and

P_π1,π0(s) = Λ(s,Sym²(τ)σ)Λ(s−1, σ)ξ(s− ³₂)

Λ(s+³₂,Sym²(τ))Λ(s+ ¹₂, τ,Ad)Λ(s+ ¹₂, σ,Ad)ξ(s+ ³₂).

Observe that Pπ1,π0

₁

2

= Λ(¹₂,Sym²(τ)σ)Λ(−¹₂, σ)ξ(−1) Λ(2,Sym²(τ))Λ(1, τ,Ad)Λ(1, σ,Ad)ξ(2)

=− Λ(¹₂,Sym²(τ)σ)Λ(³₂, σ) Λ(2,Sym²(τ))Λ(1, τ,Ad)Λ(1, σ,Ad) The main theorem of Ichino and Ikeda [15] says

|c(D)|²|G| ₂,F|²

F,F² = 2^−4κ+2D^2κ−3Λ(¹₂, τ ×τ ×σ) f, f² By using the Kohnen-Zagier formula [19]

|c(D)|²f, f

h, h = 2^κ−2D^κ−(3/2)Λ(¹₂, σ, χ), we have

|G| ₂,F|²

F,F² = 2^−5κ+4D^κ−(3/2)Λ(¹₂,Sym²(τ)σ) f, fh, h . One can show that

G,G= 2^−2κ−3D^κ+1π⁻²(4, w_K)ξ(2)Λ(2,Sym²(τ))Λ(1, f,Ad).

It is well-known that

f, f=2^−2κ+2Λ(1, σ,Ad).

The volumes of the fundamental domains are Vol(Sp₂(Z)\H2) =2ξ(2)ξ(4),

Vol(Γ⁽²⁾_K \H2) =2⁻¹D^5/2(4, w_K)ξ(2)Λ(3, χ)ξ(4).

One can showα_∞=−2π. Therefore in this case, our conjecture holds with β =−1.

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