ON THE VARIATION OF TATE-SHAFAREVICH GROUPS OF ELLIPTIC CURVES OVER HYPERELLIPTIC CURVES

MIHRAN PAPIKIAN

Abstract. Let E be an elliptic curve over F = 𝔽_{q}(t) having conductor (𝔭)·∞, where (𝔭) is a prime ideal in 𝔽_{q}[t]. Let 𝔡 ∈ 𝔽_{q}[t] be an irreducible polynomial of odd degree, and let K = F (\sqrt{𝔡}). Assume (𝔭) remains prime in K. We prove the analogue of the formula of Gross for the special value L(E ⊗_{F} K, 1). As a consequence, we obtain a formula for the order of the Tate-Shafarevich group � (E/K) when L(E ⊗_{F} K, 1) ≠ 0.

1. Introduction

Let E be an elliptic curve over the function field F := 𝔽_{q}(t) of ℙ_{𝔽_{q}}^{1}, where q is odd. Assume E has conductor (𝔭) · ∞, where ∞ is the place corresponding to 1/t and 𝔭 ∈ A := 𝔽_{q}[t] is a prime. Further assume that the reduction of E at ∞ is split multiplicative. In this situation it is known that E is a quotient of the Drinfeld Jacobian variety J := J_{0}(𝔭); see [9].

Let S := {x_{1}, x_{2}, …, x_{n}} be the set of isomorphism classes of super-singular Drinfeld modules over \overline{𝔽}_{𝔭}, where we write 𝔽_{𝔭} := A/(𝔭). It is known that n = dim(J) + 1. For each x_{i} ∈ S we let ϕ_{i} denote a super-singular Drinfeld module representing the isomorphism class corresponding to x_{i}. Let M denote the free ℤ-module on the set S, and deg : M → ℤ denote the ℤ-linear map obtained by sending each x_{i} ∈ S to 1 ∈ ℤ, and let M^{0} denote the kernel of deg. Define a symmetric, bilinear, ℤ-valued pairing on M by the formula

(1.1)  ⟨x_{i}, x_{j}⟩ = \frac{1}{q − 1}#Isom(ϕ_{i}, ϕ_{j}).

In particular, ⟨x_{i}, x_{j}⟩ = 0 for i ≠ j. It is known that Aut(ϕ_{i}) ≅ 𝔽_{q}^{×} or 𝔽_{q^{2}}^{×} (the latter case can occur only when deg 𝔭 is odd), so ⟨x_{i}, x_{i}⟩ = 1 or (q + 1).

Denote by E the Néron model of E over ℙ_{𝔽_{q}}^{1}. Let E^{0} be the relative connected component of the identity of E, i.e., the largest open subgroup-scheme of E in which all fibres are connected. Similarly, denote by J and J^{0} the Néron model of J and its relative connected component of the identity. It is known, c.f. §3, that the closed fibre J_{\overline{𝔽}_{𝔭}}^{0} is a torus and the character group Hom_{\overline{𝔽}_{𝔭}}(J_{\overline{𝔽}_{𝔭}}^{0}, 𝔾_{m,\overline{𝔽}_{𝔭}}) is canonically isomorphic to M^{0}. Moreover, the pairing in (1.1) restricted to M^{0} is Grothendieck’s monodromy pairing discussed in [12]. The character group Υ of E_{\overline{𝔽}_{𝔭}}^{0} is isomorphic to ℤ. We choose the quotient map π : J → E canonically to satisfy a ceratin minimality property; see §5.2. Let ρ be a generator of Υ. There results a functorial homomorphism between character groups π^{∗} : Υ → M^{0}. Let H_{E} := π^{∗}(ρ) ∈ M^{0}.

1991 Mathematics Subject Classification. Primary 11G05, 11G40; Secondary 11G18.

Key words and phrases. Formula of Gross, Tate-Shafarevich group, Drinfeld modular curves.

Supported in part by the European Postdoctoral Institute Fellowship.

Now let 𝔡 be an irreducible polynomial in A of odd degree. Let K = F (\sqrt{𝔡}). The field K is the function field of a hyperelliptic curve over 𝔽_{q}. The extension K/F is ramified only at (𝔡) and ∞. Let O be the integral closure of A in K. If we assume that the ideal (𝔭) remains prime in O then the endomorphism rings of some super-singular Drinfeld modules ϕ_{i} contain O as a subring. There results an action of Pic(O) on a subset of S, and one produces from this action an element H_{K} ∈ M; see §2.

Denote by E_{K} := E ⊗_{F} K the base change of E to K. The first main result of this paper is the following theorem:

Theorem 1.1. L(E_{K}, 1) = 0 if and only if ⟨H_{E}, H_{K}⟩ = 0.

If we assume L(E_{K}, 1) ≠ 0 then Tate [23] proved that E(K) and the Tate-Shafarevich group � (E/K) are finite, and the formula for L(E_{K}, 1) predicted by the conjecture of Birch and Swinnerton-Dyer holds; see also [14]. Our next main result gives a formula for the order of � (E/K) in terms of ⟨H_{E}, H_{K}⟩:

Theorem 1.2. If ⟨H_{E}, H_{K}⟩ ≠ 0 then

#� (E/K) = (⟨H_{E}, H_{K}⟩ · \frac{#E(F)}{#Φ_{E,𝔭}} · q^{(deg Ω_{E/ℙ^{1}}^{1} |_{O}−1)})^{2} ,

where Φ_{E,𝔭} := E_{\overline{𝔽}_{𝔭}}/E_{\overline{𝔽}_{𝔭}}^{0} is the group of connected components of E_{\overline{𝔽}_{𝔭}} and Ω_{E/ℙ^{1}}^{1}|_{O} is the pullback of Ω_{E/ℙ^{1}}^{1} along the relative zero section.

In particular, this theorem says that the variation of #� (E/K) over different K depends only on the relative position of H_{K} and H_{E} in M. Also, the formula in the theorem can be used to compute #� (E/K), c.f. Example 5.5.

To prove Theorem 1.1 we will prove a more general result which gives a formula for the special values of L-functions of Drinfeld cusp forms at the center of the critical strip. This formula is the function field analogue of the result of Gross over ℚ [11]. Our proof follows closely the strategy in loc.cit. The required analytic calculations involving Rankin convolutions are already carried out in [18], where the authors prove the analogue of the Gross-Zagier formula for the derivatives of L-functions. Hence we only need to explicitly compute the pairing (1.1) between certain special elements of M. Theorem 1.2 is a consequence of the analogue of Gross’ formula and the theorem of Tate [23]. We make the restriction on 𝔡 being irreducible and the characteristic being odd mainly because the analytic formulae in [18] are proven under these assumptions.

The contents of the paper are as follows. We start section 2 by proving some auxiliary results about endomorphism rings of super-singular Drinfeld modules. Using these results, later in the same section we carry out the main technical calculation of the paper, which is an explicit formula for the pairing between certain elements of M. In section 3 we prove the analogue of Eichler’s theorem over F. We show that a certain set of explicit theta series arising from quaternion algebras over F spans the whole space of Drinfeld automorphic forms. (This theorem might be of some independent interest.) In section 4 we combine our previous results with the calculations in [18] to deduce the analogue of the formula of Gross. Finally, section 5 discusses the applications of aforementioned formula to the arithmetic of elliptic curves. In particular, it contains the proof of Theorem 1.2. We close this introduction by remarking that it would be very interesting to have some cohomological explanation for our results.

Acknowledgement. The author wishes to thank Brian Conrad for numerous helpful remarks on an earlier version of this paper.

2. The arithmetic of super-singular Drinfeld modules

The proof of the main result in [11] consists of two, fairly independent, parts. The first part, which is very analytic in nature, is a calculation of a Rankin integral. The output of this calculation is the fact that the values of certain L-functions at the center of their critical strip are equal, up to a non-zero constant, to the Petersson product of two modular forms. One of those modular forms is a cusp form of weight two and the other is a modular form with very explicit Fourier expansion. The second part of the proof in [11] is an algebraic calculation involving maximal orders in quaternion algebras over ℚ. This calculation shows that the modular form obtained in the first step is a certain distinguished theta series arising from a quaternion algebra. Using these two steps, Gross deduces interesting arithmetic facts which complement his results with Zagier on the the special values of the derivatives of L-functions.

The analogue over F of the analytic portion of calculations in [11] was done by Rück and Tipp [18] (it is an intermediate step in their proof of the analogue of the Gross-Zagier formula). In this section we carry out the analogue of algebraic calculations. The main result is Corollary 2.12. Since the calculations are somewhat tedious and might seem not very motivated at this stage, we point out that the expression in Corollary 2.12 is a Fourier coefficient of a certain theta series which comes up in later sections. In particular, this theta series appears in the statement of the analogue of Gross’ formula; see Theorem 4.1. The reader might choose to simply skim though this section as the proofs in the section are not essential for understanding the main results of the paper.

2.1. Quaternion algebras over function fields. Let F = 𝔽_{q}(t) and let A = 𝔽_{q}[t]; we will assume the characteristic is not 2. A central simple algebra B over F is a quaternion algebra if dim_{F} B = 4. From Wedderburn’s structure theorem [16, (7.4)] one concludes that a quaternion algebra is either isomorphic to the matrix algebra M_{2}(F) or is a division algebra. The quaternion algebra B is said to be split at a place v of F if B_{v} := B ⊗_{F} F_{v} ≅ M_{2}(F_{v}); it is said to be ramified if B_{v} is a division algebra. The fundamental exact sequence of Brauer groups from global class field theory implies that any quaternion algebra is split at all but finitely many places and the number of places where it is ramified is even. Conversely, for any even set of places of F there is a unique quaternion algebra ramified exactly at those places and split at all the others. In particular, B ≅ M_{2}(F) if and only if it is split at all places of F. For a, b ∈ F^{×}, let H(a, b) be the F -algebra with basis 1, i, j, k (as a F -vector space) and relations i^{2} = a, j^{2} = b, ij = k = −ji. One easily checks that this is a quaternion algebra. Conversely, using the Skolem-Noether theorem [16, (7.21)], it is not hard to show that every quaternion algebra is isomorphic to H(a, b) for some (non-unique) a, b ∈ F^{×}.

For any finite dimensional F -vector space V, a full A-lattice in V is a finitely generated A-submodule M in V such that F ⊗_{A} M ≅ V. An A-order in the F - algebra B is a subring Λ of B, having the same unity element as A, and such that Λ is a full A-lattice in B. A maximal A-order in B is an A-order which is not contained in any other A-order in B. Orders exist and every order is contained in a maximal order [16, (10.4)]. Given a full A-lattice M in B, define the left order of M to be

O_{l}(M) = {a ∈ B | aM ⊆ M}.

One easily checks that this is indeed an order. Similarly to O_{l}(M), one also defines the right order O_{r}(M) of M.

Let Λ be an A-order in B. A full A-lattice I in B is called a left ideal of Λ (resp. right ideal, two-sided ideal) if it is stable under the left multiplication by Λ (resp. under the right multiplication, under multiplication on the right and on the left). Define Nr(I) ⊂ F, the reduced norm of I, to be the fractional A-ideal generated by {Nr(a) | a ∈ I}, where Nr = Nr_{B/F} is the canonical reduced norm on B.

If S is a finite-length A-module, let χ(S) be the Euler-Poincaré characteristic of S, which is an ideal of A uniquely determined by the conditions

(i) χ(S) = 𝔮 if S ≅ A/𝔮 with a prime ideal 𝔮 of A; (ii) If 0 → S_{1} → S → S_{2} → 0 is exact, then χ(S) = χ(S_{1})χ(S_{2}). The statements in the next proposition are well-known (the proofs, which are

not hard, can be found in [16]). From now on we assume B is a division algebra.

Proposition 2.1. Let I be a left ideal of some maximal order Λ in B. Let I^{−1} = {a ∈ B | IaI ⊆ I}.

(1) Let M be any full A-lattice in B. The order O_{l}(M) is maximal if and only if O_{r}(M) is maximal.

(2) If I is an integral ideal of Λ, i.e., I ⊆ Λ, then Nr(I)^{2} = χ(Λ/I).

(3) For any α ∈ Λ, Nr(α) ∈ A.

(4) I^{−1} is a right Λ-ideal, and left O_{r}(I)-ideal.

(5) II^{−1} = Λ, I^{−1}I = O_{r}(I), (I^{−1})^{−1} = I.

(6) If J is a left O_{r}(I)-ideal, then Nr(IJ) = Nr(I)Nr(J). In particular,

Nr(I^{−1}) = Nr(I)^{−1}.

(7) If J ⊂ I is another left Λ-ideal and Nr(J) = Nr(I) then J = I. In particular, if there is an element α ∈ I such that Nr(α) = Nr(I) then I = Λα.

Let Λ be a maximal A-order in B. Two left Λ-ideals I and J are said to be equivalent if there is a ∈ B with J = Ia. The number of equivalence classes of left Λ-ideals is called the class number of B. It is known that the class number is finite and is independent of the choice of Λ. If {I_{1}, I_{2}, …, I_{n}} is a set of left Λideals representing the distinct ideal classes, then each conjugacy class of maximal A-orders in B is represented in the set of right orders {O_{r}(I_{1}), O_{r}(I_{2}), …, O_{r}(I_{n})}.

Let 𝔭 ∈ A be a monic irreducible polynomial. We will denote 𝔽_{𝔭} := A/(𝔭). The finite field 𝔽_{𝔭} is the field extension of 𝔽_{q} of degree equal to the degree of 𝔭. For μ ∈ A we will denote by (μ) the principal ideal in A generated by μ.

Let 𝔡 be an irreducible polynomial in A of odd degree. Let K = F (y), where y^{2} = 𝔡. Let O := A[y]; this ring is the integral closure of A in K since we are assuming the characteristic of F is not 2, c.f. [13, §9]. The ideal D = (y) is the different of O/A. The only primes of F which ramify in K are (𝔡) and ∞.

Lemma 2.2. Let B_{𝔭} be the unique quaternion algebra over F which is ramified exactly at (𝔭) and ∞. Assume (𝔡) ≠ (𝔭). The field K embeds in B_{𝔭} if and only if (\frac{𝔡}{𝔭}) = −1. In case K ↪ B_{𝔭}, there is a canonical isomorphism B_{𝔭} = K ⊕ Kj,

where jα = \overline{α}j for α ∈ K and j^{2} = c𝔭 for an appropriate c ∈ 𝔽_{q}^{×}.

Proof. We know that B_{𝔭} ≅ H(a, b) for some a, b ∈ F^{×}. The algebra H(a, b) is split over a field L if and only if the quadratic form Q = X^{2} − aY^{2} − bZ^{2} + abT^{2} corresponding to the reduced norm on H(a, b) has a non-zero solution in L. To show that for particular a and b there is an isomorphism B_{𝔭} ≅ H(a, b), it is enough to show that H(a, b) is ramified at 𝔭 and is split at every other place not equal to ∞ (it then automatically will be ramified at ∞, since the number of ramified places must be even). Since the characteristic is not 2, using Hensel’s lemma, it is enough to show that Q has a non-zero solution in 𝔽_{𝔩} for every 𝔩 ≠ 𝔭, ∞, and has no non-zero solutions in 𝔽_{𝔭}. By an easy counting argument, any quadratic form of more than two variables has a non-zero solution over a finite field.

Suppose K ↪ B_{𝔭}. Since the completion F_{𝔭} is flat over F, there is an injection K ⊗_{F} F_{𝔭} ↪ B_{𝔭} ⊗_{F} F_{𝔭}. The algebra on the right hand-side is a division algebra by assumption. Hence K ⊗_{F} F_{𝔭} is a field, which implies that 𝔡 is not a square modulo 𝔭, i.e., (\frac{𝔡}{𝔭}) = −1. Conversely, suppose (\frac{𝔡}{𝔭}) = −1 and let c ∈ 𝔽_{q}^{×} \ (𝔽_{q}^{×})^{2}. Using

the remarks in the previous paragraph, one easily checks that in case (\frac{𝔭}{𝔡}) = 1 the algebra H(𝔡, 𝔭) is ramified exactly at 𝔭 and ∞. If (\frac{𝔭}{𝔡}) = −1 then H(𝔡, c𝔭) has the same property. In both cases we obviously have an injection K ↪ B_{𝔭} and the decomposition B_{𝔭} = K ⊕ Kj stated in the lemma. □

Lemma 2.3. Assume (\frac{𝔡}{𝔭}) = −1, so that B_{𝔭} = K ⊕ Kj. Let c ∈ 𝔽_{q}^{×} be as in

Lemma 2.2, and let ε ∈ A be a solution of the congruence ε^{2} ≡ c𝔭 (mod 𝔡). If we let Λ be the set

{α + βj | α, β ∈ D^{−1}, α ≡ εβ (mod O_{D})},

where D^{−1} is the inverse different of O/A and O_{D} is the localization of O at the prime ideal D, then Λ is a maximal order which contains O ⊕ Oj.

Proof. It is clear that Λ is a full A-lattice in B_{𝔭} and that it contains O ⊕ Oj. Let α_{1} + β_{1}j and α_{2} + β_{2}j be two elements of Λ. We have

(α_{1} + β_{1}j)(α_{2} + β_{2}j) = α′ + β′j,

where α′ = α_{1}α_{2} +c𝔭β_{1}β̄_{2} and β′ = α_{1}β_{2} +β_{1}ᾱ_{2}. To check that Λ is a ring we need to check that α′, β′ ∈ D^{−1} and α′ ≡ εβ′ (mod O_{D}). This is a straightforward but tedious calculation which we omit (one needs to use along the way that Tr_{K/F} (β_{2}) ∈ A and 𝔡β_{1}β_{2} ∈ O). Thus, Λ is an order.

Denote Δ = O ⊕ Oj; this is an order in B_{𝔭}. An A-basis for Δ is given by 1, i, j, ij, where i^{2} = 𝔡. As one easily checks, the discriminant of Δ with respect to the reduced trace on B_{𝔭} is equal to the ideal (𝔭𝔡)^{2}. The discriminant of any A-order in B_{𝔭} is divisible by (𝔭)^{2} and the maximal orders are characterized by the property that their discriminants are equal to the ideal (𝔭)^{2}. We have a strict containment Δ ⊂ Λ. Hence

(𝔭𝔡)^{2} = disc(Λ) · χ(Λ/Δ)^{2}.

Since 𝔡 is a prime, we must have disc(Λ) = (𝔭)^{2}, so Λ is maximal as claimed. □

Lemma 2.3 will be used in §2.4, but first we need few facts about the endomorphism rings of super-singular Drinfeld modules.

2.2. Super-singular Drinfeld modules. Let k be an algebraic extension of 𝔽_{𝔭}. There is a canonical A-structure on k given by γ : A → 𝔽_{𝔭} ↪ k. Let τ be the Frobenius endomorphism relative to 𝔽_{q}, i.e., the map x ↦ x^{q}. Denote by k{τ} the non-commutative polynomial algebra in τ subject to the commutation rule τx = x^{q}τ, x ∈ k.

A Drinfeld module (of rank 2) over k is a structure of A-module on k given by a ring homomorphism

ϕ : A = 𝔽_{q}[t] → k{τ}
a ↦ ϕ_{a}

such that ϕ_{t} = γ(t) + gτ + Δτ^{2} and Δ ≠ 0. A k-isogeny between two Drinfeld modules ϕ and ψ is an element u ∈ k{τ} such that u ∘ ϕ_{a} = ψ_{a} ∘ u for all a ∈ A. Each 0 ≠ f ∈ k{τ} can uniquely be written as f = f_{s} ∘ τ^{h}, where f_{s} has a non-zero constant coefficient. The number h = ht(f) is called the height of f. The height of the Drinfeld module ϕ is the height of ϕ_{𝔭} divided by deg(𝔭). It is equal to 1 or 2.

Suppose u = γ(a) + g_{1}τ + ⋯ + g_{s}τ^{s} ∈ k{τ}. We will denote by

u(X) = γ(a)X + g_{1}X^{q} + ⋯ + g_{s}X^{q^{s}} ∈ k[X]

the corresponding q-additive polynomial. The scheme-theoretic kernel ker(u) = k[X]/ϕ_{a}(X) is a commutative finite flat group-scheme of 𝔽_{q}-vector spaces. Conversely, it is easy to see that given a finite subgroup scheme of 𝔽_{q}-vector spaces H ⊂ 𝔾_{a,k}, there is a unique polynomial u ∈ k{τ} with H = ker(u)

Lemma 2.4. With notation as above, assume in addition that H has a structure of an A-module via the Drinfeld module ϕ over k. Then u is an isogeny from ϕ if and only if ht(u) is divisible by deg 𝔭.

Proof. See [10, Prop.2.5]. □

If H satisfies the conditions of Lemma 2.4, then we will say that H is a subgroupscheme of ϕ and will denote the corresponding isogeny by u_{H}. The Drinfeld module ϕ is called super-singular (s.s.) if ker(ϕ_{𝔭}) is connected (equivalent conditions are: ϕ_{𝔭}(X) is purely inseparable, or ht(ϕ) = 2). Let

End(ϕ) = {u ∈ k{τ} | u ∘ ϕ_{a} = ϕ_{a} ∘ u for all a ∈ A}.

Theorem 2.5. Let ϕ be a s.s. Drinfeld module over \overline{𝔽}_{𝔭}. The endomorphism ring End(ϕ) is a maximal order in B_{𝔭}.

Proof. See [6]. □

Fix a s.s. Drinfeld module ϕ over \overline{𝔽}_{𝔭}, and denote Λ = End(ϕ). Let I be a left integral Λ-ideal (i.e., I is a full A-lattice in B_{𝔭}, I ⊆ Λ and ΛI ⊆ I). To such an ideal one can associate another Drinfeld module ϕ^{(I)} and an isogeny u_{I} : ϕ → ϕ^{(I)} (since ϕ^{(I)} is isogenous to ϕ it is necessarily s.s.). Indeed, consider the group-scheme

H_{I} = ⋂_{i∈I} ker(i),

with the scheme-theoretic intersection taken inside of 𝔾_{a,\overline{𝔽}_{𝔭}}. It is easy to check that H_{I} satisfies the conditions in Lemma 2.4 , hence gives an isogeny u_{I}. We will denote ker(I) := H_{I}.

Define the norm 𝔫(u) of the isogeny u of ϕ by

𝔫(u) = (𝔭)^{ht(u)/ deg 𝔭} · χ(ker(u)(\overline{𝔽}_{𝔭})).

Theorem 2.6. With previous notation, we have (1) 𝔫(u_{I}) = Nr(I);

(2) Every isogeny from ϕ to another Drinfeld module has the form u_{I} for some left integral ideal I of Λ;

(3) End(ϕ^{(I)}) ≅ O_{r}(I);

(4) There is a one-to-one correspondence between the isomorphism classes of s.s. Drinfeld modules over \overline{𝔽}_{𝔭} and the left ideal classes in B_{𝔭};

(5) All s.s. Drinfeld modules over \overline{𝔽}_{𝔭} are isogenous to each other.

Proof. See [6, §§3-4]. □

Let Λ be a fixed maximal order in B_{𝔭}, {I_{1}, …, I_{n}} be integral representatives of distinct left ideal classes of Λ with I_{1} = Λ. Denote Λ_{i} = O_{r}(I_{i}). From Theorem 2.6 we know that there exist s.s. Drinfeld modules ϕ_{1}, …, ϕ_{n} with ϕ_{i} ≅ ϕ_{1}^{(I_{i})} and End(ϕ_{i}) = Λ_{i}. Denote M_{ij} = I_{j}^{−1}I_{i}. Since O_{r}(I_{j}^{−1}) = Λ, this is a lattice in B_{𝔭}, and it is naturally a left Λ_{j} and right Λ_{i} module.

Proposition 2.7.

(1) There is an isomorphism

M_{ij} ≅ Hom(ϕ_{i}, ϕ_{j})

as left Λ_{j} and right Λ_{i} modules.

(2) If v_{b} : ϕ_{i} → ϕ_{j} is the isogeny corresponding to the non-zero element b ∈ M_{ij}

then

𝔫(v_{b}) = (Nr(b)/m_{ij}),

where m_{ij} is a generator of the fractional ideal Nr(M_{ij}).

Proof. Let α ∈ I_{i}. Consider the endomorphism of ϕ := ϕ_{1} induced by α. Since ker(α) ⊇ ker(I_{i}), α : ϕ → ϕ must factor through u_{I_{i}}

\xymatrix{ϕ \ar[rr]^{α} \ar[dr]_{u_{I_{i}}} &  & ϕ \\  & ϕ_{i} \ar[ur]_{v_{α}}}

where v_{α} ∈ Hom(ϕ_{i}, ϕ). One easily checks that α ↦ v_{α} defines an injection I_{i} ↪ Hom(ϕ_{i}, ϕ) of left Λ and right Λ_{i} modules. Now let v ∈ Hom(ϕ_{i}, ϕ). The composite v ∘ u_{I_{i}} is an endomorphism w of ϕ. We claim that w ∈ I_{i}. Indeed, consider the left integral Λ-ideal generated by w and I_{i}, Λw + I_{i}. We have ker(Λw + I_{i}) = ker(w) ∩ ker(I_{i}). Since by construction ker(w) contains ker(I_{i}), we get ker(Λw+I_{i}) = ker(I_{i}). Hence by Theorem 2.6 and Proposition 2.1, w +I_{i} = I_{i}, i.e., w ∈ I_{i} as required. We conclude that the map α ↦ v_{α} constructed above is also surjective, in particular

(2.1)  I_{i} ≅ Hom(ϕ_{i}, ϕ).

Let I be a left integral Λ_{j}-ideal. The lattice I_{j}I is a left Λ-ideal. We claim that

(2.2)  ϕ^{(I_{j}I)} ≅ ϕ_{j}^{(I)}.

After scaling by an element of A, we can assume I_{j}I is an integral Λ-ideal. Since O_{r}(I_{j}) = Λ_{j} and I ⊆ Λ_{j}, we have I_{j}I ⊆ I_{j}. Hence ker(I_{j}I) ⊇ ker(I_{j}), and

ker(I_{j}I) mod ker(I_{j}) ≅ ker(I).

We conclude u_{I} ∘ u_{I_{j}} = u_{I_{j}I}, which is equivalent to the claim.

Now apply (2.1) with ϕ replaced by ϕ_{j} and ϕ_{i} replaced by ϕ_{j}^{(M_{ij})}. (Here we consider M_{ij} as a left Λ_{j}-ideal.) We have

M_{ij} ≅ Hom(ϕ_{j}^{(M_{ij})}, ϕ_{j}) (by (2.1))
≅ Hom(ϕ^{(I_{j}M_{ij})}, ϕ_{j}) (by (2.2))
≅ Hom(ϕ^{(I_{i})}, ϕ_{j}) (by Proposition 2.1)
≅ Hom(ϕ_{i}, ϕ_{j}),

where all the isomorphisms are isomorphisms of left Λ_{j} and right Λ_{i} ideals.

To prove (2), consider the left principal ideal Λ_{j}b. The isogeny v_{b} is the isogeny for which u_{Λ_{j}b} = v_{b} ∘ u_{M_{ij}}

\xymatrix{ϕ_{j} \ar[rr]^{u_{Λ_{j} b}} \ar[dr]_{u_{M_{ij}}} &  & ϕ_{j} \\  & ϕ_{i} \ar@{-->}[ur]_{v_{b}}}

From this it is easy to see that v_{b} is invariant under scaling M_{ij} by elements of A, so we can assume M_{ij} is an integral left Λ_{j}-ideal. Moreover,

𝔫(v_{b}) = 𝔫(u_{Λ_{j}b})/𝔫(u_{M_{ij}}).

The claim follows from Theorem 2.6 and Proposition 2.1. Note that the expression Nr(b)/m_{ij} is invariant under scaling M_{ij} by elements of A. □

2.3. Brandt matrices. Let ϕ_{1}, ϕ_{2}, …, ϕ_{n} be the isomorphism classes of s.s. Drinfeld modules over \overline{𝔽}_{𝔭} arranged in some fixed order. Let 𝔪 be any monic element of A. Let b_{i,j}(𝔪) be the number of subgroup-schemes H of ϕ_{i} such that u_{H}(ϕ_{i}) = ϕ_{j} and 𝔫(u_{H}) = (𝔪). The n × n matrix B(𝔪) = (b_{i,j}(𝔪))_{1≤i,j≤n} is called the 𝔪-th Brandt matrix. Using Proposition 2.7, the Brandt matrices can also be defined using only the data of the quaternion algebra B_{𝔭}.

Let M = ⊕_{i=1}^{n}ℤx_{i} be a free ℤ-module of rank n, where x_{i} corresponds to ϕ_{i}, 1 ≤ i ≤ n. For each 𝔪 we have B(𝔪) ∈ End_{ℤ}(M). Let w_{i} = #Aut(ϕ_{i})/(q − 1). It is known that w_{i} is equal either to 1 or q + 1; c.f. [4]. Define a positive definite ℤ-valued pairing

(2.3)  ⟨, ⟩ : M × M → ℤ
x_{i}, x_{j} ↦ ⟨x_{i}, x_{j}⟩ = w_{i}δ_{ij},

where δ_{ij} is the Kroneker symbol.

Theorem 2.8.

(1) The row sums ∑_{j} b_{i,j}(𝔪) are independent of i and are equal to

σ_{𝔭}(𝔪) := _{∑_{𝔪′monic} ; 𝔪′|𝔪,𝔭|̸𝔪′}q^{deg(𝔪′)}.

(2) The matrices {B(𝔪) | 𝔪 ∈ A, 𝔪 monic} generate a commutative subring 𝔹 of Mat(n, ℤ).

(3) The action of B(𝔪) is symmetric with respect to the pairing ⟨, ⟩ on M, i.e., for e, e′ ∈ M we have ⟨B(𝔪)e, e′⟩ = ⟨e, B(𝔪)e′⟩.

(4) The commutative algebra 𝔹 ⊗_{ℤ} ℚ is semi-simple and isomorphic to the product of totally real number fields.

Proof. The proof is very similar to the proof of [11, Prop.2.7]. □

Remark 2.9. The traces of B(𝔪)’s are important arithmetic invariants, which will be related to the traces of Hecke operators on the Drinfeld modular curves in §3.

2.4. Main technical calculation. Let 𝔡 be an odd degree irreducible polynomial in A. Assume (\frac{𝔡}{𝔭}) = −1. Let K = F (y), where y^{2} = 𝔡. By Lemma 2.2 K embeds

in B_{𝔭} and gives a decomposition B_{𝔭} = K ⊕ Kj, where j^{2} = c𝔭 for an appropriate c ∈ 𝔽_{q}^{×}, jα = ᾱj for all α ∈ K. Let Λ be a maximal order which contains O, and let I be an integral ideal of O. Consider a s.s. Drinfeld module ϕ over \overline{𝔽}_{𝔭} with endomorphism ring Λ. Since ΛI is a left Λ-ideal, we can form ϕ^{(ΛI)}. The left ideal class of ΛI in B_{𝔭} depends on the ideal class I ∈ Pic(O) of I rather than on I itself. Hence ϕ^{(ΛI)} is well-defined by I and we denote it by ϕ^{(I)}. In this way we get an action of the group Pic(O) on the set of isomorphism classes of s.s. Drinfeld modules over \overline{𝔽}_{𝔭} whose endomorphism rings contain O under the above fixed embedding of K into B_{𝔭}. (To see how O embeds in End(ϕ^{(I)}) = O_{r}(ΛI) note that O naturally acts on the right of ΛI.)

Now let Λ be a maximal order which contains O ⊕ Oj. Let M = ⊕_{i=1}^{n}ℤx_{i} be the free ℤ-module defined in §2.3, and assume End(ϕ_{1}) = Λ. We have defined an action of Pic(O) on ϕ_{1}, which we can transport in a formal manner into an action on x_{1}. The image of x_{1} under the action of ideals in the class I will be denoted by x_{I}. We had an action of the ℤ-algebra of Brandt matrices on M. The main result of this subsection is the calculation of

⟨x_{J} , B(𝔪)x_{IJ} ⟩,

where ⟨·, ·⟩ is the pairing in (2.3) and B(𝔪) ∈ 𝔹. From definitions

(2.4)  ⟨x_{J} , B(𝔪)x_{IJ} ⟩ = \frac{1}{q − 1}#\{u ∈ Hom(x_{IJ} , x_{J}) | 𝔫(u) = (𝔪)\}.

Proposition 2.10. Let 𝔞 and 𝔟 be integral ideals in classes I and J which are relatively prime to D = (\sqrt{𝔡}). We have a bijection

Hom(x_{IJ} , x_{J}) ≅ {α + βj | α ∈ D^{−1}𝔞, β ∈ D^{−1}𝔟^{−1}𝔟̄𝔞̄, α ≡ εβ (mod O_{D})},

where ε^{2} ≡ c𝔭 (mod 𝔡). If u corresponds to α + βj then

𝔫(u) = (Nr_{K/F} (α) + c𝔭Nr_{K/F} (β))/Nr_{K/F} (𝔞).

Here c is the constant appearing in Lemma 2.2.

Proof. By the definition of the action of Pic(O), the left ideal class of x_{IJ} is Λ𝔞𝔟, and similarly the left ideal class of x_{J} is Λ𝔟. By Proposition 2.7

Hom(x_{IJ} , x_{J}) ≅ (Λ𝔟)^{−1}(Λ𝔞𝔟) = 𝔟^{−1}Λ𝔞𝔟.

From Lemma 2.3 we get the desired expression for Hom(x_{IJ} , x_{J}). Indeed, the factors 𝔟̄𝔞̄ are due to the relation jx = x̄j for all x ∈ K, and since we assumed 𝔞 and 𝔟 to be relatively prime to D, locally at D these ideals are the unit ideals so the congruence relation is preserved.

The final statement of the proposition follows from Proposition 2.7. In fact, Nr(α + βj) = Nr_{K/F} (α) + c𝔭Nr_{K/F} (β) and

Nr(𝔟^{−1}Λ𝔞𝔟) = Nr_{K/F} (𝔟)^{−1}Nr_{K/F} (𝔟)Nr_{K/F} (𝔞) = Nr_{K/F} (𝔞).
□

For an integral ideal 𝔞 of O and a fixed element λ_{0} ∈ A define

r_{𝔞,λ_{0}}(λ) = #{μ ∈ 𝔞 | Nr_{K/F} (μ) = λ_{0}λ}.

Let L be an integral ideal of A. For an ideal class A of O define

r_{A}(L) = #{𝔞 ∈ A | 𝔞 integral with Nr_{L/K}(𝔞) = L}.

and

R(L) = ∑_{A∈Pic(O)} r_{A}(L).

For 0 ≠ μ ∈ A, define

σ(μ) = \begin{cases}q − 1 & if μ ≡ 0 mod 𝔡, \\ 1 & otherwise. \end{cases}

Let δ_{z}, z ∈ F_{∞}^{×}, be the local norm symbol at ∞, i.e., δ_{z} is equal to 1 if z is the norm of an element of F_{∞}(\sqrt{𝔡})/F_{∞} and −1 otherwise.

Proposition 2.11. We have the equality

⟨x_{J} , B(𝔪)x_{IJ} ⟩ = r_{I^{−1}}((𝔡𝔪))

+_{∑_{0≠μ∈A} deg μ≤deg(𝔪𝔡)−deg(𝔭)}r_{I^{−1}}((𝔡𝔪 − 𝔭μ)) · r_{IJ^{2}}((μ)) · σ(μ)\frac{1 − δ_{(𝔡𝔪−𝔭μ)𝔭μ}}{2} .

Proof. Let 𝔞 be a fixed ideal of I and let λ_{0} ∈ A be a fixed generator of Nr_{K/F} (𝔞). By (2.4) and Proposition 2.10 we need to count the number of solutions to the identity

(2.5)  (Nr_{K/F} (α) + c𝔭Nr_{K/F} (β)) = (𝔪λ_{0}),

with α ∈ D^{−1}𝔞, β ∈ D^{−1}𝔟^{−1}𝔟̄𝔞̄, α ≡ εβ (mod O_{D}), where ε^{2} ≡ c𝔭 (mod 𝔡). Define

ν := Nr_{K/F} (α)𝔡λ_{0}^{−1} ∈ A, and μ := cNr_{K/F} (β)𝔡λ_{0}^{−1} ∈ A.

Since (νλ_{0}𝔡^{−1} + 𝔭λ_{0}𝔡^{−1}μ) = (𝔪λ_{0}), we are looking for ν and μ such that

(ν + 𝔭μ) = (𝔡𝔪).

This is equivalent to the existence of a unique s ∈ 𝔽_{q} such that ν = s𝔡𝔪−𝔭μ. From the definition of ν, we must have Nr_{K/F} (α) = (s𝔡𝔪 − 𝔭μ)𝔡^{−1}λ_{0}. The number of such α, with μ and s being fixed, is equal to r_{𝔞,λ_{0}}(s𝔡𝔪 − 𝔭μ). To count the number of β’s, consider the integral ideal

L = (β)D𝔟𝔟̄^{−1}𝔞̄^{−1}.

We must have Nr_{K/F} (L) = (μ). If μ = 0 then β = 0, so we will assume μ ≠ 0. Since the ideal 𝔞̄𝔞 is principal, 𝔞̄ is in I^{−1}. Similarly 𝔟̄ ∈ J^{−1}. As D = (\sqrt{𝔡}) is principal, we conclude that the ideal L lies in the class of IJ^{2}. The number of integral ideals satisfying the last two properties is equal to r_{IJ^{2}}((μ)). If an element β exists at all, it is uniquely determined, up to a 𝔽_{q}^{×} multiple, by the ideal L. The existence of β is equivalent to c^{−1}μ𝔡^{−1}λ_{0} being a norm of an element in K, and by Hasse’s principle this last condition is equivalent to c^{−1}μ𝔡^{−1}λ_{0} being a norm locally at all places of F. The element 𝔡^{−1} = Nr_{K/F} (\sqrt{𝔡}^{−1}) is a global norm, so we can ignore it. Since (μλ_{0}) is the norm of the O integral ideal L𝔞, the element c^{−1}μλ_{0} is a local norm at all finite places, and the existence of β is equivalent to δ_{c^{−1}μλ_{0}} = 1. Since our quaternion is ramified at ∞, we must have δ_{c𝔭} = −1. Thus, the existence of β is equivalent to δ_{𝔭μλ_{0}} = −1. Finally, we have to take into account the congruence relation between α and β. If μ is divisible by 𝔡, then α and β are integral at D and the congruence relation is satisfied for any choice of 𝔽_{q}^{×} multiple of β. On the other hand, if μ is coprime to 𝔡, then there is a unique choice of a multiple of β, with α being fixed, for which the congruence holds. We conclude that for a given α the number of possible non-zero β’s is

σ(μ) · r_{IJ^{2}}((μ))\frac{1 − δ_{𝔭μλ_{0}}}{2} .

Hence

(q − 1)⟨x_{J} , B(𝔪)x_{IJ} ⟩ = ∑_{s∈𝔽_{q}^{×}} r_{𝔞,λ_{0}}(s𝔡𝔪)

+_{∑_{0≠μ∈A} deg μ≤deg(𝔪𝔡)−deg(𝔭)} ∑_{s∈𝔽_{q}^{×}} r_{𝔞,λ_{0}}(s𝔡𝔪 − 𝔭μ) · r_{IJ^{2}}((μ)) · σ(μ)\frac{1 − δ_{𝔭μλ_{0}}}{2} .

If r_{𝔞,λ_{0}}(s𝔡𝔪 − 𝔭μ) ≠ 0 then λ_{0}(s𝔡𝔪 − 𝔭μ) is a norm of an element in K. Hence δ_{λ_{0}(s𝔡𝔪−𝔭μ)} = δ_{λ_{0}}δ_{s𝔡𝔪−𝔭μ} = 1. In particular, δ_{λ_{0}} = δ_{s𝔡𝔪−𝔭μ} and also δ_{λ_{0}𝔭μ} = δ_{(s𝔡𝔪−𝔭μ)𝔭μ}. Under the substitution μ ↦ sμ the expression r_{IJ^{2}}((μ)) · σ(μ) clearly remain invariant. The local norm symbol becomes δ_{(s𝔡𝔪−s𝔭μ)s𝔭μ} = δ_{(𝔡𝔪−𝔭μ)𝔭μ}δ_{s^{2}} = δ_{(𝔡𝔪−𝔭μ)𝔭μ}. Hence we have

(q − 1)⟨x_{J} , B(𝔪)x_{IJ} ⟩ = ∑_{s∈𝔽_{q}^{×}} r_{𝔞,λ_{0}}(s𝔡𝔪)

+_{∑_{0≠μ∈A} deg μ≤deg(𝔪𝔡)−deg(𝔭)}r_{IJ^{2}}((μ)) · σ(μ)\frac{1 − δ_{(𝔡𝔪−𝔭μ)𝔭μ}}{2} ∑_{s∈𝔽_{q}^{×}} r_{𝔞,λ_{0}}(s(𝔡𝔪 − 𝔭μ)).

It is easy to check that for λ ∈ A

\frac{1}{q − 1} ∑_{s∈𝔽_{q}^{×}} r_{𝔞,λ_{0}}(sλ) = r_{I^{−1}}((λ)).

Finally, we get the desired expression

⟨x_{J} , B(𝔪)x_{IJ} ⟩ = r_{I^{−1}}((𝔡𝔪))

+_{∑_{0≠μ∈A} deg μ≤deg(𝔪𝔡)−deg(𝔭)}r_{I^{−1}}((𝔡𝔪 − 𝔭μ)) · r_{IJ^{2}}((μ)) · σ(μ)\frac{1 − δ_{(𝔡𝔪−𝔭μ)𝔭μ}}{2} .

□

Corollary 2.12. Let h_{O} := #Pic(O) be the class number of O. There is an equality

∑_{J ∈Pic(O)}⟨x_{J} , B(𝔪)x_{IJ} ⟩ = h_{O} · r_{I}((𝔡𝔪))

+_{∑_{0≠μ∈A} deg μ≤deg(𝔪𝔡)−deg(𝔭)}r_{I}((𝔡𝔪 − 𝔭μ)) · R((μ)) · σ(μ)\frac{1 − δ_{(𝔡𝔪−𝔭μ)𝔭μ}}{2} .

Proof. An integral ideal 𝔞 is in I if and only if its conjugate 𝔞̄ is in I^{−1}. Moreover, Nr_{K/F} (𝔞) = Nr_{K/F} (𝔞̄), so we conclude that r_{I^{−1}} = r_{I}. Next, we claim that Pic(O)[2] = 1. Assuming this for a moment, we get ∑_{J} r_{IJ^{2}} = R, and the corollary follows from Proposition 2.11.

It remains to show that h_{O} is odd. Let C/𝔽_{q} be the smooth, projective, geometrically connected curve with function field K. This is a hyperelliptic curve, and by Hurwitz’s formula its genus is equal to (deg 𝔡 − 1)/2. Denote by J the Jacobian variety of C. There is an exact sequence

0 → J(𝔽_{q}) → Pic(O) → ℤ/d_{∞}ℤ → 0,

where d_{∞} = [𝔽_{∞} : 𝔽_{q}]. Since ∞ ramifies in K/F, d_{∞} = 1. We get Pic(O)[2] = J[2](𝔽_{q}). Hence we are reduced to showing that J has no non-trivial 𝔽_{q}-rational 2-torsion. Let 𝔡 = a·∏_{i=1}^{d}(t−b_{i}) be the decomposition of the polynomial 𝔡 over \overline{𝔽}_{q}, where d = deg(𝔡), a ∈ 𝔽_{q}^{×} and b_{i} ∈ \overline{𝔽}_{q}, 1 ≤ i ≤ d. Denote by P_{i} the point on C(\overline{𝔽}_{q}) corresponding to the solution (b_{i}, 0) of y^{2} = 𝔡, 1 ≤ i ≤ d. For the Weil divisor D_{i} := P_{i} − ∞ on C we have 2D_{i} = div(t − b_{i}). Hence each D_{i} gives a 2-torsion point on J. There is one linear relation ∑_{i=1}^{d} D_{i} = div(y) = 0 in Div^{0}(C), so D_{i}’s generate J[2] ≅ (ℤ/2)^{d−1}. Since we are assuming 𝔡 is irreducible, non of these divisors is 𝔽_{q}-rational. The claim follows. □

3. The analogue of Eichler’s theorem

It is a classical result that certain theta series arising from quaternion algebras generate the space of modular forms of of weight 2 and prime level. This was conjectured by Hecke and proved by Eichler using a trace formula. In this section we will prove the analogue of Eichler’s theorem over F. Our argument will be geometric – it uses the existence of Néron models of abelian varieties and integral models of Drinfeld modular curves. The idea of this proof is due to Matthew Emerton [3]. We start with recalling the necessary facts form the theory of Drinfeld automorphic forms.

3.1. Harmonic cochains. Denote the completion of F at ∞ by F_{∞}. Let R_{∞} be the ring of integers in F_{∞}, and π_{∞} = t^{−1} be the uniformizer at ∞. Let T be the Bruhat-Tits tree of PGL_{2}(F_{∞}). The oriented edges Y (T ) of T are parametrized by the set GL_{2}(F_{∞})/I · Z(F_{∞}), where Z is the center of GL(2) and

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

We denote by X(T ) the vertices of T . For an edge e ∈ Y we denote by ē, t(e), o(e) the inversely oriented edge, the terminus of e, and the origin of e, respectively.

Multiplication from the right by (\begin{matrix}0 & 1 \\ π & 0 \end{matrix}) corresponds to the map e ↦ ē on Y (T ).

The set Y (T ) can be represented as the union of two disjoint sets: the positively oriented edges

Y^{+}(T ) := \{(\begin{matrix}π_{∞}^{k} & u \\ 0 & 1 \end{matrix}) | k ∈ ℤ, u ∈ F_{∞}/(π_{∞}^{k})\} ,

and the negatively oriented edges Y^{−}(T ) = {ē | e ∈ Y^{+}}. Denote Γ_{0}(1) := GL_{2}(A)

and Γ_{∞} = \{(\begin{matrix}a & b \\ 0 & d \end{matrix}) ∈ GL_{2}(A)\}. For 𝔫 ∈ A let

Γ_{0}(𝔫) := \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ_{0}(1)| c ∈ (𝔫)\} .

be the Hecke congruence subgroup of level 𝔫 (it is clear that Γ_{0}(𝔫) depends only on the ideal (𝔫)).

Let B be a subring of ℂ. Let Γ be a subgroup of Γ_{0}(1); for example, Γ = Γ_{0}(𝔫). Consider the following conditions on B valued functions of Y (T ):

(i) φ(ē) = −φ(e) for any e ∈ Y (T ); functions satisfying this condition will be called alternating.

(ii) ∑_{t(e)=v} φ(e) = 0 for any v ∈ X(T ); functions satisfying this condition will

be called harmonic. (iii) φ(γe) = φ(e) for any γ ∈ Γ; functions satisfying this condition will be

called Γ-invariant. (iv) φ has compact (=finite) support modulo Γ.

We will denote by \underline{H}_{!}(T , B)^{Γ} ⊂ \underline{H}(T , B)^{Γ} ⊂ \underline{H}(T , B) the spaces of B-valued functions on Y (T ) satisfying the conditions (i)-(iv), (i)-(iii), and (i)-(ii) respectively. The B-module \underline{H}(T , B) is called the space of B-valued harmonic cochains.

3.2. Fourier analysis. The theory of Fourier analysis on T was developed by Weil in [24]. We follow the exposition in [8], which gives ∞-adic formulae.

Since Γ_{∞} preserves the orientation on T , we have Γ_{∞} \ Y^{+}(T ) = Y^{+}(Γ_{∞} \ T ). Any function on Y^{+}(T ), which is invariant under Γ_{∞} acting on the left, can be regarded as a function on Y^{+}(Γ_{∞} \ T ). Any such function has a Fourier expansion. Let β be a non-negative divisor of F,

β = (α) · ∞^{deg β} = (α)_{f} · ∞^{deg β−deg α},

where (α) is the principal divisor of α ∈ A with finite part (α)_{f}.

If φ is a function on Y^{+}(Γ_{∞} \ T ) then (see [8], (2.6) to (2.8))

φ ((\begin{matrix}π_{∞}^{k} & y \\ 0 & 1 \end{matrix})) = c_{0}(φ, π_{∞}^{k}) + ∑_{deg α≤k−2_{0≠α∈A}}c(φ, (α) · ∞^{k−2})ψ_{∞}(αy),

where

(3.1)  c_{0}(φ, π_{∞}^{k}) = \begin{cases}q^{1−k} ∑_{y∈(π_{∞})/(π_{∞}^{k})} φ ((\begin{matrix}π_{∞}^{k} & y \\ 0 & 1 \end{matrix})) & k ≥ 1 \\ φ ((\begin{matrix}π_{∞}^{k} & 0 \\ 0 & 1 \end{matrix})) & k ≤ 1, \end{cases}

c(φ, β) = q^{−1−deg β} ∑_{y∈(π_{∞})/(π_{∞}^{2+deg β})} φ ((\begin{matrix}π_{∞}^{2+deg β} & y \\ 0 & 1 \end{matrix})) ψ_{∞}(−αy),

and ψ_{∞} : F_{∞} ↦ ℂ^{×} is ∑ a_{i}π_{∞}^{i} ↦ ψ(Tr_{𝔽_{q}/𝔽_{p}}(a_{−1})) with ψ a non-trivial additive character of 𝔽_{p}, for example, ψ = exp(\frac{2πi}{p}).

Lemma 3.1. Let φ be alternating and invariant under Γ_{∞}. Then φ is harmonic if and only if its Fourier coefficients satisfy:

(i) There exists a constant c such that c_{0}(φ, π_{∞}^{k}) = c · q^{−k}.

(ii) For any non-negative divisor β = β_{f} · ∞^{k}, c(φ, β) = c(φ, β_{f})q^{−k}.

Proof. See [8, Lem.2.13]. □

3.3. Eisenstein series. Following [8, (5.4)], we say that the collection

{c(β) | β is a positive divisor of F }

of Fourier coefficients is eulerian in the prime divisor ℘ with Euler factor P (X) = 1 + a_{1}X + ⋯ + a_{d}X^{d} if the following holds: Write P (X)^{−1} = ∑_{k≥0} b_{k}X^{k} as a formal power series. Then

c(β · ℘^{k}) = c(β) · b_{k}

whenever ℘ and β are coprime. For a divisor β define its norm |β| = q^{deg β}.

Let E : Y (Γ_{∞} \ T ) → ℂ be the unique alternating function defined by the collection of Fourier coefficients c_{0}, c(β), where

(i) c((1)) = 1;

(ii) c(β) is eulerian at finite primes ℘ with Euler factor

1 − (1 + |℘|^{−1})X + |℘|^{−1}X^{2};

(iii) c(β) is eulerian at ∞ with Euler factor 1 − q^{−1}X; (iv) c_{0}(π_{∞}^{k}) = −\frac{q^{2}}{q^{2}−1}q^{−k};

Let 𝔫 ∈ A be monic. Let E_{𝔫} the function on Y (T ) defined by

E_{𝔫}(e) = E(e) − E ((\begin{matrix}𝔫 & 0 \\ 0 & 1 \end{matrix}) e) .

We call E_{𝔫} the Eisenstein series of level 𝔫.

Proposition 3.2. E_{𝔫} ∈ \underline{H}(T , ℂ)^{Γ_{0}(𝔫)}.

Proof. Since E is alternating, it is clear that E_{𝔫} is also alternating. Let G(e) :=

E ((\begin{matrix}𝔫 & 0 \\ 0 & 1 \end{matrix}) e). By [8, Prop.2.10],

c(G, β) = c(E, β · (𝔫)_{f}^{−1}) and c_{0}(G, π_{∞}^{k}) = c_{0}(E, π_{∞}^{k−deg 𝔫}).

In particular, c(G, β) = 0 if 𝔫 |̸β_{f}; c.f. [8, Cor.2.11]. Hence E_{𝔫} is eulerian at ∞ (with the same Euler factor as E), and

c_{0}(E_{𝔫}, π_{∞}^{k}) = c_{0}(E, π_{∞}^{k}) − c_{0}(E, π_{∞}^{k−deg 𝔫})

= \frac{q^{2}}{q^{2} − 1} (q^{deg 𝔫} − 1) q^{−k}. (3.2)

By Lemma 3.1, E_{𝔫} is harmonic. Finally, that E_{𝔫} is Γ_{0}(𝔫)-invariant is proved in [8, Cor.6.3]. □

Remark 3.3. E may also be represented, up to a scalar factor, as conditionally convergent Eisenstein series ∑_{γ∈Γ_{∞}\Γ_{0}(1)} sgn(γe)q^{−κ(γe)}, and where the summation has to be taken in a fixed order. Here κ(e) is defined as follows: Given an edge e,

either e or ē is positively oriented, and hence is represented by (\begin{matrix}π_{∞}^{k} & u \\ 0 & 1 \end{matrix}). Take

κ(e) = k. The function sgn(e) is equal to ±1 depending on the orientation of e. The exponential q^{−κ(e)} can be thought of as the analogue of Im(z) over the complex numbers.

For a function φ ∈ \underline{H}(T , ℂ)^{Γ} there is a relation between being compactly supported modulo Γ and the Fourier coefficient c_{0}(φ). If Γ is a congruence subgroup of Γ_{0}(1) then the quotient graph Γ \ T is the edge-disjoint union of a finite graph (Γ \ T )^{0} and a finite number of half-lines h_{s} labelled by the cusps s ∈ Γ \ ℙ^{1}(F) of Γ. Hence the function φ is in \underline{H}_{!}(T , ℂ)^{Γ} if and only if it vanishes on all h_{s}. One of h_{s}, namely the one corresponding to the orbit of ∞ = (1 : 0) ∈ ℙ^{1}(F), has a

preimage in T given by the matrices (\begin{matrix}π_{∞}^{k} & 0 \\ 0 & 1 \end{matrix}), k ≤ 1. Since the values of φ on

these matrices coincides with c_{0}(φ, π_{∞}^{k}) = c · q^{−k}, c.f. (3.1) and Lemma 3.1, φ vanishes on h_{∞} if and only if the constant Fourier coefficient c_{0}(φ) = 0. It is known [20] that Γ_{0}(1) \ T = h_{∞}. Hence for every s ∈ Γ \ ℙ^{1}(F) there is γ_{s} ∈ Γ_{0}(1) such that γ_{s}s = ∞, and φ will be compactly supported if and only if c_{0}(φ ∘ γ_{s}) = 0 for all s.

Proposition 3.4. If 𝔭 ∈ A be a prime then

\underline{H}(T , ℂ)^{Γ_{0}(𝔭)} = ℂE_{𝔭} ⊕ \underline{H}_{!}(T , ℂ)^{Γ_{0}(𝔭)}.

Proof. Let Γ := Γ_{0}(𝔭). From what was said, we already know that the space on the right hand-side is a subspace of \underline{H}(T , ℂ)^{Γ}, and the sum is direct. The elements of \underline{H}(T , ℂ)^{Γ} may be considered as alternating functions on Y (Γ \ T ) with the harmonicity condition being replaced by ∑_{t(e)=v} m(e)φ(e) = 0. Here m(e) are certain multiplicities that count how many edges of T are identified modulo Γ. Using these two conditions, it is not hard to check that if φ ∈ \underline{H}(T , ℂ)^{Γ} vanishes on all but possibly one of the half-lines of Γ \ T , then in fact it must be compactly supported (i.e., has to vanish on all of the half-lines).

It is known that Γ \ T has only two cusps; see, for example, [5]. If φ ∈ \underline{H}(T , ℂ)^{Γ} then, by subtracting from φ an appropriate multiple of E_{𝔫}, we can guarantee that c_{0}(φ − cE_{𝔭}) = 0. Hence φ − cE_{𝔭} vanishes on h_{∞}, and from the previous paragraph φ − cE_{𝔭} ∈ \underline{H}_{!}(T , ℂ)^{Γ}. □

3.4. Hecke operators. In this subsection we put Γ := Γ_{0}(𝔭) with 𝔭 prime. Most of the facts stated below hold true without any restrictions on the level. We restrict to prime level to avoid discussing old cusp forms, and have Proposition 3.4 available.

Let 𝔪 be a monic polynomial in A. For a function φ on GL_{2}(F_{∞}) put

(T_{𝔪}φ)(g) = ∑ φ ((\begin{matrix}a & b \\ 0 & d \end{matrix}) g) ,

where the sum is over the triples (a, b, d) of A such that a, d are monic, ad = 𝔪, (a, 𝔭) = 1, and deg b < deg d. We call T_{𝔪} the 𝔪-th Hecke operator. These Hecke operators have the usual properties, c.f. [7], [9]:

(i) All T_{𝔪} commute;

(ii) If 𝔪 and 𝔪′ are coprime then T_{𝔪𝔪′} = T_{𝔪}T_{𝔪′};

(iii) If 𝔮 is a prime distinct form 𝔭 then

T_{𝔮^{n+1}} = T_{𝔮^{n}}T_{𝔮} − |𝔮|T_{𝔮^{n−1}};

(iv) T_{𝔪} preserves \underline{H}(T , ℂ)^{Γ}.

Denote 𝕋 := ℤ[…, T_{𝔪}, …] the ℤ-algebra generated by the Hecke operators acting on \underline{H}(T , ℂ)^{Γ}. Since 𝕋 preserves the integral structure \underline{H}(T , ℤ)^{Γ}, and it is known that \underline{H}(T , ℤ)^{Γ} is a finitely generated free ℤ-module, 𝕋 is a finitely generated free ℤ-module. One can express the Fourier expansion of T_{𝔪}φ in terms of the Fourier expansion of φ in a usual manner. Since the Fourier coefficients of E_{𝔭} are eulerian, E_{𝔭} is a 𝕋-eigenvector, with the eigenvalue of T_{𝔮}, (𝔭, 𝔮) = 1, being 1 + |𝔮|. It is also known that \underline{H}_{!}(T , ℂ)^{Γ} is preserved by 𝕋 and has a basis of simultaneous 𝕋eigenforms; c.f. [7]. In particular, 𝕋_{ℂ} := 𝕋 ⊗ ℂ is a semi-simple ℂ-algebra. We will denote the quotient of 𝕋 acting faithfully on \underline{H}_{!}(T , ℂ)^{Γ} by 𝕋^{0}, and the quotient acting faithfully on ℂE_{𝔭} by 𝕋^{E}. To abbreviate the notation, we put

M := \underline{H}(T , ℂ)^{Γ} and S := \underline{H}_{!}(T , ℂ)^{Γ}.

Theorem 3.5.

(i) There is a bilinear perfect pairing 𝕋_{ℂ} × M → ℂ given by

(T_{𝔪}, φ) ↦ c((T_{𝔪}φ), (1)).

(ii) The same pairing restricts to a perfect pairing 𝕋_{ℂ}^{0} × S → ℂ.

Proof. Part (ii) is proved in [7, Thm.3.17]. The same argument applied to the pairing 𝕋_{ℂ} ×M → ℂ shows that if the pairing is not perfect then there is φ ∈ M all whose Fourier coefficients are 0 except possibly for c_{0}(φ). In that case φ = c_{0}(φ). Hence by Lemma 3.1, up to a scalar multiple, φ(e) = sgn(e)q^{−κ(e)}, where sgn(e) and κ(e) are as in Remark 3.3. This function is in \underline{H}(T , ℂ) but it is not invariant under Γ, c.f. [8, (2.12) and (4.7)]. This gives a contradiction. □

It is clear that there is an injection 𝕋 ↪ 𝕋^{0} ⊕ 𝕋^{E} with finite cokernel.

3.5. Hecke operators as correspondences. Let, as in §3.4, 𝔭 be a monic prime. The functor which associates to an A-scheme W the set of isomorphism classes of pairs (D, Z_{𝔭}), where D is a Drinfeld module of rank 2 over W and Z_{𝔭} is a 𝔭cyclic subgroup of D, possesses a coarse moduli scheme M_{0}(𝔭)/A of pure relative dimension 1. There is a canonical compactification X_{0}(𝔭) of M_{0}(𝔭). We refer to [2], [9], and [5] for the details.

Theorem 3.6.

(a) X_{0}(𝔭) is a proper, normal, irreducible scheme of pure relative dimension 1 over Spec(A).

(b) X_{0}(𝔭) → Spec A [𝔭^{−1}] is smooth.

(c) X_{0}(𝔭)_{F} is a smooth proper geometrically connected curve over F.

(d) X_{0}(𝔭)_{\overline{𝔽}_{𝔭}} is reduced and is a union of two copies of X_{0}(1)_{\overline{𝔽}_{𝔭}} = ℙ_{\overline{𝔽}_{𝔭}}^{1} intersecting transversally at the points representing the isomorphism classes of super-singular Drinfeld modules. A super-singular point x on the first copy of ℙ_{\overline{𝔽}_{𝔭}}^{1} is glued to τ^{deg 𝔭}(x) on the second copy.

Proof. See [2] and [5]. □

Let M = ⊕_{i=1}^{n}ℤx_{i} be as in §2.3. Let deg : M → ℤ be the ℤ-linear map obtained by sending each x_{i} to 1 ∈ ℤ, and let M^{0} be the kernel of deg. Denote M_{ℂ} := M ⊗_{ℤ} ℂ. The kernel of deg : M_{ℂ} → ℂ is M_{ℂ}^{0} := M^{0} ⊗_{ℤ} ℂ.

Using moduli interpretation, one can define Hecke correspondences T_{𝔪} on X_{0}(𝔭), 𝔪 ∈ A is monic. As in the case of classical modular curves, the Hecke correspondences induce endomorphisms of M and M^{0}. Moreover, the action of T_{𝔪} ∈ End_{ℤ}(M) is given by the Brandt matrix B(𝔪). Hence the ℤ-subalgebra of End(M) generated by the Hecke operators (as correspondences) is the algebra of Brandt matrices 𝔹. In the case of classical modular curves all the previous statements are carefully explained in [15, pp. 18-22]; see also [17, pp. 443-445]. Since the arguments for Drinfeld modular curves are essentially the same, we will simply refer to loc.cit. for the proofs. Denote 𝔹_{ℂ} := 𝔹 ⊗_{ℤ} ℂ.

Theorem 3.7. There is a canonical algebra isomorphism 𝕋_{ℂ} ≅ 𝔹_{ℂ}, where on the left we have the subalgebra of End_{ℂ}(M) generated by the Hecke operators acting on M. This algebra isomorphism makes M and M_{ℂ} into isomorphic 𝕋_{ℂ}-modules.

Proof. By Theorem 2.8 (1), 𝔹 preserves M^{0}. Denote the quotients of 𝔹 acting faithfully on M^{0} and M/M^{0} ≅ ℤ by 𝔹^{0} and 𝔹^{E}, respectively. Let J := Pic_{X_{0}(𝔭)_{F} /F}^{0} be the Jacobian variety of X_{0}(𝔭)_{F}. Let J be the Néron model of J over Spec(A), and let J^{0} be the relative connected component of the identity of J. Since X_{0}(𝔭) has a degenerate 𝔽_{𝔭}-fibre, by the example 9.2/8 in [1], J_{\overline{𝔽}_{𝔭}}^{0} is a split torus. Moreover, M^{0} = Hom_{\overline{𝔽}_{𝔭}}(J_{\overline{𝔽}_{𝔭}}^{0}, 𝔾_{m,\overline{𝔽}_{𝔭}}) is the character group of J_{\overline{𝔽}_{𝔭}}^{0}. By the Néron mapping property the endomorphisms of J act on J_{\overline{𝔽}_{𝔭}}^{0}, and this action is faithful since the reduction is toric. Thus, the subalgebra of End_{F} (J) generated by the Hecke operators as correspondences acts faithfully on M^{0}.

By Drinfeld’s fundamental theorem [2, Thm. 2], there is a canonical isomorphism

(3.3)  H_{et}^{1}(X_{0}(𝔭)_{F^{sep}}, \overline{ℚ}_{ℓ}) ≅ \underline{H}_{!}(T , \overline{ℚ}_{ℓ})^{Γ_{0}(𝔭)} ⊗ sp,

where sp is the two-dimensional special ℓ-adic (ℓ ≠ p) representation of Gal(F_{∞}^{sep}/F_{∞}). Moreover, this isomorphism is compatible with the action of Hecke operators. Since

End_{F} (J) ⊗ \overline{ℚ}_{ℓ} ↪ End_{Gal}_{F^{sep}/F} (H_{et}^{1}(X_{0}(𝔭)_{F^{sep}}, \overline{ℚ}_{ℓ})^{∨}),

the Hecke correspondences generate a subalgebra in End_{F} (J) which is isomorphic to 𝕋^{0}. The action of Hecke operators on M^{0} induced by the correspondences on X_{0}(𝔭) is canonically isomorphic to the action induced by extending the action on J to J^{0}. This last fact is proved in [15]. Hence the homomorphism

(3.4)  𝕋_{ℂ} → 𝔹_{ℂ}
T_{𝔪} ↦ B(𝔪)

restricts to an isomorphism 𝕋_{ℂ}^{0} ≅ 𝔹_{ℂ}^{0}. (For all our practical purposes, we can fix an isomorphism \overline{ℚ}_{ℓ} ≅ ℂ and work with either of the fields.)

By Theorem 2.8 (1), B(𝔪) acts on M/M^{0} by multiplication by σ_{𝔭}(𝔪). The Hecke operator T_{𝔪} acts on the Eisenstein series E_{𝔭} with the same eigenvalue. Hence (3.4) restricts to an isomorphism 𝕋_{ℂ}^{E} ≅ 𝔹_{ℂ}^{E}. Since 𝕋_{ℂ} = 𝕋_{ℂ}^{0}⊕𝕋_{ℂ}^{E}, and 𝔹_{ℂ} = 𝔹_{ℂ}^{0}⊕𝔹_{ℂ}^{E}, the claim follows. □

3.6. Theta series. We keep the notation of the previous subsections. Recall the pairing on M that we defined in §2.3: Let w_{i} = #Aut(x_{i})/(q − 1), and define a positive definite ℤ-valued pairing via the formula ⟨x_{i}, x_{j}⟩ = w_{i}δ_{ij}. This pairing extends to a bilinear pairing on M_{ℂ} and the action of 𝕋 is symmetric with respect to this pairing; c.f. Theorem 2.8. Let x, y ∈ M_{ℂ} be two fixed elements. Define a set of Fourier coefficients by

c_{0}(π_{∞}^{k}) = q^{2} · deg x · deg y · q^{−k},

and for a non-negative divisor β = (𝔪)_{f} · ∞^{k} of F, 𝔪 ∈ A monic, define

c(β) = ⟨x, T_{𝔪}y⟩ · q^{−k}.

Let Θ_{x,y} be the unique alternating ℂ-valued function on Y (Γ_{∞} \ T ) defined by the collection of these Fourier coefficients. By Lemma 3.1, Θ_{x,y} ∈ \underline{H}(T , ℂ).

Proposition 3.8. Θ_{x,y} ∈ M.

Proof. The function deg on M_{ℂ} is ℂ-linear. Hence Θ : (x, y) ↦ Θ_{x,y} is a bilinear homomorphism from M_{ℂ}×M_{ℂ} into \underline{H}(T , ℂ). Since M_{ℂ} = ⊕_{i=1}^{n}ℂx_{i}, we can assume x = x_{i} for some fixed i. Consider the element x_{E} := ∑_{i=1}^{n} x_{i}/w_{i} ∈ M_{ℂ}. The image of x_{E} in M_{ℂ}/M_{ℂ}^{0} is clearly non-zero, hence it generates this one dimensional space. Every element y ∈ M_{ℂ} can be uniquely written as y = cx_{E} + x_{0}, where c ∈ ℂ and x_{0} ∈ M_{ℂ}^{0}. It is enough to prove that Θ_{x_{i},x_{E}} and Θ_{x_{i},x_{0}} are in \underline{H}(T , ℂ)^{Γ}. We have

T_{𝔪}x_{E} = ∑_{i=1}^{n} \frac{1}{w_{i}} T_{𝔪}x_{i} = ∑_{i=1}^{n} \frac{1}{w_{i}} ∑_{j=1}^{n} b_{i,j}(𝔪)x_{j}

= ∑_{j=1}^{n} x_{j} ∑_{i=1}^{n} \frac{1}{w_{i}} b_{i,j}(𝔪) = ∑_{j=1}^{n} x_{j} ∑_{i=1}^{n} \frac{1}{w_{j}} b_{j,i}(𝔪)

= ∑_{j=1}^{n} \frac{1}{w_{j}} x_{j} ∑_{i=1}^{n} b_{j,i}(𝔪) = σ_{𝔭}(𝔪) ∑_{j=1}^{n} \frac{1}{w_{j}} x_{j}

= σ_{𝔭}(𝔪)x_{E}.

Hence ⟨x_{i}, T_{𝔪}x_{E}⟩ = σ_{𝔭}(𝔪)⟨x_{i}, x_{E}⟩ = σ_{𝔭}(𝔪). In particular, for any non-negative divisor β we have c(E_{𝔭}, β) = c(Θ_{x_{i},x_{E}}, β). There is a mass-formula [4, (5.9)]:

∑_{i=1}^{n} \frac{1}{w_{i}} = \frac{q^{deg 𝔭} − 1}{q^{2} − 1}.

Thus, using (3.2), we also get c_{0}(Θ_{x_{i},x_{E}}, π_{∞}^{k}) = q^{2} deg(x_{E})q^{−k} = c_{0}(E_{𝔭}, π_{∞}^{k}). The final conclusion is that Θ_{x_{i},x_{E}} = E_{𝔭}.

Now consider Θ_{x_{i},x_{0}}. Since deg(x_{0}) = 0, c_{0}(Θ_{x_{i},x_{0}}) = 0. The map 𝕋_{ℂ}^{0} → ℂ defined by T_{𝔪} ↦ ⟨x_{i}, T_{𝔪}x_{0}⟩ is a ℂ-linear homomorphism. By Theorem 3.5, Θ_{x_{i},x_{0}} is in S. □

Theorem 3.9. The set of theta series {Θ_{x_{i},x_{j}}| 1 ≤ i, j ≤ n} generates M.

Proof. We need to show that 𝕋_{ℂ}-module homomorphism

Θ : M_{ℂ} ⊗_{𝕋_{ℂ}} M_{ℂ} → M

defined by x_{i}, x_{j} ↦ Θ_{x_{i},x_{j}} is surjective. The pairing ⟨·, ·⟩ gives an isomorphism

M_{ℂ} ≅ M_{ℂ}^{∨} := Hom_{ℂ}(M_{ℂ}, ℂ).

By Theorem 3.5, we have an isomorphism M ≅ Hom_{ℂ}(𝕋_{ℂ}, ℂ). Composing this last isomorphism with Θ, and the duality induced by ⟨·, ·⟩, we get a pairing

M_{ℂ} ⊗_{𝕋_{ℂ}} M_{ℂ}^{∨} → Hom_{ℂ}(𝕋_{ℂ}, ℂ).

This map is surjective if and only if its dual

𝕋_{ℂ} → Hom_{ℂ}(M_{ℂ} ⊗_{𝕋_{ℂ}} M_{ℂ}^{∨}, ℂ) = Hom_{𝕋_{ℂ}}(M_{ℂ}, M_{ℂ})

is injective. It is not hard to check that this last homomorphism coincides with the homomorphism induced by the natural action of 𝕋 on M. Since this action is faithful, the map is indeed injective. □

4. The analogue of a formula of Gross

In this section we will use the results of previous sections to prove the analogue of Prop. 11.2 in [11] over F.

4.1. L-series of cusp forms. Let, as in §3, Γ := Γ_{0}(𝔭). Recall that the space S := \underline{H}_{!}(T , ℂ)^{Γ} is a space of functions on the set

Y (Γ \ T ) := Γ \ GL_{2}(F_{∞})/I · Z(F_{∞}),

and as such has a natural interpretation as a space of automorphic forms in the sense of Jacquet-Langlands. We will call S the space of Drinfeld cusp forms. It is known that dim_{ℂ} S is equal to the genus of X_{0}(𝔭)_{F}; c.f. (3.3).

The Haar measure on the locally compact group GL_{2}(F_{∞}) induces a measure d on Y (Γ \ T ). Let e ∈ Y (T ), and let Stab_{Γ}(e) = {γ ∈ Γ | γ(e) = e} be the stabilizer of e in Γ. The group Stab_{Γ}(e) is finite. One can take

d(ẽ) = \frac{q − 1}{2}(#Stab_{Γ}(e))^{−1},

where e is a preimage of ẽ ∈ Y (Γ \ T ) in Y (T ). From now on we will assume that the measure d is fixed as above. Let φ, ϕ ∈ M, with φ ∈ S. We define the Petersson product with respect to measure d(e) by

(φ, ϕ) := ∑_{e∈Y (Γ\T )} φ(e)\overline{ϕ}(e)d(e).

Since φ has finite support, (φ, ϕ) is a finite sum. It is known that the action of the Hecke operators is self-adjoint with respect to the Petersson product.

Following Weil, one can attach an L-series to a cusp form. Define

L(φ, s) = ∑_{β pos. div.} c(φ, β)|β|^{−s},

where |β| = q^{deg β} denotes the norm of the divisor β, and the sum is over all non-negative divisors of F, including those with an ∞-component.

We will say that φ ∈ S is a normalized eigenform if it is an eigenform for all Hecke operators and c(φ, (1)) = 1. Since we are assuming 𝔭 is prime and it is known that dim_{ℂ} \underline{H}_{!}(T , ℂ)^{Γ_{0}(1)} = 0, the space S has a basis consisting of 𝕋-eigenforms. The Fourier coefficients of a normalized eigenform φ are eulerian. Hence the L-function of φ has Euler product expansion

L(φ, s) = ∏_{v} (1 − \frac{a_{v}}{|v|^{s}} )^{−1} (1 − \frac{b_{v}}{|v|^{s}} )^{−1} ,

where the product is over all places of F (including ∞), and for v |̸𝔭 · ∞, a_{v} = b̄_{v}, |a_{v}| = |b_{v}| = 1.

4.2. Main identity. In this subsection f will be a fixed normalized eigenform in S. We will denote c(f, β) simply by c(β). Note that c_{0}(π_{∞}^{k}) = 0 for all k as f is a cusp-form.

Let, as in §2.4, 𝔡 be an odd degree irreducible polynomial in A. Assume (\frac{𝔡}{𝔭}) =

−1, so K := F (\sqrt{𝔡}) ↪ B_{𝔭}. Without loss of generality we can assume that Λ = End(x_{1}) is the maximal order in Lemma 2.3. We had an action of Pic(O) on x_{1}, and denoted the image of x_{1} under the action of ideals in the class I by x_{I}. Define

H_{K} := ∑_{I∈Pic(O)} x_{I}.

H_{K} ∈ M has degree #Pic(O).

Let A ∈ Pic(O). We can consider the finite part β_{f} of the non-negative divisor β as an ideal in A. Define

L(f, A, s) = ∑_{β pos. div.} c(β)r_{A}(β_{f})|β|^{−s}

and

L^{(𝔭,𝔡)}(s) = \frac{1}{q − 1}_{∑_{λ∈A} ; (λ,𝔭)=1} (\frac{𝔡}{λ}) q^{−s deg λ}.

Here (\frac{𝔡}{λ}) denotes the Jacobi symbol. The series L^{(𝔭,𝔡)}(2s + 1)L(f, A, s) admits a holomorphic continuation to the whole complex plane and satisfies a functional equation [18, Thm.2.7.3]. Finally, define

L_{K}(f, s) = ∑_{A∈Pic(O)} L^{(𝔭,𝔡)}(2s + 1)L(f, A, s).

We will be interested in the special value of this function at the center s = 0 of the critical strip. Let Θ_{K} := Θ_{H_{K},H_{K}} ∈ M be the theta series we have constructed in §3.6.

Theorem 4.1.

L_{K}(f, 0) = 2q^{− \frac{1+deg 𝔡}{2}} (f, Θ_{K}).

Proof. Using Corollary 2.12, we can explicitly compute the Fourier coefficients of Θ_{K}. Using Rankin’s method, in [18] the authors show that L_{K}(f, 0) is equal, up to an explicit constant, to the Petersson product of f and some G ∈ M. Moreover, by unfolding the Rankin’s integral, they are able to compute explicitly the Fourier coefficients of G; see [18, Lem. 2.7.1 and Prop. 2.7.2]. The theorem follows from the comparison of the Fourier coefficients of G and Θ_{K}. □

If f_{1}, f_{2}, …, f_{n−1} is the basis of S consisting of normalized 𝕋-eigenforms then

M = ℂE_{𝔭} ⊕ ℂf_{1} ⊕ ⋯ ⊕ ℂf_{n−1}

is a decomposition into eigenspaces. By Theorem 3.7, M and M_{ℂ} are isomorphic 𝕋-modules, so we have a similar decomposition

M_{ℂ} = M^{E} ⊕ M^{f_{1}} ⊕ ⋯ ⊕ M^{f_{n−1}}.

Let H_{f} ∈ M_{ℂ} be a non-zero element in the f-isotypical component M^{f}.

Corollary 4.2.

L_{K}(f, 0) = 2q^{− \frac{1+deg 𝔡}{2}} (f, f)\frac{⟨H_{f}, H_{K}⟩^{2}}{⟨H_{f}, H_{f}⟩} .

In particular, L_{K}(f, 0) = 0 if and only if ⟨H_{f}, H_{K}⟩ = 0.

Proof. Let H_{K,f_{i}} be the projection of H_{K} into M^{f_{i}}, so we can write

H_{K} = H_{K,E} + ∑_{i=1}^{n−1} H_{K,f_{i}}.

Since the action of 𝕋 is symmetric with respect to ⟨·, ·⟩ and we have a multiplicity one theorem for M (i.e., two 𝕋-eigenforms having the same eigenvalues for all T_{𝔪} must differ by a non-zero scalar multiple), it is clear that ⟨H_{K}, H_{f}⟩ = ⟨H_{K,f}, H_{f}⟩. Hence

H_{K,f} = \frac{⟨H_{f}, H_{K}⟩}{⟨H_{f}, H_{f}⟩}H_{f}.

It is enough to show

L_{K}(f, 0) = 2q^{− \frac{1+deg 𝔡}{2}} (f, f)⟨H_{K,f}, H_{K,f}⟩.

As we mentioned, the action of Hecke operators is self-adjoint with respect to the Petersson norm. Thus, by repeating the above argument, we need to show that in Theorem 4.1 the f-isotypical component of Θ_{K} is ⟨H_{K,f}, H_{K,f}⟩f. The map Θ : M_{ℂ} × M_{ℂ} → M in Theorem 3.9 is 𝕋-bilinear, so Θ_{H_{K,E},H_{K,f_{i}}} = 0 and Θ_{H_{K,f_{j}} ,H_{K,f_{i}}} = 0 unless i = j. Hence, as one easily checks, the f-isotypical component is Θ_{H_{K,f} ,H_{K,f}}. On the other hand, the Fourier coefficients of Θ_{H_{K,f} ,H_{K,f}} are given by c(Θ_{H_{K,f} ,H_{K,f}} , β) = ⟨H_{K,f}, H_{K,f}⟩c(f, β). That is,

Θ_{H_{K,f} ,H_{K,f}} = ⟨H_{K,f}, H_{K,f}⟩f,

as was required. □

5. Applications to elliptic curves

In this section we compare the formula in Corollary 4.2 with the formula of Birch and Swinnerton-Dyer (BSD). First we recall a special case of the statement of the BSD conjecture for elliptic curves over function fields.

5.1. Formula of Birch and Swinnerton-Dyer. Let C be a smooth projective geometrically connected curve over 𝔽_{q}. Denote the function field of C by K = 𝔽_{q}(C), and its genus by g(C). For each place v of K denote by O_{v} the ring of integers of the completion K_{v}, and denote the residue field by k_{v}.

Let E be a non-isotrivial elliptic curve over K. Denote by E the Néron model of E over C. Denote by E^{0} be the relative connected component of the identity of E. Let E_{v} := E × k_{v} and E_{v}^{0} := E^{0} × k_{v}. The group of connected components Φ_{E,v} = E_{v}/E_{v}^{0} of E at v is a finite étale group-scheme over k_{v}. The Tamagawa number c_{v}(E) of E at v is the order of the subgroup Φ_{E,v}(k_{v}) of k_{v}-rational points in Φ_{E,v}(k̄_{v}). The group Φ_{E,v} is trivial at almost all places (for example, it is trivial at the places where E has good reduction), hence almost all c_{v} = 1. Let Ω_{E/C}^{1} be the relative canonical sheaf on E, and denote its restriction to the relative zero section by Ω_{E/C}^{1}|_{O}. Define

τ(E/K) = ∏_{v} c_{v}(E) · (q^{− deg(Ω_{E/C}^{1}|_{O})}) · q^{1−g(C)}.

Assume that the L-function of E does not vanish at 1, L(E, 1) ≠ 0. Tate proved [23] (see also [14]) that E(K) and the Tate-Shafarevich group � (E/K) of E over K are finite, and

(5.1)  L(E, 1) = \frac{τ(E/K) · #� (E/K)}{#E(K)^{2}}.

5.2. Modular elliptic curves. Let, as we have been denoting, F = 𝔽_{q}(t)(= 𝔽_{q}(ℙ_{𝔽_{q}}^{1})). Let E be an elliptic curve over F of conductor (𝔭) · ∞, where (𝔭) ⊂ A is a prime ideal. Moreover, we assume that E has a split multiplicative reduction at ∞. In this situation it is known that E can be realized as a quotient of the Drinfeld modular curve X := X_{0}(𝔭)_{F}; c.f. [9]. We will assume that the morphism π : X → E is chosen to have the minimal possible degree. Let J := Pic_{X/F}^{0} be the Drinfeld Jacobian. By Albanese functoriality, π induces a homomorphism of π_{∗} : J → E, and by Picard functoriality, it induces a homomorphism π^{∗} : E → J. The composition π_{∗} ∘ π^{∗} is the endomorphism of E given by multiplication by deg(π).

Next, let J be the Néron model of J over ℙ_{𝔽_{q}}^{1}, and let J^{0} be the relative connected component of the identity of J. As we already mentioned in §3.5, it can be shown that M^{0} = Hom_{\overline{𝔽}_{𝔭}}(J_{\overline{𝔽}_{𝔭}}^{0}, 𝔾_{m,\overline{𝔽}_{𝔭}}) is the character group of J_{\overline{𝔽}_{𝔭}}^{0}. Moreover, the pairing (2.3) on M restricted to M^{0} is Grothendieck’s monodromy pairing; see [12], [17], [15, pp.16-18].

The curve E has a multiplicative reduction at 𝔭. Let E be its Néron model over ℙ_{𝔽_{q}}^{1}, and let Υ = Hom_{\overline{𝔽}_{𝔭}}(E_{\overline{𝔽}_{𝔭}}^{0}, 𝔾_{m,\overline{𝔽}_{𝔭}}) be the character group at 𝔭. Then Υ ≅ ℤ. There is a functorial homomorphism π^{∗} : Υ → M^{0}. Choose a generator ρ of Υ and denote H_{E} := π^{∗}(ρ). It is clear that H_{E} is well-defined up to a sign.

Lemma 5.1. With previous notation,

deg(π) · #Φ_{E,𝔭} = ⟨H_{E}, H_{E}⟩.

Proof. This follows from the functorial properties of the monodromy pairing and Thm.11.5 in [12]. □

There is a unique normalized eigenform f ∈ S corresponding to E. The Lseries of E and f satisfy L(E, s + 1) = L(f, s); see [9], [7]. Let K = F (\sqrt{𝔡}) be as in §4.2. Let E_{K} := E ⊗_{F} K be the elliptic curve E over K. We also have L(E_{K}, s + 1) = L_{K}(f, s). Finally, one can take H_{f} to be H_{E}. The following is known [7]

deg(π) · #Φ_{E,∞} = (f, f).

Hence we can rewrite the formula in Corollary 4.2 as

(5.2)  L(E_{K}, 1) = 2q^{− \frac{1+deg 𝔡}{2}} \frac{#Φ_{E,∞}}{#Φ_{E,𝔭}} ⟨H_{E}, H_{K}⟩^{2}.

The nonsingular complete curve C/𝔽_{q} such that 𝔽_{q}(C) = K is a hyperelliptic curve of genus g(C) = (deg 𝔡−1)/2. By construction, the prime (𝔭) stays inert in K. Since E has multiplicative reduction at 𝔭, E_{K} necessarily will have split multiplicative reduction at 𝔭. Thus, the Tamagawa number c_{𝔭}(E_{K}) = #Φ_{E,𝔭}. The prime at ∞ ramifies in K, hence c_{∞}(E_{K}) = 2#Φ_{E,∞}. Since E_{K} has good reduction away form 𝔭 and ∞, we get

τ(E/K) = 2#Φ_{E,𝔭}#Φ_{E,∞} · (q^{− deg(Ω_{E_{K} /C}^{1}|_{O})}) · q^{\frac{3−deg 𝔡}{2}} .

If we assume ⟨H_{E}, H_{K}⟩ ≠ 0, so that L(E_{K}, 1) ≠ 0, then we can rewrite (5.1) for E_{K} as

(5.3) L(E_{K}, 1) = 2#Φ_{E,𝔭}#Φ_{E,∞} · (q^{− deg(Ω_{E_{K} /C}^{1}|_{O})}) · (q^{\frac{3−deg 𝔡}{2}} ) · \frac{#� (E/K)}{#E(K)^{2}}.

Let E_{𝔡} be the twist of E by the quadratic character of Gal(K/F).

Proposition 5.2. If L(E_{K}, 1) ≠ 0 then E(K) = E(F).

Proof. By Tate’s theorem [23], the assumption L(E_{K}, 1) ≠ 0 implies E(K) is finite. Obviously E(F) will also be finite, and we need to prove the equality of torsion groups E(F) and E(K). Let m be an integer coprime to the characteristic p of F. Let F_{m} := F (E[m]) be the smallest subextension of F^{sep} over which the finite étale group-scheme E[m] is constant. Since E has good reduction at 𝔡, by Néron-Ogg-Shafarevich criterion [21] the extension F_{m}/F is unramified at 𝔡 if m ≥ 3. This immediately implies E(F)[m] = E(K)[m] for all m ≥ 3 coprime to p. Hence we need to prove E(F)[2] = E(K)[2] and E(F)[p^{n}] = E(K)[p^{n}] for all n ≥ 1.

To prove the first statement, consider E[2]−{O}, an étale F -scheme of order 3. If this is irreducible, i.e., E(F)[2] = 1, it certainly cannot acquire a rational point over the quadratic extension K/F. Thus, we may assume it is reducible but not constant (i.e., E(F)[2] = ℤ/2). Let Δ be the minimal discriminant of E/F. Since p is odd, the quadratic field extension F_{2} of F must contain F (\sqrt{Δ}). (This follows from the fact that if E is given in terms of a Weierstrass equation y^{2} = x^{3} + ax^{2} + bx + c then the 2-torsion points besides O are exactly the points with y = 0 and Δ is essentially the discriminant of the cubic on the right hand side.) Hence F_{2} ramifies at (𝔭), as (Δ) is a power of (𝔭) (here we are using the fact that the finite part of the conductor of E is 𝔭). The only prime of A which ramifies in K is (𝔡) ≠ (𝔭). In particular, F_{2} and K are disjoint and we conclude E(F)[2] = E(K)[2] = ℤ/2.

Since p is odd, we have the eigenspace decomposition on p-primary parts

E(K)_{p} = E(F)_{p} × E_{𝔡}(F)_{p}

under the action of the non-trivial involution generating Gal(K/F). Thus, it is enough to show that E_{𝔡}(F)[p] = 1. Let E be the Néron model of E_{𝔡} over A. There is the identity on L-functions L(E_{K}, s) = L(E, s)L(E_{𝔡}, s). Our assumption clearly implies L(E, 1) ≠ 0. In particular, the sign of the functional equation of L(E, s) must be +1. Since the reduction of E at ∞ is split multiplicative, the local root number of E at ∞ is +1. Thus, the local root number at 𝔭 also must be +1. We conclude that the reduction of E at 𝔭 is split, and since 𝔭 is inert in K, E_{𝔡} has non-split multiplicative reduction at 𝔭. Hence Frob_{𝔭} induces a non-trivial automorphism of E_{𝔽_{𝔭}}^{0} such that Frob_{𝔭}^{2} = 1. This implies that the action of Frob_{𝔭} on the character group of E_{𝔽_{𝔭}}^{0} is by −1. By [12, Thm.11.5], the same is true for the action on the finite cyclic group Φ_{E_{𝔡},𝔭}(\overline{𝔽}_{𝔭}). Hence Φ_{E_{𝔡},𝔭}(𝔽_{𝔭}) = 1 or ℤ/2. Now suppose E_{𝔡}(F)[p] = ℤ/p. Denote the completion of A at 𝔭 by R, and denote the fraction field of R by L. The Néron model of E_{𝔡} over R is E_{R} := E ×_{A}R. The groupscheme G_{L} = E_{𝔡}(F)[p] is étale and constant. Let G be the schematic closure of G_{L} in E_{R}. This is a finite flat group-scheme over R whose closed fibre G_{𝔽_{𝔭}} is either étale or multiplicative, since E_{𝔽_{𝔭}} is an extension of the finite (cyclic) étale group scheme Φ_{E_{𝔡},𝔭} by the torus 𝔾_{m,𝔽_{𝔭}}. By Lemma 5.3, G_{𝔽_{𝔭}} must be an étale constant group scheme. This implies Φ_{E_{𝔡},𝔭}(𝔽_{𝔭})[p] = ℤ/p, which is a contradiction. □

Lemma 5.3. Let R be a discrete valuation ring, K be its field of fractions, and k be the residue field. Let p be the characteristic of k. Assume R is equicharacteristic. Let G_{K} be a finite étale group scheme of order p over K. If G_{K} extends to a finite flat group scheme G over R such that the closed fibre G_{k} is either étale or multiplicative, then G_{k} is necessarily étale.

Proof. Suppose G_{K} has a finite flat R-model G with multiplicative special fibre. The Cartier dual of G would be a finite flat group-scheme over R with étale special fibre, and hence has to be étale. This implies that G_{K} ≅ μ_{p}, which is a contradiction as μ_{p} is not étale in characteristic p. (Note that the statement of the lemma is false without assuming R is equicharacteristic as the example of μ_{2} over ℚ_{2} shows.) □

Theorem 5.4. If L(E_{K}, 1) ≠ 0 then

#� (E/K) = (⟨H_{E}, H_{K}⟩ · \frac{#E(F)}{#Φ_{E,𝔭}} · q^{(deg Ω_{E/ℙ^{1}}^{1} |_{O}−1)})^{2} .

Proof. The sheaf Ω_{E/A}^{1}|_{O} is isomorphic to a line bundle on Spec(A), hence must be trivial as A is a principal ideal domain (in other words, there is a holomorphic and non-vanishing relative differential form ω_{E/A}, and this is unique up to 𝔽_{q}^{×}). In particular, the divisor of Ω_{E/ℙ^{1}}^{1}|_{O} is supported at ∞. The extension O/A is ramified only at (𝔡). Since E has good reduction at 𝔡, the Néron model of E_{K} over O is E ⊗_{A} O. Hence the divisor of Ω_{E_{K}/C}^{1}|_{O} is also supported at the prime over ∞. Let R be the local ring of ∞ ∈ ℙ_{𝔽_{q}}^{1}. Let R′ be its integral closure in K. Since E has multiplicative reduction at ∞, there is an isomorphism E^{0} ⊗_{R} R′ ≅ E_{K}^{0}/R′. Hence Ω_{E_{K}/R′}^{1}|_{O} ≅ Ω_{E/R}^{1}|_{O} ⊗ R′. As R′/R is ramified, we get 2 deg(Ω_{E/ℙ^{1}}^{1}|_{O}) = deg(Ω_{E_{K}/C}^{1}|_{O}). Now the theorem follows by comparing (5.2) with (5.3) and using Proposition 5.2. □

Example 5.5. Some of the calculations below were performed using the computer package Magma.

Let F = 𝔽_{7}(T). Let 𝔭 = T^{3} − 2 and 𝔡 = T − 3. Then (\frac{𝔡}{𝔭}) = −1, so that 𝔭 and 𝔡 is a pair of primes satisfying the conditions of Lemma 2.2. Let

E/F : Y^{2} = X^{3} + aX + b,

where a = −3T (T^{3} + 2) and b = −2T^{6} + 3T^{3} + 1. One can show that E has conductor (𝔭) · ∞, and split multiplicative reduction at both primes. By computing the j-invariant of E and using Tate’s algorithm [22, IV.8-9], one also shows that #Φ_{E,𝔭} = #Φ_{E,∞} = 3. From Grothendieck’s theory of L-functions over function fields it is known that the L-function L(E, s) of an elliptic curve E over 𝔽_{q}(T) with conductor 𝔫 is a polynomial in q^{−s} of degree (deg 𝔫 − 4) and constant term 1. This immediately implies that for our curve L(E, s) = 1. Hence by Tate [23], E(F) is finite. It is not hard to show that the prime-to-p part of #E(F) is 3. On the other hand, the argument in the proof of Proposition 5.2 shows that E(F)[p] = 1, as #Φ_{E,𝔭} is coprime to p. That is, #E(F) = 3.

To compute deg(Ω_{E/ℙ^{1}}^{1}|_{O}) we need to be able to restrict to O-section. Make a change of variables X = u/v, Y = 1/v. The equation becomes

v = u^{3} + auv^{2} + bv^{3}.

This is non-singular at u = v = 0. The relative differential

ω_{E/A} = \frac{du}{1 − 2auv − 3bv^{2}}

is regular and non-zero over A restricted to O-section (u = v = 0). To compute the relative differential over ∞ we make a substitution T = 1/S. Let a′ = −3\frac{1}{S} (\frac{1}{S^{3}} + 2) and b′ = −2\frac{1}{S^{6}} + 3\frac{1}{S^{3}} + 1.The equation becomes

S^{6}v = S^{6}u^{3} + S^{6}a′uv^{2} + S^{6}b′v^{3}.

This is singular at u = v = 0, S = 0, and we have to desingularize by blowing-up (three times, as it turns out). Concretely, we make a substitution u′ = S^{3}u, v′ = S^{3}v and get v′ = S^{9}u^{′3}+S^{9}a′uv^{2}+S^{9}b′v^{3}, which is non-singular at u′ = v′ = S = 0, and the relative differential is

ω_{E/O_{∞}} = \frac{S^{3}du′}{1 − 2S^{6}a′u′v′ − 3S^{6}b′v^{′2}} .

We conclude deg(Ω_{E/ℙ^{1}}^{1}|_{O}) = 3. Now we can use (5.1) to compute the order of � (E/F); the identity is

1 = \frac{3 · 3 · 7^{−3} · 7}{3^{2}}#� (E/F).

That is, #� (E/F) = 7^{2}.

Now we proceed to verify the analogue of Gross’s formula (5.2). It is much easier to compute L-functions over F rather than over K = F (\sqrt{𝔡}), so we use the identity

L(E_{K}, s) = L(E, s)L(E_{𝔡}, s),

and compute L(E_{𝔡}, s). The conductor of E_{𝔡} is (𝔭)(𝔡)^{2}∞^{2}, and the L-function is L(E_{𝔡}, s) = 343q^{−3s} − 21q^{−3s} − 3q^{−s} + 1 (q = 7). Hence

L(E_{K}, 1) = 8/7.

The Brandt matrix calculations, necessary for determining ℤH_{E} ⊂ M^{0}, are carried out in [19, p.66]. Before stating this result, we introduce some terminology and notation.

As we already mentioned, a Drinfeld A(= 𝔽_{q}[T])-module over an A-field k is an 𝔽_{q}-linear homomorphism ϕ : A → k{τ}, a ↦ ϕ_{a} defined by ϕ_{T} = t + gτ + Δτ^{2}, where t is the image of T in k, Δ, g ∈ k, and Δ ≠ 0. The element j(ϕ) = g^{q+1}/Δ is called the j-invariant of ϕ. Two Drinfeld modules are isomorphic over k̄ if and only if they have the same j-invariant.

Now let k = \overline{𝔽}_{𝔭}. Denote

j_{α} := (t^{q} − t)(1 − (t^{q} − α)^{q−1}),

where α ∈ 𝔽_{q}. There are 8 super-singular j-invariants (that is, j-invariants of s.s. Drinfeld modules) over k, and they are given by 0 and j_{α}, α ∈ 𝔽_{7}; see [19, Prop.16]. Choose a basis of M, x_{0}, x_{j_{0}}, …, x_{j_{6}}, in this order. It is known that w(x_{0}) = 8 and w(x_{j_{α}}) = 1 for all α ∈ 𝔽_{7}. The 𝕋-eigenspace corresponding to our elliptic curve E is spanned by the vector (1, −4, −1, −1, 2, −1, 2, 2); see [19, Exm.19] (misprint in loc.cit.!) Hence H_{E} is an integer multiple of this vector. Gekeler calculated that the degree of the modular parametrization of E is 13. Using lemma 5.1, we deduce

H_{E} = (1, −4, −1, −1, 2, −1, 2, 2).

It remains to compute H_{K}. Since Pic(O) = 1, H_{K} is equal to one of the basis elements of M, and we need to find the s.s. Drinfeld module with O-action. First we construct explicitly a Drinfeld module with an action of O. Let U^{2} = T − 3. Consider the rank one O-Drinfeld module over K given by

U ↦ U + τ.

Via the embedding A ↪ B → K{τ}, we get a rank 2 A-module determined by

ϕ_{T −3} = (U + τ)(U + τ) = (T − 3) + (U + U^{q})τ + τ^{2}.

Its j-invariant is j(ϕ) = (U +U^{q})^{q+1} with q = 7. Modulo 𝔭 reduction of this module has j-invariant j(ϕ̄) = ((t − 3) + 2(t − 3)^{4} + (t − 3)^{7})^{4}, and one checks that this is equal to j_{6} in our earlier notation. (In other words, ϕ is a singular lift of the s.s. x_{j_{6}}.) Hence H_{K} = x_{j_{6}} and

⟨H_{E}, H_{K}⟩ = 2.

Substituting everything in (5.2), we get an equality

\frac{8}{7} = 2 · 7^{−1} \frac{3}{3} · 2^{2}.

Finally, from Theorem 5.4, we have #� (E/K) = 2^{2} · 7^{4}.

We briefly indicate what happens for other monic 𝔡 of degree 1, with E being the same. Let 𝔡_{β} = T − β, where β ∈ 𝔽_{q}, and let K_{β} := F (\sqrt{𝔡_{β}}). Denote E_{β} := E ⊗_{F} K_{β}. One verifies that

(\frac{𝔡_{β}}{𝔭} ) = 1 for β = 0, 1, 2, 4.

To these cases our theorems do not apply. Still, it is easy to check on a computer that in all these cases L(E_{β}, 1) = 0.

On the other hand, if β = 3, 5, 6 then (\frac{𝔡_{β}}{𝔭} ) = −1, and our theorems apply. We already discusses the case β = 3. The L-functions L(E_{β}, s) in all three cases are the same and are equal to 343q^{−3s}−21q^{−3s}−3q^{−s}+1, where q = 7. Thus, for β = 3, 5, 6 we have L(E_{β}, 1) = 8/7. A calculation similar to the case of β = 3 gives H_{K_{5}} = x_{j_{3}} and H_{K_{6}} = x_{j_{5}}. Hence in all cases ⟨H_{E}, H_{K_{β}}⟩ = 2 and #� (E_{β}) = 2^{2} · 7^{4}.

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Department of Mathematics, Stanford University, Stanford, CA 94305

   E-mail address: papikian@math.stanford.edu