A cohomological interpretation of archimedean zeta integrals for GL

A cohomological interpretation of archimedean zeta integrals for GL

× GL

Kenichi Namikawa (joint with Takashi Hara)

Faculty of Mathematics, Kyushu university

25th January, 2021

Contents

1 Motivations

2 Modular symbol method

Raghuram-Shahidi’s Whittaker periods Main theorem

3 Strategy

Sketch of proof of main theorem Sketch of proof of corollaries

4 Further expectations

1 Motivations

2 Modular symbol method

Raghuram-Shahidi’s Whittaker periods Main theorem

3 Strategy

Sketch of proof of main theorem Sketch of proof of corollaries

4 Further expectations

3 / 30

Motivations:

Special values of L-functions

We want to generalize

∑

0≤j≤k−2

(k−2 j

)√

−1^−j−1Γ_C(j+ 1)L(j+ 1, f)

Ω^±_f X^k−2−jY^j ∈Kf[X, Y], where

f =

∑∞ n=1

a(n, f)qⁿ : elliptic newform of weightk.

L(s, f) =

∑∞ n=1

a(n, f) n^s .

K_f =Q({a(n, f) | n∈N}) : Hecke field of f. K_f/Q: finite.

±= (−1)^j,0≤j≤k−2.

Ω^±_f ∈C^×/K_f^× : canonical periods for f.

Motivations:

Special values of L-functions

Remark

1 The polynomial can be understood as the Mellin transform of the image δ(f) =f(z)(X−zY)^k−2dz off via the Eichler-Shimura map over periods.

2 Assuming p is odd, we can refines the definitions of periodsΩ^±_f, depending only on f, pand p-adic units, so that we can discuss the integrality on L-values.

3 The binomial coefficients in the formula is important for the proof of Kummer congruences of L-values, i. e. a construction of p-adic L-function (Mazur-Tate-Teitelbaum).

4 ∃ generalizations toGL_n+1×GL_n due to

• (Algebraicity)Mazur-Kazhdan-Schmidt, Kasten-Schmidt, Raghuram-Shahidi, Raghuram

• (p-adic L-functions)Mazur-Kazhdan-Schmidt, Januszewski 5 / 30

Motivations:

Coates-Perrin-Riou’s ocnjecture

p : prime number. Fix C∼=Cp.

M: pure motives over Qof good ordinary at p.

Suppose that Tate motive is not a direct summand of M.

K : sufficiently large finite extension ofQp,O: the ring of integer of K.

µp^∞ ⊂C: the group of p-power roots of unity.

ϕ:Q^×\Q^×_A→C^×: algebraic Hecke character satisfying

1 the conductor c(ϕ) of ϕisp-power;

2 L(0,M(ϕ))is a critical value (M(ϕ)is the twist of Mbyϕ).

ϕb: Gal(Q(µp^∞))→C^×_p : thep-adic avatar ofϕ (e.g. εcyc=| · |dA).

Motivations:

Coates-Perrin-Riou’s conjecture

Conjecture (Coates-Perrin-Riou Existence of p-adic L-functions ) There should exist Lp(M)∈Λ :=O[[Gal(Q(µp^∞)/Q)]] such that for each algebraic Hecke character ϕ:Q^×\Q^×_A→C^× satisfying the proceeding conditions, we have

ϕ(bLp(M)) =E_∞(M(ϕ))Ep(M(ϕ))L(0,M(ϕ)) Ω(M) , where

E_∗(M(ϕ))is the modified Euler factor at∗.

Ω(M) is the period ofM, which is a product of Deligne’sc⁺(M) and a power of π.

Remark

We sometimes use a substitute of Deligne’s periods for constructions of p-adicL-functions. (e.g. canonical periods.)

7 / 30

Motivations:

Cohomological cusp. autom. rep. of GL

π⁽ⁿ⁾ : coh. cusp. autom. rep. of GLn(QA).

W_R=C^×⊔(C^×j) : Weil group ofR.

Define1 (resp. 2)-dim. rep ϕ^δ_ν (resp. ϕ_ν,l) ofW_R to be

•ϕ^δ_ν(z) = (zz^c)^ν, ϕ^δ_ν(j) = (−1)^δ,

•^ϕν,l(z) = (zz^c)^ν (

(z^c/z)^2l 0 0 (z/z^c)^2l

)

, ϕ_ν,l(j) =

(0 (−1)^l

1 0

) . Then the Langlands parameter ofπ⁽ⁿ⁾_∞ is given by





⊕m i=1ϕ

ν⁽ⁿ⁾,l⁽ⁿ⁾_i (n= 2m:even) ϕ^δ

ν⁽ⁿ⁾⊕⊕_m

i=1ϕ

ν⁽ⁿ⁾,l⁽ⁿ⁾_i (n= 2m+ 1 :odd) withl⁽ⁿ⁾₁ >· · ·> l_m⁽ⁿ⁾ Normalize ν⁽ⁿ⁾= (−l⁽ⁿ⁾₁ +n−1)/2.

Suppose that (π⁽ⁿ⁺¹⁾, π⁽ⁿ⁾) satisfies the followinginterlace condition:

l₁⁽ⁿ⁺¹⁾ > l⁽ⁿ⁾₁ > l₂⁽ⁿ⁺¹⁾ > l⁽ⁿ⁾₂ >· · ·> l⁽ⁿ⁺¹⁾_n > l⁽ⁿ⁾_n > l⁽ⁿ⁺¹⁾_n+1 . M[π⁽ⁿ⁾]: (conjectural) motive attached to π⁽ⁿ⁾ (Clozel).

L(s,M[π⁽ⁿ⁾]) =L(s−ⁿ⁻₂¹, π⁽ⁿ⁾).

M=M[π⁽ⁿ⁺¹⁾]× M[π⁽ⁿ⁾].

Motivations:

p-adic Rankin-Selberg L-functions

ϕ:Q^×\Q^×_A→C^× : algebraic Hecke character as above.

Write the infty type of ϕasz_∞^m (m∈Z).

Theorem (Januszewski p-adic L-functions for GL_n+1×GL_n ) For m∈Z as above, there exists Lp^(m)(M)∈ O[[Gal(Q(µp^∞)/Q)]] and a

period Ω(M, m) such that, for eachϕ̸=| · |^m_A, we have ϕ(bL_p^(m)(M)) =Ep(M(ϕ))L^(∞)(0,M(ϕ))

Ω(M, m) ,

Remark

1 The periodsΩ(M, m)depend onm. It is an inverse of a product of a certain weighted sum of archimedean local integrals and

Raghuram-Shahidi’s Whittaker periods.

2 It seems difficult to compareLp^(m)(M)’s for different m’s. (Kummer congruence. Januszewski (arXiv:1708.02616))

3 Ifϕ=| · |^m_A, the interpolation formula is not yet known.

=⇒ We want to refines the above interpolation formula.

9 / 30

Motivations:

1st consequence of Main Theorem

π⁽³⁾_∞ ∼= Ind^GL_P ^3(R)

2,1(R)(Dν,l3 ⊠χν,δ), π_∞⁽²⁾∼=Dν,l2.

(χν,δ(u) = sgn(u)^δ|u|^ν, Dν,l(t12) =t^2ν(t∈R>0), Dν,l|SL2(R)=D⁺_l ⊕D⁻_l .) (Interlace condition) 0< l₂< l₃.

We normalizeν₂ =−l₂ 2 +1

2, ν₃=−l₃ 2 + 1.

M=M[π⁽³⁾]× M[π⁽²⁾]has the pure weightl2+l3. L(m,M) is critical if and only if

{l3

2 + 1≤m≤ ^l₂³ +l2, (l2 ≤ ^l₂³), l2+ 1≤m≤l3, (^l₂³ < l2 < l3).

Corollary

Suppose that n= 2. The∃ a periodΩ^±(π⁽³⁾×π⁽²⁾)∈C^× so that for each ϕ̸=| · |^m_A as above,

ϕ(bL_p^(m)(M)) =E_∞(M(ϕ))Ep(M(ϕ)) L(0,M(ϕ)) Ω^±(π⁽³⁾×π⁽²⁾), where ±1 = (−1)^m+δ+l23

Motivations:

2nd consequence of Main Theorem

The period of Rankin-Selberg L-function is a product of (a refinement of) Raghuram-Shahidi’s Whittaker periods:

Ω^±(π⁽³⁾×π⁽²⁾) = Ω_π(3)×Ω^±_π(2). Here Ω^±

π⁽²⁾ is the canonical periods of π⁽²⁾ from its definition.

A priori Ω_π(3) has no relation with the motives.

2nd corollary gives a motivic back ground on Ω_π(3): Corollary

Suppose that Deligne’s conjecture forπ⁽³⁾ (existence of motives, algebraicity of critical values).

Then we have Ω_π(3) = (2π√

−1)^l23c⁺(M)c⁻(M).

The proofs are done by a PRECISE formula for L-values.

11 / 30

1 Motivations

2 Modular symbol method

Raghuram-Shahidi’s Whittaker periods Main theorem

3 Strategy

Sketch of proof of main theorem Sketch of proof of corollaries

4 Further expectations

Modular symbol method:Raghuram-Shahidi’s Whittaker periods

Field of rationality

(Field of rationality)

Q(π⁽ⁿ⁾) : field of rationality ofπ⁽ⁿ⁾, i. e., the fixed subfield of Cunder {σ ∈Aut(C) | ^σπ⁽ⁿ⁾∼=π⁽ⁿ⁾}. (Q(π⁽ⁿ⁾)/Q: finite).

E :=Q(π⁽ⁿ⁺¹⁾, π⁽ⁿ⁾).

(Rationality on (g, K) cohomology)(g=gl_n(R), K_n=R^×SO_n(R)) Y_K⁽ⁿ⁾_n = GLn(Q)\GLn(QA)/R^×SOn(R)Kn

V_ξ^∨ : contragredient of rep. ofGLn(R) of highest wtξ.

L^∨_ξ : loc. sys. on Y_K⁽ⁿ⁾

n assoc. withV_ξ^∨.

π⁽ⁿ⁾ : coh. ⇐⇒ ∃ξ s.t. H^∗(g, K;π⁽ⁿ⁾⊗V_ξ^∨)̸= 0.

Note that• H^∗(g, K;π⁽ⁿ⁾⊗V_ξ^∨)⊂H^∗(Y_K⁽ⁿ⁾

n,L^∨_ξ).

• ^bn= [n²

4 ]≤ ∗ ≤tn=bn+ [n−1

2 ]. (b3= 2, t3= 3).

V_ξ^∨ : defined overQ(π⁽ⁿ⁾)⇝ H^∗(Y_K⁽ⁿ⁾_n,L^∨_ξ)∼=H^∗(Y_K⁽ⁿ⁾_n,L^∨_ξ)_Q(π(n))⊗C.

DefineH^bn(g, K;π⁽ⁿ⁾⊗V_ξ^∨)_Q(π(n)) to be H^bn(g, K;π⁽ⁿ⁾⊗V_ξ^∨)∩H^∗(Y_K⁽ⁿ⁾

n,L^∨_ξ)_Q(π(n)).

13 / 30

Modular symbol method:Raghuram-Shahidi’s Whittaker periods

Whittaker periods

(Whittaker functions)

ψ:Q\QA−→C^× : add. char. ψ_∞(z) = exp(2π√

−1z).

W(π⁽ⁿ⁾, ψ) =W(π_∞⁽ⁿ⁾, ψ_∞)⊗ W(π_fin⁽ⁿ⁾, ψ_fin).

σ ∈Aut(C) ←→uσ ∈Zb^× s.t. σ(ψ(x)) =ψ(uσx) for ∀x∈Q^×_A,fin. DefineT_σ :W(π_fin⁽ⁿ⁾, ψ_fin)→ W(^σπ_fin⁽ⁿ⁾, ψ_fin) by

T_σW(g) =σ (

W(diag(u⁻⁽ⁿ⁻¹⁾_σ , . . . , u⁻¹_σ ,1)g) )

. (Raghuram-Shahidi’s Whittaker periods)

Fix w⁽ⁿ⁾_∞ =w(π_∞⁽ⁿ⁾,±)∈H^bn(g, K;H_π(n),K⊗V_ξ^∨)[±](1-dim.).

⇝ W(π⁽ⁿ⁾_fin, ψ_fin)→H^bn(g, K;H_π(n),K⊗V_ξ^∨)[±]⊗π⁽ⁿ⁾_fin →H^bn(Y_K⁽ⁿ⁾

nL^∨ξ).

DefineWhittaker periods p^bn(π_fin⁽ⁿ⁾,w⁽ⁿ⁾_∞ ,±)∈C^× to be Image

(W(π_fin⁽ⁿ⁾, ψ_fin)^{Aut(C/Q(π(n)))} )

=p^bn(π_fin⁽ⁿ⁾,w⁽ⁿ⁾_∞,±) (

H^bn(g, K;π⁽ⁿ⁾⊗V_ξ^∨)_Q(π(n))[±] )

. Raghuram-Shahidi’s Whittaker periods depend on the choice of w⁽ⁿ⁾_∞.

Modular symbol method:Raghuram-Shahidi’s Whittaker periods

Cup product pairing

Y_K⁽ⁿ⁾_n = GL_n(Q)\GL_n(QA)/SO_n(R)Kn pn

−→Y_K⁽ⁿ⁾

n

ι: GL_n−→GL_n+1;g7−→

(g 0 0 1 )

. (Branching rule)

H^bn+1(Y_K⁽ⁿ⁾_n, ι^∗L^∨_ξ(n+1))∋ι^∗p^∗_n+1η⁽ⁿ⁺¹⁾_π(n+1)

∇^m

7−→ ∇^mι^∗p^∗_n+1η_π⁽ⁿ⁺¹⁾(n+1)∈H^bn+1(Y_K⁽ⁿ⁾_n, ι^∗Lξ⁽ⁿ⁾(det^m))

(Cup product)

∪:H^bn+1(Y_K⁽ⁿ⁾_n, ι^∗Lξ⁽ⁿ⁾(det^m))×H^bn(Y_K⁽ⁿ⁾_n, ι^∗L^∨_ξ(n))−→H^bn+1+bn(Y_K⁽ⁿ⁾_n,det^m)

(Numerical coincidence) b_n+1+b_n= dimY_K⁽ⁿ⁾_n

=⇒I(m, π⁽ⁿ⁺¹⁾, π⁽ⁿ⁾) :=∇^mι^∗p^∗_n+1η⁽ⁿ⁺¹⁾_π(n+1)∪p^∗_nη⁽ⁿ⁾_π(n) ∈E is essentially given by the zeta-integral due to Jacquet-Piatetski-Shapiro-Shalika.

( essentially = arch. local integral includes the information of local systems.)

15 / 30

Modular symbol method:Raghuram-Shahidi’s Whittaker periods

Algebraicity of Rankin-Selberg L-functions

Theorem (Raghuram)

I(m, π⁽ⁿ⁺¹⁾, π⁽ⁿ⁾) =I_∞(m, π⁽ⁿ⁺¹⁾_∞ , π⁽ⁿ⁾_∞)

| {z }

sum of arch. loc. int.

× L^fin(m,M(π⁽ⁿ⁺¹⁾×π⁽ⁿ⁾))

p^bn+1(π_fin⁽ⁿ⁺¹⁾,w_∞⁽ⁿ⁺¹⁾,±)p^bn(π⁽ⁿ⁾_fin,w⁽ⁿ⁾_∞,±)∈E

Remark

1 The study of the algebraicity is reduced to the study of I_∞. Sun (JAMS 2017) proved the non-vanishing of I_∞.

2 Combining with Deligne’s conjecture, I_∞(m, π_∞⁽ⁿ⁺¹⁾, π⁽ⁿ⁾_∞ )

p^bn+1(π_fin⁽ⁿ⁺¹⁾,w_∞⁽ⁿ⁺¹⁾,±)p^bn(π⁽ⁿ⁾_fin,w_∞⁽ⁿ⁾,±) ∼_Q^× Γ(m,M(π⁽ⁿ⁺¹⁾×π⁽ⁿ⁾)) c⁺(M(π⁽ⁿ⁺¹⁾×π⁽ⁿ⁾)) However, ̸ ∃motivic explanation of eachI_∞and p^bn(π_fin⁽ⁿ⁾,w⁽ⁿ⁾_∞ ,±).

3 It’s difficult to study the relation between I_∞(m, π⁽ⁿ⁺¹⁾_∞ , π_∞⁽ⁿ⁾)’s for different m’s. This is one of difficulties for the Kummer congruences for p-adic Rankin-Selberg L-functions.

Modular symbol method:Main theorem

Statement

Write Ω^±_π(n) =p^bn(π⁽ⁿ⁾_fin,w_∞⁽ⁿ⁾,±) for a suitable w_∞⁽ⁿ⁾. Main Theorem ( Hara-N. (arXiv:2012.13213 ) ) Suppose that (−1)^m+δ+l23 =±1. Then

I(m, π⁽³⁾, π⁽²⁾) = (−1)^δ√

−1

l3

2−m+1( l₃ 2 −1 m−^l₂³ −1

)( l₃ 2 −1

l₃

2 +l2−m

)L(m,M) Ω_π(3)Ω^±

π⁽²⁾

.

If(−1)^m+δ+l23 ̸=±1holds, we have I(m, π⁽³⁾, π⁽²⁾) = 0.

Remark

S. Y. Chen (arXiv:2012.00625) independently proved I(m, π⁽³⁾, π⁽²⁾)∼_Q^×√

−1

l3

2−m+1L(m,M) Ω_π(3)Ω^±

π⁽²⁾

.

17 / 30

1 Motivations

2 Modular symbol method

Raghuram-Shahidi’s Whittaker periods Main theorem

3 Strategy

Sketch of proof of main theorem Sketch of proof of corollaries

4 Further expectations

Strategy:Sketch of proof of main theorem

Cohomological cusp. autom. forms on GL

Λ₃={λ= (λ, δ)|λ∈Z, λ≥0, δ∈Z/2Z}

Define an action τ_λ⁽³⁾ ofu∈O₃(R)on P ∈C[z₁, z₂, z₃]_λ by τ_λ⁽³⁾(u)P(z₁, z₂, z₃) = (detu)^δP((z₁, z₂, z₃)u).

Vλ:=

{(z₁²+z²₂+z₃²)C[z₁, z₂, z₃]_λ−2, (λ≥2),

0, (otherwise).

V_λ⁽³⁾:=C[z₁, z₂, z₃]_λ/Vλ

{

v_±^(3),λ_µ := (±z₁+√

−1z₂)^µz^λ₃^−µ }

0≤µ≤λ : basis of V_λ⁽³⁾. The minimal O₃(R)-type isτ_(l3+1,δ). (λ=l₃+ 1.) V_λ⁽³⁾,→π;v_±^(3),λ_µ 7−→f_±^λ_µ∈π(0≤µ≤l3+ 1).

f =( f_l^λ

3+1 f_l^λ

3 . . . f₋^λ_l

3−1

) : cusp form on GL₃(QA)

19 / 30

Strategy:Sketch of proof of main theorem

Whittaker functions on GL

and GL

(Normalization of f)

W_µ^λ : Whittaker function attached tof_µ^λ (−l3−1≤µ≤l3+ 1) φ⁽³⁾₊ :V_λ⁽³⁾ → W(π⁽³⁾_∞, ψ_∞): R^×SO₃(R)-homomorphism satisfying the following explicit formula on the radial parts of Whittaker functions:

(za :=z₁^a1z₂^a2z₃^a3 ∈V_λ⁽³⁾) φ⁽³⁾₊ (z_a)(diag(y₁y₂y₃, y₂y₃, y₃))

= (−1)^a1√

−1^a2y1y2(y2y3)^3ν3 (4π√

−1)² ×

∫

t2

∫

t1

Γ_C(t₁+ν₃+ ^λ3₂⁻¹)Γ_R(t₂+ν₃+a₁) Γ_R(t₁+t₂+a₁+a₃)

×Γ_C (

t₂−ν₃+λ₃−1 2

)

Γ_R(t₂−ν₃+a₃)y⁻₁^t1y⁻₂^t2dt₁dt₂. (Miyazaki (Manus. Math., 2009))

We normalizef_µ^λ so that the radial part ofW_µ^λis described as above.

Strategy:Sketch of proof of main theorem

Description of (g, K ) cohomology

H_π(3),K3 : (gl₃(R), K₃)-module attached to π_∞⁽³⁾. (K₃ =R^×SO₃(R).) Hⁱ(gl₃(R), K₃;H_π(3),K3⊗V_ξ^∨)∼= (H_π(3),K3⊗L⁽³⁾(w_λ;C)⊗∧iP_3,C^∗ )^SO3(R) A : integral domain of characteristic 0.

A[X, Y, Z;A, B, C]_w of homogenous polynomials of degreew in each variables X, Y, Z andA, B, C.

Define an action ϱ⁽³⁾w of GL3(A) onA[X, Y, Z;A, B, C]w as follows:

ϱ⁽³⁾w (g)P(X, Y, Z;A, B, C) = (detg)^wP((X, Y, Z)g; (A, B, C)^tg⁻¹).

ι_w :=_∂X∂A^∂2 +_{∂Y ∂B}^∂2 +_∂Z∂C^∂2 :A[X, Y, Z;A, B, C]_w→ A[X, Y, Z;A, B, C]_w−1. L⁽³⁾(w;A) := Kerιw.

L⁽³⁾(w;C) has the highest weight(2w, w,0).

V_ξ^∨=L⁽³⁾(wλ;C) for wλ= ^l₂³ −1.

21 / 30

Strategy:Sketch of proof of main theorem

Eichler-Shimura map for GL

P_3,^∗_C)^SO3(R), (i= 2,3) (element in L⁽³⁾(wλ;C))

(Xz1+Y z2+Zz3)^wλ ⊗(Az1+Bz2+Cz3)^wλ (

v^(3,δ)₃ . . . v₋^(3,δ)₃ )

=(

v_λ^λ₃ v_λ^λ₃₋₁ . . . v^λ_−λ3)

P(X, Y, Z, A, B, C)

Then P(X, Y, Z, A, B, C)∈M_2λ3+1,7(C[X, Y, Z;A, B, C]wλ,wλ).

(element in ∧_i P_3,^∗_C)

P3,C: the cpxif. of the complement of the fixed part of the Cartan involution ofgl₃(R).

V_(3,0)⁽³⁾ −→∧_i

P_3,^∗_C;v(3),((3,0))

±µ 7−→ωⁱ_±µ (0≤µ≤3).

Fix a coordinateg=



y1(g)y2(g) y1(g)x2(g) x3(g) 0 y₁(g) x₁(g)

0 0 1



∈Y_K⁽³⁾

3 .

Strategy:Sketch of proof of main theorem

Eichler-Shimura map for GL

ς₁ := dy₁,ς₂ := dy₂,ς₃:= dx₁, ς₄:= dx₂, ς₅ := dx₃, ς_j,j′ :=ς_j∧ς_j′. ς² =(

ς_2,1 ς_2,0 ς_2,−1 ς_2,−2 ς_1,0 ς_1,−1 ς_1,−2 ς_0,−1 ς_0,−2 ς_−1,−2) (ω²₃(g) . . . ω²₋₃(g))

=ς²Q², where

Q² =

























0 _2y¹

1y2 0 0 0 −2y¹1y2 0

0 0 ^x2+

√−1y2

2y²₁y2 0 ^x2−

√−1y2

2y₁²y2 0 0

0 −2y^√−1y¹2 0 0 0 −2y^√−1y¹2 0

0 0 −_2y¹²

1y2 0 −_2y¹²

1y2 0 0

x2+√

−1y2

8y1y₂² 0 −^x2−_8y^5√−1y2

1y²₂ 0 −^x2+5_8y^√−1y2

1y₂² 0 ^x2−

√−1y2 8y1y²₂

0 −^√_4y⁻¹2 2

0

√−1

2y²₂ 0 −^√_4y⁻¹2 2

0

−_8y¹_1y²

2

0 _8y¹

1y²₂ 0 _8y¹

1y₂² 0 −_8y¹_1y²

√ 2

−1x2−y2

8y1y₂² 0 −^3(√−_8y^1x_1y²2^+y2) 2

0 ³⁽

√−1x2−y2)

8y1y₂² 0 −^√−1x_8y1²_y^+y2 ² 2

0 0 0

√−1

y₁²y2 0 0 0

√−1

8y1y₂² 0 −³_8y^√₁⁻_y2¹ 2

0 ³

√−1

8y1y₂² 0 −_8y^√−_1y¹2 2

























.

We have a similar formula for 3-forms ω³_±µ. 23 / 30

Strategy:Sketch of proof of main theorem

Eichler-Shimura map for GL

Then the image of f via the Eichler-Shimura map for GL₃ is described as

δ^(3),i(f) :=( f_λ^λ

3 f_λ^λ

3−1 . . . f₋^λ_λ

3

)P(X, Y, Z, A, B, C)



 ωⁱ₃

... ωⁱ₋₃



,

which gives a class of H_cuspⁱ (Y_K⁽³⁾

3 ,L⁽³⁾(w_λ;C)).

This is an analogue of Eichler-Shimura map for GL₂ : δ(f) =f(z)(X−zY)^k−2dz.

Strategy:Sketch of proof of main theorem

Branching rule

A : integral domain of characteristic 0s.t. w∈ A^×.

For each 0≤k, l≤w= ^l₂³ −1,0≤l≤w₁⁻ andP ∈L⁽³⁾(w;A), set (∇k,lP)(X, Y) = 1

k!l!

∂^k+lP

∂Z^k∂C^l(X, Y,0 ;−Y, X,0).

Then ∇k,l defines the followingGL₂(A)-equivariant map:

∇= (∇k,l)_0≤k,l≤w:L⁽³⁾(w;A)|GL2(A)−→

⊕w k,l=0

L⁽²⁾(2w−k−l, l;A).

Let (k, l) = (l3−m, m−l2−1),nm= (l2−1, m−l2−1).

∇^nm :=∇k,l:L⁽³⁾(w;A)|GL2(A)−→L⁽²⁾(nm;A).

Combining these descriptions with the explicit formula of the archimedean zeta integral for GL3×GL2 due to Hirano-Ishii-Miyazaki (to appear in Mem. of AMS), we obtain the formula for I(m, π⁽³⁾×π⁽²⁾).

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Strategy:Sketch of proof of corollaries

Yoshida’s periods invariants for GL

(The refinement of Januszewski’s interpolation formula) Immediate from the main theorem.

(Period relation)

Main theorem and Deligne’s conjecture imply ^Γ(m,M) Ω_π(3)Ω^±

π⁽²⁾

∼_Q^× 1 c⁺(M(m)). By the interlace condition (0< l₂< l₃) and Yoshida’s description of periods of tensor product of motives, we observe

c^±(M)∼_Q^× (2π√

−1)^l23c⁺(M[π⁽³⁾])c⁻(M[π⁽³⁾])c^±(M[π⁽²⁾]) The modular symbol method for GL2 and Deligne’s conjecture imply

Ω^±

π⁽²⁾ ∼_Q^× c^±(M[π⁽²⁾]).

Combining these, we obtain Ω_π(3)∼_Q^×(2π√

−1)

l3

2c⁺(M[π⁽³⁾])c⁻(M[π⁽³⁾]).

1 Motivations

2 Modular symbol method

Raghuram-Shahidi’s Whittaker periods Main theorem

3 Strategy

Sketch of proof of main theorem Sketch of proof of corollaries

4 Further expectations

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Further expectations:Sketch of proof of corollaries

Yoshida’s periods invariants for GL

Expectation (Ishii-Miyazaki (arXiv:2006.04095, base field is tot. imag.)) I_∞(m, π⁽ⁿ⁺¹⁾, π⁽ⁿ⁾)∼_Q^× L_∞(m,M).

Yoshida (AJM, 2002) introduces certain period invariants c_p(M)

(p= 0, . . . , t) of M. (t:=♯of jumps to of the Hodge filt. c₀(M) =δ(M).) The interlace condition and Deligne’s conjecture yields that

Expectation

Ω^±

π⁽ⁿ⁾∼_Q^×(2π√

−1)^∑ni=1

l(n) 1 −l(n)

i 2 (n−i)

⌊∏ⁿ₂⌋ p=1

c_p(M[π⁽ⁿ⁾])×

{1, (n:odd), c^±(M[π⁽ⁿ⁾]), (n:even).

This explains the motivic back ground of Raghuram-Shahidi’s Whittaker periods for GL_n.