By the Siegel-Weil formula for “coherent” Siegel-Eisenstein series, we can relate the non-singular Fourier coefficients of the derivative in question to the arithmetic of quadratic forms

FUNCTION FIELDS

FU-TSUN WEI

Abstract. The aim of this article is to study the derivative of “incoherent” Siegel- Eisenstein series on symplectic groups over function fields. By the Siegel-Weil formula for “coherent” Siegel-Eisenstein series, we can relate the non-singular Fourier coefficients of the derivative in question to the arithmetic of quadratic forms. Restricting to the spe- cial case when the incoherent quadratic space has dimension 2, we explicitly compute all the Fourier coefficients, and connect the derivative with the special cycles on the coarse moduli schemes of rank 2 Drinfeld modules with “complex multiplication.”

Introduction

The Siegel-Eisenstein series on symplectic groups are closely related to the arithmetic of quadratic forms. By the Siegel-Weil formula (cf. [22], [11], and [20]), certain special values of “coherent” Siegel-Eisenstein series are expressed in terms of the theta series associated with coherent quadratic spaces. This can be applied to the special values of automorphicL- functions via integral representations, e.g. Rankin-Selberg method or the Garrett-type repre- sentation for triple productL-functions (cf. [24], [6], and [19]). Using the Siegel-Weil formula, the central critical derivatives of “incoherent” Siegel-Eisenstein series can be also understood by theta series. Furthermore, in the number field case, the work of Kudla, Rapoport, and Yang (cf. [8], [9], and [13]) provides evidences of the relations between the derivatives in question and arithmetic cycles on Shimura varieties of orthogonal type. More precisely, the non-singular Fourier coefficients of such derivatives are essentially the “degree” of the corre- sponding cycles. This observation can be viewed as an analogue of the Siegel-Weil formula for the derivatives, which is a key ingredient in the work of Yuan-Zhang-Zhang [23] on the central critical derivatives of triple productL-functions. The aim of this article is to study the derivatives of incoherent Siegel-Eisenstein series over function fields, and connect the non- singular Fourier coefficients with arithmetic cycles on the moduli space of Drinfeld modules in a simple case. The result obtained in this paper provides an evidence in the function field setting of the phenomenon first observed by Kudla.

Letkbe a global function field with constant fieldF^q. Supposeqis odd. Take a positive odd integernand letC={Cv}v be anincoherentquadratic space overk(defined in Section 1.2) with dimensionn+ 1. For each Schwartz function ϕonCⁿ, the associated Siegel-Eisenstein seriesE(·, s,Φϕ) on the symplectic group Sp_n(Ak) (where Ak is the adele ring ofk) always vanishes at the central critical points= 0 (cf. Theorem 3.2). To explore the central critical derivative, we first restrict ourselves to the special case when n = 1. Then the associated quadratic characterχ_C ofConk^×\A^×k must be non-trivial. LetKbe the quadratic extension ofkcorresponding to the kernel ofχ_C via class field theory. We take a place∞ofknot split inK and α∈k^× such that Cv is isomorphic to (Kv, α·N_K/k) for every place v6=∞. Here

2010Mathematics Subject Classification. 11M36, 11G09,11R58.

Key words and phrases. Function field, Eisenstein series, Drinfeld modules.

1

K_v =K⊗kk_v andk_vis the completion ofkatv;N_K/kis the norm form ofKoverk. Choose a particular Schwartz functionϕ^(α)∈S(C(Ak)) so that the associated “Siegel section” Φ_ϕ(α)

is a “new” vector in the spaceR₁(C) consisting all of the sections Φϕ forϕ∈S(C(Ak)) (see (6.1) for the precise definition ofϕ^(α)). We are interested in the central critical derivative

η^(α):= ∂

∂sE(·, s,e Φ_ϕ(α)) _s=0,

whereE(·, s,e Φ_ϕ(α)) is the Siegel-Eisenstein series associated withϕ^(α)modified by the Hecke L-function ofχC (see (6.2)). Via Rankin-Selberg method,E(·, s,e Φ_ϕ(α)) shows up in the study of Rankin-typeL-functions associated with “Drinfeld type” automorphic forms on GL₂(Ak) (cf. [2]). This is our motivation to first target at this particular function η^(α). For each β ∈k^×, we show that theβ-th Fourier coefficients ofη^(α) are expressed by representation numbers of β as the corresponding norm forms (cf. Proposition 6.9). In particular, these non-zero Fourier coefficients ofη^(α) are related to special cycles on the “compactification”

XO_K of the coarse moduli scheme of rank one Drinfeld OK-modules (whereOK is the ring of functions inK regular outside∞). To be more concrete, the main result of this paper is in the following.

Theorem 0.1. (cf. Theorem 7.14) For each y ∈ A^×k and β 6= 0, there exists an algebraic 0-cyclez(y, β)onXOK such that the Fourier coefficientη_β^(α),∗(y)is equal to:

− χ_C(y)|y|_Ak

f_∞·# Pic(A)·degz(y, β),

where f_∞ is the residue degree of ∞ in K/k, and A is the ring of functions in k regular outside∞.

The above theorem can be viewed as a function field analogue of Kudla-Rapoport-Yang’s result in [13]. The moduli problem of Drinfeld modules are recalled in Section 7.1, and we refer the readers to [3] and [14] for further details. The algebraic cyclez(y, β) is constructed via “special morphisms” defined in the following. Let (L, φ) and (L⁰, φ⁰) be rank one Drinfeld OK-modules over a schemeS. Then φ(resp.φ⁰) induces a left action of Mat2(OK) onL^⊕2 (resp.L^0⊕2). LetDαbe the quaternion algebraK+Kjαoverkwherej_α² =−αandjαa= ¯ajα

(here (a7→¯a) is the non-trivial automorphism onK overk). The reduced norm form onD_α can be expressed byN_K/k⊕(α·N_K/k). We fix aK-algebra isomorphismK⊗_kD_α∼= Mat2(K) defined by:

a⊗1 7−→

a 0 0 a

, ∀a∈K,

1⊗(a₁+a₂j_α) 7−→

a1 a2

−αa2 a1

, ∀a₁+a₂j_α∈ D_α.

A special morphism f : (L^⊕2, φ) → (L^0⊕2, φ⁰) is a morphism from L^⊕2 to L^0⊕2 (as group schemes overS) satisfying

f φ_1⊗d=φ⁰_1⊗df and f φ_a⊗1=φ⁰_a⊗1f, ∀a∈OK andd∈O_Dα :=Dα∩Mat2(OK).

When Dα splits at ∞ (i.e. Dα⊗k k_∞ ∼= Mat2(k_∞)), one can associate (L^⊕2, φ) (resp.

(L^0⊕2, φ⁰)) a Dα-elliptic sheave Eφ (resp. Eφ⁰) with “complex multiplication” by OK viaφ (resp.φ⁰, cf. [15] and [16]). Then the set of special morphisms can be identified with

{f ∈Hom(Eφ,Eφ⁰) :f φa=φaf, ∀a∈OK}.

Here the homomorphisms betweenDα-elliptic sheaves is allowed to “shift the indices” (cf. [16]

Definition 4.2). Moreover, whenS= Spec(κ) whereκis a perfect field, (L^⊕2, φ) andL^0⊕2, φ⁰) are Anderson’s “pure abelianA-modules” (cf. [1]) withO_K⊗_AO_Dα multiplication, which can

be viewed as analogue of abelian surfaces with quaternion and complex multiplication. In particular when α=−1, we have D_α ∼= Mat2(k) and the set of special morphisms can be realized by

{f ∈Hom_A (L, φ),(L⁰, φ⁰)

|f φ_a⊗1=φ⁰_a⊗1f, ∀a∈O_K}.

We remark that the constant Fourier coefficient η₀^(α),∗(y) are related to the “Taguchi height” of rank 2 DrinfeldA-modules with “complex mulitplication” by OK. The Taguchi height of a Drinfeld module (cf. [18] and [21]) is viewed as a natural analogue of the Faltings height of abelian varieties over number fields. Letφ be a Drinfeld module overk of rank 2 with complex mulitplication byOK. Comparing the formula of η₀^(α),∗(y) in Lemma 6.2 and the Taguchi height ofφin [21, Corollary 0.2], we then expressη₀^(α),∗(y) as (cf. Lemma 7.15):

η₀^(α),∗(y) = 2χ_K(y)|y|_Ak# Pic(O_K) f∞# Pic(A)

·

ln|y|_Ak−2˜h_Tag(φ)−(g_k−1) lnq−ζ_A⁰(0)

ζA(0) +(−1)^f∞q^deg∞+ 1−2^f∞

f∞(q^deg∞+ 1) ·deg∞lnq

. Here ζA(s) := Q

v6=∞(1−q^−sdegv)⁻¹ is the zeta function of A. This provides a geometric intepretation of the constant Fourier coefficientη^(α),∗₀ (y).

The description of the non-zero Fourier coefficients of η^(α) in Proposition 6.9 is in fact derived from a general pattern in Theorem 5.1. We give a short discussion in the following.

Let n be an arbitrary positive odd integer and C be an incoherent quadratic space with dimensionn+ 1. for eachβ ∈Sym_n(k) (i.e.β is symmetric) with detβ 6= 0, let

Diff(β,C) :=

place vofk

Hasse_v(C)6=χ_Cv(detβ)·(detβ,−(−1)^(n+1)/2)_v·Hasse_v(β) , where Hasse_v(C) (resp. Hasse_v(β)) is the Hasse invariant of C (resp.β) at the placev of k, and (·,·)v is the local Hilbert quadratic symbol at v. It can be shown that the cardinality of Diff(β,C) provides a lower bound for the vanishing order of β-th Fourier coefficient of E(·, s,Φϕ) at s = 0 (cf. Proposition 4.6). When Diff(β,C) = {v0}, let Vβ be the coher- ent quadratic space over k whose associated character χ_Vβ = χ_C and the Hasse invariant Hasse_v(V_β) = Hasse_v(C) for every place v 6= v₀. Then the β-th Fourier coefficient of the central critical derivative can be understood by the theta series associated withVβ:

Theorem 0.2. (cf. Theorem 5.1) Let C be an incoherent quadratic space over k with even dimensionmand taken=m−1. Forβ∈Sym_n(k)withdetβ 6= 0, supposeDiff(β,C) ={v0} and the associated quadratic space Vβ is anisotropic. Then for each pure-tensor Schwartz functionϕ=⊗vϕv∈S(C(Ak)ⁿ) anda∈GLn(Ak), the Fourier coefficient

∂

∂sE_β^∗(a, s,Φϕ) _s=0

= 2· W_v⁰

0,a_v0∗β(0,Φv₀,ϕ_v0) Wv₀,a_v0∗β(0,Φ_v0,

ϕev0)·Θ^∗_β(a,ϕ),e

wherea_v0∗β =^ta_v0βa_v0,ϕe=⊗vϕe_v ∈S(V_β(Ak)ⁿ)is any pure-tensor Schwartz function so that:

(i) forv6=v0,ϕev=ϕv (here we identify Vβ,v withCv);

(ii) the Whittaker functionWv₀,a_v0∗β(s,Φv₀,ϕe_v0)associated with ϕev₀ ∈S(V_β,vⁿ

0)does not vanish ats= 0;

andΘ^∗_β(a,ϕ)e are Fourier coefficients of the theta seriesΘ(g,ϕ)e introduced inTheorem 3.1.

We refer the readers to Section 5 for further details. The flexibility of the choice ofϕev₀ is beneficial to the calculation of the valuesWv₀,a_v0∗β(0,Φv₀,ϕe_v0) and Θ^∗_β(a,ϕ). Moreover, onee

has

Θ^∗_β(a,ϕ) =e χ_C(deta)|deta|ⁿ⁺¹²

Ak · Z

O(V_β)(k)\O(V_β)(Ak)

X

x∈V nβ, QVβ(x)=β

ϕ(he ⁻¹xa)dh,

where O(Vβ) is the orthogonal group ofVβ, QV_β denotes the quadratic form onVβ, anddh is the O(V_β)(Ak)-invariant measure having total mass 1. This observation says that theβ-th Fourier coefficients of the central critical derivative is essentially the representation number of β as the quadratic formQV_β. The assumption ofVβ being anisotropic in Theorem 0.2 is due to the restriction of the function field analogue of Siegel-Weil formula in [20]. We can remove this assumption if a stronger version of Siegel-Weil formula over function fields is verified, which will be explored in a future work.

The structure of this article is organized as follows. We fix our basic notations in Section 1.1 and review the concept of the incoherent quadratic spaces in Section 1.2. In Section 2, we recall the Weil representation on the spaces of Schwartz functions on quadratic spaces over k and the relevant results we need. In Section 3, we introduce the Siegel-Eisenstein series on symplectic groups, and recall their analytic properties. Also included in Section 3 is the Siegel-Weil formula for the anisotropic case over function fields. In Section 4, we focus on the non-singular Fourier coefficients of incoherent Siegel-Eisenstein series, and determine their vanishing order at the central critical point by Whittaker functions. In Section 5, we study the derivative of the non-singular Fourier coefficients of incoherent Siegel-Eisenstein series, and prove Theorem 0.2 via the Siegel-Weil formula. In Section 6, we concentrate on the special case when the incoherent quadratic spaceC has dimension 2, and compute explicitly all the Fourier coefficients ofη^(α). Finally, the algebraio-geometric aspects ofη^(α)is discussed in Section 7. We recall the moduli problem of Drinfeld modules in Section 7.1, and introduce the special morphisms between Drinfeld modules in Section 7.2. The proof of Theorem 0.1 and the formula of the constant Fourier coefficientη₀^(α),∗(y) are given in Section 7.3.

1. Preliminary

1.1. Basic setting. Let Fq be the finite field with q = p^r0 elements. Let k be a global function field with constant fieldFq, i.e.kis a finitely generated field extension overFq with transcendence degree one andFq is algebraically closed ink. In this article we always assume thatqisodd. For each placev ofk, letkv be the completion ofkat v, andOv denotes the valuation ring inkv. Choosing a uniformizerπv in Ov, we set Fv :=Ov/πvOv, the residue field atv, andqv to be the cardinality ofFv. Put degv := [Fv :Fq], called the degree ofv.

The absolute value onk_v is normalized to be:

|av|v:=q_v^−ordv(av)=q^{−degvordv(av)}, ∀av ∈kv. LetAk be the adele ring of k and set O_Ak :=Q

vO_v, the maximal compact subring ofAk. For each elementa= (av)v in the idele groupA^×_k, the norm|a|_Ak is defined as:

|a|_Ak :=Y

v

|a_v|_v.

Embeddingk(resp.k^×) intoAk(resp.A^×_k) diagonally, we have the product formula:|α|_Ak= 1 for everyα∈k^×. Throughout this article, fix a non-trivial additive characterψ:Ak →C^× which is trivial onk. For each placev ofk, letψv be the additive character onkvdefined by ψ_v(a_v) :=ψ(0, ...,0, a_v,0, ...) for eacha_v in k_v. We denote δ_v to be the “conductor of ψ at v,” i.e. the maximal integerrsuch that π_v^−rO_v is contained in the kernel ofψ_v. It is known

thatP

vδ_vdegv= 2g_k−2, whereg_k is the genus ofk.

Take a placevofk. Let (V, QV) be a non-degenerate quadratic space of dimensionmover kv. The bilinear form onV ×V is

hx, yiV :=QV(x+y)−QV(x)−QV(y), ∀x, y∈V.

The associated quadratic characterχV onk^×_v is defined by

χV(αv) := (αv,(−1)^m(m+1)2 detV)v, ∀αv∈k^×_v.

Here (·,·)v:k^×_v/(k_v^×)²×k^×_v/(k_v^×)²→ {±1} is the Hilbert quadratic symbol, i.e.

(av, bv)v =

(1, ifa_vX²+b_vY²=Z² has non-trivial solutions,

−1, otherwise;

and detV ∈k^×_v/(k^×_v)² is the discriminant ofV, i.e.

detV := det(hx_i, x_ji_V)_1≤i,j≤m for every basis{x₁, ..., x_m} ofV . Take a basis{x1, ..., xm} ofV such that the quadratic form becomes

QV(x) =

m

X

i=1

cia²_i, ∀x=

m

X

i=1

aixi∈V,

wherec_i∈k_v^× fori= 1, ..., m. The Hasse invariant ofV is:

Hasse_v(V) := Y

1≤i<j≤m

(c_i, c_j)_v.

For a global non-degenerate quadratic spaceW overk, setχW :=⊗χW_v :k^×\A^×_k → {±1}, whereWv=W⊗kkv, and call Hassev(Wv) the Hasse invariant of W atv.

1.2. Incoherent quadratic spaces. Fix a character χ = ⊗vχ_v : k^×\A^×k → {±1}. An incoherent quadratic spaceC={C_v}_v overkof dimension mwith characterχ is a collection of non-degenerate quadratic spacesCv overkv, indexed by the places ofk, such that

(i) dim_kvCv =mand χ_Cv =χ_v for every placev.

(ii) There exists a global non-degenerate quadratic spaceW of dimensionmoverkwith characterχ_W =χsuch thatW_v∼=C_v for almost allv.

(iii) (Incoherence condition) The product formula fails for the Hasse invariants:

Y

v

Hassev(Cv) =−1.

Remark 1.1. Let C be an incoherent quadratic space overk. Due to the incoherence condi- tion, there is no quadratic spaces W over k satisfying that Wv ∼=Cv for allv. Conversely, suppose a “coherent” collection C of quadratic spaces is given, i.e. C satisfies (i), (ii), and Q

vHassev(Cv) = 1. Then we can find a unique (up to isomorphism) quadratic spaceW over ksuch thatW_v∼=Cvfor everyv. Thus a non-degenerate quadratic space overkis also called acoherentquadratic space.

2. Weil representation on Schwartz spaces Given a positive integern, let Sp_n be the symplectic group of degreen, i.e.

Sp_n:=

(

g∈GL2n

tg

0 In

−In 0

g=

0 In

−In 0 )

.

We view Sp_n as an affine algebraic group over Fq. The Siegel-parabolic subgroup Pn is the parabolic subgroupMn·Nn, where

Mn:=

m(a) =

a 0 0 ^ta⁻¹

a∈GLn

and Nn:=

n(b) =

In b 0 I_n

b=^tb∈Matn

. Set Sym_n :={b=^tb∈Matn}. By the mapmand n, GLn and Sym_n are isomorphic to Mn

andNn respectively.

Take a placev of k. Let (V, Q_V) be a non-degenerate quadratic space overk_v with even dimensionm. The orthogonal group ofV is denoted by O(V), i.e.

O(V) ={h∈GL(V)|QV(hx) =QV(x), ∀x∈V}.

Let S(Vⁿ) be the space of Schwartz functions on Vⁿ, i.e. the space of C-valued functions onVⁿ which are locally constant and compactly supported. The (local)Weil representation ω_v(=ω_v,ψv) of Sp_n(k_v)×O(V) onS(Vⁿ) is determined by the following: for everyϕ_v∈S(Vⁿ) andx∈Vⁿ,

(ω_v(h)ϕ_v)(x) := ϕ_v(h⁻¹x₁, ..., h⁻¹x_n), ∀h∈O(V);

ωv

a 0 0 ^ta⁻¹

ϕv

(x) := χV(deta)|deta|^m/2_v ·ϕv(x·a), ∀a∈GLn(kv);

ωv

I_n b 0 I_n

ϕv

(x) := ψv

Trace b·Q⁽ⁿ⁾_V (x)

·ϕv(x), ∀b∈Sym_n(kv);

ωv

0 I_n

−In 0

ϕv

(x) := εv(V)ⁿ·ϕbv(x).

Here:

• Q⁽ⁿ⁾_V :Vⁿ →Sym_n(kv) is themoment map,i.e. for any x= (x1, ..., xn)∈Vⁿ, Q⁽ⁿ⁾_V (x) =

1

2hxi, xjiV

1≤i,j≤n

.

• ϕbv is the Fourier transform of ϕv:

ϕbv(x) :=

Z

Vⁿ

ϕv(y)·ψv(

n

X

i=1

hxi, yiiV)dy, ∀x= (x1, ..., xn)∈Vⁿ. The Haar measuredy=dy₁· · ·dy_n is chosen to beself dual, i.e.

bb

ϕ_v(x) =ϕ_v(−x), ∀x∈Vⁿ.

• εv(V) is theWeil index ofV, i.e.

εv(V) :=

Z

L

ψv QV(x) dx

for any sufficiently large O_v-latticeL in V. The Haar measure dxis also chosen to be self dual with respect to the pairingψ_v(h·,·i_V).

Given a character χ_v : k^×_v → C^×, letI_v(s, χ_v) be the space of locally constant C-valued functionsfv on Sp_n(kv) satisfying that

f_v(n(b)m(a)g) =χ_v(deta)|a|^s+v ⁿ⁺¹² f_v(g), ∀a∈GL_n(k_v), b∈Sym_n(k_v), g∈Sp_n(k_v).

The action of Sp_n(kv) on I(s, χv) is defined by right translation. Let Φv be the following Sp_n(kv)-equivariant homomorphism fromS(Vⁿ) toIv((m−n−1)/2, χV):

ϕv7−→Φv,ϕv :=

g7→ ωv(g)ϕv

(0) . Denote byRn(V) the image of Φv. We then have:

Theorem 2.1. (1)Let V be a non-degenerate quadratic space over k_v with even dimension.

ThenR_n(V)is isomorphic to S(V₀ⁿ), where V₀ⁿ ={x∈Vⁿ :Q⁽ⁿ⁾_V (x) = 0}. Moreover, the isomorphism is induced from the restriction ofΦv onS(V₀ⁿ).

(2)Let χv:k^×_v → {±1} be a quadratic character andnis a positive odd integer. Then I_v(0, χ_v) =R_n(V⁺)⊕R_n(V⁻)

whereV^±is the non-degenerate quadratic space overkvof dimensionn+1such thatχv=χ_V^± andHassev(V^±) =±1.

Proof. (1) follows from [20, Proposition B.1] and (2) follows from [10, Corollary 3.7]. Although the characteristic of the base local field is assumed to be 0 in [10], the argument actually works for the case of odd characteristic. We recall the steps of the proof for (2) in the following.

By the Frobenius reciprocity theorem, we know dim_CEnd_Sp

n(kv) I(0, χv)

≤2. In par- ticular, dim_CEndsp_n(k_v) I(0, χv)

= 1 ifχv ≡1 and dimV = 2. Using the twisted Jacquet functor onR_n(V⁺) andR_n(V⁻), we obtain thatR_n(V⁺) andR_n(V⁻) are inequivalent sub- modules ofI(0, χ_v) and

Rn(V⁺)6⊂Rn(V⁻), Rn(V⁻)6⊂Rn(V⁺).

Moreover, using the inner product (f₁, f₂) :=

Z

P_n(k_v)\Sp_n(k_v)

f₁(g)f₂(g)dg

onI(0, χv), we obtain thatI(0, χv) is completely reducible. Therefore the result holds.

Now, letW (resp.C) be a coherent (resp. incoherent) quadratic space over k with even dimension m. We have the global Weil representation ω := ⊗vωv on the Schwartz space S(W(Ak)ⁿ) (resp.S(C(Ak)ⁿ)). HereW(Ak) :=W⊗kAk=Q0

vWv andC(Ak) :=Q0

vCv. Let χ=χ_W (resp.χ_C) andI(s, χ) denotes the space of locally constantC-valued functionsf on Sp_n(Ak) satisfying that

f(n(b)m(a)g) =χ(deta)|a|^s+

n+1 2

Ak f(g), ∀a∈GL_n(Ak), b∈Sym_n(Ak), g∈Sp_n(Ak).

Let Φ be the Sp_n(Ak)-equivariant homomorphism from S(W(Ak)ⁿ) (resp. S(C(Ak)ⁿ) to Iv((m−n−1)/2, χ) defined by:

ϕ7−→Φϕ:=

g7→ ω(g)ϕ (0)

.

Denote byRn(W) (resp. Rn(C)) the image of Φ. Then Theorem 2.1 (2) implies that Corollary 2.2. Let χ : k^×\A^×k → {±1} be a quadratic character and n is a positive odd integer. Then

I(0, χ) = M

W

Rn(W)

!

⊕ M

C

Rn(C)

! ,

whereW (resp.C) runs through all the coherent (resp. incoherent) quadratic spaces of dimen- sionn+ 1 overkwith χ=χ_W (resp. χ_C).

3. Siegel-Eisenstein series

Fix a quadratic characterχ:k^×\A^×k → {±1}and a positive integern. Letf(·, s)∈I(s, χ) be a flat section, i.e. for every κ∈ Sp_n(O_Ak), f(κ, s) is independent of the chosen s. The Siegel-Eisenstein series associated withf is defined by the following:

E(g, s, f) := X

γ∈Pn(k)\Sp_n(k)

f(γg, s), ∀g∈Sp_n(Ak).

It is known that this series converges absolutely for Re(s)>(n+ 1)/2 and can be extended to a rational function inq^−s. Moreover, we have the following functional equation

E(g, s, f) =E(g,−s, M(s)(f)), ∀g∈Sp_n(Ak).

HereM(s) :I(χ, s)→I(χ,−s) is the following intertwining operator:

M(s)(f)(g) :=

Z

Sym_n(Ak)

f(wnn(b)g, s)db, wn:=

0 I_n

−I_n 0

,

and the Haar measuredbis “self-dual with respect to ψ,” i.e. viewing Sym_n(Ak) asA

n(n+1) 2

k

and writedb as Q

1≤i≤j≤ndb_ij, the Haar measuredb_ij onAk for each pair (i, j) is self-dual with respect toψ. Detecting the possible poles ofE(g, s, f) (cf. [7]), we have thatE(g, s, f) is always holomorphic at the central critical values= 0.

LetW be an anisotropic quadratic space overkwith even dimensionm(thenm≤4). Let χ=χW andn=m−1. For each Schwartz functionϕ∈S(W(Ak)ⁿ), we extend Φϕ∈I(0, χ) to be a flat section Φϕ(·, s)∈I(s, χ) (calledthe Siegel section associated withϕ) by setting

Φϕ(g, s) =|deta|^s_AkΦϕ(g)

for everyg=n(b)m(a)κwherea∈GLn(A^k), b∈Sym_n(A^k), andκ∈Sp_n(O_Ak). The Siegel- Weil formula connects the central critical value E(g,0,Φϕ) with the theta series associated with the quadratic form onW:

Theorem 3.1. (cf. [20]) For every Schwartz function ϕ ∈ S(W(Ak)ⁿ) where W is an anisotropic quadratic space overkwith even dimension m=n+ 1, set

Θ(g, ϕ) :=

Z

O(W)(k)\O(W)(Ak)

X

x∈Wⁿ

ω(g, h)ϕ (x)dh.

The measuredh is normalized so thatvol(O(W)(k)\O(W)(Ak), dh) = 1. Then E(g,0,Φϕ) = 2·Θ(g, ϕ) ∀g∈Sp_n(Ak),

whereΦϕ(·, s)∈I(s, χW)is the Siegel section associated withϕ.

On the other hand, let C be an incoherent quadratic space over k with even dimension m. For every ϕ ∈ S(C(Ak)ⁿ), by the same way we can also extend Φ_ϕ to a flat section Φ_ϕ(·, s)∈I(s, χ_C).

Theorem 3.2. For every Schwartz function ϕ∈ S(C(Ak)ⁿ) where C is an incoherent qua- dratic space overk with even dimensionm=n+ 1, we have

E(g,0,Φϕ) = 0, ∀g∈Sp_n(Ak).

Proof. Note that ϕ7→E(·,0,Φ_ϕ)

is a Sp_n(Ak)-equivariant homomorphism fromR_n(C) to the space of automorphic forms on Sp_n(Ak). In the next section, we show that every non- singular Fourier coefficients ofE(g,0,Φϕ) must be zero (see Proposition 4.6). Therefore the result follows from a similar argument of [12, Lemma 2.5].

4. Fourier coefficients of Siegel-Eisenstein series

Take a positive integern and a character χ: k^×\A^×k → {±1}. Letf(·, s)∈I(s, χ) be a flat section. Consider the Fourier expansion ofE(g, s, f):

E(g, s, f) = X

β∈Sym_n(k)

Eβ(g, s, f),

whereEβ(g, s, f), called theβ-th Fourier coefficient ofE(g, s, f), is defined by Eβ(g, s, f) :=

Z

Sym_n(k)\Sym_n(Ak)

E(n(b)g, s, f)ψ −Trace(bβ) db.

The Haar measuredbis normalized so that vol Sym_n(k)\Sym_n(Ak), db

= 1. We can choose dbto be induced from the Haar measure on Sym_n(Ak) which is self-dual with respect toψ.

It is clear that

E_β(n(b)g, s, f) =ψ_β(b)·E_β(g, s, f), ∀b∈Sym_n(Ak), whereψ_β(b) :=ψ Trace(bβ)

.

Lemma 4.1. (cf. [20, Lemma A.3])Givenβ ∈Sym_n(k)withdetβ 6= 0, Eβ(g, s, f) =

Z

Sym_n(Ak)

f(wnn(b)g, s)ψβ(−b)db.

Note that Sp_n(k)Pn(Ak) is dense in Sp_n(Ak). Forg=n(b)m(a) where a∈GLn(Ak) and b∈Sym_n(Ak),

E(g, s, f) = X

β∈Sym_n(k)

Eβ(m(a), s, f)ψβ(b).

We can focus on the Fourier coefficientsE_β^∗(a, s, f) := E_β(m(a), s, f) for a∈ GL_n(Ak) and β∈Sym_n(k). It is clear that

E_β^∗(a, s, f) =E_α?β^∗ (α⁻¹a, s, f), ∀α∈GLn(k), whereα ? β:=^tαβα.

4.1. Whittaker functions. Fix a place v ofk and a quadratic characterχ_v :k_v^× → {±1}.

Take a flat section fv(·, s) ∈ Iv(s, χv). For βv ∈ Sym_n(kv), the local Whittaker function Wv,β_v(s, fv) is defined by

Wv,βv(s, fv) :=

Z

Sym_n(k_v)

fv(wnn(bv), s)ψv,βv(−bv)dbv.

Here the Haar measuredbvis self-dual with respect toψv, andψv,β_v(bv) :=ψv Trace(bvβv) . Letρv be the left action of Sp_n(kv) onIv(s, χv) by right translation. Then it is clear that

Wv,β_v s, ρv(n(bv))fv

=ψv,β_v(bv)·Wv,β_v(s, fv), ∀bv∈Sym_n(kv).

Proposition 4.2. (cf. [17, p. 102])Letβv ∈Sym_n(kv)withdetβv 6= 0and take a flat section fv(·, s)∈Iv(s, χv).

(1) W_v,βv(s, f_v) can be extended to an entire function on the wholes-plane.

(2) Whenv is “good”, i.e.χ_v is unramified,β_v∈Sym_n(k_v)∩GL_n(O_v), the conductor of ψ_v is trivial, and f_v(κ_v) = 1 for everyκ_v∈Sp_n(O_v), we have

W_v,βv(s, f_v) =















Lv(s+ (n+ 1)/2, χv)⁻¹

(n−1)/2

Y

i=1

ζv(2s+n+ 1−2i)⁻¹, if nis odd,

n/2

Y

i=1

ζv(2s+n+ 2−2i)⁻¹, if nis even.

Here

ζ_v(s) := (1−q_v^−s)⁻¹ and L_v(s, χ_v) :=

((1−χv(πv)q_v^−s)⁻¹ if χv is unramified,

1 otherwise.

Letχ:k^×\A^×_k → {±1} be a quadratic character. Givenb0∈Sym_n(Ak) and a flat section f(·, s)∈I(s, χ), theglobal Whittaker functionWb₀(s, f) is defined by

W_b0(s, f) :=

Z

Sym_n(Ak)

f(w_nn(b), s)ψ_b0(−b)db.

Supposeb0 is also in GLn(A^k) and f is a pure-tensor, i.e.f =⊗vfv. LetS =S(χ, ψ, b0, f) be the finite subset of places ofk minimal so thatv is “good” forv /∈S (i.e. whenv /∈S,χv

is unramified, the conductor ofψv is trivial,b0,v∈Sym_n(kv)∩GLn(Ov), andfv(κv) = 1 for everyκv∈Sp_n(Ov)). Define

Λ^S_n(s, χ) :=















L^S(s+ (n+ 1)/2, χ)⁻¹

(n−1)/2

Y

i=1

ζ^S(2s+n+ 1−2i)⁻¹, ifnis odd,

n/2

Y

i=1

ζ^S(2s+n+ 2−2i)⁻¹, ifnis even,

where

L^S(s, χ) :=Y

v /∈S

Lv(s, χv) and ζ^S(s) :=Y

v /∈S

ζv(s).

Then Λ^S_n(s, χ) is holomorphic ats= 0 and non-vanishing unlessn= 1 andχ≡1. Moreover, (4.1) Wb₀(s, f) = Λ^S_n(s, χ)Y

v∈S

Wv,b_0,v(s, fv),

This gives the meromorphic continuation of the Whittaker functionWb₀(s, f).

Givenβ ∈Sym_n(k) anda∈GL_n(Ak), we seta ? β:=^taβa. Lemma 4.1 tells us that when detβ6= 0, the Fourier coefficient

E_β^∗(a, s, f) = χC(deta)|deta|^−s+_A ⁿ⁺¹²

k ·

Z

Sym_n(Ak)

f(wnn(b), s)ψa?β(−b)db

= χC(deta)|deta|^−s+

n+1 2

Ak ·Wa?β(s, f), (4.2)

Therefore the equations (4.1) and (4.2) implies the following.

Proposition 4.3. Let n be positive integer and χ :k^×\A^×k → {±1} a quadratic character.

Given a pure-tensor flat section f = ⊗vfv ∈ I(s, χ), we have that for β ∈ Sym_n(k) with detβ6= 0 anda∈GL_n(Ak),

ords=0E^∗_β(a, s, f) =X

v∈S

ords=0Wv,a_v?β(s, fv) +n,χ,

whereS=S(χ, ψ, a ? β, f)is the finite subset of places of kminimal so that v is “good” for v /∈S;_n,χ= 1 ifn= 1 andχ≡1, and_n,χ= 0otherwise.

4.2. The vanishing order of non-singular Fourier coefficients at s= 0. Take a place v of k. Let V be a non-degenerate quadratic space over kv with even dimension, and take n= dimk(V)−1. Forβv∈Sym_n(kv), we set

Ω_βv(V) :={x∈Vⁿ:Q⁽ⁿ⁾_V (x) =β_v}, whereQ⁽ⁿ⁾_V is the moment map introduced in Section 2.

Lemma 4.4. (cf. [20, Lemma A.1])

(1) Forβ_v∈Sym_n(k_v)with detβ_v 6= 0, we have W_v,βv(0,Φ_v,ϕv) = 0 for allϕ_v∈S(Vⁿ) unlessΩβ_v(V)is non-empty.

(2) When Ωβ_v(V) is non-empty, O(V) acts on Ωβ_v(V) transitively, and there exists a constantc such that forϕv∈S(Vⁿ),

Wv,β_v(0,Φv,ϕ_v) =c· Z

Ω_βv(V)

ϕv(xv)dxv, wheredxv is anO(V)-invariant measure on Ωβ_v(V).

The following ”dichotomy” plays a fundamental role.

Lemma 4.5. (cf. [8, Proposition 1.3]) For βv ∈ Symn(kv) with detβv 6= 0, we have that Ω_βv(V)is non-empty if and only if

(4.3) Hassev(V) =χV(detβv)·(detβv,−(−1)^(n+1)/2)v·Hassev(βv).

HereHasse_v(β_v)is the Hasse invariant of the quadratic space overk_v associated toβ_v. LetC be an incoherent quadratic space overk with even dimension andn= dimk(C)−1.

Forβ ∈Sym_n(k) with detβ6= 0, We set

Diff(β,C) :={placev ofk|Hassev(C)6=χC_v(detβ)·(detβ,−(−1)^(n+1)/2)v·Hassev(β)}.

The incoherence of C implies that the cardinality of Diff(β,C) must be odd. Moreover, by Proposition 4.3, Lemma 4.4 and 4.5 we have

Proposition 4.6. Let C be an incoherent quadratic space overk with even dimension and n= dimk(C)−1. Given a∈ GLn(Ak) and β ∈Sym_n(k) with detβ 6= 0, we have that for every Schwartz functionϕ∈S(C(Ak)ⁿ)

ords=0E_β^∗(a, s,Φϕ)≥# Diff(β,C)≥1.

5. Derivatives of non-singular Fourier coefficients

LetCbe an incoherent quadratic spaceCoverkwith even dimension andn= dimk(C)−1.

Givenβ ∈Sym_n(k) with detβ 6= 0, by Proposition 4.6 we know that fora∈GLn(Ak) and ϕ∈S(C(Ak)ⁿ),

∂

∂sE_β^∗(a, s,Φϕ) _s=0

= 0 if # Diff(β,C)>1.

Suppose Diff(β,C) = {v0}. Let Vβ be the non-degenerate quadratic space of dimensionm overksuch thatχ_Vβ =χ_C and

Hassev(Vβ) =

(Hassev(C), ifv6=v0,

−Hasse_v(C), ifv=v₀.

Then for every placevofkdifferent fromv₀, we haveV_β,v∼=Cv. Moreover, Hasse-Minkowski principle and Lemma 4.5 imply thatV_β representsβ, i.e. Ω_β(V_β) is non-empty.

Theorem 5.1. LetCbe an incoherent quadratic space overkwith even dimensionmand take n=m−1. Givenβ ∈Sym_n(k) withdetβ 6= 0, supposeDiff(β,C) ={v₀} and the associated quadratic space Vβ is anisotropic. Then for each pure-tensor ϕ = ⊗vϕv ∈ S(C(A^k)ⁿ) and a∈GLn(Ak), we have

∂

∂sE_β^∗(a, s,Φϕ) _s=0

= 2· W_v⁰

0,a_v0?β(0,Φv₀,ϕ_v0) W_v0,av

0?β(0,Φ_v0,

ϕev0)·Θ^∗_β(a,ϕ),e wherea_v0? β=^ta_v0βa_v0,ϕe=⊗vϕe_v∈S(V_β(Ak)ⁿ)is any pure-tensor so that:

(i) forv6=v₀,ϕe_v=ϕ_v (here we identify V_β,v withCv);

(ii) ϕe_v0 ∈S(V_β,vⁿ

0)is a Schwartz function satisfying thatW_v0,av

0?β(0,Φ_v0,

ϕev0)6= 0;

and

Θ^∗_β(a,ϕ) =e Z

Sym_n(k)\Sym_n(Ak)

Θ n(b)m(a),ϕe

ψ_β(−b)db, whereΘ(g,ϕ)e is the theta series introduced in Theorem 3.1.

Proof. Take a pure-tensorϕe=⊗vϕev∈S(Vβ(A^k)ⁿ) satisfying (i) and (ii). By the Siegel-Weil formula (Theorem 3.1) we have

E(g,0,Φ

ϕe) = 2·Θ(g,ϕ),e ∀g∈Sp_n(Ak).

Then from the equation (4.2) we obtain that fora∈GLn(A^k)

∂

∂sE_β^∗(a, s,Φϕ) _s=0

= χ_C(deta)|deta|^−s+n+12

Ak W_v⁰_0,av

0?β(0,Φv₀,ϕ_v0)· Wa?β(0,Φ

ϕe) W_v0,av

0?β(0,Φ_v0,

ϕev0)

!

= W_v⁰

0,a_v0?β(0,Φv₀,ϕ_v0) W_v0,av

0?β(0,Φ_v0,Φv

0,ϕve 0)·E_β^∗(a,0,Φ

ϕe)

= 2·W_v⁰

0,av0?β(0,Φ_v0,ϕv

0) W_v0,av

0?β(0,Φ_v0,

ϕe_v0)·Θ^∗_β(a,ϕ).e

Remark 5.2. The assumption onV_βbeing anisotropic is due to the restriction of Theorem 3.1, which can be removed as long as a stronger version of the Siegel-Weil formula is verified.

In the remaining sections, we concentrate on the special case when the incoherent quadratic space C has dimension 2, and explore the geometric interpretation of the central critical derivative of the Siegel-Eisenstein series on SL2(Ak).

6. Special case: dimk(C) = 2

Fix a place∞ofk, referred as the place at infinity, and other places are called finite places of k. Let K be a quadratic field over k which is “imaginary,” i.e. ∞ does not split inK.

TakeD ∈k such thatK=k(√

D). Choose _∞∈k^×_∞ so that the Hilbert quadratic symbol (_∞, D)_∞=−1, and putv:= 1 for every finite placevofk. For eachα∈k^×, the collection C_K^(α)={C^(α)_K,v}v, whereC_K,v^(α) := (Kv, vαN_K/k), is an incoherent quadratic space over kwith dimension 2. It is clear that

χ_C(a) =χ_K(a) :=Y

v

(a_v, D)_v, ∀a= (a_v)_v∈A^×k,

and Hasse_v(Cv) = (_vα, D)_v for every place v of k. We point out that given an arbitrary incoherent quadratic spaceC overkwith dimension 2, there always exists a triple (K,∞, α)

such thatC ∼=C_K^(α).

LetA be the ring of functions inkregular away from∞, andOK be the integral closure of A in K. For each place v of k, set OK_v := OK ⊗AOv when v 6=∞; andOK∞ denotes the integral closure ofO_∞ in K_∞. Ifv is ramified or split in K, we assume that the chosen uniformizerπ_v is inN_K/k(K_v), and pick Π_v∈K_v such thatN_K/k(Π_v) =π_v once and for all.

For each placev ofk, set

ev =ev(α, ψ) :=−δv−ordv(αv),

whereδvis the “conductor” ofψatvintroduced in Section 1.1. Choose a particular Schwartz functionϕ^(α)v ∈S(C_K,v^(α)) which is the characteristic function of the followingOv-lattice:

(6.1)

(Π^e_v^vO_Kv, ifvis ramified or split in K, π^dev^v/2eOK_v, ifvis inert in K.

Heredλe:= min{m∈Z:m≥λ}forλ∈R. Lemma 6.1. For everyκv=

a b c d

∈SL2(Ov) withc≡0 modπvOv, ωv(κv)ϕ^(α)_v =χF,v(d)ϕ^(α)_v .

In particular, suppose(i)v is inert inF ande_v is even; or (ii)v splits inF, we get further that

ωv(κv)ϕ^(α)_v =ϕ^(α)_v , ∀κv∈SL2(Ov).

Proof. One observes that the Fourier transform of ϕ^(α)v is

ϕb^(α)_v =









 1_Π^ev_{v O}

Fv, ifv is split ink,

q^−1/2·1_Πev−1

v O_Fv, ifv is ramified ink, q^{((−1)ev−1)/2}·1_πbev /2c

v O_Fv, ifv is inert ink.

Herebλc:= max{m∈Z:m≤λ}forλ∈R. By the decomposition a b

πvc d

=

1 bd⁻¹

0 1

d⁻¹ 0

0 d

0 1

−1 0

1 −πvd⁻¹c

0 1

0 −1

1 0

∈SL₂(O_v),

the result is then straightforward.

Letϕ^(α)be the pure-tensor ⊗vϕ^(α)v ∈S C_K^(α)(Ak)

. By the above lemma, Φ_ϕ(α) is viewed as a “new vector” in the representation R₁(C_K^(α)) of SL₂(Ak). Denote by FK and g_K the constant field and the genus ofK respectively. Set

L(s, χK) :=Y

v

Lv(s, χK,v) and L(s, χe K) :=q ^{[FK:Fq](gK−1)−(gk−1)} s

L(s, χK), which are extended to rational functions inq^−s with the functional equation

L(s, χe K) =L(1e −s, χK).

Consider the following (modified) Eisenstein series:

(6.2) E(g, s,e Φ_ϕ(α)) :=L(se + 1, χF)·E(g, s,Φ_ϕ(α)).

It is observed that forg∈SL2(Ak),E(g, s,e Φ_ϕ(α)) is a rational function inq^−ssatisfying:

E(g, s,e Φ_ϕ(α)) =−E(g,e −s,Φ_ϕ(α))·





 Y

vis inert andevis odd

q^−s_v L_v(1 +s, χ_F,v) Lv(1−s, χF,v)





.

We are interested in its central critical derivative η^(α)(g) := ∂

∂sE(g, s,e Φ_ϕ(α))

_s=0, ∀g∈SL2(Ak).

LetN=N(ψ, K, α) be the positive divisor ofksuch that ordv(N) =

(1, if eitherv is ramified inkor v is inert inkandev is odd, 0, otherwise.

Lemma 6.1 implies thatη^(α) is uniquely determined by its values on the representatives of the double cosets in SL2(k)\SL2(Ak)/K0(N), where

K0(N) =

a b c d

∈SL2(O_Ak)

cv ≡0 modπ_v^ordv(N)Ov for allv

.

LetB be the standard Borel subgroup of SL2, i.e.

B=P₁={n(x)m(y)|x∈Ga, y∈GL₁}.

The canonical map fromB(Ak) to the double coset space SL₂(k)\SL₂(Ak)/K0(N) is surjec- tive. Therefore to explore the derivativeη^(α), it suffices to compute the Fourier coefficients η_β^(α),∗(y) fory∈A^×k andβ ∈k.

6.1. The constant termη^(α),∗₀ . We first compute the constant termη₀^(α),∗in the following:

Lemma 6.2. Fory∈A^×k, we have η₀^(α),∗(y) = 2χK(y)|y|_AkL(0, χK)

·





ln|y|_Ak−

[FK:Fq](gK−1)−(gk−1)

lnq−L⁰(0, χK) L(0, χK) +1

2 X

vis inert andevis odd

qv−1 qv+ 1lnqv





.

Proof. Fory∈A^×k, we claim that the constant termEe₀^∗(y, s,Φ_ϕ(α)) is equal to

(6.3) χK(y)|y|_Ak





|y|^s_AkL(−s, χe K)− |y|^−s_A

kL(s, χe K)· Y

vis inert andevis odd

q_v^−sL_v(1 +s, χ_K,v) Lv(1−s, χK,v)





.

Then the result follows.

To prove the equality (6.3), we recall the fact that E₀^∗(y, s,Φ_ϕ(α)) = Φ_ϕ(α)(m(y), s) +

Z

Ak

Φ_ϕ(α)

0 1

−1 0

1 b 0 1

y 0

0 y⁻¹

, s

db.

The Haar measure db is normalized to be self-dual with respect to ψ. More precisely, if we write db as Q

vdb_v, the Haar measure db_v on k_v for each place v is choosed so that vol(Ov, dbv) =qv^−δv/2. One observes that Φ_ϕ(α)(m(y), s) =χK(y)|y|^s+1

Ak , and for each placev ofk, the local integral

Z

kv

Φ_v,ϕ(α) v

0 1

−1 0

1 bv

0 1

yv 0 0 y_v⁻¹

, s

dbv