On Definite Shimura Curves and Autormophic Forms over Function Fields

OVER FUNCTION FIELDS

FU-TSUN WEI AND JING YU

Abstract. Let k = Fq(t), with q odd. In this article we introduce denite (w.r.t. the innite place of k) Shimura curves over k, and establish Hecke module isomorphisms between their Picard groups and the spaces of Drinfeld type new forms of corresponding level. An important application is the Gross-Zagier type formula for the central critical values of L-series over imaginary quadratic ex-tensions of k. This gives, as a corollary, the central critical values of the L-series coming from elliptic curves over k with square free conductor supported at even number of places and having split multiplicative reduction at ∞.

Keywords: Function eld, quaternion algebra, Shimura curve, automorphic form on GL2, Hecke operator, special value of L-series.

MSC: 11R58, 11G18, 11F41, 11F67

Introduction

We present here a theory of denite quaternion algebras over the rational func-tion eld k := Fq(t)with q odd, denite means that the place ∞ at innity ramies

for the quaternion algebra in question. Following Gross [8], we rst give a geometric translation of Eichler's arithmetic theory of denite quaternion algebra by introducing the so-called denite Shimura curves. The geometry of these curves is simple and easy to manipulate. Basing on Eichler's trace computation, one is lead (via Jacquet-Langlands) to an explicit Hecke module isomorphism between the Picard groups of denite Shimura curves and spaces of automorphic forms of Drinfeld type over the function eld k.

Automorphic forms of Drinfeld type are very useful tools for function elds arith-metic (cf. [7], [10] and [13] for more details and applications), which can be viewed as analogue of classical modular forms of weight 2. To illustrate our approach to quaternion algebras over function eld, we give an application to the study of central critical values of certain L-series of Rankin type in the global function eld setting. These L-series include, among others, L-series coming from elliptic curves over k with square free conductor supported at even number of places and having split multi-plicative reduction at ∞. Having the extensive calculations done in [10], we obtain in particular a Gross-Zagier type formula for the central critical values of these L-series over imaginary quadratic extensions of k (with respect to ∞).

Date: September 27, 2010.

Both authors were supported by National Science Council grant (97) 2115-M-007-003. The rst author was also supported by the Graduate Students Study Abroad Program of National Science Council.

Given a denite quaternion algebra D over k and let N0 be the product of nite

ramied primes of D. We introduce the denite Shimura curve X = XN0 over k (for

maximal orders) in Ÿ1, which is a nite union of genus zero curves. Also introduced are the Gross points, which are special points on these curves associated to orders in imaginary quadratic extensions of k. With a natural choice of basis on the Picard group Pic(X), the Hecke correspondences can be expressed by Brandt matrices.

From the entries of Brandt matrices we introduce certain theta series. Taking into account the Gross height pairing on the Pic(X) (dened in Ÿ1.2), we then have at hand a construction of automorphic forms of Drinfeld type for the congruence subgroup Γ0(N0) of GL2(Fq[t]). The main theorem of this article in Ÿ2.3 is:

Theorem. There is a map Φ : Pic(X) × Pic(X)∨ _{−→ Mnew(Γ}

0(N0)) such that for

all monic polynomials m of Fq[t]

TmΦ(e, e0) = Φ(tme, e0) = Φ(e, tme0).

Here Pic(X)∨ _{is the dual group Hom(Pic(X), Z), Mnew(Γ}

0(N0))is the space of

Drin-feld type new forms for Γ0(N0), tm are Hecke correspondences on X, and Tm are

Hecke operators on Mnew_{(Γ0(N0)). Moreover, this map induces an isomorphism (as}

Hecke modules)

(Pic(X) ⊗_ZC) ⊗TC(Pic(X)

∨_⊗

ZC)∼= Mnew(Γ0(N0)).

This theorem in fact tells us that all automorphic new forms of Drinfeld type come from our theta series. In the proof of the above theorem, we claim the equality of the trace of the m-th Brandt matrix B(m) and the trace of the Hecke operator Tm on Mnew(Γ0(N0))for each monic polynomial m in Fq[t]. This claim is essentially

the Jacquet-Langlands correspondence (cf. [9]) between automorphic representations of quaternion algebras over k and automorphic cuspidal representations of GL2 over

k. Another crucial step in the proof is to show that the Hecke module Mnew(Γ0(N0))

is free of rank one, which follows from the multiplicity one theorem (cf. [3]). For the sake of completeness, we recall these results in Appendix.

In Ÿ3 we study the central critical values of Rankin type L-series associated to Drinfeld type new forms f for Γ0(N0). Given an irreducible polynomial D in Fq[t]

such that k(√D) is imaginary and P is inert in k(√D) if the prime P divides N0.

For each ideal class A of Fq[t][

√

D], we construct in Ÿ2.4 an autormophic form gA

of Drinfeld type. Comparing the Fourier coecients of gA and the calculations in

[10], the central critical value of the L-series associated to f and A is essentially the Petersson inner product of f and gA. We then deduce that when f is a normalized

Hecke eigenform, the special value of the twisted L-series Λ(f, χ, s) at s = 0 is given explicitly by the Gross height of a special divisor class ef,χ on XN0.

Notation We x the following notations:

k : the rational function eld Fq(t), q = p`0 where p is an odd prime.

A : the polynomial ring Fq[t].

∞ : the innite place, which corresponds to degree valuation v∞.

k∞: Fq((t−1)), i.e. the completion of k at ∞.

O∞: Fq[[t−1]], i.e. the valuation ring in k∞.

P : a nite prime (place) of k.

kP : the completion of k at the nite prime P .

AP : the closure of A in kP.

Ak : the adele ring of k.

ˆ k : Q0

PkP, the nite adele ring of k.

ˆ A : Q

PAP.

ψ∞: a xed additive character on k∞: for y = P_iaiπi∞∈ k∞, we dene

ψ∞(y) := exp



2π√−1

p · TrFq/Fp(−a1).

We identify non-zero ideals of A with the monic polynomials in A by using the same notation.

1. Definite Shimura curves

Given a quaternion algebra D over k ramied at ∞ (call D denite). Before introducing the denite Shimura curve for D, we start with a genus 0 curve Y over k associated with the quaternion algebra D, which is dened by the following: the points of Y over any k-algebra M are

Y (M ) = {x ∈ D ⊗kM : Tr(x) = Nr(x) = 0}/M×,

where Tr and Nr are respectively the reduced trace and the reduced norm of D. More precisely, if D = k + ku + kv + kuv where u2 _{= α, v}2 _{= β, α and β are in k}×_{, and}

uv = −vu, then Y is just the conic

αy2+ βz2= αβw2

in the projective plane P2_{. The group D}× _{acts on Y (from the right) by conjugation.}

If K is a quadratic extension of k, Y (K) is canonically identied with Hom(K, D): For each embedding f : K → D, let y = yf be the image of the unique K-line on the

quadric {x ∈ D ⊗kK : Tr(x) = Nr(x) = 0} on which conjugation by f(K×) acts by

multiplication by the character a 7→ a/¯a. Note that yf is one of the two xed points

of f(K×_{)acting on Y (K); another one is the image of the line where conjugation acts}

by the character a 7→ ¯a/a.

Let N0 be the product of the nite ramied primes of D. Choose a maximal order

R of D. For any nite prime P let RP := R ⊗AAP, DP := D ⊗kkP, and

ˆ

R := R ⊗AAˆ, ˆD := D ⊗kˆk.

Denition 1.1. (cf. [2] and [8]) The denite Shimura curve XN0 is dened as

XN0 = ˆR

×_{\ ˆD}×_{× Y}_/D×_.

We will use the notation X instead of XN0 when N0 is xed.

Proof. Let g1, ..., gn be representatives for the nite double coset space ˆR×\ ˆD×/D×, i.e. ˆ D×= n a i=1 ˆ R×giD×.

Then each coset of XN0 has a representative ( ˆR

×_g

i, y) mod D× and the map

XN0 −→

`n

i=1Y /Γi

( ˆR×gi, y) 7−→ y mod Γi

is a bijection, where Γi = gi−1Rˆ×gi∩ D× is a nite group for i = 1, ..., n. 

Denition 1.3. Let K be an imaginary quadratic extension of k (i.e. ∞ is not split in K). We call x = (g, y) ∈Image " ˆ R×\ ˆD×× Y (K) → X_N0(K) #

a Gross point on XN0 over K.

Let f : K → D be the embedding corresponding to y. Then f (K) ∩ g−1Rg = f (Oˆ d)

for some quadratic order Od := A[

√

d] where d is a non-square element in A and d /∈ k2

∞. In this case, we say x is of discriminant d. Note that the discriminant of a

Gross point is well-dened up to multiplying with elements in (F×

q)2. Set Xi := Y /Γi.

If the component g of a Gross point x is congruent to gi in ˆR×\ ˆD×/D×, then x lies

on the component Xi(K) = (Y /Γi)(K).

1.1. Actions on Gross points. Let a ∈ ˆK× where ˆK := K ⊗kˆkand x = (g, y) be

a Gross point of discriminant d. Let f : K → D be the embedding corresponding to y. This induces a homomorphism ˆf : ˆK → ˆD and we dene

xa:= (g ˆf (a), y).

Note that xa is also of discriminant d, and it is easy to check that x = xa if and only

if a ∈ ˆO_d×K× where ˆOd := Od⊗AAˆ. Hence ˆO×d\ ˆK

×_/K× _∼

= Pic(Od) acts freely on

the set Gdof Gross points of discriminant d.

The orbit space Gd/ Pic(Od) is identied with the space of double cosets

ˆ

R×\E/ ˆf ( ˆK×),

where f : K → D is a xed embedding (if any exist) and E := {g ∈ ˆD×: f (K) ∩ g−1Rg = f (Oˆ d)}. Note that ˆ R×\E/ ˆf ( ˆK×) =Y P R×_P\EP/f (K_P×)

where EP = {gP ∈ D×_P : f (KP) ∩ g_P−1RPgP = f (Od,P)}and Od,P is the closure of Od

Lemma 1.4. (cf. [12] or [13]) #(R×_P\E_P/f (K_P×)) =    1 if P 6= N0, 1 −_Pd if P |N0. Here d P

is the Eichler quadratic symbol, i.e.

 d P  =          1 if P2|d or d mod P ∈ (A/P )×
2 , −1 if d mod P ∈ (A/P )×− (A/P )×
2

, 0 if P |d but P2 - d.

Remark. The above lemma tells us that the number #(Gd) is equal to

h(d) Y P |N0  1 − d P  

where h(d) is the class number of Od.

Note that there is a natural action of QP |N0Gal(KP/kP) on Gross points over

K in the following way: Let x = (g, y) be a Gross point and fy : K ,→ D be the

embedding corresponding to y. Note that the embedding fy is uniquely determined

by the embedding ˆfy : ˆK → ˆD. Dene

xσ = (g, y)σ = (g, yσ)

where σ ∈ QP |N0Gal(KP/kP) and yσ corresponds to the embedding ˆfy ◦ σ : ˆK →

ˆ

D. If x is a Gross point of discriminant d in X_i then so is xσ_{. Moreover, let a ∈}

ˆ

O×_d\ ˆK×/K×∼= Pic(Od) and σ ∈ Gal(K/k) we have

(xσ)a= (xσ(a))σ.

Therefore we have an action of Pic(Od) o QP |N0Gal(KP/kP)

on the Gross points of discriminant d.

Assume every prime factor P of N0 is not split in K and P2 does not divides d

(i.e. the set Gd of Gross points of discriminant d is not empty). Suppose P1, . . . , Pr

are primes dividing N0and inert in K. Then Pic(Od)o

Qr

i=1Gal(KPi/kPi)acts simply

transitively on Gd. To see this, given any element (a, σ) ∈ Pic(Od)oQri=1Gal(KPi/kPi)

and a Gross point x = (g, y) of discriminant d. If x(a,σ)_{= x, i.e.}

(g, y) = (g ˆfy(a), yσ),

then fy ◦ σ = b−1fyb for some b ∈ D× with b ˆfy(a)−1 ∈ g−1Rˆ×g. Since Pi is inert

in K for each i, we get b−1_f

yb = fy and so σ = 1. As we know that Pic(Od) acts

freely on Gd, Pic(Od) o

Qr

i=1Gal(KPi/kPi) acts simply transitively by comparing the

cardinality.

1.2. Hecke correspondence and Gross height pairing. Let P be a prime of A. Let T be the Bruhat-Tits tree of PGL2(kP) as dened in [11]. The vertices are

the equivalence classes of AP-lattices in k2_P, and two such vertices are adjacent if the

distance between the lattices is 1. This is a tree where each vertex has degree qdeg P_+1.

neighbors in the tree. Identifying PGL2(AP)\ PGL2(kP) with the Bruhat-Tits tree,

we can write the Hecke correspondence for g ∈ PGL2(AP)\ PGL2(kP):

tP(g) = X deg(u)<deg P 1 u 0 P ! g + P 0 0 1 ! g. Note that XN0 can be written as

 ˆ_R×_{\ ˆD}×_/ˆk×_{× Y /D}× and ˆ R×\ ˆD×/ˆk×=Y P 0_R× P\D × P/k × P. When (P, N0) = 1, R×_P\D×_P/k×_P ∼= PGL2(AP)\ PGL2(kP)

and so we have the Hecke correspondence tP on XN0.

Now suppose P divides N0, then R×_P\D_P×/k_P× has two elements and dene the

Atkin-Lehner involution

wP(g, y) = (g0, y)

where g0 _{is another double coset in R}× P\D

× P/k

× P.

From the construction, the correspondences commute with each other. Therefore we can dene Hecke correspondence tm for every non-zero ideal (m) of A in the

following way:         

tmm0 = t_mt_m0 if m and m0 are relatively prime,

t_P` = t<sub>P</sub>`−1tP − qdeg PtP`−2 for P - N0,

t_P` = w`_P for P | N₀.

Note that X = XN0 =

`n

i=1Xi, where n is the left ideal class number of R.

Consider the Picard group Pic(X), which is isomorphic to Zn_{and is generated by the}

classes ei of degree 1 corresponding to the component Xi. Then the correspondences

tm induce endomorphisms of the group Pic(X). In fact, with respect to the basis

{e₁, ..., en}, these endomorphisms can be represented by Brandt matrices.

Let {I1, ..., In} be a set of left ideals of R representing the distinct ideal classes,

with I1 = R. Let wi := #(R×i )/(q − 1) where Ri is the right order of Ii. Consider

Mij := Ij−1Ii, which is a left ideal of Rj with right order Ri. Choose a generator

Nij ∈ kof the reduced ideal norm Nr(Mij)(:=< Nr(b) : b ∈ Mij >A)of Mij. For each

monic polynomial m in A, dene Bij(m) :=

#{b ∈ Mij : (Nr(b)/Nij) = (m)}

(q − 1)wj

and the m-th Brandt matrix

B(m) :=Bij(m)



Proposition 1.5. For all non-zero ideal (m) in A and i = 1, 2, ..., n, tmei = n X j=1 Bij(m)ej.

Proof. From the denition of tm and the recurrence relations of B(m) (cf. [12]), we

can reduce the proof to the case when m = P is a prime. According to the denition of tP, tPei =

P

jαjej where αj is the number of left

ideals I of R equivalent to Ij which are contained in Ii with Nr(I) = P Nr(Ii). It is

easy to see that αj = Bij(P )and so the proposition holds. 

We dene the Gross height pairing < ·, · > on Pic(X) with values in Z by setting 

 

< ei, ej >= 0 if i 6= j,

< ei, ei>= wi,

and extending bi-additively. Therefore Pic(X)∨ _{:= Hom(Pic(X), Z) can be viewed}

as a subgroup of Pic(X) ⊗ZQ with basis { ˇei = ei/wi : i = 1, ..., n}via this pairing.

Since wjBij(m) = wiBji(m), one has the following proposition.

Proposition 1.6. For all classes e and e0 _{in Pic(X), we have}

< tme, e0 >=< e, tme0 > .

Proof. Since wjBij(m) = wiBji(m), we have

< tmei, ej >=< ei, tmej > .

for all i, j and the result holds. 

Given a ∈ A with a /∈ k2

∞. Consider the rational divisor

ca := X a=df2_,fmonic 1 2u(d) X xd∈Gd xd.

By calculation one has

deg(ca) = 1 2 X a=df2_,fmonic   h(d) u(d)· Y P |N0 (1 − d P  )  . Let ea∈ Pic(X) ⊗ZQ be the class of the divisor ca. It can be shown that

Proposition 1.7. The class ea lies in Pic(X)∨, which is considered as a subgroup of

Pic(X) ⊗_ZQ.

Note that we can extend the Gross height pairing to Pic(X) ⊗ZC which is linear

in the rst term and conjugate linear in the second. In the next section this pairing gives us a construction of automorphic forms of Drinfeld type.

2. Automorphic forms of Drinfeld type and main theorem

2.1. Automorphic forms of Drinfeld type. Consider the open compact subgroup K0(N ∞) :=Q_PK0,P × Γ∞ of GL2(Ak), where K_0,P = ( a b c d ! ∈ GL₂(AP) : c ∈ N AP )

for nite prime P , and Γ∞= ( a b c d ! ∈ GL₂(O∞) : c ∈ π∞O∞ ) .

An automorphic form f on GL2(Ak)for K0(N ∞)(with trivial central character) is a

C-valued function on the double coset space

GL2(k)\ GL2(Ak)/K0(N ∞)k∞×.

Note that by strong approximation theorem (cf. [12]) we have the following bijection GL2(k)\ GL2(Ak)/K0(N ∞)k∞× ∼= Γ0(N )\ GL2(k∞)/Γ∞k×∞ where Γ0(N ) = ( a b c d ! ∈ GL2(A) : c ≡ 0 mod N ) .

Therefore f can be viewed as a C-valued function on Γ0(N )\ GL2(k∞)/Γ∞k×∞. From

now on given a subgroup Γ of GL2(A)with nite index, we call f an automorphic form

on GL2(k∞)for Γ if f is a function on the space of double cosets Γ\ GL2(k∞)/Γ∞k∞×.

An automorphic form f on GL2(k∞) is called a cusp form if for every g∞∈ GL2(k∞)

and γ ∈ GL2(A) Z A\k∞ f γ 1 x 0 1 ! g∞ ! dx = 0.

Note that the coset space GL2(k∞)/Γ∞k×∞ can be represented by the two disjoint

sets T+:= ( π∞r u 0 1 ! : r ∈ Z, u ∈ k∞/π∞r O∞ ) and T−:= ( π_∞r u 0 1 ! 0 1 π∞ 0 ! : r ∈ Z, u ∈ k∞/π∞r O∞ ) .

Denition 2.1. An automorphic form f on GL2(k∞)is of Drinfeld type if it satises

the following harmonic properties: for any g∞∈ GL2(k∞) we have

˜ f (g∞) := f (g∞ 0 1 π∞ 0 ! ) = −f (g∞) and X κ∈GL2(O∞)/Γ∞ f (g∞κ) = 0. Given a function f : 1 A 0 1 !

\ GL2(k∞)/Γ∞k∞× → C. Recall the Fourier expansion

of f (cf. [15]): for r ∈ Z and u ∈ k∞, 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.

Here ψ∞ is the xed additive character on k∞ in the notation table and du is a Haar

measure with R_A\k∞du = 1. Since f(gγ∞) = f (g) for all γ∞ ∈ Γ∞, f∗(r, λ) = 0 if

deg λ + 2 > r. Moreover, if f satises harmonic properties, then f∗(r, λ) = q−r+deg λ+2f∗(deg λ + 2, λ) if deg λ + 2 ≤ r.

2.1.1. Example: Theta series. Fix a denite quaternion algebra D = D(N0) where N0

is the product of nite ramied primes of D. Let R be a maximal order and n be the class number. With representatives of left ideal classes xed in Ÿ1.2, we have introduced for each (i, j), the ideal Mij of D. For 1 ≤ i, j ≤ n and g∞ ∈ GL2(k∞).

Write g∞ as γ · x y

0 1 !

γ∞z∞, where γ ∈ Γ0(N0), (x, y) ∈ k∞× × k∞, γ∞∈ Γ∞, and

z∞∈ k×∞. We introduce the theta series Θij for Mij:

Θij(g) := q−v∞(x)·    1 wj + X m∈Amonic, deg m+2≤v∞(x) Bij(m)  X ∈F×q ψ∞(my)    .

These series are Q-valued functions on Γ0(N0)\ GL2(k∞)/Γ∞k×∞which are of Drinfeld

type (cf. [14]). For each r ∈ Z and λ ∈ A with deg λ + 2 ≤ r the Fourier coecients

Θ∗_ij(r, λ) =    q−rBij(m) if (λ) = (m) 6= 0, q−r/wj if λ = 0.

2.2. Hecke operators. Given an automorphic form f on GL2(k∞) for Γ0(N ). Let

P be a prime of A. The Hecke operator TP is dened by:

TPf (g) := P u∈A/P f ( 1 u 0 P ! · g) + f ( P 0 0 1 ! · g) if P - N, TPf (g) := P u∈A/P f ( 1 u 0 P ! · g) if P |N.

Note that the Fourier coecients of TPf are of the form:

(TPf )∗(r, λ) = qdeg(P )· f∗(r + deg(P ), P λ) + f∗(r − deg(P ),_Pλ) if P - N,

(TPf )∗(r, λ) = qdeg(P )· f∗(r + deg(P ), P λ) if P |N.

Here f∗_(πr

∞,Pλ) = 0 if P - λ. Since TP and TP0 commute,we can dene Hecke

operators Tm for all monic polynomials m in A by the following way:

        

Tmm0 = T_mT_m0 if m and m0 are relatively prime,

T_P` = T<sub>P</sub>`−1T_P − qdeg PT_P`−2 for P - N,

T_P` = T<sub>P</sub>` for P | N.

We point out that if f is of Drinfeld type, then so is Tmf for any monic polynomial m.

Proposition 2.2. For any monic polynomial m in A, TmΘij = X ` Bi`(m)Θ`j = X ` B`j(m)Θi`.

Proof. The second identity will follow from the rst, as

wjΘij = wiΘji and w`Bi`(m) = wiB`i(m).

Note that the Hecke operators Tm satisfy the same relations as the matrices B(m).

Moreover, from the recurrence relations of Brandt matrices (cf. [12]) we have X ` Bi`(P )B`j(m) = Bij(mP ) + qdeg(P )Bij(m/P ) if P - N0, X ` Bi`(P )B`j(m) = Bij(mP ) if P | N0.

Comparing the Fourier coecients the result holds.  Remark. Let EN0 =

n

P

j=1

Θij (which is independent of the choice of i). For r ∈ Z and

λ ∈ A with deg λ + 2 ≤ r the Fourier coecients E_N∗ 0(r, λ) = q −r_σ(λ) N0 where σ(λ)N0 = X m|λmonic (m,N0)=1 qdeg m, and E_N∗ 0(r, 0) = q −r n X j=1 1 wj . Moreover, from Proposition 2.2 we have

TmEN0 = σ(m)N0EN0

for all monic polynomials m in A. This tell us that the function EN0, which is an

analogue of Eisenstein series, generates a one-dimensional eigenspace for all Hecke operators. We point out that suppose N0 = Q`_i=1Pi, by comparing the Fourier

coecients one gets

q2EN0(g∞) = E(g∞) + ` X i=1 (−1)i   X 1≤j1<...<ji≤` E Pj1· · · Pji 0 0 1 ! g∞ !  for g∞∈ GL2(k∞) where E is the improper Eisenstein series introduced in [6].

For each zero ideal N of A, recall the Petersson inner product, which is a non-degenerate pairing on the nite dimensional C-vector space S(Γ0(N ))of automorphic

cusp forms of Drinfeld type for Γ0(N ),

(f, g) := Z

G0(N )

f · g.

Here G0(N ) = Γ0(N )\ GL2(k∞)/Γ∞k∞×. The measure on G0(N )is taken by counting

Denition 2.3. An old form is a linear combinations of forms f0 d 0 0 1 ! g∞ !

for g∞∈ GL2(k∞), where f0 is an automorphic cusp form of Drinfeld type for Γ0(M ),

M |N, M 6= N, and d|(N/M). An automorphic cusp form f of Drinfeld type for Γ0(N )is called a new form if for any old form f0 one has

(f, f0) = 0.

If f is a new form which is also a Hecke eigenform, then f is called a newform. It is known that the dimension of Drinfeld type cusp forms for Γ0(N0) is equal

to the genus of the Drinfeld modular curve X0(N0) (cf. [7]). Let Snew(Γ0(N0)) be

the space of new forms for Γ0(N0) and hN0 be the number of left ideal classes of the

maximal order R. As in the classical case, we can deduce that hN0 = 1 q2_{− 1} Y P |N0 (qdeg P − 1) + q 2(q + 1) Y P |N0 (1 − (−1)deg P).

From the genus formula of X0(N0) in [5], the dimension of Snew(Γ0(N0)) is equal to

hN0− 1.

In the next subsection we will give our main theorem, which is essentially a con-struction of the space Snew_{(Γ0(N0)) of new forms for Γ0(N0) via the theta series}

Θij.

2.3. Main theorem. Consider the denite Shimura curve X = XN0 introduced in

Ÿ1. Recall the height pairing

< e, e0 >=X

i

aia0i

where e ∈ Pic(X) with e = Piaiei and e0∈ Pic(X)∨ with e0=

P

ia 0 ieˇi.

Let M(Γ0(N0)) be the space of automorphic forms of Drinfeld type for Γ0(N0).

Dene Φ : Pic(X) × Pic(X)∨ _{→ M (Γ}

0(N0))by

Φ(e, e0) := q2X

i,j

aia0jΘij

for any e ∈ Pic(X) with e = Piaiei and e0 ∈ Pic(X)∨ with e0 =Pia0ieˇi. Then for

r ∈ Z and u ∈ k∞ we have the following Fourier expansion

Φ(e, e0) π r ∞ u 0 1 ! = q−r+2 deg e · deg e0+ X mmonic, deg m≤r−2 < e, tme0 > X ∈F×q ψ∞(mu) ! . Since < tme, e0 >=< e, tme0>

for any monic polynomial m ∈ A, by Proposition 2.2 one has Tm Φ(e, e0)
= Φ(tme, e0) = Φ(e, tme0).

In fact, the image of Φ is in the subspace Mnew_{(Γ0(N0)) := S}new_{(Γ0(N0)) ⊕ CEN}

0.

Claim: for any monic m in A, consider tm as in End(Pic(X)) and restrict Tm to

the subspace Mnew_{(Γ0(N0)). We have}

Tr tm= Tr Tm.

This claim tells us that the C-algebra TC generated by all tm is isomorphic to

the C-algebra generated by all Hecke operators Tm. Moreover, Pic(X) ⊗ZC and

Mnew(Γ0(N0))are isomorphic as TC-modules.

According to multiplicity one theorem, which will be recalled in the Appendix ŸA.2, Mnew_{(Γ0(N0))is a free rank one T}

C-module. More precisely, Mnew(Γ0(N0))is

generated by the element f whose Fourier coecients are f∗(r, λ) = q−r+2· Tr (Tm)

for all 0 6= λ ∈ A, (λ) = (m), deg λ + 2 ≤ r. Therefore Pic(X) ⊗ZC is also a free rank

one TC-module. This shows

dim_CMnew(Γ0(N0)) = dimC

 Pic(X) ⊗_ZC
⊗TC Pic(X) ∨_⊗ ZC
 . Moreover, since n X i=1 < ei, tmˇei > = Tr (B(m)) = Tr (tm), we get Pn

i=1Φ(ei, ˇei) = f, which generates Mnew(Γ0(N0)). This also tells us that

Pn

i=1ei⊗ˇeiis a generator of the cyclic TC-module Pic(X)⊗ZC
⊗TC Pic(X)

∨_⊗ ZC

. The above argument gives us the main result:

Theorem 2.4. There is a map Φ : Pic(X) × Pic(X)∨ _{−→ M}new_(Γ

0(N0)) satisfying

that for r ∈ Z and u ∈ k∞

Φ(e, e0) π r ∞ u 0 1 ! = q−r+2 deg e · deg e0+ X mmonic, deg m≤r−2 < e, tme0 > X (λ)=(m) ψ∞(λu) ! , and for all monic polynomials m in A

TmΦ(e, e0) = Φ(tme, e0) = Φ(e, tme0).

Moreover, this map induces an isomorphism (Pic(X) ⊗_ZC) ⊗TC(Pic(X)

∨_⊗

ZC)∼= Mnew(Γ0(N0))

as TC-modules.

Remark. 1. When N0is a prime, Mnew(Γ0(N0)) = M (Γ0(N0))and so the theta series

Θij gives us a construction of all automorphic forms of Drinfeld type for Γ0(P0).

2. Since the theta series Θij are Q-valued, the map Φ in Theorem 2.4 in fact induces

an isomorphism

(Pic(X) ⊗_ZQ) ⊗TQ(Pic(X)

∨_⊗

ZQ)∼= Mnew(Γ0(N0), Q)

where TQis the Q-algebra generated by tmfor all monic m in A and Mnew(Γ0(N0), Q)

is the space of Q-valued functions in Mnew_(Γ0(N0)).

3. The Claim above is essentially Jacquet-Langlands correspondence over the func-tion eld k, which will be recalled in the Appendix ŸA.1.

2.4. Example: The function gA. Having Theorem 2.4, we exhibit automorphic

forms of Drinfeld type with nice arithmetic properties. Let D ∈ A − k2

∞be a

square-free element with the quadratic Legendre symbol D P

6= 1 for all P | N0. Let K

be the imaginary quadratic eld k(√D) and OK be the integral closure of A in K.

Recall that in Ÿ1.1 one has a free action of Pic(OK)on the set GD of Gross points of

discriminant D in the denite Shimura curve X = XN0:

GD× Pic(OK) −→ GD

(x, A) 7−→ xA.

Given a Gross point x of discriminant D in X. For each ideal class A ∈ Pic(OK),

denote eA to be the divisor class (xA) in Pic(X). Dene

gA:=

X

B∈Pic(OK)

Φ(eB, eAB).

We have a nice formula for the Fourier coecients of gA in terms of Hecke actions:

for monic m ∈ A with deg m + 2 ≤ r, g∗A(r, m) = q−r+2·

X

B∈Pic(OK)

< eB, tmeAB >,

g∗A(r, 0) = q−r+2· hOK.

Here hOK = # Pic(OK). Note that gAis independent of the choice of the Gross point

x.

From now on we assume D is irreducible with D

P
= −1 for all primes P | N0.

According to Dirichlet's theorem there exists a monic irreducible polynomial Q and 0 ∈ F×q such that deg N0QD is odd, 0N0Q ≡ 1 mod D, and

 D Q  = 1. Then there exists j ∈ D with j2 _{= }

0N0Qso that D = K + Kj and j−1αj = ¯α for α ∈ K.

Let d = (√D) be the dierent of OK. Since D is irreducible, d is a prime ideal of

OK. Note that the prime ideal (Q) is split in K and suppose (Q) = q¯q. Set

R := {α + βj : α ∈ d−1, β ∈ d−1q−1, α − β ∈ Od}.

Here Od is the localization of OK at d. Then R is a maximal order and K ∩ R = OK.

Let x be the Gross point in the denite Shimura curve X = XN0 which corresponds to

the trivial ideal R and the embedding K ,→ D. Then x is of discriminant D. Using this particular Gross point we can get an explicit formula for the Fourier coecients of gA.

Note that there is a one-to-one correspondence between the irreducible components of X and the left ideal classes of R. Let a ∈ A, b ∈ B. Then Ra and Rab are representatives of the left ideal classes of R corresponding to eA and eAB respectively.

Therefore < eB, tmeAB >= 1 q − 1#{b ∈ b −1 Rba := (Nr(b))/ Nr(a) = (m)}. Assume P0d and a are relatively prime. Then

b−1Rba = {α + βj : α ∈ d−1a, β ∈ d−1b−1¯bq−1¯a, α − (−1)ordd(b)_{β ∈ O}

d}.

We can express the Fourier coecients of gA in terms of sums of the counting

numbers

for ideals (λ) of A, by the following proposition: Proposition 2.5. Suppose D ∈ A − k2

∞ is irreducible with DP
= −1 for all primes

P | N0. Then for any monic polynomial m in A,

X B∈Pic(OK) < eB, tmeAB >= 1 2(q − 1) " 2rA((mD))(q − 1)hOK + X µ∈A,µ6=0 deg(µN0)≤deg(mD) rA((µN0− mD))(t(µ, D) + 1)(1 − δµN0(µN0−mD)) X c|µ  D c # .

Here t(µ, D) = 1 if D divides µ and 0 otherwise, and δz is the norm residue symbol

of z for z ∈ k×

∞: δz = 1 if z ∈ Nr(K∞×) and −1 otherwise.

Proof. Let a ∈ A which is a proper ideal of OK and prime to N0d. Fix a generator

λ0 of Nr(a) = a¯a. Given B ∈ Pic(OK). Let b ∈ B. For b = α + βj ∈ b−1Rba, i.e.

α ∈ d−1a, β ∈ d−1q−1b−1¯b¯a, α − (−1)orddb_{β ∈ O} d, dene: (1) c := (β)dq¯b−1_{b¯a∈ [q]B}2_A, (2) ν := − Nr(α)Dλ−1 0 ∈ A, (3) µ := −0Nr(β)DQλ−10 ∈ A.

Here [q] ∈ Pic(OK) is the ideal class containing q. Then c is integral and

Nr(α + βj) = Nr(α) − 0N0Q Nr(β) = (−ν + N0µ)D−1λ0.

Thus (Nr(α + βj)) = (mλ0)if and only if ν = N0µ − mD for a uniquely determined

 ∈ F×q.

Since β = 0 if and only if b = α ∈ a, one has #{b ∈ b−1Rba : Nr(b) = (mλ0)}

= #{b = α + βj ∈ b−1Rba : β 6= 0, Nr(b) = (mλ0)}

+#{α ∈ a : Nr(α) = (mλ0)}.

It can be shown that #{α ∈ a : Nr(α) = (mλ0)} = (q − 1)rA((mD)). Note that

β 6= 0if and only if µ 6= 0. In this case, β is uniquely determined by the integral ideal c up to multiplying elements in O_K×.

Conversely, given 0 6= µ ∈ A and  ∈ F×

q and set ν = N0µ − mD. The number of

elements α ∈ d−1_{a with Nr(α) = −νD}−1_λ

0 is ra,λ0(N0µ − mD). Here

ra,λ0(λ) := #{a ∈ a : Nr(a) = λλ0} for λ ∈ A.

In the case of ra,λ0(N0µ − mD) 6= 0, choose an element α ∈ d

−1_{a with Nr(α) =}

−νD−1_λ

0. Let c be an integral ideal which lies in a class diering from the ideal class

A[q] by a square [b]2 _{in the class group Pic(O}

K)and with ideal norm (µ). Then

c· b−1¯b¯aq−1d−1 = (β) for some β ∈ K×_{. Suppose we can nd β so that µ = −}

0Nr(β)DQλ−10 ∈ A. Since

0N0Q ≡ 1 mod D, the equality mλ0 = Nr(α) − 0N0Q Nr(β) ∈ A implies

Choose ` ∈ {0, 1} and replace b by bd` _{so that}

α − (−1)ordd(b)_{β ∈ O}

d.

Therefore b = α + βj ∈ b−1_{Rba with Nr(b) = mλ}

0. Note that if β is not in Od

(i.e. D - µ), then ` is uniquely determined. If β ∈ Od (i.e. D | µ), then we have two

choices ±β. The existence of β is equivalent to that −−1

0 DµQ−1λ0 is in Nr(K×).

Since Nr(√D) = −Dand (−1₀ µQ−1λ0) = Nr(cq−1¯a), we have −1₀ µQ−1λ0∈ Nr(K×)

if and only if δ_−1

0 µQ−1λ0 = 1. Therefore combining the above arguments we have

X B∈Pic(OK) #{b = α + βj ∈ b−1Rba : β 6= 0, Nr(b) = (mλ0)} = X 06=µ∈A X ∈F×q ra,λ0(N0µ − mD) · (t(µ, D) + 1) · R{A[q]}((µ)) · 1 + δ_−1 0 µQ−1λ0 2 .

Here R{A[q]}((µ)) is the number of integral ideals c, which lie in a class diering

from the class A[q] by a square in the class group Pic(OK) and with ideal norm (µ).

Following the proof of Lemma 3.4.9 in [10] one has Lemma 2.6. For 0 6= µ ∈ A, R_{A[q]}((µ)) · 1 + δ_−1 0 µQ−1λ0 2 = 1 q − 1 X c|µ  D c  ·1 + δ −1 0 µQ−1λ0 2 . Since δ−1₀ µQ−1_λ

0 = 1 if and only if δN0µλ0 = −1, with Lemma 2.6 we have

X B∈Pic(OK) #{b = α + βj ∈ b−1Rba : β 6= 0, Nr(b) = (mλ0)} = X 06=µ∈A X ∈F×q ra,λ0(N0µ − mD)(t(µ, D) + 1) · 1 − δN0µλ0 2 · 1 q − 1 X c|µ  D c  = X µ∈A,µ6=0 deg(µN0)≤deg(mD) rA((µN0− mD))(t(µ, D) + 1) · 1 − δµN0(µN0−λD) 2 · X c|µ  D c  . Therefore X B∈Pic(OK) < eB, tmeAB >= 1 2(q − 1) " 2rA((λD))(q − 1)hOK + X µ∈A,µ6=0 deg(µN0)≤deg(mD) rA((µN0− mD))(t(µ, D) + 1)(1 − δµN0(µN0−mD)) X c|µ  D c # .  These automorphic forms gA are closely related to the central critical value of

3. Special values of L-series

To an automorphic cusp form f of Drinfeld type for Γ0(N )one can attach an

L-series L(f, s): let m be an eective divisor of k, which can be written as div(λ)0+

(r − deg λ)∞ for a nonzero polynomial λ (= λ(m)) in A, with div(λ)0:= X nite prime P ordP(λ)P. Denote f∗(m) := Z A\k∞ f π r+2 ∞ u 0 1 ! ψ∞(−λu)du = f∗(r + 2, λ).

The L-series L(f, s) attached to f is L(f, s) := X

m≥0

f∗(m)q− deg(m)s, Re s > 1. Let D ∈ A−k2

∞be a square-free element. Consider the imaginary eld K = k(

√ D). Let OK be the integral closure of A in K and Pic(OK)be the ideal class group of OK.

Given an ideal class A ∈ Pic(OK) and a polynomial λ in A. The number of integral

ideals a in the class A with NK/k(a) = (λ)leads to the partial zeta function attached

to A:

ζA(s) :=

X

m≥0

rA(m)q− deg(m)s, Re s > 1.

Here for each eective divisor m = div(λ)0+ (r − deg λ)∞,

rA(m) := #{a ∈ A : aintegral with NK/k(a) = (λ)}.

Given an automorphic cusp form f of Drinfeld type for Γ0(N ). For an ideal class

A ∈ Pic(OK), we are interested in the Rankin product:

L(f, A, s) := X

m≥0

f∗(m)rA(m)q− deg(m)s, Re(s) > 1.

In order to study the analytic continuation and the functional equation of L(f, A, s), consider the function Λ(f, A, s) which is dened by:

Λ(f, A, s) :=   

L(N,D)(2s + 1)L(f, A, s) when deg D is odd, 1

1 + q−s−1L

(N,D)_{(2s + 1)L(f, A, s) when deg D is even.}

Here L(N,D)_{(s)is the following L-series indexed by eective divisors supported outside}

∞ L(N,D)(s) := 1 q − 1 X d∈A,(d,N )=1  D d  q−s deg d, Re(s) > 1, where D d

denotes the Legendre symbol for the polynomial ring A. Note that L(N,D)(s) = LD(s) · Y prime ideals P |N (1 − D P  q−s deg P)−1 where LD(s) is the Dirichlet L-series:

LD(s) := 1 q − 1 X d∈A,d6=0  D d  q−s deg d, Re(s) > 1.

It is known that LD(s) can be extended to a polynomial in q−s with the functional equation (cf. [1]): LD(2s + 1) = qs(−2 deg D+2)− 1 2deg D+ 1 2L_D(−2s)

if deg D is odd, and

LD(−2s + 1) = 1 + q1−2s 1 + q2s q deg D(2s−1₂)_L D(2s) if deg D is even.

When f is a new form and D is irreducible, by Rankin's method Rück and Tipp ([10]) prove the functional equation of Λ(f, A, s):

Λ(f, A, s) = − D N



q(5−2 deg D−2 deg N )sΛ(f, A, −s) when deg D is odd, and

Λ(f, A, s) = − D N



q(3−2 deg D−2 deg N )sΛ(f, A, −s) when deg D is even.

For a given character χ : Pic(OK) → C×, dene

Λ(f, χ, s) := X

A∈Pic(OK)

χ(A)Λ(f, A, s).

When χ is the trivial character and f is a newform which is normalized so that the Fourier coecient f∗_{(0) = 1, one has}

Λ(f, χ, s) = L(f, s)L(f ⊗ εD, s)

where εD is the following quadratic character on divisors of k:

εD(P ) =  D P  and εD(∞) =    −1 if deg D is even, 0 if deg D is odd; and L(f ⊗ εD, s) is the twisted L-series of f by εD:

L(f ⊗ εD, s) :=

X

m≥0

f∗(m)εD(m)q− deg ms.

We are interested in the special value of Λ(f, χ, s) at s = 0. If D N

= 1, then Λ(f, A, s) has a zero at s = 0. In this case, the derivative of Λ(f, A, s) at s = 0 is determined by heights of Heegner points, and an analogue of Gross-Zagier formula has been proved for the case when D is irreducible (cf. [10]).

In this section we focus on the special case when N = N0 and D is irreducible

with D

P
= −1 for all P | N0. From Rankin method calculations carried out in [10]

Ÿ2.7, it can be deduced that (cf. [13] Chapter 3 Theorem 3.4) given any Drinfeld type new form f for Γ0(N0), we have

Λ(f, A, 0) =        (f, gA)

q12(deg D+1) when deg D is odd,

(f, gA)

2q12deg D

Here (·, ·) is the Petersson inner product and gA is the Drinfeld type automorphic

form for Γ0(N0) canonically attached to A which is introduced in Ÿ2.4.

3.1. The special value Λ(f, χ, 0). From the denition of Λ(f, χ, s) one has

Λ(f, χ, 0) =            1 q12(deg D+1) P A∈Pic(OK)

χ(A)(f, gA) if deg D is odd,

1 2q12deg D

P

A∈Pic(OL)

χ(A)(f, gA) if deg D is even.

Note that X A∈Pic(OK) χ(A)−1gA = X A∈Pic(OK)   X B∈Pic(OK) χ(A)−1Φ(eB, eAB)   = Φ(eχ, eχ) where eχ= X A∈Pic(OK) χ(A)eA.

Suppose f is a normalized newform. By Theorem 2.4 f corresponds to an element ef ∈ Pic(X) ⊗ZR so that

f = Φ(ef, ef).

Let ef,χ be the projection of eχ to the ef-isotypical component in Pic(X) ⊗ZC with

respect to the Gross height pairing. Then the f-eigencomponent of Φ(eχ, eχ) is equal

to

Φ(ef,χ, eχ) = Φ(ef,χ, ef,χ) =< ef,χ, ef,χ > f.

Therefore we obtain the following result.

Theorem 3.1. Let f be an automorphic cusp form of Drinfeld type for Γ0(N0)which

is also a normalized newform. Then

Λ(f, χ, 0) =            (f, f ) q12(deg D+1) · < e_f,χ, ef,χ> if deg D is odd, (f, f ) 2q12deg D · < e_f,χ, ef,χ> if deg D is even.

Remark. 1. If χ is non-trivial, then deg eχ= 0 and so Φ(eχ, eχ)is a cusp form.

2. When χ is trivial, then eχ = 21−`eD where ` is the number of prime factors of

N0 and eD is introduced in Proposition 1.7.

Let E be a non-iso-trivial elliptic curve over k (i.e. E is not dened over the constant eld Fq). From the work of Weil, Jacquet-Langlands, and Deligne, one knows that

there exists an automorphic cusp form fE such that

L(E/k, s + 1) = L(fE, s).

Here L(E/k, s) is the Hasse-Weil L-series of E over k. Suppose the conductor of E is N0∞, and E has split multiplicative reduction at ∞. Then the automorphic form

Consider the Hasse-Weil L-series L(E/K, s) of E over the imaginary quadratic eld K = k(√D) where D ∈ A with D_P
= −1for all primes P | N0. One has

L(E/K, s + 1) = L(fE, s)L(fE⊗ εD, s)

where L(fE, ⊗εD, s)is the twisted L-series of fE by the quadratic character εD. Since

L(fE, s)L(fE⊗ εD, s) = Λ(fE, 1D, s)

where 1D is the trivial character on Pic(OK), from Theorem 3.1 we obtain a formula

for the special value of L(E/K, s) at s = 1 when D is irreducible.

Appendix A. J-L correspondence and the multiplicity one theorem Given a Hecke character $ on k×

\A×k. Let D be a quaternion algebra over k and

set DAk := D ⊗kAk. We embed Ak into DAk by a 7−→ 1 ⊗ a. A C-valued function f

on D×_\D×

Ak is called an automorphic form on D

×

Ak (for K) with central character $

if f is a function on the double coset space D×\D×

Ak/K

for an open compact subgroup K of D×

Ak satisfying that for all g in D

×

Ak and a in A

× k

f (ag) = $(a)f (g).

Suppose D = Mat2(k). Then D× = GL2(k) and D×_Ak = GL2(Ak). f is called a

cusp form if for all g in GL2(Ak)

Z k\Ak f 1 u 0 1 ! g ! du = 0.

We denote A0($)to be the space of automorphic cusp forms on GL2(Ak)with central

character $.

We recall Jacquet-Langlands correspondence in ŸA.1 and use newform theory to prove the claim in Ÿ2.3. In ŸA.2 we use multiplicity one theorem to show that the space Mnew_(Γ

0(N0))in Ÿ2.3 is a free TC-module of rank one.

A.1. Jacquet-Langlands correspondence. Given a denite quaternion algebra D = D_(N0)over k where N₀ is the product of nite ramied primes of D. Let A0_($)be

the space of automorphic forms on D×

Ak with central character $. Jacquet-Langlands

correspondence describes the connection between A0_{($)and A} 0($):

([9] Chapter 3, Theorem 14.4 and Theorem 16.1) If an irreducible admissible rep-resentation ρ0 _{= ⊗}

vρ0v is a constituent of A0($) and ρ0P is innite dimensional for

all nite primes P which are prime to N0, then there exist an irreducible admissible

representation ρ(=: ρ0_{JL) which is a constituent of A}

0($)so that

L(s, $0⊗ ρ) = L(s, $0⊗ ρ0) for all Hecke characters $0_.

Note that ρ = ⊗vρv where ρv = ρ0v for nite primes v not dividing N0. Moreover,

for the ramied primes v of D, ρv is determined from ρ0v via theta correspondence.

Conversely, suppose ρ = ⊗vρv is a constituent of A0($). If for every ramied

primes v of D the representation ρv is special or supercuspidal, then there is a

con-stituent ρ0 _{= ⊗ρ}0

for ramied prime v if and only if ρv is special.

Let R be a xed maximal order of D. From Jacquet-Langlands correspondence one has an isomorphism Ψ between

{ C-valued non-constant functions on ˆR×\ ˆD×/D×} and



Drinfeld type new forms on Γ0(N0)\ GL2(k∞)/Γ∞k×∞

which satises

Ψ(tmf ) = TmΨ(f )

for all non-constant functions f on ˆR×\ ˆD×_/D× _{and monic polynomials m in A. We}

briey recall the argument in the following and refer the reader to [9] for further details. Fix $ = ⊗v$v to be the trivial Hecke character on k×\A×_k. Let v be a prime of k,

O_v be the valuation ring in k_v, and π_v a uniformizer in O_v. Recall that an irreducible admissible innite-dimensional representation (ρv, Vv) of GL2(kv) with central

char-acter $v has conductor vc(v) if πc(v)v Ov is the largest ideal of Ov such that the space

of elements u ∈ Vv with

ρv(gv)u = ufor all gv ∈ Kc(v)0

is non-empty. In fact, it is one dimensional. Here Kc(v)₀ = ( a b c d ! ∈ GL₂(Ov) : c ∈ πvc(v)Ov ) . It is known that c(v) =         

0 if ρv is an unramied principal series,

1 if ρv is an unramied special representation,

≥ 2 if ρ_v is supercuspidal or ramied. Let (ρ, V ) = N0

v(ρv, Vv) be a constituent of A0($). The conductor of ρ is:

Y

v

vc(v). The space of elements f ∈ V with

ρ(g)f = f for all g ∈Y

v

K₀c(v)

is one dimensional, and called the space of new-forms of ρ. Any new-form f of ρ is a Hecke eigenform, i.e. Tvf = avf for all v where av ∈ C.

Recall that L(s, ρ) = QvL(s, ρv), where

L(s, ρv) = 1 − χv,1(πv)q−s deg v

−1

· 1 − χv,2(πv)q−s deg v

−1 if ρv is an unramied principal series π(χv,1, χv,2);

L(s, ρv) = 1 − χv(πv)q−(s+1/2) deg v

−1 if ρv is an unramied special representation sp(χv| · |1/2v , χv| · |−1/2v );

if ρv is supercuspidal or ramied. Here χv,1, χv,2, and χv are unramied characters

of k×

v with χv,1· χv,2 = 1 = χ2v. It is known that

av =    q12deg v(χ_v,1(π_v) + χ_v,2(π_v)) if ρ_v ∼= π(χ_v,1, χ_v,2), χv(πv) if ρv ∼= sp(χv| · |1/2v , χv| · |−1/2v ).

Suppose ρ = ⊗vρv is of conductor N0∞ and ρ∞ ∼= sp(| · |1/2∞ , | · |−1/2∞ ). Then

new-forms of ρ are functions on

GL2(k)\ GL2(Ak)/K0(N0∞)k∞×.

From the bijection in Ÿ2.1

GL2(k)\ GL2(Ak)/K0(N0∞)k∞× = Γ∼ 0(N0)\ GL2(k∞)/Γ∞k×∞,

new-forms of such ρ can be viewed as newforms of Drinfeld type for Γ0(N0). In fact,

the space Snew_(Γ

0(N0)) of Drinfeld type new forms for Γ0(N0) is spanned by the

new-forms of such ρ with conductor N0∞.

Since ρ is of conductor N0∞, ρP ∼= sp(χP| · |1/2P , χP| · | −1/2

P )for all P | N0 where χP

is an unramied character of k×

P with χ2P = 1. By Jacquet-Langlands correspondence

we can nd an irreducible constituent (ρ0_{, V}0_{) = ⊗}

vρ0v of A0($) so that ρ = ρ0JL. In

this case, ρ0

P = χP ◦ Nrfor P | N0 and ρ0∞ is the trivial representation. Therefore we

can nd a subspace of elements f0 _{∈ V}0 _{which are non-constant functions on}

D×\ ˆD×/ ˆR×.

This subspace is also one dimensional, called the space of new-forms of ρ0_{. Any}

new-form f0 _{of ρ}0 _{is also a Hecke eigenform, i.e. t}

vf0 = a0vf0, where a0v appears in the local

factor Lv(s, ρ0v). Since for any place v

L(s, ρv) = L(s, ρ0v),

we have av = a0v.

In fact, the space of non-constant functions on D×_{\ ˆD}×_{/ ˆR}× _{is generated by}

new-forms of such ρ0 _{= ⊗}

vρ0v where ρ0∞ is trivial and for P | N0, ρ0_P = χP ◦ Nr for an

unramied character χP of k×P with χ2P = 1. By taking inverse, we identify functions

on D×_{\ ˆD}×_{/ ˆR}× _{with functions on ˆR}×_{\ ˆD}×_/D×_{. From the dimension formula at the}

end of Ÿ2.2 we have a bijective map Ψ from

{ C-valued non-constant functions on ˆR×\ ˆD×/D×} to



Drinfeld type new forms on Γ0(N0)\ GL2(k∞)/Γ∞k×∞

so that for each monic polynomial m in A,

Ψ(tmf ) = TmΨ(f ).

Since constant functions on ˆR×\ ˆD×/D× are eigenfunctions of tm with eigenvalue

σ(m)N0, We extend Ψ by mapping constant functions into the one dimensional

Consider the denite Shimura curve X = XN0. We have a canonical bijection

between components of X and ideal classes of R and this gives the canonical isomor-phism

{ (C-valued) functions on ˆR×\ ˆD×/D×} ∼= Hom(Pic(X), C) ∼= Pic(X)∨⊗_ZC. Therefore one has:

Theorem A.1. Ψ : Pic(X)∨ _⊗

Z C ∼= Mnew(Γ0(N0)) is an isomorphism so that

Ψ(tmf ) = TmΨ(f ) for any monic polynomial m in A. Therefore

Tr (tm) = Tr (Tm)

and so the C-algebra TC generated by Hecke correspondences tm on X is isomorphic

to the C-algebra generated by Hecke operators Tm on Mnew(N0).

A.2. Multiplicity one theorem. Let $ : A× k/k

× _{be a Hecke character. Given}

two irreducible admissible representations ρ1 = ⊗vρ1,v and ρ2 = ⊗vρ2,v which are

constituents of A0($). The multiplicity one theorem (cf. [3]) tells us that ρ1 = ρ2 if

and only if

ρ1,v∼= ρ2,v

for all place v.

Fix $ to be trivial. Choose two irreducible admissible representations ρ1 = ⊗vρ1,v

and ρ2 = ⊗vρ2,v of conductor N0∞ which are constituents of A0($)satisfying

ρ1,∞∼= ρ2,∞∼= sp(| · |1/2∞ , | · |−1/2∞ )

and ρ1,P and ρ2,P are unramied special representations for P | N0. Let f1 and f2 be

new-forms of ρ1 and ρ2 respectively. Then TPfi = aP,ifi where aP,i ∈ C for i = 1, 2

and all prime P in A. If aP,1= aP,2 for all P , then LP(s, ρ1,P) = LP(s, ρ2,P) and so

ρ1,P ∼= ρ2,P

for all P . By multiplicity one theorem we have ρ1 = ρ2 and so f1, f2 are linearly

dependent.

Recall that Mnew_(Γ

0(N0)) = Snew(Γ0(N0)) ⊕ CEN0. for Γ0(P0). As a TC-module,

The space Mnew_{(Γ0(N0))is a direct sum (⊕iCfi) ⊕ CEN}

0 of one dimensional

submod-ules and each fi is a new-form of an irreducible admissible representation ρi = ⊗vρi,v

which is a constituent of A0($)with

ρi,∞∼= sp(| · |1/2∞ , | · |−1/2∞ )

and ρi,P is an unramied special representation for P | N0. According to multiplicity

one theorem, each pair of these one dimensional submodules are non-isomorphic. Therefore Mnew_{(Γ0(N0)) is a cyclic T}

C-module, which is generated by EN0 +

P

ifi.

Viewing TC as a subring of EndC Mnew(Γ0(N0))

, we have dim_CTC≤ dimCMnew(Γ0(N0)).

Therefore

Proposition A.2. The space Mnew_{(Γ0(N0))is a free T}

References

[1] Artin, E., Quadratische Körper der komplexen Multiplikation, Enzyklopädie der Math. Wiss. Band I, 2. Teil, Heft 10, Teil II.

[2] Bertolini, M. & Darmon, H., Heegner points on Mumford-Tate curves, Inv. Math., 126 (1996) 413-456.

[3] Bump, D., Automorphic Forms and Representations, Cambridge studies in advanced mathe-matics 55, (1996).

[4] Eichler, M., Zur Zahlentheorie der Quaternionen-Algebren, Crelle J. 195 (1955), 127-151. [5] Gekeler, E.-U., Invariants of Some Algebraic Curves Related to Drinfeld Modular Curves, J.

Number Theory 90 (2001) 166-183.

[6] Gekeler, E.-U., Improper Eisenstein Series on Bruhat-Tits Trees, manuscripta math. 86 (1995), 367-391.

[7] Gekeler, E.-U. & Reversat, M., Jacobians of Drinfeld modular curves, J. reine angew. Math. 476 (1996), 27-93.

[8] Gross, B. H., Heights and the Special Values of L-series, CMS Conference Proceedings, H. Kisilevsky, J. Labute, Eds., 7 (1987) 116-187.

[9] Jacquet, H. & Langlands, R., Automorphic Forms on GL(2), LNM 114, Springer 1970. [10] Rück, H.-G. & Tipp, U., Heegner Points and L-series of Automorphic Cusp Forms of Drinfeld

Type, Documenta Mathematica 5 (2000) 365-444.

[11] Serre, J.-P., Tree, Springer, Berlin-Heidelberg-New York 1980.

[12] Vignéras, M.-F., Arithmétique des Algèbres de Quaternions, LNM 800, Springer 1980.

[13] Wei, F.-T., On arithmetic of curves over function elds. Ph.D. thesis, National Tsing Hua University, 2010.

[14] Wei, F.-T. & Yu, J., On theta series from denite quaternion algebras over function elds, preprint.

[15] Weil, A., Dirichlet Series and Automorphic Forms, LNM 189, Springer 1971.

Department of Mathematics, National Tsing-Hua University, Hsinchu 30013, Taiwan E-mail address: d947205@oz.nthu.edu.tw

Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan E-mail address: yu@math.ntu.edu.tw