Via the theta series constructed from definite pure quaternions, we then establish a Shimura correspondence between Drinfeld-type forms and metaplectic forms onSLf2

OVER GLOBAL FUNCTION FIELDS

CHIH-YUN CHUANG, FU-TSUN WEI, AND JING YU

Abstract. The aim of this paper is to study the central critical value of Rankin-type L-functions coming from “Drinfeld-type” automorphic cusp forms convolved with “imag- inary” quadratic characters. Rankin-Selberg method provides us with a very explicit functional equation for these Rankin-typeL-functions. When the “root number” in ques- tion is positive, we derive a Gross-type formula over arbitrary global function field. Via the theta series constructed from definite pure quaternions, we then establish a Shimura correspondence between Drinfeld-type forms and metaplectic forms onSLf2. Having this correspondence at hand leads us to an explicit Waldspurger-type formula in this setting.

Introduction

In 1987, Gross [6] presented a formula for the central critical values of certain Rankin-type L-functions associated to a weight2 cusp form of prime level, which connects the L-values in question to the height of certain special points on the corresponding “definite” Shimura curve. Reformulating this formula via the Shimura correspondence (after Waldspurger [14]), the cusp form gives rise to a specific weight3/2form whose Fourier coefficients account for the above central critical values. There are now many arithmetic consequences of these formulas.

The purpose of this paper is to study analogous phenomena in the function field setting, with

“Drinfeld-type” automorphic forms playing the role of classical modular forms.

Letk be a global function field with finite constant field Fq. Throughout this paper, we always assume that q is odd. Fix a place ∞ of k, regarded as the place at infinity. By a Drinfeld-type automorphic formF we mean thatF is an automorphic form onGL2(A)(where Ais the adele ring ofk) satisfying a “harmonicity” condition at ∞(cf. Section 2). In other words, the∞-component of the automorphic representation associated toF is isomorphic to the special representation sp(| · |^−1/2_∞ ,| · |^1/2_∞ ). Drinfeld-type cusp forms, first introduced in Drinfeld’s paper [3], are very useful tools in function field arithmetic (cf. [4], [12], [16] and [18] for further details).

LetA be the ring of functions in k regular outside ∞. The non-zero prime ideals of A correspond bijectively to the places ofkdistinct from ∞. LetF be a “normalized” Drinfeld- type newform of square-free level. Here being a newform means thatF is a Hecke eigenform and is orthogonal to all the old forms (cf. Section 2.1 and 2.2); the level will be denoted by nF, which is a square-free ideal of A. ThenF always admits a central character denoted by ωF.

Let%F be the degree two Galois representation associated toF under the Langlands cor- respondence. We are interested in the L-function associated to the Galois representation

%_F ⊗Ind^k_K(η) of Gal(k^sep/k), where K is an imaginary quadratic extension over k (i.e.∞ does not split in K/k), and η : Gal(k^sep/K) → C^× is a continuous character. We take Langlands’ functorial point of view, and look at this type of automorphicL-functions. Under

2010Mathematics Subject Classification. 11F37, 11F41, 11F67, 11R58.

Key words and phrases. Function field, Automorphic form,L-function, Metaplectic form.

1

class field theory, we identifyη with a Hecke character on the idele class group of K. Via the theta correspondence, one can also viewη as an automorphic representation ofGL₂(A) with a central character denoted by ωη. Then it is known that the L-function in question coincides with theRankin-typeL-function L(F×η, s−1/2)associated to the pair(F, η).

The main result of this paper is to show a function field analogue of Gross formula for the central critical value ofL(F×η, s), under the following two conditions onη:

C.1 the characterη is unramified everywhere and trivial at∞, i.e.η(Frob_∞K) = 1where Frob_∞K is the Frobenius at the place∞_K ofK lying above∞;

C.2 ωF·ωη =χK, where χK is the quadratic Hecke character associated to K (cf. Sec- tion 1.2).

By the Rankin-Selberg method, we expressL(F×η, s)as azeta integral (cf. Theorem 3.10).

From the meromorphic continuation and the functional equation of the Eisenstein series in question, the L-function L(F ×η, s) extends to a meromorphic function on the complex s-plane which is holomorphic ats= 0and satisfies (cf. Theorem 3.12)

L^∗(F×η, s) = (−1)^#Σ·L^∗(F×η,−s), where:

• L^∗(F×η, s)is a modifiedL-function (cf. Section 3.3),

• Σ = Σ(F, η) :={∞} ∪

prime pdividingn_F

pis ramified inK/kandλ_p(F)·η(Frob_P) = 1

∪ {primep dividingnF

p is inert inK/k},

• λp(F)is the Hecke eigenvalue ofF corresponding top,

• FrobPis the Frobenius at the primeP lying abovep.

The value (−1)^#Σ is called the root number of L(F×η, s). When #Σ is odd, the central critical valueL(F×η,0)vanishes. Assuming that#Σis even, we present our formula for the central critical valueL(F×η,0)as follows (cf. Theorem 4.16):

Theorem 0.1. Let F be a normalized Drinfeld-type newform of square-free level nF with central characterωF. Given an imaginary quadratic extension K overk, let η be a character onGal(k^sep/K)satisfying the conditionsC.1andC.2with the root number(−1)^#Σ(F,η)= 1.

Let n⁻_F,η:=Q

p∈Σ−{∞}p and X be the definite Shimura curve of type (n_F/n⁻_F,η,n⁻_F,η). Then we have

L(F×η,0) =P(F, K)· 4·(F, F)n_F

f_K(∞)²·# Pic(A)· |heη, eFi|² he_F, e_Fi . Here

• P(F, K) is a “period constant” defined inTheorem 4.8.

• f_K(∞)is the residue degree of∞in K/k

• Pic(A)is the ideal class group ofA.

• eF ∈ Pic(X)⊗_Z C generates the one dimensional “F-eigenspace” via the Jacquet- Langlands correspondence (cf.Theorem 4.14).

• (F, F)n_F is the Petersson norm ofF (cf.Section 2.3).

• eη∈Pic(X)⊗_ZCis a divisor class coming from a “Gross point” with trivial conductor onX and the characterη. (cf.Section 4.4).

• h·,·i: Pic(X)×Pic(X)→C is the Gross height pairing onPic(X)(cf. Section 4.3).

Although the choice ofe_F is unique up to a non-zero scalar multiple, the formula in the above theorem is invariant under rescaling. The divisor classe_η =e_η(x)actually depends on the chosen Gross pointxonX; however, the Gross height ^|he_he^η(x),eFi|2

F,e_Fi is invariant asxvaries.

When the base fieldk is rational, this formula was first proved by Papikian [10] with the assumption that the leveln_F is a prime and inert inK(based upon the calculations in Rück- Tipp [12]).

To derive the formula in Theorem 0.1, one crucial step is to express the central critical value of the zeta integral in question as a Petersson inner product of the given newform F and a linear combination of “quaternionic” theta series (cf. Proposition 4.6 and Theorem 4.8).

Note that these theta series are also Drinfeld type forms. To accomplish this, our strategy is outlined as follows. Suppose the root number is positive. LetD be the quaternion algebra over k which is ramified precisely at the places in Σ(F, η). We take γ ∈ k^× so that the quadratic space(D,Nr_D/k)(whereNr_D/k is the reduced norm onD) decomposes into

(D,Nr_D/k) = (K,N_K/k)⊕(K, γN_K/k).

Here N_K/k is the norm on K/k. Applying a Siegel-Weil formula over function fields (cf.

Theorem 4.1), the central critical value of the Eisenstein series appearing in the zeta integral becomes a “theta integral” I with respect to the quadratic space (K, γN_K/k). Note that the “newform” Θ^η_K associated toη can be realized by quadratic theta series with respect to (K,NK/k). CombiningI with Θ^η_K, we are able to connect the “kernel function” in our zeta integral directly with quaternionic theta series having level higher thann_F (cf. Lemma 4.5).

The final step is to “lower” the level of these quaternionic theta series ton_F, which amounts to “enlarge” the corresponding EichlerA-order (cf. Lemma 4.7).

LetDbe the definite quaternion algebra overkramified precisely atv∈Σ. One can also vieweF as an automorphic form onD^×(A) := (D ⊗kA)^×. Normalizing the Haar measuredg onD^×(A)such that

hheF, e_Fii:=

Z

D^×A^×\D^×(A)

e_F(g)e_F(g)dg= 2

# Pic(A)heF, e_Fi, the above formula can be rewritten in an integral form:

L(F×η,0) =P(F, K)⁰·2·(F, F)n_F

hheF, e_Fii · Z

K^×A^×\A^×K

eF(a)η(a)d^×a

2

,

whereP(F, K)⁰ is defined inRemark 4.17 (3). This is parallel to the classical formula in [15, Proposition 7]. In particular, suppose that another place∞⁰ |n_F is non-split inKsuch that F, as a function onGL2(A), is also of Drinfeld-type with respect to ∞⁰ andη(FrobP⁰) = 1, whereP⁰ is the prime above ∞⁰. Replacing∞by∞⁰ gives the sameΣandD, and the level of F becomes nF∞/∞⁰. From the Jacquet-Langlands correspondence, it is clear that the correspondingeF, as an automorphic form onD^×(A), remains the same. Thus, both sides of the equality in Theorem 0.1 are invariant when switching∞and ∞⁰.

One application of Theorem 0.1 is to derive a Waldspurger-type formula, which connects the central critical values in question with the Fourier coefficients of the corresponding meta- plectic forms. Following the classical story, we first establish a Shimura-type correspondence Shbetween Drinfeld-type forms and metaplectic forms via theta series from pure quaternions (cf. Theorem 5.5). The Fourier coefficients of these metaplectic forms can be interpreted by Gross heights (cf. Lemma 5.7). Supposing that for the givenF andη, the central character

ω_F and the characterη=1_K are both trivial. The Rankin-typeL-functionL(F×1_K, s)can then be written as (cf. Lemma 3.2)

L(F×1K, s) =L(F, s)·L(F⊗χK, s),

where L(F ⊗χK, s) is the twisted L-function of F by the quadratic Hecke character χK. Applying Theorem 0.1, we then arrive at:

Theorem 0.2. Let F be a normalized Drinfeld-type newform of square-free level n_F with trivial central character. Suppose the cardinality ofΣ⁰:={primep dividingnF |λp(F) = 1}

is odd. For each imaginary quadratic fieldK satisfyingΣ(F,1K) = Σ⁰∪ {∞}, we have

L(F,0)L(F⊗χK,0) =P(F, K)· 4·(F, F)n_F

fK(∞)²·# Pic(A)·







Y

p|nF , pis unramified inK/k

4







−1

m(F, K)²,

wherem(F, K)is the “K-th” Fourier coefficient of the corresponding metaplectic formSh(F) (cf.Theorem 5.8).

The contents of this paper are as follows. Section 1 contains the basic notations used throughout this paper. In Section 2, we discuss the needed properties of Drinfeld-type auto- morphic forms. The Rankin-typeL-functionL(F×η, s)is introduced in Section 3. We choose a particular “flat section” in Section 3.1, and use the associated Eisenstein series to express L(F×η, s) as a zeta integral in Section 3.2. The explicit functional equation is verified in Section 3.3. Supposing that the root number is positive, our Gross-type formula is derived in Section 4. Making use of the Siegel-Weil formula stated in Section 4.1, the central critical valueL(F×η,0)can be written as the Petersson inner product of F and a specific quater- nionic theta series in Section 4.2. In Section 4.3 and 4.4, we briefly recall definite Shimura curves, the Gross height pairing and Gross points. In Section 4.5, we make an appeal to the key Hecke module homomorphism from the Picard group of definite Shimura curves to the space of Drinfeld type forms of the corresponding level. Also given there is a version of Jacquet-Langlands correspondence we rely on. In Section 5.2, we construct metaplectic theta series from pure quaternions. The Shimura-type correspondence between Drinfeld-type forms and metaplectic forms is established in Section 5.3. Finally, we prove our Waldspurger-type formula in Section 5.4. For completeness, we record the construction of quadratic theta series in the function field setting, with all the needed properties stated in the appendix.

1. Preliminaries 1.1. Basic setting. We fix the following notations:

• (Global)

k : a global function field with finite constant field Fq where q is odd, and Fq is algebraically closed ink.

• (Local)

kv : the completion ofkat a placev ofk.

ordv : the valuation map onkv.

Ov : the valuation ring inkv, i.e.Ov={a∈kv|ordv(a)≥0}.

πv : a chosen uniformizer inOv.

Fv : the residue field ofk_v, i.e.Fv=O_v/π_vO_v. degv : the degree ofv, i.e.degv= [Fv :Fq].

qv : the cardinality ofFv.

| · |v : the absolute value onkv, normalized by|a|v :=qv^−ordv(a)for every a∈kv.

• (Adelic)

A : the adele ring ofk, i.e.A=Q0

vkv, the restricted direct product with respect to theOv.

O_A : the maximal compact subring ofA, i.e.O_A=Q

vOv. A^× : the idele group ofk, i.e.A^×=Q0

vk_v^×, the restricted direct product with respect to theO^×_v.

| · |_A : the idele norm onA^×, i.e.|a|_A:=Q

v|a_v|_v for everya= (a_v)_v ∈A^×.

• (Divisors)

Div(k) : the divisor group ofk. We adopt the multiplicative notation so that every elementmin Div(k)is written as

m=Y

v

vordv(m). Div_≥0(k) : the monoid of effective divisors ofk.

degm : the degree ofm∈Div(k), i.e.degm=P

vdegv·ordv(m).

kmk : the norm ofm∈Div(k), i.e.kmk=q^degm.

div : the group epimorphism fromA^× ontoDiv(k)defined by a= (av)v7−→div(a) :=Y

v

v^ordv(av).

πm : the idele(π^ordv ^v(m))v∈A^× form∈Div(k).

Given divisorsm,n∈Div(k), we set (m,n) :=Y

v

v^{min{ordv(m),ordv(n)}} and [m,n] :=Y

v

v^{max{ordv(m),ordv(n)}}.

• (v-component) Letv be a place ofk. Let A^v :=Q0

v⁰6=vk_v⁰. Since A=A^v×k_v, one has the following natural embeddings

k_v ,→ A

a_v 7→ (0, a_v) and A^v ,→ A

a^v= (a_v⁰)_v⁰_6=v 7→ (a^v,0).

LetGbe a general linear groupGL_n forn≥1 or an orthogonal similitude group GO(V)of a quadratic spaceV overk. ExpressingG(A)as G(A^v)×G(k_v), one also gets two embeddings:

G(k_v) ,→ G(A)

gv 7→ (1, gv), and G(A^v) ,→ G(A) g^v 7→ (g^v,1).

Everyg∈G(A)can be written asg= (g^v, g_v)whereg^v∈G(A^v)andg_v∈G(k_v).

• (Ring of integers) Fix a place∞ofk, referred to as the place at infinity; and others are referred to as finite places ofk. LetAbe the ring of integers ofkwith respect to

∞, i.e.

A={a∈k|ord_v(a)≥0, ∀v6=∞}.

Identifying finite places ofkwith non-zero prime ideals ofA, every fractional (resp.

non-zero integral) ideal ofAcorresponds bijectively to a (resp. an effective) divisor of kwith support away from∞. Thus for every fractional ideal mofA, we still denote bym the corresponding divisor ofk(and vice versa) by abuse of notations.

SetO_A^∞:=Q

v6=∞O_v, the maximal compact subring ofA^∞, andO^×

A^∞ :=Q

v6=∞O_v^×, the maximal compact subgroup ofA^∞,×. The map div induces the following natural isomorphisms

A^∞,×/O^×_A∞∼=I(A) and k^×\A^∞,×/O_A^×∞ ∼= Pic(A),

whereI(A)is the group of fractional ideals ofA, andPic(A)is the ideal class group ofA. Herek^× embeds diagonally intoA^∞,×. Throughout this paper, every function onPic(A)is also identified as a function onI(A)via the surjectionI(A)Pic(A).

For convenience, we denote byaCAifais a non-zero integral ideal ofA.

• (Additive character) Fix a non-trivial continuous additive character ψ = A → C^× which is trivial onk. Here k embeds diagonally into A. For each place v of k, we denote by ψv : kv → C^× the restriction of ψ on kv, and let δv be the “conductor of ψ at v,” i.e. the maximal integer r such that π^−r_v O_v is contained in the kernel of ψ_v. It is known that P

vδ_vdegv = 2g_k −2, where g_k is the genus of k. By [21, Theorem 13, Section 12, chapter XIII], we may assume that the chosen additive characterψhas even conductor at every placevofk. Putδ= (π^δ_v^v)v∈A^×, and call δa differential idele belonging to ψ. It depends upon the choice of the uniformizers πv.

1.2. Imaginary quadratic function fields. LetK be a quadratic field extension over k.

We say that K is imaginary (with respect to the chosen ∞) if ∞ does not split in K/k.

LetχK be the quadratic character associated to K, i.e. χK is the non-trivial character on A^× with kernel k^× ·N_K/k(A^×K). Here AK (resp. A^×K) is the adele ring (resp. idele group) of K, andN_K/k is the norm of K/k. For each place v of k, put Kv :=K⊗vkv, and let χK,v:k^×_v → {±1} be the character defined by:

χ_K,v(a_v) =

(1, ifav∈NK/k(K_v^×),

−1, otherwise.

Then for everya= (av)v ∈A^×, one hasχK(a) =Q

vχK,v(av).

LetdK∈Div_≥0(k)be the discriminant divisor ofKoverk. SinceK/k is quadratic andq is odd, one has

dK = Y

placevofk:

vis ramified inK

v.

LetL(s, χ_K)be the HeckeL-function associated toχ_K, i.e.L(s, χ_K) =Q

vL_v(s, χ_K,v)where L_v(s, χ_K,v) :=

(1−χK,v(πv)q_v^−s)⁻¹, ifv-dK;

1, otherwise.

It is known that L(s, χK) has meromorphic continuation and functional equation with the symmetry betweens and 1−s. Let OK be the integral closure ofA in K. The ideal class group of O_K is denoted by Pic(O_K). SetFK to be the constant field of K and f_K(∞) to be the residue degree of∞in K/k. Then the class number formula (cf. [11, Theorem 5.9]) assures that

L(1, χK) = # F^×q

# F^×K

·q^1−gk· kdKk⁻¹² · # Pic(OK) fK(∞)·# Pic(A).

1.3. Definite quaternion algebras over function fields. A definite quaternion algebra D is a central simple algebra overk with dim_kD = 4 and ramified at ∞. Let n⁻CA be the product of primes ofAwhere Dis ramified, called the (finite) discriminant ofD/k. Let n⁺CA be coprime ton⁻. We call a ringR anEichler A-order of type(n⁺,n⁻) if R is an A-order of D such that Rp := R⊗AOp is a maximal Op-order inDp :=D ⊗kkp for each prime pCA with p -n⁺; and when p |n⁺, there exists an isomorphismi :Dp ∼= Mat2(kp) such that

i(Rp) = (

a b c d

∈Mat2(Op)

c∈π^ordp(n

+)

p Op

) .

Alocally-principal (fractional)right ideal I of R is anA-lattice in Dsuch thatI·R=I, and for each primepCA, there existsαp inD^×_p such that

Ip(:=I⊗AOp) =αpRp.

LetRI :={b∈ D :bI ⊂I} be the left order ofI, which is also an Eichler A-order of type (n⁺,n⁻). Two locally-principal right ideals I and I⁰ are called equivalent if there exists an elementb inD^× such thatI =b·I⁰. The left ordersR_I only depend on the ideal classes of Rup to isomorphism. LetCl(R)be the set of locally-principal right ideal classes ofR. For each classI ∈Cl(R), we setw^R_I := ^#(R

× I)

q−1 for any representativeI∈ I.

LetD_A^∞ :=D ⊗kA^∞andRb:=R⊗AO_A^∞. For each locally-principal right ideal I ofR, there exists an elementbI ∈ D^×_A∞ such that I=D ∩bIR. This induces a bijection betweenb the setCl(R)with the finite double coset spaceD^×\D^×

A^∞/Rb^×.

Given an elementbinD, its reduced trace and the reduced norm are denoted byTr_D/k(b) andNr_D/k(b), respectively. We have an involutionb7→b:= Tr_D/k(b)−bonD, which induces an order2permutationτonCl(R)defined byτ([I]) := [I⁻¹]for each class[I]∈Cl(R). Here I⁻¹:={b|b∈ Dsuch thatbI ⊂R}.

2. Drinfeld-type automorphic forms Givenm∈Div_≥0(k), letK0(m) :=Q

vK_vordv(m) where, for`≥0, K<sub>v</sub>` :=

a b c d

∈GL2(Ov)

c∈π_v^`Ov

.

FornCA, let

Y0(n) := GL2(k)\GL2(A)/k_∞^×K0(n∞).

Here the elements ink_∞^× are identified with the scalar matrices inGL2(k∞), and we embed GL2(k∞)into GL2(A)as in Section 1.1.

Definition 2.1. By aDrinfeld-type automorphic formF of level nCA,we mean aC-valued function F on the double coset space Y0(n) satisfying the following harmonicity property:

viewingF as a function onGL2(A), we have F

g

0 1 π_∞ 0

=−F(g) and X

κ∈GL2(O∞)/K∞

F(gκ) = 0, ∀g∈GL2(A).

Drinfeld-type automorphic forms can be viewed as the function field analogue of weight2 modular forms (cf. [4] and [16]). In the following, we recall the analytic properties of these forms, and refer the reader to [2, Chapter III Section 3] for further details.

2.1. Fourier expansions. LetF be a Drinfeld-type automorphic form of leveln. For each divisorm∈Div(k), them-th Fourier coefficientF^∗(m)ofF is given by

F^∗(m) :=

Z

k\A

F

δ⁻¹πm u

0 1

ψ(−u)du.

Here π_m ∈ A^× is taken in Section 1.1, andδ is the chosen differential idele of k; the Haar measureduis normalized such that the volume ofk\Ais one. Let

F₀^∗(m) :=

Z

k\A

F

δ⁻¹πm u

0 1

du.

Then it is clear thatF₀^∗(div(a)m) =F₀^∗(m)for everya∈k^×. For any (x, y)∈A^××A, one has the following Fourier expansion

F

x y 0 1

=F₀^∗(div(δx)) + X

λ∈k^×

F^∗(div(δλx))ψ(λy).

It is known thatF^∗(m) = 0unlessmis effective (cf. [20, Chapter III Proposition 1]). Moreover, put

m_∞:=∞^ord∞(m), and m^∞:= m m∞

.

Then the harmonicity ofF gives

F^∗(m) =km∞k⁻¹F^∗(m^∞), and F₀^∗(m) =km∞k⁻¹F₀^∗(m^∞).

Thus taking representativesa1, ...,ahfor the ideal classes ofA, the values ofF on

A^× A

0 1

are determined byF₀^∗(ai)andF^∗(m)fori= 1, ..., handmCA.

Given a fractional idealaofA, set ρ(a)F

(g) :=F

π_a 0 0 πa

g

, ∀g∈GL2(A).

This action only depends on the ideal class ofArepresented bya. LetB :=

∗ ∗ 0 ∗

∈GL2

, the standard Borel subgroup ofGL2. Then the values ofFonB(A)are completely determined by the Fourier coefficients

ρ(ai)F∗

(m), mCA, and ρ(ai)F∗ 0(aj).

Here a1, ...,ah are representatives for the ideal classes of A. From the surjectivity of the canonical mapB(A)Y0(n), we have thatF is also uniquely determined by these Fourier coefficients.

A Drinfeld-type automorphic formF of level nis said to have a central character ω_F if ωF :A^×→C^× is a character so that

F

a 0 0 a

g

=ωF(a)F(g), for alla∈A^×. We may identifyω_F with a character onk^×\A^×/k_∞^×O^×

A^∞ ∼= Pic(A)(and also a character on I(A)). Thenρ(a)F =ω_F(a)F for everya∈ I(A).

2.2. Hecke operators. LetM0(n)be the space of Drinfeld-type automorphic forms of level n. For each place v of k, the Hecke operator T_v onM₀(n)is defined by the following: for F∈ M0(n)andg∈GL2(A)

(TvF) (g) := X

u∈Fv

F

g

πv u 0 1

+µ_n∞(v)·F

g

1 0 0 πv

.

Hereµ_n∞(v) = 1ifv-n∞and0otherwise.

The harmonicity ofFimplies thatT_∞F =F for allF ∈ M0(n). SinceTvandTv⁰commute with each other for any placesv andv⁰, we define the Hecke operatorTm recursively for all mCA by the following:

T_mm⁰ :=T_m·T_m⁰, form andm⁰ relatively prime;

T_pl+2:=TpT_pl+1−µ_n∞(p)kpk ·ρ(p)T_pl, for any prime pCAandl∈Z≥0.

Suppose thatF is a Hecke eigenform, i.e. for each mCA, there exists λm(F)∈ C such thatT_mF=λ_m(F)·F.ThenF always has central character, and the eigenvaluesλ_m(F)can be read off by the Fourier coefficients ofF:

kmk ·F^∗(m) =λ_m(F)·F^∗(1).

We callF normalized ifF^∗(1) = 1.

2.3. Petersson inner product. GivennCA, a Drinfeld-type automorphic formF of level nis called acusp form if

Z

k\A

F

1 u 0 1

g

du= 0 for every g∈GL2(A).

Given two Drinfeld-type automorphic forms F₁ and F₂ of leveln, suppose one of them is a cusp form. ThePetersson inner product ofF1andF2 is given by:

(F1, F2)n:=

Z

k^×∞GL2(k)\GL2(A)

F1(g)F2(g)dg= X

[g]∈Y0(n)

F1(g)F2(g)µ([g]),

where for each double coset[g] ∈ Y0(n) represented by g ∈ GL2(A), the measure µ([g]) is normalized to be

µ([g]) := q−1

2·#(Pic(A))· 1

#(GL₂(k)∩gK₀(n∞)g⁻¹). SupposeFi has the central characterωi. We remark that

(F1, F2)n=







0, ifω16=ω2;

1 2·

Z

A^×GL₂(k)\GL₂(A)

F1(g)F2(g)d⁰g, ifω1=ω2.

Here the measured⁰gis induced by the Haar measure onGL2(A)normalized so thatK0(n∞) has volume1.

A Drinfeld-type cusp formF of levelnis called anold formifF is aC-linear combination of

F⁰

g

1 0 0 π_n⁰⁰

, forg∈GL2(A),

where F⁰ is a Drinfeld-type cusp form of level n⁰ with n⁰n⁰⁰ | n and n⁰ 6= n. Let F be a Drinfeld-type cusp form of levelnwhich is also a Hecke eigenform. In addition, if F is also orthogonal (with respect to Petersson inner product) to all the old forms of leveln, then we callF anewformof leveln.

2.4. L-function associated to Drinfeld-type newforms. Given a newform F of level n with central character denoted byω_F, it is known that:

(1) For primep-n, the Hecke eigenvalueλ_p(F)satisfies the so-called Ramanujan bound:

|λp(F)| ≤2· kpk¹² (cf. [3]).

(2) For primep|n, one hasλ_p(F)²=ω_F(p)iford_p(n) = 1and0otherwise.

TheL-function associated to a normalized newformF of levelnwith central characterω_F is defined by

L(F, s) := X

mCA

F^∗(m) kmk^s

which is absolutely convergent on Re(s)>1/2. SinceF is a Hecke eigenform, we can write L(F, s)as the following Euler product:

L(F, s) = Y

placesvofk

Lv(F, s),

whereL∞(F, s) := (1−q∞^−(s+1))⁻¹, and for every primepCA,

Lp(F, s) := (1−λp(F)kpk^−(s+1)+µn(p)·ωF(p)· kpk^1−2(s+1))⁻¹.

Moreover, it has analytic continuation to the whole s-plane, (in fact, a polynomial in q^−s) and satisfies the following functional equation (cf. [20, Theorem 2 in Chapter 7]):

L(F, s) =w(F)·q^(4−4gk)skn∞k^−sL(F,−s),

wherew(F), called theroot number associated to L(F, s), satisfies thatw(F)² =ωF(n). In particular, whennis square-free, one hasw(F) =−Q

p|n −λp(F) .

2.5. Whittaker functions. For each Drinfeld-type cusp form F of level n, the Whittaker functionWF associated to F is the following function on GL2(A):

WF(g) :=

Z

k\A

F

1 u 0 1

g

ψ(−u)du, ∀g∈GL2(A).

One sees that W_F

1 x 0 1

g

=ψ(x)·W_F(g) ∀x∈A, and F(g) = X

α∈k^×

W_F

α 0 0 1

g

. SupposeF is a normalized nerform. LetWF,v:=WF|_GL2(kv)(hereGL2(kv)is embedded into GL2(A)as in Section 1.1). Then

W_F(g) =Y

v

W_F,v(g_v), ∀g= (g_v)_v ∈GL₂(A).

Lemma 2.2. (cf. [1] Exercise 4.6.2) Let F be a normalized newform of square-free level n with central characterω_F. (1) Forv-n∞, one has

WF,v

π_v^r−δv 0

0 1

=







q^−r_v · α⁽¹⁾_v (F)^r+1−α⁽²⁾_v (F)^r+1

α⁽¹⁾_v (F)−α⁽²⁾_v (F) , forr≥0;

0, forr <0;

Hereα⁽¹⁾v (F) andα⁽²⁾v (F)are the two roots ofX²−λ_v(F)X+ω_F(v)q_v. (2)Forv|n∞, one has

WF,v

π^r−δ_v ^v 0

0 1

=

λ_v(F)^r·q^−r_v , forr≥0;

0, forr <0;

WF,v

π_v^r−δv 0

0 1

0 1

−1 0

=

−λv(F)^r·q^−r−1_v , forr≥ −1;

0, forr <−1.

3. Rankin-typeL-functions

Let K be an imaginary quadratic extension over k, and choose a continuous character η: Gal(k^sep/K)→C^×. Via class field theory the characterη can be identified with a Hecke character on the idele class groupK^×\A^×K (and also a character onA^×K). We embedA^× into A^×K by

A^×= 1⊗A^×⊂(K⊗_kA)^× =A^×K, and putωη :=η

A^×·χK, whereχK is the quadratic character associated toK(introduced in Section 1.2). Via the theta correspondence, the characterη corresponds to an automorphic representation onGL2(A)with central character equal to ωη.

LetF be a normalized Drinfeld-type newform of square-free leveln_F with central character ωF. Throughout of this section, we assume that:

C.1 η is unramified everywhere andη(K_∞^×) = 1, where K_∞^× ,→A^×K is the∞-component;

C.2 ωF·ωη=χK.

LetO_K be the integral closure ofAin K. By the condition C.1, we may considerη as a character onPic(O_K), and also as a character on the ideal groupI(O_K)ofO_K. The Hecke L-functionL(η, s)associated toη can then be written as the following:

L(η, s) = (1−q^−f_∞^K(∞)s)⁻¹ X

MCOK

η(M)

kMk^s = (1−q^−f_∞^K(∞)s)⁻¹ X

mCA

Rη(m) kmk^s . HerefK(∞)is the residue degree of∞in K/k, and

Rη(m) := X

A∈Pic(OK)

η(A)·R_A(m) with R_A(m) := #{ACOK|A∈ A, N_K/k(A) =m}, whereN_K/k(A)is the ideal ofAgenerated by all the normsN_K/k(x)withx∈Afor allACO_K.

Let

L(F, η, s) :=(1−q^−f_∞^K(∞)(1+s))⁻¹· X

mCA

F^∗(m)Rη(m) kmk^s

!

, Re(s)> 1 2. Put L^nF(s, χK) :=Q

v-nF∞Lv(s, χK,v). TheRankin-type L-function L(F×η, s) associated toF andη is:

L(F×η, s) :=L^nF(2s+ 1, χ_K)·L(F, η, s).

Remark 3.1. The Rankin-type L-function can be defined in complete generality via “local integrals”, cf. Jacquet & Piatetski-Shapiro & Shalika [7, (2.7)].

Note thatL(F, η, s) =Y

v

Lv(F, η, s), where

Lv(F, η, s) :=











(1−q∞^{−fK(∞)(1+s)})⁻¹, ifv=∞,

X

n≥0

F^∗(pⁿ)R_η(pⁿ)

kpk^ns , ifv=pCA.

ThusL(F×η, s)can be expressed as an Euler productQ

vLv(F×η, s), where L_v(F×η, s) =L_v(F, η, s)·

(Lv(s, χK,v) ifv-nF∞,

1 otherwise.

We want to connect the local factors ofL(F×η, s) with the local factors ofL(F, s)and L(η, s). In Section 2.4, we saw that

Lv(F, s) = 1−α_v⁽¹⁾(F)q^−(s+1)_v ⁻¹

1−α⁽²⁾_v (F)q_v^−(s+1)−1 , where:

• whenv-nF∞,α⁽¹⁾v (F)andα⁽²⁾v (F)are two complex conjugate roots of the quadratic polynomialX²−λv(F)X+ωF(v)qv;

• whenv|nF∞,α⁽¹⁾v (F) :=λv(F)andαv⁽²⁾(F) = 0.

Also writeL(η, s)as an Euler productQ

vLv(η, s)with L_v(η, s) = 1−c⁽¹⁾_v (η)q_v^−s−1

1−c⁽²⁾_v (η)q_v^−s−1

,

where (viewingηas a character on Div(K)via the natural epimorphismDiv(K)I(OK)):

• whenv=w₁w₂ splits inK/k, putc⁽ⁱ⁾v (η) :=η(w_i)fori= 1,2;

• whenvis inert in K/k, putc⁽ⁱ⁾v (η) := (−1)ⁱp

η(w)where w|v;

• whenvis ramified inK/k, putc⁽¹⁾v (η) :=η(w)wherew|v, and c⁽²⁾v (η) := 0.

Then:

Lemma 3.2. For each placev ofk, we have Lv(F×η, s) = Y

1≤i₁,i₂≤2

1−α⁽ⁱ_v¹⁾(F)c⁽ⁱ_v²⁾(η)·q_v^−(s+1)−1

.

In particular, whenωF andη=1K are both trivial, we get L(F×1_K, s) =L(F, s)·L(F⊗χ_K, s), whereL(F⊗χK, s) =Q

vLv(F⊗χK, s)with Lv(F⊗χK, s) :=

((1−λ_v(F)χ_K,v(π_v)q^−(1+s)v +µ_n∞(v)ω_F(v)q^1−2(s+1)v )⁻¹, if v-d_K,

1, otherwise.

Proof. The first statement follows from [1, Lemma 1.6.1], and the second statement is then straightforward by comparing the localL-factors of both sides.

Remark 3.3. Let%F be the degree two Galois representation associated toF. Then the first statement of Lemma 3.2 shows thatL(F×η, s−1/2)is equal to theL-function of the Galois representation%F⊗Ind^k_K(η)ofGal(k^sep/k).

We now study the analytic properties ofL(F×η, s)by the Rankin-Selberg method.

3.1. Eisenstein series. We first give a brief review of Eisenstein series onGL2(A). Further details are referred to [1, Section 3.7]. Let χ1, χ2 be two unitary Hecke characters on the idele class group k^×\A^×. For s ∈ C, let Ind_χ1,χ2(s)be the space consisting of the smooth (i.e. locally constant) functionsΦon GL₂(A)satisfying

Φ

y₁ z 0 y2

g

=|y1|^s+_A ¹²|y2|^−s−_A ¹²χ1(y1)χ2(y2)Φ(g), g∈GL2(A), z∈A, y1, y2∈A^×. For each placev ofk, putχi,v:=χi

_k×

v fori= 1,2. We can writeIndχ₁,χ₂(s)as a restricted tensor product⊗⁰_vIndχ_1,v,χ_2,v,v(s).

Take a smooth functionφon GL2(O_A)which satisfies φ

u1 x 0 u2

g

=χ1(u1)·χ2(u2)·φ(κ),

for allκ∈GL₂(O_A), x∈O_A andu₁, u₂ ∈O^×

A. We can extendφto a flat section Φ_φ(·, s):

forg=

y1 z 0 y₂

κ∈GL2(A)wherey1, y2∈A^×,z∈A, and κ∈GL2(O_A), put Φφ(g, s) :=|y1|^s+_A ¹²· |y2|^−s−_A ¹² ·χ1(y1)·χ2(y2)·φ(κ).

Then it is clear that for eachs∈C,Φφ(·, s)is a well-defined function in Indχ₁,χ₂(s)and Φφ(κ, s) =φ(κ), ∀κ∈GL2(O_A).

The Eisenstein seriesE(Φ_φ, s,·)onGL₂(k)\GL₂(A)is defined by the following:

E(Φφ, s, g) := X

γ∈B(k)\GL2(k)

Φφ(γ·g, s), ∀g∈GL2(A).

Here B is the standard Borel subgroup ofGL2 introduced in Section 2.1. It is known that E(Φφ, s, g)converges absolutely when Re(s)>1/2. Moreover,E(Φφ, s, g)has a meromorphic continuation ins∈C(in fact, a rational function inq^−s), and satisfies the following functional equation (cf. [1, Theorem 3.7.2]):

E(Φφ, s, g) = E(M(s)Φφ,−s, g), (3.1.1)

whereM(s) : Indχ₁,χ₂(s)→Indχ₂,χ₁(−s)is the intertwining operator M(s)Φ(g) :=

Z

A

Φ

0 −1 1 0

1 u 0 1

g

du.

The Haar measureduis self-dual with respect to the fixed additive characterψ. This integral is convergent forRe(s)>1/2and also has “meromorphic continuation” to the complexs-plane (cf. [1, p.355]).

For our purposes, we consider the case whenχ1 =1, the trivial character and χ2 =χK

where K is a given imaginary quadratic field extension over k. Let F be a normalized Drinfeld-type newform of square-free leveln_F. We shall partition the collection of places of kinto the following four subsets (depending onF andK):

S :={ placesv ofk

v divides neithern_F∞nord_K}, SF :={ placesv ofk

v dividesnF∞, but vdoes not dividedK}, SK :={ placesv ofk

v dividesdK, but v does not dividenF∞}, S_F,K :={ placesv ofk

v divides bothd_K andn_F∞}.

Definition 3.4. For each placevofk, we choose

φ^]_v(x) :=χK,v(d)·1Kv(x) +ξv·χK,v(c)·1K^c_v(x), ∀x=

a b c d

∈GL2(Ov), whereχK,v is the character introduced in Section 1.2,K^c_v:= GL2(Ov)− Kv and

ξv:=











1, ifv∈ S;

−q⁻¹_v , ifv∈ S_F; qv^−1/2·ε_K,v(1), ifv∈ SK;

−λv(F)·q^−1/2v ·η(w)·εK,v(1), ifv∈ SF,K.

Here ε_K,v(1) is the Weil index associated to the quadratic space(K_v,N_K/k)(cf. Appendix A.1), andw is the place ofK lying abovev.

We setφ^]:=⊗vφ^]_v, which is a function onGL₂(O_A)satisfying φ^]

u1 x 0 u2

κ

=χ_K(u₂)·φ^](κ),

for allκ∈GL2(O_A), x∈O_A and u1,u2∈O^×_A. The associated flat section to φ^] is denoted by

Φ^]:=⊗vΦ^]_v∈Ind1,χ_K(s).

For eachm∈Div_≥0(k), set K⁺₀^K(m) :=K₀(m)∩GL⁺₂^K(O_A)where GL⁺₂^K(O_A) :={κ∈GL2(O_A)|det(κ)∈N_K/k(A^×K)}.

Then:

Lemma 3.5. (1)For every g∈GL₂(A)andκ= a b

c d

∈ K⁺₀^K([n_F∞,d_K]), we have

Φ^](gκ) =



 Y

v|[nF∞,dK]

χ_K,v(d_v)



Φ^](g).

(2)For each placev of k,Mv(s)Φ^]_v(·, s) =Φe^]_v(·,−s)·^F,K_v (s)where

Φe^]_v:= (χK,v◦det)·Φ^]_v and ^F,K_v (s) :=



















qv^δv2 · Lv(2s, χK,v)

L_v(2s+ 1, χ_K,v), if v∈ S;

−qv^δv2 · Lv(−2s, χK,v)

Lv(1−2s, χK,v), if v∈ SF; q^δv

−1

v 2 ·ε_K,v(1), if v∈ S_K;

−λv(F)·η(w)·q

δv−1

v 2 ·εK,v(1), if v∈ SF,K. Proof. The argument of(1)is straightforward. For(2), we only prove the case whenv∈ SF,K

since the other cases are similar.

Takingv∈ SF,K, we have definedφ^]_v in Definition 3.4 by: for κ∈GL2(Ov), φ^]_v(κ) :=χ_K,v(d)·1_Kv(κ)−λ_v(F)·q⁻v¹² ·ε_K,v(1)·χ_K,v(c)·η(w)·1_Kc

v(κ).

Herewis the place ofK lying abovev. Giveng∈GL₂(k_v), write g=

y1 z 0 y₂

κ∈GL2(kv), wherey1, y2∈k_v^×, z∈kv andκ=

a b c d

∈GL2(Ov).

(i) Suppose thatκ∈ Kv. We have M_v(s)Φ^]_v(g, s) =

Z

k_v

Φ^]_v

0 −1

1 0

1 nv

0 1

g

dn_v

=|y2|^s+1/2_v · |y1|^−s+1/2_v ·χ_K,v(y₁)χ_K,v(a)· Z

k_v

Φ^]_v

0 −1 1 0

1 nv

0 1

dn_v

=|y₂|^s+1/2_v · |y₁|^−s+1/2_v ·χ_K,v(y₁)χ_K,v(a)· Z

k_v

Φ^]_v

0 −1 1 nv

dn_v.

It can be verified that Z

kv

Φ^]_v

0 −1 1 nv

dnv =−λv(F)·q^δv

−1

v 2 ·εK,v(1)·η(w).

Therefore

Mv(s)Φ^]_v(g, s) =− |y2|^s+1/2_v · |y1|^−s+1/2_v ·χK,v(y1)χK,v(a)·λv(F)·q^δv

−1

v 2 ·εK,v(1)·η(w)

=

−λ_v(F)·η(w)·q^δv

−1

v 2 ·ε_K,v(1)

·(χ_K,v◦det(g))·Φ^]_v(g,−s).

(ii) Supposeκ=

a b c d

∈ K^c_v. Note that κ=

1 ^a_c 0 1

0 1

−1 0

−c −d

0 −c⁻¹(det(κ))

.

ThusMv(s)Φ^]_v(g, s)is equal to

|y2|^s+1/2_v · |y1|^−s+1/2_v ·χK,v(y1)· Z

k_v

Φ^]_v

−c 0 cnv −c⁻¹det(κ)

dnv. Since

Z

kv

Φ^]_v

−c 0 cn_v −c⁻¹det(κ)

dnv=qv^δv2−1·χK,v(−1)·χK,v(c⁻¹)·χK,v(det(κ)), we obtain that

Mv(s)Φ^]_v(g, s) =|y2|^s+1/2_v |y1|^−s+1/2_v χK,v(y1)· q

δv 2−1

v ·χK,v(−1)·χK,v(c⁻¹)·χK,v(det(κ))

=

−λ_v(F)·η(w)·q^δv

−1

v 2 ·ε_K,v(1)

·(χ_K,v◦det(g))·Φ^]_v(g,−s).

The last equality comes fromχ_K,v(−1) =ε_K,v(1)² andλ_v(F)²·η(w)²= 1.

Let^F,K(s) :=Q

v^F,K_v (s). From the functional equation ofL(s, χ_K):

L(s, χK) =q^{(gk−1+degdK/2)(1−2s)}L(1−s, χK), we can express^F,K(s)as

^F,K(s) = (−1)^#Σ·q^{(2−2gk)(2s)}· k[nF∞,d_K]k^−2s· L^nF(1−2s, χ_K) L^nF(1 + 2s, χK), where Σ = Σ(F, η)

:={∞} ∪

prime pdividingn_F

pis ramified inK/kandλ_p(F)·η(P) = 1 (3.1.2)

∪ {primep dividingnF

p is inert inK/k}.

Here P is the prime ideal of OK lying above p. Therefore by the equation (3.1.1) and Lemma 3.5 (2), we obtain the explicit functional equation ofE(Φ^], s, g):

Lemma 3.6.

E(Φ^], s, g) =^F,K(s)·E(eΦ^],−s, g), ∀g∈GL2(A).

3.2. Zeta integral. Let Θ^η_K be the quadratic theta series associated to the character η (constructed in Definition A.7). Given a flat section Φ(·, s) ∈ Ind1,χK(s), the global zeta integral associated toF,η andΦis:

Z(F, η,Φ, s) :=

Z

A^×GL₂(k)\GL₂(A)

F(g)Θ^η_K(g)E(Φ, s, g)dg.

(3.2.1)

Here· is the complex conjugation. This integral is a meromorphic function on the complex s-plane.