Heegner Points and L-Series of Automorphic

Cusp Forms of Drinfeld Type

Hans-Georg Rück and Ulrich Tipp

Received: April 27, 2000

Communicated by Peter Schneider

Abstract. In their famous article [Gr-Za], Gross and Zagier proved a formula relating heights of Heegner points on modular curves and derivatives of L-series of cusp forms. We prove the function field analogue of this formula. The classical modular curves parametrizing isogenies of elliptic curves are now replaced by Drinfeld modular curves dealing with isogenies of Drinfeld modules. Cusp forms on the classical upper half plane are replaced by harmonic functions on the edges of a Bruhat-Tits tree. As a corollary we prove the conjecture of Birch and Swinnerton-Dyer for certain elliptic curves over functions fields whose analytic rank is equal to 1.

2000 Mathematics Subject Classification: 11G40 11G50 11F67 11G09 Keywords and Phrases: Heegner points, Drinfeld modular curves, Lseries, automorphic cusp forms, Gross-Zagier formula

1 Introduction

Let K = 𝔽_{q}(T) be the rational function field over a finite field 𝔽_{q} of odd characteristic. In K we distinguish the polynomial ring 𝔽_{q}[T] and the place ∞. We consider harmonic functions f on GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗}, the edges of the Bruhat-Tits tree of GL_{2}, which are invariant under Γ_{0}(N) for N ∈ 𝔽_{q}[T]. These are called automorphic cusp forms of Drinfeld type of level N (cf. section 2.1). Let L = K(\sqrt{D}), with gcd(N, D) = 1, be an imaginary quadratic extension of K (we assume that D is irreducible to make calculations technically easier). We attach to an automorphic cusp form f of Drinfeld type of level N, which is a newform, and to an element A in the class group of O_{L} = 𝔽_{q}[T][\sqrt{D}] an L-series L(f, A, s) (section 2.1).

We represent this L-series (normalized by a suitable factor L^{(N,D)}(2s + 1)) as a Petersson product of f and a function Φ_{s} on Γ_{0}(N) \ GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗} (sections 2.2 and 2.3). From this representation we get a functional equation for L(f, A, s) (Theorem 2.7.3 and Theorem 2.7.6), which shows in particular

that L(f, A, s) has a zero at s = 0, if [\frac{D}{N} ] = 1.

In this case, under the additional assumptions that N is square free and that each of its prime divisors is split in L, we evaluate the derivative of L(f, A, s) at s = 0. Since the function Φ_{s} is not harmonic in general, we apply a holomorphic projection formula (cf. section 2.4) to get

\frac{∂}{∂s}(L^{(N,D)}(2s + 1)L(f, A, s)) |_{s=0}= ∫ f · \overline{Ψ_{A}} (if deg D is odd),

resp.

\frac{∂}{∂s}(\frac{1}{1 + q^{−s−1}} L^{(N,D)}(2s + 1)L(f, A, s)) |_{s=0}= ∫ f · \overline{Ψ_{A}} (if deg D is even),

where Ψ_{A} is an automorphic cusp form of Drinfeld type of level N. The Fourier coefficients Ψ_{A}^{∗}(π_{∞}^{deg λ+2}, λ) of Ψ_{A} are evaluated in sections 2.5, 2.6 and 2.8. The results are summarized in Theorem 2.8.2 and Theorem 2.8.3. On the other hand let x be a Heegner point on the Drinfeld modular curve X_{0}(N) with complex multiplication by O_{L} = 𝔽_{q}[T][\sqrt{D}]. There exists a cusp form g_{A} of Drinfeld type of level N whose Fourier coefficients are given by (cf. Proposition 3.1.1):

g_{A}^{∗}(π_{∞}^{deg λ+2}, λ) = q^{− deg λ}⟨(x) − (∞), T_{λ}((x)^{σ_{A}} − (0))⟩,

where the automorphism σ_{A} belongs to the class A via class field theory, where T_{λ} is the Hecke operator attached to λ and where ⟨ , ⟩ denotes the global Néron-Tate height pairing of divisors on X_{0}(N) over the Hilbert class field of L. We want to compare the cusp forms Ψ_{A} and g_{A}. Therefore we have to evaluate the height of Heegner points, which is the content of chapter 3. We evaluate the heights locally at each place of K. At the places belonging to the polynomial ring 𝔽_{q}[T] we use the modular interpretation of Heegner points by Drinfeld modules. Counting homomorphisms between different Drinfeld modules (similar to calculations in [Gr-Za]) yields the formula for these local heights (Corollary 3.4.10 and Proposition 3.4.13). At the place ∞ we construct a Green’s function on the analytic upper half plane, which gives the local height in this case (Propositions 3.6.3, 3.6.5). Finally we evaluate the Fourier coefficients of g_{A} in Theorems 3.6.4 and 3.6.6. In chapter 4 we compare the results on the derivatives of the L-series, i.e. the Fourier coefficients Ψ_{A}^{∗}(π_{∞}^{deg λ+2}, λ), and the result on the heights of Heegner points, i.e. the coefficients g_{A}^{∗}(π_{∞}^{deg λ+2}, λ), and get our main result (cf. Theorem 4.1.1 and Theorem 4.1.2): If gcd(λ, N) = 1, then

Ψ_{A}^{∗}(π_{∞}^{deg λ+2}, λ) = \frac{q − 1}{2} q^{−(deg D+1)/2} g_{A}^{∗}(π_{∞}^{deg λ+2}, λ) (if deg D is odd),

Heegner Points and L-Series 367

resp.

Ψ_{A}^{∗}(π_{∞}^{deg λ+2}, λ) = \frac{q − 1}{4} q^{− deg D/2} g_{A}^{∗}(π_{∞}^{deg λ+2}, λ) (if deg D is even).

We apply this result to elliptic curves. Let E be an elliptic curve over K with conductor N · ∞, which has split multiplicative reduction at ∞, then E is modular, i.e. it belongs to an automorphic cusp form f of Drinfeld type of level N. In particular the L-series of E/K and of f satisfy L(E, s + 1) = L(f, s). The L-series of E over the field L = K(\sqrt{D}) equals L(E, s)L(E_{D}, s) and can be computed by

L(E, s + 1)L(E_{D}, s + 1) = ∑_{A∈Cl(O_{L})} L^{(N,D)}(2s + 1)L(f, A, s),

if deg D is odd, or in the even case by

L(E, s + 1)L(E_{D}, s + 1) = ∑_{A∈Cl(O_{L})} \frac{1}{1 + q^{−s−1}} L^{(N,D)}(2s + 1)L(f, A, s).

This motivates the consideration of the L-series L(f, A, s). The functional equations for all L(f, A, s) yield that L(E, s)L(E_{D}, s) has a zero at s = 1. In order to evaluate the first derivative, we consider a uniformization π : X_{0}(N) → E of the modular elliptic curve E and the Heegner point P_{L} := ∑_{A∈Cl(O_{L})} π(x^{σ_{A}}). P_{L} is an L-rational point on E. Our main result yields a formula relating the derivative of the L-series of E/L and the Néron-Tate height ĥ_{E,L}(P_{L}) of the Heegner point on E over L (Theorem 4.2.1):

\frac{∂}{∂s}(L(E, s)L(E_{D}, s)) |_{s=1}= ĥ_{E,L}(P_{L}) c(D) (deg π)^{−1} ∫_{Γ_{0}(N)\GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗}} f · \overline{f},

where the constant c(D) equals \frac{q−1}{2} q^{−(deg D+1)/2} (if deg D is odd) or \frac{q−1}{4} q^{− deg D/2} (if deg D is even). As a corollary (Corollary 4.2.2) we prove the conjecture of Birch and Swinnerton-Dyer for E/L, if its analytic rank is equal to 1. Large parts of this work were supported by the DFG-Schwerpunkt “Algorithmische Algebra und Zahlentheorie”. We are very thankful for this.

2 L-Series

2.1 Basic Definitions of L-Series

Let 𝔽_{q} be the finite field with q = p^{α} elements (p ≠ 2), and let K = 𝔽_{q}(T) be the rational function field over 𝔽_{q}. We distinguish the finite places given by the irreducible elements in the polynomial ring 𝔽_{q}[T] and the place ∞ of K. For ∞ we consider the completion K_{∞} with normalized valuation v_{∞} and valuation ring O_{∞}. We fix the prime π_{∞} = T^{−1}, then K_{∞} = 𝔽_{q}((π_{∞})). In addition we define the following additive character ψ_{∞} of K_{∞}: Take σ : 𝔽_{q} → ℂ^{∗} with σ(a) = exp(\frac{2πi}{p} T r_{𝔽_{q}/𝔽_{p}}(a)) and set ψ_{∞}(∑ a_{i}π_{∞}^{i}) = σ(−a_{1}). The oriented edges of the Bruhat-Tits tree of GL_{2} over K_{∞} are parametrized by the set GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗}, where

Γ_{∞} := \{( \begin{matrix}α & β \\ γ & δ \end{matrix} ) ∈ GL_{2}(O_{∞}) | v_{∞}(γ) > 0\}.

GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗} can be represented by the two disjoint sets

T_{+} := \{( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} ) | m ∈ ℤ, u ∈ K_{∞}/π_{∞}^{m}O_{∞}\}  (2.1.1)

and

T_{−} := \{( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} ) ( \begin{matrix}0 & 1 \\ π_{∞} & 0 \end{matrix} ) | m ∈ ℤ, u ∈ K_{∞}/π_{∞}^{m}O_{∞}\}.  (2.1.2)

Right multiplication by ( \begin{matrix}0 & 1 \\ π_{∞} & 0 \end{matrix} ) reverses the orientation of an edge.

We do not distinguish between matrices in GL_{2}(K_{∞}) and the corresponding classes in GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗}. We want to study functions on GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗}. Special functions are defined in the following way: The groups GL_{2}(𝔽_{q}[T]) and SL_{2}(𝔽_{q}[T]) operate on GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗} by left multiplication. For N ∈ 𝔽_{q}[T] let

Γ_{0}(N) := \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(𝔽_{q}[T]) | c ≡ 0 mod N\}

and Γ_{0}^{(1)}(N) := Γ_{0}(N) ∩ SL_{2}(𝔽_{q}[T]).

Definition 2.1.1 A function f : GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗} → ℂ is called an automorphic cusp form of Drinfeld type of level N if it satisfies the following conditions:

i) f is harmonic, i.e. ,

f(X ( \begin{matrix}0 & 1 \\ π_{∞} & 0 \end{matrix} )) = −f(X)

and

∑_{β∈GL_{2}(O_{∞})/Γ_{∞}} f(Xβ) = 0

for all X ∈ GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗}, ii) f is invariant under Γ_{0}(N), i.e. ,

f(AX) = f(X)

for all X ∈ GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗} and A ∈ Γ_{0}(N), iii) f has compact support modulo Γ_{0}(N), i.e. there are only finitely many elements X̄ in Γ_{0}(N)\GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗} with f(X̄) ≠ 0.

Any function f on GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗} which is invariant under ( \begin{matrix}1 & 𝔽_{q}[T] \\ 0 & 1 \end{matrix} )

has a Fourier expansion

f(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) = ∑_{λ∈𝔽_{q}[T]} f^{∗}(π_{∞}^{m}, λ)ψ_{∞}(λu)  (2.1.3)

with

f^{∗}(π_{∞}^{m}, λ) = ∫_{K_{∞}/𝔽_{q}[T]} f(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} ))ψ_{∞}(−λu) du,

where du is a Haar measure with ∫_{K_{∞}/𝔽_{q}[T]} du = 1.

Since ( \begin{matrix}1 & 𝔽_{q}[T] \\ 0 & 1 \end{matrix} ) ⊂ Γ_{0}(N) this applies to automorphic cusp forms. In this

particular case the harmonicity conditions of Definition 2.1.1 imply

f^{∗}(π_{∞}^{m}, λ) = 0, if λ = 0 or if deg λ + 2 > m,  (2.1.4)
f^{∗}(π_{∞}^{m}, λ) = q^{−m+deg λ+2}f^{∗}(π_{∞}^{deg λ+2}, λ), if λ ≠ 0 and deg λ + 2 ≤ m.

Hence we get the following:

Remark 2.1.2 All the Fourier coefficients of an automorphic cusp form f of Drinfeld type are uniquely determined by the coefficients f^{∗}(π_{∞}^{deg λ+2}, λ) for λ ∈ 𝔽_{q}[T].

To an automorphic cusp form f one can attach an L-series L(f, s) in the following way (cf. [We1], [We2]): Let 𝔪 be an effective divisor of K of degree n, then 𝔪 = (λ)_{0} + (n − deg λ)∞ with λ ∈ 𝔽_{q}[T], deg λ ≤ n. We define

f^{∗}(𝔪) = f^{∗}(π_{∞}^{n+2}, λ) and L(f, s) = ∑_{𝔪≥0} f^{∗}(𝔪)N(𝔪)^{−s},  (2.1.5)

where N(𝔪) denotes the absolute norm of the divisor 𝔪. The ℂ-vector space of automorphic cusp forms of Drinfeld type of level N is finite dimensional and it is equipped with a non-degenerate pairing, the Petersson product, given by

(f, g) ↦ ∫_{Γ_{0}(N)\GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗}} f · \overline{g}.

There is the notion of oldforms, i.e. linear combinations of forms

g(( \begin{matrix}d & 0 \\ 0 & 1 \end{matrix} ) X), where g is an automorphic cusp form of level M, M|N and M ≠ N, and d is a divisor of N/M. Automorphic cusp forms of Drinfeld type which are perpendicular under the Petersson product to all the oldforms are called newforms. Important examples of newforms are the following: Let E be an elliptic curve over K with conductor N · ∞, which has split multiplicative reduction at ∞, then E belongs to a newform f of level N such that the L-series of E satisfies ([De])

L(E, s + 1) = L(f, s).  (2.1.6)

This newform is in addition an eigenform for all Hecke operators, but we do not assume this property in general. From now on let f be an automorphic cusp form of level N which is a newform. Let L/K be an imaginary quadratic extension (i.e. a quadratic extension of K where ∞ is not split) in which each (finite) divisor of N is not ramified. Then there is a square free polynomial D ∈ 𝔽_{q}[T], prime to N with L = K(\sqrt{D}). We assume in this paper that D is an irreducible polynomial. In principle all the arguments apply to the general case, but the details are technically more complicated. We distinguish two cases. In the first case the degree of D is odd, i.e. ∞ is ramified in L/K; in the second case the degree of D is even and its leading coefficient is not a square in 𝔽_{q}^{∗}, i.e. ∞ is inert in L/K. The integral closure of 𝔽_{q}[T] in L is O_{L} = 𝔽_{q}[T][\sqrt{D}]. Let A be an element of the class group Cl(O_{L}) of O_{L}. For an effective divisor 𝔪 = (λ)_{0} + (n − deg λ)∞ (as above) we define

r_{A}(𝔪) = #{𝔞 ∈ A | 𝔞 integral with N_{L/K}(𝔞) = λ 𝔽_{q}[T]}  (2.1.7)

and hence we get the partial zeta function attached to A as

ζ_{A}(s) = ∑_{𝔪≥0} r_{A}(𝔪)N(𝔪)^{−s}.  (2.1.8)

For the calculations it is sometimes easier to define a function depending on elements of 𝔽_{q}[T] instead of divisors. We choose 𝔞_{0} ∈ A^{−1} and λ_{0} ∈ K with N_{L/K}(𝔞_{0}) = λ_{0} 𝔽_{q}[T] and define

r_{𝔞_{0},λ_{0}}(λ) = #{μ ∈ 𝔞_{0} | N_{L/K}(μ) = λ_{0}λ}.  (2.1.9)

Then

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

The theta series is defined as

Θ_{𝔞_{0},λ_{0}}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) = ∑_{deg λ+2≤m} r_{𝔞_{0},λ_{0}}(λ)ψ_{∞}(λu).  (2.1.10)

We will see later that the transformation rules of this theta series are the starting point of all our calculations. Now we combine the L-series of a newform f (cf. (2.1.5)) and the partial zeta function of A (cf. (2.1.8)) to obtain the function

L(f, A, s) = ∑_{𝔪≥0} f^{∗}(𝔪)r_{A}(𝔪)N(𝔪)^{−s}.  (2.1.11)

For technical reasons we introduce

L^{(N,D)}(2s + 1) = \frac{1}{q − 1} ∑_{\begin{matrix}k∈𝔽_{q}[T] \\ gcd(k,N)=1 \end{matrix}} [\frac{D}{k} ] q^{−(2s+1) deg k},  (2.1.12)

where [\frac{D}{k} ] denotes the Legendre resp. the Jacobi symbol for the polynomial

ring 𝔽_{q}[T]. For an irreducible k ∈ 𝔽_{q}[T] and a coprime D ∈ 𝔽_{q}[T] the Legendre

symbol [\frac{D}{k} ] is by definition equal to 1 or −1 if D is or is not a square in

(𝔽_{q}[T]/k 𝔽_{q}[T])^{∗}, respectively. If D is divisible by k, then [\frac{D}{k} ] equals 0. This

definition is multiplicatively extended to the Jacobi symbol for arbitrary, not

necessarily irreducible k, so e.g. [\frac{D}{k} ] = [\frac{D}{k_{1}} ] · [\frac{D}{k_{2}} ] if k = k_{1} · k_{2}.

In the first case, where deg D is odd, the function

L^{(N,D)}(2s + 1)L(f, A, s)

is the focus of our interest; in the case of even degree it is the function

\frac{1}{1 + q^{−s−1}} L^{(N,D)}(2s + 1)L(f, A, s).

This is motivated by the following fact:

Proposition 2.1.3 Let E be an elliptic curve with conductor N ·∞ and corresponding newform f as above and let E_{D} be its twist by D. Then the following identities hold:

L(E, s + 1)L(E_{D}, s + 1) = ∑_{A∈Cl(O_{L})} L^{(N,D)}(2s + 1)L(f, A, s)

if deg D is odd, and

L(E, s + 1)L(E_{D}, s + 1) = ∑_{A∈Cl(O_{L})} \frac{1}{1 + q^{−s−1}} L^{(N,D)}(2s + 1)L(f, A, s)

if deg D is even.

It is not difficult to prove this fact using the definitions of the coefficients f^{∗}(𝔪) (cf. (2.1.5)) and r_{A}(𝔪) (cf. (2.1.7)) and the Euler products of the L-series of the elliptic curves.

372 Hans-Georg Rück and Ulrich Tipp 2.2 Rankin’s Method The properties of the automorphic cusp form f yield

f^{∗}(π_{∞}^{m}, λ) = q^{−m+1} ∑_{u∈π_{∞}/π_{∞}^{m}} f(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} ))ψ_{∞}(−λu).  (2.2.1)

We use this to calculate

L(f, A, s) = \frac{1}{q − 1} ∑_{m=2}^{∞}( ∑_{deg λ+2≤m} f^{∗}(π_{∞}^{m}, λ)r_{𝔞_{0},λ_{0}}(λ)) q^{−(m−2)s}. (2.2.2)

Now we distinguish the two cases. 2.2.1 deg D is odd We continue with equations (2.2.1) and (2.2.2):

L(f, A, s)

= \frac{q}{q − 1} ∑_{m=2}^{∞} ∑_{u∈π_{∞}/π_{∞}^{m}} f(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} ))\overline{Θ_{𝔞_{0},λ_{0}}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} ))} q^{−m(s+1)+2s}
= \frac{q}{q − 1} ∫_{ℍ_{∞}} f(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} ))\overline{Θ_{𝔞_{0},λ_{0}}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) q^{−m(s̄+1)+2s̄}},  (2.2.3)

where

ℍ_{∞} := ( \begin{matrix}1 & 𝔽_{q}[T] \\ 0 & 1 \end{matrix} ) \ ( \begin{matrix}K_{∞}^{∗} & K_{∞} \\ 0 & 1 \end{matrix} ) / ( \begin{matrix}O_{∞}^{∗} & O_{∞} \\ 0 & 1 \end{matrix} ) .

We consider the canonical mapping

ℍ_{∞} → Γ_{0}^{(1)}(ND)\GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗} =: G(ND),

which is surjective. We take the measure on G(ND) which counts the size of the stabilizer of an element (cf. [Ge-Re], (4.8)). Then we get

L(f, A, s) = \frac{q}{2(q − 1)}  (2.2.4)

· ∫_{G(ND)}f(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} ))\overline{∑_{M} Θ_{𝔞_{0},λ_{0}}(M ( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) q^{−m^{∗}(s̄+1)+2s̄}}

where the sum is taken over those M = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ ( \begin{matrix}1 & 𝔽_{q}[T] \\ 0 & 1 \end{matrix} ) \Γ_{0}^{(1)}(ND)

with M ( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} ) ∈ T_{+}, and where m^{∗} = m − 2v_{∞}(cu + d).

Documenta Mathematica 5 (2000) 365–444

Remark 2.2.1 The definitions of T_{+} and T_{−} (cf. (2.1.1), (2.1.2)) yield:

M ( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} ) ∈ T_{+} if and only if v_{∞}(cπ_{∞}^{m}) > v_{∞}(cu + d).

In ([Rü1], Theorem 6.2) we showed that for those M satisfying v_{∞}(cπ_{∞}^{m}) > v_{∞}(cu + d) one has the following transformation rule for the theta series (cf. (2.1.10)):

Θ_{𝔞_{0},λ_{0}}(M ( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) = Θ_{𝔞_{0},λ_{0}}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) [\frac{d}{D} ] δ_{cu+d} q^{−v_{∞}(cu+d)}, (2.2.5)

where [\frac{d}{D} ] is the Legendre symbol (defined in section 2.1) and where δ_{z} denotes

the local norm symbol at ∞, i.e., δ_{z} is equal to 1 if z ∈ K_{∞}^{∗} is the norm of an element in the quadratic extension K_{∞}(\sqrt{D})/K_{∞} and −1 otherwise. Equations (2.2.4), (2.2.5) and the definition of L^{(N,D)} (cf. (2.1.12)) yield:

L^{(N,D)}(2s + 1)L(f, A, s) = \frac{q}{2(q − 1)} ∫_{G(ND)} f · \overline{Θ_{𝔞_{0},λ_{0}}H_{1,s̄}}

with

H_{1,s}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) := q^{−m(s+1)+2s} ∑_{\begin{matrix}c,d∈𝔽_{q}[T] \\ c≡0modND \\ gcd(d,N)=1 \\ v_{∞}(cπ_{∞}^{m})>v_{∞}(cu+d) \end{matrix}}[\frac{d}{D} ] δ_{cu+d} q^{v_{∞}(cu+d)(2s+1)}.

We see that Θ_{𝔞_{0},λ_{0}}H_{1,s} is a function on G(ND). Let μ : 𝔽_{q}[T] → {0, 1, −1} be the Moebius function with

∑_{\begin{matrix}e∈𝔽_{q}[T] \\ e|n \end{matrix}} μ(e) = 0 if n 𝔽_{q}[T] ≠ 𝔽_{q}[T],

and

\frac{1}{q − 1} ∑_{e∈𝔽_{q}^{∗}} μ(e) = 1,

then H_{1,s}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} ))

= \frac{q^{−m(s+1)+2s}}{q − 1} ∑_{e|N} μ(e) [\frac{e}{D} ] δ_{e} q^{−(2s+1) deg e}E_{s}^{(1)}(( 0 1^{\frac{Nπ_{∞}^{m}}{e} \frac{Nu}{e}} ))

with the Eisenstein series

E_{s}^{(1)}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) := ∑_{\begin{matrix}c,d∈𝔽_{q}[T] \\ c≡0modD \\ v_{∞}(cπ_{∞}^{m})>v_{∞}(cu+d) \end{matrix}} [\frac{d}{D} ] δ_{cu+d} q^{v_{∞}(cu+d)(2s+1)}. (2.2.6)

For a divisor e of N the function

Θ_{𝔞_{0},λ_{0}}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) q^{−m(s+1)+2s}E_{s}^{(1)}(( 0 1^{\frac{Nπ_{∞}^{m}}{e} \frac{Nu}{e}} ))

on GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗} is invariant under Γ_{0}^{(1)}(\frac{ND}{e}). Since we assume that f is a newform of level N, it is orthogonal (with respect to the Petersson product) to functions of lower level. Therefore we get

Proposition 2.2.2 Let deg D be odd, then

L^{(N,D)}(2s + 1)L(f, A, s) = \frac{q}{2(q − 1)} ∫_{G(ND)} f · \overline{Θ_{𝔞_{0},λ_{0}}H_{2,s̄}}

with H_{2,s}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} ))

:= q^{−m(s+1)+2s}E_{s}^{(1)}(( \begin{matrix}Nπ_{∞}^{m} & Nu \\ 0 & 1 \end{matrix} ))  (2.2.7)
= q^{−m(s+1)+2s}  ∑_{\begin{matrix}c,d∈𝔽_{q}[T] \\ c≡0modD \\ v_{∞}(cNπ_{∞}^{m})>v_{∞}(cNu+d) \end{matrix}}[\frac{d}{D} ] δ_{cNu+d} q^{v_{∞}(cNu+d)(2s+1)}.

.

2.2.2 deg D is even

We use equation (2.2.2) and the geometric series expansion of 1/(1 + q^{−s−1}) to evaluate

\frac{1}{1 + q^{−s−1}} L(f, A, s) = \frac{1}{q − 1}

· ∑_{m=2}^{∞} q^{−(m−2)s} ∑_{l=2}^{m}( ∑_{deg λ+2≤l} f^{∗}(π_{∞}^{l}, λ)r_{𝔞_{0},λ_{0}}(λ)) (−q^{−1})^{m−l}.

Since f is an automorphic cusp form and hence f^{∗}(π_{∞}^{l}, λ) = q^{m−l}f^{∗}(π_{∞}^{m}, λ) (cf. (2.1.4)), we get

\frac{1}{1 + q^{−s−1}} L(f, A, s) = \frac{1}{q − 1}

· ∑_{m=2}^{∞}( ∑_{deg λ+2≤m} f^{∗}(π_{∞}^{m}, λ)r_{𝔞_{0},λ_{0}}(λ)) q^{−(m−2)s} \frac{(−1)^{m−deg λ} + 1}{2}.

Documenta Mathematica 5 (2000) 365–444

If r_{𝔞_{0},λ_{0}}(λ) ≠ 0, then deg λ ≡ deg λ_{0} mod 2, because deg D is even. Now equation (2.2.1) yields

\frac{1}{1 + q^{−s−1}} L(f, A, s) = \frac{q}{q − 1}

· ∫_{ℍ_{∞}} f(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} ))\overline{Θ_{𝔞_{0},λ_{0}}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) q^{−m(s̄+1)+2s̄}} \frac{(−1)^{m−deg λ_{0}} + 1}{2}.

Thus the right side of this equation differs from (2.2.3) only by the factor ((−1)^{m−deg λ_{0}} + 1)/2. But this factor is invariant under GL_{2}(𝔽_{q}[T]) and hence causes no problems here or in the next steps. Proceeding exactly as in the case where deg D is odd gives the following result:

Proposition 2.2.3 Let deg D be even, then

\frac{1}{1 + q^{−s−1}} L^{(N,D)}(2s + 1)L(f, A, s) = \frac{q}{2(q − 1)} ·

∫_{G(ND)} f · \overline{Θ_{𝔞_{0},λ_{0}}H_{2,s̄}} \frac{(−1)^{m−deg λ_{0}} + 1}{2}

with H_{2,s} given by equation (2.2.7).

2.3 Computation of the Trace

The function Θ_{𝔞_{0},λ_{0}}H_{2,s} on GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗} is only invariant under Γ_{0}^{(1)}(ND). To make it invariant under Γ_{0}^{(1)}(N) we compute the trace with respect to the extension Γ_{0}^{(1)}(ND)\Γ_{0}^{(1)}(N). The trace from Γ_{0}^{(1)}(N) to Γ_{0}(N) is easy, this will be done at the very end of the calculations. Since N and D are relatively prime, there are μ_{1}, μ_{2} ∈ 𝔽_{q}[T] with 1 = μ_{1}N + μ_{2}D. The set

R = \{( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}1 & 1 \\ −μ_{2}D & μ_{1}N \end{matrix} ) ( \begin{matrix}0 & −1 \\ 1 & λ \end{matrix} ) (λ mod D)\}  (2.3.1)

is therefore a set of representatives of Γ_{0}^{(1)}(ND)\Γ_{0}^{(1)}(N). Here we used the assumption that D is irreducible. In order to evaluate ∑_{M∈R} Θ_{𝔞_{0},λ_{0}}H_{2,s}(M·), we treat Θ_{𝔞_{0},λ_{0}} and H_{2,s} separately. From ([Rü1], Prop. 4.4) we get, if m > v_{∞}(u):

Θ_{𝔞_{0},λ_{0}}(( 0 1^{\frac{π_{∞}^{m}}{u^{2}} \frac{1}{u}} )) = Θ_{𝔞_{0},λ_{0}}(( 0 1^{\frac{π_{∞}^{m}}{D} \frac{u}{D}} ))δ_{u} q^{−v_{∞}(u)}δ_{−λ_{0}}q^{− \frac{1}{2} deg D}ϵ_{0}^{−1}

where ϵ_{0} = 1 if deg D is even and ϵ_{0} = δ_{−t}(−1)^{α+1}γ(p)^{α} (q = p^{α} ; γ(p) = 1 if p ≡ 1 mod 4 or i otherwise) if deg D is odd. Then one evaluates

Θ_{𝔞_{0},λ_{0}}(( \begin{matrix}0 & −1 \\ 1 & λ \end{matrix} ) ( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) = Θ_{𝔞_{0},λ_{0}}(( 0  1^{\frac{π_{∞}^{m}}{D} \frac{−(u+λ)}{D}} )) · (2.3.2)

·δ_{u+λ} q^{−v_{∞}(u+λ)}δ_{λ_{0}} q^{− \frac{1}{2} deg D}ϵ_{0}^{−1}.

Now (2.2.5) and (2.3.2) yield the operation of the matrices M ∈ R (cf. (2.3.1)) on Θ_{𝔞_{0},λ_{0}}. The situation for H_{2,s} and hence for the Eisenstein series E_{s}^{(1)} (cf. (2.2.6)) is easier. Straightforward calculations (mainly transformations of the summation indices) yield:

If ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(𝔽_{q}[T]) with gcd(c, D) = 1 and if v_{∞}(cπ_{∞}^{m}) > v_{∞}(cu + d)

then

E_{s}^{(1)}(( \begin{matrix}a & b \\ c & d \end{matrix} ) ( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) = E_{s}^{(D)}(( \begin{matrix}\frac{π_{∞}^{m}}{D} & \frac{u+c^{∗}d}{D}) \\ 0 & 1 \end{matrix} )) [\frac{c}{D} ] ·

·δ_{D} q^{−(2s+1) deg D}δ_{cu+d} q^{−v_{∞}(cu+d)(2s+1)}  (2.3.3)

with c^{∗} ≡ c^{−1} mod D. Here E_{s}^{(D)} is the Eisenstein series

E_{s}^{(D)}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) := ∑_{\begin{matrix}c,d∈𝔽_{q}[T] \\ v_{∞}(cπ_{∞}^{m})>v_{∞}(cu+d) \end{matrix}} [\frac{c}{D} ] δ_{cu+d} q^{v_{∞}(cu+d)(2s+1)}. (2.3.4)

2.3.1 deg D is odd

We apply the results of this section ((2.3.2) and (2.3.3)) to Proposition 2.2.2. Let G(N) be the set Γ_{0}^{(1)}(N)\GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗}.

Proposition 2.3.1 Let deg D be odd, then

L^{(N,D)}(2s + 1)L(f, A, s) = \frac{q}{2(q − 1)} ∫_{G(N)} f · \overline{Φ(o) ; s̄}

with

Φ_{s}^{(o)}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) := ∑_{M∈R} Θ_{𝔞_{0},λ_{0}}H_{2,s}(M ( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} ))  (2.3.5)

= q^{− deg D} ∑_{λmodD} Θ_{𝔞_{0},λ_{0}}(( 0  1^{\frac{π_{∞}^{m}}{D} \frac{−(u+λ)}{D}} )) E_{s}(( 0  1^{\frac{π_{∞}^{m}}{D} \frac{u+λ}{D}} ))

where E_{s}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} ))

:= q^{(s+1) deg D+2s}q^{−m(s+1)}[E_{s}^{(1)}(( \begin{matrix}NDπ_{∞}^{m} & NDu \\ 0 & 1 \end{matrix} ))  (2.3.6)

+E_{s}^{(D)}(( \begin{matrix}Nπ_{∞}^{m} & Nu \\ 0 & 1 \end{matrix} )) δ_{λ_{0}DN} ϵ_{0}^{−1} [\frac{D}{N} ] q^{(− \frac{1}{2} −2s) deg D}].

2.3.2 deg D is even

We already mentioned that the factor ((−1)^{m−deg λ_{0}} + 1)/2 is invariant under the whole group GL_{2}(𝔽_{q}[T]). Therefore it is not affected by the trace. Proposition 2.2.3 yields the following.

Proposition 2.3.2 Let deg D be even, then

\frac{1}{1 + q^{−s−1}} L^{(N,D)}(2s + 1)L(f, A, s) = \frac{q}{2(q − 1)} ∫_{G(N)} f · \overline{Φ(e) ; s̄}

with

Φ_{s}^{(e)}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) := Φ_{s}^{(o)}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) \frac{(−1)^{m−deg λ_{0}} + 1}{2}.  (2.3.7)

2.4 Holomorphic Projection

We want to evaluate an integral ∫_{G(N)} f · \overline{Φ}, where f is our automorphic cusp form of Drinfeld type of level N (cf. section 2.1) and Φ is any function on G(N) = Γ_{0}^{(1)}(N)\GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗}. Since the Petersson product is nondegenerate on cusp forms, we find an automorphic cusp form Ψ of Drinfeld type for Γ_{0}^{(1)}(N) (one has to modify the definition of cusp forms to Γ_{0}^{(1)}(N) in an obvious way) such that

∫_{G(N)} g · \overline{Ψ} = ∫_{G(N)} g · \overline{Φ}

for all cusp forms g. If we set g = f we obtain our result. In this section we want to show how one can compute the Fourier coefficients of Ψ from those of Φ. We already noticed that only the coefficients Ψ^{∗}(π_{∞}^{deg λ+2}, λ) are important (cf. Remark 2.1.2). For this we take g = P_{λ}, where P_{λ} (λ ∈ 𝔽_{q}[T], λ ≠ 0) are the Poincaré series introduced in [Rü2], and evaluate (cf. [Rü2], Prop. 14)

∫_{G(N)} P_{λ} · \overline{Ψ} = \frac{4}{q − 1} \overline{Ψ^{∗}(π_{∞}^{deg λ+2}, λ)}.  (2.4.1)

On the other hand we calculate (with transformations as in the proof of [Rü2], Prop. 14)

∫_{G(N)} P_{λ} · \overline{Φ} = 2 lim_{σ→1} ∫_{ℍ_{∞}} g_{λ,σ} · \overline{(Φ − Φ̃)}  (2.4.2)

where

g_{λ,σ}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) := \{ \begin{matrix}0 & if deg λ + 2 > m \\ q^{−mσ}ψ_{∞}(λu) & if deg λ + 2 ≤ m \end{matrix}

Documenta Mathematica 5 (2000) 365–444

378 Hans-Georg Rück and Ulrich Tipp and where

Φ̃(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) := Φ(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} ) ( \begin{matrix}0 & 1 \\ π_{∞} & 0 \end{matrix} )).  (2.4.3)

For these calculations we used again the canonical mapping (cf. section 2.2)

ℍ_{∞} → G(N).

Since ℍ_{∞} represents only the part T_{+} of GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗} (cf. section 2.1) and since Φ is not necessarily harmonic, we also have to consider the function Φ̃. Using the Fourier expansions

Φ(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) = ∑_{μ} Φ^{∗}(π_{∞}^{m}, μ)ψ_{∞}(μu)
Φ̃(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) = ∑_{μ} Φ̃^{∗}(π_{∞}^{m}, μ)ψ_{∞}(μu)

and the character relations for ψ_{∞}, (2.4.2) yields

∫_{G(N)} P_{λ} · \overline{Φ} = \frac{2}{q} lim_{σ→0} ∑_{m=deg λ+2}^{∞} q^{−mσ}\overline{(Φ^{∗}(π_{∞}^{m}, λ) − Φ̃^{∗}(π_{∞}^{m}, λ))}.  (2.4.4)

Finally, (2.4.1) and (2.4.4) prove: Proposition 2.4.1 Let Φ : G(N) = Γ_{0}^{(1)}(N)\GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗} → ℂ be any function, then there is an automorphic cusp form Ψ of Drinfeld type for Γ_{0}^{(1)}(N) such that

∫_{G(N)} f · \overline{Ψ} = ∫_{G(N)} f · \overline{Φ}.

The Fourier coefficients of Ψ can be evaluated by the formula

Ψ^{∗}(π_{∞}^{deg λ+2}, λ) = \frac{q − 1}{2q} lim_{σ→0} ∑_{m=deg λ+2}^{∞} q^{−mσ}(Φ^{∗}(π_{∞}^{m}, λ) − Φ̃^{∗}(π_{∞}^{m}, λ)),

where Φ̃ is defined in (2.4.3). Problems could arise since the limit may not exist. We will see this in the following sections, where we apply this holomorphic projection formula to Φ_{s}^{(o)}, Φ_{s}^{(e)} (cf. (2.3.5) and (2.3.7)) or their derivatives.

Documenta Mathematica 5 (2000) 365–444

2.5 Fourier Expansions of Φ_{s}^{(o)} and Φ_{s}^{(e)}

In this section we evaluate the Fourier coefficients Φ_{s}^{(o)∗}(π_{∞}^{m}, λ) and Φ_{s}^{(e)∗}(π_{∞}^{m}, λ) (cf. (2.3.5) and (2.3.7)). The function Θ_{𝔞_{0},λ_{0}} is already defined by its coefficients r_{𝔞_{0},λ_{0}}. It remains to evaluate the coefficients of E_{s} (cf. (2.3.6)) and therefore of the Eisenstein series E_{s}^{(1)} (cf. (2.2.6)) and E_{s}^{(D)} (cf. (2.3.4)). We introduce a “basic function” on GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗}:

F_{s}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) = ∑_{λ} F_{s}^{∗}(π_{∞}^{m}, λ)ψ_{∞}(λu) := ∑_{\begin{matrix}d∈𝔽_{q}[T] \\ m>v_{∞}(u+d) \end{matrix}} δ_{u+d} q^{v_{∞}(u+d)(2s+1)}. (2.5.1)

We recall that δ_{z} is the local norm symbol of z at ∞. At first we express the Eisenstein series in terms of F_{s}. Elementary transformations give

E_{s}^{(1)}(( \begin{matrix}NDπ_{∞}^{m} & NDu \\ 0 & 1 \end{matrix} )) = ∑_{\begin{matrix}d∈𝔽_{q}[T] \\ d≠0 \end{matrix}} [\frac{D}{d} ] q^{−(2s+1) deg d} + δ_{D} q^{−(2s+1) deg D}

· ∑_{\begin{matrix}μ∈𝔽_{q}[T] \\ μ≠0 \end{matrix}} [ ∑_{\begin{matrix}c|μ \\ c≡0modD \end{matrix}} F_{s}^{∗}(cNπ_{∞}^{m}, \frac{μ}{c}) ∑_{dmodD} [\frac{d}{D} ] ψ_{∞}(\frac{μ}{c} \frac{d}{D})] ψ_{∞}(μNu).

The Gauss sum can be evaluated

∑_{dmodD} [\frac{d}{D} ] ψ_{∞}(λ\frac{d}{D}) = [\frac{λ}{D} ] ϵ_{0}^{−1}q^{\frac{1}{2} deg D},

where ϵ_{0} is as in (2.3.2). Therefore

E_{s}^{(1)}(( \begin{matrix}NDπ_{∞}^{m} & NDu \\ 0 & 1 \end{matrix} ))

= ∑_{\begin{matrix}d∈𝔽_{q}[T] \\ d≠0 \end{matrix}} [\frac{D}{d} ] q^{−(2s+1) deg d} + ϵ_{0}^{−1}δ_{D} q^{(−2s− \frac{1}{2} ) deg D} ·

· ∑_{μ≠0} ∑_{\begin{matrix}c|μ \\ c≡0modD \end{matrix}} [\frac{μ/c}{D}] F_{s}^{∗}(cNπ_{∞}^{m}, \frac{μ}{c})ψ_{∞}(μNu).  (2.5.2)

The same transformations as above yield

E_{s}^{(D)}(( \begin{matrix}Nπ_{∞}^{m} & Nu \\ 0 & 1 \end{matrix} )) = ∑_{\begin{matrix}d∈𝔽_{q}[T] \\ d≠0 \end{matrix}} [\frac{d}{D} ] F_{s}^{∗}(dNπ_{∞}^{m}, 0) +

+ ∑_{μ≠0} ∑_{c|μ} [\frac{c}{D} ] F_{s}^{∗}(cNπ_{∞}^{m}, \frac{μ}{c})ψ_{∞}(μNu).  (2.5.3)

Now we have to evaluate the Fourier coefficients of the “basic function” F_{s} (cf. (2.5.1)). This is not very difficult, though perhaps a little tedious to write down in detail. One starts with the definition of the coefficients

F_{s}^{∗}(π_{∞}^{m}, λ) = q^{−m+1} ∑_{u∈π_{∞}/π_{∞}^{m}} F_{s}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} ))ψ_{∞}(−λu)

and uses the character relations for ψ_{∞}. We do not carry it out in detail. As the local norm symbol δ_{z} behaves differently we have to distinguish again the two cases.

2.5.1 deg D is odd

L_{∞}/K_{∞} is ramified and the local norm symbol for z = e_{z}π_{∞}^{n} + … is given by δ_{z} = χ_{2}(e_{z}) δ_{T}^{n} (χ_{2} is the quadratic character on 𝔽_{q}^{∗}; we recall that π_{∞} = T^{−1}). We get:

Lemma 2.5.1 Let deg D be odd, then

F_{s}^{∗}(π_{∞}^{m}, μ) = \{ \begin{matrix}0 & , if either μ = 0 or deg μ + 2 > m \\ ϵ_{0}^{−1}q^{\frac{1}{2}} δ_{μ} q^{2s(deg μ+1)} & , if μ ≠ 0 and deg μ + 2 ≤ m. \end{matrix}

Now (2.5.2), (2.5.3), Lemma 2.5.1 and the definition of E_{s} in (2.3.6) give:

Proposition 2.5.2 Let deg D be odd, then

E_{s}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) = ∑_{\begin{matrix}μ∈𝔽_{q}[T] \\ deg(μN)+2≤m \end{matrix}} e_{s}(π_{∞}^{m}, μ)ψ_{∞}(μNu)

with

e_{s}(π_{∞}^{m}, 0) = q^{(s+1) deg D+2s−m(s+1)} ∑_{\begin{matrix}d∈𝔽_{q}[T] \\ d≠0 \end{matrix}} [\frac{D}{d} ] q^{−(2s+1) deg d}  (2.5.4)

and (μ ≠ 0)

e_{s}(π_{∞}^{m}, μ) = q^{(−s+ \frac{1}{2} ) deg D+4s+ \frac{1}{2} −m(s+1)+2s deg μ} ·  (2.5.5)

· ( ∑_{\begin{matrix}c|μ \\ c≡0modD \end{matrix}} [\frac{D}{μ/c}] q^{−2s deg c} + δ_{λ_{0}Nμ} [\frac{D}{N} ] ∑_{c|μ} [\frac{D}{c} ] q^{−2s deg c}).

2.5.2 deg D is even

L_{∞}/K_{∞} is inert and the local norm symbol for z = e_{z}π_{∞}^{n} + … is given by δ_{z} = (−1)^{n}. We get:

Heegner Points and L-Series 381

Lemma 2.5.3 Let deg D be even, then

F_{s}^{∗}(π_{∞}^{m}, 0) = \frac{1 − q}{q^{2s} + 1}(−q^{2s})^{m},

and (μ ≠ 0 with deg μ + 2 ≤ m)

F_{s}^{∗}(π_{∞}^{m}, μ) = \frac{(−q^{2s})^{deg μ+1}}{q^{2s} + 1} ((1 − q)(−q^{2s})^{m−deg μ−1} − 1 − q^{2s+1}).

Again (2.5.2), (2.5.3), Lemma 2.5.3 and the definition of E_{s} in (2.3.6) give: Proposition 2.5.4 Let deg D be even, then

E_{s}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) = ∑_{\begin{matrix}μ∈𝔽_{q}[T] \\ deg(μN)+2≤m \end{matrix}} e_{s}(π_{∞}^{m}, μ)ψ_{∞}(μNu)

with

e_{s}(π_{∞}^{m}, 0) = q^{deg D(s+1)−m(s+1)+2s}( ∑_{\begin{matrix}d∈𝔽_{q}[T] \\ d≠0 \end{matrix}} [\frac{D}{d} ] q^{−(2s+1) deg d}

+\frac{1 − q}{q^{2s} + 1}q^{deg D(− \frac{1}{2} −2s)+2sm−2s deg N}

(−1)^{deg λ_{0}+m} [\frac{D}{N} ] ∑_{\begin{matrix}d∈𝔽_{q}[T] \\ d≠0 \end{matrix}} [\frac{D}{d} ] q^{−2s deg d})  (2.5.6)

and (μ ≠ 0)

e_{s}(π_{∞}^{m}, μ) = q^{m(−s−1)+2s+deg D(−s+ \frac{1}{2} )} · (2.5.7)

(( ∑_{\begin{matrix}c|μ \\ c≡0modD \end{matrix}} [\frac{D}{μ/c}] q^{−2s deg c})(\frac{1 − q}{q^{2s} + 1}(−1)^{m−deg N−deg μ} q^{2s(m−deg N)}

+\frac{q^{2s+1} + 1}{q^{2s} + 1} q^{2s(deg μ+1)})

+ [\frac{D}{N} ] (∑_{c|μ} [\frac{D}{c} ] q^{−2s deg c})(\frac{1 − q}{q^{2s} + 1}(−1)^{deg λ_{0}+m} q^{2s(m−deg N)}

+\frac{q^{2s+1} + 1}{q^{2s} + 1}(−1)^{deg λ_{0}+deg N+deg μ} q^{2s(deg μ+1)})).

2.6 Fourier Expansions of Φ̃_{s}^{(o)} and Φ̃_{s}^{(e)} In accordance with (2.4.3) let Φ̃_{s}^{(o)} (resp. Φ̃_{s}^{(e)}) on GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗} be defined

as Φ̃_{s}^{(o)}(X) = Φ_{s}^{(o)}(X ( \begin{matrix}0 & 1 \\ π_{∞} & 0 \end{matrix} )) (resp. Φ̃_{s}^{(e)}(X) = Φ_{s}^{(e)}(X ( \begin{matrix}0 & 1 \\ π_{∞} & 0 \end{matrix} ))).

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The situation is more complicated than in the last section. To extend functions canonically from T_{+} (cf. (2.1.1)) to the whole of GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗} we need the following proposition.

Proposition 2.6.1 Let χ_{D} : (𝔽_{q}[T]/D 𝔽_{q}[T])^{∗} → ℂ^{∗} be a character modulo D and let χ_{∞} : K_{∞}^{∗} → ℂ^{∗} be a character which vanishes on the subgroup of 1-units O_{∞}^{(1)} = {x ∈ K_{∞}^{∗} | v_{∞}(x − 1) > 0}. Let F : T_{+} → ℂ be a function which satisfies

F (( \begin{matrix}a & b \\ c & d \end{matrix} ) ( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) = F (( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) χ_{D}(d) χ_{∞}(cu + d),

for all ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}^{(1)}(D) with ( \begin{matrix}a & b \\ c & d \end{matrix} ) ( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} ) ∈ T_{+}.

Then F can be defined on GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗} with

F (( \begin{matrix}a & b \\ c & d \end{matrix} ) ( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) = F (( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) χ_{D}(d) ·

· \{ \begin{matrix}χ_{∞}(cu + d) & , if v_{∞}(cπ_{∞}^{m}) > v_{∞}(cu + d) \\ χ_{∞}(c^{−1}) & , if v_{∞}(cπ_{∞}^{m}) ≤ v_{∞}(cu + d). \end{matrix}  (2.6.1)

Proof. We already know that ( \begin{matrix}a & b \\ c & d \end{matrix} ) ( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} ) ∈ T_{+} is equivalent to

v_{∞}(cπ_{∞}^{m}) > v_{∞}(cu + d) (cf. Remark 2.2.1). For each X ∈ GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗}

there is A ∈ Γ_{0}^{(1)}(D) and ( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} ) ∈ T_{+} such that X = A ( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} ) in

GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗}. Then we define F (X) by equation (2.6.1). The assumption on F guarantees that this definition is independent of the choice of A and

( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} ). □

We apply this proposition to Θ_{𝔞_{0},λ_{0}} (cf. (2.1.10)) and to the Eisenstein series. The Eisenstein series E_{s}^{(i)} (i = 1, D) (cf. (2.2.6), (2.3.4)) satisfy

E_{s}^{(i)}(( \begin{matrix}a & b \\ c & d \end{matrix} ) ( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) = E_{s}^{(i)}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) ·

· [\frac{d}{D} ] δ_{cu+d} q^{−v_{∞}(cu+d)(2s+1)}

if ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}^{(1)}(D) and v_{∞}(cπ_{∞}^{m}) > v_{∞}(cu + d). We can apply Proposition

2.6.1 with χ_{D}(d) = [\frac{d}{D} ] and χ_{∞}(z) = δ_{z} q^{−v_{∞}(z)(2s+1)}.

Hence

E_{s}^{(1)}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} ) ( \begin{matrix}0 & 1 \\ π_{∞} & 0 \end{matrix} )) = ∑_{\begin{matrix}c,d∈𝔽_{q}[T] \\ c≡0modD \\ v_{∞}(cπ_{∞}^{m})≤v_{∞}(cu+d) \end{matrix}} [\frac{d}{D} ] δ_{−c} q^{v_{∞}(c)(2s+1)}

Heegner Points and L-Series 383

and

E_{s}^{(D)}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} ) ( \begin{matrix}0 & 1 \\ π_{∞} & 0 \end{matrix} )) = ∑_{\begin{matrix}c,d∈𝔽_{q}[T] \\ v_{∞}(cπ_{∞}^{m})≤v_{∞}(cu+d) \end{matrix}} [\frac{c}{D} ] δ_{−c} q^{v_{∞}(c)(2s+1)}.

We denote these functions by Ẽ_{s}^{(1)} and { \atop E_{s}^{(D)}} as above. Starting with the definition of the Fourier coefficients we calculate

Ẽ_{s}^{(1)}(( \begin{matrix}NDπ_{∞}^{m} & NDu \\ 0 & 1 \end{matrix} ))

= ∑_{\begin{matrix}μ≠0 \\ deg(μN)+2≤m \end{matrix}} [ϵ_{0} q^{deg D(−2s− \frac{1}{2} )+deg N(−2s)+1−m} ·
·δ_{μND} ∑_{\begin{matrix}c|μ \\ c≡0modD \end{matrix}} [\frac{D}{μ/c}]q^{−2s deg c}] ψ_{∞}(μNu)  (2.6.2)

and

{ \atop E_{s}^{(D)}}(( \begin{matrix}Nπ_{∞}^{m} & Nu \\ 0 & 1 \end{matrix} )) = q^{deg N(−2s)+1−m}δ_{−N} ∑_{c≠0} [\frac{D}{c} ] q^{−2s deg c} +

+ ∑_{\begin{matrix}μ≠0 \\ deg(μN)+2≤m \end{matrix}} [q^{deg N(−2s)+1−m}δ_{−N} ∑_{c|μ} [\frac{D}{c} ] q^{−2s deg c}] ψ_{∞}(μNu). (2.6.3)

In addition we have { \atop q^{−m(s+1)}} = q^{−(1−m)(s+1)}, therefore (2.6.2) and (2.6.3) give: Proposition 2.6.2 Let deg D be odd or even, then

Ẽ_{s}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) := E_{s}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} ) ( \begin{matrix}0 & 1 \\ π_{∞} & 0 \end{matrix} )) = ∑_{\begin{matrix}μ∈𝔽_{q}[T] \\ deg(μN)+2≤m \end{matrix}} ẽ_{s}(π_{∞}^{m}, μ)ψ_{∞}(μNu)

with

ẽ_{s}(π_{∞}^{m}, 0) = q^{deg D(−s+ \frac{1}{2} )+deg N(−2s)+ms+s}ϵ_{0}^{−1} δ_{λ_{0}} [\frac{D}{N} ] ∑_{d≠0} [\frac{D}{c} ] q^{−2s deg d}

(2.6.4)

and (μ ≠ 0)

ẽ_{s}(π_{∞}^{m}, μ) = q^{deg D(−s+ \frac{1}{2} )+deg N(−2s)+ms+s}ϵ_{0}^{−1} ·  (2.6.5)

(δ_{μN} ∑_{\begin{matrix}c|μ \\ c≡0modD \end{matrix}} [\frac{D}{μ/c}] q^{−2s deg c} + δ_{λ_{0}} [\frac{D}{N} ] ∑_{c|μ} [\frac{D}{c} ] q^{−2s deg c}).

For Θ_{𝔞_{0},λ_{0}} it is not just straightforward calculation. In the following we make use of the fact that the Fourier coefficients r_{𝔞_{0},λ_{0}}(λ) of Θ_{𝔞_{0},λ_{0}} are independent of π_{∞}^{m} if deg λ + 2 ≤ m.

Θ_{𝔞_{0},λ_{0}} satisfies Proposition 2.6.1 with χ_{D}(d) = [\frac{d}{D} ] and χ_{∞}(z) = δ_{z} q^{−v_{∞}(z)}

(cf. (2.2.5)). Again we denote { \atop Θ_{𝔞_{0},λ_{0}}}(·) = Θ_{𝔞_{0},λ_{0}}(· ( \begin{matrix}0 & 1 \\ π_{∞} & 0 \end{matrix} )).

Let π_{∞}^{m} ∈ K_{∞}^{∗} and u ∈ K_{∞}. Choose c, d ∈ 𝔽_{q}[T] with c ≡ 0 mod D, gcd(c, d) = 1 and v_{∞}(u + \frac{d}{c}) ≥ m + 1 and find a, b ∈ 𝔽_{q}[T] with ad − bc = 1. Then for all k ∈ ℤ with k ≤ m + 1 there is the following identity in GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗}:

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

We use this identity for k = m and k = m + 1. Then Proposition 2.6.1 gives

{ \atop Θ_{𝔞_{0},λ_{0}}}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) − { \atop Θ_{𝔞_{0},λ_{0}}}(( \begin{matrix}π_{∞}^{m+1} & u \\ 0 & 1 \end{matrix} )) = [\frac{d}{D} ] δ_{−c} q^{v_{∞}(c)} ·

· ∑_{deg μ+2=1−m+2 deg c} r_{𝔞_{0},λ_{0}}(μ)ψ_{∞}(μ\frac{a}{c}). (2.6.6)

On the other hand we set u_{ϵ} = −\frac{d}{c} + ϵπ_{∞}^{m} for ϵ ∈ 𝔽_{q}^{∗}, we compare \frac{a}{c} with \frac{au_{ϵ}+b}{cu_{ϵ}+d} and sum over all ϵ:

(q − 1){ \atop Θ_{𝔞_{0},λ_{0}}}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) − [\frac{d}{D} ] δ_{−c} q^{v_{∞}(c)} ∑_{ϵ∈𝔽_{q}^{∗}} Θ_{𝔞_{0},λ_{0}}(( 0  1^{\frac{π_{∞}^{1−m}}{c^{2}}  \frac{au_{ϵ}+b}{cu_{ϵ}+d}} ))

= q [\frac{d}{D} ] δ_{−c} q^{v_{∞}(c)} ∑_{deg μ+2=1−m+2 deg c} r_{𝔞_{0},λ_{0}}(μ)ψ_{∞}(μ\frac{a}{c}). (2.6.7)

Now

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

in GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗}. We use this to evaluate the corresponding value of Θ_{𝔞_{0},λ_{0}}. A combination of (2.6.6) and (2.6.7) therefore gives

q { \atop Θ_{𝔞_{0},λ_{0}}}(( \begin{matrix}π_{∞}^{m+1} & u \\ 0 & 1 \end{matrix} )) − { \atop Θ_{𝔞_{0},λ_{0}}}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} ))

= δ_{−π_{∞}^{m}}q^{−m} ∑_{ϵ∈𝔽_{q}^{∗}} δ_{ϵ}Θ_{𝔞_{0},λ_{0}}(( \begin{matrix}π_{∞}^{m+1} & u + ϵπ_{∞}^{m} \\ 0 & 1 \end{matrix} )).  (2.6.8)

If we evaluate in (2.6.8) the Fourier coefficients at λ with deg λ + 2 ≤ m, we get the recursion formula

q { \atop Θ_{𝔞_{0},λ_{0}}}^{∗}(π_{∞}^{m+1}, λ) − { \atop Θ_{𝔞_{0},λ_{0}}}^{∗}(π_{∞}^{m}, λ) = δ_{π_{∞}^{m}}q^{−m} ∑_{ϵ∈𝔽_{q}^{∗}} δ_{ϵ}r_{𝔞_{0},λ_{0}}(λ). (2.6.9)

The Fourier coefficient in (2.6.8) at λ with deg λ + 2 = m + 1 yields

q { \atop Θ_{𝔞_{0},λ_{0}}}^{∗}(π_{∞}^{deg λ+2}, λ) = δ_{π_{∞}^{deg λ+1}}q^{− deg λ−1} ∑_{ϵ∈𝔽_{q}^{∗}} δ_{ϵ}ψ_{∞}(−λϵπ_{∞}^{deg λ+1})r_{𝔞_{0},λ_{0}}(λ).

(2.6.10)

For λ = 0 we calculate

{ \atop Θ_{𝔞_{0},λ_{0}}}^{∗}(π_{∞}, 0) = q^{− \frac{1}{2} deg D}δ_{λ_{0}}ϵ_{0}^{−1} ∑_{deg μ+2≤deg D} r_{𝔞_{0},λ_{0}}(μ).  (2.6.11)

It is now obvious how one evaluates { \atop Θ_{𝔞_{0},λ_{0}}}^{∗}(π_{∞}^{m}, λ) with the recursion formula (2.6.9) and the starting values (2.6.10) and (2.6.11). Here again we have to consider the two cases separately.

Proposition 2.6.3 Let deg D be odd, then

{ \atop Θ_{𝔞_{0},λ_{0}}}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) = ∑_{deg λ+2≤m} { \atop Θ_{𝔞_{0},λ_{0}}}^{∗}(π_{∞}^{m}, λ)ψ_{∞}(λu)

with

{ \atop Θ_{𝔞_{0},λ_{0}}}^{∗}(π_{∞}^{m}, λ) = q^{\frac{1}{2}} q^{−m}ϵ_{0}^{−1}δ_{λ_{0}}r_{𝔞_{0},λ_{0}}(λ).  (2.6.12)

Proposition 2.6.4 Let deg D be even, then

{ \atop Θ_{𝔞_{0},λ_{0}}}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) = ∑_{deg λ+2≤m} { \atop Θ_{𝔞_{0},λ_{0}}}^{∗}(π_{∞}^{m}, λ)ψ_{∞}(λu)

with

{ \atop Θ_{𝔞_{0},λ_{0}}}^{∗}(π_{∞}^{m}, λ) = q^{−m}(−1)^{deg λ_{0}}(\frac{q + 1}{2} + \frac{q − 1}{2}(−1)^{m+deg λ_{0}−1}) r_{𝔞_{0},λ_{0}}(λ). (2.6.13)

2.7 Functional Equations

In this section we modify the representations of the L-series of Proposition 2.3.1 and Proposition 2.3.2. With these new formulas we can prove functional equations for the L-series. Later we will use them to get our final results.

2.7.1 deg D is odd

Since f is an automorphic cusp form of Drinfeld type and therefore satisfies (cf. Definition 2.1.1)

f(X ( \begin{matrix}0 & 1 \\ π_{∞} & 0 \end{matrix} )) = −f(X) for all X ∈ GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗},

we can transform the integral in Proposition 2.3.1, and get:

Lemma 2.7.1 Let deg D be odd, then

L^{(N,D)}(2s + 1)L(f, A, s) = \frac{q}{4(q − 1)} ∫_{G(N)} f · \overline{F (o) ; s̄}

with

F_{s}^{(o)}(X) := Φ_{s}^{(o)}(X) − Φ̃_{s}^{(o)}(X),

whose Fourier coefficients are

F_{s}^{(o)∗}(π_{∞}^{m}, λ) = −F̃_{s}^{(o)}^{∗}(π_{∞}^{m}, λ) = Φ_{s}^{(o)∗}(π_{∞}^{m}, λ) − Φ̃_{s}^{(o)}^{∗}(π_{∞}^{m}, λ).

Now we evaluate F_{s}^{(o)∗}(π_{∞}^{m}, λ). We start with the definition (cf. Proposition 2.3.1)

Φ_{s}^{(o)}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) = q^{− deg D} ·
· ∑_{λmodD} Θ_{𝔞_{0},λ_{0}}(( 0  1^{\frac{π_{∞}^{m}}{D} \frac{−(u+λ)}{D}} )) E_{s}(( 0  1^{\frac{π_{∞}^{m}}{D} \frac{u+λ}{D}} ))

and use the Fourier coefficients of Θ_{𝔞_{0},λ_{0}} (cf. (2.1.10)) and E_{s} (cf. (2.3.6) and Proposition 2.5.2) to evaluate

Φ_{s}^{(o)∗}(π_{∞}^{m}, λ) = ∑_{\begin{matrix}μ∈𝔽_{q}[T] \\ deg(μN)+2≤m+deg D \end{matrix}} r_{𝔞_{0},λ_{0}}(μN − λD) e_{s}(π_{∞}^{m+deg D}, μ). (2.7.1)

On the other hand, if ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(𝔽_{q}[T]) with b, c ≡ 0 mod D and if u is

such that v_{∞}(u + d/c) ≥ m, then we have (using the transformation rules of Θ_{𝔞_{0},λ_{0}} and E_{s}):

Φ̃_{s}^{(o)}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) = Φ_{s}^{(o)}(( 0 1^{\frac{π_{∞}^{1−m}}{c^{2}} \frac{a}{c}} )).

When we expand this equation with Fourier coefficients, we get

Φ̃_{s}^{(o)}^{∗}(π_{∞}^{m}, λ) = ∑_{\begin{matrix}μ∈𝔽_{q}[T] \\ deg(μN)+2≤m+deg D \end{matrix}} { \atop Θ_{𝔞_{0},λ_{0}}}^{∗}(π_{∞}^{m+deg D}, μN − λD) ẽ_{s}(π_{∞}^{m+deg D}, μ) δ_{D}.

(2.7.2)

Now we replace e_{s}, { \atop Θ_{𝔞_{0},λ_{0}}}^{∗} and ẽ_{s} in (2.7.1) and (2.7.2) by (2.5.5), (2.6.12) and (2.6.5), and we get:

Proposition 2.7.2 Let deg D be odd and deg λ + 2 ≤ m, then

Φ_{s}^{(o)∗}(π_{∞}^{m}, λ) − Φ̃_{s}^{(o)}^{∗}(π_{∞}^{m}, λ) = r_{𝔞_{0},λ_{0}}(−λD) q^{−m− \frac{1}{2} deg D+ \frac{1}{2}} ·

(( ∑_{\begin{matrix}d∈𝔽_{q}[T] \\ d≠0 \end{matrix}} [\frac{D}{d} ] q^{−(2s+1) deg d}) q^{−ms+2s+ \frac{1}{2} deg D− \frac{1}{2}}

−( ∑_{\begin{matrix}d∈𝔽_{q}[T] \\ d≠0 \end{matrix}} [\frac{D}{d} ] q^{−2s deg d}) [\frac{D}{N} ] q^{ms−2s deg N+s})

+ ∑_{\begin{matrix}μ∈𝔽_{q}[T] , μ≠0 \\ deg(μN)+2≤m+deg D \end{matrix}} r_{𝔞_{0},λ_{0}}(μN − λD) q^{−m− \frac{1}{2} deg D+ \frac{1}{2}} ·

(( ∑_{\begin{matrix}c|μ \\ c≡0modD \end{matrix}} [\frac{D}{μ/c}] q^{−2s deg c})(q^{−2s deg D−ms+4s+2s deg μ} − δ_{λ_{0}Nμ} q^{ms−2s deg N+s})

+(∑_{c|μ} [\frac{D}{c} ] q^{−2s deg c})[\frac{D}{N} ](δ_{λ_{0}Nμ} q^{−2s deg D−ms+4s+2s deg μ} − q^{ms−2s deg N+s})).

With these formulas we prove the following result:

Theorem 2.7.3 Let deg D be odd, then

q^{(deg N+deg D− \frac{5}{2} )s} (Φ_{s}^{(o)∗}(π_{∞}^{m}, λ) − Φ̃_{s}^{(o)}^{∗}(π_{∞}^{m}, λ)) =

− [\frac{D}{N} ] q^{(deg N+deg D− \frac{5}{2} )(−s)} (Φ_{−s}^{(o)∗}(π_{∞}^{m}, λ) − Φ̃_{−s}^{(o)}^{∗}(π_{∞}^{m}, λ)),

and therefore Lemma 2.7.1 implies that

Z(s) := q^{(deg N+deg D− \frac{5}{2} )s}L^{(N,D)}(2s + 1)L(f, A, s)

satisfies the functional equation

Z(s) = − [\frac{D}{N} ] Z(−s).

Proof. One can verify the functional equation for Φ_{s}^{(o)∗}(π_{∞}^{m}, λ)−Φ̃_{s}^{(o)}^{∗}(π_{∞}^{m}, λ) independently for each summand (summation over μ ∈ 𝔽_{q}[T]) in the formula of Proposition 2.7.2 if one applies the following remarks:

a) For the first summand we mention (cf. [Ar]) that

L_{D}(s) := \frac{1}{q − 1} ∑_{\begin{matrix}d∈𝔽_{q}[T] \\ d≠0 \end{matrix}} [\frac{D}{d} ] q^{−s deg d}  (2.7.3)

is the L-series of the extension K(\sqrt{D})/K and satisfies

L_{D}(2s + 1) = q^{s(−2 deg D+2)− \frac{1}{2} deg D+ \frac{1}{2}} L_{D}(−2s).  (2.7.4)

b) Let μ ∈ 𝔽_{q}[T], μ ≠ 0 with r_{𝔞_{0},λ_{0}}(μN − λD) ≠ 0. Then there is κ ∈ L with

N_{L/K}(κ) = λ_{0}(μN − λD) (cf. (2.1.9)). Hence we get [\frac{D}{μ} ] = [\frac{D}{N} ] δ_{λ_{0}Nμ}. This

implies

∑_{c|μ} [\frac{D}{c} ] q^{−2s deg c} = q^{−2s deg μ} [\frac{D}{N} ] δ_{λ_{0}Nμ} ∑_{c|μ} [\frac{D}{c} ] q^{2s deg c}  (2.7.5)

if μ ≢ 0 mod D.

c) For μ ∈ 𝔽_{q}[T] with μ ≡ 0 mod D, it is easy to see that

∑_{c|μ} [\frac{D}{c} ] q^{−2s deg c} = q^{−2s deg μ} ∑_{\begin{matrix}c|μ \\ c≡0modD \end{matrix}} [\frac{D}{μ/c}] q^{2s deg c}. □  (2.7.6)

2.7.2 deg D is even

The automorphic cusp form f of Drinfeld type satisfies (cf. Definition 2.1.1)

∑_{β∈GL_{2}(O_{∞})/Γ_{∞}} f(Xβ) = 0 for all X ∈ GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗}.

With this identity a transformation of the integral in Proposition 2.3.2 yields immediately:

Lemma 2.7.4 Let deg D be even, then

\frac{1}{1 + q^{−s−1}} L^{(N,D)}(2s + 1)L(f, A, s) = \frac{q}{2(q − 1)} ∫_{G(N)} f · \overline{F (e) ; s̄}

with

F_{s}^{(e)}(X) := \frac{q}{q + 1} Φ_{s}^{(e)}(X) − \frac{1}{q + 1} ∑_{\begin{matrix}β∈GL_{2}(O_{∞})/Γ_{∞} \\ β≠1 \end{matrix}} Φ_{s}^{(e)}(Xβ),

whose Fourier coefficients are

F_{s}^{(e)∗}(π_{∞}^{m}, λ) = \{ \begin{matrix}\frac{q}{q+1}(Φ_{s}^{(e)∗}(π_{∞}^{m}, λ) − Φ̃_{s}^{(e)}^{∗}(π_{∞}^{m+1}, λ)) & , if m ≡ deg λ_{0} mod 2 \\ 0 & , if m ≢ deg λ_{0} mod 2 \end{matrix}

and

F̃_{s}^{(e)}^{∗}(π_{∞}^{m}, λ) = \{ \begin{matrix}0 & , if m ≡ deg λ_{0} mod 2 \\ \frac{1}{q+1}(Φ̃_{s}^{(e)}^{∗}(π_{∞}^{m}, λ) − Φ_{s}^{(e)∗}(π_{∞}^{m−1}, λ)) & , if m ≢ deg λ_{0} mod 2 \\ and deg λ + 2 < m \\ Φ̃_{s}^{(e)}^{∗}(π_{∞}^{m}, λ) & , if m ≢ deg λ_{0} mod 2 \\ and deg λ + 2 = m . \end{matrix}

The calculations of F_{s}^{(e)∗}(π_{∞}^{m}, λ) and F̃_{s}^{(e)}^{∗}(π_{∞}^{m}, λ) are similar to those above and use Propositions 2.5.4, 2.6.2 and 2.6.4 developed in the previous sections. We only give the results.

Proposition 2.7.5 Let deg D be even, then

Φ_{s}^{(e)∗}(π_{∞}^{m}, λ) − Φ̃_{s}^{(e)}^{∗}(π_{∞}^{m+1}, λ) = r_{𝔞_{0},λ_{0}}(−λD) q^{−m− \frac{1}{2} deg D} ·

(( ∑_{\begin{matrix}d∈𝔽_{q}[T] \\ d≠0 \end{matrix}} [\frac{D}{d} ] q^{−(2s+1) deg d}) q^{−ms+2s+ \frac{1}{2} deg D}

+( ∑_{\begin{matrix}d∈𝔽_{q}[T] \\ d≠0 \end{matrix}} [\frac{D}{d} ] q^{−2s deg d}) [\frac{D}{N} ] q^{ms−2s deg N+2s} \frac{−q^{1−s} − q^{s}}{q^{s} + q^{−s}} )

+ ∑_{\begin{matrix}μ∈𝔽_{q}[T] , μ≠0 \\ deg(μN)+2≤m+deg D \end{matrix}} r_{𝔞_{0},λ_{0}}(μN − λD) q^{−m− \frac{1}{2} deg D} ·

(( ∑_{\begin{matrix}c|μ \\ c≡0modD \end{matrix}} [\frac{D}{μ/c}] q^{−2s deg c})((−1)^{deg(λ_{0}Nμ)}q^{ms−2s deg N+2s} \frac{−q^{1−s} − q^{s}}{q^{s} + q^{−s}}

+q^{−ms−2s deg D+2s deg μ+4s} \frac{q^{−s} + q^{1+s}}{q^{s} + q^{−s}})

+(∑_{c|μ} [\frac{D}{c} ] q^{−2s deg c}) [\frac{D}{N} ] (q^{ms−2s deg N+2s} \frac{−q^{1−s} − q^{s}}{q^{s} + q^{−s}}

+(−1)^{deg(λ_{0}Nμ)}q^{−ms−2s deg D+2s deg μ+4s} \frac{q^{−s} + q^{1+s}}{q^{s} + q^{−s}})),

if m ≡ deg λ_{0} mod 2, and

Φ̃_{s}^{(e)}^{∗}(π_{∞}^{deg λ+2}, λ) = ∑_{\begin{matrix}μ∈𝔽_{q}[T] , μ≠0 \\ deg(μN)=deg(λD) \end{matrix}} r_{𝔞_{0},λ_{0}}(μN − λD) q^{− deg λ−1− \frac{1}{2} deg D} ·

( − ( ∑_{\begin{matrix}c|μ \\ c≡0modD \end{matrix}} [\frac{D}{μ/c}] q^{−2s deg c}) + [\frac{D}{N} ] (∑_{c|μ} [\frac{D}{c} ] q^{−2s deg c}))

· q^{s deg λ−2s deg N+3s}

if deg λ ≢ deg λ_{0} mod 2.

The proof of the following functional equation is completely analogous to the proof in the first case. Parts b) and c) in the proof of Theorem 2.7.3 are the same, part a) has to be replaced by the functional equation for deg D even 390 Hans-Georg Rück and Ulrich Tipp (cf. [Ar])

L_{D}(−2s + 1) = \frac{1 + q^{1−2s}}{1 + q^{2s}} q^{deg D(2s− \frac{1}{2} )}L_{D}(2s).  (2.7.7)

We get Theorem 2.7.6 Let deg D be even, then

q^{(deg N+deg D−3)s} (Φ_{s}^{(e)∗}(π_{∞}^{m}, λ) − Φ̃_{s}^{(e)}^{∗}(π_{∞}^{m+1}, λ)) =

− [\frac{D}{N} ] q^{(deg N+deg D−3)(−s)} (Φ_{−s}^{(e)∗}(π_{∞}^{m}, λ) − Φ̃_{−s}^{(e)}^{∗}(π_{∞}^{m+1}, λ)),

if m ≡ deg λ_{0} mod 2, and therefore Lemma 2.7.4 implies that

Z(s) := q^{(deg N+deg D−3)s} \frac{1}{1 + q^{−s−1}} L^{(N,D)}(2s + 1)L(f, A, s)

satisfies the functional equation

Z(s) = − [\frac{D}{N} ] Z(−s).

2.8 Derivatives of L-Series The functional equations in Theorem 2.7.3 and Theorem 2.7.6 show that the L-series have a zero at s = 0, if [\frac{D}{N} ] = 1. From now on we assume that [\frac{D}{N} ] = 1, and we want to compute the derivatives of the L-series at s = 0.

2.8.1 deg D is odd The first calculations are straightforward, we will only sketch this procedure. We start with the representation of L^{(N,D)}(2s + 1)L(f, A, s) in Lemma 2.7.1, then we evaluate the derivatives \frac{∂}{∂s}F_{s}^{(o)} |_{s=0} and \frac{∂}{∂s} F̃_{s}^{(o)} |_{s=0} from Proposition 2.7.2 by ordinary calculus. To simplify the formulas we introduce

t(μ, D) := \{ \begin{matrix}1 & , if μ ≡ 0 mod D \\ 0 & , if μ ≢ 0 mod D \end{matrix} ,  (2.8.1)

and we consider the function L_{D}(s) defined in equation (2.7.3). It is known that

h_{L} := #Cl(O_{L}) = L_{D}(0).

In addition we use equations (2.7.4), (2.7.5) and (2.7.6).

Documenta Mathematica 5 (2000) 365–444

Then we apply the holomorphic projection formula of Proposition 2.4.1 and evaluate

lim_{σ→0} ∑_{m=deg λ+2}^{∞} q^{−mσ}(\frac{∂}{∂s}F_{s}^{(o)∗}(π_{∞}^{m}, λ) |_{s=0} −\frac{∂}{∂s}F̃_{s}^{(o)}^{∗}(π_{∞}^{m}, λ) |_{s=0}).

In Proposition 2.7.2 there is a summation over μ ∈ 𝔽_{q}[T] with deg(μN) + 2 ≤ m + deg D. We divide this summation into two parts. The first sum is over those μ with deg(μN) ≤ deg(λD) and the second sum is over those μ with deg(μN) > deg(λD). This is done in view of the following lemma.

Lemma 2.8.1 Let μ ∈ 𝔽_{q}[T], μ ≠ 0 with r_{𝔞_{0},λ_{0}}(μN − λD) ≠ 0, then a)

\frac{1 − δ_{λ_{0}Nμ}}{2} (t(μ, D) − 1)(∑_{c|μ} [\frac{D}{c} ]) = 0

and b)

δ_{λ_{0}Nμ} = 1 if deg(μN) > deg(λD).

Proof. The proof is an immediate consequence of [\frac{D}{μ} ] = [\frac{D}{N} ] δ_{λ_{0}Nμ}, which

was shown in the proof of Theorem 2.7.3, part b). □

Now at the end of our calculations we have to apply the trace corresponding to Γ_{0}^{(1)}(N) ⊂ Γ_{0}(N) to get a cusp form of level N (and not just a cusp form for the subgroup Γ_{0}^{(1)}(N)). We recall that

∑_{ϵ∈𝔽_{q}^{∗}} r_{𝔞_{0},λ_{0}}(ϵμ) = (q − 1)r_{A}((μ)).

A heuristic consideration, based on the holomorphic projection formula of Proposition 2.4.1 and on our calculations, would then give:

\frac{∂}{∂s}(L^{(N,D)}(2s + 1)L(f, A, s)) |_{s=0}= ∫_{Γ_{0}(N)\GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗}} f · \overline{Ψ_{A}},

where Ψ_{A} is an automorphic cusp form of Drinfeld type of level N with

Ψ_{A}^{∗}(π_{∞}^{deg λ+2}, λ) = \frac{ln q}{2} q^{−(deg D+1)/2} q^{− deg λ}

·\{(q − 1) r_{A}((λ)) h_{L} (deg N − deg(λD) − \frac{q + 1}{2(q − 1)} − \frac{2}{ln q} \frac{L′_{D}(0)}{L_{D}(0)})

+ ∑_{\begin{matrix}μ≠0 \\ deg(μN)≤deg(λD) \end{matrix}} r_{A}((μN − λD))((∑_{c|μ} [\frac{D}{c} ]) (t(μ, D) + 1) \frac{1 + δ_{(μN−λD)μN}}{2} ·

· (deg(\frac{μN}{λD}) − \frac{q + 1}{2(q − 1)}) − (1 − δ_{(μN−λD)μN}) (∑_{c|μ} [\frac{D}{c} ]) deg μ

+ (1 − δ_{(μN−λD)μN}) (t(μ, D) + 1) (∑_{c|μ} [\frac{D}{c} ] deg c))

− \frac{q + 1}{2(q − 1)} lim_{σ→0} ∑_{deg(μN)>deg(λD)} r_{A}((μN − λD)) (∑_{c|μ} [\frac{D}{c} ]) (t(μ, D) + 1) ·

· q^{(−σ−1) deg( \frac{μN}{λD})} \frac{(q − 1)^{2}(q + q^{−σ})}{(q + 1)(q^{σ+1} − 1)^{2}} q^{(−σ) deg λ}\},  (2.8.2)

provided the limit exists. But unfortunately this is not the case. In order to get the final result, we proceed as follows:

1) We evaluate the pole of the limit in (2.8.2).

2) We find a function h on Γ_{0}(N)\GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗}, whose holomorphic projection formula gives the same pole part as in (2.8.2) and which is perpendicular to f under the Petersson product.

3) We replace \frac{∂}{∂s}F_{s}^{(o)} |_{s=0} by \frac{∂}{∂s}F_{s}^{(o)} |_{s=0} −h in the derivative of the equation in Lemma 2.7.1 and in our calculations. We start with 1): From section 3.5.1 we get the following result (independently of these calculations): Let C_{1} := 2(q − 1)^{2}/[GL_{2}(𝔽_{q}[T]) : Γ_{0}(N)], then the limit

lim_{σ→0} ( ∑_{deg(μN)>deg(λD)}r_{A}((μN − λD)) (∑_{c|μ} [\frac{D}{c} ]) (t(μ, D) + 1)

· q^{(−σ−1) deg( \frac{μN}{λD})} − C_{1} h_{L} (∑_{a|λ} q^{deg a})\frac{1}{1 − q^{−σ}} )

exists. But for this we have to adjust our assumptions. From now on N has

to be square free with [\frac{D}{P} ] = 1 for each prime divisor P of N and we only

consider those λ with gcd(λ, N) = 1.

We use this to calculate

lim_{σ→0} ( ∑_{deg(μN)>deg(λD)} r_{A}((μN − λD)) (∑_{c|μ} [\frac{D}{c} ]) (t(μ, D) + 1) q^{(−σ−1) deg( \frac{μN}{λD})}

·\frac{(q − 1)^{2}(q + q^{−σ})}{(q + 1)(q^{σ+1} − 1)^{2}} q^{(−σ) deg λ} − C_{1} h_{L} (∑_{a|λ} q^{deg a})\frac{q^{(−σ)(deg λ+2)}}{1 − q^{−σ}} )

= lim_{σ→0} ( ∑_{deg(μN)>deg(λD)} r_{A}((μN − λD)) (∑_{c|μ} [\frac{D}{c} ]) (t(μ, D) + 1) q^{(−σ−1) deg( \frac{μN}{λD})}

− C_{1} h_{L} (∑_{a|λ} q^{deg a})\frac{1}{1 − q^{−σ}} ) + (∑_{a|λ} q^{deg a}) · C_{2},  (2.8.3)

where C_{2} is a certain constant.

2) To find the function h we introduce for s > 1:

g_{0,s}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} )) := −g_{0,s}(( \begin{matrix}π_{∞}^{m} & u \\ 0 & 1 \end{matrix} ) ( \begin{matrix}0 & 1 \\ π_{∞} & 0 \end{matrix} )) := q^{−ms}

and the Eisenstein series

G_{s}(X) := ∑_{M} g_{0,s}(M · X),

where the sum is taken over M ∈ ( \begin{matrix}1 & 𝔽_{q}[T] \\ 0 & 1 \end{matrix} ) \SL_{2}(𝔽_{q}[T]). Then G_{s} is a

function on GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗}, which is invariant under SL_{2}(𝔽_{q}[T]) and which satisfies G̃_{s} = −G_{s}. In addition G_{s} is perpendicular to cusp forms. This can be shown analogously to calculations in the proof of [Rü2], Proposition 14 (in fact G_{s} can be seen as a Poincaré series for μ = 0). We evaluate the Fourier coefficients of G_{s} in a straightforward way (cf. proof of [Rü2], Proposition 8) and get for deg λ + 2 ≤ m, λ ≠ 0:

G_{s}^{∗}(π_{∞}^{m}, λ) = (∑_{a|λ} q^{−(2s−1) deg a})

·((1 − q^{2s}) q^{s(2 deg λ−m)−deg λ} + (q^{s} + 1)(q^{1−s} − 1) q^{s(m−2)+1−m}).

The coefficients G_{s}^{∗}(π_{∞}^{m}, 0) are not important, because they play no role in the holomorphic projection formula. Now we define the Eisenstein series G by its Fourier coefficients

G^{∗}(π_{∞}^{m}, λ) := lim_{s→1} G_{s}^{∗}(π_{∞}^{m}, λ),

and H by

H^{∗}(π_{∞}^{m}, λ) := lim_{s→1} \frac{∂}{∂s}G_{s}^{∗}(π_{∞}^{m}, λ).

In the next step we evaluate the holomorphic projection formulas for G and H and we get

lim_{σ→0} ∑_{m=deg λ+2}^{∞} q^{−mσ}(G^{∗}(π_{∞}^{m}, λ) − G̃^{∗}(π_{∞}^{m}, λ)) = −2(q + 1) q^{− deg λ−1}(∑_{a|λ} q^{deg a}),

(2.8.4)

and

lim_{σ→0} ∑_{m=deg λ+2}^{∞} q^{−mσ}(H^{∗}(π_{∞}^{m}, λ) − H̃^{∗}(π_{∞}^{m}, λ)) = −2(q + 1) q^{− deg λ−1} ln q

·( − (∑_{a|λ} q^{deg a}(deg λ − 2 deg a)) + (∑_{a|λ} q^{deg a}) lim_{σ→0} \frac{q^{(−σ)(deg λ+2)}}{1 − q^{−σ}}

−\frac{1}{q + 1}(∑_{a|λ} q^{deg a})). (2.8.5)

This construction is motivated by the fact that the limit in the last formula already occurred in equation (2.8.3).

3) Comparing (2.8.4) and (2.8.5) with (2.8.2) and (2.8.3) shows how to choose h = a · G + b · H with a, b ∈ ℂ to get the final result:

Theorem 2.8.2 Let D be irreducible of odd degree, and let N be square free

with [\frac{D}{P} ] = 1 for each prime divisor P of N. For a newform f of level N we

get:

\frac{∂}{∂s}(L^{(N,D)}(2s + 1)L(f, A, s)) |_{s=0}= ∫_{Γ_{0}(N)\GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗}} f · \overline{Ψ_{A}},

where Ψ_{A} is a cusp form of level N, whose Fourier coefficients for λ with gcd(λ, N) = 1 are given by

Ψ_{A}^{∗}(π_{∞}^{deg λ+2}, λ) = \frac{ln q}{2} q^{−(deg D+1)/2} q^{− deg λ}

·\{(q − 1) r_{A}((λ)) h_{L} (deg N − deg(λD) − \frac{q + 1}{2(q − 1)} − \frac{2}{ln q} \frac{L′_{D}(0)}{L_{D}(0)})

+ ∑_{\begin{matrix}μ≠0 \\ deg(μN)≤deg(λD) \end{matrix}}r_{A}((μN − λD))((∑_{c|μ} [\frac{D}{c} ]) (t(μ, D) + 1) \frac{1 + δ_{(μN−λD)μN}}{2}

· (deg(\frac{μN}{λD}) − \frac{q + 1}{2(q − 1)}) − (1 − δ_{(μN−λD)μN}) (∑_{c|μ} [\frac{D}{c} ]) deg μ

+ (1 − δ_{(μN−λD)μN}) (t(μ, D) + 1) (∑_{c|μ} [\frac{D}{c} ] deg c))

− \frac{q + 1}{2(q − 1)} lim_{s→0} ( ∑_{deg(μN)>deg(λD)}r_{A}((μN − λD)) (∑_{c|μ} [\frac{D}{c} ]) (t(μ, D) + 1)

· q^{(−s−1) deg( \frac{μN}{λD})} − \frac{C_{1} h_{L} (∑_{a|λ} q^{deg a})}{1 − q^{−s}})

− \frac{q + 1}{2(q − 1)} C_{1} h_{L} (∑_{a|λ} q^{deg a}(deg λ − 2 deg a)) + (∑_{a|λ} q^{deg a}) C_{2}\}.

The following notation is used: h_{L} = #Cl(O_{L}), L_{D}(s) is as in (2.7.3), t(μ, D) is as in (2.8.1),

C_{1} = 2(q − 1)^{2}/[GL_{2}(𝔽_{q}[T]) : Γ_{0}(N)],

and C_{2} is any constant (in particular independent of λ).

2.8.2 deg D is even

Of course the programme is the same as above. We start with Lemma 2.7.4, and we get the same pole as in (2.8.2) with different constants. Here we use the result (cf. section 3.5.2): Let C_{1} := (q^{2} − 1)^{2}/(2 q [GL_{2}(𝔽_{q}[T]) : Γ_{0}(N)]), then the limit

lim_{σ→0} ( ∑_{deg(μN)>deg(λD)} r_{A}((μN − λD)) (∑_{c|μ} [\frac{D}{c} ]) (t(μ, D) + 1)

· q^{(−σ−1) deg( \frac{μN}{λD})} − C_{1} h_{L} (∑_{a|λ} q^{deg a})\frac{1}{1 − q^{−σ}} )

converges.

Again we take the Eisenstein series G and H to get the final result:

Theorem 2.8.3 Let D be irreducible of even degree, and let N be square free

with [\frac{D}{P} ] = 1 for each prime divisor P of N. For a newform f of level N we

get:

\frac{∂}{∂s}(\frac{1}{1 + q^{−s−1}} L^{(N,D)}(2s + 1)L(f, A, s)) |_{s=0}= ∫_{Γ_{0}(N)\GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗}} f · \overline{Ψ_{A}},

where Ψ_{A} is a cusp form of level N, whose Fourier coefficients for λ with gcd(λ, N) = 1 are given by

Ψ_{A}^{∗}(π_{∞}^{deg λ+2}, λ) = \frac{ln q}{4} q^{− deg D/2} q^{− deg λ}

·\{r_{A}((λ)) h_{L} (q − 1) (deg N − deg(λD) − \frac{2q}{q^{2} − 1} − \frac{2}{ln q} \frac{L′_{D}(0)}{L_{D}(0)})

+ ∑_{\begin{matrix}μ≠0 \\ deg(μN)≤deg(λD) \end{matrix}}r_{A}((μN − λD))((∑_{c|μ} [\frac{D}{c} ]) (t(μ, D) + 1) \frac{1 + δ_{(μN−λD)μN}}{2}

· (deg(\frac{μN}{λD}) − \frac{2q}{q^{2} − 1}) − (1 − δ_{(μN−λD)μN}) (∑_{c|μ} [\frac{D}{c} ]) deg μ

+ (1 − δ_{(μN−λD)μN}) (t(μ, D) + 1) (∑_{c|μ} [\frac{D}{c} ] deg c))

− \frac{2q}{q^{2} − 1} lim_{s→0} ( ∑_{deg(μN)>deg(λD)}r_{A}((μN − λD)) (∑_{c|μ} [\frac{D}{c} ]) (t(μ, D) + 1)

· q^{(−s−1) deg( \frac{μN}{λD})} − C_{1} h_{L} (∑_{a|λ} q^{deg a})\frac{1}{1 − q^{−s}} )

− \frac{2q}{q^{2} − 1} C_{1} h_{L} (∑_{a|λ} q^{deg a}(deg λ − 2 deg a)) + (∑_{a|λ} q^{deg a}) C_{2}\}.

The following notation is used: h_{L} = #Cl(O_{L}), L_{D}(s) is as in (2.7.3), t(μ, D) is as in (2.8.1),

C_{1} = (q^{2} − 1)^{2}/(2 q [GL_{2}(𝔽_{q}[T]) : Γ_{0}(N)]),

and C_{2} is any constant (in particular independent of λ).

3 Heights of Heegner Points

3.1 Heegner Points

Let K = 𝔽_{q}(T) be the rational function field over 𝔽_{q} as in the previous chapters. For every N ∈ 𝔽_{q}[T] there exists a coarse moduli scheme Y_{0}(N) over 𝔽_{q}[T] parametrizing isomorphism classes of pairs (ϕ, ϕ′) of Drinfeld modules of rank 2 together with a cyclic isogeny u : ϕ → ϕ′ of degree N (cf. Lecture 2, [AB]). This means that ker u ≃ 𝔽_{q}[T]/(N). Y_{0}(N) can be compactified to a scheme X_{0}(N) by adjoining a finite number of sections. The points on these sections can be interpreted as generalized Drinfeld modules (cf. Lecture 9 [AB]). The fibres of X_{0}(N) → Spec 𝔽_{q}[T] are regular outside the divisors of N. We will also need the structure of the fibres over such places, which are known only for N square free. So we will assume this condition on N for the whole chapter. By abuse of notation we often write X_{0}(N) also for the generic fibre X_{0}(N) ⊗ K. For every λ ∈ 𝔽_{q}[T] there is a Hecke correspondence on X_{0}(N). If x ∈ X_{0}(N) is represented by two Drinfeld modules ϕ, ϕ′ and a cyclic isogeny u : ϕ → ϕ′, in which case we write x = (ϕ, ϕ′, u), then T_{λ}(x) = ∑_{C}(x_{C}), where C runs over all cyclic 𝔽_{q}[T] submodules of ϕ isomorphic to 𝔽_{q}[T]/(λ) which intersect ker u trivially. x_{C} is the point corresponding to (ϕ/C → ϕ′/u(C)). The Hecke algebra is the subalgebra of End J_{0}(N), the endomorphisms of the Jacobian of X_{0}(N), generated by the endomorphisms induced by the Hecke correspondences. For more details see for example [Ge3]. Now let L = K(\sqrt{D}) be an imaginary quadratic extension, where D is a polynomial in 𝔽_{q}[T]. In the first part of this section we prove results for general D, later we specialize to D being irreducible. We choose N ∈ 𝔽_{q}[T] such that each

of its prime divisors is split in L. Then in particular we have [\frac{D}{N} ] = 1. Suppose

that ϕ, ϕ′ are two Drinfeld modules of rank 2 for the ring 𝔽_{q}[T] with complex multiplication by an order O ⊂ O_{L} = 𝔽_{q}[T][\sqrt{D}], i.e. End ϕ = End ϕ′ = O and that u : ϕ → ϕ′ is a cyclic isogeny of degree N. Then ϕ and ϕ′ can be viewed as rank 1 Drinfeld modules over O. As explained in the paper ([Ha]) there is a natural action on rank 1 Drinfeld modules: If 𝔫 ⊂ O is an invertible ideal and ϕ is a rank 1 Drinfeld module then there is a Drinfeld module 𝔫 ∗ ϕ with an isogeny ϕ_{𝔫} : ϕ → 𝔫 ∗ ϕ. As was remarked in ([Ha]) just before Proposition 8.3, every isogeny is of this form up to isomorphism. The explicit class field theory ([Ha]) shows that ϕ, ϕ′ and the isogeny u can be defined (modulo isomorphisms) over the class field H_{O} of O, which is unramified outside the conductor 𝔣 := {α ∈ L : αO_{L} ⊂ O} of O and where ∞ is totally split. (For the maximal order O_{L} we will simply write H instead of H_{O_{L}}.) Therefore the triple (ϕ, ϕ′, u) defines an H_{O}−rational point x on X_{0}(N). This holds even though X_{0}(N) is not a fine moduli space. These rational points x are called Heegner points. We will primarily consider Heegner points for the maximal order O_{L} but Heegner points corresponding to non-maximal orders will occur naturally, when we consider the operation of Hecke operators on the Heegner points.

Heegner points corresponding to a maximal order can also be described by the following data: Let K_{∞} be the completion of K at ∞ and let C_{∞} be the completion of the algebraic closure of K_{∞}. The category of Drinfeld modules of rank 2 over C_{∞} is equivalent to the category of rank 2 lattices in C_{∞}. If ϕ, ϕ′ correspond to lattices Λ, Λ′, the isogenies are described by {c ∈ C_{∞}^{∗} : cΛ ⊂ Λ′}. If ϕ has complex multiplication by O_{L}, the corresponding lattice is isomorphic to an ideal 𝔞 in O_{L}. Now let 𝔫|N be an ideal of O_{L} which contains exactly one prime divisor of every conjugated pair over the primes dividing N. If 𝔫|𝔞, the ideal 𝔫^{−1}𝔞 is integral and corresponds to another Drinfeld module ϕ′ with complex multiplication. The inclusion 𝔞 ⊂ 𝔫^{−1}𝔞 defines a cyclic isogeny of degree N, because 𝔫^{−1}𝔞/𝔞 ≃ O_{L}/𝔫 ≃ 𝔽_{q}[T]/(N). The data describing the Heegner point x is the ideal class of 𝔞 and the ideal 𝔫. We get the following analytic realization of the Heegner point x. Let Ω = C_{∞}−K_{∞} be the Drinfeld upper half plane. Then X_{0}(N) is analytically given by the quotient Γ_{0}(N) \ Ω compactified by adjoining finitely many cusps. Let z ∈ Ω with

z = \frac{B + \sqrt{D}}{2A} , N|A, B^{2} ≡ D mod A.

Then the lattice ⟨z, 1⟩ is isomorphic to the ideal 𝔞 = A𝔽_{q}[T] + (B + \sqrt{D})𝔽_{q}[T], which defines together with the ideal 𝔫 = N𝔽_{q}[T] + (β + \sqrt{D})𝔽_{q}[T] with β ≡ B mod N a Heegner point. Now we consider the global Néron-Tate height pairing on the H-rational points of the Jacobian J_{0}(N) of X_{0}(N). There is an embedding of J_{0}(N) in the projective space ℙ^{2^{g}−1} (Kummer embedding), where g is the genus of X_{0}(N). The naive height on points in the projective space defines a height function h on J_{0}(N)(H). The Néron-Tate height is the unique function ĥ, which differs from h by a bounded function and such that the map ⟨ . ⟩ : J_{0}(N)×J_{0}(N) → ℝ defined by ⟨D, E⟩ = (1/2)(ĥ(D + E) − ĥ(D) − ĥ(E)) is bilinear. ⟨ . ⟩ is called the Néron-Tate height pairing (cf. [Gr1]). The pairing depends on H although we omit this in the notation. Whenever we consider height pairings over other fields than H, it will be explicitly mentioned. The global pairing can be written as a sum ∑_{v}⟨ . ⟩_{v} running over all places v of H. For an irreducible polynomial P ∈ 𝔽_{q}[T] we write ⟨ . ⟩_{P} for ∑_{v|P} ⟨ . ⟩_{v}. For the definition of the local pairing see [Gr1, 2.5]. We recall the relation of the local pairing at non-archimedian primes with the intersection product on a regular model (see [Gr1, 3]). Let v be some place of H and let H_{v} be the completion with valuation ring O_{v}. We write q_{v} for the cardinality of the residue field. Let X/H_{v} be a curve and X /O_{v} be a regular model of X. Suppose D, E are divisors of degree 0 on X_{0}(N) with disjoint support. Let F_{i} be the fibre components of the special fibre of the regular model X and let D̃, Ẽ be the horizontal divisors to D, E. (The horizontal divisor of a point in the generic fibre is just the Zariski closure of it in X.) Let ( .)_{v} be the intersection product on X, which is defined in the following way. Let x ≠ y be two distinct points on X and x̃, ỹ their closure in X. For a point z in the special fibre we consider the stalk O_{X ,z} of the structure sheaf in z. Let f_{x}, f_{y} be local equations for x̃, ỹ in z. Then O_{X ,z}/(f_{x}, f_{y}) is a module of finite length. The intersection number (x̃ . ỹ)_{v,z} is defined to be the length of the module O_{X ,z}/(f_{x}, f_{y}) and it is zero for almost all z. Let deg z be the degree of the residue field in z over the residue field of v. The intersection number is then (x̃ . ỹ)_{v} = ∑_{z}(x̃ . ỹ)_{v,z} · deg z. Now return to the divisors D, E of degree 0. There exist α_{i} ∈ ℚ such that

⟨D, E⟩_{v} = − ln q_{v}[(D̃ . Ẽ)_{v} + ∑_{i} α_{i}(F_{i} . Ẽ)_{v}]  (3.1.1)

cf. [Gr1, 3]. The elements α_{i} are unique up to an additive constant, independent of i. In particular if (D̃ . F_{i})_{v} = 0 for all i, the equation (3.1.1) is satisfied with α_{i} = 0 for all i. Let x = (ϕ, ϕ′, u) be a Heegner point on X_{0}(N) for the maximal order O_{L}. We denote by σ_{A} the element in the Galois group of H/L which corresponds via class field theory to A ∈ Cl(O_{L}). Then x^{σ_{A}} is again a Heegner point for the maximal order. The cusps are given by the cosets Γ_{0}(N) \ ℙ^{1}(K) and they are K-rational. If deg N > 0 we have at least the two different cusps 0 and ∞. We get the divisors (x) − (∞) and (x)^{σ_{A}} − (0) of degree 0 on X_{0}(N). Let T_{λ} be a Hecke operator and let g be an automorphic cusp form of Drinfeld type of level N (cf. Definition (2.1.1). If we associate to (T_{λ}, g) the Fourier coefficient (T_{λ}g)^{∗}(π_{∞}^{2}, 1), we get a bilinear map between the Hecke algebra and the space of cusp forms of level N. This map is a non-degenerate pairing ([Ge3, Thm. 3.17]). For gcd(λ, N) = 1 we have

(T_{λ}g)^{∗}(π_{∞}^{2}, 1) = q^{deg λ}g^{∗}(π_{∞}^{deg λ+2}, λ).

This is the key to the proof of the following proposition as in [Gr-Za, V 1]

Proposition 3.1.1 There is an automorphic cusp form g_{A} of Drinfeld type of level N such that

⟨(x) − (∞), T_{λ}((x)^{σ_{A}} − (0))⟩ = q^{deg λ}g_{A}^{∗}(π_{∞}^{deg λ+2}, λ)

for all λ ∈ 𝔽_{q}[T] with gcd(λ, N) = 1.

We want to compare g_{A} with the cusp form Ψ_{A} of the previous section. Therefore we have to evaluate this global height pairing. As we compare the cusp forms only up to old forms, it suffices to calculate the height pairings above only for the Hecke operators with gcd(λ, N) = 1. Thus we restrict to this case in the whole section. The first objective of this section is to express the intersection number of the Heegner divisors on X_{0}(N) at the finite places, i.e., those places corresponding to irreducible polynomials in 𝔽_{q}[T], by numbers of homomorphisms between the corresponding Drinfeld modules (Theorem 3.3.4). For a place v of H we write H_{v} for the completion at v and O_{v} for the valuation ring. Let W be the completion of the maximal unramified extension of O_{v} and π a uniformizing element of O_{v} (W, resp.). In order to calculate the local pairings

⟨(x) − (∞) , T_{λ}((x)^{σ_{A}} − (0))⟩_{v}

we first describe the divisor T_{λ}x^{σ_{A}}.

3.2 The divisor T_{λ}x^{σ_{A}}

Definition 3.2.1 If x = (ϕ, ϕ′, u) and y = (ψ, ψ′, v) are two points on X_{0}(N), where ϕ, ϕ′, ψ, ψ′ are Drinfeld modules of rank 2 and u : ϕ → ϕ′ and v : ψ → ψ′ are cyclic isogenies of degree N, we define

Hom _{R}(x, y) := {(f, f′) ∈ Hom _{R}(ϕ, ψ) × Hom _{R}(ϕ′, ψ′) : vf = f′u}

for any ring R where this is well defined, e.g. for R a local ring with algebraically closed residue field.

Consider a finite place v of H. Let H_{v} be the completion of H at v and let O_{v} be the valuation ring. Let W again be the completion of the maximal unramified extension of O_{v}.

Lemma 3.2.2 Let x = (ϕ, ϕ′, ϕ_{𝔫}) be a Heegner point for the maximal order and 𝔞 an integral ideal in the class A, which corresponds to σ_{A} ∈ Gal (H/L) under the Artin homomorphism. Then

Hom _{W} (x^{σ_{A}}, x) ≃ End _{W} (x) · 𝔞  (3.2.1)

and

Hom _{W/π^{n}}(x^{σ_{A}}, x) ≃ End _{W/π^{n}}(x) · 𝔞  (3.2.2)

for every n ≥ 1 as left modules over the prevailing ring of endomorphisms.

Proof. It is enough to show the second assertion for all n, because for n sufficiently big the second and the first assertion coincide. We show the assertion for ϕ instead of x. To show it for x one only has to remark that the morphism defined below is compatible with the morphism ϕ_{𝔫}. We again assume that ϕ is defined over W and has leading coefficients in \overline{𝔽_{q}}^{∗}. For brevity we write R_{n} := End _{W/π^{n}}(ϕ). Let Λ be the fraction field of W and I_{𝔞} be the left ideal in Λ{τ} generated by all ϕ_{a} with a ∈ 𝔞. This ideal is left principal and generated by some ϕ_{𝔞} ∈ W {τ} ([Ha], Prop. 7.5). So I_{𝔞} ∩ R_{n} is a left ideal in R_{n} and we shall show that it is equal to the left ideal R_{n}𝔞. The inclusion R_{n}𝔞 ⊂ I_{𝔞} ∩ R_{n} is trivial. For the other inclusion we shall show (I_{𝔞} ∩ R_{n})𝔞^{−1} ⊂ R_{n}. Without loss of generality we shall assume that the image under the natural inclusion of 𝔞 is not divisible by π. Then for every b ∈ 𝔞^{−1} there is a twisted power series ϕ_{b} in W {{τ}} such that for a ∈ 𝔞 we get ϕ_{a}ϕ_{b} = ϕ_{ab}. Now let f ∈ I_{𝔞} ∩ R_{n} and b ∈ 𝔞^{−1}, then f = ∑ f_{i}ϕ_{a_{i}}, for some f_{i} ∈ W {τ}. So fϕ_{b} = ∑ f_{i}ϕ_{a_{i}b}, which is a polynomial, because a_{i}b ∈ O. We also have ϕ_{b}ϕ_{a} = ϕ_{a}ϕ_{b} for every a ∈ 𝔽_{q}[T], and therefore fϕ_{b}ϕ_{a} ≡ ϕ_{a}fϕ_{b} mod π^{n}. This implies f · b ∈ R_{n}. We know from (Thm. 8.5[Ha]) that there exists a w ∈ W^{∗} such that

ϕ_{𝔞}ϕ_{a} = w^{−1}ϕ_{a}^{σ_{A}}wϕ_{𝔞}

holds for every a ∈ O, i.e. wϕ_{𝔞} ∈ Hom _{W} (ϕ, ϕ^{σ}). Now define an R_{n}−module homomorphism from Hom _{W/π^{n}}(ϕ^{σ_{A}}, ϕ) to R_{n} ∩ I_{𝔞} by the assignment f ↦ f · wϕ_{𝔞}. On the other hand if g ∈ W/π^{n}{τ} such that gwϕ_{𝔞} =: u ∈ R_{n}, then we have to show that g ∈ Hom _{W/π^{n}}(ϕ^{σ}, ϕ). We have

gϕ_{a}^{σ}wϕ_{𝔞} = gwϕ_{𝔞}ϕ_{a} = uϕ_{a} = ϕ_{a}u

for all a ∈ 𝔽_{q}[T], where the last equality holds, because 𝔽_{q}[T] is central in R_{1} and therefore also in R_{n} for every n ≥ 1. But ϕ_{a}u = ϕ_{a}gwϕ_{𝔞} and so

(gϕ_{a}^{σ} − ϕ_{a}g)wϕ_{𝔞} = 0.

wϕ_{𝔞} cannot be a zero divisor, because the leading coefficient of ϕ_{𝔞} is 1. This finishes the proof of the lemma. □ From this lemma we get the following result about the multiplicity of x in T_{λ}x^{σ_{A}}. The proof is exactly the same as in the characteristic 0 case ([Gr-Za, (4.3)]).

Proposition 3.2.3 Let σ_{A} ∈ Gal (H/L), let A be the ideal class corresponding to σ_{A} and let λ ∈ 𝔽_{q}[T]. Then the multiplicity of x in the divisor T_{λ}x^{σ_{A}} is equal to the number r_{A}((λ)) of integral ideals in the class of A with norm (λ).

The points y ∈ T_{λ}x^{σ_{A}} are Heegner points for orders O_{y} ⊂ O_{L}. Let 𝔣 = {α ∈ L : αO_{L} ⊂ O_{y}} be the conductor of the order O_{y}. Let P be an irreducible, monic polynomial in 𝔽_{q}[T] and let s = s(y, P) be the greatest integer with P^{s}|𝔣. We call s the level of y at P. If P ∤ λ, we get s = 0, because 𝔣|(λ). If λ = P^{t} · R with P ∤ R and t > 0, then

T_{λ}x^{σ_{A}} = ∑_{z∈T_{R}x^{σ_{A}}} T_{P^{t}}z.

The following proposition tells us how often each level occurs in the divisor T_{P^{t}}z. For a proof see [Ti1].

Proposition 3.2.4 Let P ∈ 𝔽_{q}[T] be irreducible and let z be a Heegner point of level 0 at P. Set d = deg P and q_{P} = q^{d}. Then the number of points of level s in the divisor T_{P^{t}}z is equal to

\begin{matrix}(t − s + 1)(q_{P}^{s} − q_{P}^{s−1}) & for t ≥ s ≥ 1 \\ t + 1 & for s = 0 \end{matrix} \} if P is split in L/K;
\begin{matrix}q_{P}^{s} + q_{P}^{s−1} & for t ≥ s ≥ 1, s ≡ t mod 2 \\ 1 & for s = 0, t ≡ 0 mod 2 \end{matrix} \} if P is inert in L/K;
q_{P}^{s} for t ≥ s ≥ 0 if P is ramified in L/K.

The next proposition shows where the points with level s are defined. The proof is given by D. Hayes ([Ha, Thm 8.10, Thm. 1.5])

Proposition 3.2.5 Let P be any irreducible polynomial and let z be a Heegner point for an order O with conductor prime to N. Suppose that z has level 0 at P. Then

1. z is defined over H_{O}, the ring class field of O, which is unramified over H at P. The Galois group of H_{O}/H is isomorphic to the group of ideals in O_{L} modulo the principal ideals generated by some element a ∈ O which is prime to the conductor of O.

2. Every y ∈ T_{P^{t}}z of level s at P is defined over another class field H_{s} with [H_{s} : H_{O}] = |(O_{L}/P^{s}O_{L})^{∗}|/|(𝔽_{q}[T]/P^{s}𝔽_{q}[T])^{∗}|, which is totally ramified at P over H_{O}.

3.3 The finite places

For the calculations of height pairings at the finite places we want to make use of the modular interpretation of the points on the modular curve in every fibre including the fibres over the divisors of N. In contrast to the elliptic curve case, we do not know how these fibres look like if N is not square free. This is one reason why we confine ourselves to this case. The first step is to describe the pairings at a finite place v by intersection products on a regular model of X_{0}(N) ⊗ K. When v ∤ N then X_{0}(N) ⊗ O_{v} is a regular model and when v|N we take a regularization of X_{0}(N) ⊗ O_{v}, which can be done by finitely many blow ups at the singular points. After that we use the modular interpretation to describe the intersection numbers by numbers of homomorphisms. First we recall the structure of the fibres of X_{0}(N) at the places over N (see [Ge2]).

Proposition 3.3.1 For N ∈ 𝔽_{q}[T] square free, N ∉ 𝔽_{q}, the modular curve X_{0}(N) over 𝔽_{q}[T] is regular outside N and outside the supersingular points in the fibres above prime divisors of N. Let P be any prime divisor of N of degree d. Then the special fibre over P consists of two copies of X_{0}(N/P), which intersect transversally in the supersingular points. One of the components is the image of the map

X_{0}(N/P) × 𝔽_{q}[T]/P ⟶ X_{0}(N) × 𝔽_{q}[T]/P
(ϕ, ϕ′, u) mod P ⟼ (ϕ, ϕ′′, τ^{d}u) mod P,

where τ^{d} is the Frobenius of 𝔽_{q}[T]/P regarded as isogeny of degree P. This component is the “local component”, the other one is the “reduced component”. The cusp 0 lies on the reduced component and ∞ lies on the local component.

Remarks. 1. We need not know what the resolutions of singular points are, because our horizontal divisors always intersect the fibres over N outside the supersingular points and the next proposition will show that no contribution from the fibre components of the regular model will occur.

2. Because gcd(D, N) = 1, the regular model remains regular under base change to the Hilbert class field H/L as well as over the completion of the maximal unramified extension W of some completion O_{v} for a place v of H.

Proposition 3.3.2 Let x = (ϕ, ϕ′, ϕ_{𝔫}) be a Heegner point for the maximal order, σ_{A} ∈ Gal (H/L), A the corresponding ideal class, λ ∈ 𝔽_{q}[T], v a finite place of H of residue cardinality q_{v}. Suppose that r_{A}((λ)) = 0, then

⟨(x) − (∞), T_{λ}((x)^{σ_{A}} − (0))⟩_{v} = −(x . T_{λ}x^{σ_{A}})_{v} ln q_{v}.

Proof. At first we check that the horizontal extension of one of the divisors in the pairing has zero intersection with both fibre components if v|N. It follows that the values α_{i} in 3.1.1 all vanish. Let 𝔫 be the ideal, such that ϕ_{𝔫} is the cyclic isogeny defining x. If v|𝔫, x intersects the fibre in the local component. If v|\overline{𝔫} then it intersects in the reduced component. Thus one of the divisors (x) − (∞), (x) − (0) has zero intersection with both fibre components. But the points in T_{λ}x^{σ_{A}} reduce to the same component as x. This shows that one of the divisors has zero intersection with all fibre components. This is trivially true for the places of good reduction (v ∤ N). The result now follows by linearity of the pairing and the fact that x can be represented by a Drinfeld module with good reduction, so it does not intersect with the cusps. □

Proposition 3.3.3 Let x = (ϕ → ϕ′), y = (ψ → ψ′) be two W −rational sections, i.e. horizontal divisors on X_{0}(N) over W, which intersect properly and which reduce to regular points outside the cusps in the special fibre. Then

(y . x)_{v} = \frac{1}{q − 1} ∑_{n≥1} #Isom _{W/π^{n}}(y, x).

Proof. Let f : Y ⟶ X_{0}(N) be a fine moduli scheme (e.g. a supplementary full level N′− structure with gcd(N′, N) = 1 and N′ with at least two different prime factors.) Let y_{0} be a pre-image of y and x_{i} the different pre-images of x, i.e. f_{∗}(y_{0}) = y , f^{∗}(x) = ∑ x_{i}. Because f is proper, the projection formula [Sha, Lect.6,(7)] implies that

(y . x)_{v} = (f_{∗}y_{0} . x)_{v} = (y_{0} . f^{∗}x)_{v} = ∑_{i}(y_{0} . x_{i})_{v}.

If (ϕ, ϕ′, u) is a representative of x, all the x_{i} are represented by (ϕ, ϕ′, u, P, Q), where P, Q generates the N′ torsion module. Every such point occurs with multiplicity #Aut(x)/(q − 1) in f^{∗}(x). The q − 1 trivial automorphisms all give the same point in f^{∗}(x). Now let (ψ, ψ′, v) be a representative of y and (ϕ, ϕ′, u) a representative of x. Let y_{0} be represented by (ψ, ψ′, v, P, Q).

Then an isomorphism f : (ψ, ψ′, v) → (ϕ, ϕ′, u) defines an isomorphism f̂ : (ψ, ψ′, v, P, Q) → (ϕ, ϕ′, u, f(P), f(Q)) and this is uniquely determined. If x_{i_{0}} is the class of (ϕ, ϕ′, u, f(P), f(Q)), we have

#Isom _{W/π^{n}}(y_{0}, x_{i}) = \{ \begin{matrix}1 & , if x_{i} = x_{i_{0}} \\ 0 & , otherwise. \end{matrix}

Now x_{i_{0}} occurs in f^{∗}(x) with multiplicity #Aut(x)/(q − 1) and therefore

∑_{i} #Isom _{W/π^{n}}(y_{0}, x_{i}) = \{ \begin{matrix}\frac{#Aut(x)}{q − 1} & , if #Isom _{W/π^{n}}(y, x) ≠ 0 \\ 0 & , otherwise \end{matrix}

= \frac{1}{q − 1}#Isom _{W/π^{n}}(y, x).

Therefore we only have to show that

(y_{0} . x_{i})_{v} = ∑_{n≥1} #Isom _{W/π^{n}}(y_{0}, x_{i}).

Let Y ↪ ℙ_{W}^{r} be a projective embedding. Let 𝔸_{W}^{r} be an affine part, which contains the intersection point s of x_{i} and y_{0}. The coordinates with respect to this affine part are denoted by y_{0} = (η_{1}, …, η_{r}) and x_{i} = (ξ_{i1}, …, ξ_{ir}). In the local ring O_{Y,s} we have the local functions z_{j} − η_{j}, z_{j} − ξ_{ij}, respectively. The ideal generated by these functions contains all differences (η_{j} − ξ_{ij}) and therefore is the ideal (π)^{k} with k = min_{j} v(η_{j} − ξ_{ij}). From the definition of the intersection number we get

(y_{0} . x_{i})_{v} = dim_{W/π}(O_{Y,s}/(z_{j} − η_{j}, z_{j} − ξ_{ij})) = dim_{W/π}(W/π^{k}W) = k.

On the other hand

#Isom _{W/π^{n}}(y_{0}, x_{i}) = \{ \begin{matrix}0, & η_{j} ≢ ξ_{ij} mod π^{n} for some j \\ 1, & η_{j} ≡ ξ_{ij} mod π^{n} for all j, \end{matrix}

because Y is a fine moduli scheme. It follows that

∑_{n≥1} #Isom _{W/π^{n}}(y_{0}, x_{i}) = k.

□

The degree of an isogeny u between two Drinfeld modules ϕ, ϕ′ is by definition the ideal IJ, if ker u ≃ 𝔽_{q}[T]/I ⊕ 𝔽_{q}[T]/J.

Theorem 3.3.4 Let P ∈ 𝔽_{q}[T] be irreducible , v|P a place of H with local parameter π and W the completion of the maximal unramified extension of O_{v}. Let x = (ϕ, ϕ′, u) be a Heegner point for the maximal order O_{L}. Let σ_{A} ∈ Gal (H/L). For λ, N without common divisor and r_{A}((λ)) = 0 we get

(x . T_{λ}x^{σ_{A}})_{v} = \frac{1}{q − 1} ∑_{n≥1} #Hom _{W/π^{n}}(x^{σ_{A}}, x)_{deg λ}.

The subscript deg λ indicates that only homomorphisms of degree λ are counted. The sum is finite because #Hom _{W/π^{n}}(x^{σ_{A}}, x)_{deg λ} = 0 for n sufficiently large, because Hom _{W} (x^{σ_{A}}, x) = ⋂_{n≥1} Hom _{W/π^{n}}(x^{σ_{A}}, x) and #Hom _{W} (x^{σ_{A}}, x)_{deg λ} = r_{A}((λ)) = 0 by assumption.

The rest of this section is used to prove this theorem. First of all we consider the easiest case, namely P ∤ λ. For the case P |λ we need the Eichler-Shimura congruence and a congruence between points of level 0 and points of higher level. After that the formula of the theorem is proved at first for P split, then for P inert and finally for P ramified. Suppose now that P ∤ λ. We have

\frac{d}{dt}ϕ_{λ}(t) = λ ≢ 0 mod P,

and so the zeroes of ϕ_{λ}(t) are pairwise disjoint mod P. If u : x^{σ_{A}} → x is an isogeny over W/π^{n} of degree λ, then u is uniquely determined up to automorphism of x by ker u(t) ⊂ ker ϕ_{λ}(t). For a fixed u we have a unique lifting to a submodule of ker ϕ_{λ}(t) over W, i.e., there exists y and an isogeny x^{σ_{A}} → y of degree λ, such that

[figure: x^{σ_{A}}; x; y; ≃]

commutes. Therefore

#Hom _{W/π^{n}}(x^{σ_{A}}, x)_{deg λ} = ∑_{y∈T_{λ}x^{σ_{A}}} #Isom _{W/π^{n}}(y, x).

By summation over n together with Proposition 3.3.3 the assertion of the theorem follows. Now let λ = P^{t}R with t ≥ 1 and P ∤ R. The elements y ∈ T_{λ}x^{σ_{A}} are Heegner points of different levels and are defined over some extension H_{s}/H which is ramified at P. The analogue of the Eichler-Shimura congruence holds, i.e.

T_{P} ≡ F^{∗} + F mod P,

where F is the Frobenius correspondence and F^{∗} is its dual correspondence on X_{0}(N). This can be shown in the following way. Let u : ϕ → ϕ′ be a cyclic isogeny of degree P given as a τ−polynomial. Then there exists a dual isogeny v : ϕ′ → ϕ, such that ϕ_{P} = v · u. Then either v ≡ τ^{d} mod P and consequently ϕ′ ≡ ϕ^{F} mod P or u ≡ τ^{d} mod P and consequently ϕ ≡ ϕ^{′F} mod P. This proves the Eichler-Shimura congruence. By a simple induction we get

T_{P^{t}} ≡ F^{∗t} + F^{∗(t−1)}F + ⋯ + F^{∗}F^{t−1} + F^{t} mod P.

Lemma 3.3.5 Let y ∈ T_{P^{t}}z. If s is the level of y, then y is defined over a class field H_{s} (cf. Proposition 3.2.5). Let H_{s,v} be the completion at v|P and W_{s} the valuation ring of the maximal unramified extension of H_{s,v} with local parameter π_{s}. y is defined over W_{s}. If z has level 0 at P then it is defined over an unramified extension H_{O}/H (cf. Proposition 3.2.5), thus also over W. There exists a y_{0} of level 0 defined over W, with y ≡ y_{0} mod π_{s}.

Proof. For P ramified or split in L/K let σ_{𝔭} ∈ Gal (H_{O}/L) be the Frobenius of 𝔭|P over L. For P inert let σ_{𝔭} = σ_{P} ∈ Gal (H_{O}/K) be the Frobenius of P over K. Then σ_{𝔭} operates on ϕ. The definition of Frobenius yields ϕ^{σ_{𝔭}} ≡ ϕ^{F} mod π_{s} and ϕ^{σ_{𝔭}^{−1}} ≡ ϕ′ mod π_{s} with ϕ^{′F} ≡ ϕ mod π_{s}. Now let y ∈ T_{P^{t}}z. From the Eichler-Shimura congruence we get the existence of y′ and i with 0 ≤ i ≤ t, such that y^{′F^{i}} ≡ z mod π_{s} and y ≡ y^{′F^{t−i}} mod π_{s} therefore y ≡ y^{′σ_{𝔭}^{t−i}} ≡ z^{σ_{𝔭}^{t−2i}} mod π_{s}, so we can take y_{0} = z^{σ_{𝔭}^{t−2i}}. □ Remark. In the ramified and in the inert case we have y_{0} = z^{σ_{𝔭}} for t odd or y_{0} = z for t even. This holds because , if P is inert in L/K it is a principal ideal generated by an element which does not divide the conductor of O. This implies, that σ_{𝔭}^{2} = 1 for σ_{𝔭} ∈ Gal(H_{O}/K). If P is ramified in L/K we have that 𝔭^{2} = (P) is a principal ideal prime to the conductor. Therefore σ_{𝔭}^{2} = 1 also in this case.

Lemma 3.3.6 Let y ∈ T_{P^{t}}z with level s ≥ 1 and y_{0}, π_{s} as in Lemma 3.3.5. Then

y ≢ y_{0} mod π_{s}^{2}.

Proof. The assertion is even true for the associated formal modules [Gr2, Prop. 5.3]. The formal module associated to a Drinfeld module is an extension of ϕ to a homomorphism ϕ^{(P)} : 𝔽_{q}[T]_{P} → W/π{{τ}} where 𝔽_{q}[T]_{P} is the completion at P and W/π{{τ}} is the twisted power series ring. □ Now we can go on with the proof of Theorem 3.3.4. We treat the cases P split, P inert and P ramified separately. Suppose at first that P is split. Then ϕ has ordinary reduction and therefore Hom _{W} (x^{σ_{A}}, x) = Hom _{W/π^{n}}(x^{σ_{A}}, x) for all n ≥ 1, because Hom _{W/π^{n}}(x, x) = Hom _{W} (x, x) = O_{L} and Hom _{W/π^{n}}(x^{σ_{A}}, x) = 𝔞 . By assumption we have r_{A}((λ)) = 0, so Hom _{W/π^{n}}(x^{σ_{A}}, x)_{deg λ} = ∅.

On the other hand let y ∈ T_{λ}x^{σ_{A}} with x ≡ y mod π_{s}. Then Lemma 3.3.5 gives y ≡ y_{0} mod π_{s} for a y_{0} ∈ T_{λ}x^{σ_{A}} of level 0. Because x ≃ y_{0} over W/π and therefore also over W, we get x ≃ y_{0} over W, and so x ∈ T_{λ}x^{σ_{A}}, which contradicts the assumption r_{A}((λ)) = 0. Therefore we get

(x . T_{λ}x^{σ_{A}})_{v} = 0.

If P is inert then let y_{0} = z^{σ_{𝔭}} for t odd and y_{0} = z for t even, respectively, as in the remark following Lemma 3.3.5. Then Lemma 3.3.5 and Lemma 3.3.6 yield y ≡ y_{0} mod π_{s}, y ≢ y_{0} mod π_{s}^{2}.

Each y of level s = s(y) is defined over W_{s}, which is ramified of degree e_{s} = q_{P}^{(s−1)}(q_{P} + 1) (cf. Proposition 3.2.5(2)).

We distinguish between the intersection pairing over W and W_{s}. For the latter we write ( .)_{v,s}. From the definition of the intersection multiplicity we get

(T_{P^{t}}z . x)_{v} = \frac{1}{e_{s}} (T_{P^{t}}z . x)_{v,s}

and further

(T_{P^{t}}z . x)_{v} = \frac{1}{e_{s}} ∑_{y∈T_{P^{t}} z}(y . x)_{v,s} =  (3.3.1)

= ∑_{{s=0 \atop s≡tmod2}}^{t} ∑_{{y∈T_{P^{t}} z \atop s(y)=s}} \frac{1}{(q − 1)e_{s}} ∑_{n≥1} #Isom _{W_{s}/π_{s}^{n}}(y, x)

= \{ \begin{matrix}\frac{1}{q − 1} ∑_{n≥1} #Isom _{W/π^{n}}(z, x) \\ + ∑_{{s=1 \atop s≡tmod2}}^{t} \frac{#\{y ∈ T_{P^{t}}z : s(y) = s\}}{(q − 1) · e_{s}}#Isom _{W/π}(y_{0}, x) & , if t is even \\ ∑_{{s=1 \atop s≡tmod2}}^{t} \frac{#\{y ∈ T_{P^{t}}z : s(y) = s\}}{(q − 1) · e_{s}}#Isom _{W/π}(y_{0}, x) & , if t is odd \end{matrix}

= \{ \begin{matrix}\frac{1}{q − 1} (∑_{n≥1} #Isom _{W/π^{n}}(z, x) + ·\frac{t}{2}#Isom _{W/π}(z, x)) & , if t is even \\ \frac{1}{q − 1} · \frac{t + 1}{2}#Hom _{W/π}(z, x)_{deg P} & , if t is odd. \end{matrix}

Documenta Mathematica 5 (2000) 365–444

Summing over all z ∈ T_{R}x we obtain (P ∤ R)

(T_{λ}x^{σ_{A}} . x)_{v} = ∑_{z∈T_{R}x^{σ_{A}}}(T_{P^{t}}z . x)_{v} =

= \{ \begin{matrix}\frac{1}{q − 1} ∑_{n≥1} #Hom _{W/π^{n}}(x^{σ_{A}}, x)_{deg R} \\ +\frac{t}{2(q − 1)}#Hom _{W/π}(x^{σ_{A}}, x)_{deg R} & , if t is even \\ \frac{t + 1}{2(q − 1)}#Hom _{W/π}(x^{σ_{A}}, x)_{deg RP} & , if t is odd. \end{matrix}

Lemma 3.3.7 a) If t is even:

#Hom _{W/π^{n}}(x^{σ_{A}}, x)_{deg R} = #Hom _{W/π^{n+t/2}}(x^{σ_{A}}, x)_{deg λ} for n ≥ 1
#Hom _{W/π}(x^{σ_{A}}, x)_{deg R} = #Hom _{W/π^{n}}(x^{σ_{A}}, x)_{deg λ}  for n ≤ t/2.

b) If t is odd:

#Hom _{W/π}(x^{σ_{A}}, x)_{deg RP} = #Hom _{W/π^{(t+1)/2}}(x^{σ_{A}}, x)_{deg λ}
#Hom _{W/π}(x^{σ_{A}}, x)_{deg RP} = #Hom _{W/π^{n}}(x^{σ_{A}}, x)_{deg λ}

for n ≤ (t + 1)/2.

Proof. a) We have that ϕ_{P^{t/2}} ≡ τ^{dt} mod π^{t/2} is an isogeny of degree P^{t}. If u ∈ Hom _{W/π^{n}}(x^{σ_{A}}, x)_{deg R}, then ϕ_{P^{t/2}}(uϕ_{x}^{σ_{A}} − ϕ_{x}u) ≡ 0 mod π^{n+t/2} and therefore ϕ_{P^{t/2}}uϕ_{x}^{σ_{A}} ≡ ϕ_{x}ϕ_{P^{t/2}}u mod π^{n+t/2}, i.e.

ϕ_{P^{t/2}}u ∈ Hom _{W/π^{n+t/2}}(x^{σ_{A}}, x)_{deg λ}.

Now P ∤ R, thus π ∤ u_{0} for u ≠ 0, and ϕ_{P^{t/2}}u ≡ 0 mod π^{n+t/2} implies u ≡ 0 mod π^{n}, i.e. the map is injective. Now let u ∈ Hom _{W/π^{n+t/2}}(x^{σ_{A}}, x)_{deg λ}, then there is a splitting u = u_{1} · u_{2} with an isogeny u_{1} of degree P^{t} and an isogeny u_{2} of degree R. We have u_{1} ≡ τ^{dt} mod π^{t/2}, therefore the map is also surjective. This also shows that

Hom _{W/π}(x^{σ_{A}}, x)_{deg R} ⟶ Hom _{W/π^{n}}(x^{σ_{A}}, x)_{deg λ}

is bijective for n ≤ t/2, which implies a).

b) Analogous to a) with

Hom _{W/π}(x^{σ_{A}}, x)_{deg RP} ⟶ Hom _{W/π^{(t+1)/2}}(x^{σ_{A}}, x)_{deg λ}
f ⟼ ϕ_{P^{(t−1)/2}} · f.

This completes the proof of Theorem 3.3.4, if P is inert.

Now let P be ramified. 𝔭 a prime in L over P. Then y_{0} = z^{σ_{𝔭}} for t odd and y_{0} = z for t even, respectively, where σ_{𝔭} is now the Frobenius over L. Lemma 3.3.5 and Lemma 3.3.6 tell us again that y ≡ y_{0} mod π_{s}, y ≢ y_{0} mod π_{s}^{2}.

(T_{λ}x^{σ_{A}} . x)_{v} =  (3.3.2)

= \{ \begin{matrix}\frac{1}{q − 1} ∑_{n≥1} #Isom _{W/π^{n}}(z, x) + \frac{t}{q − 1}#Isom _{W/π}(z, x) \\ \frac{1}{q − 1} ∑_{n≥1} #Isom _{W/π^{n}}(z^{σ_{𝔭}}, x) + \frac{t}{q − 1}#Hom _{W/π}(z^{σ_{𝔭}}, x)_{deg P} \end{matrix}

for t even or odd, resp. Summing over all z ∈ T_{R}x^{σ_{A}} yields

(T_{λ}x^{σ_{A}} . x)_{v} = ∑_{z∈T_{R}x^{σ_{A}}}(T_{P^{t}}z . x)_{v} =

= \{ \begin{matrix}\frac{1}{q − 1} ∑_{n≥1} #Hom _{W/π^{n}}(x^{σ_{A}}, x)_{deg R} \\ +\frac{t}{q − 1}#Hom _{W/π}(x^{σ_{A}}, x)_{deg R} & , if t is even \\ \frac{1}{q − 1} ∑_{n≥1} #Hom _{W/π^{n}}(x^{σ_{A}σ_{𝔭}}, x)_{deg R} \\ +\frac{t}{q − 1}#Hom _{W/π}(x^{σ_{A}σ_{𝔭}}, x)_{deg R} & , if t is odd. \end{matrix}

Lemma 3.3.8 a) For t even

#Hom _{W/π^{n}}(x^{σ_{A}}, x)_{deg R} = #Hom _{W/π^{n+t}}(x^{σ_{A}}, x)_{deg λ} for n ≥ 1
#Hom _{W/π}(x^{σ_{A}}, x)_{deg R} = #Hom _{W/π^{n}}(x^{σ_{A}}, x)_{deg λ}  for n ≤ t.

b) For t odd

#Hom _{W/π^{n}}(x^{σ_{A}σ_{𝔭}}, x)_{deg R} = #Hom _{W/π^{(t+n)}}(x^{σ_{A}}, x)_{deg λ}
#Hom _{W/π}(x^{σ_{A}σ_{𝔭}}, x)_{deg R} = #Hom _{W/π^{n}}(x^{σ_{A}}, x)_{deg λ}

where the first equality holds for all n ≥ 1 and the second one for all n ≤ t + 1.

Proof. The proof is analogous to the proof of the previous lemma. Using the facts that f ⟼ ϕ_{P^{t/2}}f for a) and f ⟼ ϕ_{𝔭^{t}}f for b) are bijections of the sets.□ Now Theorem 3.3.4 is also completely proved.

3.4 Quaternions

Assume that P is a prime which is non split in L. Then ϕ has supersingular reduction and therefore End _{W/π}(ϕ) is a maximal order in a quaternion algebra B over K with maximal subfield L, unramified outside P and ∞ and having invariants inv_{P} = 1/2 and inv_{∞} = −1/2 (cf. [Ge4]). The reduced norm of B/K will be denoted by nr and the reduced trace by tr. The norm of L/K will be denoted by N_{L/K}(·). Let x = (ϕ, ϕ′, ϕ_{𝔫}) be a Heegner point. Then

R := End _{W/π}(x) = {f ∈ End _{W/π}(ϕ) : ϕ_{𝔫}f = gϕ_{𝔫}, for some g ∈ End _{W/π}(ϕ)}

which is the same as R = End _{W/π}(ϕ) ∩ End _{W/π}(ϕ′) in B. This is also an order in B, but is not a maximal one in general. Let 𝔽_{q}[T]_{P} be the completion of 𝔽_{q}[T] at P. Then ϕ extends to a formal module

ϕ^{(P)} : 𝔽_{q}[T]_{P} ⟶ W/π{{τ}}

where W/π{{τ}} is the twisted power series ring. Then we have the following analogue of the theorem of Serre-Tate [Dr] with x^{(P)} = (ϕ^{(P)}, ϕ^{′(P)}, ϕ_{𝔫}^{(P)}):

Lemma 3.4.1

End _{W/π^{n}}(x) = End _{W/π}(x) ∩ End _{W/π^{n}}(x^{(P)}).

Proof. It suffices to show that

End _{W/π^{n}}(ϕ) = End _{W/π}(ϕ) ∩ End _{W/π^{n}}(ϕ^{(P)}).

We use induction. Let f ∈ End _{W/π^{n}}(ϕ) ∩ End _{W/π^{n+1}}(ϕ^{(P)}). We have to show that f ∈ End _{W/π^{n+1}}(ϕ). Therefore let

f = f_{0} + f_{1}τ + … + f_{i}τ^{i} + … ∈ W/π^{n+1}{{τ}}

with fϕ_{a} ≡ ϕ_{a}f mod π^{n+1} for all a ∈ 𝔽_{q}[T] and assume that there is an M ∈ ℕ, such that f_{i} ≡ 0 mod π^{n} for all i ≥ M. Now ϕ has supersingular reduction at π, therefore

ϕ_{P} = a_{0} + a_{1}τ + … + a_{2d}τ^{2d} ≡ a_{2d}τ^{2d} mod π,

if d = deg P. Now if k > M + 2d then the k−th coefficient of fϕ_{P} is

f_{k−2d}a_{2d}^{q^{k−2d}} + f_{k−2d+1}a_{2d−1}^{q^{k−2d+1}} + ⋯ + f_{k}a_{0}^{q^{k}} ≡ f_{k−2d}a_{2d}^{q^{k−2d}} mod π^{n+1},

because f_{i} ≡ 0 mod π^{n} and a_{i} ≡ 0 mod π for i < 2d. On the other hand this coefficient is equal to the k−th coefficient of ϕ_{P} f which is

a_{0}f_{k} + a_{1}f_{k−1}^{q} + ⋯ + a_{2d−1}f_{k−2d+1}^{q^{2d−1}} + a_{2d}f_{k−2d}^{q^{2d}} ≡ 0 mod π^{n+1}.

Here a_{2d} occurs only together with f_{k−2d}^{q^{2d}} which vanishes modulo π^{n+1} because d ≥ 1. Comparing both yields the assertion, namely f_{k−2d} ≡ 0 mod π^{n+1} for all k > M + 2d. □ From Lemma 3.4.1 and the corresponding statement for formal groups ([Gr2]) we immediately get for the order R in the quaternion algebra B:

Proposition 3.4.2 Let 𝔭|P be a prime ideal in L and j in R with B = L+Lj. Then End _{W/π^{n}}(x) =

{b = b_{1} + b_{2}j ∈ R : D · N_{L/K}(b_{2})nr(j) ≡ 0 mod P (N_{L/K}(𝔭))^{n−1}}.

Together with 3.3.4 we get

Proposition 3.4.3 Let x be a Heegner point for the maximal order O_{L}, σ_{A} ∈ Gal (H/L) and 𝔞 an ideal from the ideal class A corresponding to σ_{A}. Let R = End _{W/π}(x) and suppose gcd(λ, N) = 1. Then

(x . T_{λ}x^{σ_{A}})_{v} = \frac{1}{q − 1} ∑_{{b∈R𝔞,b∉L \atop nr(b)=λN_{L/K} (𝔞)}} \{ \begin{matrix}\frac{1}{2}(1 + ord_{P} (nr(j)N_{L/K}(b_{2}))) & (P inert) \\ ord_{P} (D nr(j)N_{L/K}(b_{2})) & (P ramified). \end{matrix}

Proof. Theorem 3.3.4 yields

(x . T_{λ}x^{σ_{A}})_{v} = \frac{1}{q − 1} ∑_{n≥1} #Hom _{W/π^{n}}(x^{σ_{A}}, x)_{deg λ}.

With Lemma 3.2.2 and Proposition 3.4.2 we obtain

(q − 1)(x . T_{λ}x^{σ_{A}})_{v}

= ∑_{n≥1} #{b = b_{1} + b_{2}j ∈ R𝔞 :

(nr(b)) = (λN_{L/K}(𝔞)), D · N_{L/K}(b_{2})nr(j) ≡ 0 mod P (N_{L/K}(𝔭))^{n−1}}

= ∑_{{b∈R𝔞 \atop (nr(b))=(λN_{L/K} (𝔞))}} \{ \begin{matrix}#\{n : nr(j)N_{L/K}(b_{2}) ≡ 0 mod P^{2n−1}\}, & (P inert) \\ #\{n : D nr(j)N_{L/K}(b_{2}) ≡ 0 mod P^{n}\}, & (P ramified) \end{matrix}

= ∑_{{b∈R𝔞 \atop (nr(b))=(λN_{L/K} (𝔞))}} \{ \begin{matrix}\frac{1}{2}(1 + ord_{P} (nr(j)N_{L/K}(b_{2}))) & (P inert) \\ ord_{P} (D nr(j)N_{L/K}(b_{2})) & (P ramified). \end{matrix}

In the assertion of the proposition we sum only over b ∉ L or equivalently b_{2} ≠ 0. As we assume that r_{A}((λ)) = 0 this makes no difference because the elements with b_{2} = 0 correspond to homomorphisms defined over W. □ The next step towards our final formulae is to describe R𝔞 explicitly. This can be done in almost the same way as in the paper of Gross and Zagier, therefore we omit the details. First of all we want to describe the quaternion algebra by Hilbert symbols. This is obtained by class field theory.

Proposition 3.4.4 Let P be monic and inert. Let ϵ_{D} be the leading coefficient of D. Then there exists a monic, irreducible polynomial Q ≠ P and ϵ ∈ 𝔽_{q}^{∗}−𝔽_{q}^{2}, such that 1. deg P QD is odd,

2. ϵP Q ≡ 1 mod l for all l|D.

In terms of the Hilbert symbol this means

(\frac{D, ϵP Q}{l}) = \{ \begin{matrix}1 & for l ∤ P ∞ \\ −1 & for l = P or l = ∞. \end{matrix}

Furthermore D is a quadratic residue modulo Q, i.e. Q is split in L/K.

Corollary 3.4.5 B is described by

B ≃ (D, ϵP Q), B = L + Lj

with j^{2} = ϵP Q.

We recall the following definition.

Definition 3.4.6 The level (or reduced discriminant) rd of an order J in a quaternion algebra B is defined by

rd := n(J̃)^{−1},

where J̃ = {b ∈ B : tr(bJ) ⊂ 𝔽_{q}[T]} is the complement of J and n(J̃) is the gcd of the norms of elements in J̃.

Then we can show that R has level NP and O_{L} is optimally embedded in R, i.e. R ∩ L = O_{L}. The next step is to identify the order R in B.

Proposition 3.4.7 The set

S = {α + βj : α ∈ 𝔡^{−1}, β ∈ 𝔡^{−1}𝔮^{−1}𝔫, α ≡ β mod O_{𝔣} ∀𝔣|𝔡}

is an order in (D, ϵP Q) of level NP and O_{L} is optimally embedded in S. Here 𝔡 = (\sqrt{D}) is the different, 𝔮|Q is a prime of L and O_{𝔣} is the localization of O_{L} at 𝔣.

The proof is given by straightforward calculations (cf. Satz 3.18, [Ti1]). Now R, S are both orders in which O_{L} is optimally embedded and sharing the same level. A Theorem of Eichler [Ei, Satz 7] states the existence of an ideal 𝔟 of O_{L} with R𝔟 = 𝔟S. So if 𝔞 is an ideal in the class A corresponding to σ_{A} ∈ Gal (H/L), and without loss of generality we assume that P is not a divisor of 𝔞, then

R𝔞 = 𝔟S𝔟^{−1}𝔞 =

= \{α + βj : α ∈ 𝔡^{−1}𝔞, β ∈ 𝔡^{−1}𝔮^{−1}𝔫𝔟\overline{𝔟}^{−1}\overline{𝔞}, α ≡ (−1)^{ord_{𝔣}(𝔟)}β mod O_{𝔣} ∀𝔣|𝔡\}.

The ideal class B of 𝔟 depends on the place v|P. But P is inert, therefore it is a principal prime ideal in L, and so P is totally split in H/L. The places over P are permuted transitively by the Galois group. If τ ∈ Gal (H/L) and W_{τ} is the maximal unramified extension of O_{H,v^{τ}} and π_{τ} is a uniformizing parameter and R_{τ} = End _{W_{τ} /π_{τ}} (x), then R_{τ} = 𝔠_{τ} R𝔠_{τ}^{−1}, where 𝔠_{τ} lies in the ideal class corresponding to τ. If 𝔟_{τ} is defined by R_{τ} 𝔟_{τ} = 𝔟_{τ} S, it follows that 𝔟_{τ} = 𝔟𝔠_{τ}. Now we can give a more explicit formula for the height pairing at inert primes. We define d(μ, D) to be the number of common prime factors of μ and D.

Proposition 3.4.8 For P inert we get the formula:

⟨(x) − (∞) , T_{λ}((x)^{σ_{A}} − (0))⟩_{P}

= ∑_{v|P} −(x . T_{λ}x^{σ_{A}})_{v} ln q_{v}

= −u^{2} \frac{1}{q − 1} ln q deg P ∑_{μ∈𝔽q [T]−\{0\} deg μ≤deg λD−deg NP} ord_{P} (P^{2}μ) · r_{A}((NP μ − λD)) ·

2^{d(μ,D)}R_{\{A[𝔮𝔫]\}}((μ))\frac{1 − δ_{(NP μ−λD)NP μ}}{2}

with

u = |O_{L}^{∗}/𝔽_{q}^{∗}| = \{ \begin{matrix}q + 1, & if deg D = 0 \\ 1, & otherwise. \end{matrix}

Here δ is again the local norm symbol at ∞ (cf. the definition of δ in section 2.2 following equation (2.2.5) ). R_{{A[𝔮𝔫]}}(μ) denotes the number of integral ideals 𝔠, which lie in a class differing from the class A[𝔮𝔫] by a square in the class group and with norm (μ).

Proof. Let 𝔞 be a fixed ideal in A and let λ_{0} be a fixed generator of N_{L/K}(𝔞). We calculate the height pairing using Proposition 3.4.3 together with the explicit description of R𝔞. If b = α + βj ∈ R_{τ} 𝔞, i.e. α ∈ 𝔡^{−1}𝔞, β ∈ 𝔡^{−1}𝔮^{−1}𝔫𝔟_{τ} \overline{𝔟}_{τ}^{−1}\overline{𝔞}, α ≡ (−1)^{ord_{𝔣}(𝔟)}β mod O_{𝔣}, we define

𝔠 := (β)𝔡𝔮𝔫^{−1}𝔟_{τ}^{−1}\overline{𝔟_{τ}} \overline{𝔞}^{−1} ∈ [𝔮𝔫^{−1}]B^{2}A

and

ν := −N_{L/K}(α)Dλ_{0}^{−1} ∈ 𝔽_{q}[T]

μ := −ϵN_{L/K}(β)DQN^{−1}λ_{0}^{−1} ∈ 𝔽_{q}[T].

Then 𝔠 is integral and

nr(α + βj) = N_{L/K}(α) − ϵP QN_{L/K}(β) = (−ν + NP μ)D^{−1}λ_{0}

thus

(nr(α + βj)) = (λλ_{0}) ⟺ (−ν + NP μ) = (Dλ) ⟺ ν = NP μ − ϵ̃Dλ

for a uniquely determined ϵ̃ ∈ 𝔽_{q}^{∗}. Now if μ ∈ 𝔽_{q}[T] and ϵ ∈ 𝔽_{q}^{∗} are given, then the number of α ∈ 𝔡^{−1}𝔞 with N_{L/K}(α) = −νD^{−1}λ_{0} is r_{𝔞,λ_{0}}(NP μ − ϵ̃Dλ). β is determined by the integral ideal 𝔠 up to multiplication with elements of O_{L}^{∗}. If deg D = 0 there are no further restrictions on α, β. Now suppose deg D > 0. We have that ϵ̃λλ_{0} = N_{L/K}(α) − ϵP QN_{L/K}(β) is integral and that ϵP Q ≡ 1 mod f for all f|D. Therefore α ≡ ±β mod O_{𝔣}. Let (\sqrt{D}) = 𝔭_{1} ⋯ 𝔭_{t}, we can modify 𝔟_{τ} modulo squares of classes to find 𝔟 with

𝔟 = 𝔟_{τ} · 𝔭_{1}^{ϵ_{1}} ⋯ 𝔭_{t}^{ϵ_{t}}

with ϵ_{i} ∈ {0, 1} such that

α ≡ (−1)^{ord_{𝔭_{i}} (𝔟)}β mod O_{𝔭_{i}}.

The ϵ_{i} are uniquely determined if β ∉ O_{𝔣}, which is the case exactly when 𝔭_{i} ∤ μ. If β ∈ O_{𝔣} then both choices of ϵ_{i} give the correct congruence. Thus there are 2^{d(μ,D)} ideal classes which differ from the class of 𝔟_{τ} only by classes of order 2 and which have the given congruence for α and β. The only exception to this is when D|μ. In this case all ϵ_{i} can be chosen arbitrarily, so for each dtuple (ϵ_{1}, …, ϵ_{d}) also (−ϵ_{1}, …, −ϵ_{d}) is possible but both give the same class. The number of classes is therefore divided by two. On the other hand the congruences fix β, except when all congruences are trivial. So the number of pairs (α, β) doubles in the latter case. The existence of β is equivalent to ϵ^{−1}μQ^{−1}Nλ_{0} being a norm of an element in L^{∗}. As we already know that it is the norm of an ideal, we get the following local condition:

ϵ^{−1}μQ^{−1}Nλ_{0} ∈ N_{L/K}(L^{∗}) ⟺ δ_{ϵ^{−1}μQ^{−1}Nλ_{0}} = 1.

By definition of Q we have (\frac{D, ϵP Q}{∞}) = −1. Therefore the condition is equiv-

alent to δ_{μP Nλ_{0}} = −1. For a given α the number of β in some class 𝔟_{τ} is then

2^{d(μ,D)} R_{\{A[𝔮𝔫]\}}(μ)\frac{1 − δ_{μP Nλ_{0}}}{2} .

This shows that

⟨(x) − (∞) , T_{λ}((x)^{σ_{A}} − (0))⟩_{P}
= −u\frac{1}{q − 1} ln q_{v} ∑_{μ∈𝔽q [T]−\{0\} deg μ≤deg λD−deg NP} ∑_{ϵ̃∈𝔽_{q}^{∗}} \frac{1}{2}(1 + ord_{P} (P μ)) ·

r_{𝔞,λ_{0}}(NP μ − ϵ̃λD) · 2^{d(μ,D)}R_{\{A[𝔮𝔫]\}}((μ))\frac{1 − δ_{NP μλ_{0}}}{2} .

If r_{𝔞,λ_{0}}((NP μ − ϵ̃λD)) ≠ 0 then δ_{λ_{0}} = δ_{NP μ−ϵ̃λD}, ln q_{v} = 2 deg P ln q. If we substitute this and change μ ↦ ϵ̃μ we get

⟨(x) − (∞) , T_{λ}((x)^{σ_{A}} − (0))⟩_{P}

= −u deg P ln q ∑_{μ∈𝔽q [T]−{0} deg μ≤deg λD−deg NP} ord_{P} (P^{2}μ) ∑_{ϵ̃∈𝔽_{q}^{∗}} r_{𝔞,λ_{0}}(ϵ̃(NP μ − λD)) ·

2^{d(μ,D)}R_{\{A[𝔮𝔫]\}}((μ))\frac{1 − δ_{NP μ(NP μ−λD)}}{2} .

Using the identity

\frac{1}{q − 1} ∑_{ϵ̃∈𝔽_{q}^{∗}} r_{𝔞,λ_{0}}(ϵ̃(NP μ − λD)) = ur_{A}((NP μ − λD))

we get the formula of the proposition. □ We specialize this result to the case where D is irreducible. Then u = 1 because deg D > 0. d(μ, D) = t(μ, D) = 0 or 1 for t(μ, D) defined in (2.8.1) and therefore 2^{d(μ,D)} = t(μ, D) + 1.

Lemma 3.4.9 If D is irreducible, then

R_{\{A[𝔮𝔫]\}}((μ))\frac{1 − δ_{−NQμλ_{0}}}{2} = \frac{1}{q − 1} ∑_{c|μ} [\frac{D}{c} ] \frac{1 − δ_{−NQμλ_{0}}}{2} .

Proof. One has to show that

R_{\{A[𝔮𝔫]\}}((μ))\frac{1 − δ_{−NQμλ_{0}}}{2} = #\{𝔠 integral ideal : N_{L/K}(𝔠) = (μ)\}.

Then the assertion follows by comparing the coefficients of both sides of ζ_{L}(s) = ζ_{K}(s)L_{D}(s). If deg D is odd then every class is a square in the class group and we are done. If deg D is even and if −NQμλ_{0} is a norm, then deg N_{L/K}(𝔞_{0}𝔮𝔫) ≡ deg μ mod 2. #{𝔠 integral ideal : N_{L/K}(𝔠) = (μ)} is the sum of rÃ((μ)) over all classes Ã, which is equal to the sum over all square classes if μ ≡ 0 mod 2 and equal to the sum over all non-square classes if μ ≢ 0 mod 2. In any case this is R_{A[𝔮𝔫]}(μ). □ From this lemma the following corollary follows.

Corollary 3.4.10 Let P be inert and D be irreducible. Then

⟨(x) − (∞) , T_{λ}((x)^{σ_{A}} − (0))⟩_{P}
= ∑_{v|P} −(x . T_{λ}x^{σ_{A}})_{v} ln q_{v}

= −\frac{ln q}{q − 1} ∑_{μ∈𝔽q [T]−\{0\} deg μ≤deg λD−deg NP} deg P ord_{P} (P^{2}μ) · r_{A}((NP μ − λD)) ·

(t(μ, D) + 1) (∑_{c|μ} [\frac{D}{c} ]) \frac{1 − δ_{(NP μ−λD)NP μ}}{2} .

If P is ramified we can get a similar formula arguing in the same way as for the inert case. Let 𝔭|P be a prime over P and let f (= 1 or 2) be the order of the place in the class group. Then 𝔭 splits in H into h/f factors all of which have residue degree f over the residue field of 𝔭.

Proposition 3.4.11 There exists ϵ ∈ 𝔽_{q}^{∗} − 𝔽_{q}^{2} and a monic polynomial Q ∈ 𝔽_{q}[T] with deg QD odd, such that ϵQ ≡ 1 mod l for all l|D, l ≠ P and

(\frac{ϵQ}{P}) = −1.

Also Q is split in L/K, B ≃ (D, ϵQ) and B = L + Lj with j^{2} = ϵQ.

Proposition 3.4.12 The order

S = {α + βj : α ∈ 𝔭𝔡^{−1}, β ∈ 𝔭𝔡^{−1}𝔮^{−1}𝔫, α ≡ β mod O_{𝔣} ∀𝔣|𝔡}

in (D, ϵQ) has level N · P and O_{L} is optimally embedded in S.

From this and the Theorem of Eichler we get

R_{τ} 𝔞 = 𝔟_{τ} S𝔟_{τ}^{−1}𝔞 =

= \{α + βj : α ∈ 𝔭𝔡^{−1}𝔞, β ∈ 𝔭𝔡^{−1}𝔮^{−1}𝔫𝔟_{τ} \overline{𝔟}_{τ}^{−1}\overline{𝔞}, α ≡ (−1)^{ord_{𝔣}(𝔟_{τ})}β mod O_{𝔣}\}.

In the same way as for the inert primes we can show:

Proposition 3.4.13 Assume again that r_{A}((λ)) = 0. Let P be ramified. Then deg D > 0 and u = 1. We have:

⟨(x) − (∞) , T_{λ}((x)^{σ_{A}} − (0))⟩_{P}
= ∑_{v|P} −(x . T_{λ}x^{σ_{A}})_{v} ln q_{v}

= −\frac{ln q}{q − 1} deg P ∑_{μ∈𝔽q [T]−\{0\} deg μ≤deg λD−deg NP} ord_{P} (P μ) · r_{A}((NP μ − λD)) ·

2^{d(μ,D)}R_{\{A[𝔮𝔫]\}}(P μ)\frac{1 − δ_{(NP μ−λD)NP μ}}{2} .

Documenta Mathematica 5 (2000) 365–444

If D is irreducible we get the formula:

⟨(x) − (∞) , T_{λ}((x)^{σ_{A}} − (0))⟩_{P}
= −\frac{ln q}{q − 1} deg P ∑_{μ∈𝔽q [T]−\{0\} deg μ≤deg λD−deg NP} ord_{P} (P μ) · r_{A}((NP μ − λD)) ·

2 (∑_{μ|c} [\frac{D}{c} ]) \frac{1 − δ_{(NP μ−λD)NP μ}}{2} .

Proof. The proof is just as in the inert case. The only thing we want to mention is that all classes for 𝔟 are counted. But the sum runs only over the v|P, so only over the h/f classes mod 𝔭. On the other hand there is a factor f from ln q_{v} = f · deg P ln q which compensates for this. □ Now we sum up the formulae for all finite P in the case that D is irreducible.

Theorem 3.4.14 Let N ∈ 𝔽_{q}[T] square free, D ∈ 𝔽_{q}[T] irreducible and D ≡ b^{2} mod N for some b ∈ 𝔽_{q}[T]. Let L = K(\sqrt{D}) and let H denote the Hilbert class field of L. Let σ_{A} ∈ Gal(H/L) and suppose that A is the corresponding ideal class. Let λ ∈ 𝔽_{q}[T] be such that gcd(λ, N) = 1 and r_{A}((λ)) = 0. Then

∑_{P ≠∞}⟨(x) − (∞) , T_{λ}((x)^{σ_{A}} − (0))⟩_{P}

= −\frac{ln q}{q − 1} ∑_{{μ∈𝔽q [T]−\{0\} \atop deg μN≤deg λD}} r_{A}((μN − λD)) · (1 − δ_{(μN−λD)μN})

[ − (t(μ, D) + 1) (∑_{c|μ} [\frac{D}{c} ] deg c) + (∑_{c|μ} [\frac{D}{c} ] deg μ) ].

Proof. The sum over all P ≠ ∞ of the formulae in Proposition 3.4.8 and in Proposition 3.4.13 gives

∑_{P ≠∞}⟨(x) − (∞) , T_{λ}((x)^{σ_{A}} − (0))⟩_{P}

= −\frac{ln q}{q − 1} ∑_{{μ∈𝔽q [T]−\{0\} \atop deg μN≤deg λD}} r_{A}((μN − λD)) · (1 − δ_{(μN−λD)μN})

[(t(μ, D) + 1) ∑_{{P |μ \atop P inert}} deg P ord_{P} (P μ) (∑_{c| \frac{μ}{p}} [\frac{D}{c} ])

+2 ord_{D}(μ) deg D (∑_{c| \frac{μ}{D}} [\frac{D}{c} ]) ].

Some calculations with the Dirichlet character show that

∑_{{P |μ \atop P inert}} deg P ord_{P} (P μ) (∑_{c| \frac{μ}{p}} [\frac{D}{c} ])

= (∑_{c|μ} [\frac{D}{c} ]) deg μ − ord_{D}(μ) deg D (∑_{c|μ} [\frac{D}{c} ])

−2 (∑_{c|μ} [\frac{D}{c} ] deg c) .

Substituting this yields

∑_{P ≠∞}⟨(x) − (∞) , T_{λ}((x)^{σ_{A}} − (0))⟩_{P}

= −\frac{ln q}{q − 1} ∑_{{μ∈𝔽q [T]−\{0\} \atop deg μN≤deg λD}} r_{A}((μN − λD)) · (1 − δ_{(μN−λD)μN})

·[(t(μ, D) + 1) (∑_{c|μ} [\frac{D}{c} ]) deg μ

− deg D (t(μ, D) + 1) ord_{D}(μ) (∑_{c|μ} [\frac{D}{c} ])

−2(t(μ, D) + 1) (∑_{c|μ} [\frac{D}{c} ] deg c) + 2 ord_{D}(μ) deg D (∑_{c|μ} [\frac{D}{c} ]) ].

For r_{A}((μN − λD)) ≠ 0 we observe that (cf. Lemma 2.8.1):

(1 − δ_{(μN−λD)μN})(t(μ, D) − 1) (∑_{c| \frac{μ}{D}} [\frac{D}{c} ]) = 0

from which the theorem follows. □

3.5 The local height pairing at ∞

At first we assume that r_{A}((λ)) = 0. The local height pairing at places over ∞ can be calculated by Green’s functions as described in [Ti3]. This approach is based on the general formula (3.1.1). This means that there are contributions coming from the intersection of horizontal divisors and from the intersection with the fibre components. In contrast to [Ti3] here we always consider Γ as a subgroup of GL_{2}(𝔽_{q}[T]) instead of P GL_{2}(𝔽_{q}[T]), so the formulae differ by a factor q − 1.

The cases deg D odd and deg D even will be treated separately starting with the former case. In the whole section we again assume that D is irreducible. We write |z|_{i} = min{|z − y| : y ∈ K_{∞}} and

d(z, z′) = log_{q} \frac{|z − z′|^{2}}{|z|_{i}|z′|_{i}} .

3.5.1 deg D odd

If z, z′ are two elements in Ω with log_{q} |z|_{i}, log_{q} |z′|_{i} ∉ ℤ and which represent L_{∞}− rational points on the algebraic curve X_{0}(N) then by definition of the Green’s function G ([Ti3, Def 2]) and Theorem 2 together with Proposition 8 of [Ti3] we have

⟨(z) − (∞), (z′) − (0)⟩_{L_{∞}}
= (− ln q)(G(z, z′) − G(z′, ∞) − G(0, z) + G(0, ∞))
= \frac{− ln q}{q − 1} [ ∑_{{γ∈Γ \atop d(γz,z′)≤0}} (\frac{q + 1}{2(q − 1)} − d(γz, z′))

+ lim_{s→1} [\frac{q + 1}{2(q − 1)} ∑_{{γ∈Γ \atop d(γz,z′)>0}} q^{−d(γz,z′)s} − \frac{2κ(q − 1)}{1 − q^{1−s}}]

− lim_{s→1} [q^{1/2}(q^{2} − 1)(Ei_{s}^{(N)}(z′) + Ei_{s}^{(N)}(\frac{1}{Nz})) − \frac{4κ(q − 1)}{1 − q^{1−s}}]]

with κ := \frac{q^{2} − 1}{2 [GL_{2}(𝔽_{q}[T]) : Γ_{0}(N)]}. Here Ei_{s}^{(N)}(z) is the Eisenstein series

Ei_{s}^{(N)}(z) = |z|_{i}^{s} ∑_{{(c,d), N|c \atop gcd(c,d)=1}} |cz + d|^{−2s}.

We define

R_{N} := \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Mat_{2×2}(𝔽_{q}[T]) : N|c, det ( \begin{matrix}a & b \\ c & d \end{matrix} ) ≠ 0\}.

If λ ∈ 𝔽_{q}[T], λ ≠ 0 we get

⟨(z) − (∞), T_{λ}((z′) − (0))⟩_{L_{∞}}

= \frac{− ln q}{q − 1} [ ∑_{{γ∈R_{N} , (det γ)=(λ) \atop d(z,γz′)≤0}} (\frac{q + 1}{2(q − 1)} − d(z, γz′))

+ lim_{s→1} [\frac{q + 1}{2(q − 1)} ∑_{{γ∈R_{N} , (det γ)=(λ) \atop d(z,γz′)>0}} q^{−d(z,γz′)s} − \frac{2κσ_{1}(λ)}{1 − q^{1−s}} ]

− lim_{s→1} [q^{1/2}(q + 1)(q^{deg λs}σ_{1−2s}(λ)Ei_{s}^{(N)}(z′) + σ_{1}(λ)Ei_{s}^{(N)}(\frac{1}{Nz}))

−\frac{4κσ_{1}(λ)}{1 − q^{1−s}} ]]

with σ_{s}(λ) := ∑_{a|λ} q^{deg a s} for any s. Now we specialize z to be a Heegner point and z′ to be a conjugate under the Galois group.

Let λ ∈ 𝔽_{q}[T] \ \{0\} with r_{A}((λ)) = 0. Let 𝔫 be an ideal with 𝔫\overline{𝔫} = (N). For j = 1, 2 let 𝔞_{j} = A_{j}𝔽_{q}[T] + (B_{j} + \sqrt{D})𝔽_{q}[T] be two ideals contained in 𝔫 with N_{L/K}(𝔞_{j}) = (A_{j}) and let A_{j} be the corresponding ideal classes. Then this data defines two Heegner points which are represented in the upper half plane by

τ_{j} = \frac{−B_{j} + \sqrt{D}}{2A_{j}}. We have that log_{q} |τ_{j}|_{i} = log_{q} |\sqrt{D}/A_{j}| ∉ ℤ.

If A is an ideal class and σ_{A} ∈ Gal(H/L) the corresponding automorphism we get

⟨(τ) − (∞), T_{λ}((τ)^{σ_{A}} − (0))⟩_{∞}
= ∑_{v|∞}⟨(τ) − (∞), T_{λ}((τ)^{σ_{A}} − (0))⟩_{v}
= ∑_{{A_{1},A_{2}∈Cl(O_{L}) \atop A_{1}A{−1 \atop 2}=A}}⟨(τ_{A_{1}}) − (∞), T_{λ}((τ_{A_{2}}) − (0))⟩_{L_{∞}}

= \frac{− ln q}{q − 1} [ lim_{s→1} [F_{1}(A, s) − \frac{2κh_{L}σ_{1}(λ)}{1 − q^{1−s}}]

− lim_{s→1} [F_{2}(A, s) − \frac{4κh_{L}σ_{1}(λ)}{1 − q^{1−s}}] ]  (3.5.1)

where τ is one of the τ_{A_{j}}, κ = \frac{(q^{2} − 1)}{2[GL_{2}(𝔽_{q}[T]) : Γ_{0}(N)]} and

F_{1}(A, s) := ∑_{{A_{1},A_{2}∈Cl(O_{L}) \atop A_{1}A{−1 \atop 2}=A}} [ ∑_{{γ∈R_{N} , (det γ)=(λ) \atop d(γτ_{A_{1}} ,τ_{A_{2}} )≤0}} (\frac{q + 1}{2(q − 1)} − d(γτ_{A_{1}}, τ_{A_{2}}))

+\frac{q + 1}{2(q − 1)} ∑_{{γ∈R_{N} , (det γ)=(λ) \atop d(γτ_{A_{1}} ,τ_{A_{2}} )>0}} q^{−d(γτ_{A_{1}} ,τ_{A_{2}} )s}]

and

F_{2}(A, s) := ∑_{{A_{1},A_{2}∈Cl(O_{L}) \atop A_{1}A{−1 \atop 2}=A}} q^{1/2}(q + 1) ·

·[q^{deg λs}σ_{1−2s}(λ)Ei_{s}^{(N)}(τ_{A_{2}}) + σ_{1}(λ)Ei_{s}^{(N)}(\frac{1}{Nτ_{A_{1}}} )].  (3.5.2)

At first we calculate the function F_{1}(A, s). The following proposition combined with the convergence of the limits in (3.5.1) implies the existence of the limits in section 2.8 (cf. the corresponding remark there).

Proposition 3.5.1 The following equation for F_{1} holds:

F_{1}(A, s) = ∑_{{μ≠0 \atop deg(μN)≤deg(λD)}} r_{A}((μN − λD)) (∑_{c|μ} [\frac{D}{c} ]) (t(μ, D) + 1)

·\frac{1 + δ_{(μN−λD)μN}}{2} (− deg μN + deg λD + \frac{q + 1}{2(q − 1)})

+\frac{q + 1}{2(q − 1)} ∑_{{μ∈𝔽q [T]−\{0\} \atop deg μN>deg λD}} r_{A}((μN − λD)) (∑_{c|μ} [\frac{D}{c} ])

·(t(μ, D) + 1)q^{−(deg μN−deg λD)s}.

Proof. We define

M(𝔞_{1}, 𝔞_{2}, 𝔫) = \{(α, β) ∈ 𝔞_{1}^{−1}\overline{𝔞}_{2}^{−1} × 𝔞_{1}^{−1}𝔞_{2}^{−1}𝔫| A_{1}A_{2}(α − β) ∈ \sqrt{D}𝔽_{q}[T][\sqrt{D}]\}

By calculations analogous to [Gr-Za], II (3.6)-(3.10) the map

R_{N} ⟶ M(𝔞_{1}, 𝔞_{2}, 𝔫)
( \begin{matrix}a & b \\ c & d \end{matrix} ) ⟼ (α = cτ_{1}\overline{τ}_{2} + d\overline{τ}_{2} − aτ_{1} − b, β = cτ_{1}τ_{2} + dτ_{2} − aτ_{1} − b)

is a bijection and det ( \begin{matrix}a & b \\ c & d \end{matrix} ) = −A_{1}A_{2}(N_{L/K}(α) − N_{L/K}(β))D^{−1}.

For λ ∈ 𝔽_{q}[T], λ ≠ 0 we get

\{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ R_{N}, (ad − bc) = (λ)\}

≃ \{(α, β) ∈ M(𝔞_{1}, 𝔞_{2}, 𝔫)|(\frac{−A_{1}A_{2}(N_{L/K}(α) − N_{L/K}(β))}{D}) = (λ)\}.

We set μ = N_{L/K}(β)/A_{1}^{−1}A_{2}^{−1}N ∈ 𝔽_{q}[T], then d(γτ_{A_{1}}, τ_{A_{2}}) = deg μN − deg λD. Then it follows that

∑_{{γ∈R_{N} \atop (det γ)=(λ)}} (\frac{q + 1}{2(q − 1)} − d(γτ_{A_{1}}, τ_{A_{2}}))

= ∑_{μ∈𝔽_{q}[T]−\{0\}} (\frac{q + 1}{2(q − 1)} − deg μN + deg λD) ·

#{(α, β) ∈ M(𝔞_{1}, 𝔞_{2}, 𝔫)|

(\frac{−A_{1}A_{2}(N_{L/K}(α) − N_{L/K}(β))}{D}) = (λ), \frac{N_{L/K}(β)}{A_{1}^{−1}A_{2}^{−1}N} = μ\}.

We see that

#{(α, β) ∈ M(𝔞_{1}, 𝔞_{2}, 𝔫)|

(\frac{−A_{1}A_{2}(N_{L/K}(α) − N_{L/K}(β))}{D}) = (λ), \frac{N_{L/K}(β)}{A_{1}^{−1}A_{2}^{−1}N} = μ\}

= #\{β ∈ 𝔞_{1}^{−1}𝔞_{2}^{−1}𝔫| \frac{N_{L/K}(β)}{A_{1}^{−1}A_{2}^{−1}N} = μ\} ·

#\{α ∈ 𝔞_{1}^{−1}\overline{𝔞}_{2}^{−1}| (−A_{1}A_{2}(N_{L/K}(α) − N_{L/K}(β))) = (λD)\} ·

·\frac{1}{2}(t(μ, D) + 1)

= r_{𝔞_{1}^{−1}𝔞_{2}^{−1}𝔫,A_{1}^{−1}A_{2}^{−1}N}(μ) · ∑_{ϵ∈𝔽_{q}^{∗}} r_{𝔞_{1}^{−1}\overline{𝔞}_{2}^{−1},A_{1}^{−1}A_{2}^{−1}}(μN − ϵλD) · \frac{1}{2}(t(μ, D) + 1).

Now we set 𝔞_{2} = \overline{𝔞}_{1}^{−1}\overline{𝔞}_{0}^{−1} and A_{2} = A_{1}^{−1}λ_{0}^{−1}, summing over all classes we get for the first part of the formula in the proposition:

∑_{{A_{1},A_{2}∈Cl(O_{L}) \atop A_{1}A{−1 \atop 2}=A}} ∑_{{γ∈R_{N} , \atop (det γ)=(λ)}} (\frac{q + 1}{2(q − 1)} − d(γτ_{A_{1}}, τ_{A_{2}}))

= ∑_{μ∈𝔽_{q}[T]−\{0\}} (\frac{q + 1}{2(q − 1)} − deg μN + deg λD) ·

( ∑_{A_{1}∈Cl(O_{L})} r_{𝔞_{1}^{−1}\overline{𝔞}_{1}\overline{𝔞}_{0}𝔫,λ_{0}N}(μ)) · ( ∑_{ϵ∈𝔽_{q}^{∗}} r_{𝔞_{0},λ_{0}}(μN − ϵλD)) \frac{1}{2}(t(μ, D) + 1).

Since deg D is odd we see that

r_{𝔞_{1}^{−1}\overline{𝔞}_{1}\overline{𝔞}_{0}𝔫,λ_{0}N}(μ) = r_{A_{1}^{2}[𝔞_{0}\overline{𝔫}]}((μ))(δ_{λ_{0}Nμ} + 1).

The class number is odd, and therefore every class is a square. Hence

∑_{A_{1}∈Cl(O_{L})}r_{𝔞_{1}^{−1}\overline{𝔞}_{1}\overline{𝔞}_{0}𝔫,λ_{0}N}(μ) = ∑_{B∈Cl(O_{L})}r_{B}((μ))(δ_{λ_{0}Nμ} + 1) = \frac{δ_{λ_{0}Nμ} + 1}{q − 1} ∑_{c|μ} [\frac{D}{c} ].

We use this equation, we change the order of the summation, and we continue with our formula

∑_{{A_{1},A_{2}∈Cl(O_{L}) \atop A_{1}A{−1 \atop 2}=A}} ∑_{{γ∈R_{N} , \atop (det γ)=(λ)}} (\frac{q + 1}{2(q − 1)} − d(γτ_{A_{1}}, τ_{A_{2}}))

= ∑_{ϵ∈𝔽_{q}^{∗}} ∑_{μ∈𝔽_{q}[T]−\{0\}} (\frac{q + 1}{2(q − 1)} − deg μN + deg λD) ·

\frac{1}{q − 1}( ∑_{c|μ} [\frac{D}{c} ])(δ_{λ_{0}Nϵμ} + 1) r_{𝔞_{0},λ_{0}} (ϵ(μN − λD)) \frac{1}{2}(t(μ, D) + 1).

If r_{𝔞_{0},λ_{0}} (ϵ(μN − λD)) ≠ 0, then λ_{0}ϵ(μN − λD) is a norm and δ_{λ_{0}Nϵμ} = δ_{μN(μN−λD)}. In addition we use the relation

∑_{ϵ∈𝔽_{q}^{∗}} r_{𝔞_{0},λ_{0}} (ϵ(μN − λD)) = (q − 1)r_{A}((μN − λD)).

This proves the first part of the formula in the proposition. The same calculations hold with q^{−d(γτ_{A_{1}} ,τ_{A_{2}} )s} instead of d(γτ_{A_{1}}, τ_{A_{2}}). If deg μN > deg λD we have δ_{μN(μN−λD)} = 1. Therefore the second part of the formula is also true. □ Now we continue with the calculation of the function F_{2}(A, s) defined in equation (3.5.2).

Proposition 3.5.2 For the function F_{2}(A, s) the following formula holds

lim_{s→1} [F_{2}(A, s) − \frac{4κh_{L}σ_{1}(λ)}{1 − q^{1−s}}]
= C( ∑_{a|λ} q^{deg a}) + 2κh_{L} ∑_{a|λ}(deg λ − 2 deg a)q^{deg a}

with

C := −4κh_{L}( deg N − ∑_{{(P)|(N) \atop (P) prime}} deg P (1 + q^{deg P})^{−1}

−\frac{deg D}{2} − \frac{2}{q − 1} − \frac{1}{ln q} \frac{L′_{D}(1)}{L_{D}(1)}).

Proof. Ei_{s}^{(N)}(τ) is invariant under the non trivial automorphism of L/K. From \overline{τ}_{A,𝔫} = τ_{A^{−1},\overline{𝔫}} and 1/(Nτ_{A_{1},𝔫}) = τ_{A_{1}[𝔫],\overline{𝔫}} it follows that

∑_{A∈Cl(O_{L})} Ei_{s}^{(N)}(\frac{1}{Nτ_{A}}) = ∑_{A∈Cl(O_{L})} Ei_{s}^{(N)}(τ_{A}).

Ei^{(N)} can be expressed through the Eisenstein series Ei^{(1)} by (cf. Lemma 7, [Ti3])

Ei_{s}^{(N)}(τ_{A}) = |N|^{−s} ∏_{{(P)|(N) \atop (P) prime}}(1 − |P |^{−2s})^{−1}( ∑_{{δ|N \atop δmod𝔽^{∗}_{q}}} μ(δ)|δ|^{−s})Ei_{s}^{(1)}(\frac{N}{δ} τ_{A}).

(N/δ)τ_{A} are Heegner points for δ instead of N with the same discriminant. Immediately from the definitions we get

Ei_{s}^{(1)}(τ) = (1 − q^{1−2s})|\sqrt{D}|^{s}ζ_{L}(A, s)

where ζ_{L}(A, s) is the partial ζ−function to the class A. This yields

∑_{A∈Cl(O_{L})} Ei_{s}^{(N)}(τ_{A})

= |N|^{−s} ∏_{{(P)|(N) \atop (P) prime}}(1 − |P |^{−2s})^{−1}( ∑_{{δ|N \atop δmod𝔽^{∗}_{q}}} μ(δ)|δ|^{−s})

(1 − q^{1−2s})|\sqrt{D}|^{s} ∑_{A} ζ_{L}(A, s).

We have ∑_{A} ζ_{L}(A, s) = ζ_{L}(s)(1 − q^{−s}) = L_{D}(s)/(1 − q^{1−s}). This gives

F_{2}(A, s) = |N|^{−s} ∏_{{P |N \atop P mod𝔽^{∗}_{q}}} (1 + |P |^{−s})^{−1}q^{1/2}(q + 1)\frac{1 − q^{1−2s}}{1 − q^{1−s}}

·|\sqrt{D}|^{s}L_{D}(s)(σ_{1}(λ) + |λ|^{s}σ_{1−2s}(λ)).

Now a straightforward calculation gives the desired result. □

3.5.2 deg D even

For the case where the degree of D is even we proceed in almost the same way as for the case of odd degree, so we only need mention here the statements and the differences in the proofs. We start again with the general formula for the local height pairing at infinity

Heegner Points and L-Series 425

for two points given by z, z′ ∈ Ω of [Ti3] (Thm.1, Prop. 8,9):

⟨(z) − (∞), (z′) − (0)⟩_{L_{∞}}

= \frac{−2 ln q}{q − 1} [ ∑_{{γ∈Γ \atop d(γz,z′)≤0}} (\frac{q}{q^{2} − 1} − \frac{1}{2}d(γz, z′))

+ lim_{s→1} [\frac{q}{q^{2} − 1} ∑_{{γ∈Γ \atop d(γz,z′)>0}} q^{−d(γz,z′)s} − \frac{κ(q − 1)}{1 − q^{1−s}} ]

− lim_{s→1} [q(q − 1)(Ei_{s}^{(N)}(z′) + Ei_{s}^{(N)}(\frac{1}{Nz})) − \frac{2κ(q − 1)}{1 − q^{1−s}}]].

Again we take τ_{A_{j}} ∈ Ω to be elements corresponding to the different ideal classes A_{j}. If τ is one of these τ_{A_{j}} we get

⟨(τ) − (∞), T_{λ}((τ)^{σ_{A}} − (0))⟩_{∞}

= \frac{− ln q}{q − 1} [ lim_{s→1} [F_{1}(A, s) − \frac{2κh_{L}σ_{1}(λ)}{1 − q^{1−s}}]

− lim_{s→1} [F_{2}(A, s) − \frac{4κh_{L}σ_{1}(λ)}{1 − q^{1−s}}] ]

with κ := \frac{q^{2} − 1}{2 [GL_{2}(𝔽_{q}[T]) : Γ_{0}(N)]} as above and the modified functions F_{1}, F_{2}

F_{1}(A, s) := ∑_{{A_{1},A_{2}∈Cl(O_{L}) \atop A_{1}A{−1 \atop 2}=A}} [ ∑_{{γ∈R_{N} , (det γ)=(λ) \atop d(γτ_{A_{1}} ,τ_{A_{2}} )≤0}} (\frac{2q}{q^{2} − 1} − d(γτ_{A_{1}}, τ_{A_{2}}))

+\frac{2q}{q^{2} − 1} ∑_{{γ∈R_{N} , (det γ)=(λ) \atop d(γτ_{A_{1}} ,τ_{A_{2}} )>0}} q^{−d(γτ_{A_{1}} ,τ_{A_{2}} )s}]

and

F_{2}(A, s) := ∑_{{A_{1},A_{2}∈Cl(O_{L}) \atop A_{1}A{−1 \atop 2}=A}} q[q^{deg λs}σ_{1−2s}(λ)Ei_{s}^{(N)}(τ_{A_{2}}) + σ_{1}(λ)Ei_{s}^{(N)}(\frac{1}{Nτ_{A_{1}}} )].

With these definitions we get:

Documenta Mathematica 5 (2000) 365–444

Proposition 3.5.3 The following equation for F_{1} holds

F_{1}(A, s) = ∑_{{μ≠0 \atop deg(μN)≤deg(λD)}} r_{A}((μN − λD)) (∑_{c|μ} [\frac{D}{c} ]) (t(μ, D) + 1)

·\frac{1 + δ_{(μN−λD)Nμ}}{2} (− deg μN + deg λD + \frac{2q}{q^{2} − 1})

+\frac{2q}{q^{2} − 1} ∑_{{μ∈𝔽q [T]−\{0\} \atop deg μN>deg λD}} r_{A}((μN − λD)) (∑_{c|μ} [\frac{D}{c} ])

·(t(μ, D) + 1)q^{−(deg μN−deg λD)s}.

Proof. The proof of this proposition differs from the corresponding Proposition 3.5.1 only slightly. We start with

∑_{μ≠0} (\frac{2q}{q^{2} − 1} − deg μN + deg λD) r_{A}((μN − λD)) ·

(∑_{c|μ} [\frac{D}{c} ]) (δ_{μN(μN−λD)} + 1) \frac{1}{2}(t(μ, D) + 1).

Since D is irreducible with even degree, the ideal class number is divisible by 2 exactly once. Hence the set {A^{2}|A ∈ Cl(O_{L})} is equal to the set

{B ∈ Cl(O_{L})| deg N_{L/K}(𝔟) is even for all 𝔟 ∈ B}.

This yields:

∑_{A_{1}∈Cl(O_{L})}r_{A_{1}^{2}[\overline{𝔞}_{0}𝔫]}((μ)) = ∑_{B∈Cl(O_{L})}r_{B}((μ))(δ_{λ_{0}Nμ} + 1)

= \frac{1}{q − 1}( ∑_{c|μ} [\frac{D}{c} ])(δ_{λ_{0}Nμ} + 1).

Using similar arguments as in Proposition 3.5.1 we get for our first sum:

∑_{μ≠0} (\frac{2q}{q^{2} − 1} − deg μN + deg λD)( ∑_{A_{1}∈Cl(O_{L})}\frac{1}{q − 1} ∑_{ϵ_{1}∈𝔽_{q}^{∗}} r_{𝔞_{1}^{−1}\overline{𝔞}_{1}\overline{𝔞}_{0}𝔫,λ_{0}N}(ϵ_{1}μ))

·( ∑_{ϵ∈𝔽_{q}^{∗}} r_{𝔞_{0},λ_{0}}(μN − ϵλD)) \frac{1}{2}(t(μ, D) + 1).

Each ϵ_{1} ∈ 𝔽_{q}^{∗} is norm at the extension L/K, i.e., ϵ_{1} = N_{L/K}(κ). The divisor of κ is of the form (κ) = 𝔟^{−1}\overline{𝔟}. This proves

r_{𝔞_{1}^{−1}\overline{𝔞}_{1}\overline{𝔞}_{0}𝔫,λ_{0}N}(μ) = r_{(𝔞_{1}𝔟)^{−1}\overline{𝔞_{1}𝔟}\overline{𝔞}_{0}𝔫,λ_{0}N}(ϵ_{1}μ).

Therefore an appropriate choice of the ideals 𝔞_{1} yields for our sum:

∑_{μ≠0} (\frac{2q}{q^{2} − 1} − deg μN + deg λD)( ∑_{A_{1}∈Cl(O_{L})} r_{𝔞_{1}^{−1}\overline{𝔞}_{1}\overline{𝔞}_{0}𝔫,λ_{0}N}(μ)) ·

( ∑_{ϵ∈𝔽_{q}^{∗}} r_{𝔞_{0},λ_{0}}(μN − ϵλD)) \frac{1}{2}(t(μ, D) + 1).

The rest follows in the same way as in the proof of Proposition 3.5.1. □ The formula for F_{2}(A, s) can be calculated in exactly the same way, so we only write down the result.

Proposition 3.5.4 For the function F_{2}(A, s) the following formula holds

lim_{s→1} [F_{2}(A, s) − \frac{4κh_{L}σ_{1}(λ)}{1 − q^{1−s}}]
= C( ∑_{a|λ} q^{deg a}) + 2κh_{L} ∑_{a|λ}(deg λ − 2 deg a)q^{deg a}

with

C := −4κh_{L}(deg N − ∑_{{(P)|(N) \atop (P) prime}} deg P (1 + q^{deg P})^{−1}

−\frac{deg D}{2} − \frac{q + 3}{q^{2} − 1} − \frac{1}{ln q} \frac{L′_{D}(1)}{L_{D}(1)}).

3.6 Modification if r_{A}((λ)) ≠ 0

So far we have only evaluated heights and intersection numbers, when the divisors involved have a disjoint support. In order to get a final result we must also define and compute self-intersection numbers. Let X be a complete, non-singular, irreducible curve defined over a global function field F over 𝔽_{q}, and let x be a F -rational point on X. Let X be a regular model of X over ℙ_{F}^{1} . We call l_{x} a local parameter at x if l_{x} generates the prime ideal corresponding to x in the local ring at x in the generic fibre. Let x̃ be the Zariski closure of x in X. If π is a local parameter of a fibre corresponding to a valuation v, we call l_{x} a local parameter at x for the valuation v, if l_{x} together with π generate the maximal ideal corresponding to the intersection point of x̃ with the fibre over v. Now fix a local parameter l_{x} at x. Then we define for each normalized valuation v of F the local self-intersection number of x as

(x . x)_{v} := lim_{y→x}((x . y)_{v} − v(l_{x}(y))) = lim_{y→x}((x . y)_{v} + \frac{1}{deg v} log_{q} |l_{x}(y)|_{v}),

(3.6.1) where deg v is defined as usual, and where the absolute value is given by |α|_{v} := q^{− deg v·v(α)}, according to the product formula of the function field F. The definition (3.6.1) and the definition of the ordinary intersection number (x . y)_{v} (cf. section 3.1) show immediately that (x . x)_{v} = 0, if l_{x} is a local parameter at x for the valuation v. In the next step we have to choose the local parameter l_{x} in our situation. The curve X_{0}(1) is the projective line parametrized by the j-invariant of a Drinfeld module of rank 2. We recall that a Drinfeld module of rank 2 over 𝔽_{q}[T] is given by an additive polynomial Φ_{T} (X) = T X +gX^{q} +ΔX^{q^{2}} with discriminant Δ and j-invariant j = g^{q+1}/Δ. We let Y_{1} be the projective line given by the parameter u with u^{q+1} = j. Then Y_{1}/X_{0}(1) is an extension of degree q + 1, where only the elliptic points and cusps (i.e. zeroes and poles of j) are ramified (These facts and the definition of elliptic points can be found in [Ge1] or in other textbooks on Drinfeld modules). Let Y_{N} be the composite of Y_{1} and X_{0}(N), we get the following diagram:

\xymatrix{ & Y_{N} \ar[d] \ar[dl] \\ Y_{1} \ar[d] & X_{0}(N) \ar[dl] \\ X_{0}(1)}

On Y_{N} we choose for a point y the local parameter l_{y} := u − u(y). The selfintersection numbers on X_{0}(N) will then be evaluated with this local parameter on Y_{N} and with the projection formula for the extension Y_{N}/X_{0}(N). We distinguish different cases for the valuations v:

3.6.1 v ∤ N · ∞

Let x be a Heegner point on X_{0}(N), defined locally over W as in section 3.1, and let y_{1}, …, y_{t} be the points on Y_{N} lying over x. Then the projection formula yields

(x . x)_{v} = (y_{1} . y_{1})_{v} + (y_{1} . y_{2})_{v} + ⋯ + (y_{1} . y_{t})_{v}.

Since the covering Y_{N}/X_{0}(N) is unramified outside the elliptic points and cusps and outside the divisors of N · ∞, we see that u − u(y_{1}) is a local parameter of y_{1} at v. Hence (y_{1} . y_{1})_{v} = 0 by the above remark. Since

(y_{1} . y_{j})_{v} = \frac{1}{q − 1} ∑_{k=1}^{∞} #Isom_{W/π^{k}}(y_{1}, y_{j})

for j ≠ 1 (Proposition 3.3.3), we therefore get

(x . x)_{v} = \frac{1}{q − 1} ∑_{k=1}^{∞}(#Aut_{W/π^{k}}(x) − #Aut_{W} (x)).  (3.6.2)

As mentioned at the end of the proof of Proposition 3.4.3 the automorphisms not defined over W correspond to the elements b ∈ R𝔞, b = b_{1} + b_{2}j with b_{2} ≠ 0 which corresponds to μ ≠ 0 in the formulae of Corollary 3.4.10 and Proposition 3.4.13. So these formulae already count only the “new” part, i.e. without counting homomorphisms over W. Thus if λ ∈ 𝔽_{q}[T] is prime to P the formulae for the local height pairings ⟨(x) − (∞), T_{λ}(x)^{σ_{A}} − (0)⟩_{P} of Corollary 3.4.10 and Proposition 3.4.13 remain valid. This is not true however, if v|λ. We write as before λ = P^{t}R with P ∤ R. For the points of level s > 0 it is not correct to take only the ”new” part. So we have to add the contribution from homomorphisms over W for these points to the formulae of Corollary 3.4.10 and Proposition 3.4.13. For P inert we look at the last line of (3.3.1) We get a contribution of

\frac{1}{q − 1} \{ \begin{matrix}\frac{t}{2}# Isom_{W} (z, x) #\{z ∈ T_{R}x^{σ_{A}}\} & if t is even \\ \frac{t + 1}{2}# Hom_{W} (z, x)_{deg P} #\{z ∈ T_{R}x^{σ_{A}}\} & if t is odd \end{matrix}

which is (t/2)r_{1}((P^{t}))r_{A}((R)) if t is even and 0 if t is odd. In both cases this is equal to (ord_{P} (λ)/2)r_{A}((λ)). If P is ramified we get in a similar way from (3.3.2) a contribution of ord_{P} (λ)r_{A}((λ)). If P is split we have t + 1 points of level 0 in T_{λ}x^{σ_{A}}, where x is just one of them (cf. Proposition 3.2.4). From the t − s + 1 divisors of points of level s > 0 there is at most one whose points are congruent to x. Summing over all levels shows that the correction term in this case is ord_{P} (λ)r_{A}((λ))k_{𝔭}, where k_{𝔭} is a number less or equal to t, and k_{𝔭} + k𝔭̄ = ord_{P} (λ).

3.6.2 v|N

Let x be a Heegner point on X_{0}(N) represented by the pair of ideals (𝔞, 𝔞𝔫^{−1}), where 𝔫 is a divisor of N in L (cf. section 3.1).

a) Suppose that v|𝔫, in particular let v divide the prime divisor 𝔭 of norm N_{L/K}(𝔭). The Artin reciprocity law in explicit class field theory ([Ha, (8.7)]) uses the fundamental congruence

j(𝔞𝔭^{−1}) ≡ j(𝔞)^{N_{L/K}(𝔭)} mod v.

From this we see that u − u(y_{1}) is again a local parameter of y_{1} at v for a point y_{1} on Y_{N} lying over x. Hence the calculations of the previous section, in particular equation (3.6.2), remain true in this situation.

b) Suppose that v|𝔫̄. Then the calculations of a) show that w_{N}(u − u(y_{1})) is a local parameter of y_{1} at v, where w_{N} denotes the canonical involution on X_{0}(N) and Y_{N}. Hence

(y_{1} . y_{1})_{v} = v(\frac{∂w_{N}(u)}{∂u}(u(y_{1}))).  (3.6.3)

Using the fact that u^{q+1} = j we get

(\frac{∂w_{N}(u)}{∂u})^{q+1} = (\frac{∂w_{N}(j)}{∂j})^{q+1}(\frac{j}{w_{N}(j)})^{q}.  (3.6.4)

If the Heegner point x is represented by τ ∈ Ω, then w_{N}(j)(τ) = j(Nτ). And

we can evaluate the right hand side of (3.6.4) with \frac{∂w_{N}(j)}{∂z}(τ) and \frac{∂j}{∂z} (τ). For

\frac{∂j}{∂z} we use the definition j = g^{q+1}/Δ and get

\frac{∂j}{∂z} = \frac{g^{q}}{Δ^{2}} (\frac{∂g}{∂z} Δ − g \frac{∂Δ}{∂z}).

\frac{∂g}{∂z} Δ − g \frac{∂Δ}{∂z} can be expressed in terms of Δ (cf. equation (3.6.11)). For

\frac{∂w_{N}(j)}{∂z} we perform similar calculations. Hence we get

(\frac{∂w_{N}(u)}{∂u}(u(y_{1})))^{q^{2}−1} = N^{q^{2}−1} (\frac{Δ(Nτ)}{Δ(τ)})^{2}.  (3.6.5)

Δ(τ)/Δ(Nτ) is algebraic over L and its divisor is equal to 𝔫̄^{q^{2}−1} (we get this by calculations analogous to those in [Deu], sect. 13). With this fact and with (3.6.5) we can evaluate the value in (3.6.3). Together we get

Lemma 3.6.1 If v|𝔫, then

(x . x)_{v} = 0,

and if v|𝔫̄, then

(x . x)_{v} = −v(N) = −1.

We can now summarize the results of the first two cases. We want to evaluate the height of Heegner points as in section 3.4, but without any restriction on r_{A}((λ)). We combine the calculations in section 3.4 with the contributions from subsection 3.6.1 and Lemma 3.6.1, and we get

Proposition 3.6.2

∑_{P ≠∞}⟨(x) − (∞), T_{λ}((x)^{σ_{A}} − (0))⟩_{P} = \frac{ln q}{q − 1}

·{(q − 1) r_{A}((λ)) h_{L} (deg N − deg λ)

+ ∑_{\begin{matrix}μ≠0 \\ deg(μN)≤deg(λD) \end{matrix}}r_{A}((μN − λD)) (1 − δ_{(μN−λD)μN})

· ((t(μ, D) + 1)(∑_{c|μ} [\frac{D}{c} ] deg c) − (∑_{c|μ} [\frac{D}{c} ]) deg μ) \}.

3.6.3 v|∞

Let x be a Heegner point on X_{0}(N) represented by τ ∈ Ω, and let y_{1}, …, y_{t} be the points on Y_{N} lying over x. As above the projection formula yields

(x . x)_{v} = (y_{1} . y_{1})_{v} + (y_{1} . y_{2})_{v} + ⋯ + (y_{1} . y_{t})_{v}.

The self-intersection number on Y_{N} is by definition given as

(y_{1} . y_{1})_{v} := lim_{ỹ→y_{1}}((y_{1} . ỹ)_{v} − v(u(ỹ) − u(y_{1}))).

Therefore, if ỹ on Y_{N} is mapped to x̃ on X_{0}(N), we get

(x . x)_{v} := lim_{x̃→x}((x . x̃)_{v} − v(u(ỹ) − u(y_{1}))).  (3.6.6)

The point x is represented by τ ∈ Ω, let in addition x̃ be represented by τ̃ ∈ Ω. At first we treat the case where deg D is odd. The local height pairing of x and x̃ at v is given by the Green’s function G(τ̃, τ) ([Ti3], cf. also section 3.5):

G(τ̃, τ) = \frac{1}{q − 1} ∑_{{γ∈Γ_{0}(N) \atop |τ−γτ̃|^{2}≤|τ|_{i}|γτ̃|_{i}}} (\frac{q + 1}{2(q − 1)} − log_{q} \frac{|τ − γτ̃|^{2}}{|τ|_{i}|γτ̃|_{i}} )

+ \frac{q + 1}{2(q − 1)^{2}} lim_{σ→1} ( ∑_{{γ∈Γ_{0}(N) \atop |τ−γτ̃|^{2}>|τ|_{i}|γτ̃|_{i}}} (\frac{|τ − γτ̃|^{2}}{|τ|_{i}|γτ̃|_{i}})^{−σ} − \frac{C_{1}}{1 − q^{1−σ}} ),(3.6.7)

where we normalize the absolute value such that |f| = q^{deg f} = q^{−v(f)} for f ∈ 𝔽_{q}[T]. The Green’s function G(τ̃, τ) contains two parts, the intersection number (x̃ . x)_{v} and the contribution of the fibre components (cf. (3.1.1)). We must replace (x̃ . x)_{v} by the self intersection number (x . x)_{v}. The contribution of the fibre components remains unchanged. We have u^{q+1} = j and j = j(z) for z ∈ Ω, this yields

lim_{ỹ→y_{1}}(v(u(ỹ) − u(y_{1}))) = v(\frac{∂u}{∂τ}) + lim_{τ̃→τ}(v(τ̃ − τ)).  (3.6.8)

Here \frac{∂u}{∂τ} only represents the two derivatives \frac{∂u}{∂j} and \frac{∂j}{∂τ}, we do not assume

that Y_{N} is a quotient of Ω. Now (3.6.6), (3.6.7) and (3.6.8) show that we have to replace G(τ̃, τ) by

G(τ, τ) := \frac{1}{q − 1} ∑_{{γ∈Γ_{0}(N),γτ≠τ \atop |τ−γτ|^{2}≤|τ|_{i}|γτ|_{i}}} (\frac{q + 1}{2(q − 1)} − log_{q} \frac{|τ − γτ|^{2}}{|τ|_{i}|γτ|_{i}} )

+ \frac{q + 1}{2(q − 1)^{2}} lim_{σ→1} ( ∑_{{γ∈Γ_{0}(N) \atop |τ−γτ|^{2}>|τ|_{i}|γτ|_{i}}} (\frac{|τ − γτ|^{2}}{|τ|_{i}|γτ|_{i}})^{−σ} − \frac{C_{1}}{1 − q^{1−σ}} )

+ \frac{q + 1}{2(q − 1)} + 2 log_{q}(|τ|_{i}|\frac{∂u}{∂τ}|),

When we compare the results in section 3.5 with these formulas, we get

⟨(x) − (∞), T_{λ}((x)^{σ_{A}} − (0))⟩_{∞} = right hand side of eq. (3.5.1)
−2 ln q r_{A}((λ)) ∑_{τ} ( log_{q}(|τ|_{i}|\frac{∂u}{∂τ}|) + \frac{q + 1}{4(q − 1)}), (3.6.9)

where we sum over all τ corresponding to the classes in O_{L}. We denote the second sum in (3.6.9) by S, which we will evaluate now. We use the definitions u^{q+1} = j and j = g^{q+1}/Δ to evaluate

(\frac{∂u}{∂z})^{q^{2}−1} = Δ^{2−q^{2}−q}(\frac{∂g}{∂z} Δ − g \frac{∂Δ}{∂z})^{q^{2}−1}.  (3.6.10)

(\frac{∂g}{∂z} Δ − g \frac{∂Δ}{∂z})^{q−1} is a modular form of weight q(q^{2} − 1) for the group GL_{2}(𝔽_{q}[T]), and it is therefore a polynomial in g and Δ (cf. [Go]). The evaluation of the expansion around the cusp yields the identity

(\frac{∂g}{∂z} Δ − g \frac{∂Δ}{∂z})^{q−1} = −π̄^{1−q}Δ^{q},  (3.6.11)

where π̄ is a well-defined element (cf. [Ge3]) with log_{q} |π̄| = q/(q − 1). Now (3.6.10) and (3.6.11) yield

log_{q}(|τ|_{i}|\frac{∂u}{∂τ}|) = log_{q}(|Δ(τ)|^{2/(q^{2}−1)}|τ|_{i}) − \frac{q}{q − 1}.

Since |Δ(τ)|^{2/(q^{2}−1)}|τ|_{i} is invariant under GL_{2}(𝔽_{q}[T]), we can assume that τ satisfies |τ| = |τ|_{i} > 1. For these τ we use the product formula for Δ (for all the details concerning the product formula we refer to [Ge3]):

Δ(τ) = −π̄^{q^{2}−1}t(τ)^{q−1} ∏_{{a∈𝔽q [T] \atop monic}} f_{a}(t(τ))^{(q^{2}−1)(q−1)},

where

t(τ) = (π̄τ ∏_{{l∈𝔽q [T] \atop l≠0}}(1 − \frac{τ}{l}))^{−1},

and where f_{a} are well-defined polynomials. Using the definitions of f_{a} and t(τ) we can show that in our case (i.e. τ ∈ K_{∞}(\sqrt{D}), deg D odd, |τ| = |τ|_{i} > 1)

log_{q} |t(τ)| = −|τ|_{i}q^{1/2} \frac{q + 1}{2(q − 1)}

and

log_{q} |f_{a}(t(τ))| = 0.

Therefore

log_{q} |Δ(τ)| = q(q + 1) − \frac{1}{2}q^{1/2}(q + 1)|τ|_{i}.  (3.6.12)

Hence we get

log_{q}(|τ|_{i}|\frac{∂u}{∂τ}|) = \frac{q}{q − 1} − \frac{q^{1/2}}{q − 1}|τ|_{i} + log_{q} |τ|_{i}.  (3.6.13)

Now the definition of S in (3.6.9) and equation (3.6.13) yield

S = −2 ln q r_{A}((λ))(\frac{5q + 1}{4(q − 1)}h_{L} + ∑_{τ}(−\frac{q^{1/2}|τ|_{i}}{q − 1} + log_{q} |τ|_{i})). (3.6.14)

On the other hand we consider the Eisenstein series

Ei_{s}(τ) := ∑_{{c,d∈𝔽q [T] \atop (c,d)≠(0,0)}} \frac{|τ|_{i}^{s}}{|cτ + d|^{2s}} .  (3.6.15)

We see that

∑_{τ} Ei_{s}(τ) = \frac{(q − 1)|\sqrt{D}|^{s}}{1 − q^{1−s}} L_{D}(s),  (3.6.16)

where L_{D}(s) again is the non-trivial L-series of the extension L/K. Straightforward calculations of the sum in (3.6.15) show that Ei_{s}(τ) can be expressed as a rational function:

Ei_{s}(τ) = (q − 1)\frac{|τ|_{i}^{s}}{1 − q^{1−2s}} + \frac{q^{1/2}|τ|_{i}^{1−s}}{1 − q^{2−2s}} ((q − 1)^{2} \frac{q^{−s}}{1 − q^{1−2s}} + q − 1). (3.6.17)

With (3.6.16) and (3.6.17) we can evaluate the following term

\frac{2}{ln q} \frac{L′_{D}(0)}{L_{D}(0)} = − deg D + \frac{2q}{q − 1} − \frac{2}{h_{L}} ∑_{τ} (\frac{q^{1/2}|τ|_{i}}{q − 1} − log_{q} |τ|_{i}).  (3.6.18)

We compare equation (3.6.14) coming from values of Δ and equation (3.6.18) dealing with Eisenstein series, to get

S = ln q r_{A}((λ)) h_{L}( − deg D − \frac{2}{ln q} \frac{L′_{D}(0)}{L_{D}(0)} − \frac{q + 1}{2(q − 1)}).  (3.6.19)

This result can be seen as the Kronecker limit formula for function fields. We summarize Propositions 3.5.1 and 3.5.2 and the result (3.6.9), (3.6.19) about S in the following proposition.

434 Hans-Georg Rück and Ulrich Tipp Proposition 3.6.3 Let deg D be odd, then

⟨(x) − (∞), T_{λ}((x)^{σ_{A}} − (0))⟩_{∞} = \frac{ln q}{q − 1}

·\{(q − 1) r_{A}((λ)) h_{L} (− deg D − \frac{q + 1}{2(q − 1)} − \frac{2}{ln q} \frac{L′_{D}(0)}{L_{D}(0)})

+ ∑_{\begin{matrix}μ≠0 \\ deg(μN)≤deg(λD) \end{matrix}}r_{A}((μN − λD))(∑_{c|μ} [\frac{D}{c} ]) (t(μ, D) + 1) \frac{1 + δ_{(μN−λD)μN}}{2}

· (deg(\frac{μN}{λD}) − \frac{q + 1}{2(q − 1)})

− \frac{q + 1}{2(q − 1)} lim_{σ→0} ( ∑_{deg(μN)>deg(λD)}r_{A}((μN − λD)) (∑_{c|μ} [\frac{D}{c} ]) (t(μ, D) + 1)

· q^{(−σ−1) deg( \frac{μN}{λD})} − \frac{C_{1} h_{L} (∑_{a|λ} q^{deg a})}{1 − q^{−σ}})

− \frac{q + 1}{2(q − 1)} C_{1} h_{L} (∑_{a|λ} q^{deg a}(deg λ − 2 deg a)) + (∑_{a|λ} q^{deg a}) C_{2}\}

with

C_{1} := \frac{2(q − 1)^{2}}{[GL_{2}(𝔽_{q}[T]) : Γ_{0}(N)]}

and

C_{2} := −\frac{2(q^{2} − 1)}{[GL_{2}(𝔽_{q}[T]) : Γ_{0}(N)]} h_{L} (deg N − ∑_{{(P)|(N) \atop (P) prime}} \frac{deg P}{q^{deg P} + 1} − \frac{deg D}{2}

−\frac{2}{q − 1} − \frac{1}{ln q} \frac{L′_{D}(1)}{L_{D}(1)}) .

Combining this with the results for the finite primes finally gives:

Theorem 3.6.4 Let deg D be odd and let g_{A} be the Drinfeld automorphic cusp form of Proposition 3.1.1. Then g_{A} has the Fourier coefficients for all λ ∈ 𝔽_{q}[T]

Documenta Mathematica 5 (2000) 365–444

with gcd(λ, N) = 1:

g_{A}^{∗}(π_{∞}^{deg λ+2}, λ) = \frac{ln q}{q − 1} q^{− deg λ}

·\{(q − 1) r_{A}((λ)) h_{L} (deg N − deg(λD) − \frac{q + 1}{2(q − 1)} − \frac{2}{ln q} \frac{L′_{D}(0)}{L_{D}(0)})

+ ∑_{\begin{matrix}μ≠0 \\ deg(μN)≤deg(λD) \end{matrix}}r_{A}((μN − λD))((∑_{c|μ} [\frac{D}{c} ]) (t(μ, D) + 1) \frac{1 + δ_{(μN−λD)μN}}{2}

· (deg(\frac{μN}{λD}) − \frac{q + 1}{2(q − 1)}) − (1 − δ_{(μN−λD)μN}) (∑_{c|μ} [\frac{D}{c} ]) deg μ

+ (1 − δ_{(μN−λD)μN}) (t(μ, D) + 1) (∑_{c|μ} [\frac{D}{c} ] deg c))

− \frac{q + 1}{2(q − 1)} lim_{σ→0} ( ∑_{deg(μN)>deg(λD)}r_{A}((μN − λD)) (∑_{c|μ} [\frac{D}{c} ]) (t(μ, D) + 1)

· q^{(−σ−1) deg( \frac{μN}{λD})} − \frac{C_{1} h_{L} (∑_{a|λ} q^{deg a})}{1 − q^{−σ}})

− \frac{q + 1}{2(q − 1)} C_{1} h_{L} (∑_{a|λ} q^{deg a}(deg λ − 2 deg a)) + (∑_{a|λ} q^{deg a}) C_{2}\}

with

C_{1} := \frac{2(q − 1)^{2}}{[GL_{2}(𝔽_{q}[T]) : Γ_{0}(N)]}

and

C_{2} := −\frac{2(q^{2} − 1)}{[GL_{2}(𝔽_{q}[T]) : Γ_{0}(N)]} h_{L} (deg N − ∑_{{(P)|(N) \atop (P) prime}} \frac{deg P}{q^{deg P} + 1} − \frac{deg D}{2}

−\frac{2}{q − 1} − \frac{1}{ln q} \frac{L′_{D}(1)}{L_{D}(1)}) .

If deg D is even, the calculations are the same. We will present only the differences in the formulas to the first case. The calculations with the corresponding Green’s function (cf. (3.6.9)) give

S = −2 ln q r_{A}((λ)) ∑_{τ} ( log_{q}(|τ|_{i}|\frac{∂u}{∂τ}|) + \frac{q}{q^{2} − 1}).

Equation (3.6.12) has to be replaced by

log_{q} |Δ(τ)| = q(q + 1) − q |τ|_{i}.

436 Hans-Georg Rück and Ulrich Tipp Hence (3.6.14) has the form

S = −2 ln q r_{A}((λ))(\frac{q^{2} + 2q}{q^{2} − 1} h_{L} + ∑_{τ}(−\frac{2q|τ|_{i}}{q^{2} − 1} + log_{q} |τ|_{i})).

The definition (3.6.15) and the relation (3.6.16) remain unchanged, but the rational expression (3.6.17) becomes

Ei_{s}(τ) = (q − 1)\frac{|τ|_{i}^{s}}{1 − q^{1−2s}} + \frac{q|τ|_{i}^{1−s}}{1 − q^{2−2s}} ((q − 1)^{2} \frac{q^{−2s}}{1 − q^{1−2s}} + q − 1).

Equation (3.6.18) has to be replaced by

\frac{2}{ln q} \frac{L′_{D}(0)}{L_{D}(0)} = − deg D + \frac{2q^{2} + 2q}{q^{2} − 1} − \frac{2}{h_{L}} ∑_{τ} (\frac{2q|τ|_{i}}{q^{2} − 1} − log_{q} |τ|_{i}).

And finally we get: Proposition 3.6.5 Let deg D be even, then

⟨(x) − (∞), T_{λ}((x)^{σ_{A}} − (0))⟩_{∞} = \frac{ln q}{q − 1}

·\{(q − 1) r_{A}((λ)) h_{L} (− deg D − \frac{2q}{q^{2} − 1} − \frac{2}{ln q} \frac{L′_{D}(0)}{L_{D}(0)})

+ ∑_{\begin{matrix}μ≠0 \\ deg(μN)≤deg(λD) \end{matrix}}r_{A}((μN − λD))(∑_{c|μ} [\frac{D}{c} ]) (t(μ, D) + 1) \frac{1 + δ_{(μN−λD)μN}}{2}

· (deg(\frac{μN}{λD}) − \frac{2q}{q^{2} − 1})

− \frac{2q}{q^{2} − 1} lim_{σ→0} ( ∑_{deg(μN)>deg(λD)}r_{A}((μN − λD)) (∑_{c|μ} [\frac{D}{c} ]) (t(μ, D) + 1)

· q^{(−σ−1) deg( \frac{μN}{λD})} − \frac{C_{1} h_{L} (∑_{a|λ} q^{deg a})}{1 − q^{−σ}})

− \frac{2q}{q^{2} − 1} C_{1} h_{L} (∑_{a|λ} q^{deg a}(deg λ − 2 deg a)) + (∑_{a|λ} q^{deg a}) C_{2}\}

with

C_{1} := \frac{(q^{2} − 1)^{2}}{2q[GL_{2}(𝔽_{q}[T]) : Γ_{0}(N)]}

Documenta Mathematica 5 (2000) 365–444

and

C_{2} := −\frac{2(q^{2} − 1)}{[GL_{2}(𝔽_{q}[T]) : Γ_{0}(N)]} h_{L}( deg N − ∑_{{(P)|(N) \atop (P) prime}} \frac{deg P}{q^{deg P} + 1} − \frac{deg D}{2}

−\frac{q + 3}{q^{2} − 1} − \frac{1}{ln q} \frac{L′_{D}(1)}{L_{D}(1)}).

Combining this with the results for the finite places yields:

Theorem 3.6.6 Let deg D be even and let g_{A} be the Drinfeld automorphic cusp form of Proposition 3.1.1. Then g_{A} has the Fourier coefficients for all λ ∈ 𝔽_{q}[T] with gcd(λ, N) = 1:

g_{A}^{∗}(π_{∞}^{deg λ+2}, λ) = \frac{ln q}{q − 1} q^{− deg λ}

·\{r_{A}((λ)) h_{L} (q − 1) (deg N − deg(λD) − \frac{2q}{q^{2} − 1} − \frac{2}{ln q} \frac{L′_{D}(0)}{L_{D}(0)})

+ ∑_{\begin{matrix}μ≠0 \\ deg(μN)≤deg(λD) \end{matrix}}r_{A}((μN − λD))((∑_{c|μ} [\frac{D}{c} ]) (t(μ, D) + 1) \frac{1 + δ_{(μN−λD)μN}}{2}

· (deg(\frac{μN}{λD}) − \frac{2q}{q^{2} − 1}) − (1 − δ_{(μN−λD)μN}) (∑_{c|μ} [\frac{D}{c} ]) deg μ

+ (1 − δ_{(μN−λD)μN}) (t(μ, D) + 1) (∑_{c|μ} [\frac{D}{c} ] deg c))

− \frac{2q}{q^{2} − 1} lim_{σ→0} ( ∑_{deg(μN)>deg(λD)}r_{A}((μN − λD)) (∑_{c|μ} [\frac{D}{c} ]) (t(μ, D) + 1)

· q^{(−σ−1) deg( \frac{μN}{λD})} − C_{1} h_{L} (∑_{a|λ} q^{deg a})\frac{1}{1 − q^{−σ}} )

− \frac{2q}{q^{2} − 1} C_{1} h_{L} (∑_{a|λ} q^{deg a}(deg λ − 2 deg a)) + (∑_{a|λ} q^{deg a}) C_{2}\}

with

C_{1} := \frac{(q^{2} − 1)^{2}}{2q[GL_{2}(𝔽_{q}[T]) : Γ_{0}(N)]}

Documenta Mathematica 5 (2000) 365–444

and

C_{2} := −\frac{2(q^{2} − 1)}{[GL_{2}(𝔽_{q}[T]) : Γ_{0}(N)]} h_{L}( deg N − ∑_{{(P)|(N) \atop (P) prime}} \frac{deg P}{q^{deg P} + 1} − \frac{deg D}{2}

−\frac{q + 3}{q^{2} − 1} − \frac{1}{ln q} \frac{L′_{D}(1)}{L_{D}(1)}).

4 Conclusion

4.1 Main Results

Now we combine the previous chapters on L-series (chapter 2) and on Heegner points (chapter 3). We recall the assumptions: D ∈ 𝔽_{q}[T] is an irreducible polynomial and N ∈ 𝔽_{q}[T] is a square free polynomial, whose prime divisors are split in the imaginary quadratic extension K(\sqrt{D})/K. If deg D is odd, we evaluated in Theorem 2.8.2 the Fourier coefficients Ψ_{A}^{∗}(π_{∞}^{deg λ+2}, λ) of an automorphic cusp form Ψ_{A} of Drinfeld type with

\frac{∂}{∂s}(L^{(N,D)}(2s + 1)L(f, A, s)) |_{s=0}= ∫_{Γ_{0}(N)\GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗}} f · \overline{Ψ_{A}}.

On the other hand, in Theorem 3.6.4 we obtained the Fourier coefficients of g_{A}, which are defined as

g_{A}^{∗}(π_{∞}^{deg λ+2}, λ) = q^{− deg λ}⟨(x) − (∞), T_{λ}((x)^{σ_{A}} − (0))⟩.

If we compare the two formulas, we see that

Ψ_{A}^{∗}(π_{∞}^{deg λ+2}, λ) = \frac{q − 1}{2} q^{−(deg D+1)/2} g_{A}^{∗}(π_{∞}^{deg λ+2}, λ)

for all λ ∈ 𝔽_{q}[T] with gcd(λ, N) = 1. Hence the two automorphic cusp forms Ψ_{A} and (q − 1)/2 · q^{−(deg D+1)/2} g_{A} differ only by an old form. Since f is a newform, the occurring old form does not affect the integral. And this can be summarized by the following main result:

Theorem 4.1.1 Let deg D be odd. Let x be a Heegner point on X_{0}(N) with complex multiplication by O_{L} = 𝔽_{q}[T][\sqrt{D}], let A ∈ Cl(O_{L}), and let g_{A} be the automorphic cusp form of Drinfeld type of level N, which is given by

(T_{λ}g_{A})^{∗}(π_{∞}^{2}, 1) = ⟨(x) − (∞), T_{λ}((x)^{σ_{A}} − (0))⟩

for all λ ∈ 𝔽_{q}[T]. Let f be a newform of level N, then

\frac{∂}{∂s}(L^{(N,D)}(2s + 1)L(f, A, s)) |_{s=0}= \frac{q − 1}{2} q^{−(deg D+1)/2} ∫_{Γ_{0}(N)\GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗}} f · \overline{g_{A}}.

If deg D is even, we have to compare Theorem 2.8.3 and Theorem 3.6.6. Let Ψ_{A} be defined by

\frac{∂}{∂s}(\frac{1}{1 + q^{−s−1}} L^{(N,D)}(2s + 1)L(f, A, s)) |_{s=0}= ∫_{Γ_{0}(N)\GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗}} f · \overline{Ψ_{A}}.

The Fourier coefficients of Ψ_{A} and g_{A} satisfy

Ψ_{A}^{∗}(π_{∞}^{deg λ+2}, λ) = \frac{q − 1}{4} q^{− deg D/2} g_{A}^{∗}(π_{∞}^{deg λ+2}, λ)

for all λ ∈ 𝔽_{q}[T] with gcd(λ, N) = 1. Hence we have

Theorem 4.1.2 Let deg D be even. Let x be a Heegner point on X_{0}(N) with complex multiplication by O_{L} = 𝔽_{q}[T][\sqrt{D}], let A ∈ Cl(O_{L}), and let g_{A} be the automorphic cusp form of Drinfeld type of level N, which is given by

(T_{λ}g_{A})^{∗}(π_{∞}^{2}, 1) = ⟨(x) − (∞), T_{λ}((x)^{σ_{A}} − (0))⟩

for all λ ∈ 𝔽_{q}[T]. Let f be a newform of level N, then

\frac{∂}{∂s}(\frac{1}{1 + q^{−s−1}} L^{(N,D)}(2s+1)L(f, A, s)) |_{s=0}= \frac{q − 1}{4} q^{− deg D/2} ∫_{Γ_{0}(N)\GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗}} f ·\overline{g_{A}}.

4.2 Application to Elliptic Curves

We want to apply our main results to elliptic curves. Therefore we assume in addition that the newform f is an eigenform for all Hecke operators. So far we haven’t required this condition in our calculations. Let χ be a character of the class group Cl(O_{L}). If deg D is odd, we define

L(f, χ, s) := ∑_{A∈Cl(O_{L})} χ(A) L^{(N,D)}(2s + 1)L(f, A, s).

Then Theorem 4.1.1 yields immediately

\frac{∂}{∂s}L(f, χ, s) |_{s=0}= \frac{q − 1}{2} q^{−(deg D+1)/2} ∫_{Γ_{0}(N)\GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗}} f · \overline{∑_{A} χ(A)g_{A}}. (4.2.1)

Note that ∑_{A} χ(A)g_{A} is an automorphic cusp form which satisfies (cf. the definition of g_{A} in Proposition 3.1.1)

(T ∑_{A} χ(A)g_{A})^{∗}(π_{∞}^{2}, 1) = ∑_{A} χ(A)⟨(x) − (∞), T ((x)^{σ_{A}} − (0))⟩ (4.2.2)

for each Hecke operator T. Exactly the same calculations as in [Gr-Za], p. 308 show that (4.2.2) can be computed as

(T ∑_{A} χ(A)g_{A})^{∗}(π_{∞}^{2}, 1) = h_{L}^{−1}⟨c_{χ}, T c_{χ}⟩,

where c_{χ} = ∑_{A} χ^{−1}(A)((x) − (∞))^{σ_{A}} is an element in the Jacobian J_{0}(N)(H) ⊗ ℂ. Here we used the fact that (0) − (∞) is an element of finite order in J_{0}(N) (cf. [Ge2, Satz 4.1]). Let {f_{i}} be a basis of the space of automorphic cusp forms of Drinfeld type of level N which consists of normalized newforms together with a basis of the space of oldforms. We assume that f_{1} = f. And let c_{χ} = ∑_{i} c_{χ}^{(i)} be the decomposition of c_{χ} in f_{i}-isotypical components (i.e. components, where the Hecke operators act by multiplication of the corresponding Hecke eigenvalues). Then again as in [Gr-Za], p. 308 we get

∑_{A} χ(A)g_{A} = h_{L}^{−1} ∑_{i,j}⟨c_{χ}^{(i)}, c_{χ}^{(j)}⟩f_{j}.  (4.2.3)

Since f is a newform, we have ⟨c_{χ}^{(i)}, c_{χ}^{(1)}⟩ = 0 for i ≠ 1. Then equations (4.2.1) and (4.2.3) yield: Corollary of Theorem 4.1.1 If deg D is odd, then

\frac{∂}{∂s}L(f, χ, s) |_{s=0} = \frac{q − 1}{2} q^{−(deg D+1)/2} ·

·h_{L}^{−1} ⟨c_{χ}^{(1)}, c_{χ}^{(1)}⟩ ∫_{Γ_{0}(N)\GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗}} f · \overline{f}.

If deg D is even we define

L(f, χ, s) := ∑_{A∈Cl(O_{L})} χ(A) \frac{1}{1 + q^{−s−1}} L^{(N,D)}(2s + 1)L(f, A, s),

and we get analogously: Corollary of Theorem 4.1.2 If deg D is even, then

\frac{∂}{∂s}L(f, χ, s) |_{s=0} = \frac{q − 1}{4} q^{− deg D/2} ·

h_{L}^{−1} ⟨c_{χ}^{(1)}, c_{χ}^{(1)}⟩ ∫_{Γ_{0}(N)\GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗}} f · \overline{f}.

Now let E be an elliptic curve with conductor N ·∞, which has split multiplicative reduction at ∞, and let f be the corresponding newform as in section 2.1. We have already seen that the L-series of E over the imaginary quadratic field L satisfies (with the notations of this section) L(E, s+1)L(E_{D}, s+1) = L(f, χ_{0}, s), where χ_{0} is the trivial character of Cl(O_{L}). Let π : X_{0}(N) → E be a uniformization (cf. [Ge-Re], (8)) which maps the point ∞ on X_{0}(N) to the zero on E. The two homomorphisms π_{∗} : J_{0}(N) → E and π^{∗} : E → J_{0}(N) are related by the formula π_{∗} ∘ π^{∗} = deg π. On J_{0}(N) we consider the elliptic curve E′ = π^{∗}(E). Then π_{∗|E′} and π^{∗} are dual isogenies of E and E′, in particular we get

π^{∗} ∘ π_{∗|E′} = deg π.  (4.2.4)

For a Heegner point x on X_{0}(N) let P_{L} := ∑_{A∈Cl(O_{L})} π(x^{σ_{A}}) be the corre-

sponding Heegner point on E. The component c_{χ_{0}}^{(1)} lies on E′ and we have

π_{∗|E′}(c_{χ_{0}}^{(1)}) = P_{L}.  (4.2.5)

The points P_{L} on E and c_{χ_{0}}^{(1)} on J_{0}(N) are both defined over the field L. Let ĥ_{E,L} be the canonical Néron-Tate height of E over L, analogously we consider ĥ_{J_{0}(N),L}. If we apply the projection formula and equations (4.2.4) and (4.2.5) we get

deg π · ĥ_{J_{0}(N),L}(c_{χ_{0}}^{(1)}) = ⟨deg π · c_{χ_{0}}^{(1)}, c_{χ_{0}}^{(1)}⟩_{J_{0}(N),L}
= ⟨π^{∗} ∘ π_{∗|E′}(c_{χ_{0}}^{(1)}), c_{χ_{0}}^{(1)}⟩_{J_{0}(N),L}
= ⟨π_{∗|E′}(c_{χ_{0}}^{(1)}), π_{∗|E′}(c_{χ_{0}}^{(1)})⟩_{E,L}
= ĥ_{E,L}(P_{L}).  (4.2.6)

Since the height pairing ⟨ , ⟩ is normalized for the Hilbert class field H (cf. section 3.1), we have

⟨c_{χ_{0}}^{(1)}, c_{χ_{0}}^{(1)}⟩ = h_{L} · ĥ_{J_{0}(N),L}(c_{χ_{0}}^{(1)}).  (4.2.7)

Now (4.2.6), (4.2.7) and the two corollaries yield in the case of elliptic curves:

Theorem 4.2.1 Let E be an elliptic curve with conductor N · ∞, which has split multiplicative reduction at ∞, with corresponding newform f as above, let P_{L} ∈ E(L) be the Heegner point given by the parametrization π : X_{0}(N) → E. Then the derivative of the L-series of E over L and the canonical height ĥ_{E,L}(P_{L}) are related by the formula

\frac{∂}{∂s}(L(E, s)L(E_{D}, s)) |_{s=1}= ĥ_{E,L}(P_{L}) c(D) (deg π)^{−1} ∫_{Γ_{0}(N)\GL_{2}(K_{∞})/Γ_{∞}K_{∞}^{∗}} f · \overline{f},

where the constant

c(D) := \{ \begin{matrix}\frac{q−1}{2} q^{−(deg D+1)/2} & if deg D is odd, \\ \frac{q−1}{4} q^{− deg D/2} & if deg D is even. \end{matrix}

Finally we mention just one consequence of Theorem 4.2.1. The L-series L(E, s) · L(E_{D}, s) of E over the field L has a zero at s = 1 according to the functional equations of section 2.7. In the function field case it is known ([Ta], [Sh]) that the analytic rank of E/L is not smaller than the Mordell-Weil rank of E(L). Therefore Theorem 4.2.1 implies

Corollary 4.2.2 If

\frac{∂}{∂s}(L(E, s)L(E_{D}, s)) |_{s=1}≠ 0,

then the Birch and Swinnerton-Dyer conjecture is true for E, i.e. the analytic rank and the Mordell-Weil rank of E/L are both equal to 1.

Remarks.

1) In [Br] Brown proved the Birch and Swinnerton-Dyer conjecture in the case that the Heegner point has infinite order. And he conjectured that this assumption is true if and only if the first derivative of the L-series does not vanish at the point 1. Theorem 4.2.1 proves his conjecture.

2) Milne ([Mi]) showed that the equality of the analytic rank and the Mordell-Weil rank implies even the strong Birch and Swinnerton-Dyer conjecture. Therefore in our case the assumption of Corollary 4.2.2 implies

\frac{∂}{∂s}L^{∗}(E/L, s) |_{s=1}= \frac{#� · ĥ_{E,L}(P_{0})}{(#E(L)_{tors})^{2}},

where L^{∗}(E/L, s) is the modified L-series of the elliptic curve E over the field L (cf. [Ta], [Mi]), P_{0} is a generator of the free part of the Mordell-Weil group E(L) and � is the Tate-Shafarevich group of E/L.

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Hans-Georg Rück Fachbereich 17 D-34109 Kassel, Germany rueck@mathematik.uni-kassel.de Ulrich Tipp secunet AG Im Teelbruch 116 D-45219 Essen tipp@secunet.de