Let F = f1⊗f2⊗f3, where f1, f2, and f3 are normalized Drinfeld type newforms for the congruence subgroupΓ0(N0)ofGL2(Fq[t

CENTRAL CRITICAL VALUES

FU-TSUN WEI

Abstract. The aim of this article is to study the special values of Rankin triple product L- functions associated to Drinfeld type newforms of equal square-free levels. The functional equa- tion of theseL-functions is deduced from a Garrett-type integral representation and the functional equation of Eisenstein series on the group of similitudes of a symplectic vector space of dimension 6. When the associated root number is positive, we present a function field analogue of Gross- Kudla formula for the central critical value. This formula is then applied to the non-vanishing of L-functions coming from elliptic curves over function fields.

Keywords: Function field, Automorphic form, Rankin triple product, Special value ofL- function

MSC (2000): 11R58, 11F41, 11F67

Introduction

The purpose of this manuscript is to explore the Rankin triple product L-functions associated to automorphic cusp forms of Drinfeld type. These automorphic forms can be viewed as function field analogue of classical weight2 modular forms (cf. [7] and [25]). Let N0 be a square-free ideal of Fq[t] (with q odd). Let F = f1⊗f2⊗f3, where f1, f2, and f3 are normalized Drinfeld type newforms for the congruence subgroupΓ0(N0)ofGL2(Fq[t]). The Rankin triple productL-function L(F, s)is defined by a convergent Euler product in the half-planeRe(s)>5/2 (cf. §2). Following idea of Piatetski-Shapiro and Rallis [16], we have a Garrett-type integral representation ofL(F, s), i.e. L(F, s) can be expressed essentially by an integral of F times the Eisenstein series associated to a suitable section in the Siegel-parabolic induced representation of GSp₃. From the functional equation of Eisenstein series, we get (cf. Theorem 2.1):

L(F, s) =ε·q(2−s)·(5 degN0−11)·L(F,4−s).

Here ε = ε(F) = −Q

primeP|N0εP is called the root number, where εP = −cP(f1)cP(f2)cP(f3) (∈ {±1})andcP(fi)is the eigenvalue of the Hecke operatorTP associated tofi.

The root numberεdetermines the parity of the vanishing order at the central critical points= 2.

It is natural to study first the central critical valueL(F,2)when the root numberεis positive. The main result of this article is the following analogue of Gross-Kudla formula:

Theorem 0.1. Let N0 be a square-free ideal of Fq[t] and let γN₀ be the number of prime factors of N0. LetF =f1⊗f2⊗f3, where fi is a normalized Drinfeld type newform forΓ0(N0)for each i. Suppose the root number ε(F) = 1. Let N₀⁻ =Q

ε_P=−1P andN₀⁺ =N0/N₀⁻. Then the central critical value L(F,2) is equal to

(F, F)^⊗3

q|N0|∞2^γN0−1·<∆F,∆F >^⊗3.

Here (F, F)^⊗3 is the "Petersson norm" of F; ∆F is the F-component of the "diagonal cycle" in Pic(X_N+

0,N₀⁻)^⊗3;X_N+

0,N₀⁻ is the "definite" Shimura curve of type (N₀⁺, N₀⁻); and<·,·>^⊗3 is the Gross height pairing onPic(X_N+

0,N₀⁻)^⊗3.

Each object in the above formula is defined in §3.4. One ingredient in the proof is a Siegel-Weil formula over function fields. This formula connects the Eisenstein series appearing in the integral

1

representation of L(F, s) and theta series from the associated definite quaternion algebra B. By strong multiplicity one theorem and Jacquet-Langlands correspondence betweenGL2 and B^× (cf.

[24]), the integral representation ofL(F, s)is then expressed in terms of the "periods" coming from F and the Gross height of the corresponding cycle ∆F in the definite Shimura curves X_N+

0,N₀⁻. An immediate consequence from the above formula is thatL(F,2)is always non-negative, and the Gross height of∆_F determines the non-vanishing ofL(F,2). In the case when the root numberεis negative, the central critical derivativeL⁰(F,2) will be treated in a subsequent paper.

Let E be an elliptic curve over Fq(t) which is of conductor N₀∞ and has split multiplicative reduction at ∞. From the works of Weil, Jacquet-Langlands, and Deligne, it is well known that there exists a normalized Drinfeld type newformfE forΓ0(N0)such that the Hasse-WeilL-function L(E, s)is equal toL(fE, s−1). LetFE=fE⊗fE⊗fE. Then we have

L(F_E, s) =L(E, s−1)²·L(Sym³E, s),

whereL(Sym³E, s)is theL-function associated to the symmetric cube representation Sym³E. The works of Deligne [3] and Lafforgue [14] implies thatL(Sym³E, s)is entire. Suppose the root number ε(F_E)is positive. Then the non-vanishing of the Gross height of∆_FE guarantees the non-vanishing of L(E,1). Note that the Gross height of ∆F_E is only determined by the elliptic curve E. We expect that, after further works, the value<∆F_E,∆F_E >^⊗3 could be interpreted geometrically by the invariants ofE.

The structure of this article is organized as follows. We set up the general notation in the first section, and review basic facts about automorphic forms of Drinfeld type which are needed for our purpose. The second section consists of analytic properties of the Rankin triple productL-functions associated to Drinfeld type newforms with square-free level. The functional equation is formulated in §2.1, and the proof is given at the end of §2 by using the local results in §2.2 and §2.3. In §3, we establish the analogue of Gross-Kudla formula for the central critical value. After a brief review of the Weil representation in §3.1, we recall the Siegel-Eisenstein series and state the Siegel-Weil formula in §3.2. The central critical value L(F,2) is then expressed as an integral of F times a theta series in §3.3. In §3.4, we introduce a Hecke module homomorphism from the Picard group of definite Shimura curves to the space of Drinfeld type automorphic forms. This homomorphism relates the theta series to the diagonal cycle of the associated definite Shimura curveX_N+

0,N₀⁻, which leads us to the main result in Theorem 3.10. An application to the non-vanishing of Hasse-Weil L-values associated to elliptic curves is given in §4. Finally, two examples from the elliptic curves is given in §4.1.

1. Preliminary

In this section, we start with the general setting, and give a brief review of Drinfeld type auto- morphic forms. For further details, we refer to Gekeler-Revesat [7], also Weil [27].

1.1. Notation. LetF^q be the finite field withqelements and the characteristic ofF^q is denoted by p. We always assume that pis odd. Letk be the rational function fieldFq(t) with one variablet, and denote byAthe polynomial ringFq[t]. We denote by∞the place ofkat infinity, i.e. the place corresponding to the degree valuation. Recall the degree valuationord_∞(a)of any elementainAis

−dega. Letk_∞ be the completion ofkat∞andO_∞the valuation ring ink_∞. Setπ_∞ to bet⁻¹, which is a uniformizer inO_∞. ThenO_∞=Fq[[π_∞]]andk_∞=Fq((π_∞)). For any elementα∈k_∞, the absolute value|α|_∞:=q^{−ord∞(α)}. We fix the following additive characterψ_∞ fromk_∞ toC^×:

ψ_∞(X

i

aiπⁱ_∞) := exp 2π√

−1

p Tr_Fq/Fp(−a1)

.

1.2. Automorphic forms of Drinfeld type. LetK∞ be the Iwahori subgroup ofGL2(O∞), i.e.

K_∞:=

a b c d

∈GL₂(O_∞)

c≡0 modπ_∞O_∞

.

For each non-zero idealN ofA, let Γ₀(N) :=

a b c d

∈GL₂(A)

c≡0 modN

.

Anautomorphic form of Drinfeld type forΓ0(N)is aC-valued functionf on the double coset space Y0(N) := Γ₀(N)\GL₂(k_∞)/Z(k_∞)K_∞

(whereZ is the center ofGL2) satisfying the so-calledharmonic property: forg∈GL2(k∞), f(g) +f

g

0 1 π∞ 0

= 0and X

κ∈GL2(O∞)/K∞

f(gκ) = 0.

LetT∞ be the Bruhat-Tits tree corrsponding to the equivalent classes of rank 2 lattices in the vector spacek²_∞ (cf. [20] or [7] (1.3)). Then the double coset space Y0(N) can be identified with the set of oriented edges in the quotient graphΓ0(N)\T∞. Under this identification, automorphic forms of Drinfeld type forΓ0(N)are also calledC-valued harmonic cochains on Γ0(N)\T∞ (cf. [7]

§3).

1.3. Petersson inner product. An automorphic form f of Drinfeld type for Γ0(N) is called a cusp form if f is compactly supported modulo Z(k∞)·Γ0(N), i.e. f vanishes except for finitely many double cosets inY0(N). Suppose two Drinfeld type automorphic forms f1 andf2 for Γ0(N) are given. If one of them is a cusp form, thePetersson inner product off₁ andf₂ is

(f1, f2) :=

Z

Y0(N)

f1f2= X

[g]∈Y0(N)

f1(g)f2(g)µ([g]).

Here the measureµ([g])for eachg∈GL₂(k_∞)is defined by µ([g]) := q−1

2 · 1

# (g⁻¹Γ0(N)g∩ K_∞).

A Drinfeld type cusp form f forΓ0(N)is called an old formiff is a linear combination of the forms

f⁰

d 0 0 1

g_∞

forg_∞∈GL₂(k_∞), wheref⁰ is a Drinfeld type cusp form forΓ₀(M),M|N,M 6=N, andd|(N/M).

A Drinfeld type cusp formf forΓ₀(N)is called anew form iff is orthogonal (with respect to the Petersson inner product) to any old form forΓ₀(N).

1.4. Fourier expansion and L-functions. Let f be an automorphic form of Drinfeld type for Γ₀(N). Forr∈Zandu∈k_∞, recall theFourier expansion

f

π_∞^r u

0 1

=X

λ∈A

f^∗(r, λ)ψ∞(λu), where

f^∗(r, λ) = Z

A\k∞

f

π_∞^r u

0 1

ψ_∞(−λu)du.

Note that f^∗(r, λ) = f^∗(r, λ) for any ∈ F^×q and f^∗(r, λ) vanishes when degλ > r+ 2. If degλ≤r+ 2, the harmonic property off implies thatf^∗(r+ 1, λ) =q⁻¹·f^∗(r, λ).

Supposef is a cusp form. The L-function associated tof is L(f, s) := (1−q^−(s+1))⁻¹· X

m∈A,monic

f^∗(degm+ 2, m)

|m|^s_∞ , Re(s)>1.

ThisL-function can be extended to an entire function onC(which is in fact a polynomial inq^−s).

Moreover,

L(f, s) =−q^(3−degN)s·L(f⁰,−s),

wheref⁰ is the Drinfeld type cusp form forΓ0(N)defined by f⁰(g) :=f

0 1 N 0

g

.

1.5. Hecke operators. Letf be an automorphic form of Drinfeld type forΓ₀(N). For each monic irreducible polynomialP ofA, theHecke operatorT_P is defined by:

TPf(g) := P

degu<degP

f







 1 u 0 P



·g



+µN(P)·f







 P 0

0 1



·g



.

HereµN(P) = 1ifP -N and0otherwise. It is clear thatTPf still satisfies the harmonic property, and the Fourier coefficients ofTPf are of the form:

(TPf)^∗(r, λ) =|P|∞·f^∗(r+ degP, P λ) +µN(P)·f^∗(r−degP, λ/P).

Here f^∗(π_∞^r , λ/P) = 0 if P - λ. Note thatT_P and T_P⁰ commute to each other, we define the Hecke operatorT_mfor each monic polynomialmin Aas follows:

(T_mm⁰ =T_mT_m⁰ ifmandm⁰ are relatively prime, T_P` =T<sub>P</sub>`−1TP−µN(P)· |P|_∞T_P`−2.

Now, supposef is a cusp form. We callf aHecke eigenformifTmf =cm(f)·f wherecm(f)∈C for all monic polynomialminA. In this case, we must have

c_m(f)·f^∗(2,1) =|m|_∞·f^∗(degm+ 2, m).

SinceTP is self-adjoint with respect to the Petersson inner product for P -N, we have cP(f)∈R forP - N. If f is normalized, i.e. f^∗(2,1) = 1, thenL(f, s) can be written as the following Euler product:

(1−q^−(1+s))⁻¹· Y

monic irrduciblePofA

(1−c_P(f)|P|^−(1+s)_∞ +µ_N(P)|P|^1−2(1+s)_∞ )⁻¹.

Suppose the Hecke eigenformf is a new form (called anewform). It is known for a newformf that:

(1) For P | N, cP(f)∈ {±1} if P k N and 0 otherwise. Therefore cm(f)∈ R for all monic polynomialsm, which implies thatf is anR-valued function if f is normalized.

(2) f⁰=ε_N(f)·f where ε_N(f)∈ {±1}.

(3) ForP -N, the quadratic polynomialX²−c_P(f)X+|P|∞has two complex conjugate roots (i.e.c_P satisfies the so-called Ramanujan bound: |c_P(f)| ≤2|P|^1/2∞ ).

1.6. Adelic language. For each placev of k, the completion of k atv is denoted by kv, andOv

is the valuation ring in kv. We call v a finite place of k if v 6= ∞. For any finite place v, there exists a unique monic irreducible polynomialPvinAwhich is a uniformizer inOv. We setπv:=Pv

and Fv :=Ov/πvOv. The cardinality ofFv is denoted byqv (which is equal to |Pv|∞). For each α∈kv,|α|v:=qv^−ordv(α). For the infinite place∞, we have chosen a uniformizerπ∞=t⁻¹, and the cardinalityq_∞of the residue fieldF∞:=O_∞/π_∞O_∞ is equal to q. The adele ring of kis denoted byAk, with the maximal compact subringQ

vO_v=:O_Ak. Consider the compact subgroup K₀(N∞) := Q

vK_v of GL₂(Ak), where for v = ∞, we have definedK_∞in §1.2; for v-N∞,K_v:= GL₂(O_v); forv|N,

Kv:=

a b c d

∈GL2(Ov)

c≡0 modπ^ord_v ^v(N)Ov

.

The strong approximation theorm (cf. [23] Chapter III Theorem 4.3) tells that the natural map from the double coset spaceY0(N)to GL₂(k)\GL₂(Ak)/Z(Ak)K0(N∞) is a bijection. Therefore every automorphic form f of Drinfeld type for Γ₀(N) can be viewed as a function on the double coset spaceGL₂(k)\GL₂(Ak)/Z(Ak)K₀(N∞). The harmonic property off is equivalent to say that (cf.

[7] §4) the space generated byfg(·) :=f(·g)for allg∈GL2(k_∞)is isomorphic (as a representation ofGL2(k∞)) to the special representationσ(| · |^1/2∞ ,| · |^−1/2∞ ).

1.6.1. Whittaker functions. Fix the additive character ψ on Ak defined byψ(a) := Q

vψv(av) for a= (av)v ∈Ak, where for each placev ofk,

ψ_v(a_v) := exp 2π√

−1

p Tr_Fv/Fp Res_v(a_vdt)

.

For each Drinfeld type cusp formf for Γ₀(N), the Whittaker function W_f associated to f is the following function onGL2(A^k):

Wf(g) :=

Z

k\Ak

f

1 u 0 1

g

ψ(−u)du.

Here the Haar measure is normalized so that R

k\Ak1du = 1. The adelic version of the “Fourier expansion” off is

f(g) = X

α∈k^×

W_f

α 0 0 1

g

∀g∈GL₂(Ak).

Supposef is a newform. LetWf,v:=Wf |_GL2(kv). Then Wf(g) =Y

v

Wf,v(gv) ∀g= (gv)v ∈GL2(Ak).

2. Rankin triple product

Let N₀ be a square-free ideal of A. Given f₁, f₂, and f₃ be three normalized Drinfeld type newforms forΓ₀(N₀), letF =f₁⊗f₂⊗f₃ be the function onGL₂(k_∞)³ defined by

F(g₁, g₂, g₃) :=f₁(g₁)f₂(g₂)f₃(g₃).

Thetriple productL-functionL(F, s)associated to f1,f2, andf3 is the Euler product L(F, s) :=L_∞(F, s)· Y

monic irreduciblePinA

LP(F, s),

where each local factor is defined by the following:

(1) L_∞(F, s) := (1−q^−s)⁻¹(1−q^1−s)⁻².

(2) For P|N₀, we set ε_P :=−c_P(f₁)c_P(f₂)c_P(f₃)∈ {±1}and L_P(F, s) := (1 +ε_P|P|^−s_∞)⁻¹(1 +ε_P|P|^1−s_∞ )⁻².

(3) For P -N₀, let α⁽¹⁾_P,i andα⁽²⁾_P,i be two complex conjugate roots of the quadratic polynomial X²−c_P(f_i)X+|P|∞. Then we set

LP(F, s) := Y

1≤j1,j2,j3≤2

1−α^(j_P,1¹⁾α^(j_P,2²⁾α^(j_P,3³⁾|P|^−s_∞−1

.

The Ramanujan bound of cP(fi)implies that L(F, s)converges absolutely for Re(s)>5/2. We remark that the localL-factorLv(F, s)for each placev is in fact the localL-function associated to ρf₁,v⊗ρf₂,v⊗ρf₃,v. Here for1≤i≤3,ρf_i,vis the Weil-Deligne representation corresponding tofi

atv via local Langlands correspondence (cf. [2] Chapter 7, 8).

We setε∞:=−1. The root number ofL(F, s)is, by definition, equal toε:=ε∞·Q

P|N0εP.Let Λ(F, s) :=q^−8(s−32)·L(F, s). Then

Theorem 2.1. The function Λ(F, s) can be extended to an entire function (in fact, a polynomial inq^−s), and satisfies the following functional equation:

Λ(F, s) =ε·(|N0|_∞·q)^5·(2−s)·Λ(F,4−s).

Remark. The functional equation implies thatL(F, s)is a polynomial inq^−sof degree5 degN0−11 and the constant coefficient is1. Moreover,

ε= (−1)^ords=2L(F,s),

which means that the root numberεtells us the parity of the vanishing order ofL(F, s)ats= 2.

The proof of Theorem 2.1 is in §2.3, by using the local results in §2.1 and §2.2.

2.1. Zeta integrals. LetG:= GSp₃, i.e. the set ofR-points ofGfor any algebraR is GSp₃(R) :=

g∈GL6(R)

tg·

0 I3

−I3 0

·g=`g·

0 I3

−I3 0

for some`g∈R^×

. The centerZG ofGconsists of scalar matrices inGL6. There is a canonical embedding from

H :={(g1, g₂, g₃)∈(GL₂)³|detg₁= detg₂= detg₃} intoG:

a₁ b₁ c₁ d₁

,

a₂ b₂ c₂ d₂

,

a₃ b₃ c₃ d₃

7−→

















a1 b1

a2 b2

a₃ b₃

c₁ d₁

c₂ d₂

c₃ d₃















 .

LetPG=NG·MG be the Siegel parabolic subgroup ofG, where the set ofR-points ofNG is NG(R) :=

n(b) :=

I3 b 0 I3

b=^tb∈Mat3(R)

; the set ofR-points ofM_G is

MG(R) :=

m(a, `) :=

a 0 0 `·^ta⁻¹

a∈GL3(R), `∈R^×

. LetKG :=G(O_Ak). The Bruhat decomposition ofGsays that

G(Ak) =P(Ak)·KG.

For s ∈C, let I_Ak(s) be the representation of G(A^k)consisting of smooth functions Φ onG(A^k) such that

Φ(n·m(a, `)·g) =|deta|^2s+2

A · |`|^−3s−3

A Φ(g) for allg∈G(Ak),n∈NG(Ak), andm(a, `)∈MG(Ak). Here|α|_A:=Q

v|αv|vfor allα= (αv)∈A^×_k. Letφbe a function onKG such that

φ(n·m(a, `)·g) =φ(g), for allg∈KG,n∈NG(O<sub>Ak</sub>), m(a, `)∈MG(O_Ak).

Then φ gives us a flat section Φ_φ, i.e. for each s ∈ C, φ can be extended uniquely to a function Φ_φ(·, s) in I_Ak(s)such thatΦ_φ |_G(O

Ak))=φ. We call Φa meromorphic (respectively, holomorphic) section ifΦis a linear combination of flat sections where the coefficients are rational functions in q^−s (respectively, the coefficients are inC[q^−s, q^s]).

LetΦbe a meromorphic section. TheEisenstein series E(Φ, s,·)onG(Ak)is defined by:

E(Φ, s, g) := X

γ∈PG(k)\G(k)

Φ(γ·g, s), ∀g∈G(Ak).

It is well-known that this series converges for Re(s) sufficiently large. Moreover, E(Φ, s, g) has a meromorphic continuation ins∈C(in fact, a rational function inq^−s, cf. [15] IV.1.12).

Definition 2.2. The(global)zeta integral associated toF and a meromorphic sectionΦis:

Z(F,Φ, s) :=

Z

Z_G(Ak)H(k)\H(Ak)

F(h)·E(Φ, s, h)dh.

HereF is viewed as a function onGL2(Ak)³, and the measuredhis induced from the Haar measure onZG(Ak)\H(Ak)normalized so that the volume of ZG(Ak)\ZG(Ak)H(O_Ak)is1.

LetU0be the following algebraic subgroup of H:

U0:=

1 u1

0 1

,

1 u2

0 1

, 1 u3

0 1

∈(GL2)³

u1+u2+u3= 0

. Following Garrett and Harris [6], we choose a particular element

δ:=

















1 1 1 −1 0 0 0 1 0 −1 1 0 0 0 1 −1 0 1

1 1 1 0 0 0

0 0 0 −1 1 0 0 0 0 −1 0 1

















∈GSp₃.

Then we have

Proposition 2.3. (cf. [16])ForRe(s)sufficiently large, Z(F,Φ, s) =q⁻²·

Z

Z_G(Ak)U₀(Ak)\H(Ak)

W_F(h)·Φ(δ·h, s)dh

whereW_F(h₁, h₂, h₃) =W_f1(h₁)·W_f2(h₂)·W_f3(h₃)for any h= (h₁, h₂, h₃)∈H(Ak), and W_fi is the Whittaker function associated tof_i for1≤i≤3 intoduced in §1.6.

For each placevofk, we setI_v(s)to be the space of smooth functionsϕonG(k_v)satisfying that ϕ(nvm(av, `v)gv) =|detav|<sup>2s+2</sup><sub>v</sub> · |`v|^−3s−3_v ·ϕ(gv)

for allnv ∈NG(kv), m(av, `v)∈MG(kv), gv ∈G(kv). Let Φφ_G(Ov)(·, s) ∈Iv(s) be the flat section associated to φ_G(Ov) where φ_G(Ov) ≡ 1 on G(O_v). Then I_Ak(s) is the restricted tensor product

⊗⁰_vI_v(s)(w. r. t.{Φφ_G(Ov)}v). We call a meromorphic sectionΦ∈I_Ak(s)is apure-tensorifΦ =⊗vΦ_v where for eachv,Φ_v(·, s)∈I_v(s)is a meromorphic section, andΦ_v= Φ_φ

G(Ov) for almost allv.

Lemma 2.4. For any pure-tensorΦ =⊗vΦv∈I_Ak(s), we have Z(F,Φ, s) =q⁻²·Q

vZv(F,Φv, s), where

Zv(F,Φv, s) :=

Z

Z_G(k_v)U₀(k_v)\H(kv)

WF,v(hv)Φv(δhv, s)dhv

andW_F,v(h_v,1, h_v,2, h_v,3) :=W_f1,v(h_v,1)W_f2,v(h_v,2)W_f3,v(h_v,3).

2.2. Local factors. When v - N₀∞, the conductor of the fixed additive characterψ_v is trivial.

Takeφ_v=φ_G(Ov) whereφ_G(Ov)≡1onG(O_v). Then (cf. [16] Theorem 3.1) Z_v(F,Φ_φv, s) =

Z

Z_G(k_v)U₀(k_v)\H(kv)

W_F,v(h_v)·Φ_φv(δh_v, s)dh_v= 1

bv(s)·L_v(F, s+ 2) whereb_v(s) := (1−q^−2s−2_v )⁻¹(1−q_v^−4s−2)⁻¹.

Now, supposev|N₀∞. LetK₀(v)be the following compact subgroup in G(k_v):

K₀(v) :=

A B

C D

∈G(O_v)

A, B, C, D∈Mat₃(O_v)andC≡0 modπ_vO_v

.

For0≤i≤3, let

w_i:=









I_3−i 0 0 0

0 0 0 Ii

0 0 I_3−i 0

0 −Ii 0 0







 .

Then the Iwasawa decomposition ofGimplies G(O_v) = a

0≤i≤3

K₀(v)w_iK₀(v).

For 0 ≤ i ≤ 3, we set φ⁽ⁱ⁾v to be the characteristic function of K0(v)wiK0(v) on G(Ov). Then P

0≤i≤3

φ⁽ⁱ⁾v =φG(O_v). These four functionsΦ_φ(i)

v (·, s),0≤i≤3, form a basis for the spaceIv(s)^K0(v) ofK0(v)-fixed functions inIv(s).

For eachΦ∈Iv(s)^K0(v), we define(ωvΦ)(g) := Φ(gηv), where η_v :=

0 I3

−πvI3 0

∈G(k_v).

It is observed thatωvΦ_φ(i)

v =q(2i−3)(s+1)

v Φ_φ(3−i)

v .SetΦe_φ(i)

v (·, s) :=qv^−i(s+1)·Φ_φ(i)

v (·, s)∈Iv(s), then one hasωvΦe_φ(i)

v =Φe_φ(3−i)

v . Note that Φφ_G(Ov) = X

0≤i≤3

q_v^i(s+1)·Φe_φ(i)

v andΦ⁰_φ

G(Ov):=ωvΦφ_G(Ov) = X

0≤i≤3

q_v^(3−i)(s+1)·Φe_φ(i) v .

We choose two more functionsΦ^±_v(·, s)in Iv(s)^K0(v)which are defined by Φ^±_v(·, s) := X

0≤i≤3

(±1)ⁱΦe_φ(i) v (·, s).

ThenωvΦ^±_v =±Φ^±_v. Suppose s6=−1. Then{Φφ_G(Ov),Φ⁰_φ

G(Ov),Φ^±_v} also form a basis ofIv(s)^K0(v). Next, we calculate the local zeta integral associated to these four functions.

For eachhv∈H(kv), we haveωvWF,v(hv) :=WF,v(hvηv) =εvWF,v(hv). Therefore Zv(F, ωvΦ, s) =εvZv(F,Φ, s) ∀Φ∈Iv(s)^K0(v).

This tells us thatZ_v(F,Φ^−ε_v ^v, s) = 0. we also deduce that Zv(F,Φφ_G(Ov), s) =Zv(F,Φ⁰_φ

G(Ov), s) = 0.

The remaining case is the zeta integralZv(F,Φ^ε_v^v, s). SinceωvΦe_φ(i)

v =Φe_φ(3−i)

v , we get Z_v(F,Φ^ε_v^v, s) =−2ε_vq^s+1_v ·(1−ε_vq^−(s+1)_v )²·Z_v(F,Φe_φ(0)

v , s).

Proposition 2.5. If v|N0∞, then we have Zv(F,Φ_φ(0)

v , s) =−q_v^(1−s)δv ·(qv+ 1)⁻³·q_v^−2s−2·(1 +εvq^−s−2_v )⁻¹·(1 +εvq_v^−s−1)⁻² whereπ^δ_v^v is the conductor of the additive characterψ_v.

Remark. The conductor ofψ_∞ is not trivial. Hence this result is not covered by Proposition 4.2 in [8]. We rework the proof here accordingly.

Proof. Let

U :=

1 x1

0 1

,

1 x2

0 1

,

1 x3

0 1

⊂GL³₂ and

T :=

a₁ 0 0 a⁻¹₁

,

a₂ 0 0 a⁻¹₂

,

a₃ 0 0 a⁻¹₃

⊂GL³₂. Fix a placev|N₀∞. LetK_Hv :=G(O_v)∩H(k_v), and

H_v⁰:={h∈H(kv) : ordv(deth)is even}.

Then ZG(kv), U(kv), T(kv), KH_v are subgroups in H_v⁰. Let d^×z be the Haar measure on ZG(kv) normalized such that vol(ZG(Ov)) = 1, du be the Haar measure on U(kv) normalized such that vol(U(Ov)) = 1, d^×a(= d^×a1·d^×a2·d^×a3) be the Haar measure on T(kv)normalized such that vol(T(O_v)) = 1, anddκ be the Haar measure onK_Hv such that vol(K_Hv) = 1. Thend^×z_|a|^du₂

v

d^×adκ

is a Haar measure onH_v⁰, where |a|v:=|a1a2a3|v. We can extend this measure to a Haar measure onH(kv), as

H(kv) =H_v⁰∪

πvI3 0 0 I3

H_v⁰. Now, we embedkv into U(kv) byx7→u(x) :=

1 x 0 1

, I2, I2

. Then the invariant measure dhonZG(kv)U0(kv)\H_v⁰ is

dh0:= 1

|a|²_vdxd^×adκ, and onZ_G(k_v)U₀(k_v)\

π_vI₃ 0 0 I₃

H_v⁰ isq_v²dh₀. HenceZ_v(F,Φ_φ(0)

v , s) =Z₁+Z₂, where Z1=

Z

ZG(kv)U0(kv)\H⁰_v

WF(h0)Φ_φ(0)

v (δh0, s)dh0, and

Z₂=q²_v Z

Z_G(k_v)U₀(k_v)\H⁰_v

W_F

π_vI₃ 0 0 I3

h₀

Φ_φ(0)

v

δ

π_vI₃ 0 0 I3

h₀, s

dh₀. SinceΦ_φ(0)

v is right invariant byK0(v),Z1 is equal to vol(K0(v)∩H(kv))· X

κ∈K_Hv/K₀(v)∩H(kv)

Z1(κ),

where

Z₁(κ) :=

Z

(k_v^×)³

Z

k_v

W_F u(x)aκ Φ_φ(0)

v δu(x)aκ, sdx

|a|²_vd^×a.

Choose the following coset representatives ofKHv/K0(v)∩H(kv):



















1, κ(x1) :=

1 x1

0 1

0 1

−1 0

, I2, I2

, κ(x2) :=

I2,

1 x2

0 1

0 1

−1 0

, I2

, κ(x3) :=

I2, I2,

1 x3

0 1

0 1

−1 0

x1, x2, x3∈Fv

















 .

Whenκ= 1, Φ_φ(0)

v (δu(x)aκ, s) =

(|a|^2s+2_v · |x|^−2s−2_v if|x|v>|ai|²_v for1≤i≤3,

0 otherwise.

We then deduce thatZ1(1)is equal to

(−ε_v)^δvq_v^3δv(1−q^−2s−2_v )⁻³(1−q_v^−2s−4)⁻¹

·

−q^δ_v^v(2s+1)q_v^−2s−2q^{(−6s−6)dδ}_v ^v/2e(1−q^−2s−4_v ) +(1−q⁻¹_v )q^−6s−6_v q^{(−2s−4)dδ}_v ^v/2e

+(1−q⁻¹_v )q^−4s−5_v q^{(−2s−4)bδ}_v ^v/2c

Whenκ=κ(xi)for somexi ∈Fv, Φ_φ(0)

v (δu(x)aκ, s) =

(|a|^2s+2_v |ai|^−2s−4_v if|ai|²_v>|x+a²_ix_i|v and|ai|v >|aj|v forj6=i,

0 otherwise.

Therefore we get

Z1(κ(xi)) =−q⁻¹_v (−εv)^δvq^3δ_v ^vq^−4s−5_v (1−q_v^−2s−2)⁻²(1−q^−2s−4_v )⁻¹·q^{(−2s−4)bδ}_v ^v/2c.

Since the volume vol(K0(v)∩H(kv))is(qv+ 1)⁻³,Z1 is equal to (−εv)^δvq^3δ_v ^v(qv+ 1)⁻³q^−2s−2_v (1−q_v^−2s−2)⁻³(1−q_v^−2s−4)⁻¹

·

−3·q_v^−2s−3q_v^{(−2s−4)bδv/2c}(1−q_v^−2s−2) + (−1)q_v^(2s+1)δvq_v^{(−6s−6)dδv/2e}(1−q_v^−2s−4)

+(1−q_v⁻¹)q_v^−4s−4q_v^{(−2s−4)dδv/2e}+ (1−q⁻¹_v )q^−2s−3_v q^{(−2s−4)bδ}_v ^v/2c

Next, we considerZ₂. It is clear that Φ_φ(0)

v

δ

πvI3 0 0 I3

h0, s

=q^−s−1_v Φ_φ(0)

v (δh0, s).

Note thatb(δv−1)/2c=bδv/2c −¹⁺⁽⁻¹⁾₂ ^δv. By the same argument, we obtain thatZ2is equal to (−εv)^δvq_v^3δv(qv+ 1)⁻³q_v^−2s−2(1−q_v^−2s−2)⁻³(1−q^−2s−4_v )⁻¹q^{(−2s−4)bδ}_v ^v/2c

· (−εv)q⁻³_v ·q^−s−1_v ·q²_v·q^(2s+4)(

1+(−1)δv

2 )

v

·

−2q_v^−2s−3+q^−2s−4_v

q^(−4s−5)(

1+(−1)δv

2 )

v −1

+q^−4s−4_v q^(−2s−4)(

1+(−1)δv

2 )

v

+q_v^−4s−5

3−q^(−2s−4)(

1+(−1)δv

2 )

v

−q^(−4s−5)(

1+(−1)δv

2 )

v

. Therefore

Z1+Z2=−(qv+ 1)⁻³q_v^(1−s)δvq_v^−2s−2(1 +εvq_v^−s−2)⁻¹(1 +εvq^−s−1_v )⁻²

whereπ^δ_v^v is the conductor of the additive characterψv, which is the desired conclusion.

Forv|N0∞, let

ξ_v(s) := 2ε_v(q_v+ 1)⁻³·q_v^−s−1·(1−ε_vq^−s−1_v )²·b_v(s).

Moreover, we setb(s) :=Q

vbv(s), andb^∗(s) :=q^−6s−4·b(s). Then one has Corollary 2.6. (1)Whenv|N₀∞,

Z_v(F,Φ^ε_v^v, s) =q_v^(1−s)δv·ξ_v(s)· 1

bv(s)·L_v(F, s+ 2), whereπ^δ_v^v is the conductor of the additive characterψv.

(2)Take Φ^\=⊗vΦ^\_v ∈I_Ak(s), where (Φ^\_v= Φ_φ

G(Ov) forv-N₀∞;

Φ^\_v:=ξv(s)⁻¹·Φ^ε_v^v forv|N0∞.

Then

Z(F,Φ^\, s) =q⁻²·q^−2s+2· 1

b(s)·L(F, s+ 2) = 1

b^∗(s)Λ(F, s+ 2).

Proof. Forv-N0∞, we have mentioned that

Zv(F,Φφ_G(Ov), s) = 1

b_v(s)·Lv(F, s+ 2).

Therefore(2)follows from Proposition 2.3, Lemma 2.4 and(1). Forv|N0∞, recall that Zv(F,Φ^ε_v^v, s) =−2εvq^s+1_v ·(1−εvq^−(s+1)_v )²·Zv(F,Φe_φ(0)

v , s).

Therefore(1)follows from Proposition 2.5.

Remark. We point out that the zeta integral Z(F,Φ^\, s) can be extended to a rational function in q^−s. Thus the above corollary gives us immediately the meromorphic continuation of L(F, s).

In fact, by the main theorem of [10], the triple product L-fucntion L(F, s) is entire. In the next subsection, we shall give the functional equation ofL(F, s).

2.3. Intertwining operator M(s) and the functional equation. For Re(s) > 1, the global intertwining operatorM(s) :I_Ak(s)→I_Ak(−s)is given by the following integral

M(s)Φ(g) :=

Z

NG(Ak)

Φ(w3ng)dn.

The Haar measure dn is normalized so that vol(NG(k)\NG(Ak)) = 1. It is known that (cf. [16]

§4) this integral operator has a meromorphic continuation to the whole s-plane, and it gives the following functional equation of the Eisenstein seriesE(Φ, s, g)(cf. [15] IV.1.10):

E(Φ, s, g) =E(M(s)Φ,−s, g).

ReplacingΦby the functionΦ^\in Corollary 2.6 (2), the above functional equation implies Z(F,Φ^\, s) =Z(F, M(s)Φ^\,−s).

Since Φ^\ =⊗vΦ^\_v where Φ^\_v ∈Iv(s), it is clear that for Re(s) >1, M(s)Φ^\ =⊗vMv(s)Φ^\_v where Mv(s) :Iv(s)→Iv(−s)is defined by

Mv(s)Φv(gv) = Z

NG(kv)

Φv(w3nvgv)dnv, ∀Φv∈Iv(s).

For each placev, the Haar measurednv can be normalized so that vol(NG(Ov)) =q_v^3δv.

For each place v of k, let av(s) := (1−q^−2s+1_v )⁻¹·(1−q^−4s+1_v )⁻¹. Note that for v - N0∞, Φ^\_v= Φφ_G(Ov). Hence (cf. [16] §4)

MvΦ^\_v(s) = av(s)

b_v(s)Φ^\_v(−s).

SinceM_v isG(k_v)intertwining, it carriesI_v(s)^K0(v) toI_v(−s)^K0(v). Thus whenv|N₀∞, Zv(F, Mv(s)Φ^ε_v^v,−s) =αv(s)·Zv(F,Φ^ε_v^v,−s)

whereαv(s)is a meromorphic function. By the same argument in [8] Proposition 5.1, one gets Proposition 2.7. Whenv|N0∞,

αv(s) =εv·q^3δv ·q_v^−3s−2·(1−εvq_v^−s−1)²(1 +εvq^1−s_v )(1 +q_v^1−2s) (1−εvq^1−sv )(1−q^1−4sv ) , whereπ^δ_v^v is the conductor of the additive characterψv.

For each placev ofk, we set ηv(s) :=







1 ifv-N₀∞;

q^−3δ_v ^v ·ξv(−s) ξv(s) · bv(s)

av(s)·α_v(s) =ε_vq_v^−5s ifv|N₀∞.

Then

Corollary 2.8. For each placev ofk, we have

Z_v(F, M_v(s)Φ^\_v,−s) =q^3δ_v ^v·η_v(s)·a_v(s)

bv(s)Z_v(F,Φ^\_v,−s).

Proof of Theorem 2.1. The remark at the end of §2.2 has already told us that Λ(F, s) can be extended to a polynomial inq^−s. Letη(s) :=Q

v|N0∞ηv(s)(which is equal toε·(|N0|_∞q)^−5s) and a(s) :=Q

vav(s). By Corollary 2.6 and 2.8, we obtain that 1

b^∗(s)Λ(F, s+ 2) =q⁶·δ(s)· a(s) b(s)

1

b^∗(−s)Λ(F,−s+ 2).

Letζ_k^∗(s) :=q^−sQ

v(1−q^−s_v )⁻¹. Then the functional equationζ_k^∗(s) =ζ_k^∗(1−s)implies that q⁶·a(s)

b(s) · 1

b^∗(−s) = 1 b^∗(s).

Therefore we get the functional equation ofΛ(F, s)in Theorem 2.1:

Λ(F, s+ 2) =δ(s)·Λ(F,−s+ 2) =ε·(|N0|_∞q)^−5s·Λ(F,−s+ 2).

3. Function field analogue of Gross-Kudla formula

From now on, we assume the root numberεis positive, i.e.L(F, s)does not vanish automatically at s= 2. In this section, we explore the central critical valueL(F,2) and present an analogue of Gross-Kudla formula.

3.1. Weil representation. For convenience, we setε_v = 1forv-N₀∞. LetSbe the set consisting of the places v of k such that ε_v = −1. Then the cardinality of S is even. LetB be the unique quaternion algebra overkwhich is ramified at the places inSand unramified elsewhere. Let(V, Q_V) be the quadratic space(B,Nr_B)overkwhereNr_B= Nris the reduced norm form onB. Forx, yin V, the bilinear form< x, y >V associated toQV is TrB(x¯y), whereTrB = Tris the reduced trace onB andy¯:= Tr(y)−y is the main involution ofB. We denote byO(V)the orthogonal group of V. LetG¹:= Sp₃, i.e. the following algebraic subgroup ofG:

g∈GL6

tg·

0 I₃

−I3 0

·g=

0 I₃

−I3 0

.

For each placev ofk, we have fixed an additive characterψ_v onk_v in §1.6. LetV(k_v) :=V ⊗_kk_v and letS(V(k_v))be the space ofSchwartz functions on V(k_v), i.e. the space of functions onV(k_v) which are locally constant and compactly supported. The(local) Weil representation ω_v(= ω_v,ψv) ofG¹(kv)×O(V)(kv)on the space

S(V(kv))⊗_CS(V(kv))⊗_CS(V(kv)) =S(V(kv)³)

is defined by the following: forϕ=ϕ1⊗ϕ2⊗ϕ3∈S(V(kv)³)andx= (x1, x2, x3)∈V(kv)³, (ωv(h)ϕ)(x) := ϕ(h⁻¹x1, h⁻¹x2, h⁻¹x3), ∀h∈O(V)(kv),;

ωv

a 0 0 ^ta⁻¹

ϕ

(x) := |deta|²_v·ϕ((x1, x2, x3)·a), ∀a∈GL3(kv);

ωv

I3 b 0 I3

ϕ

(x) := ψv

Tr b·Q(x)

·ϕ(x), ∀b=^tb∈Mat3(kv);

ω_v(w_i)ϕ := (ε_v)ⁱ·ϕ₁⊗ · · · ⊗ϕ_3−i⊗ϕb_3−i+1⊗ · · · ⊗ϕb₃, 0≤i≤3.

HereQ(x)is the3-by-3 matrix whose(i, j)-entry is ¹₂ < xi, xj >V; andϕbi is the Fourier transform ofϕi (with respect toψv)

ϕb_i(x_i) :=

Z

V(k_v)

ϕ_i(y)ψ_v(< x_i, y >_V)dy.

The Haar measure dy is chosen to be self dual, i.e. b

ϕb_i(xi) = ϕi(−xi). Let V(Ak) := V ⊗k Ak

and letS(V(Ak)) be the space of Schwartz functions on V(Ak). Then we have the (global) Weil representationω=⊗vωv ofG¹(Ak)×O(V)(Ak)on the space

S(V(Ak))⊗_CS(V(Ak))⊗_CS(V(Ak)) =S(V(Ak)³).

LetN₀⁻be the product of primesP ofAwithε_P =−1, andN₀⁺:=N₀/N₀⁻. LetRbe an Eichler A-order of B of type(N₀⁺, N₀⁻), i.e. for each finite placev, Rv :=R⊗AOv is a maximalOv-order inBv:=B⊗kkv ifv-N₀⁺; and

Rv ∼=

a b c d

∈GL2(Ov)

c≡0 modN₀⁺Ov

ifv|N₀⁺. Letϕ=⊗vϕv be the Schwartz function inS(V(Ak))where for eachv6=∞,

ϕv= the characteristic function ofRv;