Theta Series and Function Field Analogue

of Gross Formula

Fu-Tsun Wei and Jing Yu^{1}

Received: October 14, 2010 Revised: July 6, 2011

Communicated by Peter Schneider

Abstract. Let k = 𝔽_{q}(t), with q odd. In this article we introduce “definite” (with respect to the infinite 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 a function field analogue of Gross formula for the central critical values of Rankin type L-series coming from automorphic cusp forms of Drinfeld type.

2010 Mathematics Subject Classification: 11R58 11G18 11F41 11F67 Keywords and Phrases: function field, quaternion algebra, Shimura curve, automorphic form, Hecke operator, special value of L-series.

Introduction

We present here a theory of “definite” quaternion algebras over the rational function field k := 𝔽_{q}(t) with q odd, “definite” means that the place ∞ at infinity ramifies for the quaternion algebra in question. Following Gross [8], we first give a geometric translation of Eichler’s arithmetic theory of definite quaternion algebra by introducing the so-called “definite” 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 definite Shimura curves and spaces of automorphic forms of Drinfeld type over the function field k.

1Both authors were supported by National Science Council grant (100)-2119-M002-007.

The first author was also supported by the Graduate Students Study Abroad Program of National Science Council.

Documenta Mathematica 16 (2011) 723–765

Automorphic forms of Drinfeld type are very useful tools for function fields arithmetic (cf. [7], [12] and [17] for more details and applications), which can be viewed as an analogue of classical modular forms of weight 2. To illustrate our approach to quaternion algebras over function field, we give an application to the study of central critical values of certain L-series of “Rankin type” in the global function field 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 multiplicative reduction at ∞. Having the extensive calculations done in [12], we obtain in particular a function field analogue of Gross formula for the central critical values of these L-series over “imaginary” quadratic extensions of k (with respect to ∞).

The structure of this article is modelled on [8]. Let 𝒟 be a “definite” quaternion algebra over k and let N_{0} be the product of finite ramified primes of 𝒟. We introduce the definite Shimura curve X = X_{N_{0}} over k (for maximal orders) in §1, which is a finite 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) (defined in §1.2), we then have at hand a construction of automorphic forms of Drinfeld type for the congruence subgroup Γ_{0}(N_{0}) of GL_{2}(𝔽_{q}[t]). The main theorem of this article in §2.3 is:

Theorem. There is a map Φ : Pic(X) × Pic(X)^{∨} ⟶ M^{new}(Γ_{0}(N_{0})) such that for all monic polynomials m of 𝔽_{q}[t]

T_{m}Φ(e, e′) = Φ(t_{m}e, e′) = Φ(e, t_{m}e′).

Here Pic(X)^{∨} is the dual group Hom(Pic(X), ℤ), M^{new}(Γ_{0}(N_{0})) is the space of Drinfeld type “new” forms for Γ_{0}(N_{0}), t_{m} are Hecke correspondences on X, and T_{m} are Hecke operators on M^{new}(Γ_{0}(N_{0})). Moreover, this map induces an isomorphism (as Hecke modules)

(Pic(X) ⊗_{ℤ} ℂ) ⊗_{𝕋_{ℂ}} (Pic(X)^{∨} ⊗_{ℤ} ℂ) ≅ M^{new}(Γ_{0}(N_{0})).

This theorem in fact tells us that all automorphic “new” forms of Drinfeld type come from our theta series. The special case of our theorem when N_{0} is single prime is also obtained in Papikian [10] §3, by a different geometric method using Néron models of jacobians of Drinfeld modular curve X_{0}(N_{0}). In our proof of the above theorem, we use the explicit construction of theta series and claim the equality of the trace of the m-th Brandt matrix B(m) and the trace of the Hecke operator T_{m} on M^{new}(Γ_{0}(N_{0})) for each monic polynomial m in 𝔽_{q}[t]. This claim is essentially the Jacquet-Langlands correspondence (cf.

[9]) between automorphic representations of quaternion algebras over k and automorphic cuspidal representations of GL_{2} over k. Another crucial step in the proof is to show that the Hecke module M^{new}(Γ_{0}(N_{0})) 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.

Let D be an irreducible polynomial in 𝔽_{q}[t] such that K = k(\sqrt{D}) is imaginary and P is inert in K if the prime P divides N_{0}. For each ideal class 𝒜 of 𝔽_{q}[t][\sqrt{D}] = O_{K}, we construct in §2.4 an automorphic form g_{𝒜} of Drinfeld type with its Fourier coefficients worked out. In §3.1 we recall Rankin product of L-series Λ(f, 𝒜, s) associated to Drinfeld type new form f for Γ_{0}(N_{0}) and partial zeta function ζ_{𝒜} . In §3.2 we express the central critical value Λ(f, 𝒜, 0) as the Petersson inner product of f and g_{𝒜} . Furthermore, when f is a “normalized” Hecke eigenform and χ is a character of ideal class group Pic(O_{K}) of O_{K}, we give the twisted critical value Λ(f, χ, 0) explicitly in terms of the Gross height of a special divisor class e_{f,χ} on the definite Shimura curve X_{N_{0}}. This is our analogue of Gross formula.

Let E be an elliptic curves over k with conductor N_{0}∞ and split multiplicative reduction at ∞. From the work of Weil, Jacquet-Langlands, and Deligne, it is well known that there exists a Drinfeld type cusp form f_{E} such that

L(E/k, s + 1) = L(f_{E}, s).

Here L(E/k, s) is the Hasse-Weil L-series of E over k. After doing base change to the quadratic field K, one gets

L(E/K, s + 1) = Λ(f, 1_{D}, s)

where 1_{D} is the trivial character of Pic(O_{K}). Our formula can certainly be applied to these elliptic curves. An example is given in §3.4.

Notation

We fix the following notations:

k : the rational function field 𝔽_{q}(t), q = p^{ℓ_{0}} where p is an odd prime.
A : the polynomial ring 𝔽_{q}[t].
∞ : the infinite place, which corresponds to degree valuation v_{∞}.
π_{∞} : t^{−1}, a fixed uniformizer of ∞.
k_{∞} : 𝔽_{q}((t^{−1})), i.e. the completion of k at ∞.
𝒪_{∞} : 𝔽_{q}[[t^{−1}]], i.e. the valuation ring in k_{∞}.
P : a finite prime (place) of k.
k_{P} : the completion of k at the finite prime P.

A_{P} : the closure of A in k_{P}.
𝔸_{k} : the adele ring of k.
k̂ : ∏′_{P} k_{P}, the finite adele ring of k.
Â : ∏_{P} A_{P}.

ψ_{∞} : a fixed additive character on k_{∞}: for y = ∑_{i} a_{i}π_{∞}^{i} ∈ k_{∞}, we define ψ_{∞}(y) := exp (\frac{2π\sqrt{−1}}{p} · Tr_{𝔽_{q}/𝔽_{p}}(−a_{1})).

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

1 Definite Shimura curves

Let 𝒟 be a quaternion algebra over k ramified at ∞ (call 𝒟 “definite”). Before introducing the definite Shimura curve for 𝒟, we start with a genus 0 curve Y over k associated with the quaternion algebra 𝒟, which is defined by the following: the points of Y over any k-algebra M are

Y (M) = {x ∈ 𝒟 ⊗_{k} M : x ≠ 0, Tr(x) = Nr(x) = 0}/M^{×},

where the action of M^{×} on 𝒟 ⊗_{k} M is by multiplication on M, Tr and Nr are respectively the reduced trace and the reduced norm of 𝒟. More precisely, if 𝒟 = k+ku+kv+kuv where u^{2} = α, v^{2} = β, α and β are in k^{×}, and uv = −vu, then Y is just the conic

αy^{2} + βz^{2} = αβw^{2}

in the projective plane ℙ^{2}. The group 𝒟^{×} acts on Y (from the right) by conjugation. If K is a quadratic extension of k, Y (K) is canonically identified with the set Hom(K, 𝒟) of embeddings: for each embedding f : K → 𝒟, let y = y_{f} be the image of the unique K-line on the quadric {x ∈ 𝒟 ⊗_{k} K : Tr(x) = Nr(x) = 0} on which conjugation by f(K^{×}) acts by multiplication by the character a ↦ a/ā. Note that y_{f} is one of the two fixed points of f(K^{×}) acting on Y (K); another one is the image of the line where conjugation acts by the character a ↦ ā/a. Let N_{0} be the product of the finite ramified primes of 𝒟. Choose a maximal A-order R of 𝒟. For any finite prime P let R_{P} := R ⊗_{A} A_{P}, 𝒟_{P} := 𝒟 ⊗_{k} k_{P}, and

R̂ := R ⊗_{A} Â, 𝒟̂ := 𝒟 ⊗_{k} k̂.

Definition 1.1. (cf. [2] and [8]) The definite Shimura curve X_{N_{0}} is defined as

X_{N_{0}} = (R̂^{×}\𝒟̂^{×} × Y ) /𝒟^{×}.

We will use the notation X instead of X_{N_{0}} when N_{0} is fixed.

Lemma 1.2. X_{N_{0}} is a finite union of curves of genus 0.

Proof. Let g_{1}, ..., g_{n} be representatives for the finite double coset space R̂^{×}\𝒟̂^{×}/𝒟^{×}, i.e.

𝒟̂^{×} = ∐_{i=1}^{n} R̂^{×}g_{i}𝒟^{×}.

Then each coset of X_{N_{0}} has a representative (R̂^{×}g_{i}, y) mod 𝒟^{×} and the map

\begin{matrix}X_{N_{0}} & ⟶ & ∐_{i=1}^{n} Y/Γ_{i} \\ (R̂^{×}g_{i}, y) & ⟼ & y mod Γ_{i} \end{matrix}

is a bijection, where Γ_{i} = g_{i}^{−1}R̂^{×}g_{i} ∩ 𝒟^{×} is a finite group for i = 1, ..., n.

Definition 1.3. Let K be an imaginary quadratic extension of k (i.e. ∞ is not split in K). We call

x = (g, y) ∈ Image[R̂^{×}\𝒟̂^{×} × Y (K) → X_{N_{0}}(K)]

a Gross point on X_{N_{0}} over K.

Let f : K → 𝒟 be the embedding corresponding to y. Then

f(K) ∩ g^{−1}R̂g = f(O_{d})

for some quadratic order O_{d} := A[\sqrt{d}] where d is an element in A with d ∉ k_{∞}^{2}. In this case, we say x is of discriminant d. Note that the discriminant of a Gross point is well-defined up to multiplying with elements in (𝔽_{q}^{×})^{2}. Set X_{i} := Y/Γ_{i}. If the component g of a Gross point x is congruent to g_{i} in R̂^{×}\𝒟̂^{×}/𝒟^{×}, then x lies on the component X_{i}(K) = (Y/Γ_{i})(K).

1.1 Actions on Gross points

Let a ∈ K̂^{×} where K̂ := K ⊗_{k} k̂ and x = (g, y) be a Gross point of discriminant d. Let f : K → 𝒟 be the embedding corresponding to y. This induces a homomorphism f̂: K̂ → 𝒟̂ and we define

x_{a} := (gf̂(a), y).

Note that x_{a} is also of discriminant d, and it is easy to check that x = x_{a} if and only if a ∈ Ô_{d}^{×}K^{×} where Ô_{d} := O_{d} ⊗_{A} Â. Hence Ô_{d}^{×}\K̂^{×}/K^{×} ≅ Pic(O_{d}) acts freely on the set G_{d} of Gross points of discriminant d.

The orbit space G_{d}/ Pic(O_{d}) is identified with the space of double cosets

R̂^{×}\ℰ/f̂(K̂^{×}),

where f : K → 𝒟 is a fixed embedding (if any exist) and

ℰ := {g ∈ 𝒟̂^{×} : f(K) ∩ g^{−1}R̂g = f(O_{d})}.

Note that

R̂^{×}\ℰ/f̂(K̂^{×}) = ∏_{P} R_{P}^{×}\ℰ_{P} /f(K_{P}^{×})

where ℰ_{P} := {g_{P} ∈ 𝒟_{P}^{×} : f(K_{P})∩g_{P}^{−1}R_{P} g_{P} = f(O_{d,P})} and O_{d,P} is the closure of O_{d} in K_{P} := K ⊗_{k} k_{P}.

Lemma 1.4. (cf. [16] or [17])

#(R_{P}^{×}\ℰ_{P} /f(K_{P}^{×})) = \begin{cases}1 & if P ∤ N_{0}, \\ 1 − \{\frac{d}{P} \} & if P | N_{0}. \end{cases}

Here \{\frac{d}{P} \} is the Eichler quadratic symbol, i.e.

\{\frac{d}{P} \} = \begin{cases}1 & if P^{2}|d or d mod P ∈ ((A/P)^{×})^{2}, \\ −1 & if d mod P ∈ (A/P)^{×} − ((A/P)^{×})^{2}, \\ 0 & if P |d but P^{2} ∤ d. \end{cases}

Remark. The above lemma tells us that the number #(G_{d}) is equal to

h(d) ∏_{P |N_{0}} (1 − \{\frac{d}{P} \} )

where h(d) is the class number of O_{d}.

There is a natural action of Gal(K/k) on Gross points in the following way: let x = (g, y) be a Gross point and f_{y} : K ↪ 𝒟 be the embedding corresponding to y. Define

x^{σ} = (g, y)^{σ} = (g, y_{σ})

where σ ∈ Gal(K/k) and y_{σ} corresponds to the embedding f_{y}∘σ. If x is a Gross point of discriminant d in X_{i} then so is x^{σ}. Moreover, let a ∈ Ô_{d}^{×}\K̂^{×}/K^{×} ≅ Pic(O_{d}) and σ ∈ Gal(K/k) one has

(x^{σ})_{a} = (x_{σ(a)})^{σ}.

Therefore we have an action of Pic(O_{d}) ⋊ Gal(K/k) on the set G_{d} of Gross points of discriminant d.

1.2 Hecke correspondences and Gross height pairing

Let P be a prime of A. Let 𝒯 be the Bruhat-Tits tree of PGL_{2}(k_{P}) as defined in [14]. The vertices are the equivalence classes of A_{P} -lattices L in k_{P}^{2} , and two such vertices [L] and [L′] are adjacent if there exists an integer r such that

P^{r+1}L ⊊ L′ ⊊ P^{r}L.

This is a tree where each vertex has degree q^{deg P} + 1. For a vertex v, the Hecke correspondence t_{P} sends v to the formal sum of its q^{deg P} + 1 neighbors in the tree. Identifying PGL_{2}(A_{P})\ PGL_{2}(k_{P}) with the Bruhat-Tits tree, we can write the Hecke correspondence for g ∈ PGL_{2}(A_{P})\ PGL_{2}(k_{P}):

t_{P} (g) := ∑_{deg(u)<deg P} (\begin{matrix}1 & u \\ 0 & P \end{matrix}) g + (\begin{matrix}P & 0 \\ 0 & 1 \end{matrix}) g.

Note that X_{N_{0}} can be written as

(R̂^{×}\𝒟̂^{×}/k̂^{×}) × Y/𝒟^{×}

and

R̂^{×}\𝒟̂^{×}/k̂^{×} = ∏_{P}′R_{P}^{×}\𝒟_{P}^{×}/k_{P}^{×}.

When (P, N_{0}) = 1,

R_{P}^{×}\𝒟_{P}^{×}/k_{P}^{×} ≅ PGL_{2}(A_{P})\ PGL_{2}(k_{P})

and so we have the Hecke correspondence t_{P} on X_{N_{0}}.

Now suppose P divides N_{0}, then R_{P}^{×}\𝒟_{P}^{×}/k_{P}^{×} has two elements and define the Atkin-Lehner involution

w_{P} (g, y) := (g′, y)

where g′ is another double coset in R_{P}^{×}\𝒟_{P}^{×}/k_{P}^{×}.

From the construction, these correspondences commute with each other. Therefore we can define Hecke correspondence t_{m} for every non-zero ideal (m) of A in the following way:

t_{mm′} = t_{m}t_{m′}  if m and m′ are relatively prime,
t_{P^{ℓ}} = t_{P^{ℓ−1}}t_{P} − q^{deg P} t_{P^{ℓ−2}} for P ∤ N_{0},
t_{P^{ℓ}} = w_{P}^{ℓ}  for P | N_{0}.

Note that X = X_{N_{0}} = ∐_{i=1}^{n} X_{i}, where n is the left ideal class number of R. Consider the Picard group Pic(X), which is isomorphic to ℤ^{n} and is generated by the classes e_{i} of degree 1 corresponding to the component X_{i}. Then the correspondences t_{m} induce endomorphisms of the group Pic(X). In fact, with respect to the basis {e_{1}, ..., e_{n}}, these endomorphisms can be represented by Brandt matrices. Let {I_{1}, ..., I_{n}} be a set of left ideals of R representing the distinct ideal classes, with I_{1} = R. Let w_{i} := #(R_{i}^{×})/(q − 1) where R_{i} is the right order of I_{i}. Consider M_{ij} := I_{j}^{−1}I_{i}, which is a left ideal of R_{j} with right order R_{i}. Choose a generator N_{ij} ∈ k of the reduced ideal norm Nr(M_{ij})(:=< Nr(b) : b ∈ M_{ij} >_{A}) of M_{ij}. For each monic polynomial m in A, define

B_{ij}(m) := \frac{#\{b ∈ M_{ij} : (Nr(b)/N_{ij}) = (m)\}}{(q − 1)w_{j}}

and the m-th Brandt matrix

B(m) := (B_{ij}(m))_{1≤i,j≤n} ∈ Mat_{n}(ℤ).

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

t_{m}e_{i} = ∑_{j=1}^{n} B_{ij}(m)e_{j}.

Proof. From the definition of t_{m} and the recurrence relations of B(m) (cf. [16]), we can reduce the proof to the case when m = P is a prime. From the following bijection

R̂^{×}\𝒟̂^{×} ≅ {left ideals of R}
R̂^{×}g ↔ I_{g} := R̂g ∩ 𝒟,

for any element g in 𝒟̂^{×} we can identify the following set

\{R̂^{×} (\begin{matrix}1 & u \\ 0 & P \end{matrix}) g : deg u < deg P \} ∪ \{R̂^{×} (\begin{matrix}P & 0 \\ 0 & 1 \end{matrix}) g\}

with

{left ideal I of R contained in I_{g} with Nr(I) = P Nr(I_{g})}.

According to the definition of t_{P}, t_{P} e_{i} = ∑_{j} α_{j}e_{j} where α_{j} is the number of left ideals I of R equivalent to I_{j} which are contained in I_{i} with Nr(I) = P Nr(I_{i}). It is easy to see that α_{j} = B_{ij}(P) and so the proposition holds.

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

\begin{cases}< e_{i}, e_{j} >:= 0 & if i ≠ j, \\ < e_{i}, e_{i} >:= w_{i}, \end{cases}

and extending bi-additively. Therefore Pic(X)^{∨} := Hom(Pic(X), ℤ) can be viewed as a subgroup of Pic(X) ⊗_{ℤ} ℚ with basis {ě_{i} := e_{i}/w_{i} : i = 1, ..., n} via this pairing. Since w_{j}B_{ij}(m) = w_{i}B_{ji}(m), one has the following proposition.

Proposition 1.6. For all classes e and e′ in Pic(X), we have

< t_{m}e, e′ >=< e, t_{m}e′ > .

Proof. Since w_{j}B_{ij}(m) = w_{i}B_{ji}(m), we have

< t_{m}e_{i}, e_{j} >=< e_{i}, t_{m}e_{j} > .

for all i, j and the result holds.

Let d ∈ A with d ∉ k_{∞}^{2}. Assume every prime factor P of N_{0} is not split in K and P^{2} does not divides d (i.e. the set G_{d} of Gross points of discriminant d is not empty). For any prime P | N_{0}, one has w_{P} (G_{d}) = G_{d}. Suppose P_{1}, …, P_{r} are primes dividing N_{0} and inert in K. We have in fact a free action of Pic(O_{d}) × ∏_{i=1}^{r}⟨w_{P_{i}}⟩ on G_{d}. Since w_{P_{i}} are of order 2 for all i, Pic(O_{d}) × ∏_{i=1}^{r}⟨w_{P_{i}}⟩ acts simply transitively on G_{d}.

Let a ∈ A with a ∉ k_{∞}^{2}. Consider the rational divisor

c_{a} := ∑_{a=df^{2},f monic} \frac{1}{2u(d)} ∑_{x_{d}∈G_{d}} x_{d}.

Here u(d) = #(O_{d}^{×}). By calculation one has

deg(c_{a}) = \frac{1}{2} ∑_{a=df^{2},f monic} [\frac{h(d)}{u(d)} · ∏_{P |N_{0}}(1 − \{\frac{d}{P} \})] .

Let e_{a} ∈ Pic(X) ⊗_{ℤ} ℚ be the class of the divisor c_{a}. It can be shown that

Proposition 1.7. The class e_{a} lies in Pic(X)^{∨}, which is considered as a subgroup of Pic(X) ⊗_{ℤ} ℚ.

Note that we can extend the Gross height pairing to Pic(X) ⊗_{ℤ} ℂ which is linear in the first 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 𝒦_{0}(N∞) := ∏_{P} 𝒦_{0,P} × Γ_{∞} of GL_{2}(𝔸_{k}), where

𝒦_{0,P} := \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL_{2}(A_{P}) : c ∈ NA_{P} \}

for finite prime P, and

Γ_{∞} := \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL_{2}(𝒪_{∞}) : c ∈ π_{∞}𝒪_{∞}\} .

An automorphic form f on GL_{2}(𝔸_{k}) for 𝒦_{0}(N∞) (with trivial central character) is a ℂ-valued function on the double coset space

GL_{2}(k)\ GL_{2}(𝔸_{k})/𝒦_{0}(N∞)k_{∞}^{×}.

Note that by strong approximation theorem (cf. [16]) we have the following bijection

GL_{2}(k)\ GL_{2}(𝔸_{k})/K_{0}(N∞)k_{∞}^{×} ≅ Γ_{0}(N)\ GL_{2}(k_{∞})/Γ_{∞}k_{∞}^{×}

where

Γ_{0}(N) = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL_{2}(A) : c ≡ 0 mod N\} .

Therefore f can be viewed as a ℂ-valued function on Γ_{0}(N)\ GL_{2}(k_{∞})/Γ_{∞}k_{∞}^{×}. From now on, we call f an automorphic form for Γ_{0}(N) if f is a function on the space of double cosets Γ_{0}(N)\ GL_{2}(k_{∞})/Γ_{∞}k_{∞}^{×}. An automorphic form f for Γ_{0}(N) is called a cusp form if for every g_{∞} ∈ GL_{2}(k_{∞}) and γ ∈ GL_{2}(A)

∫_{A\k_{∞}} f (γ (\begin{matrix}1 & h_{γ}x \\ 0 & 1 \end{matrix}) g_{∞}) dx = 0.

Here du is a Haar measure with ∫_{A\k_{∞}} du = 1 and h_{γ} is a generator of the ideal of A which is maximal for the property that

γ (\begin{matrix}1 & h_{γ}A \\ 0 & 1 \end{matrix}) γ^{−1} ⊂ Γ_{0}(N).

Note that the coset space GL_{2}(k_{∞})/Γ_{∞}k_{∞}^{×} can be represented by the two disjoint sets

𝒯_{+} := \{(\begin{matrix}π_{∞}^{r} & u \\ 0 & 1 \end{matrix}) : r ∈ ℤ, u ∈ k_{∞}/π_{∞}^{r}𝒪_{∞}\}

and

𝒯_{−} := \{(\begin{matrix}π_{∞}^{r} & u \\ 0 & 1 \end{matrix}) (\begin{matrix}0 & 1 \\ π_{∞} & 0 \end{matrix}) : r ∈ ℤ, u ∈ k_{∞}/π_{∞}^{r}𝒪_{∞}\} .

Definition 2.1. An automorphic form f on GL_{2}(k_{∞}) is of Drinfeld type if it satisfies the following harmonic properties: for any g_{∞} ∈ GL_{2}(k_{∞}) we have

f̃(g_{∞}) := f(g_{∞} (\begin{matrix}0 & 1 \\ π_{∞} & 0 \end{matrix})) = −f(g_{∞}) and ∑_{κ∈GL_{2}(𝒪_{∞})/Γ_{∞}} f(g_{∞}κ) = 0.

Suppose a function f : (\begin{matrix}1 & A \\ 0 & 1 \end{matrix}) \ GL_{2}(k_{∞})/Γ_{∞}k_{∞}^{×} → ℂ is given. Recall the

Fourier expansion of f (cf. [18]): for r ∈ ℤ and u ∈ k_{∞},

f (\begin{matrix}π_{∞}^{r} & u \\ 0 & 1 \end{matrix}) = ∑_{λ∈A} f^{∗}(r, λ)ψ_{∞}(λu)

where

f^{∗}(r, λ) := ∫_{A\k_{∞}} f (\begin{matrix}π_{∞}^{r} & u \\ 0 & 1 \end{matrix}) ψ_{∞}(−λu)du.

Here ψ_{∞} is the fixed additive character on k_{∞} in the notation table. Since f(gγ_{∞}) = f(g) for all γ_{∞} ∈ Γ_{∞}, f^{∗}(r, λ) = 0 if deg λ + 2 > r. Moreover, if f satisfies harmonic properties, then

f^{∗}(r, λ) = q^{−r+deg λ+2}f^{∗}(deg λ + 2, λ)

if deg λ + 2 ≤ r.

2.1.1 Example: Theta series

Fix a definite quaternion algebra 𝒟 = 𝒟_{(N_{0})} where N_{0} is the product of finite ramified primes of 𝒟. Let R be a maximal order and n be the class number. With representatives of left ideal classes fixed in §1.2, we have introduced for each (i, j), the ideal M_{ij} of 𝒟 and chose a generator N_{ij} of the fractional ideal Nr(M_{ij}). For 1 ≤ i, j ≤ n and (x, y) ∈ k_{∞}^{×} × k_{∞}, define

θ_{ij}(x, y) := ∑_{b∈M_{ij}} ϕ_{∞}(\frac{Nr(b)}{N_{ij}}xt^{2}) · ψ_{∞}(\frac{Nr(b)}{N_{ij}}y),

where ϕ_{∞} is the characteristic function of 𝒪_{∞}. It is easy to obtain the following properties: (1)

θ_{ij}(x, y) = ∑_{λ∈A,deg λ+2≤v_{∞}(x)} B′_{ij}(λ)ψ_{∞}(λy)

where for each λ ∈ A,

B′_{ij}(λ) = #{b ∈ M_{ij} : Nr(b)/N_{ij} = λ}.

(2) θ_{ij}(x, y + h) = θ_{ij}(x, y) for h ∈ A.

(3) θ_{ij}(αx, βx + y) = θ_{ij}(x, y) for α ∈ 𝒪_{∞}^{×}, β ∈ 𝒪_{∞}.

Basing on Poisson summation formula, we have the following transformation law for θ_{ij} (cf. Appendix B):

Proposition 2.2. Let (x, y) ∈ k_{∞}^{×} × k_{∞} and γ = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL_{2}(A). Suppose

v_{∞}(cx) > v_{∞}(cy + d) and c ≡ 0 mod N_{0}. Then for 1 ≤ i, j ≤ n,

θ_{ij} (\frac{x}{(cy + d)^{2}} , \frac{ay + b}{cy + d}) = q^{−2v_{∞}(cy+d)} · θ_{ij}(x, y).

For g_{∞} ∈ GL_{2}(k_{∞}), write g_{∞} as γ (\begin{matrix}x & y \\ 0 & 1 \end{matrix}) γ_{∞}z_{∞}, where γ is in Γ_{0}(N_{0}), (x, y)

is in k_{∞}^{×} × k_{∞}, γ_{∞} is in Γ_{∞}, and z_{∞} is in k_{∞}^{×}. We introduce the theta series Θ_{ij} for M_{ij}:

Θ_{ij}(g_{∞}) := \frac{1}{(q − 1)w_{j}} · q^{−v_{∞}(x)} · ( ∑_{ϵ∈𝔽_{q}^{×}} θ_{ij}(x, ϵy))

= q^{−v_{∞}(x)} · [\frac{1}{w_{j}} + ∑_{m∈A monic, deg m+2≤v∞(x)} B_{ij}(m)( ∑_{ϵ∈𝔽_{q}^{×}} ψ_{∞}(ϵmy))] .

The last equality follows from B′_{ij}(0) = 1 and for each monic polynomial m ∈ A,

(q − 1)w_{j} · B_{ij}(m) = ∑_{ϵ∈𝔽_{q}^{×}} B′_{ij}(ϵm).

The transformation law of θ_{ij} tells us that

Lemma 2.3. Θ_{ij} is a well-defined ℚ-valued function on the double coset space Γ_{0}(N_{0})\ GL_{2}(k_{∞})/Γ_{∞}k_{∞}^{×}.

Proof. Let g_{∞} be an element in GL_{2}(k_{∞}). Suppose

g_{∞} = γ_{1} (\begin{matrix}x_{1} & y_{1} \\ 0 & 1 \end{matrix}) γ_{∞,1}z_{1} = γ_{2} (\begin{matrix}x_{2} & y_{2} \\ 0 & 1 \end{matrix}) γ_{∞,2}z_{2},

where for i = 1, 2, γ_{i} ∈ Γ_{0}(N_{0}), (x_{i}, y_{i}) ∈ k_{∞} × k_{∞}^{×}, γ_{∞,i} ∈ Γ_{∞}, z_{i} ∈ k_{∞}^{×}. We need to show that

q^{−v_{∞}(x_{1})} · ( ∑_{ϵ∈𝔽_{q}^{×}} θ_{ij}(x_{1}, ϵy_{1})) = q^{−v_{∞}(x_{2})} · ( ∑_{ϵ∈𝔽_{q}^{×}} θ_{ij}(x_{2}, ϵy_{2})) .

Set γ = γ_{2}^{−1}γ_{1} = (\begin{matrix}a & b \\ c & d \end{matrix}), z = z_{1}^{−1}z_{2}, and γ_{∞} = γ_{∞,1}^{−1}γ_{∞,2}. Then one has

v_{∞}(cx_{1}) > v_{∞}(cy_{1} + d) and

γ (\begin{matrix}x_{1} & y_{1} \\ 0 & 1 \end{matrix}) = ( 0  1^{\frac{det γ·x_{1}}{(cy_{1}+d)^{2}}  \frac{ay_{1}+b}{cy_{1}+d}}) (\begin{matrix}1 & 0 \\ \frac{cx_{1}}{cy_{1}+d} & 1 \end{matrix}) (\begin{matrix}cy_{1} + d & 0 \\ 0 & cy_{1} + d \end{matrix})

= (\begin{matrix}x_{2} & y_{2} \\ 0 & 1 \end{matrix}) γ_{∞}z.

Therefore v_{∞}(x_{2}) = v_{∞}(x_{1}) − 2v_{∞}(cy_{1} + d), and the properties of θ_{ij} implies

θ_{ij}(x_{2}, ϵy_{2}) = θ_{ij} (\frac{det γ · x_{1}}{(cy_{1} + d)^{2}} , ϵ\frac{ay_{1} + b}{cy_{1} + d}) .

for each ϵ ∈ 𝔽_{q}^{×}. Hence the transformation law of θ_{ij} in Proposition 2.2 shows

q^{−v_{∞}(x_{2})} · ( ∑_{ϵ∈𝔽_{q}^{×}} θ_{ij}(x_{2}, ϵy_{2})) = q^{−v_{∞}(x_{1})} · ( ∑_{ϵ∈𝔽_{q}^{×}} θ_{ij}(det γ · x_{1}, ϵ det γ · y_{1}))

= q^{−v_{∞}(x_{1})} · ( ∑_{ϵ∈𝔽_{q}^{×}} θ_{ij}(x_{1}, ϵy_{1})) .

The Fourier coefficients of Θ_{ij} can be easily read off from Brandt matrices: for each r ∈ ℤ and λ ∈ A with deg λ + 2 ≤ r the Fourier coefficients

Θ_{ij}^{∗}(r, λ) = \begin{cases}q^{−r}B_{ij}(m) & if (λ) = (m) ≠ 0, \\ q^{−r}/w_{j} & if λ = 0. \end{cases}

Therefore Θ_{ij}^{∗}(r + 1, λ) = q^{−1}Θ_{ij}^{∗}(r, λ) for all λ ∈ A with deg λ + 2 ≤ r.

In fact, Θ_{ij} are of Drinfeld type for all 1 ≤ i, j ≤ n. To show the harmonicity of Θ_{ij}, by [6] Lemma 2.13, it is enough to prove that for all g_{∞} ∈ GL_{2}(k_{∞})

Θ̃_{ij}(g_{∞}) = −Θ_{ij}(g_{∞}).

Let π_{∞}^{r} ∈ k_{∞}^{×} and u ∈ k_{∞}. Choose c, d ∈ A with c ≡ 0 mod N_{0}, (c, d) = 1, v_{∞}(u + \frac{d}{c}) ≥ r + 1, and find a, b ∈ A with ad − bc = 1. Then for ℓ ∈ ℤ with ℓ ≤ r + 1 the following two matrices:

(\begin{matrix}π_{∞}^{ℓ} & u \\ 0 & 1 \end{matrix}) (\begin{matrix}0 & 1 \\ π_{∞} & 0 \end{matrix}) and (\begin{matrix}d & −b \\ −c & a \end{matrix}) ( 0 1^{\frac{π_{∞}^{1−ℓ}}{c^{2}}  \frac{a}{c}})

represent the same coset in GL_{2}(k_{∞})/Γ_{∞}k_{∞}^{×}. Using this fact for ℓ = r and ℓ = r + 1 one obtains

Θ̃_{ij} (\begin{matrix}π_{∞}^{r} & u \\ 0 & 1 \end{matrix}) − q^{−1}Θ̃_{ij} (\begin{matrix}π_{∞}^{r+1} & u \\ 0 & 1 \end{matrix})

= ∑_{deg μ+2=1−r+2 deg c} Θ_{ij}^{∗}(1 − r + 2 deg c, μ)ψ_{∞}(μ\frac{a}{c}).

Set u_{ϵ} := −\frac{d}{c} + ϵπ_{∞}^{r} for ϵ ∈ 𝔽_{q}^{×}. From the identity

\frac{a}{c} − \frac{1}{c^{2}ϵπ_{∞}^{r}} = \frac{au_{ϵ} + b}{cu_{ϵ} + d},

and summing over all ϵ we get:

(q − 1)Θ̃_{ij} (\begin{matrix}π_{∞}^{r} & u \\ 0 & 1 \end{matrix}) − ∑_{ϵ∈𝔽_{q}^{×}} Θ_{ij} ( 0  1^{\frac{π_{∞}^{1−r}}{c^{2}}  \frac{au_{ϵ}+b}{cu_{ϵ}+d}})

= q ∑_{deg μ+2=1−r+2 deg c} Θ_{ij}^{∗}(1 − r + 2 deg c, μ)ψ_{∞}(μ\frac{a}{c}).

Note that

( 0  1^{\frac{π_{∞}^{1−r}}{c^{2}}  \frac{au_{ϵ}+b}{cu_{ϵ}+d}}) and (\begin{matrix}a & b \\ c & d \end{matrix}) (\begin{matrix}π_{∞}^{r+1} & u_{ϵ} \\ 0 & 1 \end{matrix})

represent the same coset in GL_{2}(k_{∞})/Γ_{∞}k_{∞}^{×}. Thus one has

Θ̃_{ij} (\begin{matrix}π_{∞}^{r+1} & u \\ 0 & 1 \end{matrix}) − Θ̃_{ij} (\begin{matrix}π_{∞}^{r} & u \\ 0 & 1 \end{matrix}) = ∑_{ϵ∈𝔽_{q}^{×}} Θ_{ij} (\begin{matrix}π_{∞}^{r+1} & u + ϵπ_{∞}^{r} \\ 0 & 1 \end{matrix}) .

From the Fourier expansion of Θ̃_{ij} and Θ_{ij} we have that for λ ∈ A with deg λ + 2 ≤ r,

Θ̃_{ij}^{∗}(r + 1, λ) − Θ̃_{ij}^{∗}(r, λ) = (q − 1)Θ_{ij}^{∗}(r + 1, λ),

and for deg λ + 2 = r + 1,

Θ̃_{ij}^{∗}(deg λ + 2, λ) = −Θ_{ij}^{∗}(r + 1, λ).

Therefore Θ̃_{ij}^{∗}(r, λ) = −Θ_{ij}^{∗}(r, λ) for λ ∈ A with λ ≠ 0 and r ≥ deg λ + 2.

To compute Θ̃_{ij}^{∗}(r, 0), note that

Θ̃_{ij} (\begin{matrix}π_{∞}^{r} & 0 \\ 0 & 1 \end{matrix}) = ∑_{deg λ≤r−2} Θ̃_{ij}^{∗}(r, λ)
= Θ̃_{ij}^{∗}(r, 0) + ∑_{λ≠0,deg λ≤r−2} −Θ_{ij}^{∗}(r, λ).

On the other hand, for any ϵ ∈ 𝔽_{q}^{×} and ℓ ≥ 0 the following two matrices

(\begin{matrix}π_{∞}^{deg N_{0}+ℓ} & 0 \\ 0 & 1 \end{matrix}) (\begin{matrix}0 & 1 \\ π_{∞} & 0 \end{matrix}), (\begin{matrix}ϵ^{−1} & −1 \\ −t^{ℓ}N_{0} & ϵ(t^{ℓ}N_{0} + 1) \end{matrix}) (  0  1^{\frac{π_{∞}^{1−deg N_{0}−ℓ}}{(t^{ℓ}N_{0})^{2}}  \frac{ϵ(t^{ℓ}N_{0}+1)}{t^{ℓ}N_{0}}})

represent the same coset in GL_{2}(k_{∞})/Γ_{∞}k_{∞}^{×}. Therefore

Θ̃_{ij} (\begin{matrix}π_{∞}^{deg N_{0}+ℓ} & 0 \\ 0 & 1 \end{matrix}) = ∑_{deg λ≤deg N_{0}+ℓ−1} Θ_{ij}^{∗}(deg N_{0} + ℓ + 1, λ)ψ_{∞}(λ\frac{ϵ}{t^{ℓ}N_{0}} )

=  ∑_{deg λ≤deg N_{0}+ℓ−2}Θ_{ij}^{∗}(deg N_{0}+ℓ+1, λ)− \frac{1}{q − 1}  ∑_{deg λ=deg N_{0}+ℓ−1}Θ_{ij}^{∗}(deg N_{0}+ℓ+1, λ).

This gives

Θ̃_{ij}^{∗}(deg N_{0} + ℓ, 0) =

= (Θ_{ij}^{∗}(deg N_{0} + ℓ + 1, 0) + (1 + q) ∑_{λ≠0,deg λ≤deg N_{0}+ℓ−2} Θ_{ij}^{∗}(deg N_{0} + ℓ + 1, λ)

−\frac{1}{q − 1} ∑_{deg λ=deg N_{0}+ℓ−1} Θ_{ij}^{∗}(deg N_{0} + ℓ + 1, λ))

= −Θ_{ij}^{∗}(deg N_{0}+ℓ, 0)+ \frac{1}{q − 1} ·[qΘ_{ij} (\begin{matrix}π_{∞}^{deg N_{0}+ℓ} & 0 \\ 0 & 1 \end{matrix}) − Θ_{ij} (\begin{matrix}π_{∞}^{deg N_{0}+ℓ+1} & 0 \\ 0 & 1 \end{matrix})]

Using the fact that M_{ij} is discrete and cocompact in 𝒟_{∞} = 𝒟 ⊗_{k} k_{∞}, it can be deduced that for sufficiently large s one has

qΘ_{ij} (\begin{matrix}π_{∞}^{s} & 0 \\ 0 & 1 \end{matrix}) = Θ_{ij} (\begin{matrix}π_{∞}^{s+1} & 0 \\ 0 & 1 \end{matrix}) .

Thus from the equality Θ̃_{ij}^{∗}(r + 1, 0) − Θ̃_{ij}^{∗}(r, 0) = (q − 1)Θ_{ij}^{∗}(r + 1, 0) for all r ∈ ℤ one has

Θ̃_{ij}^{∗}(r, 0) = −Θ_{ij}^{∗}(r, 0).

Comparing the Fourier coefficients we obtain Θ̃_{ij} = −Θ_{ij} and hence Θ_{ij} is of Drinfeld type for any 1 ≤ i, j ≤ n.

2.2 Hecke operators

Let f be an automorphic form on GL_{2}(k_{∞}) for Γ_{0}(N). For each prime P of A, the Hecke operator T_{P} is defined by:

T_{P} f(g) := ∑_{deg u<deg P} f((\begin{matrix}1 & u \\ 0 & P \end{matrix}) · g) + f((\begin{matrix}P & 0 \\ 0 & 1 \end{matrix}) · g) if P ∤ N,

T_{P} f(g) := ∑_{deg u<deg P} f((\begin{matrix}1 & u \\ 0 & P \end{matrix}) · g)  if P |N.

Note that the Fourier coefficients of T_{P} f are of the form:

(T_{P} f)^{∗}(r, λ) = q^{deg(P)} · f^{∗}(r + deg(P), P λ) + f^{∗}(r − deg(P), \frac{λ}{P}) if P ∤ N,
(T_{P} f)^{∗}(r, λ) = q^{deg(P)} · f^{∗}(r + deg(P), P λ)  if P |N.

Here f^{∗}(π_{∞}^{r}, \frac{λ}{P}) = 0 if P ∤ λ. Since T_{P} and T_{P′} commute,we can define Hecke operators T_{m} for monic polynomial m in A as follows:

T_{mm′} = T_{m}T_{m′}  if m and m′ are relatively prime,
T_{P^{ℓ}} = T_{P^{ℓ−1}}T_{P} − q^{deg P} T_{P^{ℓ−2}} for P ∤ N,
T_{P^{ℓ}} = T_{P}^{ℓ}  for P | N.

We point out that if f is of Drinfeld type, then so is T_{m}f for any monic polynomial m (cf. [7] Section 4.9).

When T_{m} acts on Θ_{ij}, we get:

Proposition 2.4. For any monic polynomial m in A,

T_{m}Θ_{ij} = ∑_{ℓ} B_{iℓ}(m)Θ_{ℓj} = ∑_{ℓ} B_{ℓj}(m)Θ_{iℓ}.

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

w_{j}Θ_{ij} = w_{i}Θ_{ji} and w_{ℓ}B_{iℓ}(m) = w_{i}B_{ℓi}(m).

Note that the Hecke operators T_{m} satisfy the same relations as the matrices B(m). Moreover, from the recurrence relations of Brandt matrices (cf. [16]) we have

∑_{ℓ} B_{iℓ}(P)B_{ℓj}(m) = B_{ij}(mP) + q^{deg(P)}B_{ij}(m/P) if P ∤ N_{0},
∑_{ℓ} B_{iℓ}(P)B_{ℓj}(m) = B_{ij}(mP)  if P | N_{0}.

Comparing the Fourier coefficients the result holds.

Remark. Let ℰ_{N_{0}} := ∑_{j=1}^{n} Θ_{ij} (which is independent of the choice of i). For

r ∈ ℤ and λ ∈ A with deg λ + 2 ≤ r the Fourier coefficients are

ℰ_{N_{0}}^{∗}(r, λ) = q^{−r}σ(λ)_{N_{0}}

where

σ(λ)_{N_{0}} = ∑_{{m|λ monic \atop (m,N_{0})=1}} q^{deg m},

and

ℰ_{N_{0}}^{∗}(r, 0) = q^{−r} ∑_{j=1}^{n} \frac{1}{w_{j}} .

Moreover, from Proposition 2.4 we have

T_{m}ℰ_{N_{0}} = σ(m)_{N_{0}}ℰ_{N_{0}}

for all monic polynomials m in A. This tell us that the function ℰ_{N_{0}}, which is an analogue of Eisenstein series, generates a one-dimensional eigenspace for all Hecke operators. We point out that suppose N_{0} = ∏_{i=1}^{ℓ} P_{i}, by comparing the Fourier coefficients one gets

q^{2}ℰ_{N_{0}}(g_{∞}) = E(g_{∞}) + ∑_{i=1}^{ℓ}(−1)^{i} [ ∑_{1≤j_{1}<...<j_{i}≤ℓ} E ((\begin{matrix}P_{j_{1}} ⋯ P_{j_{i}} & 0 \\ 0 & 1 \end{matrix}) g_{∞})]

for g_{∞} ∈ GL_{2}(k_{∞}) where E is the improper Eisenstein series introduced in [6]. For each non-zero ideal N of A, recall the Petersson inner product, which is a non-degenerate pairing on the finite dimensional ℂ-vector space S(Γ_{0}(N)) of automorphic cusp forms of Drinfeld type for Γ_{0}(N),

(f, g) := ∫_{G_{0}(N)} f · \overline{g}.

Here G_{0}(N) = Γ_{0}(N)\ GL_{2}(k_{∞})/Γ_{∞}k_{∞}^{×}. The measure on G_{0}(N) is taken by counting the size of the stablizer of an element (cf. [7] §4.8). More precisely, let Γ be a congruence subgroup and e ∈ GL_{2}(k_{∞})/Γ_{∞}k_{∞}^{×}. We denote Stab_{Γ}(e) the stabilizer of e in Γ, which is a finite subgroup in Γ. One takes the measure d([e]) of each double coset [e] in Γ\ GL_{2}(k_{∞})/Γ_{∞}k_{∞}^{×} where

d([e]) := \frac{#(Z(Γ))}{#(Stab_{Γ}(e))}.

Here Z(Γ) is the subgroup of scalar matrices in Γ. When Γ = Γ_{0}(N), for f and g in S(Γ_{0}(N)),

(f, g) = ∑_{[e]∈G_{0}(N)} f(e)\overline{g(e)}d([e]).

Definition 2.5. An old form is a linear combinations of forms

f′ ((\begin{matrix}d & 0 \\ 0 & 1 \end{matrix}) g_{∞})

for g_{∞} ∈ GL_{2}(k_{∞}), where f′ is an automorphic cusp form of Drinfeld type for Γ_{0}(M), M|N, M ≠ N, and d|(N/M). An automorphic cusp form f of Drinfeld type for Γ_{0}(N) is called a new form if for any old form f′ one has

(f, f′) = 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 Γ_{0}(N_{0}) is equal to the genus of the Drinfeld modular curve X_{0}(N_{0}) (cf. [7]). Let S^{new}(Γ_{0}(N_{0})) be the space of new forms for Γ_{0}(N_{0}) and h_{N_{0}} be the number of left ideal classes of the maximal order R. As in the classical case, we can deduce that

h_{N_{0}} = \frac{1}{q^{2} − 1} ∏_{P |N_{0}}(q^{deg P} − 1) + \frac{q}{2(q + 1)} ∏_{P |N_{0}}(1 − (−1)^{deg P}).

From the genus formula of X_{0}(N_{0}) in [5], the dimension of S^{new}(Γ_{0}(N_{0})) is equal to h_{N_{0}} − 1. In the next subsection we will give our main theorem, which is essentially a construction of the space S^{new}(Γ_{0}(N_{0})) of new forms for Γ_{0}(N_{0}) via the theta series Θ_{ij}.

2.3 Main theorem

Consider the definite Shimura curve X = X_{N_{0}} introduced in §1. Recall the height pairing

< e, e′ >= ∑_{i} a_{i}a′_{i}

where e ∈ Pic(X) with e = ∑_{i} a_{i}e_{i} and e′ ∈ Pic(X)^{∨} with e′ = ∑_{i} a′_{i}ě_{i}.

Let M(Γ_{0}(N_{0})) be the space of automorphic forms of Drinfeld type for Γ_{0}(N_{0}). Define Φ : Pic(X) × Pic(X)^{∨} → M(Γ_{0}(N_{0})) by

Φ(e, e′) := q^{2} ∑_{i,j} a_{i}a′_{j}Θ_{ij}

for any e ∈ Pic(X) with e = ∑_{i} a_{i}e_{i} and e′ ∈ Pic(X)^{∨} with e′ = ∑_{i} a′_{i}ě_{i}. Then for r ∈ ℤ and u ∈ k_{∞} we have the following Fourier expansion

Φ(e, e′) (\begin{matrix}π_{∞}^{r} & u \\ 0 & 1 \end{matrix}) = q^{−r+2}( deg e · deg e′ + ∑_{m monic, deg m≤r−2} < e, t_{m}e′ > ∑_{ϵ∈𝔽_{q}^{×}} ψ_{∞}(ϵmu)).

Since

< t_{m}e, e′ >=< e, t_{m}e′ >

for any monic polynomial m ∈ A, by Proposition 2.4 one has

T_{m}(Φ(e, e′)) = Φ(t_{m}e, e′) = Φ(e, t_{m}e′).

In fact, the image of Φ is in M^{new}(Γ_{0}(N_{0})) := S^{new}(Γ_{0}(N_{0})) ⊕ ℂℰ_{N_{0}}. To see this, we need the following claim.

Claim: for any monic m in A, consider t_{m} as in End(Pic(X)) and restrict T_{m} to the subspace M^{new}(Γ_{0}(N_{0})). We have

Tr t_{m} = Tr T_{m}.

This claim tells us that the ℂ-algebra 𝕋_{ℂ} generated by all t_{m} is isomorphic to the ℂ-algebra generated by all Hecke operators T_{m}. Moreover, Pic(X) ⊗_{ℤ} ℂ and M^{new}(Γ_{0}(N_{0})) are isomorphic as 𝕋_{ℂ}-modules. According to multiplicity one theorem, which will be recalled in the Appendix §A.2, M^{new}(Γ_{0}(N_{0})) is a free rank one 𝕋_{ℂ}-module. More precisely, M^{new}(Γ_{0}(N_{0})) is generated by the element f whose Fourier coefficients are

f^{∗}(r, λ) = q^{−r+2} · Tr(T_{m})

for all 0 ≠ λ ∈ A, (λ) = (m), deg λ + 2 ≤ r. Therefore Pic(X) ⊗_{ℤ} ℂ is also a free rank one 𝕋_{ℂ}-module. This shows

dim_{ℂ} M^{new}(Γ_{0}(N_{0})) = dim_{ℂ} [( Pic(X) ⊗_{ℤ} ℂ) ⊗_{𝕋_{ℂ}} ( Pic(X)^{∨} ⊗_{ℤ} ℂ)].

Moreover, since

∑_{i=1}^{n} < e_{i}, t_{m}ě_{i} > = Tr(B(m)) = Tr(t_{m}),

we get ∑_{i=1}^{n} Φ(e_{i}, ě_{i}) = f, which generates M^{new}(Γ_{0}(N_{0})). This also tells us that ∑_{i=1}^{n} e_{i} ⊗ ě_{i} is a generator of the cyclic 𝕋_{ℂ}-module ( Pic(X) ⊗_{ℤ} ℂ) ⊗_{𝕋_{ℂ}} ( Pic(X)^{∨} ⊗_{ℤ} ℂ).

The above argument gives us the main result:

Theorem 2.6. There is a map Φ : Pic(X) × Pic(X)^{∨} ⟶ M^{new}(Γ_{0}(N_{0})) satisfying that for r ∈ ℤ and u ∈ k_{∞}

Φ(e, e′) (\begin{matrix}π_{∞}^{r} & u \\ 0 & 1 \end{matrix}) = q^{−r+2}( deg e · deg e′ + ∑_{m monic, deg m≤r−2} < e, t_{m}e′ > ∑_{(λ)=(m)} ψ_{∞}(λu)),

and for all monic polynomials m in A

T_{m}Φ(e, e′) = Φ(t_{m}e, e′) = Φ(e, t_{m}e′).

Moreover, this map induces an isomorphism

(Pic(X) ⊗_{ℤ} ℂ) ⊗_{𝕋_{ℂ}} (Pic(X)^{∨} ⊗_{ℤ} ℂ) ≅ M^{new}(Γ_{0}(N_{0}))

as 𝕋_{ℂ}-modules.

Remark. 1. When N_{0} is a prime, M^{new}(Γ_{0}(N_{0})) = M(Γ_{0}(N_{0})) and so the theta series Θ_{ij} gives us a construction of all automorphic forms of Drinfeld type for Γ_{0}(N_{0}). This case was proven by Papikian [10] via a geometric approach.

2. Since the theta series Θ_{ij} are ℚ-valued, the map Φ in Theorem 2.6 in fact induces an isomorphism

(Pic(X) ⊗_{ℤ} ℚ) ⊗_{𝕋_{ℚ}} (Pic(X)^{∨} ⊗_{ℤ} ℚ) ≅ M^{new}(Γ_{0}(N_{0}), ℚ)

where 𝕋_{ℚ} is the ℚ-algebra generated by t_{m} for all monic m in A and M^{new}(Γ_{0}(N_{0}), ℚ) is the space of ℚ-valued functions in M^{new}(Γ_{0}(N_{0})).

3. The Claim above is essentially Jacquet-Langlands correspondence over the function field k, which will be recalled in the Appendix §A.1.

2.4 Example: The function g_{𝒜}

Having Theorem 2.6, we exhibit automorphic forms of Drinfeld type with nice arithmetic properties. Let D ∈ A − k_{∞}^{2} be a square-free element with the quadratic Legendre symbol (\frac{D}{P} ) ≠ 1 for all P | N_{0}. Let K be the imaginary quadratic field k(\sqrt{D}) and O_{K} be the integral closure of A in K. Recall that in §1.1 one has a free action of Pic(O_{K}) on the set G_{D} of Gross points of discriminant D in the definite Shimura curve X = X_{N_{0}}:

\begin{matrix}G_{D} × Pic(O_{K}) & ⟶ & G_{D} \\ (x, 𝒜) & ⟼ & x_{𝒜} . \end{matrix}

Suppose a Gross point x of discriminant D in X is given. For each ideal class 𝒜 in Pic(O_{K}), denote e_{𝒜} to be the divisor class (x_{𝒜} ) in Pic(X). Define

g_{𝒜} := ∑_{ℬ∈Pic(O_{K})} Φ(e_{ℬ} , e_{𝒜ℬ} ).

We have a nice formula for the Fourier coefficients of g_{𝒜} in terms of Hecke actions: for monic m ∈ A with deg m + 2 ≤ r,

g_{𝒜}^{∗} (r, m) = q^{−r+2} · ∑_{ℬ∈Pic(O_{K})} < e_{ℬ} , t_{m}e_{𝒜ℬ} >,

g_{𝒜}^{∗} (r, 0) = q^{−r+2} · h_{O_{K}} .

Here h_{O_{K}} = # Pic(O_{K}). Note that g_{𝒜} is independent of the choice of the Gross point x.

From now on we assume D is irreducible with (\frac{D}{P} ) = −1 for all primes P | N_{0}. According to Dirichlet’s theorem there exists a monic irreducible polynomial Q prime to N_{0} and ϵ_{0} ∈ 𝔽_{q}^{×} − 𝔽_{q}^{2} such that deg N_{0}QD is odd and ϵ_{0}N_{0}Q ≡ 1 mod D. Then there exists j ∈ 𝒟 with j^{2} = ϵ_{0}N_{0}Q so that 𝒟 = K + Kj and j^{−1}αj = ᾱ for α ∈ K. Let 𝔡 = (\sqrt{D}) be the different of O_{K}, which is a prime ideal in O_{K}. Since ϵ_{0}N_{0}Q ≡ 1 mod D, one has (\frac{ϵN_{0}Q}{D}) = 1. From the reciprocity law we get

(\frac{D}{Q} ) = 1 and so the prime ideal (Q) is split in K. Suppose (Q) = 𝔮𝔮̄ and set

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

Here O_{𝔡} is the localization of O_{K} at 𝔡. It is clear that R is an A-lattice in 𝒟 containing 1. In fact, R is a maximal A-order and K ∩ R = O_{K}. To show R is an A-order, let α_{1} + β_{1}j and α_{2} + β_{2}j be two elements in R. Then

(α_{1} + β_{1}j)(α_{2} + β_{2}j) = (α_{1}α_{2} + β_{1}β̄_{2}ϵ_{0}N_{0}Q) + (α_{1}β_{2} + β_{1}ᾱ_{2})j.

For i = 1, 2, write β_{i} as α_{i} + δ_{i} with δ_{i} ∈ O_{𝔡}. Then

α_{1}α_{2} + β_{1}β̄_{2}ϵ_{0}N_{0}Q = α_{1}(α_{2} + ᾱ_{2}) + (δ_{1}β̄_{2} + β_{1}δ̄_{2} + δ_{1}δ̄_{2})ϵ_{0}N_{0}Q.

Since α_{2} ∈ 𝔡^{−1} = A + \sqrt{D^{−1}}A, one has

α_{2} + ᾱ_{2} ∈ A.

Hence

α_{1}α_{2} + β_{1}β̄_{2}ϵ_{0}N_{0}Q ∈ 𝔡^{−2} ∩ \sqrt{D}^{−1}O_{𝔡} = 𝔡^{−1}.

Similarly,

α_{1}β_{2} + β_{1}ᾱ_{2} ∈ 𝔡^{−2}𝔮^{−1} ∩ \sqrt{D}^{−1}O_{𝔡} = 𝔡^{−1}𝔮^{−1}.

From the condition that ϵ_{0}N_{0}Q ≡ 1 mod D, one can check that

α_{1}α_{2} + β_{1}β̄_{2}ϵ_{0}N_{0}Q − (α_{1}β_{2} + β_{1}ᾱ_{2}) ∈ O_{𝔡}.

Therefore R is an A-order. The discriminant of R is (N_{0})^{2}, which can be checked locally. This implies that R is maximal.

Let x be the Gross point in the definite Shimura curve X = X_{N_{0}} 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 coefficients of g_{𝒜} .

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 e_{𝒜} and e_{𝒜ℬ} respectively. Therefore

< e_{ℬ} , t_{m}e_{𝒜ℬ} >= \frac{1}{q − 1}#\{b ∈ 𝔟^{−1}R𝔟𝔞 := (Nr(b))/ Nr(𝔞) = (m)\}.

Assume N_{0}𝔡 and 𝔞 are relatively prime. Then

𝔟^{−1}R𝔟𝔞 = {α + βj : α ∈ 𝔡^{−1}𝔞, β ∈ 𝔡^{−1}𝔟^{−1}𝔟̄𝔮^{−1}𝔞̄, α − (−1)^{ord_{𝔡}(𝔟)}β ∈ O_{𝔡}}.

We can express the Fourier coefficients of g_{𝒜} in terms of sums of the counting numbers

r_{𝒜} ((λ)) := #{𝔞 ∈ 𝒜 : 𝔞 integral with Nr(𝔞) = (λ)},

for ideals (λ) of A, by the following proposition:

Proposition 2.7. Suppose D ∈ A − k_{∞}^{2} is irreducible with (\frac{D}{P} ) = −1 for all primes P | N_{0}. Then for any monic polynomial m in A,

∑_{ℬ∈Pic(O_{K})} < e_{ℬ} , t_{m}e_{𝒜ℬ} >= \frac{1}{2(q − 1)}[2r_{𝒜} ((mD))(q − 1)h_{O_{K}}

+ ∑_{{μ∈A,μ≠0 \atop deg(μN_{0})≤deg(mD)}} r_{𝒜} ((μN_{0} − mD))(t(μ, D) + 1)(1 − δ_{μN_{0}(μN_{0}−mD)}) ∑_{c|μ} (\frac{D}{c} ) ].

Here t(μ, D) = 1 if D divides μ and 0 otherwise, and δ_{z} is the norm residue symbol of z for z ∈ k_{∞}^{×} : δ_{z} = 1 if z ∈ Nr(K_{∞}^{×}) and −1 otherwise.

Proof. Let 𝔞 ∈ 𝒜 which is a proper ideal of O_{K} and prime to N_{0}𝔡. Fix a generator λ_{0} of Nr(𝔞) = 𝔞𝔞̄. Given ℬ ∈ Pic(O_{K}). Let 𝔟 ∈ ℬ. For b = α + βj ∈ 𝔟^{−1}R𝔟𝔞, i.e. α ∈ 𝔡^{−1}𝔞, β ∈ 𝔡^{−1}𝔮^{−1}𝔟^{−1}𝔟̄𝔞̄, α − (−1)^{ord_{𝔡} 𝔟}β ∈ O_{𝔡}, define:

(1) 𝔠 := (β)𝔡𝔮𝔟̄^{−1}𝔟𝔞̄ ∈ [𝔮]ℬ^{2}𝒜,

(2) ν := − Nr(α)Dλ_{0}^{−1} ∈ A,

(3) μ := −ϵ_{0} Nr(β)DQλ_{0}^{−1} ∈ A.

Here [𝔮] ∈ Pic(O_{K}) is the ideal class containing 𝔮. Then 𝔠 is integral and

Nr(α + βj) = Nr(α) − ϵ_{0}N_{0}Q Nr(β) = (−ν + N_{0}μ)D^{−1}λ_{0}.

Thus (Nr(α + βj)) = (mλ_{0}) if and only if ν = N_{0}μ − ϵmD for a uniquely determined ϵ ∈ 𝔽_{q}^{×}.

Since β = 0 if and only if b = α ∈ 𝔞, one has

#{b ∈ 𝔟^{−1}R𝔟𝔞 : Nr(b) = (mλ_{0})}
= #{b = α + βj ∈ 𝔟^{−1}R𝔟𝔞 : β ≠ 0, Nr(b) = (mλ_{0})}
+#{α ∈ 𝔞 : Nr(α) = (mλ_{0})}.

It can be shown that #{α ∈ 𝔞 : Nr(α) = (mλ_{0})} = (q − 1)r_{𝒜} ((mD)). Note that β ≠ 0 if and only if μ ≠ 0. In this case, β is uniquely determined by the integral ideal 𝔠 up to multiplying elements in O_{K}^{×}.

Conversely, given 0 ≠ μ ∈ A and ϵ ∈ 𝔽_{q}^{×} and set ν = N_{0}μ − ϵmD. The number of elements α ∈ 𝔡^{−1}𝔞 with Nr(α) = −νD^{−1}λ_{0} is r_{𝔞,λ_{0}}(N_{0}μ − ϵmD). Here

r_{𝔞,λ_{0}}(λ) := #{a ∈ 𝔞 : Nr(a) = λλ_{0}} for λ ∈ A.

In the case of r_{𝔞,λ_{0}}(N_{0}μ−ϵmD) ≠ 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 differing 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}𝔟̄𝔞̄𝔮^{−1}𝔡^{−1} = (β)

for some β ∈ K^{×}. Suppose we can find β so that μ = −ϵ_{0} Nr(β)DQλ_{0}^{−1} ∈ A. Since ϵ_{0}N_{0}Q ≡ 1 mod D, the equality ϵmλ_{0} = Nr(α) − ϵ_{0}N_{0}Q Nr(β) ∈ A implies

α ± β ∈ O_{𝔡}.

Choose ℓ ∈ {0, 1} and replace 𝔟 by 𝔟𝔡^{ℓ} so that α−(−1)^{ord_{𝔡}(𝔟)}β ∈ O_{𝔡}. Therefore b = α + βj ∈ 𝔟^{−1}R𝔟𝔞 with Nr(b) = ϵmλ_{0}. Note that if β is not in O_{𝔡} (i.e. D ∤ μ), then ℓ is uniquely determined. If β ∈ O_{𝔡} (i.e. D | μ), then we have two choices ±β. The existence of β is equivalent to that −ϵ_{0}^{−1}DμQ^{−1}λ_{0} is in Nr(K^{×}). Since Nr(\sqrt{D}) = −D and (ϵ_{0}^{−1}μQ^{−1}λ_{0}) = Nr(𝔠𝔮^{−1}𝔞̄), we have ϵ_{0}^{−1}μQ^{−1}λ_{0} ∈ Nr(K^{×}) if and only if δ_{ϵ_{0}^{−1}μQ^{−1}λ_{0}} = 1. Therefore combining the above arguments we have

∑_{ℬ∈Pic(O_{K})} #{b = α + βj ∈ 𝔟^{−1}R𝔟𝔞 : β ≠ 0, Nr(b) = (mλ_{0})}

= ∑_{0≠μ∈A} ∑_{ϵ∈𝔽_{q}^{×}} r_{𝔞,λ_{0}}(N_{0}μ − ϵmD) · (t(μ, D) + 1) · ℛ_{\{𝒜[𝔮]\}} ((μ)) · \frac{1 + δ_{ϵ_{0}^{−1}μQ^{−1}λ_{0}}}{2} .

Here ℛ_{{𝒜[𝔮]}} ((μ)) is the number of integral ideals 𝔠, which lie in a class differing from the class 𝒜[𝔮] by a square in the class group Pic(O_{K}) and with ideal norm (μ). Following the proof of Lemma 3.4.9 in [12] one has

Lemma 2.8. For 0 ≠ μ ∈ A,

ℛ_{\{𝒜[𝔮]\}} ((μ)) · \frac{1 + δ_{ϵ_{0}^{−1}μQ^{−1}λ_{0}}}{2} = \frac{1}{q − 1} ∑_{c|μ} (\frac{D}{c} ) · \frac{1 + δ_{ϵ_{0}^{−1}μQ^{−1}λ_{0}}}{2} .

Documenta Mathematica 16 (2011) 723–765

Since δ_{ϵ_{0}^{−1}μQ^{−1}λ_{0}} = 1 if and only if δ_{N_{0}μλ_{0}} = −1, with Lemma 2.6 we have

∑_{ℬ∈Pic(O_{K})} #{b = α + βj ∈ 𝔟^{−1}R𝔟𝔞 : β ≠ 0, Nr(b) = (mλ_{0})}

= ∑_{0≠μ∈A} ∑_{ϵ∈𝔽_{q}^{×}} r_{𝔞,λ_{0}}(N_{0}μ − ϵmD)(t(μ, D) + 1) · \frac{1 − δ_{N_{0}μλ_{0}}}{2} · \frac{1}{q − 1} ∑_{c|μ} (\frac{D}{c} )

= ∑_{{μ∈A,μ≠0 \atop deg(μN_{0})≤deg(mD)}} r_{𝒜} ((μN_{0} − mD))(t(μ, D) + 1) · \frac{1 − δ_{μN_{0}(μN_{0}−mD)}}{2} · ∑_{c|μ} (\frac{D}{c} ) .

Therefore

∑_{ℬ∈Pic(O_{K})} < e_{ℬ} , t_{m}e_{𝒜ℬ} >= \frac{1}{2(q − 1)}[2r_{𝒜} ((mD))(q − 1)h_{O_{K}}

+ ∑_{{μ∈A,μ≠0 \atop deg(μN_{0})≤deg(mD)}} r_{𝒜} ((μN_{0} − mD))(t(μ, D) + 1)(1 − δ_{μN_{0}(μN_{0}−mD)}) ∑_{c|μ} (\frac{D}{c} ) ].

3 Special values of L-series

3.1 Rankin product

To an automorphic cusp form f of Drinfeld type for Γ_{0}(N) one can attach an L-series L(f, s): let 𝔪 be an effective 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} := ∑_{finite prime P} ord_{P} (λ)P.

Denote

f^{∗}(𝔪) := ∫_{A\k_{∞}} f (\begin{matrix}π_{∞}^{r+2} & u \\ 0 & 1 \end{matrix}) ψ_{∞}(−λu)du = f^{∗}(r + 2, λ).

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

L(f, s) := ∑_{𝔪≥0} f^{∗}(𝔪)q^{− deg(𝔪)s}, Re s > 1.

Let D ∈ A − k_{∞}^{2} be a square-free element. Consider the imaginary field K = k(\sqrt{D}). Let O_{K} be the integral closure of A in K and Pic(O_{K}) be the ideal class group of O_{K}. Given an ideal class 𝒜 ∈ Pic(O_{K}) and a polynomial λ in A. The number of integral ideals 𝔞 in the class 𝒜 with N_{K/k}(𝔞) = (λ) leads to the partial zeta function attached to 𝒜:

ζ_{𝒜} (s) := ∑_{𝔪≥0} r_{𝒜} (𝔪)q^{− deg(𝔪)s}, Re s > 1.

Here for each effective divisor 𝔪 = div(λ)_{0} + (r − deg λ)∞,

r_{𝒜} (𝔪) := #{𝔞 ∈ 𝒜 : 𝔞 integral with N_{K/k}(𝔞) = (λ)}.

Let f be an automorphic cusp form of Drinfeld type for Γ_{0}(N). For each ideal class 𝒜 ∈ Pic(O_{K}), we are interested in the Rankin product:

L(f, 𝒜, s) := ∑_{𝔪≥0} f^{∗}(𝔪)r_{𝒜} (𝔪)q^{− deg(𝔪)s}, Re(s) > 1.

To study the analytic continuation and the functional equation of L(f, 𝒜, s), consider the function Λ(f, 𝒜, s) which is defined by:

Λ(f, 𝒜, s) := \begin{cases}L^{(N,D)}(2s + 1)L(f, 𝒜, s) & when deg D is odd, \\ \frac{1}{1 + q^{−s−1}} L^{(N,D)}(2s + 1)L(f, 𝒜, s) & when deg D is even. \end{cases}

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

L^{(N,D)}(s) := \frac{1}{q − 1} ∑_{d∈A,(d,N)=1} (\frac{D}{d} ) q^{−s deg d}, Re(s) > 1,

where (\frac{D}{d} ) denotes the Legendre symbol for the polynomial ring A. Note that

L^{(N,D)}(s) = L_{D}(s) · ∏_{prime ideals P |N}(1 − (\frac{D}{P} ) q^{−s deg P})^{−1}

where L_{D}(s) is the Dirichlet L-series:

L_{D}(s) := \frac{1}{q − 1} ∑_{d∈A,d≠0} (\frac{D}{d} ) q^{−s deg d}, Re(s) > 1.

It is known that L_{D}(s) can be extended to a polynomial in q^{−s} with the functional equation (cf. [1]):

L_{D}(2s + 1) = q^{s(−2 deg D+2)− \frac{1}{2} deg D+ \frac{1}{2}} L_{D}(−2s)

if deg D is odd, and

L_{D}(−2s + 1) = \frac{1 + q^{1−2s}}{1 + q^{2s}} q^{deg D(2s− \frac{1}{2} )}L_{D}(2s)

if deg D is even. When f is a new form and D is irreducible, Rück and Tipp ([12]) prove the following functional equation of Λ(f, 𝒜, s):

Λ(f, 𝒜, s) = − (\frac{D}{N} ) q^{(5−2 deg D−2 deg N)s}Λ(f, 𝒜, −s)

when deg D is odd, and

Λ(f, 𝒜, s) = − (\frac{D}{N} ) q^{(6−2 deg D−2 deg N)s}Λ(f, 𝒜, −s)

when deg D is even.

3.2 Central critical values of Λ(f, 𝒜, s)

We are interested in the special value of Λ(f, 𝒜, s) at s = 0. Note that if (\frac{D}{N} ) = 1, then Λ(f, 𝒜, s) has a zero at s = 0. We focus here on the special case when (\frac{D}{P} ) = −1 for all primes P | N_{0}. Adapting Rankin’s method (cf. [12]), we can establish the following theorem.

Theorem 3.1. Let f be a Drinfeld type new form for Γ_{0}(N_{0}) and let D be an irreducible polynomial in A − k_{∞}^{2} with (\frac{D}{P} ) = −1 for all primes P | N_{0}. One has

Λ(f, 𝒜, 0) = \begin{cases}\frac{(f, g_{𝒜} )}{q^{\frac{1}{2} (deg D+1)}} & when deg D is odd, \\ \frac{(f, g_{𝒜} )}{2q^{\frac{1}{2} deg D}} & when deg D is even. \end{cases}

Here (·, ·) is the Petersson inner product and g_{𝒜} is the Drinfeld type automorphic form for Γ_{0}(N_{0}) canonically attached to 𝒜 in §2.4.

3.2.1 Review of Rankin’s method

Given 𝒜 ∈ Pic(O_{K}). Choose 𝔞_{0} ∈ 𝒜^{−1} and λ_{0} ∈ k such that N_{K/k}(𝔞_{0}) = (λ_{0}) Recall the counting number

r_{𝔞_{0},λ_{0}}(λ) = #{μ ∈ 𝔞_{0} : N_{K/k}(μ) = λ_{0}λ}.

Note that r_{𝔞_{0},λ_{0}}(λ) = r_{𝔞_{0}^{−1},λ_{0}^{−1}}(λ), and for effective divisor 𝔪 = div(λ)_{0} + (deg 𝔪 − deg λ)∞ we have

r_{𝒜} (𝔪) = \frac{1}{q − 1} ∑_{ϵ∈𝔽_{q}^{×}} r_{𝔞_{0},λ_{0}}(ϵλ).

We consider the following theta series θ_{𝔞_{0},λ_{0}} (introduced in [11]) defined on k_{∞}^{×} × k_{∞}:

θ_{𝔞_{0},λ_{0}}(π_{∞}^{r}, u) := ∑_{deg λ+2≤r} r_{𝔞_{0},λ_{0}}(λ)ψ_{∞}(λu).

It satisfies the following transformation law:

θ_{𝔞_{0},λ_{0}} (\frac{π_{∞}^{r}}{(cu + d)^{2}} , \frac{au + b}{cu + d}) = δ_{cu+d} (\frac{d}{D} ) q^{−v_{∞}(cu+d)}θ_{𝔞_{0},λ_{0}}(π_{∞}^{r}, u)

for all γ = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ_{0}^{(1)}(N) := Γ_{0}(N)∩SL_{2}(A) with v_{∞}(cπ_{∞}^{r}) > v_{∞}(cu+d).

Here δ is the local norm symbol at ∞, i.e. δ_{z} = 1 if z ∈ k_{∞}^{×} is a norm of an element in K_{∞} = k_{∞}(\sqrt{D}) and −1 otherwise.

Viewing θ_{𝔞_{0},λ_{0}} as a function on

ℍ_{∞} := (\begin{matrix}1 & A \\ 0 & 1 \end{matrix}) \ (\begin{matrix}k_{∞}^{×} & k_{∞} \\ 0 & 1 \end{matrix}) / (\begin{matrix}𝒪_{∞}^{×} & 𝒪_{∞} \\ 0 & 1 \end{matrix}) ,

one can write

L(f, 𝒜, s) = \frac{q}{q − 1} ∑_{r=2}^{∞} [ ∑_{u∈π_{∞}O_{∞}/π_{∞}^{r}O_{∞}} f · \overline{θ_{𝔞_{0},λ_{0}}} (\begin{matrix}π_{∞}^{r} & u \\ 0 & 1 \end{matrix}) q^{−r(s+1)+2s}]

= \frac{q}{q − 1} ∫_{ℍ_{∞}} f(h)\overline{θ_{𝔞_{0},λ_{0}}(h)q^{−r(s̄+1)+2s̄}}dh.

For every monic polynomial M in A, the canonical map

ℍ_{∞} ⟶ G(M) := Γ_{0}^{(1)}(M)\ GL_{2}(k_{∞})/Γ_{∞}k_{∞}^{×}

is surjective. Following [12], we consider the “Eisenstein series”

E_{s} (\begin{matrix}π_{∞}^{r} & u \\ 0 & 1 \end{matrix}) := ∑_{{c,d∈A,c≡0 mod D \atop v∞(cπ^{r}_{∞})>v∞(cu+d)}} (\frac{d}{D} )δ_{cu+d}q^{v_{∞}(cu+d)(2s+1)}

and let H_{s} (\begin{matrix}π_{∞}^{r} & u \\ 0 & 1 \end{matrix}) :=

q^{−r(s+1)+2s}E_{s} (\begin{matrix}Nπ_{∞}^{r} & Nu \\ 0 & 1 \end{matrix})  when deg D is odd,

(\frac{(−1)^{r−deg λ_{0}} +1}{2}) · q^{−r(s+1)+2s}E_{s} (\begin{matrix}Nπ_{∞}^{r} & Nu \\ 0 & 1 \end{matrix}) when deg D is even.

Then θ_{𝔞_{0},λ_{0}}Hs̄ can be viewed as a function on G(ND). By [12] Proposition 2.2.2 and Proposition 2.3.2

Λ(f, 𝒜, s) = \frac{q}{2(q − 1)} ∫_{G(ND)} f · \overline{θ_{𝔞_{0},λ_{0}}Hs̄}.

750 Fu-Tsun Wei and Jing Yu Given M ∈ A. Let ℱ(M) be the space of functions on G(M). The trace map from ℱ(ND) to ℱ(N) is given by

f ⟶ Tr_{N}^{ND} f(g) := ∑_{γ∈Γ_{0}^{(1)}(ND)\Γ_{0}^{(1)}(N)} f(γg).

Set Φ_{s} := Tr_{N}^{ND}(θ_{𝔞_{0},λ_{0}}Hs̄). Then

Λ(f, 𝒜, s) = \frac{q}{2(q − 1)} ∫_{G(N)} f · \overline{Φs̄}.

From the harmonicity of f one has

Λ(f, 𝒜, s) = \frac{q}{4(q − 1)} ∫_{G(N)} f · \overline{Fs̄}

where for g ∈ GL_{2}(k_{∞}),

F_{s}(g) := \frac{q}{q + 1}(Φ_{s}(g) − Φ̃_{s}(g)) − \frac{1}{q + 1} ∑_{{β∈GL_{2}(O∞)/Γ∞, \atop β≠1}} (Φ_{s}(gβ) − Φ̃_{s}(gβ)).

Note that F_{s} depends on the choice of 𝔞_{0} and λ_{0}. 3.2.2 Proof of Theorem 3.1 Let Ψ be the average map from functions F on G(N) to functions on G_{0}(N):

Ψ(F)(g) := \frac{1}{q − 1} ∑_{ϵ∈𝔽_{q}^{×}} F ((\begin{matrix}ϵ & 0 \\ 0 & 1 \end{matrix}) g).

Define

Ψ_{𝒜} := Ψ(F_{0}).

Note that Ψ_{𝒜} now depends only on 𝒜. Taking the formulas in Proposition 2.7.2 and Proposition 2.7.5 in [12] and specializing at s = 0 we deduce that for any λ ∈ A with deg λ + 2 ≤ r

Ψ_{𝒜}^{∗} (r, λ) = \frac{3 − (−1)^{deg D}}{4} · q^{−r+1−⌈ \frac{deg D}{2} ⌉} · [2r_{𝒜} ((λD))(q − 1)L_{D}(0)

+ ∑_{{μ∈A,μ≠0 \atop deg(μN)≤deg(λD)}} r_{𝒜} ((μN − λD))(t(μ, D) + 1)(1 − δ_{μN(μN−λD)}) ∑_{c|μ} (\frac{D}{c} ) ].

Moreover, one has Proposition 3.2.

Documenta Mathematica 16 (2011) 723–765

Λ(f, 𝒜, 0) = \frac{q}{2(q − 1)} ∫_{G_{0}(N)} f · \overline{Ψ_{𝒜}} .

Let N = N_{0}. Note that L_{D}(0) = h_{O_{K}}. Comparing the Fourier coefficients of Ψ_{𝒜} with that of g_{𝒜} we obtain

Ψ_{𝒜} = g_{𝒜} · \begin{cases}q^{− \frac{1}{2} deg D+ \frac{1}{2}} · q^{−2} · (q − 1) · 2 & when deg D is odd, \\ q^{−1} · q^{− \frac{1}{2} deg D} · (q − 1) & when deg D is even. \end{cases}

Therefor Theorem 3.1 holds. □

3.3 A function field analogue of Gross formula

Now given a character χ : Pic(O_{K}) → ℂ^{×}, define

Λ(f, χ, s) := ∑_{𝒜∈Pic(O_{K})} χ(𝒜)Λ(f, 𝒜, s).

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

Λ(f, χ, s) = L(f, s)L(f ⊗ ε_{D}, s)

where ε_{D} is the following quadratic character on divisors of k:

ε_{D}(P) = (\frac{D}{P} ) and ε_{D}(∞) = \begin{cases}−1 & if deg D is even, \\ 0 & if deg D is odd; \end{cases}

and L(f ⊗ ε_{D}, s) is the twisted L-series of f by ε_{D}:

L(f ⊗ ε_{D}, s) := ∑_{𝔪≥0} f^{∗}(𝔪)ε_{D}(𝔪)q^{− deg 𝔪s}.

From the definition of Λ(f, χ, s) and Theorem 3.1 one has

Λ(f, χ, 0) = ( ∑_{𝒜∈Pic(O_{K})} χ(𝒜)(f, g_{𝒜} )) · \begin{cases}\frac{1}{q^{\frac{1}{2} (deg D+1)}} & if deg D is odd, \\ \frac{1}{2q^{\frac{1}{2} deg D}} & if deg D is even. \end{cases}

Note that

∑_{𝒜∈Pic(O_{K})} χ(𝒜)^{−1}g_{𝒜} = ∑_{𝒜∈Pic(O_{K})} ( ∑_{ℬ∈Pic(O_{K})} χ(𝒜)^{−1}Φ(e_{ℬ} , e_{𝒜ℬ} ))
= Φ(e_{χ}, e_{χ})

where Φ is the map in Theorem 2.6 and

e_{χ} = ∑_{𝒜∈Pic(O_{K})} χ(𝒜)e_{𝒜} .

Suppose f is a normalized newform. Then from Theorem 2.6 f corresponds to a particular element e_{f} ∈ Pic(X) ⊗_{ℤ} ℝ such that

f = Φ(e_{f}, e_{f}).

Let e_{f,χ} be the projection of e_{χ} to the e_{f}-isotypical component in the space Pic(X) ⊗_{ℤ} ℂ with respect to the Gross height pairing. Then the feigencomponent of Φ(e_{χ}, e_{χ}) is equal to

Φ(e_{f,χ}, e_{χ}) = Φ(e_{f,χ}, e_{f,χ}) =< e_{f,χ}, e_{f,χ} > f.

The last equality holds as f is normalized (i.e. f^{∗}(0) = 1) and the Fourier coefficient Φ(e_{f,χ}, e_{f,χ})^{∗}(0) =< e_{f,χ}, e_{f,χ} >. Therefore we obtain

Theorem 3.3. Let f be an automorphic cusp form of Drinfeld type for Γ_{0}(N_{0}) which is also a normalized newform. Then

Λ(f, χ, 0) = \begin{cases}\frac{(f, f)}{q^{\frac{1}{2} (deg D+1)}} · < e_{f,χ}, e_{f,χ} > & if deg D is odd, \\ \frac{(f, f)}{2q^{\frac{1}{2} deg D}} · < e_{f,χ}, e_{f,χ} > & if deg D is even. \end{cases}

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

2. When χ is trivial, then

∑_{monic m|N_{0}} t_{m}e_{χ} = 2e_{D}

where e_{D} is the divisor class introduced in Proposition 1.7.

3. The special case when N_{0} is a prime and deg D is odd, the above formula coincides with the result in [10] §4 (be aware of the different choices of measures for the Petersson inner product).

4. When irreducible D ∈ A−k_{∞}^{2} satisfies (\frac{D}{N_{0}} ) = 1, the derivative of Λ(f, χ, s) at s = 0 is given by Néron-Tate height of Heegner points on the Drinfeld modular curve X_{0}(N_{0}), and an analogue of Gross-Zagier formula has been proved by Rück and Tipp in the case D is irreducible (cf. [12]).

3.4 Example and application to elliptic curves

Let E be a non-iso-trivial elliptic curve over k (i.e. E is not defined over the constant field 𝔽_{q}). From the work of Weil, Jacquet-Langlands, and Deligne, one knows that there exists an automorphic cusp form f_{E} such that

L(E/k, s + 1) = L(f_{E}, s).

Here L(E/k, s) is the Hasse-Weil L-series of E over k. Suppose the conductor of E is N_{0}∞, and E has split multiplicative reduction at ∞. Then the automorphic form f_{E} is of Drinfeld type for Γ_{0}(N_{0}), which is a normalized newform (cf. [7]).

Consider the Hasse-Weil L-series L(E/K, s) of E over the imaginary quadratic field K = k(\sqrt{D}) where D ∈ A with (\frac{D}{P} ) = −1 for all primes P | N_{0}. One has

L(E/K, s + 1) = L(f_{E}, s)L(f_{E} ⊗ ε_{D}, s)

where L(f_{E}, ⊗ε_{D}, s) is the twisted L-series of f_{E} by the quadratic character ε_{D}. Since

L(f_{E}, s)L(f_{E} ⊗ ε_{D}, s) = Λ(f_{E}, 1_{D}, s)

where 1_{D} is the trivial character on Pic(O_{K}), from Theorem 3.3 we obtain a formula for the special value of L(E/K, s) at s = 1 when D is irreducible.

Now, let k = 𝔽_{3}(t) (i.e. q = 3). Let E be the following elliptic curve over k:

E : y^{2} = x^{3} + (t^{2} + 1)x^{2} + t^{2}x = x(x + 1)(x + t^{2}).

The conductor of E is (t)(t + 1)(t − 1)∞. More precisely, E has split multiplicative reduction at (t) and ∞, and has non-split multiplicative reduction at (t + 1) and (t − 1). Let N_{0} = t(t + 1)(t − 1) = t^{3} − t. Let f_{E} be the normalized Drinfeld type cusp form for Γ(N_{0}) associated to E. Since the L-series L(E/k, s) of E over k is a polynomial in q^{−s} of degree (deg N_{0} + 1) − 4 with constant term 1, this implies that L(E/k, s) = L(f_{E}, s − 1) = 1. Let D = t^{3} − t − 1 and K = k(\sqrt{D}). Then

(\frac{D}{t} ) = (\frac{D}{t + 1}) = (\frac{D}{t − 1}) = −1.

The twist E_{D} of E by D is the following elliptic curve over k:

y^{2} = x^{3} + (t^{2} + 1)Dx^{2} + t^{2}D^{2}x.

The conductor of E_{D} is (D)^{2}(t)(t + 1)(t − 1)∞^{2}, and the L-series L(E_{D}/k, s) is

1 + q^{−s} + 4q^{−2s} + 108q^{−5s} + 243q^{−6s} + 2187q^{−7s}.

Since L(E/K, s) = L(E/k, s) · L(E_{D}/k, s), we have

L(E/K, s) = 1 + q^{−s} + 4q^{−2s} + 108q^{−5s} + 243q^{−6s} + 2187q^{−7s}

and L(E/K, 1) = \frac{32}{9}.

On the other hand, from a formula of Gekeler (cf. [13] Theorem 1.1) we immediately get

(f_{E}, f_{E}) = 32.

We point out that our choice of the measure is twice of the one in [13]. Such computation can be also checked via the algorithm in [15].

The only remaining term is the Gross height of the corresponding point e_{f_{E}} in Pic(X_{N_{0}}) ⊗_{ℤ} ℚ. Let 𝒟 be the definite quaternion algebra over k ramified at (t), (t + 1), and (t − 1). Then

𝒟 = k + kα + kβ + kαβ

where α^{2} = −1, β^{2} = N_{0} = t^{3}−t, and βα = −αβ. Let R = A+Aα+Aβ+Aαβ, which is a maximal order in 𝒟. The cardinality of R^{×} is 8, and the class number (of left ideal classes of R) is 4. We choose the following representatives of left ideal classes of R:

\begin{aligned}I_{1} & = & R, \\ I_{2} & = & At + Atα + Aβ + Aαβ, \\ I_{3} & = & A(t + 1) + A(t + 1)α + Aβ + Aαβ, \\ I_{4} & = & A(t − 1) + A(t − 1)α + Aβ + Aαβ. \end{aligned}

Note that these ideals are in fact two-sided, and the norm form on each of them can be easily written down. We calculate the following Brandt matrices:

B(t) = (\begin{matrix}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{matrix}) , B(t+1) = (\begin{matrix}0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix}) , B(t−1) = (\begin{matrix}0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix}) .

Since we have T_{t}f_{E} = f_{E}, T_{t+1}f_{E} = −f_{E}, T_{t−1}f_{E} = −f_{E}, and the Gross height < e_{f_{E}}, e_{f_{E}} >= f_{E}^{∗}(0) = 1, the corresponding point e_{f_{E}} in Pic(X_{N_{0}}) ⊗_{ℤ} ℚ can only be

±[1/4, 1/4, −1/4, −1/4].

The class number of O_{K}(= A[\sqrt{D}]) is 1. Choose the Gross point x in the first component of X_{N_{0}} corresponding to the embedding K ↪ 𝒟 which maps \sqrt{D} to α + β. Then e_{x} = [1, 0, 0, 0] in Pic(X_{N_{0}}) ⊗_{ℤ} ℚ. Therefore

< e_{f_{E},1_{D}}, e_{f_{E},1_{D}} >=< e_{f_{E}}, e_{x} >^{2}= (4 · 1/4)^{2} = 1

and

\frac{(f_{E}, f_{E})}{q^{\frac{1}{2} (deg D+1)}} < e_{f_{E},1_{D}}, e_{f_{E},1_{D}} >= \frac{32}{9} = L(E/K, 1).

Appendix A Jacquet-Langlands correspondence and multiplicity one the-

orem

Let ϖ be a Hecke character on k^{×}\𝔸_{k}^{×}. Let 𝒟 be a quaternion algebra over k and set 𝒟_{𝔸_{k}} := 𝒟 ⊗_{k} 𝔸_{k}. We embed 𝔸_{k} into 𝒟_{𝔸_{k}} by a ⟼ 1 ⊗ a. A ℂ-valued function f on 𝒟^{×}\𝒟_{𝔸_{k}}^{×} is called an automorphic form on 𝒟_{𝔸_{k}}^{×} (for 𝒦) with central character ϖ if f is a function on the double coset space

𝒟^{×}\𝒟_{𝔸_{k}}^{×}/𝒦

for an open compact subgroup 𝒦 of 𝒟_{𝔸_{k}}^{×} satisfying that for all g in 𝒟_{𝔸_{k}}^{×} and a in 𝔸_{k}^{×}

f(ag) = ϖ(a)f(g).

Suppose 𝒟 = Mat_{2}(k). Then 𝒟^{×} = GL_{2}(k) and 𝒟_{𝔸_{k}}^{×} = GL_{2}(𝔸_{k}). f is called a cusp form if for all g in GL_{2}(𝔸_{k})

∫_{k\𝔸_{k}} f ((\begin{matrix}1 & u \\ 0 & 1 \end{matrix}) g) du = 0.

We denote A_{0}(ϖ) to be the space of automorphic cusp forms on GL_{2}(𝔸_{k}) with central character ϖ.

We recall Jacquet-Langlands correspondence in §A.1 and use newform theory to explain the claim in §2.3. In §A.2 we use multiplicity one theorem to show that the space M^{new}(Γ_{0}(N_{0})) in §2.3 is a free 𝕋_{ℂ}-module of rank one.

A.1 Jacquet-Langlands correspondence

Let 𝒟 = 𝒟_{(N_{0})} be a definite quaternion algebra over k where N_{0} is the product of finite ramified primes of 𝒟. Let A′(ϖ) be the space of automorphic forms on 𝒟_{𝔸_{k}}^{×} with central character ϖ. Jacquet-Langlands correspondence describes the connection between A′(ϖ) and A_{0}(ϖ):

([9] Chapter 3, Theorem 14.4 and Theorem 16.1) If an irreducible admissible representation ρ′ = ⊗_{v}ρ′_{v} is a constituent of A′(ϖ) and ρ′_{P} is infinite dimensional for all finite primes P which are prime to N_{0}, then there exist an irreducible admissible representation ρ(=: ρ^{′JL}) which is a constituent of A_{0}(ϖ) so that

L(s, ϖ′ ⊗ ρ) = L(s, ϖ′ ⊗ ρ′)

for all Hecke characters ϖ′. Note that ρ = ⊗_{v}ρ_{v} where ρ_{v} = ρ′_{v} for finite primes v not dividing N_{0}. Moreover, for the ramified primes v of 𝒟, ρ_{v} is determined from ρ′_{v} via theta correspondence. Conversely, suppose ρ = ⊗_{v}ρ_{v} is a constituent of A_{0}(ϖ). If for every ramified primes v of 𝒟 the representation ρ_{v} is special or supercuspidal, then there is a constituent ρ′ = ⊗ρ′_{v} of A′(ϖ) such that ρ_{v} = ρ_{v}^{′JL} . In particular, ρ′_{v} is one dimensional for ramified prime v if and only if ρ_{v} is special.

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

{ ℂ-valued non-constant functions on R̂^{×}\𝒟̂^{×}/𝒟^{×}}

and

{Drinfeld type new forms on Γ_{0}(N_{0})\ GL_{2}(k_{∞})/Γ_{∞}k_{∞}^{×}}

which satisfies

Ψ(t_{m}f) = T_{m}Ψ(f)

for all non-constant functions f on R̂^{×}\𝒟̂^{×}/𝒟^{×} and monic polynomials m in A. We briefly recall the argument in the following and refer the reader to [9] for further details.

Fix ϖ = ⊗_{v}ϖ_{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 infinite-dimensional representation (ρ_{v}, V_{v}) of GL_{2}(k_{v}) with central character ϖ_{v} has conductor v^{c(v)} if π_{v}^{c(v)}𝒪_{v} is the largest ideal of 𝒪_{v} such that the space of elements u ∈ V_{v} with

ρ_{v}(g_{v})u = u for all g_{v} ∈ 𝒦_{0}^{c(v)}

is non-empty. In fact, it is one dimensional. Here

𝒦_{0}^{c(v)} = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL_{2}(𝒪_{v}) : c ∈ π_{v}^{c(v)}𝒪_{v}\} .

It is known that

c(v) = \begin{cases}0 & if ρ_{v} is an unramified principal series, \\ 1 & if ρ_{v} is an unramified special representation, \\ ≥ 2 & if ρ_{v} is supercuspidal or ramified. \end{cases}

Let (ρ, V) = ⨂′_{v}(ρ_{v}, V_{v}) be a constituent of 𝒜_{0}(ϖ). The conductor of ρ is:

∏_{v} v^{c(v)}.

The space of elements f ∈ V with

ρ(g)f = f for all g ∈ ∏_{v} 𝒦_{0}^{c(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. T_{v}f = a_{v}f for all v where a_{v} ∈ ℂ. Recall that L(s, ρ) = ∏_{v} L(s, ρ_{v}), where

L(s, ρ_{v}) = (1 − χ_{v,1}(π_{v})q^{−s deg v})^{−1} · (1 − χ_{v,2}(π_{v})q^{−s deg v})^{−1}

if ρ_{v} is an unramified principal series π(χ_{v,1}, χ_{v,2});

L(s, ρ_{v}) = (1 − χ_{v}(π_{v})q^{−(s+1/2) deg v})^{−1}

if ρ_{v} is an unramified special representation sp(χ_{v}| · |_{v}^{1/2}, χ_{v}| · |_{v}^{−1/2});

L(s, ρ_{v}) = 1

if ρ_{v} is supercuspidal or ramified. Here χ_{v,1}, χ_{v,2}, and χ_{v} are unramified characters of k_{v}^{×} with χ_{v,1} · χ_{v,2} = 1 = χ_{v}^{2}. It is known that

a_{v} = \begin{cases}q^{\frac{1}{2} deg v}(χ_{v,1}(π_{v}) + χ_{v,2}(π_{v})) & if ρ_{v} ≅ π(χ_{v,1}, χ_{v,2}), \\ χ_{v}(π_{v}) & if ρ_{v} ≅ sp(χ_{v}| · |_{v}^{1/2}, χ_{v}| · |_{v}^{−1/2}). \end{cases}

Suppose ρ = ⊗_{v}ρ_{v} is of conductor N_{0}∞ and ρ_{∞} ≅ sp(| · |_{∞}^{1/2}, | · |_{∞}^{−1/2}). Then new-forms of ρ are functions on

GL_{2}(k)\ GL_{2}(𝔸_{k})/𝒦_{0}(N_{0}∞)k_{∞}^{×}.

From the bijection in §2.1

GL_{2}(k)\ GL_{2}(𝔸_{k})/𝒦_{0}(N_{0}∞)k_{∞}^{×} ≅ Γ_{0}(N_{0})\ GL_{2}(k_{∞})/Γ_{∞}k_{∞}^{×},

new-forms of such ρ can be viewed as newforms of Drinfeld type for Γ_{0}(N_{0}). In fact, the space S^{new}(Γ_{0}(N_{0})) of Drinfeld type new forms for Γ_{0}(N_{0}) is spanned by the new-forms of such ρ with conductor N_{0}∞.

Since ρ is of conductor N_{0}∞, ρ_{P} ≅ sp(χ_{P} | · |_{P}^{1/2}, χ_{P} | · |_{P}^{−1/2}) for all P | N_{0} where χ_{P} is an unramified character of k_{P}^{×} with χ_{P}^{2} = 1. By Jacquet-Langlands correspondence we can find an irreducible constituent (ρ′, V′) = ⊗_{v}ρ′_{v} of A′(ϖ) so that ρ = ρ^{′JL}. In this case, ρ′_{P} = χ_{P} ∘ Nr for P | N_{0} and ρ′_{∞} is the trivial representation. Therefore we can find a subspace of elements f′ ∈ V′ which are non-constant functions on

𝒟^{×}\𝒟̂^{×}/R̂^{×}.

This subspace is also one dimensional, called the space of new-forms of ρ′. Any new-form f′ of ρ′ is also a Hecke eigenform, i.e. t_{v}f′ = a′_{v}f′, where a′_{v} appears in the local factor L_{v}(s, ρ′_{v}). Since for any place v

L(s, ρ_{v}) = L(s, ρ′_{v}),

we have a_{v} = a′_{v}.

In fact, the space of non-constant functions on 𝒟^{×}\𝒟̂^{×}/R̂^{×} is generated by new-forms such that ρ′ = ⊗_{v}ρ′_{v} where ρ′_{∞} is trivial and for P | N_{0}, ρ′_{P} = χ_{P} ∘Nr for an unramified character χ_{P} of k_{P}^{×} with χ_{P}^{2} = 1. By taking congugate, we identify functions on 𝒟^{×}\𝒟̂^{×}/R̂^{×} with functions on R̂^{×}\𝒟̂^{×}/𝒟^{×}. From the dimension formula at the end of §2.2 we have a bijective map Ψ from

{ ℂ-valued non-constant functions on R̂^{×}\𝒟̂^{×}/𝒟^{×}}

to

{Drinfeld type new forms on Γ_{0}(N_{0})\ GL_{2}(k_{∞})/Γ_{∞}k_{∞}^{×}}

so that for each monic polynomial m in A,

Ψ(t_{m}f) = T_{m}Ψ(f).

Since constant functions on R̂^{×}\𝒟̂^{×}/𝒟^{×} are eigenfunctions of t_{m} with eigenvalue σ(m)_{N_{0}}, we extend Ψ by mapping constant functions into the one dimensional subspace ℂℰ_{N_{0}} of M^{new}(Γ_{0}(P_{0})).

Consider the definite Shimura curve X = X_{N_{0}}. We have a canonical bijection between components of X and ideal classes of R and this gives the canonical isomorphism

{ (ℂ-valued) functions on R̂^{×}\𝒟̂^{×}/𝒟^{×}} ≅ Hom(Pic(X), ℂ) ≅ Pic(X)^{∨} ⊗_{ℤ} ℂ.

Therefore one has:

Theorem A.1. Ψ : Pic(X)^{∨} ⊗_{ℤ} ℂ ≅ M^{new}(Γ_{0}(N_{0})) is an isomorphism so that Ψ(t_{m}f) = T_{m}Ψ(f) for any monic polynomial m in A. Moreover,

Tr(t_{m}) = Tr(T_{m})

and so the ℂ-algebra 𝕋_{ℂ} generated by Hecke correspondences t_{m} on X is isomorphic to the ℂ-algebra generated by Hecke operators T_{m} on M^{new}(Γ_{0}(N_{0})).

A.2 Multiplicity one theorem

Let ϖ : 𝔸_{k}^{×}/k^{×} be a Hecke character. Let ρ_{1} = ⊗_{v}ρ_{1,v} and ρ_{2} = ⊗_{v}ρ_{2,v} be two irreducible admissible representations which are constituents of A_{0}(ϖ). The multiplicity one theorem (cf. [3]) tells us that ρ_{1} = ρ_{2} if and only if

ρ_{1,v} ≅ ρ_{2,v}

for all place v. Fix ϖ to be trivial. Choose two irreducible admissible representations ρ_{1} = ⊗_{v}ρ_{1,v} and ρ_{2} = ⊗_{v}ρ_{2,v} of conductor N_{0}∞ which are constituents of A_{0}(ϖ) satisfying

ρ_{1,∞} ≅ ρ_{2,∞} ≅ sp(| · |_{∞}^{1/2}, | · |_{∞}^{−1/2})

and ρ_{1,P} and ρ_{2,P} are unramified special representations for P | N_{0}. Let f_{1} and f_{2} be new-forms of ρ_{1} and ρ_{2} respectively. Then T_{P} f_{i} = a_{P,i}f_{i} where a_{P,i} ∈ ℂ for i = 1, 2 and all prime P in A. If a_{P,1} = a_{P,2} for all P, then L_{P} (s, ρ_{1,P}) = L_{P} (s, ρ_{2,P}) and so

ρ_{1,P} ≅ ρ_{2,P}

for all P. By multiplicity one theorem we have ρ_{1} = ρ_{2} and so f_{1}, f_{2} are linearly dependent.

Recall that M^{new}(Γ_{0}(N_{0})) = S^{new}(Γ_{0}(N_{0})) ⊕ ℂℰ_{N_{0}}. for Γ_{0}(P_{0}). As a 𝕋_{ℂ}- module, the space M^{new}(Γ_{0}(N_{0})) is a direct sum (⊕_{i}ℂf_{i}) ⊕ ℂℰ_{N_{0}} of one dimensional submodules and each f_{i} is a new-form of an irreducible admissible representation ρ_{i} = ⊗_{v}ρ_{i,v} which is a constituent of A_{0}(ϖ) with

ρ_{i,∞} ≅ sp(| · |_{∞}^{1/2}, | · |_{∞}^{−1/2})

and ρ_{i,P} is an unramified special representation for P | N_{0}. According to multiplicity one theorem, each pair of these one dimensional submodules are non-isomorphic. Therefore M^{new}(Γ_{0}(N_{0})) is a cyclic 𝕋_{ℂ}-module, which is generated by ℰ_{N_{0}} + ∑_{i} f_{i}. Viewing 𝕋_{ℂ} as a subring of End_{ℂ} (M^{new}(Γ_{0}(N_{0}))), we have

dim_{ℂ} 𝕋_{ℂ} ≤ dim_{ℂ} M^{new}(Γ_{0}(N_{0})).

Therefore

Proposition A.2. The space M^{new}(Γ_{0}(N_{0})) is a free 𝕋_{ℂ}-module of rank one.

B Transformation law of theta series

Fix a definite quaternion algebra 𝒟 = 𝒟_{(N_{0})} where N_{0} is the product of finite ramified primes of 𝒟. Let R be a maximal order and n be the class number. In this section we deduce the transformation law of the theta series θ_{ij} for 1 ≤ i, j ≤ n introduced in §2.1.1. Recall that for each (i, j), theta series θ_{ij} is a function on k_{∞}^{×} × k_{∞}:

θ_{ij}(x, y) = ∑_{b∈M_{ij}} ϕ_{∞}(\frac{Nr(b)}{N_{ij}}xt^{2}) · ψ_{∞}(\frac{Nr(b)}{N_{ij}}y),

where ϕ_{∞} is the characteristic function of 𝒪_{∞} and ψ_{∞} is the fixed additive character on k_{∞}.

B.1 Fourier Transform

Let 𝒟_{∞} = 𝒟 ⊗_{k} k_{∞},. For α, β ∈ k_{∞}^{×} with v_{∞}(α) > v_{∞}(β) − 2, let

\begin{aligned}Φ_{α,β} : & 𝒟_{∞} & ⟶ & ℂ \\ & w & ⟼ & ϕ_{∞}( Nr(w)α)ψ_{∞}( Nr(w)β). \end{aligned}

Define [·, ·] : 𝒟_{∞} × 𝒟_{∞} → ℂ^{×} by [w, w^{∗}] := ψ_{∞}( Tr(ww^{∗})). The Fourier transform of Φ_{α,β} is given by:

Φ_{α,β}^{∗}(w^{∗}) := ∫_{𝒟_{∞}} Φ_{α,β}(w)[w, w^{∗}]dw, for all w^{∗} in k_{∞}

where dw is a Haar measure on 𝒟_{∞}.

We define

S(α, β, dw) := ∫_{𝒟_{∞}} ϕ_{∞}( Nr(w)α)ψ_{∞}(Nr(w)β)dw.

Then Φ_{α,β}^{∗}(w^{∗}) is equal to

S(α, β, dw)ϕ_{∞}( Nr(w^{∗})\frac{α}{β^{2}} )ψ_{∞}( Nr(w^{∗})\frac{−1}{β}).

More generally, take h ∈ k_{∞}^{×}, ρ ∈ 𝒟_{∞}. For α, β ∈ k_{∞}^{×} with v_{∞}(α) > v_{∞}(β) − 2, let Ψ_{α,β}(w) := Φ_{α,β}(ρ + hw). Then Ψ_{α,β}^{∗}(w^{∗}) is equal to

q^{4v_{∞}(h)} · S(α, β, dw)ϕ_{∞}( Nr(\frac{w^{∗}}{h})\frac{α}{β^{2}} )ψ_{∞}( Nr(\frac{w^{∗}}{h})\frac{−1}{β})ψ_{∞}( Tr(−\frac{ρw^{∗}}{h})).

B.2 Poisson summation

Let 𝒪_{𝒟_{∞}} be the maximal order of 𝒟_{∞}. For v_{∞}(α) > v_{∞}(β) − 2, we have

S(α, β, dw) = −q^{2v_{∞}(β)−3} · dw(𝒪_{𝒟_{∞}} ).

For the pair (i, j), 1 ≤ i, j ≤ n, we choose Haar measure dw with dw(𝒟_{∞}/M_{ij}) = 1 and denote the integral S(α, β, dw) by S(α, β, M_{ij}). Then

S(α, β, M_{ij}) = −q^{2v_{∞}(β)−deg(N_{0})} · q^{2v_{∞}(N_{ij})}.

Let M̃_{ij} be the dual lattice of M_{ij}, i.e.,

M̃_{ij} = {w ∈ 𝒟_{∞} : Tr(wμ) ∈ A for all μ ∈ M_{ij}}.

We apply the Poisson summation formula

∑_{μ∈M_{ij}} Ψ_{α,β}(μ) = ∑_{μ^{∗}∈M̃_{ij}} Ψ_{α,β}^{∗}(μ^{∗})

and get

Proposition B.1. Let α, β ∈ k_{∞}^{∗} with v_{∞}(α) > v_{∞}(β) − 2, h ∈ k_{∞}^{×}, ρ ∈ 𝒟_{∞}. Then

∑_{μ∈M_{ij}} ϕ_{∞}( Nr(ρ + hμ)α)ψ_{∞}( Nr(ρ + hμ)β)

= q^{4v_{∞}(h)}S(α, β, M_{ij}) ∑_{μ^{∗}∈M̃_{ij}}ϕ_{∞}( Nr(\frac{μ^{∗}}{h})\frac{α}{β^{2}} )ψ_{∞}( Nr(\frac{μ^{∗}}{h})\frac{−1}{β})ψ_{∞}( Tr(\frac{ρμ^{∗}}{h})).

Let x ∈ k_{∞}^{×}, y ∈ k_{∞}, M ⊂ 𝒟_{∞} a discrete A-lattice, N_{M} ∈ k such that N_{M} · A is the fractional ideal of A generated by Nr(μ) for μ ∈ M. For h ∈ A with h ≠ 0, ρ ∈ M, define “partial theta” series:

θ(x, y, M, N_{M}, h, ρ) := ∑_{μ∈M,μ≡ρ mod hM} ϕ_{∞} (\frac{Nr(μ)xt^{2}}{N_{M}h} ) ψ_{∞} (\frac{Nr(μ)y}{N_{M}h} ) .

Note that θ_{ij}(x, y) = θ(x, y, M_{ij}, N_{ij}, 1, 0), and

θ(x, y, M, N_{M}, h, ρ) = ∑_{μ∈M} ϕ_{∞}( Nr(ρ + hμ)α)ψ_{∞}( Nr(ρ + hμ)β)

where α = \frac{xt^{2}}{N_{M} h}, β = \frac{y}{N_{M} h}.

Proposition B.2. Let x, y ∈ k_{∞}^{×}, v_{∞}(x) > v_{∞}(y), 0 ≠ h ∈ A, κ ∈ M̃_{ij}. Then

θ(\frac{x}{y^{2}} , \frac{−1}{y}, M̃_{ij}, N_{ij}^{−1}N_{0}^{−1}, h, κ)

= S(\frac{xt^{2}}{N_{ij}N_{0}h}, \frac{y}{N_{ij}N_{0}h}, M_{ij})^{−1} ∑_{ρ∈M_{ij}/hM_{ij}} ψ_{∞}( Tr(\frac{ρκ}{h}))θ(\frac{x}{N_{0}} , \frac{y}{N_{0}} , M_{ij}, N_{ij}, h, ρ).

Proof. By Proposition B.1 we have

θ(x, y, M_{ij}, N_{ij}, h, ρ)
= q^{4v_{∞}(h)}S(α, β, M_{ij}) ∑_{μ^{∗}∈M̃_{ij}}ϕ_{∞}( Nr(\frac{μ^{∗}}{h})\frac{α}{β^{2}} )ψ_{∞}( Nr(\frac{μ^{∗}}{h})\frac{−1}{β})ψ_{∞}( Tr(\frac{−ρμ^{∗}}{h})).

Multiply this by ψ_{∞}( Tr(\frac{ρκ}{h})) for κ ∈ M̃_{ij} and sum over ρ ∈ M_{ij}/hM_{ij}, we obtain

∑_{ρ∈M_{ij}/hM_{ij}} ψ_{∞}(Tr(\frac{ρκ}{h})) · θ(x, y, M_{ij}, N_{ij}, h, ρ)

= q^{4v_{∞}(h)}S(α, β, M_{ij}) ∑_{μ^{∗}∈M̃_{ij}} \{ϕ_{∞}( Nr(\frac{μ^{∗}}{h})\frac{α}{β^{2}} )ψ_{∞}( Nr(\frac{μ^{∗}}{h})\frac{−1}{β})

·[ ∑_{ρ∈M_{ij}/hM_{ij}} ψ_{∞}( Tr (\frac{ρ}{h}(κ − μ^{∗})))]\}.

Since

∑_{ρ∈M_{ij}/hM_{ij}} ψ_{∞}(Tr(\frac{ρ}{h}(κ − μ^{∗}))) = \begin{cases}0 & if μ^{∗} − κ ∉ hM̃_{ij}, \\ q^{−4v_{∞}(h)} & if μ^{∗} − κ ∈ hM̃_{ij}. \end{cases}

The proposition follows by replacing x with \frac{x}{N_{0}} , and y with \frac{y}{N_{0}} .

762 Fu-Tsun Wei and Jing Yu B.3 Transformation law Let (x, y) ∈ k_{∞}^{×} × k_{∞}. Suppose a matrix γ = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL_{2}(A) is given such that cy + d ≠ 0. We define

γ ∘ (x, y) := (\frac{x(ad − bc)}{(cy + d)^{2}}, \frac{ay + b}{cy + d}) .

Lemma B.3. Suppose γ = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL_{2}(A), c ≡ 0 mod N_{0}, v_{∞}(x) > v_{∞}(y), and v_{∞}(cx) > v_{∞}(cy + d). Let 1 ≤ i, j, ≤ n. Then

θ_{ij}(γ ∘ (x, y)) = S(\frac{N_{ij}xt^{2}}{y^{2}} , \frac{−N_{ij}(cy + d)}{dy}, M̃_{ij})^{−1} · S(\frac{xt^{2}}{N_{ij}} , \frac{y}{N_{ij}} , M_{ij})^{−1}

· ( ∑_{κ∈M_{ij}/dM_{ij}} ψ_{∞}(\frac{Nr(κ)b}{N_{ij}d})) θ_{ij}(x, y).

Proof. Put u = \frac{x}{y^{2}} , v = \frac{−1}{y}. Then

θ_{ij}(γ ∘ (x, y)) = θ(\frac{u}{(c − dv)^{2}} , \frac{b}{d} + \frac{1}{d(c − dv)}, M_{ij}, N_{ij}, 1, 0)
= ∑_{κ∈M_{ij}/dM_{ij}} θ(\frac{du}{(c − dv)^{2}} , b + \frac{1}{c − dv} , M_{ij}, N_{ij}, d, κ)
= ∑_{κ∈M_{ij}/dM_{ij}} ψ_{∞}(\frac{Nr(κ)b}{N_{ij}d})θ(\frac{du}{(dv − c)^{2}} , \frac{−1}{dv − c}, M_{ij}, N_{ij}, d, κ).

Since v_{∞}(cx) > v_{∞}(cy + d), we have v_{∞}(du) > v_{∞}(dv − c) and

θ_{ij}(γ ∘ (x, y)) = S(N_{ij}ut^{2}, N_{ij}(v − c/d), M̃_{ij})^{−1} · ∑_{κ∈M_{ij}/dM_{ij}} [ψ_{∞}(\frac{Nr(κ)b}{N_{ij}d})

· ∑_{ρ∈M̃_{ij}/dM̃_{ij}} ψ_{∞}( Tr(\frac{ρκ}{d}))θ(\frac{du}{N_{0}} , \frac{dv − c}{N_{0}}, M̃_{ij}, N_{ij}^{−1}N_{0}^{−1}, d, ρ)].

Since −c/N_{0} ∈ A, we have

θ_{ij}(γ ∘ (x, y)) = S(N_{ij}ut^{2}, N_{ij}(v − c/d), M̃_{ij})^{−1}

· ∑_{ρ∈M̃_{ij}/dM̃_{ij}} [θ(\frac{du}{N_{0}} , \frac{dv}{N_{0}} , M̃_{ij}, N_{ij}^{−1}N_{0}^{−1}, d, ρ)

· ∑_{κ∈M_{ij}/dM_{ij}} ψ_{∞}(\frac{Nr(κ)b}{N_{ij}d} + \frac{Tr(ρκ)}{d} − \frac{Nr(ρ)cN_{ij}}{d} )].

Documenta Mathematica 16 (2011) 723–765

Theta series and function field analogue … 763

Note that cN_{ij}ρ̄ ∈ M_{ij}. Replacing κ by κ + cN_{ij}ρ̄ the last summand equals to

\frac{Nr(κ)b}{N_{ij}d} + a Tr(ρκ) + N_{ij}ac Nr(ρ).

Since a Tr(ρκ) + N_{ij}ac Nr(ρ) ∈ A, we have

θ_{ij}(γ ∘ (x, y)) = S(N_{ij}ut^{2}, N_{ij}(v − c/d), M̃_{ij})^{−1} · ( ∑_{κ∈M_{ij}/dM_{ij}} ψ_{∞}(\frac{Nr(κ)b}{N_{ij}d}))

·θ(\frac{u}{N_{0}} , \frac{v}{N_{0}} , M̃_{ij}, N_{ij}^{−1}N_{0}^{−1}, 1, 0).

Recall that u = \frac{x}{y^{2}} , v = \frac{−1}{y}. By Proposition B.2 we have

θ_{ij}(g ∘ (x, y)) = S(\frac{N_{ij}xt^{2}}{y^{2}} , \frac{−N_{ij}(cy + d)}{dy}, M̃_{ij})^{−1} · S(\frac{xt^{2}}{N_{ij}} , \frac{y}{N_{ij}} , M_{ij})^{−1}

· ( ∑_{κ∈M_{ij}/dM_{ij}} ψ_{∞}(\frac{Nr(κ)b}{N_{ij}d})) θ_{ij}(x, y).

Note that

S(\frac{N_{ij}xt^{2}}{y^{2}} , \frac{−N_{ij}(cy + d)}{dy}, M̃_{ij}) · S(\frac{xt^{2}}{N_{ij}} , \frac{y}{N_{ij}} , M_{ij}) = q^{2v_{∞}(cy+d)+2 deg d}.

By standard argument we get ∑_{κ∈M_{ij}/dM_{ij}} ψ_{∞}(\frac{Nr(κ)b}{N_{ij}d}) = q^{2 deg(d)}. Since θ_{ij}(x, y) = θ_{ij}(x, y + h) for any h ∈ A, we can drop the assumption v_{∞}(x) > v_{∞}(y) and obtain the transformation law of θ_{ij}: Theorem B.4. For 1 ≤ i, j ≤ n. Let x ∈ k_{∞}^{×}, y ∈ k_{∞}, γ = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL_{2}(A). Assume v_{∞}(cx) > v_{∞}(cy + d), and c ≡ 0 mod N_{0}. Then

θ_{ij}(γ ∘ (x, y)) = q^{−2v_{∞}(cy+d)} · θ_{ij}(x, y).

Acknowledgements We would like to thank the referee for referring us to the work of Papikian [10].

Documenta Mathematica 16 (2011) 723–765

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Fu-Tsun Wei Department of Mathematics National Tsing Hua University Hsinchu 30013 Taiwan Jing Yu TIMS and the Department of Mathematics National Taiwan University Taipei 10617 Taiwan 766

Documenta Mathematica 16 (2011)