The Eisenstein Ideal and Jacquet-Langlands Isogeny

over Function Fields

Mihran Papikian, Fu-Tsun Wei

Received: May 4, 2014 Revised: March 13, 2015

Communicated by Peter Schneider

Abstract. Let 𝔭 and 𝔮 be two distinct prime ideals of 𝔽_{q}[T]. We use the Eisenstein ideal of the Hecke algebra of the Drinfeld modular curve X_{0}(𝔭𝔮) to compare the rational torsion subgroup of the Jacobian J_{0}(𝔭𝔮) with its subgroup generated by the cuspidal divisors, and to produce explicit examples of Jacquet-Langlands isogenies. Our results are stronger than what is currently known about the analogues of these problems over ℚ.

2010 Mathematics Subject Classification: 11G09, 11G18, 11F12 Keywords and Phrases: Drinfeld modular curves; Cuspidal divisor group; Shimura subgroup; Eisenstein ideal; Jacquet-Langlands isogeny

1

Contents

1. Introduction  552
1.1. Motivation  552
1.2. Main results  554
1.3. Notation  557
2. Harmonic cochains and Hecke operators  558
2.1. Harmonic cochains  558
2.2. Hecke operators and Atkin-Lehner involutions  564
2.3. Fourier expansion  566
2.4. Atkin-Lehner method  570
3. Eisenstein harmonic cochains  572
3.1. Eisenstein series  572
3.2. Cuspidal Eisenstein harmonic cochains  575

1The first author was supported in part by the Simons Foundation. The second author was partially supported by National Science Council and Max Planck Institute for Mathematics.

3.3. Special case  579
4. Drinfeld modules and modular curves  583
5. Component groups  585
6. Cuspidal divisor group  590
7. Rational torsion subgroup  596
7.1. Main theorem  596
7.2. Special case  597
8. Kernel of the Eisenstein ideal  601
8.1. Shimura subgroup  601
8.2. Special case  606
9. Jacquet-Langlands isogeny  611
9.1. Modular curves of 𝒟-elliptic sheaves  611
9.2. Rigid-analytic uniformization  612
9.3. Explicit Jacquet-Langlands isogeny conjecture  613
9.4. Special case  616
10. Computing the action of Hecke operators  619
10.1. Action on H  619
10.2. Action on H′  620
10.3. Computation of Brandt matrices  624
Acknowledgements  626
References  627

1. Introduction

1.1. Motivation. Let 𝔽_{q} be a finite field with q elements, where q is a power of a prime number p. Let A = 𝔽_{q}[T] be the ring of polynomials in indeterminate T with coefficients in 𝔽_{q}, and F = 𝔽_{q}(T) the field of fractions of A. The degree map deg : F → ℤ ∪ {−∞}, which associates to a non-zero polynomial its degree in T and deg(0) = −∞, defines a norm on F by |a| := q^{deg(a)}. The corresponding place of F is usually called the place at infinity, and is denoted by ∞. We also define a norm and degree on the ideals of A by |𝔫| := #(A/𝔫) and deg(𝔫) := log_{q} |𝔫|. Let F_{∞} denote the completion of F at ∞, and ℂ_{∞} denote the completion of an algebraic closure of F_{∞}. Let Ω := ℂ_{∞} − F_{∞} be the Drinfeld half-plane. Let 𝔫⊲A be a non-zero ideal. The level-𝔫 Hecke congruence subgroup of GL_{2}(A)

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

The Eisenstein Ideal and Jacquet-Langlands Isogeny 553

plays a central role in this paper. This group acts on Ω via linear fractional transformations. Drinfeld proved in [6] that the quotient Γ_{0}(𝔫) \ Ω is the space of ℂ_{∞}-points of an affine curve Y_{0}(𝔫) defined over F, which is a moduli space of rank-2 Drinfeld modules (we give a more formal discussion of Drinfeld modules and their moduli schemes in Section 4). The unique smooth projective curve over F containing Y_{0}(𝔫) as an open subvariety is denoted by X_{0}(𝔫). The cusps of X_{0}(𝔫) are the finitely many points of the complement of Y_{0}(𝔫) in X_{0}(𝔫); the cusps generate a finite subgroup C(𝔫) of the Jacobian variety J_{0}(𝔫) of X_{0}(𝔫), called the cuspidal divisor group. By the Lang-Néron theorem, the group of F -rational points of J_{0}(𝔫) is finitely generated, in particular, its torsion subgroup T (𝔫) := J_{0}(𝔫)(F)_{tor} is finite. It is known that when 𝔫 is square-free C(𝔫) ⊆ T (𝔫). For a square-free ideal 𝔫 ⊲ A divisible by an even number of primes, let D be the division quaternion algebra over F with discriminant 𝔫. The group of units Γ^{𝔫} of a maximal A-order in D acts on Ω, and the quotient Γ^{𝔫} \ Ω is the space of ℂ_{∞}-points of a smooth projective curve X^{𝔫} defined over F; this curve is a moduli space of 𝒟-elliptic sheaves introduced in [28]. Let J^{𝔫} be the Jacobian variety of X^{𝔫}. The analogy between X_{0}(𝔫) and the classical modular curves X_{0}(N) over ℚ classifying elliptic curves with Γ_{0}(N)-structures is well-known and has been extensively studied over the last 35 years. Similarly, the modular curves X^{𝔫} are the function field analogues of Shimura curves X^{N} parametrizing abelian surfaces equipped with an action of the indefinite quaternion algebra over ℚ with discriminant N. Let 𝕋(𝔫) be the ℤ-algebra generated by the Hecke operators T_{𝔪}, 𝔪 ⊲ A, acting on the group H_{0}(𝒯 , ℤ)^{Γ_{0}(𝔫)} of ℤ-valued Γ_{0}(𝔫)-invariant cuspidal harmonic cochains on the Bruhat-Tits tree 𝒯 of PGL_{2}(F_{∞}). The Eisentein ideal 𝔈(𝔫) of 𝕋(𝔫) is the ideal generated by the elements T_{𝔭} −|𝔭|−1, where 𝔭 ∤ 𝔫 is prime. In this paper we study the Eisenstein ideal in the case when 𝔫 = 𝔭𝔮 is a product of two distinct primes, with the goal of applying this theory to two important arithmetic problem: 1) comparing T (𝔫) with C(𝔫), and 2) constructing explicit homomorphisms J_{0}(𝔫) → J^{𝔫}. Our proofs use the rigid-analytic uniformizations of J_{0}(𝔫) and J^{𝔫} over F_{∞}. It seems that the existence of actual geometric fibres at ∞ allows one to prove stronger results than what is currently known about either of these problems in the classical setting; this is specific to function fields since the analogue of ∞ for ℚ is the archimedean place. Our initial motivation for studying 𝔈(𝔭𝔮) came from an attempt to prove a function field analogue of Ogg’s conjecture [37] about the so-called Jacquet-Langlands isogenies. We briefly recall what this is about. A geometric consequence of the Jacquet-Langlands correspondence [25] is the existence of Heckeequivariant ℚ-rational isogenies between the new quotient J_{0}(N)^{new} of J_{0}(N) and the Jacobian J^{N} of X^{N}; see [45]. (Here N is a square-free integer with an even number of prime factors.) The proof of the existence of aforementioned isogenies relies on Faltings’ isogeny theorem, so provides no information about them beyond the existence. It is a major open problem in this area to make the isogenies more canonical (cf. [24]). In [37], Ogg made several predictions about the kernel of an isogeny J_{0}(N)^{new} → J^{N} when N = pp′ is a product of two distinct primes and p = 2, 3, 5, 7, 13. As far as the authors are aware, Ogg’s conjecture remains open except for the special cases when J^{N} has dimension 1 (N = 14, 15, 21, 33, 34) or dimension 2 (N = 26, 38, 58). In these cases, J^{N} and J_{0}(N)^{new} are either elliptic curves or, up to isogeny, decompose into a product of two elliptic curves given by explicit Weierstrass equations. One can then find an isogeny J_{0}(N)^{new} → J^{N} by studying the isogenies between these elliptic curves; see the proof of Theorem 3.1 in [21]. This argument does not generalize to J^{N} of dimension ≥ 3 because they contain absolutely simple abelian varieties of dimension ≥ 2, and one’s hold on such abelian varieties is decidedly more fleeting. Now returning to the setting of function fields, let 𝔫 ⊲ A be a square-free ideal with an even number of prime factors. The global Jacquet-Langlands correspondence over F, combined with the main results in [6] and [28], and Zarhin’s isogeny theorem, implies the existence of a Hecke-equivariant F -rational isogeny J_{0}(𝔫)^{new} → J^{𝔫}. In Section 9, by studying the groups of connected components of the Néron models of J_{0}(𝔫) and J^{𝔫}, we propose a function field analogue of Ogg’s conjecture (see Conjecture 9.3). This conjecture predicts that, when 𝔫 = 𝔭𝔮 is a product of two distinct primes with deg(𝔭) ≤ 2, there is a Jacquet-Langlands isogeny whose kernel comes from cuspidal divisors and is isomorphic to a specific abelian group. Our approach to proving this conjecture starts with the observation that C(𝔫) is annihilated by the Eisenstein ideal 𝔈(𝔫) acting on J_{0}(𝔫), so we first try to show that there is a Jacquet-Langlands isogeny whose kernel is annihilated by 𝔈(𝔫), and then try to describe the kernel of the Eisenstein ideal J[𝔈(𝔫)] in J_{0}(𝔫) explicitly enough to pin down the kernel of the isogeny. This naturally leads to the study of J[𝔈(𝔫)] for composite 𝔫. On the other hand, J[𝔈(𝔫)] also plays an important role in the analysis of T (𝔫), as was first demonstrated by Mazur in his seminal paper [33] in the case of classical modular Jacobian J_{0}(p) of prime level. These two applications of the theory of the Eisenstein ideal constitute the main theme of this paper.

1.2. Main results. The Shimura subgroup S(𝔫) of J_{0}(𝔫) is the kernel of the homomorphism J_{0}(𝔫) → J_{1}(𝔫) induced by the natural morphism X_{1}(𝔫) → X_{0}(𝔫) of modular curves (see Section 8.1). Assume 𝔭 ⊲ A is prime. Define N(𝔭) = \frac{|𝔭|−1}{q−1} if deg(𝔭) is odd, and define

N(𝔭) = \frac{|𝔭|−1}{q^{2}−1}, otherwise. In [38], Pál developed a theory of the Eisenstein ideal 𝔈(𝔭) in parallel with Mazur’s paper [33]. In particular, he showed that J[𝔈(𝔭)] is everywhere unramified of order N(𝔭)^{2}, and is essentially generated by C(𝔭) and S(𝔭), both of which are cyclic of order N(𝔭). Moreover, C(𝔭) = T (𝔭) and S(𝔭) is the largest μ-type subgroup scheme of J_{0}(𝔭). These results are the analogues of some of the deepest results from [33], whose proof first establishes that the completion of the Hecke algebra 𝕋(𝔭) at any maximal ideal in the support of 𝔈(𝔭) is Gorenstein. As we will see in Section 8, even in the simplest composite level case the kernel of the Eisenstein ideal J[𝔈(𝔫)] has properties quite different from its prime level counterpart. For example, J[𝔈(𝔫)] can be ramified, generally S(𝔫) has smaller order than C(𝔫), neither of these groups is cyclic, and S(𝔫) is not the largest μ-type subgroup scheme of J_{0}(𝔫). First, we discuss our results about C(𝔫), S(𝔫), and T (𝔫):

Theorem 1.1.

(1) We give a complete description of C(𝔭𝔮) as an abelian group; see Theorem 6.11.

(2) For an arbitrary square-free 𝔫 we show that the group scheme S(𝔫) is μ-type, and therefore annihilated by 𝔈(𝔫), and we give a complete description of S(𝔫) as an abelian group; see Proposition 8.5 and Theorem 8.6.

(3) If ℓ ≠ p is a prime number which does not divide

(q − 1) · gcd(|𝔭| + 1, |𝔮| + 1),

then the ℓ-primary subgroups of C(𝔭𝔮) and T (𝔭𝔮) are equal; see Theorem 7.3.

Usually, many of the primes dividing the order of C(𝔭𝔮) satisfy the condition in (3), so, aside from a relatively small explicit set of primes, we can determine the ℓ-primary subgroup T (𝔭𝔮)_{ℓ} of T (𝔭𝔮). For example, (1) and (3) imply that if ℓ does not divide (|𝔭|^{2}−1)(|𝔮|^{2}−1), then T (𝔭𝔮)_{ℓ} = 0. The most advantageous case for applying (3) is when deg(𝔮) = deg(𝔭)+1, since then gcd(|𝔭|+1, |𝔮|+1) divides q − 1. In particular, if q = 2 and deg(𝔮) = deg(𝔭) + 1, then we conclude that the odd part of T (𝔭𝔮) coincides with C(𝔭𝔮). These results are qualitatively stronger than what is currently known about the rational torsion subgroup J_{0}(N)(ℚ)_{tor} of classical modular Jacobians of composite square-free levels (cf. [4]).

Outline of the Proof of Theorem 1.1. Although it was known that C(𝔫) is finite for any 𝔫 (see Theorem 6.1), there were no general results about its structure, besides the prime level case 𝔫 = 𝔭. The curve X_{0}(𝔭) has two cusps, so C(𝔭) is cyclic; its order was computed by Gekeler in [10]. The first obvious difference between the prime level and the composite level 𝔫 = 𝔭𝔮 is that X_{0}(𝔭𝔮) has 4 cusps, so C(𝔭𝔮) is usually not cyclic and is generated by 3 elements. To prove the result mentioned in (1), i.e., to compute the group structure of C(𝔭𝔮), we follow the strategy in [10], but the calculations become much more complicated. The idea is to use Drinfeld discriminant function to obtain upper bounds on the orders of cuspidal divisors, and then use canonical specializations of C(𝔭𝔮) into the component groups of J_{0}(𝔭𝔮) at 𝔭 and 𝔮 to obtain lower bounds on these orders. To deduce the group structure of S(𝔫) mentioned in (2) we use the rigid-analytic uniformizations of J_{0}(𝔫) and J_{1}(𝔫) over F_{∞}, and the “changing levels” result from [18], to reduce the problem to a calculation with finite groups. The proof of (3) is similar to the proof of Theorem 7.19 in [38], although there are some important differences, too. Suppose ℓ is a prime that does not divide q(q − 1). Since J_{0}(𝔭𝔮) has split toric reduction at ∞, the ℓ-primary subgroup T (𝔭𝔮)_{ℓ} maps injectively into the component group Φ_{∞} of J_{0}(𝔭𝔮) at ∞. Using the Eichler-Shimura relations, one shows that the image of T (𝔭𝔮)_{ℓ} in Φ_{∞} can be identified with a subspace of H_{0}(𝒯 , ℤ)^{Γ_{0}(𝔭𝔮)} ⊗ ℤ/ℓ^{n}ℤ annihilated by the Eisenstein ideal 𝔈(𝔭𝔮) for any sufficiently large n ∈ ℕ. Denote by E_{00}(𝔭𝔮, ℤ/ℓ^{n}ℤ) the subspace of H_{0}(𝒯 , ℤ)^{Γ_{0}(𝔭𝔮)} ⊗ ℤ/ℓ^{n}ℤ annihilated by 𝔈(𝔭𝔮). Then we have the inclusions

C(𝔭𝔮)_{ℓ} ↪ T (𝔭𝔮)_{ℓ} ↪ E_{00}(𝔭𝔮, ℤ/ℓ^{n}ℤ).

The space E_{00}(𝔭𝔮, ℤ/ℓ^{n}ℤ) contains the reductions modulo ℓ^{n} of certain Eisenstein series. We prove that if ℓ does not divide q(q − 1)gcd(|𝔭| + 1, |𝔮| + 1), then the whole E_{00}(𝔭𝔮, ℤ/ℓ^{n}ℤ) is generated by the reductions of these Eisenstein series (see Theorem 3.9 and Lemma 3.10). This allows us to compute E_{00}(𝔭𝔮, ℤ/ℓ^{n}ℤ). It turns out that E_{00}(𝔭𝔮, ℤ/ℓ^{n}ℤ) ≅ C(𝔭𝔮)_{ℓ}, and consequently C(𝔭𝔮)_{ℓ} = T (𝔭𝔮)_{ℓ}. To prove Theorem 3.9, we first prove a version of the key Theorem 1 in the famous paper by Atkin and Lehner [1] for ℤ/ℓ^{n}ℤ-valued harmonic cochains (see Theorem 2.26). The fact that we need to work with ℤ/ℓ^{n}ℤ rather than ℂ leads to technical difficulties, which results in the restriction ℓ ∤ q(q − 1)gcd(|𝔭| + 1, |𝔮| + 1). Note that in our definition the Hecke algebra 𝕋(𝔭𝔮) includes the operators U_{𝔭} and U_{𝔮}. This is important since we need to deal systematically with “old” forms of level 𝔭 and 𝔮. The smaller algebra 𝕋(𝔭𝔮)^{0} generated by the Hecke operators T_{𝔪} with 𝔪 coprime to 𝔭𝔮 used by Pál in [38] and [39] is not sufficient for getting a handle on E_{00}(𝔭𝔮, ℤ/ℓ^{n}ℤ). □

Now we concentrate on the case where we investigate the Jacquet-Langlands isogenies. We fix two primes x and y of A of degree 1 and 2, respectively. This differs from our usual 𝔉𝔯𝔞𝔨𝔱𝔲𝔯 notation for ideals of A. This is done primarily to make it easy for the reader to distinguish the theorems which assume that the level is xy. Several sections in the paper are titled “Special case” and deal exclusively with the case 𝔭𝔮 = xy. Note that X_{0}(𝔭𝔮) has genus 0 if 𝔭 and 𝔮 are distinct primes with deg(𝔭𝔮) ≤ 2. The genus of X_{0}(xy) is q, so this curve is the simplest example of a Drinfeld modular curve of composite level and positive genus. Also, by a theorem of Schweizer [49], X_{0}(𝔭𝔮) is hyperelliptic if and only if 𝔭 = x and 𝔮 = y, so one can think of this case as the hyperelliptic case. The cusps of X_{0}(xy) can be naturally labelled [x], [y], [1], [∞]; see Lemma 2.14. Let c_{x} and c_{y} denote the classes of divisors [x] − [∞] and [y] − [∞] in J_{0}(xy). First, we show that (see Theorem 7.13)

T (xy) = C(xy) = ⟨c_{x}⟩ ⊕ ⟨c_{y}⟩ ≅ ℤ/(q + 1)ℤ ⊕ ℤ/(q^{2} + 1)ℤ.

The reason we can prove this stronger result compared to (3) of Theorem 1.1 is that we can compute E_{00}(xy, ℤ/ℓ^{n}ℤ) without any restrictions on ℓ, and we can deal with the 2-primary torsion T (xy)_{2} using the fact that X_{0}(xy) is hyperelliptic. To simplify the notation, for the rest of this section denote 𝕋 = 𝕋(xy), 𝔈 = 𝔈(xy), H := H_{0}(𝒯 , ℤ)^{Γ_{0}(xy)}, H′ := H(𝒯 , ℤ)^{Γ^{xy}}, where this last group is the group of ℤ-valued Γ^{xy}-invariant harmonic cochains on 𝒯 . We show that (see Corollary 3.18)

𝕋/𝔈 ≅ ℤ/(q^{2} + 1)(q + 1)ℤ,

so the residue characteristic of any maximal ideal of 𝕋 containing 𝔈 divides (q^{2} + 1)(q + 1). The Jacquet-Langlands correspondence over F implies that there is an isomorphism H ⊗ ℚ ≅ H′ ⊗ ℚ which is compatible with the action of 𝕋.

Theorem 1.2 (See Theorems 9.5 and 9.6).

(1) If H ≅ H′ as 𝕋-modules, then there is an isogeny J_{0}(xy) → J^{xy} defined over F whose kernel is cyclic of order q^{2} + 1 and is annihilated by 𝔈.

(2) If H ≅ H′ as 𝕋-modules and for every prime ℓ|(q^{2}+1) the completion of 𝕋 ⊗ ℤ_{ℓ} at 𝔐 = (𝔈, ℓ) is Gorenstein, then there is an isogeny J_{0}(xy) → J^{xy} whose kernel is ⟨c_{y}⟩ ≅ ℤ/(q^{2} + 1)ℤ.

Remark 1.3. An isogeny J_{0}(xy) → J^{xy} with kernel ⟨c_{y}⟩ does not respect the canonical principal polarizations on the Jacobians since ⟨c_{y}⟩ is not a maximal isotropic subgroup of J_{0}(xy) with respect to the Weil pairing.

Outline of the Proof of Theorem 1.2. Both J_{0}(xy) and J^{xy} have rigid-analytic uniformization over F_{∞}. The assumption that H and H′ are isomorphic 𝕋modules allows us to identify the uniformizing tori of both Jacobians with 𝕋 ⊗ ℂ_{∞}^{×}. Next, we show that the groups of connected components of the Néron models of J_{0}(xy) and J^{xy} at ∞ are annihilated by 𝔈. This allows us to identify the uniformizing lattices of the Jacobians with ideals in 𝕋. These two observations, combined with a theorem of Gerritzen, imply (1). If in addition we assume that 𝕋_{𝔐} is Gorenstein, then we get an explicit description of the kernel of the Eisenstein ideal from which (2) follows. □

Proving that the assumptions in Theorem 1.2 hold seems difficult. First, even though H ⊗ ℚ and H′ ⊗ ℚ are isomorphic 𝕋-modules, the integral isomorphism is much more subtle. It is related to a classical problem about the conjugacy classes of matrices in Mat_{n}(ℤ); cf. [27]. Second, when ℓ|(q^{2} + 1) the kernel of 𝔐 in J_{0}(xy) is ramified, and Mazur’s Eisenstein descent arguments for proving 𝕋_{𝔐} is Gorenstein do not work in this ramified situation. (Both versions of Mazur’s descent discussed in [38, §§10,11] rely on subtle arithmetic properties of J_{0}(𝔭) which are valid only for prime level.) Nevertheless, both assumptions in Theorem 1.2 can be verified computationally; Section 10 is devoted to these calculations. We were able to check the assumptions for several cases for each prime q ≤ 7. In particular, we were able to go beyond dimension 2, which is currently the only dimension where the Ogg’s conjecture is known to be true over ℚ. Section 10 is also of independent interest since it provides an algorithm for computing the action of Hecke operators on H′; this should be useful in other arithmetic problems dealing with X^{xy}. (An algorithm for computing the Hecke action on H was already known from the work of Gekeler; see Remark 10.2.)

1.3. Notation. Aside from ∞, the places of F are in bijection with non-zero prime ideals of A. Given a place v of F, we denote by F_{v} the completion of F at v, by O_{v} the ring of integers of F_{v}, and by 𝔽_{v} the residue field of O_{v}. The valuation ord_{v} : F_{v} → ℤ is assumed to be normalized by ord_{v}(π_{v}) = 1, where π_{v} is a uniformizer of O_{v}. The normalized absolute value on F_{∞} is denoted by | · |.

Given a field K, we denote by K̄ an algebraic closure of K and K^{sep} a separable closure in K̄. The absolute Galois group Gal(K^{sep}/K) is denoted by G_{K}. Moreover, F_{v}^{nr} and O_{v}^{nr} will denote the maximal unramified extension of F_{v} and its ring of integers, respectively. Let R be a commutative ring with identity. We denote by R^{×} the group of multiplicative units of R. Let Mat_{n}(R) be the ring of n × n matrices over R, GL_{n}(R) the group of matrices whose determinant is in R^{×}, and Z(R) ≅ R^{×} the subgroup of GL_{n}(R) consisting of scalar matrices. If X is a scheme over a base S and S′ → S any base change, X_{S′} denotes the pullback of X to S′. If S′ = Spec(R) is affine, we may also denote this scheme by X_{R}. By X(S′) we mean the S′-rational points of the S-scheme X, and again, if S′ = Spec(R), we may also denote this set by X(R). Given a commutative finite flat group scheme G over a base S (or just an abelian group G, or a ring G) and an integer n, G[n] is the kernel of multiplication by n in G, and G_{ℓ} is the maximal ℓ-primary subgroup of G. The Cartier dual of G is denoted by G^{∗}. Given an ideal 𝔫 ⊲ A, by abuse of notation, we denote by the same symbol the unique monic polynomial in A generating 𝔫. It will always be clear from the context in which capacity 𝔫 is used; for example, if 𝔫 appears in a matrix, column vector, or a polynomial equation, then the monic polynomial is implied. The prime ideals 𝔭 ⊲ A are always assumed to be non-zero.

2. Harmonic cochains and Hecke operators

2.1. Harmonic cochains. Let G be an oriented connected graph in the sense of Definition 1 of §2.1 in [50]. We denote by V (G) and E(G) its set of vertices and edges, respectively. For an edge e ∈ E(G), let o(e), t(e) ∈ V (G) and ē ∈ E(G) be its origin, terminus and inversely oriented edge, respectively. In particular, t(ē) = o(e) and o(ē) = t(e). We will assume that for any v ∈ V (G) the number of edges with t(e) = v is finite, and t(e) ≠ o(e) for any e ∈ E(G) (i.e., G has no loops). A path in G is a sequence of edges {e_{i}}_{i∈I} indexed by the set I where I = ℤ, I = ℕ or I = {1, …, m} for some m ∈ ℕ such that t(e_{i}) = o(e_{i+1}) for every i, i + 1 ∈ I. We say that the path is without backtracking if e_{i} ≠ ē_{i+1} for every i, i + 1 ∈ I. We say that the path without backtracking {e_{i}}_{i∈ℕ} is a half-line if for every vertex v of G there is at most one index n ∈ ℕ such that v = o(e_{n}). Let Γ be a group acting on a graph G, i.e., Γ acts via automorphisms. We say that Γ acts with inversion if there is γ ∈ Γ and e ∈ E(G) such that γe = ē. If Γ acts without inversion, then we have a natural quotient graph Γ \ G such that V (Γ \ G) = Γ \ V (G) and E(Γ \ G) = Γ \ E(G), cf. [50, p. 25].

Definition 2.1. Fix a commutative ring R with identity. An R-valued harmonic cochain on G is a function f : E(G) → R that satisfies

(i)

f(e) + f(ē) = 0 for all e ∈ E(G),

(ii)

∑_{\begin{matrix}e∈E(G) \\ t(e)=v \end{matrix}} f(e) = 0 for all v ∈ V (G).

Denote by H(G, R) the group of R-valued harmonic cochains on G.

The most important graphs in this paper are the Bruhat-Tits tree 𝒯 of PGL_{2}(F_{∞}), and the quotients of 𝒯 . We recall the definition and introduce some notation for later use. Fix a uniformizer π_{∞} of F_{∞}. The sets of vertices V (𝒯 ) and edges E(𝒯 ) are the cosets GL_{2}(F_{∞})/Z(F_{∞})GL_{2}(O_{∞}) and GL_{2}(F_{∞})/Z(F_{∞})I_{∞}, respectively, where I_{∞} is the Iwahori group:

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

The matrix (\begin{matrix}0 & 1 \\ π_{∞} & 0 \end{matrix}) normalizes I_{∞}, so the multiplication from the right by

this matrix on GL_{2}(F_{∞}) induces an involution on E(𝒯 ); this involution is e ↦ ē. The matrices

(2.1)  E(𝒯 )^{+} = \{(\begin{matrix}π_{∞}^{k} & u \\ 0 & 1 \end{matrix}) | \begin{matrix}k ∈ ℤ \\ u ∈ F_{∞}, u mod π_{∞}^{k}O_{∞} \end{matrix}\}

are in distinct left cosets of I_{∞}Z(F_{∞}), and there is a disjoint decomposition (cf. [12, (1.6)])

E(𝒯 ) = E(𝒯 )^{+} ⨆ E(𝒯 )^{+} (\begin{matrix}0 & 1 \\ π_{∞} & 0 \end{matrix}) .

We call the edges in E(𝒯 )^{+} positively oriented. The group GL_{2}(F_{∞}) naturally acts on E(𝒯 ) by left multiplication. This induces an action on the group of R-valued functions on E(𝒯 ): for a function f on E(𝒯 ) and γ ∈ GL_{2}(F_{∞}) we define the function f|γ on E(𝒯 ) by (f|γ)(e) = f(γe). It is clear from the definition that f|γ is harmonic if f is harmonic, and for any γ, σ ∈ GL_{2}(F_{∞}) we have (f|γ)|σ = f|(γσ). Let Γ be a subgroup of GL_{2}(F_{∞}) which acts on 𝒯 without inversions. Denote by H(𝒯 , R)^{Γ} the subgroup of Γ-invariant harmonic cochains, i.e., f|γ = f for all γ ∈ Γ. It is clear that f ∈ H(𝒯 , R)^{Γ} defines a function f′ on the quotient graph Γ \ 𝒯 , and f itself can be uniquely recovered from this function: If e ∈ E(𝒯 ) maps to ẽ ∈ E(Γ \ 𝒯 ) under the quotient map, then f(e) = f′(ẽ). The conditions of harmonicity (i) and (ii) can be formulated in terms of f′ as follows. Since Γ acts without inversion, (i) is equivalent to

(i′)

f′(ẽ) + f′(¯ẽ) = 0 for all ẽ ∈ E(Γ \ 𝒯 ) .

Let v ∈ V (𝒯 ) and ṽ ∈ V (Γ \ 𝒯 ) be its image. The stabilizer group

Γ_{v} = {γ ∈ Γ | γv = v}

acts on the set {e ∈ E(𝒯 ) | t(e) = v}, and the orbits correspond to

{ẽ ∈ E(Γ \ 𝒯 ) | t(ẽ) = ṽ}.

Let Γ_{e} := {γ ∈ Γ | γe = e}; clearly Γ_{e} is a subgroup of Γ_{t(e)}. The weight of e

w(e) := [Γ_{t(e)} : Γ_{e}]

is the length of the orbit corresponding to e. Since w(e) depends only on its image ẽ in Γ \ 𝒯 , we can define w(ẽ) := w(e). Note that ∑_{t(ẽ)=ṽ} w(ẽ) = q + 1. We stress that, in general, w(e) depends on the orientation, i.e., w(e) ≠ w(ē). With this notation, condition (ii) is equivalent to

(ii′)

∑_{\begin{matrix}ẽ∈E(Γ\𝒯 ) \\ t(ẽ)=ṽ \end{matrix}} w(ẽ)f′(ẽ) = 0 for all ṽ ∈ V (Γ \ 𝒯 ) ,

cf. [18, (3.1)].

Definition 2.2. The group of R-valued cuspidal harmonic cochains for Γ, denoted H_{0}(𝒯 , R)^{Γ}, is the subgroup of H(𝒯 , R)^{Γ} consisting of functions which have compact support as functions on Γ \ 𝒯 , i.e., functions which have value 0 on all but finitely many edges of Γ \ 𝒯 . Let H_{00}(𝒯 , R)^{Γ} denote the image of H_{0}(𝒯 , ℤ)^{Γ} ⊗ R in H_{0}(𝒯 , R)^{Γ}.

Definition 2.3. It is known that the quotient graph Γ_{0}(𝔫) \ 𝒯 is the edge disjoint union

Γ_{0}(𝔫) \ 𝒯 = (Γ_{0}(𝔫) \ 𝒯 )^{0} ∪ ⋃_{s∈Γ_{0}(𝔫)\ℙ^{1}(F)} h_{s}

of a finite graph (Γ_{0}(𝔫)\𝒯 )^{0} with a finite number of half-lines h_{s}, called cusps; cf. Theorem 2 on page 106 of [50]. The cusps are in bijection with the orbits of the natural action of Γ_{0}(𝔫) on ℙ^{1}(F); cf. Remark 2 on page 110 of [50].

To simplify the notation, we put

H(𝔫, R) := H(𝒯 , R)^{Γ_{0}(𝔫)}
H_{0}(𝔫, R) := H_{0}(𝒯 , R)^{Γ_{0}(𝔫)}
H_{00}(𝔫, R) the image of H_{0}(𝔫, ℤ) ⊗ R in H_{0}(𝔫, R).

One can show that H_{0}(𝔫, ℤ) and H(𝔫, ℤ) are finitely generated free ℤ-modules of rank g(𝔫) and g(𝔫) + c(𝔫) − 1, respectively, where g(𝔫) is the genus of X_{0}(𝔫) and c(𝔫) is the number of cusps. From the above description it is clear that f is in H_{0}(𝔫, R) if and only if it eventually vanishes on each h_{s}. It is also clear that if R is flat over ℤ, then H_{0}(𝔫, R) = H_{00}(𝔫, R). On the other hand, it is easy to construct examples where this equality does not hold.

Example 2.4. The quotient graph GL_{2}(A) \ 𝒯 is a half-line; see Figure 1. Denote the edge with origin v_{i} and terminus v_{i+1} by e_{i}. The stabilizers of vertices and edges of GL_{2}(A) \ 𝒯 are well-known, cf. [17, p. 691]. From this one computes w(e_{i}) = q for all i, w(ē_{0}) = q + 1, and w(ē_{i}) = 1 for i ≥ 1. Therefore, if φ ∈ H(1, R), then φ(e_{i}) = q^{i}α (i ≥ 0) for some fixed α ∈ R[q + 1]. Now it is clear that H(1, R) = R[q + 1] and H_{0}(1, R) = H_{00}(1, R) = 0.

\begin{matrix}v_{0} \\ v_{1} \\ v_{2} \\ v_{3} \end{matrix}

Figure 1. GL_{2}(A) \ 𝒯

\begin{matrix}v_{0} \\ v_{1} \\ v_{2} \\ v_{3} \\ v_{−1} \\ v_{−2} \\ v_{−3} \end{matrix} ; Figure 2. Γ_{0}(x) \ 𝒯

\begin{matrix}v_{0} \\ v_{1} \\ v_{2} \\ v_{3} \\ v_{−1} \\ v_{−2} \\ v_{−3} \\ u \end{matrix} ; Figure 3. Γ_{0}(y) \ 𝒯

Example 2.5. The graph of Γ_{0}(x) \ T is given in Figure 2, where the vertex v_{i} (i ∈ ℤ) is the image of (\begin{matrix}T^{i} & 0 \\ 0 & 1 \end{matrix}) ∈ V (𝒯 ); the positive orientation is induced from E(𝒯 )^{+}. Denote by e_{i} the edge with origin v_{i−1} and terminus v_{i}. Since (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) v_{−i} = v_{i} and the stabilizers of v_{i} (i ≥ 0) in GL_{2}(A) are well-known (cf. [17, p. 691]), one easily computes

w(e_{i}) = \begin{cases}q & if i ≥ 0 \\ 1 & if i ≤ −1 \end{cases} w(ē_{i}) = \begin{cases}1 & if i ≥ −1 \\ q & if i ≤ −2 \end{cases}

Suppose φ ∈ H(x, R) and denote α = φ(e_{−1}). Since w(e_{i})φ(e_{i}) = w(ē_{i+1})φ(e_{i+1}), we get

φ(e_{i}) = \begin{cases}αq^{i+1} & if i ≥ −1 \\ α & if i = −2 \\ αq^{−i−3} & if i ≤ −3. \end{cases}

We conclude that H(x, R) = R, H_{0}(x, R) = R_{p}, and H_{00}(x, R) = 0. (Recall that R_{p} denotes the p-primary subgroup of R.)

Example 2.6. The graph Γ_{0}(y) \ 𝒯 is given in Figure 3, where v_{i} is the image of (\begin{matrix}T^{i} & 0 \\ 0 & 1 \end{matrix}) ∈ V (𝒯 ) and u is the image of (\begin{matrix}T^{−2} & T^{−1} \\ 0 & 1 \end{matrix}). We denote the edge with origin v_{i−1} and terminus v_{i} by e_{i}, and the edge with terminus u by e_{u}. One computes

w(e_{i}) = \begin{cases}q & if i ≥ 0 \\ 1 & if i ≤ −1 \end{cases} w(ē_{i}) = \begin{cases}1 & if i ≥ 0 \\ q & if i ≤ −1 \end{cases}

w(e_{u}) = q + 1, w(ē_{u}) = q − 1.

Let φ ∈ H(y, R). Denote φ(e_{0}) = α and φ(e_{u}) = β. Then (q + 1)β = 0 and

φ(e_{i}) = \begin{cases}αq^{i} & if i ≥ 0 \\ q^{−i−1}(α + (q − 1)β) & if i ≤ −1. \end{cases}

This implies that H(y, R) ≅ R ⊕ R[q + 1]. For φ to be cuspidal we must have q^{n}α = 0 and q^{n}(q − 1)β = 0 for some n ≥ 1. Thus, α ∈ R_{p} and β ∈ R[2] (resp. β = 0) if p is odd (resp. 2). We get an isomorphism H_{0}(y, R) ≅ R_{p} ⊕ R[2] if p is odd and H_{0}(y, R) ≅ R_{2} if p = 2. Note that H_{00}(y, R) = 0.

Lemma 2.7. The following holds:

(1) If 𝔫 ⊲ A has a prime divisor of odd degree, assume q(q − 1) ∈ R^{×}. Otherwise, assume q(q^{2} − 1) ∈ R^{×}. Then H_{0}(𝔫, R) = H_{00}(𝔫, R).

(2) If 𝔫 = 𝔭 is prime and q(q − 1) ∈ R^{×}, then H_{0}(𝔫, R) = H_{00}(𝔫, R).

Proof. Our proof relies on the results in [17], and is partly motivated by the proof of Theorem 3.3 in [17]. Let Γ := Γ_{0}(𝔫). By 1.11 and 2.10 in [17], the stabilizer Γ_{v} for any v ∈ V (𝒯 ) is finite, contains the scalar matrices Z(𝔽_{q}), and n(v) := #Γ_{v}/𝔽_{q}^{×} either divides (q − 1)q^{m} for some m ≥ 0, or is equal to q + 1. Moreover, n(v) = q + 1 is possible only if all prime divisors of 𝔫 have even degrees. Overall, we see that our assumptions in (1) imply that n(v) is invertible in R for any v ∈ V (𝒯 ). Since the stabilizer Γ_{e} of any e ∈ V (𝒯 ) is a subgroup of Γ_{t(e)} containing Z(𝔽_{q}), we also have n(e) := #Γ_{e}/𝔽_{q}^{×} ∈ R^{×}. Note that n(e) does not depend on the orientation of e and depends only on its image ẽ in Γ \ 𝒯 , so we can define n(ẽ) = n(e). Let H_{0}(Γ \ 𝒯 , R) be the subgroup of H(Γ \ 𝒯 , R) consisting of compactly supported harmonic cochains on Γ \ 𝒯 . There is an injective homomorphism

(2.2)  H_{0}(Γ \ 𝒯 , R) → H_{0}(𝔫, R)
φ ↦ φ^{†}

defined by φ^{†}(ẽ) = n(ẽ)φ(ẽ). Indeed, since n(ẽ) does not depend on the orientation of e, φ^{†} clearly satisfies (i′). As for (ii′), we have

∑_{\begin{matrix}ẽ∈E(Γ\𝒯 ) \\ t(ẽ)=ṽ \end{matrix}} w(ẽ)φ^{†}(ẽ) = ∑_{\begin{matrix}ẽ∈E(Γ\𝒯 ) \\ t(ẽ)=ṽ \end{matrix}} \frac{n(ṽ)}{n(ẽ)}n(ẽ)φ(ẽ) = n(ṽ) ∑_{\begin{matrix}ẽ∈E(Γ\𝒯 ) \\ t(ẽ)=ṽ \end{matrix}} φ(ẽ) = 0.

The map (2.2) is also defined over ℤ, and by [17, Thm. 3.3] gives an isomorphism H_{0}(Γ \ 𝒯 , ℤ) ⟶^{∼} H_{0}(𝔫, ℤ). Next, there is an isomorphism

H_{0}(Γ \ 𝒯 , R) ≅ H_{0}(Γ \ 𝒯 , ℤ) ⊗_{ℤ} R, which follows, for example, by observing that H_{1}(Γ \ 𝒯 , R) ≅ H_{0}(Γ \ 𝒯 , R) and applying the universal coefficient theorem for simplicial homology. Hence

H_{0}(Γ \ 𝒯 , R) ≅ H_{0}(Γ \ 𝒯 , ℤ) ⊗_{ℤ} R ≅ H_{0}(𝔫, ℤ) ⊗_{ℤ} R.

Let g = rank_{ℤ}H_{0}(Γ \ 𝒯 , ℤ). Thinking of the elements of H_{0}(Γ \ 𝒯 , ℤ) as 1cycles, it is easy to show by induction on g that one can choose e_{1}, …, e_{g} ∈ E(Γ\𝒯 ) and a ℤ-basis φ_{1}, …, φ_{g} of H_{0}(Γ\𝒯 , ℤ) such that Γ\𝒯 −{e_{1}, …, e_{g}} is a tree, and φ_{i}(e_{j}) = δ_{ij} =(Kronecker’s delta), 1 ≤ i, j ≤ g. By slight abuse of notation, denote the image of φ_{i}^{†} in H_{00}(𝔫, R) by the same symbol. Let ψ ∈ H_{0}(𝔫, R). Then

ψ′ := ψ − ∑_{i=1}^{g} \frac{ψ(e_{i})}{n(e_{i})}φ_{i}^{†}

is supported on a finite subtree S of Γ \ 𝒯 . Let v ∈ V (S) be a vertex such that there is a unique e ∈ E(S) with t(e) = v. Note that w(e) ∈ R^{×}. Condition (ii′) gives w(e)ψ′(e) = 0, so ψ′(e) = 0. This process can be iterated to show that ψ′ = 0. This implies that the natural map H_{0}(𝔫, ℤ) ⊗_{ℤ} R → H_{0}(𝔫, R) is surjective, which is part (1). To prove part (2), we can assume that deg(𝔭) is even. A consequence of 2.7 and 2.8 in [17] is that there is a unique v_{0} ∈ V (Γ \ 𝒯 ) with n(v_{0}) = q + 1 and a unique e_{0} ∈ E(Γ\𝒯 ) with o(e_{0}) = v_{0}. For any other v ∈ V (Γ\𝒯 ), n(v) divides (q − 1)q^{m}. Since the stabilizer of any edge e ∈ E(Γ \ 𝒯 ) is a subgroup of the stabilizers of both t(e) and o(e), we have n(e) ∈ R^{×}. After this observation, we can repeat the argument used to prove (1) to reduces the problem to showing that ψ ∈ H_{0}(𝔭, R) supported on a finite tree S is identically 0. We can always choose v ∈ V (S) to be a vertex different from v_{0} but such that there is a unique e ∈ E(S) with t(e) = v. Since w(e) is a unit in R, we can also finish as in part (1). □

The conclusion in Example 2.6 that H_{0}(y, R) ≠ H_{00}(y, R) if R[2] ≠ 0 is a special case of a general fact:

Lemma 2.8. Assume p is odd and invertible in R. Let 𝔭 ⊲ A be prime of even degree. If R[2] ≠ 0, then H_{0}(𝔭, R) ≠ H_{00}(𝔭, R).

Proof. Let Γ := Γ_{0}(𝔭). As in Lemma 2.7, let v_{0} be the unique vertex of Γ \ 𝒯 with n(v_{0}) = q + 1, and let e_{0} ∈ E(Γ \ 𝒯 ) be the unique edge with o(e_{0}) = v_{0}. Note that w(ē_{0}) = q + 1. As we already mentioned in the proof of Lemma 2.7, for any other vertex v in Γ \ 𝒯 , n(v) divides (q − 1)q^{m}. Moreover, it is easy to see, for example by case (a) of Lemma 2.7 in [17], that there is at least one vertex v such that n(v) is divisible by q − 1. Consider all the paths without backtracking connecting v_{0} to such a vertex, and fix a path of shortest length {e_{0}, e_{1}, …, e_{m}}. Then w(ē_{i}) (1 ≤ i ≤ m) is invertible in R, but w(e_{m}) is divisible by q − 1. For a fixed non-zero α ∈ R[2], define f on E(Γ \ 𝒯 ) by f(e_{0}) = α, f(e_{i}) = \frac{w(e_{i−1})}{w(ē_{i})}f(e_{i−1}) (1 ≤ i ≤ m), f(ē_{j}) = f(e_{j}) (0 ≤ j ≤ m), and f(e) = 0 for all other edges. It is easy to see that f ∈ H_{0}(𝔭, R). On the other hand, any function φ ∈ H_{0}(𝔭, ℤ) must be zero on e_{0}, since condition (ii′) for v_{0} gives (q + 1)φ(ē_{0}) = 0. Therefore, f ∉ H_{00}(𝔭, R). □

Remark 2.9. The fact stated in Lemma 2.8 is deduced in [38] by different (algebro-geometric) methods. Our combinatorial proof seems to answer the question in Remark 11.9 in [38].

2.2. Hecke operators and Atkin-Lehner involutions. Assume 𝔫 ⊲ A is fixed. Given a non-zero ideal 𝔪 ⊲ A, define an R-linear transformation of the space of R-valued functions on E(𝒯 ) by

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

where f|γ for γ ∈ GL_{2}(F_{∞}) is defined in Section 2.1, and the above sum is over a, b, d ∈ A such that a, d are monic, (ad) = 𝔪, (a) + 𝔫 = A, and deg(b) < deg(d). This transformation is the 𝔪-th Hecke operator. Following a common convention, for a prime divisor 𝔭 of 𝔫 we often write U_{𝔭} instead of T_{𝔭}.

Proposition 2.10. The Hecke operators preserve the spaces H(𝔫, R) and H_{0}(𝔫, R), and satisfy the recursive formulas:

T_{𝔪𝔪′} = T_{𝔪}T_{𝔪′} if 𝔪 + 𝔪′ = A,
T_{𝔭^{i}} = T_{𝔭^{i−1}}T_{𝔭} − |𝔭|T_{𝔭^{i−2}} if 𝔭 ∤ 𝔫,
T_{𝔭^{i}} = T_{𝔭}^{i} if 𝔭|𝔫.

Proof. The group-theoretic proofs of the analogous statement for the Hecke operators acting on classical modular forms work also in this setting; cf. [34, §4.5]. □

Definition 2.11. Let 𝕋(𝔫) be the commutative subalgebra of End_{ℤ}(H_{0}(𝔫, ℤ)) with the same unity element generated by all Hecke operators. Let 𝕋(𝔫)^{0} to be the subalgebra of 𝕋(𝔫) generated by the Hecke operators T_{𝔪} with 𝔪 coprime to 𝔫.

For every ideal 𝔪 dividing 𝔫 with gcd(𝔪, 𝔫/𝔪) = 1, let W_{𝔪} be any matrix in Mat_{2}(A) of the form

(2.3)  (\begin{matrix}a𝔪 & b \\ c𝔫 & d𝔪 \end{matrix})

such that a, b, c, d, ∈ A and the ideal generated by det(W_{𝔪}) in A is 𝔪. It is not hard to check that for f ∈ H(𝔫, R), f|W_{𝔪} does not depend on the choice of the matrix for W_{𝔪} and f|W_{𝔪} ∈ H(𝔫, R). Moreover, as R-linear endomorphisms of H(𝔫, R), W_{𝔪}’s satisfy

(2.4)  W_{𝔪_{1}}W_{𝔪_{2}} = W_{𝔪_{3}}, where 𝔪_{3} = \frac{𝔪_{1}𝔪_{2}}{gcd(𝔪_{1}, 𝔪_{2})^{2}} .

Therefore, the matrices W_{𝔪} acting on the R-module H(𝔫, R) generate an abelian group 𝕎 ≅ (ℤ/2ℤ)^{s}, called the group of Atkin-Lehner involutions, where s is the number of prime divisors of 𝔫. The following proposition, whose proof we omit, follows from calculations similar to those in [1, §2].

Proposition 2.12. Let

B_{𝔪} = (\begin{matrix}𝔪 & 0 \\ 0 & 1 \end{matrix}) .

(1) If 𝔫 is coprime to 𝔪 and f ∈ H(𝔫, R), then

(f|B_{𝔪})|W_{𝔪} = f,

where W_{𝔪} is the Atkin-Lehner involution acting on H(𝔫𝔪, R). (Note that by Lemma 2.25, f|B_{𝔪} ∈ H(𝔫𝔪, R).)

(2) Let 𝔪|𝔫 with gcd(𝔪, 𝔫/𝔪) = 1, and 𝔟 be coprime to 𝔪. If f ∈ H(𝔫, R),

then

(f|B_{𝔟})|W_{𝔪} = (f|W_{𝔪})|B_{𝔟},

where on the left hand-side W_{𝔪} denotes the Atkin-Lehner involution acting on H(𝔫𝔟, R) and on the right hand-side W_{𝔪} denotes the involution acting on H(𝔫, R).

(3) Let f ∈ H(𝔫, R). If 𝔮 is a prime ideal which divides 𝔫 but does not divide 𝔫/𝔮, then f|(U_{𝔮} + W_{𝔮}) ∈ H(𝔫/𝔮, R).

The vector space H_{0}(𝔫, ℚ) is equipped with a natural (Petersson) inner product

⟨f, g⟩ = ∑_{e∈E(Γ_{0}(𝔫)\𝒯 )} n(e)^{−1}f(e)g(e),

where n(e) is defined in the proof of Lemma 2.7. The Hecke operator T_{𝔪} is self-adjoint with respect to this inner product if 𝔪 is coprime to 𝔫; one can prove this by an argument similar to the proof of Lemma 13 in [1].

Definition 2.13. Let 𝔪 be a divisor of 𝔫 and 𝔡 be a divisor of 𝔫/𝔪. By Lemma 2.25, the map φ ↦ φ|B_{𝔡} gives an injective homomorphism

i_{𝔡,𝔪} : H_{0}(𝔪, ℚ) → H_{0}(𝔫, ℚ).

We denote the subspace generated by the images of all i_{𝔡,𝔪} (𝔪 ≠ 𝔫) by H_{0}(𝔫, ℚ)^{old}. The orthogonal complement of H_{0}(𝔫, ℚ)^{old} with respect to the Petersson product is the new subspace of H_{0}(𝔫, ℚ), and will be denoted by H_{0}(𝔫, ℚ)^{new}. The new subspace of H_{0}(𝔫, ℚ) is invariant under the action 𝕋(𝔫) (this again can be proven as in [1]). We denote by 𝕋(𝔫)^{new} the quotient of 𝕋(𝔫) through which 𝕋(𝔫) acts on H_{0}(𝔫, ℚ)^{new}.

As we mentioned, the cusps of Γ_{0}(𝔫) are in bijection with the orbits of the action of Γ_{0}(𝔫) on

ℙ^{1}(F) = ℙ^{1}(A) = \{(\begin{matrix}a \\ b \end{matrix}) | a, b ∈ A, gcd(a, b) = 1, a is monic\} ,

where Γ_{0}(𝔫) acts on ℙ^{1}(F) from the left as on column vectors. We leave the proof of the following lemma to the reader.

Lemma 2.14. Assume 𝔫 is square-free.

(1) For 𝔪|𝔫 let [𝔪] be the orbit of (\begin{matrix}1 \\ 𝔪 \end{matrix}) under the action of Γ_{0}(𝔫). Then

[𝔪] ≠ [𝔪′] if 𝔪 ≠ 𝔪′, and the set {[𝔪] | 𝔪|𝔫} is the set of cusps of Γ_{0}(𝔫). In particular, there are 2^{s} cusps, where s is the number of prime divisors of 𝔫.

(2) Since W_{𝔪} normalizes Γ_{0}(𝔫), it acts on the set of cusps of Γ_{0}(𝔫). There

is the formula

W_{𝔪}[𝔫] = [𝔫/𝔪].

The cusp [𝔫] is usually called the cusp at infinity. We will denote it by [∞].

2.3. Fourier expansion. An important observation in [38] is that the theory of Fourier expansions of automorphic forms over function fields developed in [57] works over more general rings than ℂ. Here we follow Gekeler’s reinterpretation [12] of Weil’s adelic approach as analysis on the Bruhat-Tits tree, but we will extend [12] to the setting of these more general rings.

Definition 2.15. Following [38] we say that R is a coefficient ring if p ∈ R^{×} and R is a quotient of a discrete valuation ring R̃ which contains p-th roots of unity. Note that the image of the p-th roots of unity of R̃ in R is exactly the set of p-th roots of unity of R. For example, any algebraically closed field of characteristic different from p is a coefficient ring.

Let

η : F_{∞} → R^{×}
∑ a_{i}π_{∞}^{i} ↦ η_{0}(Trace_{𝔽_{q}/𝔽_{p}}(a_{1}))

where η_{0} : 𝔽_{p} → R^{×} is a non-trivial additive character fixed once and for all. Let f be an R-valued function on E(𝒯 ), which is invariant under the action of

Γ_{∞} := \{(\begin{matrix}a & b \\ 0 & d \end{matrix}) ∈ GL_{2}(A)\} ,

and is alternating (i.e., satisfies f(e) = −f(ē) for all e ∈ E(𝒯 )). The constant Fourier coefficient of f is the R-valued function f^{0} on π_{∞}^{ℤ} defined by

f^{0}(π_{∞}^{k}) = \begin{cases}q^{1−k} ∑_{u∈(π_{∞})/(π_{∞}^{k})} f ((\begin{matrix}π_{∞}^{k} & u \\ 0 & 1 \end{matrix})) & if k ≥ 1 \\ f ((\begin{matrix}π_{∞}^{k} & 0 \\ 0 & 1 \end{matrix})) & if k ≤ 1. \end{cases}

For a divisor 𝔪 on F, the 𝔪-th Fourier coefficient f^{∗}(𝔪) of f is

f^{∗}(𝔪) = q^{−1−deg(𝔪)} ∑_{u∈(π_{∞})/(π_{∞}^{2+deg(𝔪)})} f ((\begin{matrix}π_{∞}^{2+deg(𝔪)} & u \\ 0 & 1 \end{matrix})) η(−mu),

if 𝔪 is non-negative, and f^{∗}(𝔪) = 0, otherwise; here m ∈ A is the monic polynomial such that 𝔪 = div(m) · ∞^{deg(𝔪)}.

Theorem 2.16. Let f be an R-valued function on E(𝒯 ), which is Γ_{∞}-invariant and alternating. Then

f ((\begin{matrix}π_{∞}^{k} & y \\ 0 & 1 \end{matrix})) = f^{0}(π_{∞}^{k}) + ∑_{\begin{matrix}0≠m∈A \\ deg(m)≤k−2 \end{matrix}} f^{∗}(div(m) · ∞^{k−2}) · η(my).

In particular, f is uniquely determined by the functions f^{0} and f^{∗}.

Proof. This follows from [38, §2] and [12, §2]. □

Lemma 2.17. Assume f is alternating and Γ_{∞}-invariant. Then f is a harmonic cochain if and only if

(i) f^{0}(π_{∞}^{k}) = f^{0}(1)q^{−k} for any k ∈ ℤ; (ii) f^{∗}(𝔪∞^{k}) = f^{∗}(𝔪)q^{−k} for any non-negative divisor 𝔪 and k ∈ ℤ_{≥0}.

Proof. See Lemma 2.13 in [12]. □

Lemma 2.18. For an ideal 𝔪 ⊲ A and f ∈ H(𝔫, ℤ) we have

(f|T_{𝔪})^{∗}(𝔯) = ∑_{a monic ; a| gcd(𝔪,𝔯) ; (a)+𝔫=A} \frac{|𝔪|}{|a|}f^{∗} (\frac{𝔯𝔪}{a^{2}}) .

In particular,

(f|T_{𝔪})^{∗}(1) = |𝔪|f^{∗}(𝔪).

Proof. See Lemma 3.2 in [38]. □

Lemma 2.19. Assume 𝔫 is square-free. A harmonic cochain f ∈ H(𝔫, R) is cuspidal if and only if (f|W)^{0}(1) = 0 for all W ∈ 𝕎.

Proof. By definition, f is cuspidal if and only if it vanishes on all but finitely many edges of each cusp [𝔪]. The positively oriented edges of the cusp [∞] are

given by the matrices (\begin{matrix}π_{∞}^{k} & 0 \\ 0 & 1 \end{matrix}), k ≤ 1. By definition of f^{0} and Lemma 2.17,

f ((\begin{matrix}π_{∞}^{k} & 0 \\ 0 & 1 \end{matrix})) = f^{0}(π_{∞}^{k}) = q^{−k}f^{0}(1).

Since q is invertible in R, we see that f eventually vanishes on [∞] if and only if f^{0}(1) = 0. Next, by Lemma 2.14, f vanishes on [𝔫/𝔪] if and only if f|W_{𝔪} vanishes on [∞], which is equivalent to (f|W_{𝔪})^{0}(1) = 0. □

Theorem 2.20. If R is a coefficient ring, then the bilinear 𝕋(𝔫)⊗R-equivariant pairing

(𝕋(𝔫) ⊗ R) × H_{00}(𝔫, R) → R
T, f ↦ (f|T)^{∗}(1)

is perfect.

\begin{aligned}[x] & [∞] & [y] & [1] & ⋮ b_{u} & \begin{matrix}c_{3} \\ a_{2} = b_{0} \\ a_{1} \\ c_{1} \\ c_{4} \\ a_{3} \\ a_{4} \\ a_{5} \\ a_{6} \\ c_{2} \end{matrix} \\ & Figure 4. Γ_{0}(xy) \ 𝒯 \end{aligned}

Proof. Theorem 3.17 in [11] says that the pairing

𝕋(𝔫) × H_{0}(𝔫, ℤ) → ℤ (2.5)

T, f ↦ (f|T)^{∗}(1)

is non-degenerate and becomes a perfect pairing after tensoring with ℤ[p^{−1}]. Since p is invertible in R by assumption, the claim follows. □

It is not known if in general the pairing (2.5) is perfect. This is in contrast to the situation over ℚ where the analogous pairing between the Hecke algebra and the space of weight-2 cusp forms on Γ_{0}(N) with integral Fourier expansions is perfect (cf. [46, Thm. 2.2]). This dichotomy comes from the formula (f|T_{𝔪})^{∗}(1) = |𝔪|f^{∗}(𝔪); in the classical situation the first Fourier coefficient of f|T_{m} is just the mth Fourier coefficient of f.

Proposition 2.21. In the special case 𝔫 = xy, the pairing (2.5)

𝕋(xy) × H_{0}(xy, ℤ) → ℤ

is perfect. Moreover, as ℤ-modules,

𝕋(xy)^{0} = 𝕋(xy) ≅ ℤ ⊕ ⨁_{\begin{matrix}deg(𝔭)=1 \\ 𝔭≠x \end{matrix}} ℤT_{𝔭}.

Proof. Take α_{x}, β_{x} ∈ 𝔽_{q} such that y = x^{2} +α_{x}x+β_{x}. Let ϖ_{x} := x^{−1}, which is also a uniformizer at ∞. The quotient graph Γ_{0}(xy)\𝒯 is depicted in Figure 4 with positively oriented edges

c_{1} = (\begin{matrix}ϖ_{x} & 0 \\ 0 & 1 \end{matrix}) , c_{2} = (\begin{matrix}ϖ_{x}^{3} & 0 \\ 0 & 1 \end{matrix}) , c_{3} = (\begin{matrix}ϖ_{x}^{4} & ϖ_{x} \\ 0 & 1 \end{matrix}) , c_{4} = (\begin{matrix}ϖ_{x}^{5} & y^{−1} \\ 0 & 1 \end{matrix});

a_{1} = (\begin{matrix}ϖ_{x}^{2} & ϖ_{x} \\ 0 & 1 \end{matrix}) , a_{2} = (\begin{matrix}ϖ_{x}^{3} & ϖ_{x} \\ 0 & 1 \end{matrix}) , a_{3} = (\begin{matrix}ϖ_{x}^{4} & y^{−1} \\ 0 & 1 \end{matrix}) , a_{4} = (\begin{matrix}ϖ_{x}^{3} & ϖ_{x}^{2} \\ 0 & 1 \end{matrix});

a_{5} = (\begin{matrix}ϖ_{x}^{2} & 0 \\ 0 & 1 \end{matrix}) , a_{6} = (\begin{matrix}ϖ_{x}^{4} & ϖ_{x} − β_{x}ϖ_{x}^{3} \\ 0 & 1 \end{matrix});

b_{u} = (\begin{matrix}ϖ_{x}^{3} & ϖ_{x} + uϖ_{x}^{2} \\ 0 & 1 \end{matrix}) , u ∈ 𝔽_{q}.

Note that in this notation a_{2} = b_{0}. A small calculation shows that

w(a_{1}) = w(ā_{2}) = w(ā_{3}) = w(a_{4}) = q − 1,

and the weights of all other edges in (Γ_{0}(xy) \ 𝒯 )^{0} are 1. It is easy to see that the map

H_{0}(xy, ℤ) → ⨁_{u∈𝔽_{q}} ℤ
f ↦ (f(b_{u}))_{u∈𝔽_{q}}

is an isomorphism, so the harmonic cochains f_{v} ∈ H_{0}(xy, ℤ), v ∈ 𝔽_{q}, defined by f_{v}(b_{u}) = δ_{v,u}=(Kronecker’s delta) form a ℤ-basis. Let f ∈ H_{0}(xy, ℤ) and κ ∈ 𝔽_{q}. By Lemma 2.18

q(f|T_{x−κ})^{∗}(1) = q^{2}f^{∗}(x − κ) = ∑_{w∈ϖ_{x}O_{∞}/ϖ_{x}^{3}O_{∞}} f ((\begin{matrix}ϖ_{x}^{3} & w \\ 0 & 1 \end{matrix})) η (−(x − κ)w)

=f ((\begin{matrix}ϖ_{x}^{3} & 0 \\ 0 & 1 \end{matrix})) + ∑_{β∈𝔽_{q}^{×}} f ((\begin{matrix}ϖ_{x}^{3} & βϖ_{x}^{2} \\ 0 & 1 \end{matrix})) η (−(ϖ_{x}^{−1} − κ)βϖ_{x}^{2})

+ ∑_{u∈𝔽_{q}} ∑_{β∈𝔽_{q}^{×}} f ((\begin{matrix}ϖ_{x}^{3} & β(ϖ_{x} + uϖ_{x}^{2}) \\ 0 & 1 \end{matrix})) η (−(ϖ_{x}^{−1} − κ)β(ϖ_{x} + uϖ_{x}^{2})) .

Since the double class of (\begin{matrix}ϖ_{x}^{3} & w \\ 0 & 1 \end{matrix}) does not change if w is replaced by βw

(β ∈ 𝔽_{q}^{×}), f ((\begin{matrix}ϖ_{x}^{3} & 0 \\ 0 & 1 \end{matrix})) = f(c_{2}) = 0, and ∑_{β∈𝔽_{q}^{×}} η(βϖ_{x}) = −1, the above

sum reduces to

−f(a_{4}) + ∑_{u∈𝔽_{q}} f(b_{u})(qδ_{u,κ} − 1).

Using (ii′),

(q − 1)f(a_{1}) + f(a_{5}) = 0, (q − 1)f(a_{4}) + f(ā_{5}) = 0, f(a_{1}) = ∑_{u∈𝔽_{q}} f(b_{u}).

Therefore, f(a_{4}) = − ∑_{u∈𝔽_{q}} f(b_{u}) and we get

(f|T_{x−κ})^{∗}(1) = f(b_{κ}).

In particular, (f_{v}|T_{x−κ})^{∗}(1) = δ_{κ,v}. This implies that the homomorphism

(2.6)  𝕋(xy) → Hom(H_{0}(xy, ℤ), ℤ)

induced by the pairing (2.5) is surjective. Comparing the ranks of both sides, we conclude that this map is in fact an isomorphism, which is equivalent to the pairing being perfect. Let M be the ℤ-submodule of 𝕋(xy) generated by {T_{x−κ} | κ ∈ 𝔽_{q}}. The composition of M ↪ 𝕋(xy) with (2.6) gives a surjection M → Hom(H_{0}(xy, ℤ), ℤ). This implies that M = 𝕋(xy) and M ≅ ⨁_{κ∈𝔽_{q}} ℤT_{x−κ}. An easy consequence of the definitions is that f^{∗}(1) = −f(a_{1}), cf. [11, (3.16)]. If we denote S = ∑_{κ∈𝔽_{q}} T_{x−κ}, then

(2.7)  (f|S)^{∗}(1) = ∑_{κ∈𝔽_{q}} f(b_{κ}) = f(a_{1}) = −f^{∗}(1).

The non-degeneracy of the pairing implies that S = −1. Therefore

𝕋(xy) = ℤ ⊕ ⨁_{κ∈𝔽_{q}^{×}} ℤT_{x−κ} ⊆ 𝕋(xy)^{0},

which implies 𝕋(xy) = 𝕋(xy)^{0}. □

Remark 2.22. In [44], we have extended the statement of Proposition 2.21 to arbitrary 𝔫 ⊲ A of degree 3. More precisely, we proved that the pairing (2.5) is perfect if deg(𝔫) = 3. Moreover, if 𝔫 has degree 3 but is not a product of three distinct primes of degree 1, then 𝕋(𝔫) = 𝕋(𝔫)^{0}. Finally, if 𝔫 is a product of three distinct primes of degree 1, then 𝕋(𝔫)/𝕋(𝔫)^{0} is finite but non-zero.

2.4. Atkin-Lehner method. For b ∈ A, let S_{b} = (\begin{matrix}1 & b \\ 0 & 1 \end{matrix}). Define a linear

operator U_{𝔭} on the space of R-valued functions on E(𝒯 ) by

f|U_{𝔭} = ∑_{\begin{matrix}b∈A \\ deg(b)<deg(𝔭) \end{matrix}} f|B_{𝔭}^{−1}S_{b}.

Note that the action of B_{𝔪}^{−1} on functions on E(𝒯 ) is the same as the action of

the matrix (\begin{matrix}1 & 0 \\ 0 & 𝔪 \end{matrix}) (since the diagonal matrices act trivially), so this operator

agrees with the Hecke operator U_{𝔭} when restricted to H(𝔫, R) for any 𝔫 divisible by 𝔭.

Lemma 2.23. Let 𝔭 and 𝔮 be two distinct prime ideals of A. If f ∈ H(𝒯 , R)^{Γ_{∞}}, then

(f|B_{𝔭})|U_{𝔭} = |𝔭| · f,
(f|B_{𝔭})|U_{𝔮} = (f|U_{𝔮})|B_{𝔭}.

Proof. We have

(f|B_{𝔭})|U_{𝔭} = ∑_{\begin{matrix}b∈A \\ deg(b)<deg(𝔭) \end{matrix}}(f|B_{𝔭})|B_{𝔭}^{−1}S_{b} = ∑_{\begin{matrix}b∈A \\ deg(b)<deg(𝔭) \end{matrix}} f|S_{b}.

Since S_{b} ∈ Γ_{∞}, we have f|S_{b} = f for all b, so the last sum is equal to |𝔭|f. Next, for b ∈ A representing a residue modulo 𝔮 we have

B_{𝔭}B_{𝔮}^{−1}S_{b} = (\begin{matrix}𝔭 & b𝔭 \\ 0 & 𝔮 \end{matrix}) .

By the division algorithm there is a ∈ A and b′ ∈ A with deg(b′) < deg(𝔮) such that b𝔭 = a𝔮 + b′. Now

(\begin{matrix}1 & a \\ 0 & 1 \end{matrix}) (\begin{matrix}𝔭 & b𝔭 \\ 0 & 𝔮 \end{matrix}) = (\begin{matrix}𝔭 & b′ \\ 0 & 𝔮 \end{matrix}) = B_{𝔮}^{−1}S_{b′}B_{𝔭}.

As b runs over the residues modulo 𝔮, b′ runs over the same set since 𝔭 ≠ 𝔮. Thus, using Γ_{∞}-invariance of f, we get (f|B_{𝔭})|U_{𝔮} = (f|U_{𝔮})|B_{𝔭}. □

Lemma 2.24. For any non-zero ideal 𝔪 ⊲ A and f ∈ H(𝒯 , R)^{Γ_{∞}}

(f|B_{𝔪})^{0}(π_{∞}^{k}) = f^{0}(π_{∞}^{k−deg(𝔪)}), (f|B_{𝔪})^{∗}(𝔫) = f^{∗}(𝔫/𝔪).

Proof. See Proposition 2.10 in [12]. □

Given ideals 𝔫, 𝔪 ⊲ A, denote

Γ_{0}(𝔫, 𝔪) = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL_{2}(A) | c ∈ 𝔫, b ∈ 𝔪\} .

Lemma 2.25. If f ∈ H(𝔫, R), then f|B_{𝔪} is Γ_{0}(𝔫𝔪)-invariant and f|B_{𝔪}^{−1} is Γ_{0}(𝔫/ gcd(𝔫, 𝔪), 𝔪)-invariant.

Proof. This follows from a straightforward manipulation with matrices. □

Theorem 2.26. Let 𝔭 and 𝔮 be two distinct primes such that 𝔭𝔮 divides 𝔫, and 𝔭𝔮 is coprime to 𝔫/𝔭𝔮. Let φ ∈ H(𝔫, R). Assume φ^{∗}(𝔪) = 0 unless 𝔭 or 𝔮 divides 𝔪. Then there exist ψ_{1} ∈ H(𝔫/𝔭, R) and ψ_{2} ∈ H(𝔫/𝔮, R) such that

s_{𝔭,𝔮} · φ = ψ_{1}|B_{𝔭} + ψ_{2}|B_{𝔮},

where s_{𝔭,𝔮} = gcd(|𝔭| + 1, |𝔮| + 1).

Proof. Take ϕ_{2} := |q|^{−1} · φ|U_{𝔮} ∈ H(𝔫, R). We have

(ϕ_{2})^{0}(π_{∞}^{k}) = φ^{0}(π_{∞}^{k+deg(𝔮)}), ϕ_{2}^{∗}(𝔪) = φ^{∗}(𝔪𝔮).

Let φ_{1} := φ − ϕ_{2}|B_{𝔮} ∈ H(𝔫𝔮, R). Then by Lemma 2.24,

(φ_{1})^{0}(π_{∞}^{k}) = 0, φ_{1}^{∗}(𝔪) = φ^{∗}(𝔪) if 𝔮 ∤ 𝔪, φ_{1}^{∗}(𝔪) = 0 if 𝔮|𝔪.

Let ϕ_{1} := φ_{1}|B_{𝔭}^{−1}, which is Γ_{0}(𝔫𝔮/𝔭, 𝔭)-invariant by Lemma 2.25. In particular, φ_{1}^{∗}(𝔪) = 0 unless 𝔭|𝔪, which implies that ϕ_{1} is Γ_{∞}-invariant. Since Γ_{∞} and Γ_{0}(𝔫𝔮/𝔭, 𝔭) generates Γ_{0}(𝔫𝔮/𝔭), we get ϕ_{1} ∈ H(𝔫𝔮/𝔭, R) with

(ϕ_{1})^{0}(π_{∞}^{k}) = 0, ϕ_{1}^{∗}(𝔪) = φ^{∗}(𝔪𝔭) if 𝔮 ∤ 𝔪, ϕ_{1}^{∗}(𝔪) = 0 if 𝔮|𝔪,

and

φ = ϕ_{1}|B_{𝔭} + ϕ_{2}|B_{𝔮}.

By Proposition 2.12, ψ_{1} := φ|(U_{𝔭} + W_{𝔭}) ∈ H(𝔫/𝔭, R). Using Proposition 2.12 and Lemma 2.23,

(ϕ_{1}|B_{𝔭})|(U_{𝔭} + W_{𝔭}) = ϕ_{1}|B_{𝔭}|U_{𝔭} + ϕ_{1}|B_{𝔭}|W_{𝔭} = |𝔭|ϕ_{1} + ϕ_{1} = (|𝔭| + 1)ϕ_{1}.

On the other hand, using the fact that ϕ_{2} ∈ H(𝔫, R), we have

(ϕ_{2}|B_{𝔮})|(U_{𝔭} + W_{𝔭}) = ϕ_{2}|(U_{𝔭} + W_{𝔭})|B_{𝔮}.

If we denote ψ := ϕ_{2}|(U_{𝔭} + W_{𝔭}), then we proved that

ψ_{1} = (|𝔭| + 1)ϕ_{1} + ψ|B_{𝔮} ∈ H(𝔫/𝔭, R).

Therefore,

(|𝔭| + 1)φ = (|𝔭| + 1)ϕ_{1}|B_{𝔭} + (|𝔭| + 1)ϕ_{2}|B_{𝔮}
= ((|𝔭| + 1)ϕ_{1} + ψ|B_{𝔮})|B_{𝔭} + ((|𝔭| + 1)ϕ_{2} − ψ|B_{𝔭})|B_{𝔮} = ψ_{1}|B_{𝔭} + ψ_{2}|B_{𝔮},

where ψ_{2} := (|𝔭| + 1)ϕ_{2} − ψ|B_{𝔭}. We already proved that ψ_{1} ∈ H(𝔫/𝔭, R). Obviously ψ_{2}|B_{𝔮} ∈ H(𝔫, R). By Lemma 2.25, ψ_{2} is Γ_{0}(𝔫/𝔮, 𝔮)-invariant. Since it is also Γ_{∞}-invariant, we conclude ψ_{2} ∈ H(𝔫/𝔮, R). Finally, interchanging the roles of 𝔭 and 𝔮 we obtain

(|𝔮| + 1)φ = ψ′_{1}|B_{𝔭} + ψ′_{2}|B_{𝔮}

with ψ′_{1} ∈ H(𝔫/𝔭, R) and ψ′_{2} ∈ H(𝔫/𝔮, R). This implies the claim of the theorem. □

3. Eisenstein harmonic cochains

3.1. Eisenstein series. In this section R always denotes a coefficient ring, in particular, p is invertible in R. We say that a harmonic cochain φ ∈ H(𝔫, R) is Eisenstein if φ|T_{𝔭} = (|𝔭| + 1)φ for every prime ideal 𝔭 ⊲ A not dividing 𝔫. It is clear that the Eisenstein harmonic cochains form an R-submodule of H(𝔫, R) which we denote by E(𝔫, R). The Drinfeld half-plane

Ω = ℙ^{1}(ℂ_{∞}) − ℙ^{1}(F_{∞}) = ℂ_{∞} − F_{∞}

has a natural structure of a smooth connected rigid-analytic space over F_{∞}; see [18, §1]. The group Γ_{0}(𝔫) acts on Ω via linear fractional transformations:

(\begin{matrix}a & b \\ c & d \end{matrix}) z = \frac{az + b}{cz + d}.

This action is discrete, so the quotient

(3.1)  Y_{0}(𝔫)(ℂ_{∞}) = Γ_{0}(𝔫) \ Ω

has a natural structure of a rigid-analytic curve over F_{∞}, which is in fact an affine algebraic curve; cf. [6, Prop. 6.6]. If we denote \overline{Ω} = Ω ∪ ℙ^{1}(F), then

X_{0}(𝔫)(ℂ_{∞}) = Γ_{0}(𝔫) \ \overline{Ω}

is the projective closure of Y_{0}(𝔫). The points X_{0}(𝔫)(ℂ_{∞}) − Y_{0}(𝔫)(ℂ_{∞}) are called the cusps of X_{0}(𝔫), and they are in natural bijection with the cusps of Γ_{0}(𝔫) \ 𝒯 in Definition 2.3. The Hecke operator T_{𝔭} induces a correspondence on X_{0}(𝔫)(ℂ_{∞})

(3.2)  T_{𝔭} : z ↦ ∑_{\begin{matrix}b∈A \\ deg(b)<deg(𝔭) \end{matrix}} \frac{z + b}{𝔭} + 𝔭z mod Γ_{0}(𝔫);

U_{𝔭}(z) is given by the same sum but without the last summand. Let O(Ω)^{×} be the group of nowhere vanishing holomorphic functions on Ω. The group GL_{2}(F_{∞}) act on O(Ω)^{×} via (f|γ)(z) = f(γz). To each f ∈ O(Ω)^{×} van der Put associated a harmonic cochain r(f) ∈ H(𝒯 , ℤ) so that the sequence

(3.3)  0 → ℂ_{∞}^{×} → O(Ω)^{×} ⟶^{r} H(T , ℤ) → 0

is exact and GL_{2}(F_{∞})-equivariant. As is explained in [18], the map r plays the role of a logarithmic derivation.

Lemma 3.1. Assume 𝔫 is square-free and f ∈ O(Ω)^{×} is Γ_{0}(𝔫)-invariant. Then r(f) is Eisenstein.

Proof. Put

e_{𝔫}(z) = z ∏_{0≠a∈𝔫} (1 − \frac{z}{a}) and Γ_{∞}^{u} = \{(\begin{matrix}1 & a \\ 0 & 1 \end{matrix}) |a ∈ 𝔫\} .

For z ∈ Ω, let |z|_{i} = inf{|z − s| | s ∈ F_{∞}} be its “imaginary” absolute value. The subspace Ω_{d} = {z ∈ Ω | |z|_{i} ≥ d} of Ω is stable under Γ_{∞}^{u}, and for d ≫ 0, the function t(z) = e_{A}(z)^{−1} identifies Γ_{∞}^{u} \ Ω_{d} with a small punctured disc D_{ε}^{0} = {t ∈ ℂ_{∞} | 0 < |t| ≤ ε}. The function f is Γ_{∞}^{u}-invariant, so can be considered as a holomorphic non-vanishing function on D_{ε}^{0}. By the nonarchimedean analogue of Picard’s Big Theorem [55, (1.3)], f has at worst a pole at t = 0, or equivalently, at the cusp [∞]. Now let [c] be any other cusp of Γ_{0}(𝔫) and γ ∈ GL_{2}(A) be such that γ[∞] = [c]. The function f|γ ∈ O(Ω)^{×} is invariant under Γ′ = γ^{−1}Γ_{0}(𝔫)γ. The stabilizer of [∞] in Γ′ contains Γ_{∞}^{u}, so the previous argument shows that f|γ is meromorphic at [∞]. Thus, f is meromorphic at [c]. We conclude that f descends to a rational function on X_{0}(𝔫) whose divisor is supported at the cusps. Now we use an idea from the proof of Lemma 6.2 in [38]. The Hecke correspondence T_{𝔭} defines a map from the group of divisors on X_{0}(𝔫) supported at the cusps to itself (cf. (3.2)):

T_{𝔭}[𝔡] = (\begin{matrix}𝔭 & 0 \\ 0 & 1 \end{matrix}) (\begin{matrix}1 \\ 𝔡 \end{matrix})+ ∑_{\begin{matrix}b≠0 \\ deg(b)<deg(𝔭) \end{matrix}} (\begin{matrix}1 & b \\ 0 & 𝔭 \end{matrix}) (\begin{matrix}1 \\ 𝔡 \end{matrix}) = (\begin{matrix}𝔭 \\ 𝔡 \end{matrix})+ ∑_{\begin{matrix}b≠0 \\ deg(b)<deg(𝔭) \end{matrix}}(\begin{matrix}1 + b𝔡 \\ 𝔭𝔡 \end{matrix}) .

The orbit of the cusp [𝔡] consists exactly of those (\begin{matrix}α \\ β \end{matrix}) ∈ ℙ^{1}(A) such that β

is divisible by 𝔡 and is coprime to 𝔫/𝔡, so T_{𝔭}[𝔡] = (1 + |𝔭|)[𝔡]. Thus, f|T_{𝔭} and f^{|𝔭|+1}, as rational functions on X_{0}(𝔫), have the same divisor. This implies that (f|T_{𝔭})/f^{|𝔭|+1} is a constant function. Applying r, we get

r(f|T_{𝔭}) = (|𝔭| + 1)r(f).

Since r is GL_{2}(F_{∞})-equivariant, r(f|T_{𝔭}) = r(f)|T_{𝔭}, which finishes the proof. □

Lemma 3.1 gives a natural source of ℤ-valued Eisenstein harmonic cochains. Let Δ(z) be the Drinfeld discriminant function on Ω defined on page 183 of [14]. This is a Drinfeld modular form of weight (q^{2} − 1) and type 0 for GL_{2}(A), which vanishes nowhere on Ω. Let Δ_{𝔫} := Δ|B_{𝔫} = Δ(𝔫z). By page 194 of [14], Δ/Δ_{𝔫} is a Γ_{0}(𝔫)-invariant function in O(Ω)^{×}. Hence r(Δ/Δ_{𝔫}) ∈ E(𝔫, ℤ). Define

ν(𝔫) = \begin{cases}1, & if deg(𝔫) is even \\ q + 1, & if deg(𝔫) is odd \end{cases}

and

(3.4)  E_{𝔫} = \frac{ν(𝔫)}{(q − 1)(q^{2} − 1)}r(Δ/Δ_{𝔫}).

By [14, (3.18)], E_{𝔫} is ℤ-valued and primitive (i.e., E_{𝔫} is not a scalar multiple of another harmonic cochain in H(𝔫, ℤ) except for ±E_{𝔫}). We call E_{𝔫} ∈ E(𝔫, ℤ) the Eisenstein series. The Fourier expansion of E_{𝔫} can be deduced from [14]:

E_{𝔫} ((\begin{matrix}π_{∞}^{k} & y \\ 0 & 1 \end{matrix})) = ν(𝔫) · q^{−k+1} · [\frac{1 − |𝔫|}{1 − q^{2}} + ∑_{{0≠m∈A, \atop deg(m)≤k−2}} σ_{𝔫}(m)η(my)] ,

where σ_{𝔫}(m) := σ(m) − |𝔫| · σ(m/𝔫), and σ is the divisor function

σ(m) := \begin{cases}∑_{{monic m′∈A, \atop m′|m}} |m′|, & if m ∈ A, \\ 0, & otherwise. \end{cases}

Remark 3.2. Note that for each prime 𝔭 ⊲ A and m ∈ A,

σ(m𝔭) = σ(𝔭)σ(m) − |𝔭|σ(m/𝔭).

Therefore the Fourier expansion of E_{𝔫}|T_{𝔭} for each prime 𝔭 not dividing 𝔫 also tells us that E_{𝔫} ∈ E(𝔫, ℤ).

Example 3.3. Let E_{0} be the R-valued function on E(𝒯 ) defined by

E_{0} ((\begin{matrix}π_{∞}^{k} & u \\ 0 & 1 \end{matrix})) = −E_{0} ((\begin{matrix}π_{∞}^{k} & u \\ 0 & 1 \end{matrix}) (\begin{matrix}0 & 1 \\ π_{∞} & 0 \end{matrix})) = q^{−k}.

This function is alternating and Γ_{∞}-invariant. For γ = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL_{2}(A), let

ω := ord_{∞}(cu + d). By the calculations in [12, p. 379]

(E_{0}|γ) ((\begin{matrix}π^{k} & u \\ 0 & 1 \end{matrix})) = \begin{cases}−q^{k−2 deg(c)−1} & if ω ≥ k − deg(c) \\ q^{2ω−k} & if ω < k − deg(c). \end{cases}

Now it is easy to see that if α ∈ R[q + 1], then αE_{0} is the function in H(1, R) discussed in Example 2.4. The Hecke operator T_{𝔭} acts on αE_{0} by

(αE_{0}|T_{𝔭}) ((\begin{matrix}π^{k} & u \\ 0 & 1 \end{matrix})) =αE_{0} ((\begin{matrix}π^{k}𝔭 & u𝔭 \\ 0 & 1 \end{matrix}))

+ ∑_{deg(b)<deg(𝔭)} αE_{0} ((\begin{matrix}π^{k}/𝔭 & (u + b)/𝔭 \\ 0 & 1 \end{matrix}))

= α(−1)^{k−deg(𝔭)} + q^{deg(𝔭)}α(−1)^{k+deg(𝔭)} = (1 + |𝔭|)αE_{0} ((\begin{matrix}π^{k} & u \\ 0 & 1 \end{matrix})) .

Therefore, αE_{0} is Eisenstein and H(1, R) = E(1, R) ≅ R[q + 1].

Example 3.4. Let φ ∈ H_{0}(x, R). As we saw in Example 2.5, φ is uniquely determined by φ(e_{0}) = α. Note that φ^{0}(1) = φ(e_{1}) = qα. On the other hand, E_{x}^{0}(1) = q, so H(x, R) = E(x, R) ≅ R is generated by E_{x}.

Example 3.5. Finally, we return to the setting of Example 2.6. The Eisenstein series E_{y}, as a function on Γ_{0}(y) \ 𝒯 , can be explicitly described by E_{y}(e_{i}) = E_{y}(e_{−i−1}) = q^{i} for i ≥ 0, and E_{y}(e_{u}) = 0. Next, the function αE_{0}, for any α ∈ R[q+1], can be considered as a function on Γ_{0}(y)\𝒯 , and as such it is given by αE_{0}(e_{u}) = −α and αE_{0}(e_{i}) = α(−1)^{i+1} (∀i ∈ ℤ). Since any f ∈ H(y, R) is uniquely determined by its values on e_{u} and e_{0}, we see that

H(y, R) = E(y, R) = RE_{y} ⊕ {αE_{0} | α ∈ R[q + 1]} ≅ R ⊕ R[q + 1].

3.2. Cuspidal Eisenstein harmonic cochains. We set

E_{0}(𝔫, R) := E(𝔫, R) ∩ H_{0}(𝔫, R), E_{00}(𝔫, R) := E(𝔫, R) ∩ H_{00}(𝔫, R).

Let 𝔭 ⊲ A be a prime. Theorem 6.6 in [38] states that E_{0}(𝔭, R) ≅ R [\frac{|𝔭|−1}{q−1}],

if deg(𝔭) is odd, and E_{0}(𝔭, R) ≅ R [2\frac{|𝔭|−1}{q^{2}−1}], if deg(𝔭) is even. Note that Examples 2.6 and 3.5 imply that E_{0}(y, R) ≅ R[2], which is a special case of this theorem. The main result of this section is a similar description of E_{0}(𝔭𝔮, R) and E_{00}(𝔭𝔮, R) under certain assumptions. Note that as a consequence of the Ramanujan-Petersson conjecture over function fields E_{0}(𝔫, ℂ) = 0 for any 𝔫. Therefore, E_{0}(𝔫, R) can be non-trivial only if R has non-trivial additive torsion.

Definition 3.6. The Eisenstein ideal 𝔈(𝔫) of 𝕋(𝔫) is the ideal generated by the elements {T_{𝔭} − |𝔭| − 1 | 𝔭 is prime, 𝔭 ∤ 𝔫}. We say that a maximal ideal 𝔐 ⊲ 𝕋(𝔫) is Eisenstein if 𝔈(𝔫) ⊂ 𝔐.

Lemma 3.7. Let 𝔭⊲A be a prime, and S be a set of prime ideals of A of density one that does not contain 𝔭. A cochain f ∈ H_{00}(𝔭, R) is Eisenstein if and only if

f|T_{𝔮} = (|𝔮| + 1)f, ∀𝔮 ∈ S.

Proof. Let ℑ(𝔭) be the ideal of 𝕋(𝔭) generated by the elements T_{𝔮} − |𝔮| − 1, where 𝔮 ∈ S. It is enough to show that 𝔈(𝔭) ⊗ R = ℑ(𝔭) ⊗ R in 𝕋(𝔭) ⊗ R. The proof of the analogous statement over ℚ can be found in [4, Lem. 4]. We briefly sketch the argument over F. Let 𝔐 ⊲ 𝕋(𝔭) be a maximal ideal such that the characteristic ℓ of 𝕋(𝔭)/𝔐 is different from p. There is a unique semi-simple representation

ρ_{𝔐} : G_{F} → GL_{2}(𝕋(𝔭)/𝔐),

which is unramified away from 𝔭 and ∞, and such that for all primes 𝔮 ⊲ A, 𝔮 ≠ 𝔭, the following relations hold:

Trρ_{𝔐}(Frob_{𝔮}) = T_{𝔮} mod 𝔐, det ρ_{𝔐}(Frob_{𝔮}) = |𝔮| mod 𝔐.

The existence of such residual representations for G_{ℚ} is well-known. The corresponding statement over F can be proved along the same lines (cf. [40, Prop. 2.6]); this relies on Drinfeld’s fundamental results in [6]. If 𝔐 ⊃ ℑ(𝔭), then T_{𝔮} ≡ (1 + |𝔮|) (mod 𝔐) for all 𝔮 ∈ S. In view of the Chebotarev density and the Brauer-Nesbitt theorems, we conclude that ρ_{𝔐} is the direct sum 1⊕χ_{ℓ} of the trivial and cyclotomic characters. But this means that T_{𝔮} ≡ (1 + |𝔮|) (mod 𝔐) for all 𝔮 ≠ 𝔭, and therefore 𝔐 is Eisenstein. Now it suffices to show that ℑ(𝔭) ⊗ R ⊆ 𝔈(𝔭) ⊗ R is an equality in the completion (𝕋(𝔭) ⊗ R)_{𝔐} at any maximal Eisenstein 𝔐 of residue characteristic ≠ p. (Recall that p is invertible in R.) A consequence of the proof of Theorem 2.20 in [11], the argument in the proof of Theorem 2.26, and the fact that H_{0}(1, R) = 0 is that 𝕋(𝔭)^{0} ⊗ R = 𝕋(𝔭) ⊗ R. On the other hand, by Theorem 5.13 in [39], the ideal 𝔈(𝔭)_{𝔐} in 𝕋(𝔭)_{𝔐}^{0} is principal, generated by T_{𝔮} − |𝔮| − 1 for any “good” prime 𝔮. Since the density of these good primes is positive (see [38, p. 186]), there is a good prime in S. Therefore (ℑ(𝔭) ⊗ R)_{𝔐} contains a generator of (𝔈(𝔭) ⊗ R)_{𝔐} and must be equal to it. □

Fix two distinct primes 𝔭 and 𝔮. Set

ν(𝔭, 𝔮) = \begin{cases}1, & if deg(𝔭) or deg(𝔮) is even, \\ q + 1, & otherwise. \end{cases}

Let E_{(𝔭,𝔮)} ∈ E(𝔭𝔮, ℤ) be the Eisenstein series defined by

E_{(𝔭,𝔮)} ((\begin{matrix}π_{∞}^{k} & u \\ 0 & 1 \end{matrix})) = ν(𝔭, 𝔮)·q^{−k+1}·[\frac{(1 − |𝔭|)(1 − |𝔮|)}{1 − q^{2}} + ∑_{{0≠m∈A, \atop deg(m)≤k−2}} σ′_{𝔭𝔮}(m)η(mu)] ,

where

σ′_{𝔫}(m) := ∑_{monic m′∈A, (m′,𝔫)=1, and m′|m} |m′|.

It is clear that E_{(𝔭,𝔮)} = E_{(𝔮,𝔭)}, and comparing the Fourier expansions we get

(3.5)  \frac{ν(𝔭)}{ν(𝔭, 𝔮)} · E_{(𝔭,𝔮)} = (E_{𝔭} − E_{𝔭}|B_{𝔮}).

Documenta Mathematica 20 (2015) 551–629

Lemma 3.8.

(1) Viewing E_{𝔭} as a harmonic cochain in H(𝔭𝔮, ℤ), we get

E_{𝔭}|U_{𝔭} = E_{𝔭} = −E_{𝔭}|W_{𝔭} and E_{𝔭}|W_{𝔮} = E_{𝔭}|B_{𝔮}.

(2)

E_{(𝔭,𝔮)} = E_{(𝔭,𝔮)}|U_{𝔭} = E_{(𝔭,𝔮)}|U_{𝔮} = E_{(𝔭,𝔮)}|W_{𝔭𝔮} = −E_{(𝔭,𝔮)}|W_{𝔭} = −E_{(𝔭,𝔮)}|W_{𝔮}.

Proof. The proof is straightforward. □

Let E′(𝔭𝔮, R) be the R-submodule of E(𝔭𝔮, R) spanned by E_{𝔭}, E_{𝔮} and E_{(𝔭,𝔮)}, i.e., E′(𝔭𝔮, R) = RE_{𝔭} + RE_{𝔮} + RE_{(𝔭,𝔮)}. Denote

E′_{0}(𝔭𝔮, R) = E′(𝔭𝔮, R) ∩ H_{0}(𝔭𝔮, R) and E′_{00}(𝔭𝔮, R) = E′(𝔭𝔮, R) ∩ H_{00}(𝔭𝔮, R).

Theorem 3.9. The following holds:

(1) If ν(𝔭, 𝔮) is invertible in R, then E′(𝔭𝔮, R) is a free R-module of rank 3.

(2) If q − 1 and s_{𝔭,𝔮} = gcd(|𝔭| + 1, |𝔮| + 1) are both invertible in R, then

E_{00}(𝔭𝔮, R) = E′_{00}(𝔭𝔮, R).

Proof. By (3.5), E_{𝔭}|B_{𝔮} and E_{𝔮}|B_{𝔭} are both in E′(𝔭𝔮, R), and

E′(𝔭𝔮, R) = R(E_{𝔭}|B_{𝔮}) + R(E_{𝔮}|B_{𝔭}) + RE_{(𝔭,𝔮)}.

Suppose f = aE_{𝔭}|B_{𝔮} + bE_{𝔮}|B_{𝔭} + cE_{(𝔭,𝔮)} = 0. Then f^{∗}(1) = cE_{(𝔭,𝔮)}^{∗}(1) = c · ν(𝔭, 𝔮) = 0. Since ν(𝔭, 𝔮) is invertible in R under our assumption, we get c = 0. Without loss of generality we can assume that either deg(𝔭) is even, or deg(𝔭) and deg(𝔮) are both odd. Now E_{𝔭}^{∗}(1) = (E_{𝔭}|B_{𝔮})^{∗}(𝔮) = ν(𝔭) is invertible in R. Therefore from f^{∗}(𝔮) = a · ν(𝔭) = 0 we get a = 0. From the fact that E_{𝔭} is primitive, we have b = 0. The proof of Part (1) is complete. To prove Part (2), it suffices to show that E_{00}(𝔭𝔮, R) is contained in E′(𝔭𝔮, R). Given an Eisenstein harmonic cochain f ∈ E_{00}(𝔭𝔮, R), let

f_{1} := f − f^{∗}(1)ν(𝔭, 𝔮)^{−1}E_{(𝔭,𝔮)} ∈ E(𝔭𝔮, R).

Then for every ideal 𝔪 ⊲ A, f_{1}^{∗}(𝔪) = 0 unless 𝔭 or 𝔮 divides 𝔪. By the method of Section 2, (|𝔭|+1)f_{1} = ψ_{1}|B_{𝔭}+ψ_{2}|B_{𝔮}, where ψ_{1} ∈ H(𝔮, R) and ψ_{2} ∈ H(𝔭, R). Moreover, by the proof of Theorem 2.26, we can take

ψ_{1} = f_{1}|(U_{𝔭} + W_{𝔭}) and ψ_{2} = |𝔮|^{−1}[(|𝔭| + 1)f_{1}|U_{𝔮} − f_{1}|U_{𝔮}(U_{𝔭} + W_{𝔭})B_{𝔭}].

Since Lemma 3.8 (2) implies that E_{(𝔭,𝔮)}|(U_{𝔭} + W_{𝔭}) = 0, we get ψ_{1} = f|(U_{𝔭} + W_{𝔭}) ∈ H_{00}(𝔮, R) and

ψ_{2} = |𝔮|^{−1}[(|𝔭| + 1)f|U_{𝔮} − f|U_{𝔮}(U_{𝔭} + W_{𝔭})B_{𝔭}]
−|𝔮|^{−1}(|𝔭| + 1)f^{∗}(1)ν(𝔭, 𝔮)^{−1}E_{(𝔭,𝔮)} ∈ H(𝔮, R).

The constant term ψ_{2}^{0}(1) is equal to

−|𝔮|^{−1}(|𝔭| + 1)f^{∗}(1) · \frac{(1 − |𝔭|)(1 − |𝔮|)}{q(1 − q^{2})} = −|𝔮|^{−1}(|𝔭| + 1)f^{∗}(1) · \frac{1 − |𝔮|}{ν(𝔭)}E_{𝔭}^{0}(1)

Documenta Mathematica 20 (2015) 551–629

and (ψ_{2}|W_{𝔭})^{0}(1) = −ψ_{2}^{0}(1). Let

ψ′_{2} := ψ_{2} + |𝔮|^{−1}(|𝔭| + 1)f^{∗}(1) · \frac{1 − |𝔮|}{ν(𝔭)}E_{𝔭} ∈ H(𝔭, R).

Since q − 1 is invertible in R by assumption, Lemma 2.7 and 2.19 show that

ψ′_{2} ∈ H_{0}(𝔭, R) = H_{00}(𝔭, R).

Note that for every prime 𝔭′ ⊲ A different from 𝔭 and 𝔮, we have

ψ_{1}|T_{𝔭′} = (|𝔭′| + 1)ψ_{1} and ψ′_{2}|T_{𝔭′} = (|𝔭′| + 1)ψ′_{2}.

Lemma 3.7 implies that ψ_{1} ∈ E_{00}(𝔮, R) and ψ′_{2} ∈ E_{00}(𝔭, R). By Theorem 6.6 in [38], we can find a_{1}, a_{2} ∈ R such that ψ_{1} = a_{1}E_{𝔮} and ψ′_{2} = a_{2}E_{𝔭} (as 2 is invertible in R by our assumption). We conclude that

(|𝔭| + 1)f = ψ_{1}|B_{𝔭} + ψ_{2}|B_{𝔮} + f^{∗}(1)ν(𝔭, 𝔮)^{−1}E_{(𝔭,𝔮)}

= a_{2}E_{𝔮}|B_{𝔭} \begin{matrix}+ (a_{2} − |𝔮|^{−1}(|𝔭| + 1)f^{∗}(1) · \frac{1 − |𝔮|}{ν(𝔭)})E_{𝔭}|B_{𝔮} \\ + f^{∗}(1)ν(𝔭, 𝔮)^{−1}E_{(𝔭,𝔮)} \end{matrix}

is in E′(𝔭𝔮, R). Similarly, we also get (|𝔮| + 1)f ∈ E′(𝔭𝔮, R) by interchanging 𝔭 and 𝔮. Since s_{𝔭,𝔮} is invertible in R, f must be in E′(𝔭𝔮, R), which completes the proof of Part (2). □

Lemma 3.10. The following holds:

(1) Suppose ν(𝔭, 𝔮) is invertible in R. The R-module E′_{0}(𝔭𝔮, R) is torsion and isomorphic to the submodule of R^{3} consisting of elements (a, b, c) with

\frac{(|𝔭| − 1)(|𝔮| + 1)}{q^{2} − 1}ν(𝔭)a = 0,

\frac{(|𝔭| + 1)(|𝔮| − 1)}{q^{2} − 1}ν(𝔮)b = 0,

\frac{1 − |𝔭|}{1 − q^{2}}ν(𝔭)a + \frac{1 − |𝔮|}{1 − q^{2}}ν(𝔮)b + \frac{(1 − |𝔭|)(1 − |𝔮|)}{1 − q^{2}}ν(𝔭, 𝔮)c = 0.

(2) Suppose further that 2 is also invertible in R. Then E′_{0}(𝔭𝔮, R) is isomorphic to

R [\frac{(|𝔭| − 1)(|𝔮| + 1)}{q^{2} − 1}ν(𝔭)]⊕R [\frac{(|𝔭| + 1)(|𝔮| − 1)}{q^{2} − 1}ν(𝔮)]⊕R [\frac{(|𝔭| − 1)(|𝔮| − 1)}{q^{2} − 1}] .

Proof. Let f = aE_{𝔭} + bE_{𝔮} + cE_{(𝔭,𝔮)} with a, b, c ∈ R. By Lemma 2.19, f ∈ E′_{0}(𝔭𝔮, R) if and only if

f^{0}(1) = (f|W_{𝔭})^{0}(1) = (f|W_{𝔮})^{0}(1) = (f|W_{𝔭𝔮})^{0}(1) = 0.

This gives us the following equations:

(3.6)  \frac{1 − |𝔭|}{1 − q^{2}}ν(𝔭)a + \frac{1 − |𝔮|}{1 − q^{2}}ν(𝔮)b + \frac{(1 − |𝔭|)(1 − |𝔮|)}{1 − q^{2}}ν(𝔭, 𝔮)c = 0,
(3.7) −\frac{1 − |𝔭|}{1 − q^{2}}ν(𝔭)a + |𝔭|\frac{1 − |𝔮|}{1 − q^{2}}ν(𝔮)b − \frac{(1 − |𝔭|)(1 − |𝔮|)}{1 − q^{2}}ν(𝔭, 𝔮)c = 0,
(3.8) |𝔮|\frac{1 − |𝔭|}{1 − q^{2}}ν(𝔭)a − \frac{1 − |𝔮|}{1 − q^{2}}ν(𝔮)b − \frac{(1 − |𝔭|)(1 − |𝔮|)}{1 − q^{2}}ν(𝔭, 𝔮)c = 0,
−|𝔮|\frac{1 − |𝔭|}{1 − q^{2}}ν(𝔭)a − |𝔭|\frac{1 − |𝔮|}{1 − q^{2}}ν(𝔮)b + \frac{(1 − |𝔭|)(1 − |𝔮|)}{1 − q^{2}}ν(𝔭, 𝔮)c = 0. (3.9)

We remark that

Equation (3.9) = −( Equation (3.6) + (3.7) + (3.8)).

Considering the equation (3.6)+(3.7), (3.6)+(3.8), and (3.6)+(3.9), we get

\frac{(|𝔭| + 1)(|𝔮| − 1)}{q^{2} − 1}ν(𝔮)b = 0, (3.10)

\frac{(|𝔭| − 1)(|𝔮| + 1)}{q^{2} − 1}ν(𝔭)a = 0, (3.11)

\frac{(|𝔭| − 1)(|𝔮| − 1)}{q^{2} − 1}(ν(𝔭)a + ν(𝔮)b + 2ν(𝔭, 𝔮)c) = 0. (3.12)

Since the conditions of (a, b, c) described by Equation (3.6)∼(3.9) is equivalent to those described by Equation (3.6), (3.10), and (3.11) combined, the proof of Part (1) is complete. To prove Part (2), note that 2ν(𝔭, 𝔮) is invertible in R by assumption. Therefore the conditions of (a, b, c) described by Equation (3.6)∼(3.9) is equivalent to those described by Equation (3.10)∼(3.12). Let E′ := ν(𝔭)E_{𝔭} + ν(𝔮)E_{𝔮} + 2ν(𝔭, 𝔮)E_{(𝔭,𝔮)}. By Theorem 3.9 (1), we also have E′(𝔭𝔮, R) = RE_{𝔭}⊕RE_{𝔮}⊕RE′. Therefore Equation (3.10)∼(3.12) assures the result. □

In fact, when q − 1 and s_{𝔭,𝔮} are invertible in the coefficient ring R, one can show that E_{00}(𝔭𝔮, R) = E′_{0}(𝔭𝔮, R); see Remark 7.4.

3.3. Special case. In this subsection we give a concrete description of E_{0}(xy, R) and E_{00}(xy, R) for an arbitrary coefficient ring R. Recall that E_{0} is the function on E(𝒯 ) satisfying

E_{0} ((\begin{matrix}π_{∞}^{k} & u \\ 0 & 1 \end{matrix})) = −E_{0} ((\begin{matrix}π_{∞}^{k} & u \\ 0 & 1 \end{matrix}) (\begin{matrix}0 & 1 \\ π_{∞} & 0 \end{matrix})) = q^{−k}.

Lemma 3.11. Given a ∈ R[q + 1], we have aE_{0} = −aE_{𝔫} ∈ H(1, R) if deg(𝔫) is odd. Moreover, aE_{0}|B_{𝔪} = (−1)^{deg(𝔪)}aE_{0}.

Proof. The proof is straightforward. □

According to Examples 3.4 and 3.5 we have:

Lemma 3.12. H(x, R) = RE_{x} and H(y, R) = RE_{y} ⊕ R[q + 1]E_{0}.

Note that ν(x, y) = ν(y) = 1 and ν(x) = q + 1. Given f ∈ E(xy, R), let f′ := f − f^{∗}(1)E_{(x,y)}. Then f^{′∗}(𝔪) = 0 unless x or y divides 𝔪. By Theorem 2.26 and Lemma 3.12, there exists a, b ∈ R and b′ ∈ R[q + 1] such that

2f′ = (bE_{y} + b′E_{0})|B_{x} + aE_{x}|B_{y}.

When q is even, 2 is invertible in R and we have f ∈ E′(xy, R). Suppose q is odd. Note that b′E_{0}|B_{x} = b′E_{x} by Lemma 3.11. Hence

2f = (a + b′)E_{x}|B_{y} + bE_{y}|B_{x} + 2f^{∗}(1)E_{(x,y)} ∈ E′(xy, R).

In fact, there exists a′′, b′′ ∈ R such that a + b′ = 2a′′ and b = 2b′′. Indeed, by Lemma 2.24 we have

2f^{∗}(x) = b+2f^{∗}(1) and 2f^{0}(π_{∞}) = 2q^{−1}f^{0}(1) = |y|(a+b′)+|x|b+2f^{∗}(1)(q−1).

Set f′′ := f − (a′′E_{x}|B_{y} + b′′E_{y}|B_{x} + f^{∗}(1)E_{(x,y)}) ∈ E(xy, R). Then the above discussion shows that 2f′′ = 0. Take f′′′ := f′′ − (f′′)^{0}(1)E_{0}. Then

f′′′ ∈ E(xy, R)[2] and (f′′′)^{∗}(1) = (f′′′)^{0}(1) = 0.

The following lemma shows that f′′′ ∈ E′(xy, R), which also implies that f ∈ E′(xy, R).

Lemma 3.13. Suppose q is odd. Given φ ∈ E(xy, R)[2] with φ^{∗}(1) = φ^{0}(1) = 0, there exists α ∈ R[2] such that

φ = α(E_{x} + E_{y}|B_{x}).

Proof. We use the notation in the proof of Proposition 2.21. Given φ ∈ E(xy, R)[2] with φ^{0}(1) = 0, we have that for each prime 𝔭 with 𝔭 ∤ xy,

φ|T_{𝔭} = (|𝔭| + 1)φ = 0.

Therefore for every prime 𝔭 with 𝔭 ∤ xy and a non-zero ideal 𝔪 ⊲ A,

|𝔭|φ^{∗}(𝔭𝔪) + φ^{∗}(𝔪/𝔭) = (φ|T_{𝔭})^{∗}(𝔪) = 0. (3.13)

By Equation (3.13) and the Fourier expansion of φ, we get

φ ((\begin{matrix}ϖ_{x}^{k} & 0 \\ 0 & 1 \end{matrix})) = 0, ∀k ∈ ℤ,

φ(a_{1}) = φ(a_{2}) = 0,

φ(b_{u}) = φ(a_{4}) = φ^{∗}(x), ∀u ∈ 𝔽_{q}^{×},

φ(c_{3}) = φ^{∗}(y), φ(a_{3}) = φ^{∗}(x) + φ^{∗}(x^{2}), φ(a_{6}) = φ^{∗}(x^{2}).

The harmonicity of φ gives us that

0 = φ(a_{4}) + ∑_{u∈𝔽_{q}^{×}} φ(b_{u}) − φ(a_{3}) = φ^{∗}(x^{2}),

0 = φ(a_{6}) + φ(c_{3}) + (1 − q)φ(a_{2}) = φ^{∗}(x^{2}) + φ^{∗}(y).
0 = φ(a_{6}) + (q − 1)φ(a_{3}) − φ(c_{4}) = φ^{∗}(x^{2}) + φ(c_{4}).

Hence φ(e) = 0 for e = c_{1}, c_{2}, c_{3}, c_{4}, a_{1}, a_{2}, a_{5}, a_{6}, and

φ(a_{3}) = φ(a_{4}) = φ(b_{u}) = φ^{∗}(x) ∈ R[2], for u ∈ 𝔽_{q}^{×}.

On the other hand, for α ∈ R[2],

α(E_{x} + E_{y}|B_{x})(e) = 0 for e = c_{1}, c_{2}, c_{3}, c_{4}, a_{1}, a_{2}, a_{5}, a_{6}, and

α(E_{x}+E_{y}|B_{x})(a_{3}) = α(E_{x}+E_{y}|B_{x})(a_{4}) = α(E_{x}+E_{y}|B_{x})(b_{u}) = α, for u ∈ 𝔽_{q}^{×}.

Therefore φ = φ(a_{3}) · (E_{x} + E_{y}|B_{x}) and the proof is complete. □

From the above discussion, we conclude that

Corollary 3.14. E(xy, R) = E′(xy, R) for every coefficient ring R. In other words, every Eisenstein harmonic cochain of level xy can be generated by Eisenstein series.

By Lemma 3.10 and Corollary 3.14, we immediately get

Corollary 3.15. The space E_{0}(xy, R) is isomorphic to the torsion R-module

{(a, b, c) ∈ R^{3} | (q^{2} + 1)a = (q + 1)b = a + b + (1 − q)c = 0}.

In particular,

E_{0}(xy, R) ≅ R[\frac{(q − 1)}{2}(q^{2} + 1)(q + 1)] ⊕ R[2].

From the graph in Figure 4, an alternating R-valued function f on E(Γ_{0}(xy)\𝒯 ) is in H_{0}(xy, R) if and only if f vanishes on the cusps c_{1}, c_{2}, c_{3}, c_{4} and

(q − 1)f(a_{1}) + f(a_{5}) = 0, (q − 1)f(a_{2}) − f(a_{6}) = 0, (q − 1)f(a_{3}) + f(a_{6}) = 0, (q − 1)f(a_{4}) − f(a_{5}) = 0,

f(a_{2}) + ∑_{u∈𝔽_{q}^{×}} f(b_{u}) = f(a_{1}), f(a_{3}) − ∑_{u∈𝔽_{q}^{×}} f(b_{u}) = f(a_{4}).

Moreover, f is in H_{00}(xy, R) if and only if f satisfies an extra equation:

f(a_{1}) + f(a_{4}) = 0.

In particular, every harmonic cochain f ∈ H_{00}(xy, R) is determined uniquely by the values

f(a_{1}), f(b_{u}) for u ∈ 𝔽_{q}^{×}.

Let f = aE_{x} + bE_{y} + cE_{(x,y)} ∈ E_{0}(xy, R). By Corollary 3.15 we have

(q^{2} + 1)a = (q + 1)b = a + b + (1 − q)c = 0.

It is observed that

\begin{matrix}E_{x}(a_{1}) = −1, & E_{x}(a_{4}) = −q, \\ E_{y}(a_{1}) = 0, & E_{y}(a_{4}) = −1, \\ E_{(x,y)}(a_{1}) = −1, & E_{(x,y)}(a_{4}) = −1. \end{matrix}

We then get f(a_{1}) + f(a_{4}) = −((q + 1)a + b + 2c).

Hence f ∈ E_{00}(xy, R) if and only if

c ∈ R[(q^{2} + 1)(q + 1)], a = −q^{−1}(q + 1)c, b = q^{−1}(q^{2} + 1)c.

We conclude that

Proposition 3.16. The module E_{00}(xy, R) is isomorphic to R[(q^{2} + 1)(q + 1)]. More precisely, every harmonic cochain in E_{00}(xy, R) must be of the form

c · ( − (q + 1)E_{x} + (q^{2} + 1)E_{y} + qE_{(x,y)})

where c ∈ R[(q^{2} + 1)(q + 1)].

Corollary 3.17. For every natural number n relatively prime to p the module E_{00}(xy, ℤ/nℤ) is isomorphic to ℤ/nℤ[(q^{2} + 1)(q + 1)].

Proof. The difference between this claim and Proposition 3.16 is that ℤ/nℤ is not a coefficient ring in general. Still, one can deduce this from Proposition 3.16 by arguing as in the proof of Corollary 6.9 in [38]. First, one easily reduces to the case when n is a power of some prime ℓ ≠ p, and applies Proposition 3.16 with R = ℤ_{ℓ}[ζ_{p}]/nℤ_{ℓ}[ζ_{p}], where ζ_{p} is the primitive pth root of unity. The claim follows by observing that R is a free ℤ/nℤ-module. □

Corollary 3.18. 𝕋(xy)/𝔈(xy) ≅ ℤ/(q^{2} + 1)(q + 1)ℤ.

Proof. Let (𝕋(𝔫)^{0})^{new} be the quotient of 𝕋(𝔫)^{0} with which it acts on H_{0}(𝔫, ℚ)^{new}; cf. Definition 2.13. By Lemma 4.2 and Theorem 4.5 in [39], for any square-free 𝔫⊲A the quotient ring (𝕋(𝔫)^{0})^{new}/𝔈(𝔫) is a finite cyclic group of order coprime to p; here with abuse of notation 𝔈(𝔫) denotes the ideal generated by the images of T_{𝔭} − |𝔭| − 1 in (𝕋(𝔫)^{0})^{new}. Since H_{0}(xy, ℚ) = H_{0}(xy, ℚ)^{new}, we have 𝕋(xy)^{0} = (𝕋(xy)^{0})^{new}. On the other hand, by Proposition 2.21, 𝕋(xy)^{0} = 𝕋(xy). Hence 𝕋(xy)/𝔈(xy) ≅ ℤ/nℤ for some n coprime to p. The perfectness of the pairing (2.5) implies

Hom_{ℤ/nℤ}(𝕋(xy) ⊗_{ℤ} ℤ/nℤ, ℤ/nℤ) ≅ H_{00}(xy, ℤ/nℤ).

Hence

E_{00}(xy, ℤ/nℤ) ≅ Hom_{ℤ/nℤ}(𝕋(xy) ⊗_{ℤ} ℤ/nℤ, ℤ/nℤ)[𝔈(xy)]
≅ Hom_{ℤ/nℤ}((𝕋(xy)/𝔈(xy)) ⊗_{ℤ} ℤ/nℤ, ℤ/nℤ)
≅ Hom_{ℤ/nℤ}(ℤ/nℤ ⊗_{ℤ} ℤ/nℤ, ℤ/nℤ) ≅ ℤ/nℤ.

Applying Corollary 3.17, we conclude that n must divide (q + 1)(q^{2} + 1). Later in this paper we will prove that the component group Φ_{∞} ≅ ℤ/(q^{2} + 1)(q + 1)ℤ of J_{0}(xy) is annihilated by 𝔈(xy) (see Lemma 8.16). This implies that n is divisible by (q^{2} + 1)(q + 1). Therefore, n = (q^{2} + 1)(q + 1). □

Remark 3.19. In [44], we extended our calculation of 𝕋(𝔫)/𝔈(𝔫) to arbitrary 𝔫 of degree 3. Up to an affine transformation T ↦ aT + b with a ∈ 𝔽_{q}^{×} and b ∈ 𝔽_{q}, there are 5 different cases, namely

(1) If 𝔫 = T^{3}, then 𝕋(𝔫)/𝔈(𝔫) ≅ ℤ/q^{2}ℤ;

(2) If 𝔫 = T^{2}(T − 1), then 𝕋(𝔫)/𝔈(𝔫) ≅ ℤ/q(q^{2} − 1)ℤ;

(3) If 𝔫 is irreducible, then 𝕋(𝔫)/𝔈(𝔫) ≅ ℤ/(q^{2} + q + 1)ℤ;
(4) If 𝔫 = xy, then 𝕋(𝔫)/𝔈(𝔫) ≅ ℤ/(q^{2} + 1)(q + 1)ℤ;

(5) If 𝔫 = T (T − 1)(T − c), where c ∈ 𝔽_{q}, c ≠ 0, 1 (here we must have q > 2), then

𝕋(𝔫)/𝔈(𝔫) ≅ ℤ/(q + 1)ℤ × ℤ/(q + 1)ℤ × ℤ/(q − 1)^{2}(q + 1)ℤ.

4. Drinfeld modules and modular curves

In this section we collect some facts about Drinfeld modules and their moduli schemes that will be used later in the paper. Let S be an A-scheme and L a line bundle over S. Let L{τ} be the noncommutative ring ⊕_{i≥0}L^{⊗(1−q^{i})}(S)τ^{i}, where τ stands for the qth power Frobenius mapping. The multiplication in this ring is given by α_{i}τ^{i} · α_{j}τ^{j} = (α_{i} ⊗ α_{j}^{⊗q^{j}})τ^{i+j}. A Drinfeld A-module of rank r (in standard form) over S is given by a line bundle L over S together with a ring homomorphism ϕ^{L} : A → End_{𝔽_{q}}(L) = L{τ}, a ↦ ϕ_{a}^{L}, such that ϕ_{a}^{L} = ∑_{i=0}^{m(a)} α_{i}(a)τ^{i}, where m(a) = −r · ord_{∞}(a), α_{m(a)}(a) is a nowhere vanishing section of L^{⊗(1−m(a))}, and α_{0} coincides with the map ∂ : A → H^{0}(S, O_{S}) giving the structure of an A-scheme to S; cf. [6, p. 575]. The kernel of ∂ is called the A-characteristic of S. A Drinfeld A-module over S is clearly an A-module scheme over S, and a homomorphism of Drinfeld modules is a homomorphism of these A-module schemes. A homomorphism of Drinfeld modules over a connected scheme is either the zero homomorphism, or it has finite kernel, in which case it is usually called an isogeny. When S is the spectrum of a field K, we will omit mention of L and write ϕ : A → K{τ}. Let 𝔫 ⊲ A be a non-zero ideal. A cyclic subgroup of order 𝔫 of ϕ^{L} is an Asubmodule scheme C_{𝔫} of L which is finite and flat over S, and such that there is a homomorphism of A-modules ι : A/𝔫 → L(S) giving an equality of relative

effective Cartier divisors ∑_{a∈A/𝔫} ι(a) = C_{𝔫}. Denote ϕ^{L}[𝔫] = ker(L ⟶^{ϕ_{n}^{L}} L), where n is a generator of the ideal 𝔫. It is clear that the A-submodule scheme ϕ^{L}[𝔫] of L does not depend on the choice of n, and C_{𝔫} ⊂ ϕ^{L}[𝔫]. To each C_{𝔫} ⊂ ϕ one can associate a unique, up to isomorphism, Drinfeld module ϕ′ := ϕ/C_{𝔫} such that there is an isogeny ϕ → ϕ′ whose kernel is C_{𝔫}. Now assume S = Spec(K), where K is a field. Explicitly, a homomorphism of Drinfeld modules u : ϕ → ψ is u ∈ K{τ} such that ϕ_{a}u = uψ_{a} for all a ∈ A, and u is an isomorphism if u ∈ K^{×}. Let End(ϕ) denote the centralizer of ϕ(A) in K̄{τ}, i.e., the ring of all homomorphisms ϕ → ϕ over K̄. The automorphism group Aut(ϕ) is the group of units End(ϕ)^{×}. It is known that End(ϕ) is a free A-module of rank ≤ r^{2}; cf. [6]. From now on we also assume that r = 2. Note that ϕ is uniquely determined by the image of T:

ϕ_{T} = ∂(T) + gτ + Δτ^{2},

where g ∈ K and Δ ∈ K^{×}. The j-invariant of ϕ is j(ϕ) = g^{q+1}/Δ. It is easy to check that if K is algebraically closed, then ϕ ≅ ψ if and only if j(ϕ) = j(ψ).

It is also easy to check that

(4.1)  Aut(ϕ) = \begin{cases}𝔽_{q}^{×} & if j(ϕ) ≠ 0; \\ 𝔽_{q^{2}}^{×} & if j(ϕ) = 0. \end{cases}

If 𝔫 is coprime to the A-characteristic of K, then ϕ[𝔫](K̄) ≅ (A/𝔫)^{2}. On the other hand, if 𝔭 = ker(∂) ≠ 0, then ϕ[𝔭](K̄) ≅ (A/𝔭) or 0; when ϕ[𝔭](K̄) = 0, ϕ is called supersingular. The following is Theorem 5.9 in [8]:

Theorem 4.1. Let 𝔭⊲A be a prime ideal. The number of isomorphism classes of supersingular rank-2 Drinfeld A-modules over \overline{𝔽}_{𝔭} is

\begin{cases}\frac{|𝔭|−1}{q^{2}−1} & if deg(𝔭) is even; \\ \frac{|𝔭|−q}{q^{2}−1} + 1 & if deg(𝔭) is odd. \end{cases}

The Drinfeld module with j(ϕ) = 0 is supersingular if and only if deg(𝔭) is odd.

The functor from the category of A-schemes to the category of sets, which associates to an A-scheme S the set of isomorphism classes of pairs (ϕ^{L}, C_{𝔫}), where ϕ^{L} is a Drinfeld module of rank 2 and C_{𝔫} is a cyclic subgroup of order 𝔫 has a coarse moduli scheme Y_{0}(𝔫). The scheme Y_{0}(𝔫) is affine, finite type, of relative dimension 1 over Spec(A), and smooth over Spec(A[𝔫^{−1}]). This is well-known and can be deduced from the results in [6]. The rigid-analytic uniformization of Y_{0}(𝔫) over F_{∞} is given by (3.1). The scheme Y_{0}(𝔫) has a canonical compactification over Spec(A):

Theorem 4.2. There is a proper normal geometrically irreducible scheme X_{0}(𝔫) of pure relative dimension 1 over Spec(A) which contains Y_{0}(𝔫) as an open dense subscheme. The complement X_{0}(𝔫) − Y_{0}(𝔫) is a disjoint union of irreducible schemes. Finally, X_{0}(𝔫) is smooth over Spec(A[𝔫^{−1}]).

Proof. See [6, §9] and [29, Prop. V.3.5]. □

Denote the Jacobian variety of X_{0}(𝔫)_{F} by J_{0}(𝔫). Let 𝔭⊲A be prime. There are two natural degeneracy morphisms α, β : Y_{0}(𝔫𝔭) → Y_{0}(𝔫) with moduli-theoretic interpretation:

α : (ϕ, C_{𝔫𝔭}) ↦ (ϕ, C_{𝔫}), β : (ϕ, C_{𝔫𝔭}) ↦ (ϕ/C_{𝔭}, C_{𝔫𝔭}/C_{𝔭}),

where C_{𝔫} and C_{𝔭} are the subgroups of C_{𝔫𝔭} of order 𝔫 and 𝔭, respectively. These morphisms are proper, and hence uniquely extend to morphisms α, β : X_{0}(𝔫𝔭) → X_{0}(𝔫). By Picard functoriality, α and β induce two homomorphisms α_{∗}, β_{∗} : J_{0}(𝔫) → J_{0}(𝔭𝔫). The Hecke endomorphism of J_{0}(𝔫) is T_{𝔭} := α^{∗} ∘ β_{∗} , where α^{∗} : J_{0}(𝔭𝔫) → J_{0}(𝔫) is the dual of α_{∗}. The ℤ-subalgebra of End(J_{0}(𝔫)) generated by all Hecke endomorphisms is canonically isomorphic to 𝕋(𝔫). This is a consequence of Drinfeld’s reciprocity law [6, Thm. 2]. The Jacobian J_{0}(𝔫) has a rigid-analytic uniformization over F_{∞} as a quotient of a multiplicative torus by a discrete lattice. To simplify the notation denote Γ := Γ_{0}(𝔫) and let \overline{Γ} be the maximal torsion-free abelian quotient of Γ. In [18] and [13], Gekeler and Reversat associate a meromorphic theta function θ(ω, η, ·) on Ω with each pair ω, η ∈ \overline{Ω} := Ω ∪ ℙ^{1}(F). The theta function θ(ω, η, ·) satisfies a functional equation

θ(ω, η, γz) = c(ω, η, γ)θ(ω, η, z), ∀γ ∈ Γ,

where c(ω, η, ·) : Γ → ℂ_{∞}^{×} is a homomorphism that factors through \overline{Γ}. The divisor of θ(ω, η, ·) is Γ-invariant and, as a divisor on X_{0}(𝔫)(ℂ_{∞}), equals [ω]−[η], where [ω] is the class of ω ∈ \overline{Ω} in Γ \ \overline{Ω} = X_{0}(𝔫)(ℂ_{∞}). For a fixed α ∈ Γ, the function u_{α}(z) = θ(ω, αω, z) is holomorphic and invertible on Ω. Moreover, u_{α} is independent of the choice of ω ∈ \overline{Ω}, and depends only on the class ᾱ of α in \overline{Γ}. Let c_{α}(·) = c(ω, αω, ·) be the multiplier of u_{α}. It induces a pairing

\overline{Γ} × \overline{Γ} → F_{∞}^{×}
(α, β) ↦ c_{α}(β)

which is bilinear, symmetric, and

⟨·, ·⟩ : \overline{Γ} × \overline{Γ} → ℤ (4.2)

⟨α, β⟩ = ord_{∞}(c_{α}(β))

is positive definite. One of the main result of [18] is that there is an exact sequence

(4.3)  0 → \overline{Γ} ⟶^{α↦c_{α}(·)} Hom(\overline{Γ}, ℂ_{∞}^{×}) → J_{0}(𝔫)(ℂ_{∞}) → 0.

One can define Hecke operators T_{𝔭} as endomorphisms of \overline{Γ} in purely grouptheoretical terms as some sort of Verlagerung (see [18, (9.3)]). These operators then also act on the torus Hom(\overline{Γ}, ℂ_{∞}^{×}) through their action on the first argument \overline{Γ}. By [18, (3.3.3)] and [17], there is a canonical isomorphism

(4.4)  j : \overline{Γ} ⟶^{∼} H_{0}(𝔫, ℤ)

which is compatible with the action of Hecke operators. Through this construction, the Hecke algebra 𝕋(𝔫) in Definition 2.11 acts faithfully on \overline{Γ} and Hom(\overline{Γ}, ℂ_{∞}^{×}). The sequence (4.3) is compatible with the action of 𝕋(𝔫)^{0} on its three terms; see [18, (9.4)]. Assume 𝔫 is square-free. The matrix (2.3) representing the Atkin-Lehner involution W_{𝔪} for 𝔪|𝔫 is in the normalizer of Γ in GL_{2}(F_{∞}), and the induced involution of X_{0}(𝔫)_{F_{∞}} does not depend on the choice of this matrix. In terms on the moduli problem, the involution W_{𝔪} on X_{0}(𝔫) is given by

W_{𝔪} : (ϕ, C_{𝔫}) ↦ (ϕ/C_{𝔪}, (ϕ[𝔪] + C_{𝔫/𝔪})/C_{𝔪}),

where C_{𝔪} and C_{𝔫/𝔪} are the subgroups of C_{𝔫} of order 𝔪 and 𝔫/𝔪, respectively.

5. Component groups

Let 𝔫 ⊲ A be a non-zero ideal. Let J := J_{0}(𝔫) and J denote the Néron model of J over ℙ_{𝔽_{q}}^{1}. Let J^{0} denote the relative connected component of the identity of J, that is, the largest open subscheme of J containing the identity section which has connected fibres. The group of connected components (or component group) of J at a place v of F is Φ_{v} := J_{𝔽_{v}}/J_{𝔽_{v}}^{0}. This is a finite abelian group equipped with an action of the absolute Galois group G_{𝔽_{v}}. The homomorphism ℘_{v} : J(F_{v}^{nr}) → Φ_{v} obtained from the composition

℘_{v} : J(F_{v}^{nr}) = J (O_{v}^{nr}) → J_{𝔽_{v}}(\overline{𝔽}_{v}) → Φ_{v}

will be called the canonical specialization map. Assume 𝔭⊲A is a prime not dividing 𝔪. Then the curve X_{0}(𝔪)_{𝔽_{𝔭}} is smooth. We call a point P ∈ X_{0}(𝔪)(\overline{𝔽}_{𝔭}) supersingular if it corresponds to the isomorphism class of a pair (ϕ, C_{𝔪}) with ϕ supersingular over \overline{𝔽}_{𝔭}. For a Drinfeld A-module ϕ over \overline{𝔽}_{𝔭} given by ϕ_{T} = ∂(T)+gτ +Δτ^{2}, let ϕ^{(𝔭)} : A → \overline{𝔽}_{𝔭}\{τ\} be the Drinfeld module given by ϕ_{T}^{(𝔭)} = ∂(T) + g^{|𝔭|}τ + Δ^{|𝔭|}τ^{2}. Since ∂(A) ⊆ 𝔽_{𝔭}, we see that τ^{|𝔭|}ϕ_{a} = ϕ_{a}^{(𝔭)}τ^{|𝔭|} for all a ∈ A, so τ^{|𝔭|} is an isogeny ϕ → ϕ^{(𝔭)}. Denote the image of C_{𝔪} in ϕ^{(𝔭)} under τ^{|𝔭|} by C_{𝔪}^{(𝔭)}. The map from X_{0}(𝔪)(\overline{𝔽}_{𝔭}) to itself given by (ϕ, C_{𝔪}) ↦ (ϕ^{(𝔭)}, C_{𝔪}^{(𝔭)}) restricts to an involution on the finite set of supersingular points; cf. [8, Thm. 5.3].

Theorem 5.1. Assume 𝔫 = 𝔭𝔪, with 𝔭 prime not dividing 𝔪. The curve X_{0}(𝔫)_{F_{𝔭}} is smooth and extends to a proper flat scheme X_{0}(𝔫)_{O_{𝔭}} over O_{𝔭} such that the special fibre X_{0}(𝔫)_{𝔽_{𝔭}} is geometrically reduced and consists of two irreducible components, both isomorphic to X_{0}(𝔪)_{𝔽_{𝔭}}, intersecting transversally at the supersingular points. More precisely, a supersingular point (ϕ, C_{𝔪}) on the first copy of X_{0}(𝔪)_{𝔽_{𝔭}} is glued to (ϕ^{(𝔭)}, C_{𝔪}^{(𝔭)}) on the second copy. The curve X_{0}(𝔫)_{𝔽_{𝔭}} is smooth outside of the locus of supersingular points. Denote by Aut(ϕ, C_{𝔪}) the subgroup of automorphisms of ϕ which map C_{𝔪} to itself. For a supersingular point P ∈ X_{0}(𝔪)_{𝔽_{𝔭}} corresponding to (ϕ, C_{𝔪}), let m(P) := \frac{1}{q−1}#Aut(ϕ, C_{𝔪}). Then, locally at P for the étale topology, X_{0}(𝔫)_{O_{𝔭}} is given by the equation XY = 𝔭^{m(P)}.

Proof. This is proven in [10, §5] for 𝔫 = 𝔭, but the proof easily extends to this more general case. □

We will compute the component group Φ_{𝔭} using a classical theorem of Raynaud, but first we need to determine the number of singular points on X_{0}(𝔫)_{𝔽_{𝔭}}, and the integers m(P) ≥ 1 defined in Theorem 5.1. We call m(P) the thickness of P. Let 𝔪 = ∏_{1≤i≤s} 𝔭_{i}^{r_{i}} be the prime decomposition of 𝔪. Define

R(𝔪) = \begin{cases}1 & if deg(𝔭_{i}) is even for all 1 ≤ i ≤ s \\ 0 & otherwise \end{cases}

L(𝔪) = #ℙ^{1}(A/𝔪) = ∏_{1≤i≤s} |𝔭_{i}|^{r_{i}−1}(|𝔭_{i}| + 1).

If 𝔪 = A, we put s = 0 and L(𝔪) = R(𝔪) = 1.

Lemma 5.2. The number of supersingular points on X_{0}(𝔪)_{𝔽_{𝔭}} is

S(𝔭, 𝔪) = \begin{cases}\frac{|𝔭|−1}{q^{2}−1}L(𝔪) & if deg(𝔭) is even; \\ \frac{|𝔭|−q}{q^{2}−1} L(𝔪) + \frac{L(𝔪)+q2^{s}R(𝔪)}{q+1} & if deg(𝔭) is odd. \end{cases}

The thickness of a supersingular point on X_{0}(𝔪)_{𝔽_{𝔭}} is either 1 or q + 1. Supersingular points with thickness q + 1 can exist only if deg(𝔭) is odd and their number is 2^{s}R(𝔪).

Proof. Let ϕ be a fixed Drinfeld module of rank 2 over \overline{𝔽}_{𝔭}. Since 𝔪 is assumed to be coprime to 𝔭, the number of distinct cyclic subgroups C_{𝔪} ⊂ ϕ[𝔪] is L(𝔪). If Aut(ϕ) = 𝔽_{q}^{×}, then all pairs (ϕ, C_{𝔪}) are non-isomorphic, since 𝔽_{q}^{×} fixes each of them. Therefore, all these pairs correspond to distinct points on X_{0}(𝔪)_{𝔽_{𝔭}}. We know from (4.1) that Aut(ϕ) ≠ 𝔽_{q}^{×} if and only if j(ϕ) = 0. Since Theorem 4.1 gives the number of isomorphism classes of supersingular Drinfeld modules, the claim of the lemma follows if we exclude the case with j(ϕ) = 0. Now assume j(ϕ) = 0. Then Aut(ϕ) ≅ 𝔽_{q^{2}}^{×}. We can identify the set of cyclic subgroups of ϕ of order 𝔪 with ℙ^{1}(A/𝔪). From this perspective, the action of Aut(ϕ) on this set is induced from an embedding

𝔽_{q^{2}}^{×} ↪ GL_{2}(A/𝔪) ≅ Aut(ϕ[𝔪]),

with GL_{2}(A/𝔪) acting on ℙ^{1}(A/𝔪) in the usual manner. We can decompose

ℙ^{1}(A/𝔪) ≅ ∏_{1≤i≤s} ℙ^{1}(A/𝔭_{i}^{r_{i}})

with GL_{2}(A/𝔪) acting on ℙ^{1}(A/𝔭_{i}^{r_{i}}) via its quotient GL_{2}(A/𝔭_{i}^{r_{i}}). The image of 𝔽_{q^{2}}^{×} in GL_{2}(A/𝔪) is a maximal non-split torus in GL_{2}(𝔽_{q}). If the stabilizer Stab_{𝔽_{q^{2}}^{×}} (P) of P ∈ ℙ^{1}(A/𝔭_{i}^{r_{i}}) is strictly larger than 𝔽_{q}^{×}, then it is easy to

see that in fact Stab_{𝔽_{q^{2}}^{×}} (P) = 𝔽_{q^{2}}^{×}. Moreover, this is possible if and only if

𝔽_{q^{2}} ↪ A/𝔭_{i}^{r_{i}}, in which case there are exactly 2 fixed points in ℙ^{1}(A/𝔭_{i}^{r_{i}}) under the action of 𝔽_{q^{2}}; cf. [17, p. 695]. Note that the existence of an embedding 𝔽_{q^{2}} ↪ A/𝔭_{i}^{r_{i}} is equivalent to deg(𝔭_{i}) being even. We conclude that 𝔽_{q^{2}}^{×} acting on the set of L(𝔪) pairs (ϕ, C_{𝔪}) has 2^{s}R(𝔪) fixed elements, and the orbit of any other element has length q + 1. This implies that there are

\frac{L(𝔪) − 2^{s}R(𝔪)}{q + 1} + 2^{s}R(𝔪) = \frac{L(𝔪) + q2^{s}R(𝔪)}{q + 1}

points on X_{0}(𝔪)_{𝔽_{𝔭}} corresponding to ϕ with j(ϕ) = 0. Finally, by Theorem 4.1, the Drinfeld module with j(ϕ) = 0 is supersingular if and only if deg(𝔭) is odd. □

Theorem 5.3. Assume 𝔫 = 𝔭𝔪, with 𝔭 prime not dividing 𝔪. If deg(𝔭) is odd and 𝔪 = A, then

Φ_{𝔭} ≅ ℤ/ ((q + 1)(S(𝔭, A) − 1) + 1) ℤ.

If deg(𝔭) is even or 𝔪 has a prime divisor of odd degree, then

Φ_{𝔭} ≅ ℤ/S(𝔭, 𝔪)ℤ.

Finally, if deg(𝔭) is odd, 𝔪 ≠ A, and all prime divisors of 𝔪 have even degrees, then

Φ_{𝔭} ≅ ℤ/ ((q + 1)^{2}S(𝔭, 𝔪) − q(q + 1)2^{s}) ℤ ⨁_{1≤i≤2^{s}−2} ℤ/(q + 1)ℤ.

(The isomorphisms above are meant only as isomorphisms of groups, not group schemes, so Φ_{𝔭} can have non-trivial G_{𝔽_{𝔭}}-action.)

Proof. This follows from Corollary 11 on page 285 in [3], combined with Lemma 5.2 and Theorem 5.1. This result for 𝔫 = 𝔭 is also in [10]. □

Proposition 5.4. Assume 𝔫 = 𝔭𝔮 is a product of two distinct primes. Denote

n(𝔭, 𝔮) = \frac{(|𝔭| − 1)(|𝔮| + 1)}{q^{2} − 1}.

Let Φ_{𝔭}(𝔽_{𝔭}) be the subgroup of Φ_{𝔭} fixed by G_{𝔽_{𝔭}}. If deg(𝔭) is odd and deg(𝔮) is even, then

Φ_{𝔭} ≅ ℤ/(q + 1)^{2}n(𝔭, 𝔮)ℤ, Φ_{𝔭}(𝔽_{𝔭}) ≅ ℤ/(q + 1)n(𝔭, 𝔮)ℤ.

Otherwise,

Φ_{𝔭}(𝔽_{𝔭}) = Φ_{𝔭} ≅ ℤ/n(𝔭, 𝔮)ℤ.

Proof. To simplify the notation, denote X = X_{0}(𝔭𝔮)_{O_{𝔭}} and k = 𝔽_{𝔭}. Let X̃ → X be the minimal resolution of X. We know that X_{k} consists of two irreducible components, both isomorphic to X_{0}(𝔮)_{k}. If deg(𝔭) is even or deg(𝔮) is odd, then X = X̃. On the other hand, if deg(𝔭) is odd and deg(𝔮) is even, then there are two points P and Q of thickness q + 1. To obtain the minimal regular model one performs a sequence of q blows-ups at P and Q. With the notation of [41, §4.2], let E_{1}, …, E_{q} and G_{1}, …, G_{q} be the chain of projective lines resulting from these blow-ups at P and Q, respectively. Let Z and Z′ denote the two copies of X_{0}(𝔮)_{k}, with the convention that E_{1} and G_{1} intersect Z. Let B(X̃_{k}) be the free abelian group generated by the set of geometrically irreducible components of X̃_{k}. Let B^{0}(X̃_{k}) be the subgroup of degree-0 elements in B(X̃_{k}). The theorem of Raynaud that we mentioned earlier gives an explicit description of Φ_{𝔭} as a quotient of B^{0}(X̃_{k}); cf. [41, §4.2]. Let Frob_{𝔭} : α → α^{|𝔭|} be the usual topological generator of G_{k}. The Frobenius Frob_{𝔭} naturally acts on the geometrically irreducible components of X̃_{k}, and therefore on B^{0}(X̃_{k}). Since X_{0}(𝔮)_{k} is defined over k, the action of Frob_{𝔭} fixes z := Z − Z′ ∈ B^{0}(X̃_{k}). If deg(𝔭) is even or deg(𝔮) is odd, then z generates B^{0}(X̃_{k}), so G_{k} acts trivially on Φ_{𝔭}, which by Theorem 5.3 is isomorphic to ℤ/n(𝔭, 𝔮)ℤ. From now on we assume deg(𝔭) is odd and deg(𝔮) is even. The claim Φ_{𝔭} ≅ ℤ/(q + 1)^{2}n(𝔭, 𝔮)ℤ follows from Theorem 5.3. Let ϕ be given by ϕ_{T} = T + τ^{2}.

Note that j(ϕ) = 0 and ϕ is defined over k, so Frob_{𝔭} acts on the set of cyclic subgroups C_{𝔮} ⊂ ϕ. Denote A′ = 𝔽_{q^{2}}[T]. It is easy to see that A′ ⊆ End(ϕ). Since 𝔮 = 𝔮_{1}𝔮_{2} splits in A′ into a product of two irreducible polynomials of the same degree, we have

ϕ[𝔮] ≅ A′/𝔮 ≅ A′/𝔮_{1} ⊕ A′/𝔮_{2} ≅ A/𝔮 ⊕ A/𝔮 ≅ ϕ[𝔮_{1}] ⊕ ϕ[𝔮_{2}].

The above decomposition is preserves under the action of 𝔽_{q^{2}}. In particular, Aut(ϕ, ϕ[𝔮_{i}]) ≅ 𝔽_{q^{2}}^{×}. On the other hand, since 𝔭 has odd degree, this decomposition is not defined over k and Frob_{𝔭}(ϕ[𝔮_{1}]) = ϕ[𝔮_{2}]. We conclude that P = (ϕ, ϕ[𝔮_{1}]), Q = (ϕ, ϕ[𝔮_{2}]) and Frob_{𝔭}(P) = Q. Since the action of Frob_{𝔭} preserves the incidence relations of the irreducible components of X̃_{k}, we have Frob_{𝔭}(E_{i}) = G_{i}, 1 ≤ i ≤ q. Following the notation of Theorem 4.1 in [41], let e_{q} = E_{q} − Z′ and g_{q} = G_{q} − Z′. According to that theorem, the image of e_{q} in Φ_{𝔭} generates Φ_{𝔭}, and in the component group we have g_{q} = − ((q + 1)(S(𝔭, 𝔮) − 2) + 1) e_{q}. Thus, the action of Frob_{𝔭} on Φ_{𝔭} in terms of the generator e_{q} is given by Frob_{𝔭}(e_{q}) = − ((q + 1)(S(𝔭, 𝔮) − 2) + 1) e_{q}. This implies that ae_{q} ∈ Φ_{𝔭}(𝔽_{𝔭}) if and only if

ae_{q} = Frob_{𝔭}(ae_{q}) = −a ((q + 1)(S(𝔭, 𝔮) − 2) + 1) e_{q},

which is equivalent to a ((q + 1)S(𝔭, 𝔮) − 2q) e_{q} = 0. Hence a must be a multiple of q + 1, and

Φ_{𝔭}(𝔽_{𝔭}) = ⟨(q + 1)⟩ ≅ ℤ/(q + 1)n(𝔭, 𝔮)ℤ.
□

The existence of rigid-analytic uniformization (4.3) of J implies that J has totally degenerate reduction at ∞, i.e., J_{𝔽_{∞}}^{0} is a split algebraic torus over 𝔽_{∞}. The problem of explicitly describing Φ_{∞} is closely related to the problem of describing Γ_{0}(𝔫) \ 𝒯 together with the stabilizers of its edges; cf. [41, §5.2]. Since the graph Γ_{0}(𝔫) \ 𝒯 becomes very complicated as |𝔫| grows, no explicit description of Φ_{∞} is known in general. On the other hand, Φ_{∞} has a description in terms of the uniformization of J. Let ⟨·, ·⟩ : \overline{Γ} × \overline{Γ} → ℤ be the pairing (4.2). This pairing is bilinear, symmetric, and positive-definite, so it induces an injection ι : \overline{Γ} → Hom(\overline{Γ}, ℤ), γ ↦ ⟨γ, ·⟩. Identifying \overline{Γ} with H_{0}(𝔫, ℤ) via (4.4), we get an injection ι : H_{0}(𝔫, ℤ) → Hom(H_{0}(𝔫, ℤ), ℤ). The Hecke algebra 𝕋(𝔫) acts on Hom(H_{0}(𝔫, ℤ), ℤ) through its action on the first argument. The action of 𝕋(𝔫) on J, by the Néron mapping property, canonically extends to an action on J. Thus, 𝕋(𝔫) functorially acts on Φ_{∞}.

Theorem 5.5. The absolute Galois group G_{𝔽_{∞}} acts trivially on Φ_{∞}, and there is a 𝕋(𝔫)^{0}-equivariant exact sequence

0 → H_{0}(𝔫, ℤ) ⟶^{ι} Hom(H_{0}(𝔫, ℤ), ℤ) → Φ_{∞} → 0.

Proof. The exact sequence is the statement of Corollary 2.11 in [11]. The 𝕋(𝔫)^{0}- equivariance of this sequence follows from the 𝕋(𝔫)^{0}-equivariance of (4.3). The fact that G_{𝔽_{∞}} acts trivially on Φ_{∞} is a consequence of J_{𝔽_{∞}}^{0} being a split torus. □

6. Cuspidal divisor group

The cuspidal divisor group C(𝔫) of J := J_{0}(𝔫) is the subgroup of J generated by the classes of divisors [c] − [c′], where c, c′ run through the set of cusps of X_{0}(𝔫)_{F}.

Theorem 6.1. C(𝔫) is finite and if 𝔫 is square-free then C(𝔫) ⊂ J(F)_{tor}.

Proof. This theorem is due to Gekeler [15], where the finiteness of C(𝔫) is proven for general congruence subgroups over general function fields. Since the proof is fairly short, for the sake of completeness, we give a sketch. The space H_{0}(𝔫, ℂ) has an interpretation as a space of automorphic cusp forms (cf. [18]), so the Ramanujan-Petersson conjecture over function fields (proved by Drinfeld) implies that the eigenvalues of T_{𝔭} are algebraic integers of absolute value ≤ 2\sqrt{|𝔭|} for any prime 𝔭 ∤ 𝔫. This implies that the operator η_{𝔭} = T_{𝔭} − |𝔭| − 1 is invertible on H_{0}(𝔫, ℂ). Hence η_{𝔭} : H_{0}(𝔫, ℤ) → H_{0}(𝔫, ℤ) is injective with finite cokernel. Now using the T_{𝔭} equivariance of (4.3) and (4.4), we see that η_{𝔭} : J → J is an isogeny. Assume for simplicity that 𝔫 is squarefree. In the proof of Lemma 3.1 we showed that η_{𝔭}([c] − [c′]) = 0. Hence C(𝔫) ⊆ ker(η_{𝔭}), which is finite. Finally, when 𝔫 is square-free, all the cusps of X_{0}(𝔫)_{F} are rational over F, so the cuspidal divisor group is in J(F); see [16, Prop. 6.7]. □

Proposition 6.2. With notation of §3.1, let μ(𝔫) be the largest integer ℓ such that there exists an ℓ-th root of Δ/Δ_{𝔫} in O(Ω)^{×}. Then μ(𝔫) = (q − 1)^{2} if deg 𝔫 is odd and μ(𝔫) = (q − 1)(q^{2} − 1) if deg 𝔫 is even. Let D_{𝔫} be a μ(𝔫)-th root of Δ/Δ_{𝔫}. There exists a character ω_{𝔫} : Γ_{0}(𝔫) → 𝔽_{q}^{×} such that for each γ ∈ Γ_{0}(𝔫),

D_{𝔫}(γz) = ω_{𝔫}(γ)D_{𝔫}(z).

Proof. See Corollary 3.5 and 3.21 in [14]. □

If 𝔫 = ∏_{i} 𝔭_{i}^{r_{i}} ⊲ A is the prime decomposition of 𝔫, define a character χ_{𝔫} :

Γ_{0}(𝔫) → 𝔽_{q}^{×} by the following: for γ = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ_{0}(𝔫),

χ_{𝔫}(γ) := ∏_{i} N_{i}(d mod 𝔭_{i})^{−r_{i}},

where N_{i} : (A/𝔭_{i}A)^{×} → 𝔽_{q}^{×} is the norm map. Then ω_{𝔫} = χ_{𝔫} · det^{deg(𝔫)/2} if deg(𝔫) is even and ω_{𝔫} = χ_{𝔫}^{2} · det^{deg(𝔫)} if deg(𝔫) is odd. In particular, the order of ω_{𝔫} is q − 1 when 𝔫 is square-free (cf. [14] Proposition 3.22). By Proposition 6.2, we immediately get:

Corollary 6.3.

(1) D′_{𝔫} := D_{𝔫}^{q−1} is a meromorphic function on the Drinfeld modular curve X_{0}(𝔫) satisfying

(D′_{𝔫})^{\frac{μ(𝔫)}{q−1}} = \frac{Δ}{Δ_{𝔫}} .

(2) Given two ideals 𝔪 and 𝔫 of A, we set D_{𝔫,𝔪}(z) := D_{𝔫}(z)/D_{𝔫}(𝔪z), ∀z ∈ Ω. Then

(D_{𝔫,𝔪})^{μ(𝔫)} = \frac{ΔΔ_{𝔪𝔫}}{Δ_{𝔫}Δ_{𝔪}} = (D_{𝔪,𝔫})^{μ(𝔪)}

and for every γ ∈ Γ_{0}(𝔪𝔫), we have

D_{𝔫,𝔪}(γz) = D_{𝔫,𝔪}(z).

In other words, D_{𝔫,𝔪} and D_{𝔪,𝔫} can be viewed as meromorphic functions on X_{0}(𝔪𝔫).

Remark 6.4. Take two coprime ideals 𝔪 and 𝔫 of A. By Corollary 6.3 (1), the (q − 1)-th roots of (ΔΔ_{𝔪})/(Δ_{𝔫}Δ_{𝔪𝔫}) always exist in the function field of X_{0}(𝔪𝔫)_{ℂ_{∞}}. In fact, we can find a (q^{2} − 1)-th root of (ΔΔ_{𝔪})/(Δ_{𝔫}Δ_{𝔪𝔫}) when deg 𝔫 or deg 𝔪𝔫 is even. Indeed, we notice that

\frac{Δ(z)Δ_{𝔪}(z)}{Δ_{𝔫}(z)Δ_{𝔪𝔫}(z)} = \frac{Δ(z)}{Δ_{𝔫}(z)} · \frac{Δ(z′)}{Δ_{𝔫}(z′)} = \frac{Δ(z)}{Δ_{𝔪𝔫}(z)} · \frac{Δ(z′′)}{Δ_{𝔪𝔫}(z′′)} · 𝔪^{−(q^{2}−1)}.

Here

z′ = 𝔪z, z′′ = \frac{𝔪z + 1}{b𝔪𝔫z + a𝔪} = (\begin{matrix}𝔪 & 1 \\ b𝔪𝔫 & a𝔪 \end{matrix}) · z,

and a, b ∈ A such that a𝔪 − b𝔫 = 1. In particular, when q is odd,

D′′_{𝔫}(z) := (D_{𝔫}(z) · D_{𝔫}(mz))^{\frac{q−1}{2}}

is a 2(q^{2} − 1)-th (resp. 2(q − 1)-th) root of (ΔΔ_{𝔪})/(Δ_{𝔫}Δ_{𝔪𝔫}) in the function field of X_{0}(𝔪𝔫)_{ℂ_{∞}} when deg 𝔫 is even (resp. odd).

The following lemma is immediate from the definitions.

Lemma 6.5. Let r be the van der Put derivative (3.3) and E_{𝔫} ∈ H(𝔫, ℤ) the Eisenstein series (3.4). Then r(D_{𝔫}) = E_{𝔫}, and for any two distinct prime ideals 𝔭 and 𝔮,

r(D_{𝔭,𝔮}) = \begin{cases}(q + 1) · E_{(𝔭,𝔮)}, & if deg 𝔭 is odd and deg 𝔮 is even, \\ E_{(𝔭,𝔮)}, & otherwise. \end{cases}

Take two distinct prime ideals 𝔭 and 𝔮, let ℓ be the largest number such that there exists an ℓ-th root ξ of (ΔΔ_{𝔭𝔮})/(Δ_{𝔭}Δ_{𝔮}) in the function field of X_{0}(𝔭𝔮)_{ℂ_{∞}}. Comparing the first Fourier coefficient r(ξ)^{∗}(1) with r((ΔΔ_{𝔭𝔮})/(Δ_{𝔭}Δ_{𝔮}))^{∗}(1), we must have ℓ|(q − 1)(q^{2} − 1). By Corollary 3.17 in [14], for every non-zero ideal 𝔫 of A

\frac{Δ}{Δ_{𝔫}} = const.\frac{G_{𝔫}^{(q−1)(q^{2}−1)}}{Δ^{(q^{deg 𝔫}−1)}} ,

where G_{𝔫} is a Drinfeld modular form on Ω such that for any γ = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈

Γ_{0}(𝔫)

G_{𝔫}(γz) = χ_{𝔫}(γ)(cz + d)^{(|𝔫|−1)/(q−1)}G_{𝔫}(z).

Then

ξ^{ℓ} = \frac{Δ(z)Δ_{𝔭𝔮}(z)}{Δ_{𝔭}(z)Δ_{𝔮}(z)} = const. (\frac{G_{𝔭}(z)}{G_{𝔭}(𝔮z)})^{(q−1)(q^{2}−1)} · (\frac{Δ_{𝔮}(z)}{Δ(z)})^{|𝔭|−1}

= const. (\frac{G_{𝔭}(z)}{G_{𝔭}(𝔮z)})^{(q−1)(q^{2}−1)} · D_{𝔮}(z)^{−μ(𝔮)(|𝔭|−1)}.

Since G_{𝔭}(z)/G_{𝔭}(𝔮z) is a meromorphic function on X_{0}(𝔭𝔮)_{ℂ_{∞}},

D_{𝔮}(z)^{\frac{μ(𝔮)(|𝔭|−1)}{ℓ}} = const. (\frac{G_{𝔭}(z)}{G_{𝔭}(𝔮z)})^{\frac{(q−1)(q^{2}−1)}{ℓ}} · ξ^{−1}

is also in the function field of X_{0}(𝔭𝔮)_{ℂ_{∞}}. Note that the character ω_{𝔮} has order q − 1. Set

μ(𝔭, 𝔮) := \begin{cases}(q − 1)(q^{2} − 1), & if deg(𝔭) or deg(𝔮) is even, \\ (q − 1)^{2}, & otherwise. \end{cases}

We then have:

Lemma 6.6. The largest number ℓ such that there exists an ℓ-th roots of \frac{ΔΔ_{𝔭𝔮}}{Δ_{𝔭}Δ_{𝔮}} in the function field of X_{0}(𝔭𝔮)_{ℂ_{∞}} is μ(𝔭, 𝔮).

From now on we assume that 𝔫 = 𝔭𝔮 is a product of two distinct primes. In this case, X_{0}(𝔫)_{F} has 4 cusps, which in the notation of Lemma 2.14 are [∞], [1], [𝔭], [𝔮]. Let c_{1}, c_{𝔭}, c_{𝔮} ∈ J(F) be the classes of divisors [1] − [∞], [𝔭]−[∞], and [𝔮]−[∞], respectively. To simplify the notation, we put C := C(𝔭𝔮). The cuspidal divisor group C is generated by c_{1}, c_{𝔭}, and c_{𝔮}.

Proposition 6.7. The order of c_{1} and c_{𝔮} is divisible by

\begin{cases}\frac{(|𝔭|−1)(|𝔮|+1)}{q−1} & if deg(𝔭) is odd and deg(𝔮) is even, \\ \frac{(|𝔭|−1)(|𝔮|+1)}{q^{2}−1} & otherwise. \end{cases}

Proof. We use the notation in the proof of Proposition 5.4. Let ℘_{𝔭} : J(F_{𝔭}) → Φ_{𝔭} be the canonical specialization map. This map can be explicitly described as follows. Let D = ∑_{i} n_{i}P_{i} be a degree-0 divisor, where all P_{i} ∈ X(F_{𝔭}). Denote by D also the linear equivalence class of D in J(F_{𝔭}). Since X and X̃ are proper, X(F_{𝔭}) = X(O_{𝔭}) = X̃(O_{𝔭}). Since X̃ is regular, each P_{i} specializes to a unique irreducible component c(P_{i}) of X̃_{k}. Then ℘_{𝔭}(D) is the image of ∑_{i} n_{i}c(P_{i}) ∈ B^{0}(X̃_{k}) in Φ_{𝔭}. By Theorem 4.2, the cusps reduce to distinct points in the smooth locus of X_{k}. The Atkin-Lehner involution W_{𝔭} interchanges the two irreducible components Z and Z′ of X_{k}. Since W_{𝔭}([∞]) = [𝔮], the reductions of [∞] and [𝔮] lie on distinct components. On the other hand, W_{𝔮} acts on X_{k} by acting on each component Z and Z′ separately, without interchanging them. Since W_{𝔮}([∞]) = [𝔭], the reductions of [∞] and [𝔭] lie on the same component of X_{k}. Let Z′ be the component containing [∞] and [𝔭]. Let z be the image of Z − Z′ in Φ_{𝔭}. Then ℘_{𝔭}(c_{1}) = ℘_{𝔭}(c_{𝔮}) = z and ℘_{𝔭}(c_{𝔭}) = 0.

By Theorem 4.1 in [41], Φ_{𝔭}/⟨z⟩ ≅ ℤ/(q + 1)ℤ. Now the claim follows from Theorem 5.3. □

Remark 6.8. Suppose 𝔫 = 𝔭𝔪, with 𝔭 prime not dividing 𝔪. Let C(𝔫)(F) denote the F -rational cuspidal subgroup of J_{0}(𝔫). Generalizing the argument in the proof of Proposition 6.7 and using Theorem 5.3, it is not hard to show that the exponent of the group Φ_{𝔭}/℘_{𝔭}(C(𝔫)(F)) divides (q + 1). In particular, the map

℘_{𝔭} : C(𝔫)(F)_{ℓ} → (Φ_{𝔭})_{ℓ}

is surjective for ℓ ∤ (q + 1).

The orders of Δ at the cusps are known (cf. [14, (3.10)]):

ord_{[∞]}Δ = 1, ord_{[1]}Δ = |𝔭||𝔮|, ord_{[𝔭]}Δ = |𝔭|, ord_{[𝔮]}Δ = |𝔮|.

Using the action of the Aktin-Lehner involutions on the cusps (Lemma 2.14) and on the functions Δ, Δ_{𝔭}, and Δ_{𝔭𝔮}, one obtains the divisors of the following functions on X_{0}(𝔫)_{ℂ_{∞}} (cf. [41]):

div(Δ/Δ_{𝔭}) = (|𝔭| − 1)(|𝔮|([1] − [∞]) − |𝔮|([𝔭] − [∞]) + ([𝔮] − [∞])),
div(Δ/Δ_{𝔮}) = (|𝔮| − 1)(|𝔭|([1] − [∞]) + ([𝔭] − [∞]) − |𝔭|([𝔮] − [∞])),
div(Δ/Δ_{𝔭𝔮}) = (|𝔭𝔮| − 1)([1] − [∞]) + (|𝔮| − |𝔭|)([𝔭] − [∞])+
+ (|𝔭| − |𝔮|)([𝔮] − [∞]).

Hence

(6.1)  div (\frac{ΔΔ_{𝔭𝔮}}{Δ_{𝔭}Δ_{𝔮}} ) = (|𝔭| − 1)(|𝔮| − 1)([∞] + [1] − [𝔭] − [𝔮]),
(6.2) div (\frac{ΔΔ_{𝔮}}{Δ_{𝔭}Δ_{𝔭𝔮}} ) = (|𝔭| − 1)(|𝔮| + 1)([∞] + [1] − [𝔭] + [𝔮]),
(6.3) div (\frac{ΔΔ_{𝔭}}{Δ_{𝔮}Δ_{𝔭𝔮}} ) = (|𝔭| + 1)(|𝔮| − 1)([∞] + [1] + [𝔭] − [𝔮]).

From these equations we get:

Lemma 6.9.

0 = (|𝔭|^{2} − 1)(|𝔮|^{2} − 1)c_{1} = (|𝔭|^{2} − 1)(|𝔮| − 1)c_{𝔭} = (|𝔭| − 1)(|𝔮|^{2} − 1)c_{𝔮}.

In particular, the order of C is not divisible by p.

Let

c_{(−,−)} := c_{1} − c_{𝔭} − c_{𝔮}, c_{(−,+)} := c_{1} − c_{𝔭} + c_{𝔮}, c_{(+,−)} := c_{1} + c_{𝔭} − c_{𝔮}.

The subgroup C′ of C generated by c_{(−,−)}, c_{(−,+)}, and c_{(+,−)} has index 1, 2, or 4. Set

ϵ(𝔭, 𝔮) := \begin{cases}q − 1, & if q is even, deg 𝔭 is odd and deg 𝔮 is even, \\ 2(q − 1), & if q is odd, deg 𝔭 is odd, and deg 𝔮 is even, \\ q^{2} − 1, & if (i) deg 𝔭 and deg 𝔮 are both odd or \\ & (ii) q is even and deg 𝔭 is even, \\ 2(q^{2} − 1), & otherwise, \end{cases}

and

N_{(−,−)} := \frac{(|𝔭| − 1)(|𝔮| − 1)}{μ(𝔭, 𝔮)},

N_{(−,+)} := \frac{(|𝔭| − 1)(|𝔮| + 1)}{ϵ(𝔭, 𝔮)}, N_{(+,−)} := \frac{(|𝔭| + 1)(|𝔮| − 1)}{ϵ(𝔮, 𝔭)}.

Proposition 6.10. The orders of c_{(−,−)}, c_{(−,+)}, and c_{(+,−)} are N_{(−,−)}, N_{(−,+)}, and N_{(+,−)}, respectively. The group C′ is isomorphic to

\frac{ℤ}{N_{(−,−)}ℤ} × \frac{ℤ}{N_{(−,+)}ℤ} × \frac{ℤ}{N_{(+,−)}ℤ}.

Proof. Lemma 6.6 and Equation (6.1) tell us immediately that the order of c_{(−,−)} is N_{(−,−)}. In Remark 6.4, we have constructed ϵ(𝔭, 𝔮)-th root of \frac{ΔΔ_{𝔮}}{Δ_{𝔭}Δ_{𝔭𝔮}} in the function field of X_{0}(𝔭𝔮)_{ℂ_{∞}}. Therefore Equation (6.2) and (6.3) imply that

N_{(−,+)}c_{(−,+)} = N_{(+,−)}c_{(+,−)} = 0.

Recall from the proof of Proposition 6.7 that ℘_{𝔭}(c_{1}) = ℘_{𝔭}(c_{𝔮}) = z and ℘_{𝔭}(c_{𝔭}) = 0, where ℘_{𝔭} : J(F_{𝔭}) → Φ_{𝔭} is the canonical specialization map. Then ℘_{𝔭}(c_{(−,+)}) = 2z, and the order of c_{(−,+)} must be divisible by N_{(−,+)}. Therefore the order of c_{(−,+)} is N_{(−,+)}, which is also the order of ℘_{𝔭}(c_{(−,+)}). By interchanging 𝔭 and 𝔮 we obtain that the order of c_{(+,−)} is N_{(+,−)}. This proves the first claim. Now, suppose c = α_{1}c_{(−,−)} +α_{2}c_{(−,+)} +α_{3}c_{(+,−)} = 0 for α_{1}, α_{2}, α_{3} ∈ ℤ. Then ℘_{𝔭}(c) = α_{2}℘_{𝔭}(c_{(−,+)}) = 0, which implies that N_{(−,+)} | α_{2}. Similarly, we have N_{(+,−)} | α_{3}. Hence c = α_{1}c_{(−,−)} = 0, and N_{(−,−)} | α_{1}. This implies the second claim. □

When q is even, Lemma 6.9 implies that C = C′. Suppose q is odd. Note that the quotient group C/C′ is generated by c_{𝔭} and c_{𝔮}, where 2c_{𝔭} ≡ 2c_{𝔮} ≡ 0 mod C′. When deg(𝔭) and deg(𝔮) are both odd, we can find meromorphic functions φ_{𝔭} and φ_{𝔮} on X_{0}(𝔭𝔮)_{ℂ_{∞}} such that

div(φ_{𝔭}) = \frac{(|𝔭|^{2} − 1)(|𝔮| − 1)}{(q − 1)(q^{2} − 1)}([𝔭] − [∞]),

div(φ_{𝔮}) = \frac{(|𝔭| − 1)(|𝔮|^{2} − 1)}{(q − 1)(q^{2} − 1)}([𝔮] − [∞]).

Documenta Mathematica 20 (2015) 551–629

Indeed, let

φ_{𝔭}(z) := \frac{D_{𝔮}(z)^{\frac{|𝔭|−q}{q^{2}−1}} · D_{𝔮}(𝔭z)^{\frac{|𝔭|q−1}{q^{2}−1}}}{(D_{𝔭𝔮}(z) · D_{𝔭𝔮}(W_{𝔭}z))^{\frac{|𝔭|−1}{q−1}}} , φ_{𝔮}(z) := \frac{D_{𝔭}(z)^{\frac{|𝔮|−q}{q^{2}−1}} · D_{𝔭}(𝔮z)^{\frac{|𝔮|q−1}{q^{2}−1}}}{(D_{𝔭𝔮}(z) · D_{𝔭𝔮}(W_{𝔮}z))^{\frac{|𝔮|−1}{q−1}}}

(Proposition 6.2 implies that φ_{𝔭} and φ_{𝔮} are Γ_{0}(𝔭𝔮)-invariant.) We conclude that the orders of c_{𝔭} and c_{𝔮} are odd, and therefore C = C′. Suppose q is odd and deg(𝔭) · deg(𝔮) is even. Then Proposition 6.7 tells us that the order of c_{𝔭} and c_{𝔮} are both even. Thus from the canonical specialization maps ℘_{𝔭} and ℘_{𝔮}, we have the following exact sequence:

0 ⟶ C′ ⟶ C ⟶ \frac{ℤ}{2ℤ} × \frac{ℤ}{2ℤ} ⟶ 0.

Since 2c_{𝔭} = c_{(+,−)} − c_{(−,−)} and 2c_{𝔮} = c_{(−,+)} − c_{(−,−)}, the order of c_{𝔭} is 2 · lcm(N_{(−,−)}, N_{(+,−)}), and the order of c_{𝔮} is 2 · lcm(N_{(−,−)}, N_{(−,+)}). From the above discussion, we finally conclude that:

Theorem 6.11.

(1) The order of c_{𝔭} is \frac{(|𝔭|^{2}−1)(|𝔮|−1)}{(q−1)(q^{2}−1)} and the order of c_{𝔮} is \frac{(|𝔭|−1)(|𝔮|^{2}−1)}{(q−1)(q^{2}−1)}.

(2) The odd part of C is isomorphic to

\frac{ℤ}{N_{(−,−)}^{odd}ℤ} × \frac{ℤ}{N_{(−,+)}^{odd}ℤ} × \frac{ℤ}{N_{(+,−)}^{odd}ℤ},

where N^{odd} is the odd part of a positive integer N.

(3) When q is even or q · deg(𝔭) · deg(𝔮) is odd, we have C = C′.

(4) Suppose q is odd and deg(𝔭) · deg(𝔮) is even. Let C_{2} (resp. C′_{2}) be the 2-primary part of C (resp. C′). Let r_{1}, r_{2}, r_{3} ∈ ℤ_{≥0} with r_{1} ≥ r_{2} ≥ r_{3} such that

C′_{2} ≅ ℤ/2^{r_{1}}ℤ × ℤ/2^{r_{2}}ℤ × ℤ/2^{r_{3}}ℤ.

Then

C_{2} ≅ ℤ/2^{r_{1}+1}ℤ × ℤ/2^{r_{2}+1}ℤ × ℤ/2^{r_{3}}ℤ.

Example 6.12. If q is even, then C′(xy) = C(xy) ≅ ℤ/(q + 1)ℤ × ℤ/(q^{2} + 1)ℤ. If q is odd, then C′(xy) ≅ _{\frac{ℤ}{\frac{q+1}{2} ℤ}} × _{\frac{ℤ}{\frac{q^{2}+1}{2} ℤ}} and C(xy) ≅ ℤ/(q + 1)ℤ × ℤ/(q^{2} + 1)ℤ.

Example 6.13. Assume deg(𝔭) = deg(𝔮) = 2. If q is even, then

C′(𝔭𝔮) = C(𝔭𝔮) ≅ \frac{ℤ}{(q^{2} + 1)ℤ} × \frac{ℤ}{(q + 1)(q^{2} + 1)ℤ}.

If q is odd,

C′(𝔭𝔮) ≅ \frac{ℤ}{(q + 1)ℤ} × \frac{ℤ}{\frac{q^{2}+1}{2}ℤ} × \frac{ℤ}{\frac{q^{2}+1}{2}ℤ},

C(𝔭𝔮) ≅ \frac{ℤ}{(q^{2} + 1)ℤ} × \frac{ℤ}{(q + 1)(q^{2} + 1)ℤ}.

Documenta Mathematica 20 (2015) 551–629

7. Rational torsion subgroup

7.1. Main theorem. Let 𝔫 ⊲ A be a non-zero ideal. To simplify the notation in this section we denote J = J_{0}(𝔫). Let J denote the Néron model of J over ℙ_{𝔽_{q}}^{1}. Let T (𝔫) the torsion subgroup of J(F).

Lemma 7.1. Let ℓ be a prime not equal to p. Then T (𝔫)_{ℓ} is annihilated by the Eisenstein ideal 𝔈(𝔫), i.e., (T_{𝔭} − |𝔭| − 1)P = 0 for every prime ideal 𝔭 not dividing 𝔫 and P ∈ T (𝔫)_{ℓ}.

Proof. It follows from Theorem 4.2 that J has good reduction at 𝔭 ∤ 𝔫. By the Néron mapping property, T (𝔫)_{ℓ} extends to an étale subgroup scheme of J. This implies that there is a canonical injective homomorphism T (𝔫)_{ℓ} ↪ J_{𝔽_{𝔭}}(𝔽_{𝔭}). The action of 𝕋(𝔫) on J canonically extends to an action on J. Since the reduction map commutes with the action of 𝕋(𝔫), it is enough to show that (T_{𝔭} − |𝔭| − 1) annihilates T (𝔫)_{ℓ} over 𝔽_{𝔭}. Let Frob_{𝔭} be the Frobenius endomorphism of J_{𝔽_{𝔭}}. The Hecke operator T_{𝔭} satisfies the Eichler-Shimura relation:

Frob_{𝔭}^{2} − T_{𝔭} · Frob_{𝔭} + |𝔭| = 0.

Since Frob_{𝔭} acts trivially on T (𝔫)_{ℓ}, the claim follows. □

Lemma 7.2. Suppose ℓ is a prime not dividing q(q − 1). There is a natural injective homomorphism T (𝔫)_{ℓ} ↪ E_{00}(𝔫, ℤ/ℓ^{r}ℤ) for any r ∈ ℤ_{≥0} with ℓ^{r} ≥ #(Φ_{∞,ℓ}).

Proof. Since J has split toric reduction at ∞, J_{𝔽_{∞}}^{0}(𝔽_{∞}) ≅ ∏_{i=1}^{g} 𝔽_{q}^{×}, where g = dim(J). Under the assumption that ℓ does not divide q(q − 1), we see that T (𝔫)_{ℓ} has trivial intersection with J_{𝔽_{∞}}^{0}, so the canonical specialization T (𝔫)_{ℓ} → (Φ_{∞})_{ℓ} is injective. Since this map is 𝕋(𝔫)-equivariant, by Lemma 7.1 we get an injection T (𝔫)_{ℓ} → (Φ_{∞})_{ℓ}[𝔈]. Fix some r ∈ ℤ_{≥0} with ℓ^{r} · (Φ_{∞})_{ℓ} = 0. Multiplying the sequence in Theorem 5.5 by ℓ^{r} and applying the snake lemma, we get an injection (Φ_{∞})_{ℓ} ↪ H_{00}(𝔫, ℤ/ℓ^{r}ℤ). Since this map is again 𝕋(𝔫)-equivariant, restricting to the kernels of 𝔈(𝔫), we get (Φ_{∞})_{ℓ}[𝔈(𝔫)] ↪ E_{00}(𝔫, ℤ/ℓ^{r}ℤ). Therefore, T (𝔫)_{ℓ} ↪ E_{00}(𝔫, ℤ/ℓ^{r}ℤ) as was required to show. □

Theorem 7.3. Suppose 𝔫 = 𝔭𝔮 is a product of two distinct primes 𝔭 and 𝔮. Let s_{𝔭,𝔮} = gcd(|𝔭| + 1, |𝔮| + 1). If ℓ does not divide q(q − 1)s_{𝔭,𝔮}, then C(𝔫)_{ℓ} = T (𝔫)_{ℓ}.

Proof. First of all, since 𝔫 is square-free, C(𝔫) is rational over F, so C(𝔫)_{ℓ} ⊆ T (𝔫)_{ℓ}; see Theorem 6.1. Next, note that ℓ is odd by our assumption, so by Proposition 6.10

C(𝔫)_{ℓ} = C(𝔫)_{ℓ}^{odd} ≅ ℤ/ℓ^{r_{(−,−)}}ℤ × ℤ/ℓ^{r_{(−,+)}}ℤ × ℤ/ℓ^{r_{(+,−)}}ℤ,

where r_{(±,±)} := ord_{ℓ}(N_{(±,±)}). Since ℓ is coprime to q(q − 1)s_{𝔭,𝔮}, Theorem 3.9 and Lemma 3.10 give the inclusion

E_{00}(𝔫, ℤ/ℓ^{r}ℤ) = E′_{00}(𝔫, ℤ/ℓ^{r}ℤ)
⊆ E′_{0}(𝔫, ℤ/ℓ^{r}ℤ) ≅ ℤ/ℓ^{r_{(−,−)}}ℤ × ℤ/ℓ^{r_{(−,+)}}ℤ × ℤ/ℓ^{r_{(+,−)}}ℤ.

Finally, by Lemma 7.2 we have an injection C(𝔫)_{ℓ} ↪ T (𝔫)_{ℓ} ↪ E_{00}(𝔫, ℤ/ℓ^{r}ℤ). Comparing the orders of these groups, we conclude that

C(𝔫)_{ℓ} = T (𝔫)_{ℓ} = E_{00}(𝔫, ℤ/ℓ^{r}ℤ) = E′_{0}(𝔫, ℤ/ℓ^{r}ℤ).
□

Remark 7.4. The previous proof shows that E_{00}(𝔭𝔮, ℤ/ℓ^{r}ℤ) = E′_{0}(𝔭𝔮, ℤ/ℓ^{r}ℤ). A generalization of this argument gives the equality E_{00}(𝔭𝔮, R) = E′_{0}(𝔭𝔮, R) for any coefficient ring in which q − 1 and s_{𝔭,𝔮} are invertible.

Corollary 7.5. If ℓ does not divide q(|𝔭|^{2} − 1)(|𝔮|^{2} − 1), then T (𝔭𝔮)_{ℓ} = 0.

Lemma 7.6. Assume 𝔫 = 𝔪𝔭 is square-free, 𝔭 is prime, and deg(𝔪) ≤ 2. Then T (𝔫)_{p} = 0.

Proof. If deg(𝔪) ≤ 2, then X_{0}(𝔪)_{𝔽_{𝔭}} ≅ ℙ_{𝔽_{𝔭}}^{1}. It follows from Theorem 5.1 and [3, p. 246] that J_{𝔽_{𝔭}}^{0} is a torus. Since J has toric reduction at 𝔭, the p-primary torsion subgroup T (𝔫)_{p} injects into Φ_{𝔭}; see Lemma 7.13 in [38]. Finally, as is easy to see from Theorem 5.3, if 𝔫 is square-free, the order of Φ_{𝔭} is coprime to p. Thus, T (𝔫)_{p} = 0. □

7.2. Special case. Here we focus on the case 𝔫 = xy and prove that C(𝔫) = T (𝔫). To simplify the notation, let C := C(𝔫) and T := T (𝔫). By Theorem 7.3 and Lemma 7.6, we know that C_{ℓ} = T_{ℓ} for any ℓ ∤ (q − 1). Let

N = (q + 1)(q^{2} + 1).

By Corollary 3.18, 𝕋(xy)/𝔈(xy) ≅ ℤ/Nℤ, so N ∈ 𝔈(xy). On the other hand, 𝔈(xy) annihilates T_{ℓ} for ℓ ≠ p. Therefore, the exponent of T_{ℓ} divides N. Since gcd(q − 1, N) divides 4, T_{ℓ} = 0 when ℓ | (q − 1) is an odd prime. Therefore, we are reduced to showing that C_{2} = T_{2} in the case when q is odd. To prove this we will use the fact that X_{0}(xy)_{F} is hyperelliptic. Let C be a hyperelliptic curve of genus g over a field F of characteristic not equal to 2. Let B ⊂ C(F̄) be the set of fixed points of the hyperelliptic involution of C. The cardinality of B is 2g + 2. Let J be the Jacobian variety of C. Let G be the set of subsets of even cardinality of B modulo the equivalence relation defined by S_{1} ∼ S_{2} if S_{1} = S_{2} or S_{1} = B − S_{2} (= the complement of S_{2}). Denote the equivalence class of S ⊂ B by [S]. Define a binary operation on G by

[S_{1}] ∘ [S_{2}] = [(S_{1} ∪ S_{2}) − (S_{1} ∩ S_{2})].

Then G is an abelian group isomorphic to (ℤ/2ℤ)^{2g} (the identity is [∅]). There is an obvious action of the absolute Galois group G_{F} on G, induced from the action on B.

Theorem 7.7. There is a canonical isomorphism J[2] ≅ G of Galois modules.

Proof. Follows from Lemma 2.4 in [36], page 3.32. □

Now let F = 𝔽_{q}(T) with q odd. Let f(T) be a monic square-free polynomial of degree 3. Let 𝔫 be the ideal in A generated by f(T). The Drinfeld modular curve X_{0}(𝔫)_{F} is hyperelliptic with the Atkin-Lehner involution W_{𝔫} being the hyperelliptic involution; see [49]. Let e ∈ 𝔽_{q}^{×} be a non-square. Denote K_{1} = F (\sqrt{f(T)}), K_{2} = F (\sqrt{ef(T)}), O_{1} = A[\sqrt{f(T)}], O_{2} = A[\sqrt{ef(T)}]. Note that since ∞ does not split in K_{i}/F, O_{i} is a maximal order in K_{i} (i = 1, 2).

Theorem 7.8. The fixed points of W_{𝔫} on X_{0}(𝔫)_{F} correspond to the isomorphism classes of Drinfeld modules with complex multiplication by O_{1} or O_{2}.

Proof. See (3.5) in [10]. □

Theorem 7.9. Let K be an imaginary quadratic extension of F, i.e., ∞ does not split in K/F. Let O be the integral closure of A in K. Let H be the Hilbert class field of K, i.e., the maximal abelian unramified extension of K in which ∞ splits completely.

(1) Gal(H/K) ≅ Pic(O).

(2) There is a unique irreducible monic polynomial H_{K}(z) ∈ A[z] whose roots are the j-invariants of various non-isomorphic rank-2 Drinfeld A-modules over ℂ_{∞} with CM by O.

(3) The degree of H_{K}(z) is the class number of O, and H is its splitting field.

(4) The field F′ = F [z]/H_{K}(z) is linearly disjoint from K, and H is the composition of F′ and K.

Proof. Follows from Corollary 2.5 in [8]. □

Denote H_{i}(z) = H_{K_{i}}(z), F_{i} = F [z]/H_{i}(z), h_{i} = #Pic(O_{i}), and H_{i} the Hilbert class field of K_{i} (i = 1, 2). Let B be the set of fixed points of W_{𝔫} on X_{0}(𝔫). Since the action of G_{F} on X_{0}(𝔫)(F̄) commutes with the action of W_{𝔫}, the set B is stable under the action of G_{F}. The previous two theorems imply that B can be identified with the set of roots of the polynomial H_{1}(z)H_{2}(z) compatibly with the action of G_{F}. Let J := J_{0}(𝔫).

Lemma 7.10. If f(T) is irreducible, then J(F)[2] = 0.

Proof. Let X be the smooth, projective curve over 𝔽_{q} with function field K_{1}. It is well-known that there is an exact sequence

0 → Jac_{X}(𝔽_{q}) → Pic(O_{1}) → ℤ/d_{∞}ℤ → 0,

where d_{∞} is the degree of ∞ on X. Since f(T) has degree 3, ∞ ramifies in K_{1}/F, so d_{∞} = 1 and X is isomorphic to the elliptic curve E_{1} defined by the equation Y^{2} = f(T). Thus, Pic(O_{1}) ≅ E_{1}(𝔽_{q}). Similarly, Pic(O_{2}) ≅ E_{2}(𝔽_{q}), where E_{2} is defined by Y^{2} = ef(T). In particular,

h_{1} + h_{2} = #E_{1}(𝔽_{q}) + #E_{2}(𝔽_{q}) = 2q + 2.

The last equality follows from the observation that for any α ∈ 𝔽_{q} either f(α) = 0, in which case we get one 𝔽_{q}-rational point on both E_{1} and E_{2}, or exactly one of f(α), ef(α) is a square in 𝔽_{q}^{×}, in which case we get two 𝔽_{q}-rational on one of the elliptic curves and no points on the other. Now suppose f(T) is irreducible. Then f(T) has no 𝔽_{q}-rational roots, and therefore E_{1}(𝔽_{q})[2] = O. This implies that h_{1} and h_{2} are both odd. Denote the set of roots of H_{1} (resp. H_{2}) by B_{1} (resp. B_{2}), so that B = B_{1} ∪ B_{2}. Note that B_{1} and B_{2} are stable under the action of G_{F}, but have no non-trivial G_{F} stable subsets. Since #B_{1} and #B_{2} are odd, by Mumford’s theorem the only possibility for having an F -rational 2-torsion on J is to have a subset S ⊂ B of order q + 1 such that σS = S or σS = B − S for any σ ∈ G_{F}. Denote S_{1} = S ∩ B_{1}. Without loss of generality we can assume that S_{1} ≠ ∅, B_{1}. We must have σS_{1} = S_{1} or σS_{1} = B_{1} − S_{1} for any σ ∈ G_{F}. But #S_{1} and #(B_{1} − S_{1}) have different parity, and therefore σS_{1} = S_{1} for any σ ∈ G_{F}, which is a contradiction. □

Remark 7.11. Lemma 7.10 also follows from Theorem 1.2 in [38]. Indeed, if f(T) is irreducible, then according to [38], J(F)_{tor} ≅ ℤ/(q^{2} + q + 1)ℤ, so J(F)[2] = 0.

Lemma 7.12. If f(T) decomposes into a product f_{1}(T)f_{2}(T), where f_{2}(T) is irreducible of degree 2, then J(F)[2] ≅ ℤ/2ℤ ⊕ ℤ/2ℤ.

Proof. We retain the notation of the proof of Lemma 7.10. The first part of the proof of that lemma implies that Pic(O_{i})[2] ≅ ℤ/2ℤ, since both elliptic curves E_{1} and E_{2} have exactly one non-trivial 𝔽_{q}-rational 2-torsion point corresponding to (α, 0), where f_{1}(α) = 0. In particular, h_{1} and h_{2} are even, and H_{i}/K_{i} has a unique quadratic subextension. We conclude that [B_{1}] = [B_{2}] is F -rational. The other F -rational 2-torsion points on J are in bijection with the disjoint decompositions B_{1} = R_{1} ∪ R_{2} and B_{2} = R′_{1} ∪ R′_{2} such that #R_{1} = #R_{2}, #R′_{1} = #R′_{2}, and for any σ ∈ G_{F} either

σR_{1} = R_{1} and σR′_{1} = R′_{1}

or

σR_{1} = R_{2} and σR′_{1} = R′_{2}.

In that case, P = [R_{1} ∪ R′_{1}] = [R_{2} ∪ R′_{2}] and Q = [R_{1} ∪ R′_{2}] = [R_{2} ∪ R′_{1}] give two distinct non-trivial 2-torsion points on J which are rational over F. Note that the subgroup generated by P, Q and [B_{1}] is isomorphic to ℤ/2ℤ ⊕ ℤ/2ℤ. On the other hand, such disjoint decompositions are in bijection with the quadratic extensions L of F which simultaneously embed into both F_{1} and F_{2}. Note that over a quadratic subextension L of F_{1}/F the polynomial H_{1} decomposes into a product G_{1}(z)G_{2}(z) of two irreducible polynomials in L[z] of the same degree. In that case, R_{1} and R_{2} are the sets of roots of G_{1} and G_{2}, respectively. Now if there are two distinct quadratic subextensions L and L′ of F_{1}, then H_{1} = F_{1}K_{1} also contains two distinct quadratic subextensions. As we indicated above, this is not the case, hence there is at most one L. Consider L = F (\sqrt{f_{2}}). Since f_{2} is monic of degree 2, ∞ splits in L/F. Since the only place that ramifies in L/F is the place corresponding to f_{2}, which also ramifies in K/F with the same ramification index, LK is a quadratic subextension of H_{1}/K. Hence the composition LF_{1} is a subfield of H_{1}, and if L and F_{1} are linearly disjoint over F, then by comparing the degrees we see that LF_{1} = H_{1}. Since H_{1}(z) has even degree, F_{1} embeds into the completion F_{∞}. The same is true for L. Thus, if LF_{1} = H_{1}, then H_{1} embeds into F_{∞}, which is not the case as K/F is imaginary quadratic. We conclude that L and F_{1} cannot be linearly disjoint, and therefore L embeds into F_{1}. The same argument works also with F_{2}, and this finishes the proof of the lemma. □

Theorem 7.13. C = T ≅ ℤ/(q + 1)ℤ × ℤ/(q^{2} + 1)ℤ.

Proof. As we already discussed, it is enough to show that C_{2} = T_{2} when q is odd. By Example 6.12,

C = ⟨c_{x}⟩ ⊕ ⟨c_{y}⟩ ≅ ℤ/(q + 1)ℤ ⊕ ℤ/(q^{2} + 1)ℤ,

and c_{1} = c_{x} + c_{y}. The component group Φ_{∞} (for the case 𝔫 = xy) and the canonical specialization map ℘_{∞} : C → Φ_{∞} are computed in [41, Thm. 5.5]:

Φ_{∞} = Φ_{∞}(𝔽_{∞}) ≅ ℤ/Nℤ, ℘_{∞}(c_{x}) = q^{2} + 1, ℘_{∞}(c_{y}) = −q(q + 1).

In particular, if we denote C^{0} = ker(℘_{∞} : C → Φ_{∞}), then C^{0} ≅ ℤ/2ℤ when q is odd. On the other hand, by Lemma 7.12, T [2] ≅ ℤ/2ℤ × ℤ/2ℤ. Hence C[2] = T [2]. Let T^{0} := ker(℘_{∞} : T → Φ_{∞}). As we discussed at the beginning of this section, T^{0} is a subgroup of (ℤ/(q − 1)ℤ)^{⊕q} annihilated by N. We have

gcd((q − 1), N) = \begin{cases}2 & if q ≡ 3 mod 4 \\ 4 & if q ≡ 1 mod 4. \end{cases}

Assume q ≡ 3 mod 4. Then T^{0} ⊂ T [2] = C[2]. This implies T^{0} = C^{0} ≅ ℤ/2ℤ, and is generated by

c := \frac{(q + 1)}{2}c_{x} + \frac{(q^{2} + 1)}{2}c_{y}.

In this case ℘_{∞} is injective on ⟨c_{1}⟩, and ℘_{∞}(c_{1}) generates ℘_{∞}(C). If T_{2} ≠ C_{2}, then there is an element t ∈ T such that 2℘_{∞}(t) = ℘_{∞}(c_{1}). Thus, 2t = c_{1} + c or 2t = c_{1}. We know from the proof of Proposition 6.7 that ℘_{y}(c_{y}) = 0, and ℘_{y}(c_{x}) generates Φ_{y}(𝔽_{y}) ≅ Φ_{y} ≅ ℤ/(q + 1)ℤ. If 2t = c_{1}, then 2℘_{y}(t) generates Φ_{y}. Since 2 divides q + 1, the multiplication by 2 map is not surjective on Φ_{y}, so we get a contradiction. If 2t = c_{1} + c, then 2℘_{y}(t) = ℘_{y}(c_{x}) + \frac{(q+1)}{2}℘_{y}(c_{x}), which is still a generator of Φ_{y}, and we again arrive at a contradiction. Now assume q ≡ 1 mod 4. In this case

C_{2} ≅ (ℤ/(q + 1)ℤ)_{2} × (ℤ/(q^{2} + 1)ℤ)_{2} = ℤ/2ℤ × ℤ/2ℤ ≅ C[2]

is generated by \frac{q+1}{2}c_{x} and \frac{q^{2}+1}{2}c_{y}. If T ≠ C, then there is t ∈ T of order 4 such that 2t ∈ C[2]. If 2t = c or 2t = \frac{q+1}{2}c_{x}, then applying ℘_{y} we see that 2℘_{y}(t) ≠ 0. On the other hand, ℘_{y}(t) ∈ (Φ_{y})_{2} ≅ ℤ/2ℤ, so 2℘_{y}(t) = 0, which is a contradiction. Finally, suppose 2t = \frac{q^{2}+1}{2}c_{y}, which implies 2℘_{x}(t) ≠ 0. Since t is a rational point, ℘_{x}(t) ∈ Φ_{x}(𝔽_{x})_{2}. On the other hand, by Proposition 5.4, Φ_{x}(𝔽_{x})_{2} ≅ (ℤ/(q^{2} +1)ℤ)_{2} ≅ ℤ/2ℤ, which again leads to a contradiction. (Note that Φ_{x} ≅ ℤ/(q +1)(q^{2} +1)ℤ, so here we cannot just rely on Theorem 5.3.) □

Corollary 7.14. J(F) ≅ ℤ/(q + 1)ℤ × ℤ/(q^{2} + 1)ℤ. For any ℓ ≠ p, the ℓ-primary part of the Tate-Shafarevich group � (J) is trivial.

Proof. Denote by L(J, s) the L-function of J; see [26] for the definition. Let f ∈ H_{0}(𝔫, ℂ) be an eigenform for 𝕋(𝔫), normalized by f^{∗}(1) = 1. The Lfunction of f is the sum

L(f, s) = ∑_{𝔪 pos. div.} f^{∗}(𝔪)q^{−s·deg(𝔪)}

over all positive divisors on ℙ_{𝔽_{q}}^{1}; here s ∈ ℂ. Drinfeld’s fundamental result [6, Thm. 2] implies that L(J, s) = ∏_{f} L(f, s), where the product is over normalized 𝕋(xy)-eigenforms in H_{0}(xy, ℂ). It is known that L(f, s) is a polynomial in q^{−s} of degree deg(𝔫) − 3; cf. [54, p. 227]. Thus, L(J, s) = 1. From the main theorem in [26] we conclude that J(F) = T , � (J) is a finite group, and the Birch and Swinnerton-Dyer conjecture holds for J. The claim J(F) ≅ ℤ/(q + 1)ℤ × ℤ/(q^{2} + 1)ℤ follows from Theorem 7.13. The ℓ-primary part of the Birch and Swinnerton-Dyer formula becomes the equality

1 = \frac{(#� (J)_{ℓ})(#Φ_{x}(𝔽_{x})_{ℓ})(#Φ_{y}(𝔽_{y})_{ℓ})(#Φ_{∞}(𝔽_{∞})_{ℓ})}{(#T_{ℓ})^{2}}.

We know all entries of this formula from Theorem 7.13 and its proof, except #� (J)_{ℓ}. This implies that � (J)_{ℓ} = 1. □

8. Kernel of the Eisenstein ideal

8.1. Shimura subgroup. Let 𝔫 ⊲ A be a non-zero ideal. Consider the subgroup Γ_{1}(𝔫) of GL_{2}(A) consisting of matrices

Γ_{1}(𝔫) = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL_{2}(A) | a ≡ 1 mod 𝔫, c ≡ 0 mod 𝔫\} .

The quotient Γ_{1}(𝔫) \ Ω is the rigid-analytic space associated to a smooth affine curve Y_{1}(𝔫)_{F_{∞}} over F_{∞}; cf. (3.1). This curve is the modular curve of isomorphism classes of pairs (ϕ, P), where ϕ is a Drinfeld A-module of rank 2 and P ∈ ϕ[𝔫] is an element of exact order 𝔫. Let Y_{1}(𝔫)_{F} be the canonical model of Y_{1}(𝔫)_{F_{∞}} over F, and X_{1}(𝔫)_{F} be the smooth projective curve containing Y_{1}(𝔫)_{F} as a Zariski open subvariety. Denote by J_{1}(𝔫) the Jacobian variety of X_{1}(𝔫)_{F}. To simplify the notation, in this section we denote Γ := Γ_{0}(𝔫) and Δ :=

Γ_{1}(𝔫). The map w : Γ → (A/𝔫)^{×} given by (\begin{matrix}a & b \\ c & d \end{matrix}) ↦ a mod 𝔫 is a surjective

homomorphism whose kernel is Δ. Hence Δ is a normal subgroup of Γ and Γ/Δ ≅ (A/𝔫)^{×}. One deduces from the action of Γ on Ω ∪ ℙ^{1}(F) an action of Γ on X_{1}(𝔫)_{ℂ_{∞}}. The group Δ and the scalar matrices act trivially on X_{1}(𝔫)_{ℂ_{∞}}, hence we have an action of the group (A/𝔫)^{×}/𝔽_{q}^{×} on X_{1}(𝔫)_{ℂ_{∞}}. This implies that there is a natural morphism X_{1}(𝔫)_{ℂ_{∞}} → X_{0}(𝔫)_{ℂ_{∞}} which is a Galois covering with Galois group (A/𝔫)^{×}/𝔽_{q}^{×}. This covering is in fact defined over F, as in terms of the moduli problems it is induced by (ϕ, P) ↦ (ϕ, ⟨P ⟩), where ⟨P ⟩ is the order-𝔫 cyclic subgroup of ϕ generated by P. By the Picard functoriality we get a homomorphism π : J_{0}(𝔫) → J_{1}(𝔫) defined over F, whose kernel S(𝔫) is the Shimura subgroup of J_{0}(𝔫).

Definition 8.1. Let Q be the subgroup of (A/𝔫)^{×} generated by the elements which satisfy a^{2} − ta + κ = 0 mod 𝔫 for some t ∈ 𝔽_{q} and κ ∈ 𝔽_{q}^{×}. Denote U = (A/𝔫)^{×}/Q.

Theorem 8.2. The Shimura subgroup S(𝔫), as a group scheme over F_{∞}, is canonically isomorphic to the Cartier dual U^{∗} of U viewed as a constant group scheme.

Proof. Denote by Γ^{ab} the abelianization of Γ and let \overline{Γ} := Γ^{ab}/(Γ^{ab})_{tor} be the maximal abelian torsion-free quotient of Γ. The inclusion Δ ↪ Γ induces a homomorphism I : \overline{Δ} → \overline{Γ} (which is not necessarily injective). There is also a homomorphism V : \overline{Γ} → \overline{Δ}, the transfer map, such that I ∘ V : \overline{Γ} → \overline{Γ} is the multiplication by [Γ : Δ]/(q − 1); see [18, p. 71]. First, we note that the homomorphism V : \overline{Γ} → \overline{Δ} is injective with torsion-free cokernel. Indeed, by [18, p. 72], there is a commutative diagram

\xymatrix{\overline{Γ} \ar[r]^{j_{Γ}} \ar[d]^{V} & H_{0}(T , ℤ)^{Γ} \ar[d] \\ \overline{Δ} \ar[r]^{j_{Δ}} & H_{0}(T , ℤ)^{Δ}}

where the right vertical map is the natural injection. This last homomorphism obviously has torsion-free cokernel. Since by [17] j_{Γ} and j_{Δ} are isomorphisms, the claim follows. Next, by the results in Sections 6 and 7 of [18], there is a commutative diagram

\xymatrix{0 \ar[r] & \overline{Γ} \ar[r] \ar[d]^{V} & Hom(\overline{Γ}, ℂ_{∞}^{×}) \ar[r] \ar[d]^{I^{∗}} & J_{0}(𝔫) \ar[r] \ar[d]^{π} & 0 \\ 0 \ar[r] & \overline{Δ} \ar[r] & Hom(\overline{Δ}, ℂ_{∞}^{×}) \ar[r] & J_{1}(𝔫) \ar[r] & 0,}

where the top row is the uniformization in (4.3), the bottom row is a similar uniformization for J_{1}(𝔫) constructed in [18] for an arbitrary congruence group, and the middle vertical map I^{∗} is the dual of I : \overline{Δ} → \overline{Γ}. This diagram gives an exact sequence of group schemes

0 → (\overline{Γ}/I\overline{Δ})^{∗} → S(𝔫) → \overline{Δ}/V \overline{Γ}.

Since there are no non-trivial maps from an abelian variety to a discrete group, S(𝔫) is finite. On the other hand, by the previous paragraph, \overline{Δ}/V \overline{Γ} is torsionfree, so

S(𝔫) ≅ (\overline{Γ}/I\overline{Δ})^{∗}.

Denote by Γ^{c} the commutator subgroup of Γ. The fact that Γ/Δ is abelian implies Γ^{c} ⊂ Δ. Hence Γ/Δ ≅ Γ^{ab}/(Δ/Γ^{c}). It follows from Corollary 1 on page 55 of [50] that (Γ^{ab})_{tor} is generated by the images of finite order elements of Γ in Γ^{ab}. Since \overline{Γ} is the quotient of Γ^{ab} by the torsion subgroup (Γ^{ab})_{tor}, we conclude that \overline{Γ}/I\overline{Δ} is the quotient of Γ/Δ by the subgroup generated by the images of finite order elements of Γ. An element γ ∈ GL_{2}(A) has finite order if and only if Tr(γ) ∈ 𝔽_{q}. Therefore, if γ ∈ Γ has finite order then w(γ) satisfies the equation a^{2} − ta + κ = 0, where t = Tr(γ) ∈ 𝔽_{q} and κ = det(γ) ∈ 𝔽_{q}^{×}. Conversely, suppose ā ∈ (A/𝔫)^{×} satisfies ā^{2} − tā + κ = 0 for some t ∈ 𝔽_{q} and κ ∈ 𝔽_{q}^{×}. Fix some a ∈ A reducing to ā modulo 𝔫. Since a(t − a) ≡ κ (mod 𝔫), there exists c ∈ A such that

a(t − a) = κ + c𝔫. The matrix γ = (\begin{matrix}a & 1 \\ c𝔫 & t − a \end{matrix}) has determinant κ and trace

t. It is clear that γ ∈ Γ is a torsion element, and w(γ) = ā. We conclude that \overline{Γ}/I\overline{Δ} ≅ (A/𝔫)^{×}/Q. □

Remark 8.3. The previous theorem is the function field analogue of Theorem 1 in [30].

Lemma 8.4. Assume 𝔫 is square-free, so that the order of (A/𝔫)^{×} is not divisible by p and S(𝔫) is étale over F_{∞}. The Shimura subgroup S(𝔫) extends to a finite flat subgroup scheme of J_{O_{∞}}^{0}.

Proof. From the proof of Theorem 8.2, S(𝔫) is canonically a subgroup of Hom(\overline{Γ}, ℂ_{∞}^{×}). It is easy to see that S(𝔫) actually lies in Hom(\overline{Γ}, (O_{∞}^{nr})^{×}). On the other hand, as is implicit in the proof of Corollary 2.11 in [11], there is a canonical isomorphism J^{0}(O_{∞}^{nr}) ≅ Hom(\overline{Γ}, (O_{∞}^{nr})^{×}). □

Proposition 8.5. Assume 𝔫 is square-free. The Shimura subgroup S(𝔫), as a group scheme over F, is an étale group scheme whose Cartier dual is canonically isomorphic to U viewed as a constant group scheme. The endomorphisms T_{𝔭} − |𝔭| − 1 and W_{𝔫} + 1 of J_{0}(𝔫) annihilate S(𝔫); here 𝔭 ⊲ A is any prime not dividing 𝔫.

Proof. The covering π : X_{1}(𝔫)_{F} → X_{0}(𝔫)_{F} can ramify only at the elliptic points and the cusps of X_{0}(𝔫)_{F}. (By definition, an elliptic point on X_{0}(𝔫)_{ℂ_{∞}} is the image of z ∈ Ω whose stabilizer in GL_{2}(A) is strictly larger than 𝔽_{q}^{×}.) The proof of Theorem 8.2 shows that U is the Galois group of the maximal abelian unramified covering X_{F} → X_{0}(𝔫)_{F} through which π factorizes. Now [30, Prop. 6] implies that S(𝔫) as a group scheme over F is isomorphic to Hom(U, F̄^{×}). The Jacobian J_{0}(𝔫) has good reduction at 𝔭. Since S(𝔫) has order coprime to p and is unramified at 𝔭, the reduction map injects S(𝔫) into J_{0}(𝔫)(\overline{𝔽}_{𝔭}). Let Frob_{𝔭} be the Frobenius endomorphism of J_{0}(𝔫)_{𝔽_{𝔭}}. The Hecke operator T_{𝔭} satisfies the Eichler-Shimura relation:

Frob_{𝔭}^{2} − T_{𝔭} · Frob_{𝔭} + |𝔭| = 0.

Since S(𝔫)^{∗} is constant, Frob_{𝔭} acts on S(𝔫) by multiplication by |𝔭|. Therefore, the endomorphisms |𝔭|(T_{𝔭} − |𝔭| − 1) annihilates S(𝔫). Since the reduction map commutes with the action of Hecke algebra and the multiplication by |𝔭| is an automorphism of S(𝔫), we conclude that T_{𝔭} − |𝔭| − 1 annihilates S(𝔫).

Note that for (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ we have

(\begin{matrix}0 & −1 \\ 𝔫 & 0 \end{matrix}) (\begin{matrix}a & b \\ c & d \end{matrix}) (\begin{matrix}0 & 1/𝔫 \\ −1 & 0 \end{matrix}) = (\begin{matrix}d & −c/𝔫 \\ −b𝔫 & a \end{matrix}) .

Since d, up to an element of 𝔽_{q}^{×}, is the inverse of a modulo 𝔫, this calculation shows that W_{𝔫} acts on the group (Γ/Δ)/𝔽_{q}^{×} by u ↦ u^{−1}, so it acts as −1 on S(𝔫). □

Theorem 8.6. Assume 𝔫 is square-free, and let 𝔫 = 𝔭_{1} ⋯ 𝔭_{s} be its prime decomposition. As an abelian group, S(𝔫) is isomorphic to

∏_{i=1}^{s} (ℤ/\frac{|𝔭_{i}|−1}{q−1}ℤ) ,  if some 𝔭_{i} has odd degree
∏_{i=1}^{s} (ℤ/\frac{|𝔭_{i}|−1}{q^{2}−1}ℤ) ,  if q is even and all 𝔭_{i} have even degrees
∏_{i=1}^{s} (ℤ/\frac{2(|𝔭_{i}|−1)}{q^{2}−1}ℤ) /diag (ℤ/2ℤ) , if q is odd and all 𝔭_{i} have even degrees

where diag : ℤ/2ℤ → ∏_{i=1}^{s} (ℤ/\frac{2(|𝔭_{i}|−1)}{q^{2}−1}ℤ) is the diagonal embedding.

Proof. First, we claim that the image of Q under the isomorphism (A/𝔫)^{×} ≅ ∏_{i=1}^{s} 𝔽_{𝔭_{i}}^{×} given by

a ↦ (a mod 𝔭_{1}, ⋯, a mod 𝔭_{s})

contains the subgroup ∏_{i=1}^{s} 𝔽_{q}^{×}. Indeed, let α ∈ 𝔽_{q}^{×} be arbitrary, and consider (1, …, α, …, 1) with α in the ith position. This element is in the image of Q since a ∈ (A/𝔫)^{×} mapping to it satisfies the quadratic equation (x−α)(x−1) = 0 modulo 𝔫. Suppose a ∈ Q is a zero of f(x) := x^{2} − tx + κ. Then a satisfies the same equation modulo 𝔭_{i}. If deg(𝔭_{i}) is odd, then f(x) must be reducible over 𝔽_{q}. This implies that the image of a in 𝔽_{𝔭_{i}}^{×} lies in the subfield 𝔽_{q}. On the other hand, since a mod 𝔭_{j} (1 ≤ j ≤ s) satisfies the same reducible quadratic equation, the image of a in 𝔽_{𝔭_{j}} also lies in its subfield 𝔽_{q}, and we conclude that Q ≅ ∏_{i=1}^{s} 𝔽_{q}^{×}. Now suppose all 𝔭_{i}’s have even degrees. Let α ∈ 𝔽_{q^{2}}^{×} be arbitrary. The elements (α, …, α) and (α, …, α^{q}, …, α), with α^{q} in the ith position (1 ≤ i ≤ s), are in the image of Q. Indeed, a ∈ (A/𝔫)^{×} which maps to such an element satisfies (x − α)(x − α^{q}) = 0, and this polynomial has coefficients in 𝔽_{q}. Therefore, (1, …, α^{q−1}, …, 1) ∈ Q. Since α^{q+1} ∈ 𝔽_{q}^{×}, we get that (1, …, α^{2}, …, 1) ∈ Q. Thus, diag(𝔽_{q^{2}}^{×})·∏_{i=1}^{s}(𝔽_{q^{2}}^{×})^{2} is a subgroup of Q, where (𝔽_{q^{2}}^{×})^{2} = {α^{2} | α ∈ 𝔽_{q^{2}}^{×}} and diag : 𝔽_{q^{2}}^{×} ↪ ∏_{i=1}^{s} 𝔽_{𝔭_{i}}^{×} is the diagonal embedding. If f(x) is reducible, then a ∈ ∏_{i=1}^{s} 𝔽_{q}^{×}. If f(x) is irreducible, then a = (α_{i_{1}}, …, α_{i_{s}}) with α_{i_{1}}, …, α_{i_{s}} ∈ {α, α^{q}}, where α, α^{q} are the roots of f(x). We can write

a = (α_{i_{1}}, …, α_{i_{1}}) · (1, \frac{α_{i_{2}}}{α_{i_{1}}} , …, \frac{α_{i_{s}}}{α_{i_{1}}} ) .

The first element in this product is in diag(𝔽_{q^{2}}^{×}) and the second is in ∏_{i=1}^{s}(𝔽_{q^{2}}^{×})^{2} since each α_{i_{j}}/α_{i_{1}} is either 1 or α^{±(q−1)}. We conclude that diag(𝔽_{q^{2}}^{×}) · ∏_{i=1}^{s}(𝔽_{q^{2}}^{×})^{2} = Q. Since S(𝔫) is isomorphic to (A/𝔫)^{×}/Q, the claim follows. □

Example 8.7. If 𝔭 ⊲ A is prime, then S(𝔭) is cyclic of order \frac{|𝔭|−1}{q−1} (resp. \frac{|𝔭|−1}{q^{2}−1}) is deg(𝔭) if odd (resp. even). This is also proved in [38] by a different method.

Example 8.8. S(xy) is cyclic of order q + 1.

Example 8.9. Assume 𝔭 and 𝔮 are two distinct primes of degree 2. Then S(𝔭𝔮) ≅ ℤ/2ℤ (resp. 0) if q is odd (resp. even).

Definition 8.10. Let 𝔫 ⊲ A be a non-zero ideal and denote J = J_{0}(𝔫). The kernel of the Eisenstein ideal J[𝔈(𝔫)] is the intersection of the kernels of all elements of 𝔈(𝔫) acting on J(F̄). Given a prime ℓ ≠ p, the action of 𝕋(𝔫) on J induces an action on J[ℓ^{n}] (n ≥ 1) and the ℓ-divisible group J_{ℓ} := lim_{→} J[ℓ^{n}], so one can also define J_{ℓ}[𝔈(𝔫)]. From the proof of Theorem 6.1, we see that J[𝔈(𝔫)] is a finite group scheme over F, as J[𝔈(𝔫)] ⊆ J[η_{𝔭}] for any 𝔭 ∤ 𝔫, where η_{𝔭} = T_{𝔭} − |𝔭| − 1. By Lemma 7.1 and Proposition 8.5, if 𝔫 is square-free then S(𝔫)_{ℓ} and C(𝔫)_{ℓ} are in J_{ℓ}[𝔈(𝔫)].

Definition 8.11. We say that a group scheme H over the base S is μ-type if it is finite, flat and its Cartier dual is a constant group scheme over S; H is pure if it is the direct sum of a constant and μ-type group schemes; H is admissible if it is finite, étale and has a filtration by groups schemes such that the successive quotients are pure.

Lemma 8.12. Assume 𝔫 = 𝔭𝔮 is a product of two distinct primes. Let s_{𝔭,𝔮} = gcd(|𝔭|+1, |𝔮|+1). If ℓ does not divide q(q −1)s_{𝔭,𝔮}, then J_{ℓ}[𝔈(𝔫)] is unramified except possibly at 𝔭 and 𝔮, and there is an exact sequence of group schemes over F

0 → C(𝔫)_{ℓ} → J_{ℓ}[𝔈(𝔫)] → M_{ℓ} → 0,

where M_{ℓ} is μ-type and contains S(𝔫)_{ℓ}.

Proof. Since C(𝔫)_{ℓ} is fixed by G_{F} and is invariant under the action of 𝕋(𝔫), the quotient M_{ℓ} := J_{ℓ}[𝔈(𝔫)]/C(𝔫)_{ℓ} is naturally a 𝕋(𝔫) × G_{F} -module . From the uniformization sequence (4.3) and the isomorphism (4.4), for any n ≥ 1 we obtain the exact sequence of 𝕋(𝔫) × G_{F_{∞}}-modules

0 → D[ℓ^{n}] → J[ℓ^{n}] → H_{00}(𝔫, ℤ/ℓ^{n}ℤ) → 0,

where D := Hom(\overline{Γ}, ℂ_{∞}^{×}). Considering the parts annihilated by 𝔈(𝔫), we obtain the exact sequence

0 → D[ℓ^{n}, 𝔈(𝔫)] → J[ℓ^{n}, 𝔈(𝔫)] → E_{00}(𝔫, ℤ/ℓ^{n}ℤ).

From the proof of Theorem 7.3 we know that if ℓ does not divide q(q−1)s_{𝔭,𝔮} and n is sufficiently large, then C(𝔫)_{ℓ} ⊆ J[ℓ^{n}, 𝔈(𝔫)] and C(𝔫)_{ℓ} maps isomorphically onto E_{00}(𝔫, ℤ/ℓ^{n}ℤ). Hence for n ≫ 0 the above sequence is exact also on the right:

0 → D[ℓ^{n}, 𝔈(𝔫)] → J[ℓ^{n}, 𝔈(𝔫)] → C(𝔫)_{ℓ} → 0.

Since the map of 𝕋(𝔫) × G_{F_{∞}}-modules J[ℓ^{n}, 𝔈(𝔫)] → C(𝔫)_{ℓ} has a retraction, we get the splitting J_{ℓ}[𝔈(𝔫)] ≅ D_{ℓ}[𝔈(𝔫)] ⊕ C(𝔫)_{ℓ} over F_{∞}. This shows that M_{ℓ} ≅ D_{ℓ}[𝔈(𝔫)] is μ-type over F_{∞}, and J_{ℓ}[𝔈(𝔫)] is unramified at ∞. Since S(𝔫)_{ℓ} ⊆ D_{ℓ}[𝔈(𝔫)] (cf. Lemma 8.4), we see that S(𝔫)_{ℓ} is a subgroup scheme of M_{ℓ}. By the Néron-Ogg-Shafarevich criterion J_{ℓ}[𝔈(𝔫)] is unramified at all finite places except possibly at 𝔭 and 𝔮. Let 𝔩 ⊲ A be any prime not equal to 𝔭 or 𝔮. Since T_{𝔩} acts on J_{ℓ}[𝔈(𝔫)] by multiplication by |𝔩| + 1, the Eichler-Shimura congruence relation implies that the action of Frob_{𝔩} on J_{ℓ}[𝔈(𝔫)] satisfies the relation (Frob_{𝔩}−1)(Frob_{𝔩}−|𝔩|) = 0. Now one can use Mazur’s argument [33, Prop. 14.1] to show that J_{ℓ}[𝔈(𝔫)] is admissible over U = ℙ_{𝔽_{q}}^{1} − 𝔭 − 𝔮; cf. the proof of Proposition 10.8 in [38]. Since the quotient map J_{ℓ}[𝔈(𝔫)] → M_{ℓ} is compatible with the action of G_{F} and 𝕋(𝔫), M_{ℓ} is also admissible over U = ℙ_{𝔽_{q}}^{1} − 𝔭 − 𝔮. On the other hand, M_{ℓ} is μ-type over F_{∞}, so all Jordan-Hölder components of M_{ℓ} over U must be isomorphic to μ_{ℓ}. We say that 𝔩_{1}, 𝔩_{2} is a pair of good primes, if 𝔩_{i} ≠ 𝔭, 𝔮, |𝔩_{i}| − 1 is not divisible by ℓ, and the images of (𝔩_{1}, 𝔩_{1}) and (𝔩_{2}, 𝔩_{2}) in (𝔽_{𝔭}^{×}/(𝔽_{𝔭}^{×})^{ℓ} × 𝔽_{𝔮}^{×}/(𝔽_{𝔮}^{×})^{ℓ})/𝔽_{q}^{×} generate this group; here (𝔽_{𝔭}^{×})^{ℓ} is the subgroup of 𝔽_{𝔭}^{×} consisting of ℓth powers, and 𝔽_{q}^{×} is embedded diagonally into 𝔽_{𝔭}^{×} × 𝔽_{𝔮}^{×}. The Chebotarev density theorem shows that a pair of good primes exists. The operator (Frob_{𝔩_{i}} − |𝔩_{i}|)(Frob_{𝔩_{i}} − 1) annihilates M_{ℓ}. On the other hand, since the semi-simplification of M_{ℓ} is isomorphic to (μ_{ℓ})^{n} for some n and ℓ does not divide |𝔩_{i}|−1, the operator Frob_{𝔩_{i}} −1 must be invertible on M_{ℓ}. Therefore, M_{ℓ} is annihilated by Frob_{𝔩_{i}} − |𝔩_{i}|. This implies that M_{ℓ} is μ-type over F_{𝔩_{i}}. Finally, one can argue as in Proposition 10.7 in [38] to conclude that M_{ℓ} is μ-type over F. □

Remark 8.13. When 𝔫 = 𝔭 is prime and ℓ does not divide q −1, then J_{ℓ}[𝔈(𝔭)] = C(𝔭)_{ℓ} ⊕ S(𝔭)_{ℓ}; see [38]. As we will see in the next section, for a composite level, M_{ℓ} can be larger than S(𝔫)_{ℓ} and the sequence in Lemma 8.12 need not split over F.

8.2. Special case. Let 𝔭 ⊲ A be a prime of degree 3. Then

(1) The rational torsion subgroup T (𝔭) of J_{0}(𝔭) is equal to the cuspidal divisor group C(𝔭) ≅ ℤ/(q^{2} + q + 1)ℤ;

(2) The maximal μ-type étale subgroup scheme of J_{0}(𝔭) is the Shimura subgroup S(𝔭) ≅ (ℤ/(q^{2} + q + 1)ℤ)^{∗};

(3) The kernel J[𝔈(𝔭)] is everywhere unramified and has order (q^{2}+q+1)^{2}. Here (1) and (2) follow from the main results in [38], and (3) follows from [38] and [40]. We proved that (see Theorem 7.13)

T (xy) = C(xy) ≅ ℤ/(q + 1)ℤ × ℤ/(q^{2} + 1)ℤ, so the analogue of (1) is true for 𝔫 = xy, although C(xy) is not cyclic if q is odd. Interestingly, even for small composite levels such as 𝔫 = xy, the analogues of (2) and (3) are no longer true. For simplicity, we will omit the level xy from notation. First of all, J_{ℓ}[𝔈] can be ramified. To see this, let q = 2^{s} with s odd, x = T + 1, y = T^{2} + T + 1. Consider the elliptic curve E over F given by the Weierstrass equation

E : Y^{2} + T XY + Y = X^{3} + X^{2} + T.

This curve embeds into J according to [41, Prop. 7.10]. There is an exact sequence of G_{F} -modules

0 → ℤ/5ℤ → E[5] → μ_{5} → 0.

The component group of E at y is trivial, so E[5] is ramified at y and the above sequence does not split over F. On the other hand, from the Eichler-Shimura congruence relations we see that E[5] ⊆ J_{5}[𝔈]. Thus, J_{5}[𝔈] is ramified at y. Next, the Shimura subgroup S ≅ (ℤ/(q + 1)ℤ)^{∗} has smaller order than the cuspidal divisor group, and S is not the maximal μ-type étale subgroup scheme of J when q is odd. Indeed, C[2] ≅ ℤ/2ℤ × ℤ/2ℤ is μ-type but is not cyclic. In the remainder of this section we will show that even though (2) and (3) fail for the xy-level, these properties do not fail “too much”.

Proposition 8.14. Assume q is odd. Then S_{2} + C[2] ≅ S_{2} × ℤ/2ℤ is the maximal 2-primary μ-type subgroup of J(F̄).

Proof. First, we show that the canonical specialization map ℘_{y} : S → Φ_{y} is an isomorphism. Since S and Φ_{y} have the same order, it is enough to show that the induced morphism J_{0}(xy)_{\overline{𝔽}_{y}}^{0} → J_{1}(xy)_{\overline{𝔽}_{y}}^{0} on the connected components of the identity of the special fibres of the Néron models of J_{0}(xy) and J_{1}(xy) at y is injective. Here we argue as in Proposition 11.9 in [33]. Let Gr_{0} be the dual graph of X_{0}(xy)_{\overline{𝔽}_{y}}; this graph has two vertices corresponding to the two irreducible components of X_{0}(xy)_{\overline{𝔽}_{y}} joined by q + 1 edges corresponding to the supersingular points. Let Gr_{1} be the dual graph of the special fibre of the model of X_{1}(xy) over O_{y} constructed in [51]. The underlying reduced subscheme X_{1}(xy)_{\overline{𝔽}_{y}}^{red} has two irreducible components intersecting at the supersingular points. Since the Jacobian J_{0}(xy) has toric reduction at y, there is a canonical isomorphism (cf. [3, p. 246])

J_{0}(xy)_{\overline{𝔽}_{y}}^{0} ≅ H^{1}(Gr_{0}, ℤ) ⊗ 𝔾_{m,\overline{𝔽}_{y}}.

Similarly, the toric part of J_{1}(xy)_{\overline{𝔽}_{y}}^{0} is canonically isomorphic to H^{1}(Gr_{1}, ℤ) ⊗ 𝔾_{m,\overline{𝔽}_{y}}. There is an obvious map Gr_{1} → Gr_{0} and our claim reduces to showing that this map induces a surjection H_{1}(Gr_{1}, ℤ) → H_{1}(Gr_{0}, ℤ). This is clear from the description of the graphs Gr_{0} and Gr_{1}. Let M_{2} be the maximal 2-primary μ-type subgroup of J(F̄). It is clear that S_{2} + T [2] ⊆ M_{2} and M_{2}[2] = T [2]. Similar to the notation in the proof of Theorem 7.13, let M_{2}^{0} denote the intersection of M_{2} with Hom(\overline{Γ}, ℂ_{∞}^{×}).

⋮ ; \begin{matrix}φ_{q} \\ φ_{1} \end{matrix} ; φ_{2} ; φ_{q−1} ; Figure 5. (Γ_{0}(xy) \ 𝒯 )^{0}

Note that S_{2} ⊆ M_{2}^{0}. Moreover, we must have S[2] = M_{2}^{0}[2] since M_{2}^{0}[2] ⊆ T^{0}[2] = C^{0}[2] = S[2]. Thus, if S_{2} ≠ M_{2}^{0}, then there exists P ∈ M_{2}^{0} such that 2P generates S_{2} ≅ (ℤ/(q + 1)ℤ)_{2}^{∗}. Since ℘_{y} : S → Φ_{y} is an isomorphism, 2℘_{y}(P) generates ℤ/(q + 1)ℤ. On the other hand, 2 divides q + 1, so we get a contradiction. We conclude that there is an exact sequence

0 → S_{2} → M_{2} → ℤ/2^{n}ℤ → 0

for some n ≥ 1, where ℤ/2^{n}ℤ is the image of M_{2} in Φ_{∞} ≅ ℤ/(q^{2} + 1)(q + 1)ℤ (see Corollary 8.15). Since M_{2}[2] = T [2] ≅ ℤ/2ℤ × ℤ/2ℤ, the above sequence splits as a sequence of abelian groups. Moreover, since S_{2} is G_{F} -invariant and M_{2}^{∗} is constant, the sequence splits as a sequence of F -groups schemes: M_{2} ≅ S_{2} ⊕ ℤ/2^{n}ℤ. If q ≡ 3 (mod 4), then ℤ/2^{n}ℤ is μ-type if and only if n ≤ 1, which implies M_{2} = S_{2} + T [2]. If q ≡ 1 (mod 4) and n ≥ 2, then T = C contains a point of order 4, which is a contradiction. □

The point of the previous proposition is that even though S_{2} is not the maximal 2-primary μ-type subgroup of J, it is not far from it. Next, we will show that under a reasonable assumption S_{ℓ} is the maximal ℓ-primary μ-type subgroup of J for any odd ℓ ≠ p, and the order of J_{ℓ}[𝔈] is (#C_{ℓ})^{2} for any ℓ ≠ p, which is of course very similar to (2) and (3). We start by showing that Φ_{∞} is annihilated by 𝔈. This almost follows from Theorem 5.5 in [41], since according to that theorem the canonical specialization map ℘_{∞} : C → Φ_{∞} is an isomorphism if q is even and has cokernel ℤ/2ℤ if q is odd. On the other hand, ℘_{∞} is 𝕋-equivariant and C is annihilated by 𝔈. Below we give an alternative direct argument which works for any characteristic and does not use the cuspidal divisor group C. Choose the cycles φ_{1}, ..., φ_{q} as a basis of H_{1}(Γ_{0}(xy) \ 𝒯 , ℤ) ≅ H_{0}(xy, ℤ); see Figure 5. We assume that all these cycles are oriented counterclockwise. Let ψ_{1}, ..., ψ_{q} be the dual basis of Hom(H_{0}(xy, ℤ), ℤ). The map ι in Theorem 5.5

sends φ ∈ H_{0}(xy, ℤ) to ∑_{i=1}^{q}(φ, φ_{i})ψ_{i}, where for 1 ≤ j ≤ i ≤ q

(φ_{i}, φ_{j}) = \begin{cases}q + 2, & if i = j = 1 or q, \\ 2, & if 1 < i = j < q, \\ −1, & if j = i − 1, \\ 0, & otherwise. \end{cases}

This follows from [11, Thm. 2.8] and a calculation of the stabilizers of edges of (Γ_{0}(xy) \ 𝒯 )^{0} (see the proof of Proposition 2.21). Hence the component group Φ_{∞} can be explicitly described as follows: It is the quotient of the free abelian group ⨁_{i=1}^{q} ℤψ_{i} by the relations

(q + 2)ψ_{q} − ψ_{q−1} = 0,
−ψ_{i+1} + 2ψ_{i} − ψ_{i−1} = 0, for 1 ≤ i ≤ q − 1,
−ψ_{2} + (q + 2)ψ_{1} = 0.

From the last q − 1 relations we get

ψ_{i} = ((i − 1)q + i)ψ_{1}, 1 ≤ i ≤ q.

On the other hand, by the first relation, ψ_{q−1} = (q + 2)ψ_{q} = q^{2}(q + 2)ψ_{1}. Therefore q^{2}(q + 2)ψ_{1} = (q^{2} − q − 1)ψ_{1} and

(q^{2} + 1)(q + 1)ψ_{1} = 0.

We conclude that

Corollary 8.15. Φ_{∞} ≅ ℤ/(q^{2} + 1)(q + 1)ℤ is generated by the image of ψ_{1}.

Multiplication by N = (q^{2} + 1)(q + 1) on the exact sequence in Theorem 5.5 and the snake lemma give an embedding δ_{N} : Φ_{∞} ↪ H_{00}(xy, ℤ/Nℤ).

Lemma 8.16. Let ψ̄_{1} be the image of ψ_{1} in Φ_{∞}. Then δ_{N}(ψ̄_{1}) = −(q + 1)E_{x} + (q^{2} + 1)E_{y} + qE_{(x,y)}, the generator of E_{00}(xy, ℤ/Nℤ). In particular, the component group Φ_{∞} is annihilated by the Eisenstein ideal.

Proof. Note that

Nψ_{1} = ∑_{i=1}^{q} ((i − 1)q + i)ι(φ_{i}) ∈ Hom(H_{0}(xy, ℤ), ℤ).

Hence δ_{N}(ψ̄_{1}) = ∑_{i=1}^{q} ((i − 1)q + i)φ_{i} mod N ∈ H_{00}(xy, ℤ/Nℤ). Viewing the cycles φ_{1}, ..., φ_{q} as harmonic cochains in H_{0}(xy, ℤ), it is observed that

(i)

φ_{1}(a_{1}) = 1, φ_{1}(a_{4}) = −1, φ_{1}(a_{6}) = 1 − q, φ_{1}(b_{1}) = 1,
φ_{1}(a_{2}) = φ_{1}(a_{3}) = φ_{1}(a_{5}) = φ_{1}(b_{u}) = 0, 2 ≤ u ≤ q − 1;

(ii)

φ_{q}(a_{1}) = φ_{q}(a_{4}) = φ_{q}(a_{6}) = φ_{q}(b_{u}) = 0, 1 ≤ u ≤ q − 2,
φ_{q}(a_{2}) = 1, φ_{q}(a_{3}) = −1, φ_{q}(a_{5}) = q − 1, φ_{1}(b_{q−1}) = −1.

(iii) for 2 ≤ j ≤ q − 1,

φ_{j}(a_{i}) = φ_{j}(b_{u}) = 0, 1 ≤ i ≤ 6, u ≠ j − 1, j; φ_{j}(b_{j}) = −φ_{j}(b_{j−1}) = 1.

Here a_{i}, b_{u} are the oriented edges of the graph (Γ_{0}(xy) \ 𝒯 )^{0} in Figure 4. Therefore δ_{N}(ψ̄_{1})(a_{1}) = 1 and δ_{N}(ψ̄_{1})(b_{u}) = −(q + 1), 1 ≤ u ≤ q − 1. On the other hand, let f = −(q +1)E_{x} +(q^{2} +1)E_{y} +qE_{(x,y)} ∈ E_{00}(xy, ℤ/Nℤ). From the Fourier expansion of E_{x}, E_{y}, and E_{(x,y)}, we also get

f(a_{1}) = 1 and f(b_{u}) = −(q + 1) for 1 ≤ u ≤ q − 1.

Since every harmonic cochain in H_{00}(xy, ℤ/Nℤ) is determined uniquely by its values at a_{1} and b_{u} for 1 ≤ u ≤ q − 1, we get δ_{N}(ψ̄_{1}) = f and the proof is complete. □

Definition 8.17. Let 𝔫 ⊲ A be a non-zero ideal and ℓ a prime number. Let 𝕋(𝔫)_{ℓ} := 𝕋(𝔫) ⊗_{ℤ} ℤ_{ℓ}. Given a maximal ideal 𝔐 ⊲ 𝕋(𝔫)_{ℓ}, let 𝕋(𝔫)_{𝔐} be the completion of 𝕋(𝔫)_{ℓ} at 𝔐. We say that 𝕋(𝔫)_{𝔐} is a Gorenstein ℤ_{ℓ}-algebra if 𝕋(𝔫)_{𝔐}^{∨} := Hom_{ℤ_{ℓ}}(𝕋(𝔫)_{𝔐}, ℤ_{ℓ}) is a free 𝕋(𝔫)_{𝔐} module of rank 1. In [38], following Mazur [33], Pál proved that 𝕋(𝔭)_{𝔐} is Gorenstein for the maximal ideals containing 𝔈(𝔭).

Let 𝔈_{ℓ} be the ideal generated by 𝔈(xy) in 𝕋_{ℓ}. We know that 𝕋_{ℓ}/𝔈_{ℓ} ≅ ℤ_{ℓ}/Nℤ_{ℓ}; see Corollary 3.18. Thus, N annihilates J_{ℓ}[𝔈].

Proposition 8.18. Assume ℓ | N. The finite group-scheme J_{ℓ}[𝔈] is unramified at ∞. If 𝕋_{𝔐} is Gorenstein for 𝔐 = (𝔈_{ℓ}, ℓ), then there is an exact sequence of G_{F_{∞}}-modules

0 → (ℤ_{ℓ}/Nℤ_{ℓ})^{∗} → J_{ℓ}[𝔈] → ℤ_{ℓ}/Nℤ_{ℓ} → 0.

Proof. The argument in the proof of Lemma 8.12 and the isomorphism of Hecke modules (Φ_{∞})_{ℓ} ≅ E_{00}(xy, ℤ/ℓ^{n}ℤ) that we just proved (n ≫ 0) give the exact sequence

0 → D_{ℓ}[𝔈] → J_{ℓ}[𝔈] → (Φ_{∞})_{ℓ} → 0.

Since J[ℓ^{n}]^{I_{∞}} ≅ D[ℓ^{n}] × (Φ_{∞})[ℓ^{n}], we see that J_{ℓ}[𝔈] is unramified at ∞. Next, using Proposition 2.21, (4.3) and (4.4), we have

D_{ℓ}[𝔈] ≅ Hom(H_{0}(xy, ℤ_{ℓ})/𝔈_{ℓ}H_{0}(xy, ℤ_{ℓ}), ℂ_{∞}^{×})
≅ Hom(𝕋_{ℓ}^{∨}/𝔈_{ℓ}𝕋_{ℓ}^{∨}, ℂ_{∞}^{×}) ≅ Hom(𝕋_{𝔐}^{∨}/𝔈_{ℓ}𝕋_{𝔐}^{∨}, ℂ_{∞}^{×}).

If 𝕋_{𝔐} is Gorenstein, then

Hom(𝕋_{𝔐}^{∨}/𝔈_{ℓ}𝕋_{𝔐}^{∨}, ℂ_{∞}^{×}) ≅ Hom(𝕋_{𝔐}/𝔈_{ℓ}, ℂ_{∞}^{×}) ≅ Hom(ℤ_{ℓ}/Nℤ_{ℓ}, ℂ_{∞}^{×}).
□

If ℓ is odd and divides q +1 then, under the assumption that 𝕋_{𝔐} is Gorenstein, Proposition 8.18 implies that S_{ℓ} is the maximal μ-type subgroup scheme of J_{ℓ} and J_{ℓ}[𝔈] ≅ C_{ℓ} ⊕ S_{ℓ} is pure. Indeed, since Φ_{∞} is constant and gcd(q + 1, q − 1) divides 2, the maximal μ-type subgroup scheme of J_{ℓ} specializes to the connected component J_{𝔽_{∞}}^{0} of the Néron model, so must be isomorphic to (ℤ_{ℓ}/(q + 1)ℤ_{ℓ})^{∗}. Since S_{ℓ} is μ-type, it must be maximal by comparing the orders. The fact that 𝕋_{𝔐} is Gorenstein in this case can be proved by Mazur’s Eisenstein descent discussed in Section 10 of [38]. Since this fact is not central for our paper, and the proof closely follows the argument in [38], we omit the details. Now assume ℓ is odd and divides q^{2} + 1. Assume 𝕋_{𝔐} is Gorenstein. Since s_{x,y} divides 2, Lemma 8.12 and Proposition 8.18 imply that there is an exact sequence of G_{F} -modules

(8.1)  0 → C_{ℓ} → J_{ℓ}[𝔈] → M_{ℓ} → 0,

where M_{ℓ} ≅ (ℤ_{ℓ}/(q^{2} + 1)ℤ_{ℓ})^{∗}. If J_{ℓ}[𝔈]^{I_{y}} is strictly larger than C_{ℓ}, then the Galois representation ρ_{𝔐} : G_{F} → Aut(J_{ℓ}[𝔐]) ≅ GL_{2}(𝔽_{ℓ}) is unramified at y. On the other hand, (Φ_{y})_{ℓ} = 0, so J_{ℓ}[𝔐] is isomorphic to a 𝕋_{ℓ} ×G_{𝔽_{y}}-submodule of the torus J_{𝔽_{y}}^{0}. Now the same argument that proves Proposition 3.8 in [47] implies that Frob_{y} acts on J_{ℓ}[𝔐] as a scalar ±|y|^{2}. Hence det(ρ_{𝔐}) = q^{4}. On the other hand, the sequence (8.1) shows that the eigenvalues of Frob_{y} are 1 and |y| = q^{2}. Thus, det(ρ_{𝔐}) = q^{2}. This forces q^{4} ≡ q^{2} (mod ℓ), which is a contradiction since gcd(q^{2} + 1, q^{2}(q^{2} − 1)) divides 2. We conclude that if 𝕋_{𝔐} is Gorenstein then the ℓ-primary μ-type subgroup scheme of J is trivial. This “explains” why S is smaller than C – the missing part corresponds to the ramified portion of J[𝔈].

9. Jacquet-Langlands isogeny

9.1. Modular curves of 𝒟-elliptic sheaves. Let D be a division quaternion algebra over F. Assume D is split at ∞, i.e., D ⊗_{F} F_{∞} ≅ Mat_{2}(F_{∞}); this is the analogue of “indefinite” over ℚ. Let 𝔫⊲A be the discriminant of D. It is known that 𝔫 is square-free with an even number of prime factors. Moreover, any 𝔫 with this property is a discriminant of some D, and, up to isomorphism, 𝔫 determines D; cf. [56]. Let C := ℙ_{𝔽_{q}}^{1}. Fix a locally free sheaf 𝒟 of O_{C}-algebras with stalk at the generic point equal to D and such that 𝒟_{v} := 𝒟 ⊗_{O_{C}} O_{v} is a maximal order in D_{v} := D ⊗_{F} F_{v} for every place v. Let S be an 𝔽_{q}-scheme. Denote by Frob_{S} its Frobenius endomorphism, which is the identity on the points and the qth power map on the functions. Denote by C × S the fiber product C ×_{Spec(𝔽_{q})} S. Let z : S → Spec(A[𝔫^{−1}]) be a morphism of 𝔽_{q}-schemes. A 𝒟-elliptic sheaf over S, with pole ∞ and zero z, is a sequence 𝔼 = (E_{i}, j_{i}, t_{i})_{i∈ℤ}, where each E_{i} is a locally free sheaf of O_{C×S}-modules of rank 4 equipped with a right action of 𝒟 compatible with the O_{C}-action, and where

j_{i} : E_{i} → E_{i+1}
t_{i} : ^{τ} E_{i} := (Id_{C} × Frob_{S})^{∗}E_{i} → E_{i+1}

are injective O_{C×S}-linear homomorphisms compatible with the 𝒟-action. The maps j_{i} and t_{i} are sheaf modifications at ∞ and z, respectively, which satisfy certain conditions, and it is assumed that for each closed point P of S, the Euler-Poincaré characteristic χ(E_{0}|_{C×P}) is in the interval [0, 2); we refer to [28, §2] for the precise definition. Denote by Eℓℓ^{𝒟} (S) the set of isomorphism classes of 𝒟-elliptic sheaves over S. The following theorem can be deduced from (4.1), (5.1) and (6.2) in [28]:

Theorem 9.1. The functor S ↦ Eℓℓ^{𝒟} (S) has a coarse moduli scheme X^{𝔫}, which is proper and smooth of pure relative dimension 1 over Spec(A[𝔫^{−1}]).

For each prime 𝔭 ⊲ A not in R, there is a Hecke correspondence T_{𝔭} on X^{𝔫}; we refer to [28, §7] for the definition.

9.2. Rigid-analytic uniformization. Let D be a maximal A-order in D. Let Γ^{𝔫} := D^{×} be the group of units of D. By fixing an isomorphism D⊗_{F} F_{∞} ≅ Mat_{2}(F_{∞}), we get an embedding Γ^{𝔫} ↪ GL_{2}(F_{∞}). Hence Γ^{𝔫} acts on the Drinfeld half-plane Ω = ℂ_{∞} − F_{∞}. We have the following uniformization theorem

(9.1)  Γ^{𝔫} \ Ω ≅ (X_{F_{∞}}^{𝔫})^{an},

which follows by applying Raynaud’s “generic fibre” functor to Theorem 4.4.11 in [2]. The proof of this theorem is only outlined in [2]. Nevertheless, as is shown in [52, Prop. 4.28], (9.1) can be deduced from Hausberger’s version [23] of the Cherednik-Drinfeld theorem for X^{𝔫}. An alternative proof of (9.1) is given in [53, Thm. 4.6]. Assume for simplicity that 𝔫 is divisible by a prime of even degree. In this case, the normal subgroup Γ_{f}^{𝔫} of Γ^{𝔫} generated by torsion elements is just the center 𝔽_{q}^{×} of Γ^{𝔫}, cf. [42, Thm. 5.6]. This implies that the image of Γ^{𝔫} in PGL_{2}(F_{∞}) is a discrete, finitely generated free group, and (9.1) is a Mumford uniformization [20], [35]. Denote Γ = Γ^{𝔫}/𝔽_{q}^{×} and let \overline{Γ} be the abelianization of Γ. The group Γ acts without inversions on the Bruhat-Tits tree 𝒯 , and the quotient graph Γ^{𝔫}\𝒯 = Γ \ 𝒯 is finite; cf. [42, Lem. 5.1]. Fix a vertex v_{0} ∈ 𝒯 . For any γ ∈ Γ there is a unique path in 𝒯 without backtracking from v_{0} to γv_{0}. The map Γ → H_{1}(Γ \ 𝒯 , ℤ) which sends γ to the homology class of the image of this path in Γ\𝒯 does not depend on the choice of v_{0}, and induces an isomorphism

(9.2)  \overline{Γ} ≅ H_{1}(Γ \ 𝒯 , ℤ).

Since Γ is a free group and Γ \ 𝒯 is a finite graph,

H(𝔫, ℤ)′ := H_{0}(𝒯 , ℤ)^{Γ^{𝔫}} = H_{0}(𝒯 , ℤ)^{Γ} = H(𝒯 , ℤ)^{Γ}
≅ H(Γ \ 𝒯 , ℤ) ≅ H_{1}(Γ \ 𝒯 , ℤ),

cf. Definitions 2.1 and 2.2. The space H(𝔫, ℤ)′ is equipped with a natural action of Hecke operators T_{𝔪} (𝔪 ⊲ A), which generate a commutative ℤ-algebra; cf. [34, §5.3]. The map ⟨·, ·⟩ : E(𝒯 ) × E(𝒯 ) → ℤ

⟨e, f⟩ = \{ \begin{matrix}1 & if f = e \\ −1 & if f = ē \\ 0 & otherwise \end{matrix}

induces a ℤ-valued bilinear symmetric positive-definite pairing on H_{1}(Γ\𝒯 , ℤ), which we denote by the same symbol. Using (9.2) we get a pairing

(9.3)  \overline{Γ} × \overline{Γ} → ℤ
γ, δ ↦ ⟨γ, δ⟩.

Let J^{𝔫} denote the Jacobian variety of X_{F}^{𝔫}, and J^{𝔫} denote the Néron model of J^{𝔫} over O_{∞}. Since X^{𝔫} is a Mumford curve over F_{∞}, (J^{𝔫})_{𝔽_{∞}}^{0} is a split algebraic torus. Let M := Hom((J^{𝔫})_{\overline{𝔽}_{∞}}^{0}, 𝔾_{m,\overline{𝔽}_{∞}}) be the character group of this torus. By the mapping property of Néron models, each endomorphism of J^{𝔫} defined over F_{∞} canonically extends to J^{𝔫}, hence acts functorially on M. Since J^{𝔫} has purely toric reduction, End_{F_{∞}}(J^{𝔫}) operates faithfully on M. By [35, p. 132], the graph Γ \ 𝒯 is the dual graph of the special fibre of the minimal regular model of X^{𝔫} over O_{∞}. On the other hand, by [3, p. 246], there is a canonical isomorphism M ≅ H_{1}(Γ \ 𝒯 , ℤ). The Hecke correspondence T_{𝔭} induces an endomorphism of J^{𝔫} by Picard functoriality, which we denote by the same symbol. Via the isomorphisms mentioned in this paragraph and (9.2), we get a canonical action of T_{𝔭} on \overline{Γ}. This action agrees with the action of T_{𝔭} on H(𝔫, ℤ)′. Using rigid-analytic theta functions, one constructs a symmetric bilinear pairing

\overline{Γ} × \overline{Γ} → F_{∞}^{×}
γ, δ ↦ [γ, δ],

such that ord_{∞}[γ, δ] = ⟨γ, δ⟩; see [31, Thm. 5]. The main result of [31] gives a rigid-analytic uniformization of J^{𝔫}:

(9.4)  0 → \overline{Γ} ⟶^{γ↦[γ,·]} Hom(\overline{Γ}, ℂ_{∞}) → J^{𝔫}(ℂ_{∞}) → 0.

This sequence is equivariant with respect to the action of T_{𝔭} (𝔭 ∉ R); this was proven by Ryan Flynn [7] following the method in [18]. Finally, we have the following quaternionic analogue of Theorem 5.5. Let Φ′_{∞} be the component group of J^{𝔫} at ∞. After identifying \overline{Γ} with the character group M, the pairing (9.3) becomes Grothendieck’s monodromy pairing on M, so there is an exact sequence (see [22, §§11-12] and [43])

(9.5)  0 → \overline{Γ} ⟶^{γ↦⟨γ,·⟩} Hom(\overline{Γ}, ℤ) → Φ′_{∞} → 0.

This sequence is equivariant with respect to the action of T_{𝔭}.

9.3. Explicit Jacquet-Langlands isogeny conjecture. Let 𝔫 ⊲ A be a product of an even number of distinct primes. The space H_{0}(𝔫, ℚ) can be interpreted as a space of automorphic forms on Γ_{0}(𝔫); see [18, §4]. A similar argument can be used to show that H(𝔫, ℚ)′ has an interpretation as a space of automorphic forms on Γ^{𝔫}; cf. [43, §8]. The Jacquet-Langlands correspondence over F implies that there is an isomorphism

JL : H_{0}(𝔫, ℚ)^{new} ≅ H(𝔫, ℚ)′

deg(𝔮) is even  deg(𝔮) is odd
deg(𝔭) = 1 ℤ/(q + 1)M(𝔮)ℤ  ℤ/M(𝔮)ℤ
deg(𝔭) = 2 ℤ/M(𝔮)ℤ ℤ/(q + 1)M(𝔮)ℤ

Table 1. Component group Φ′_{𝔮} of J^{𝔭𝔮}

which is compatible with the action of Hecke operators T_{𝔭}, 𝔭 ∤ 𝔫; cf. [25, Ch. III].

Remark 9.2. In fact, JL is U_{𝔭}-equivariant also for 𝔭 | 𝔫. The analogous statement over ℚ follows from the results of Ribet [47, §4] (cf. [24, Thm. 2.3]), which rely on the geometry of the Jacobians of modular curves. The Néron models of J_{0}(𝔫) and J^{𝔫} have the necessary properties for Ribet’s arguments to apply. Since this fact will not be essential for our purposes, we do not discuss the details further. The U_{𝔭}-equivariance of JL can also be deduced from [25]; the necessary input is contained in [25, Thm. 4.2] and its proof.

Let J_{0}(𝔫)^{old} be the abelian subvariety of J_{0}(𝔫) generated by the images of J_{0}(𝔪) under the maps J_{0}(𝔪) → J_{0}(𝔫) induced by the degeneracy morphisms X_{0}(𝔫)_{F} → X_{0}(𝔪)_{F} for all 𝔪 ⊋ 𝔫. Let J_{0}(𝔫)^{new} be the quotient of J_{0}(𝔫) by J_{0}(𝔫)^{old}. Combining JL with the main results in [6] and [28], and Zarhin’s isogeny theorem, one concludes that there is a 𝕋(𝔫)^{0}-equivariant isogeny J_{0}(𝔫)^{new} → J^{𝔫} defined over F, which we call a Jacquet-Langlands isogeny; cf. [41, Cor. 7.2]. (This isogeny is in fact 𝕋(𝔫)-equivariant by the previous remark; see [24, Cor. 2.4] for the corresponding statement over ℚ.) Zarhin’s isogeny theorem provides no information about the Jacquet-Langlands isogenies beyond their existence. One possible approach to making these isogenies more explicit is via the study of component groups. More precisely, since J_{0}(𝔫)^{new} and J^{𝔫} have purely toric reduction at the primes dividing 𝔫 and at ∞, one can deduce restrictions on possible kernels of isogenies J_{0}(𝔫)^{new} → J^{𝔫} from the component groups of these abelian varieties using [41, Thm. 4.3]. Unfortunately, the explicit structure of these component groups is not known in general. From now on we restrict to the case when 𝔫 = 𝔭𝔮 is a product of two distinct primes and deg(𝔭) ≤ 2. In this case, the structure of X_{𝔽_{𝔮}}^{𝔫} is fairly simple and can be deduced from [41, Prop. 6.2]. Using this and Raynaud’s theorem, one computes that the component group Φ′_{𝔮} of J^{𝔫} at 𝔮 is given by Table 1, where

M(𝔮) = \begin{cases}|𝔮| + 1, & if deg(𝔮) is even; \\ \frac{|𝔮|+1}{q+1}, & if deg(𝔮) is odd. \end{cases}

The cuspidal divisor group of J_{0}(𝔮) is generated by [1] − [∞], which has order

N(𝔮) = \begin{cases}\frac{|𝔮|−1}{q^{2}−1}, & if deg(𝔮) is even; \\ \frac{|𝔮|−1}{q−1}, & if deg(𝔮) is odd. \end{cases}

deg(𝔮) is even  deg(𝔮) is odd
deg(𝔭) = 1 ℤ/(q + 1)ℤ  0
deg(𝔭) = 2 ℤ/(q^{2} + 1)ℤ ℤ/(q^{2} + 1)(q + 1)ℤ
Table 2. Φ̃_{𝔮}

Let α : X_{0}(𝔭𝔮)_{F} → X_{0}(𝔮)_{F} be the degeneracy morphism discussed in Section 4. The image of C(𝔮) in J_{0}(𝔭𝔮) under the induced map J_{0}(𝔮) → J_{0}(𝔭𝔮) is generated by

c := α^{∗}([1] − [∞]) = |𝔭|[1] + [𝔭] − |𝔭|[𝔮] − [∞].

By examining the specializations of the cusps in X_{0}(𝔭𝔮)_{𝔽_{𝔮}}, we see that

℘_{𝔮}(c) = (|𝔭| + 1)z,

where z ∈ Φ_{𝔮} is the element from the proof of Proposition 5.4. Let Φ̃_{𝔮} := Φ_{𝔮}/℘_{𝔮}(c). The order of z in Φ_{𝔮} is given in [41, Thm. 4.1], and the order of Φ_{𝔮} itself is given in Proposition 5.4. From this one easily computes that Φ̃_{𝔮} is the group in Table 2. Since c ∈ J_{0}(𝔫)^{old}, the map of component groups Φ_{𝔮} → Φ_{𝔮}^{new} induced by the quotient J_{0}(𝔫) → J_{0}(𝔫)^{new} must factor through Φ̃_{𝔮} (here Φ_{𝔮}^{new} denotes the component group of J_{0}(𝔫)^{new} at 𝔮). Assume deg(𝔭) = 1. Then the cuspidal divisor c_{𝔮} := [𝔮]−[∞] ∈ C(𝔭𝔮) has order N(𝔮)M(𝔮) (see Theorem 6.11) and specializes to the connected component of identity J_{𝔽_{𝔮}}^{0} of the Néron model of J_{0}(𝔫). Theorem 4.3 in [41] describes how the component groups of abelian varieties with toric reduction over a local field change under isogenies, depending on the specialization of the kernel of the isogeny in the closed fibre. After comparing the groups Φ̃_{𝔮} and Φ′_{𝔮}, and the orders of c and c_{𝔮}, this theorem suggests that there is an isogeny J_{0}(𝔫)^{new} → J^{𝔫} whose kernel is isomorphic to ℤ/M(𝔮)ℤ and is generated by the image of c_{𝔮} in J_{0}(𝔫)^{new}. The case deg(𝔭) = 2 can be analysed similarly. The cuspidal divisor c_{𝔮} has order (q + 1)N(𝔮)M(𝔮). The image of c_{𝔭} in Φ̃_{𝔮} generates its cyclic subgroup of order q^{2} + 1. Hence there might be an isogeny J_{0}(𝔫)^{new} → J^{𝔫} whose kernel is isomorphic to \frac{ℤ}{M(𝔮)ℤ} × \frac{ℤ}{(q^{2}+1)ℤ}.

Conjecture 9.3. Assume 𝔫 = 𝔭𝔮, where 𝔭, 𝔮 are distinct primes and deg(𝔭) ≤ 2. There is an isogeny J_{0}(𝔭𝔮)^{new} → J^{𝔭𝔮} whose kernel K is generated by cuspidal divisors and

K ≅ \begin{cases}\frac{ℤ}{M(𝔮)ℤ} & if deg(𝔭) = 1; \\ \frac{ℤ}{M(𝔮)ℤ} × \frac{ℤ}{(q^{2}+1)ℤ} & if deg(𝔭) = 2. \end{cases}

This conjecture is the function field analogue of Ogg’s conjectures about Jacquet-Langlands isogenies over ℚ; see [37, pp. 212-216]. There are only two cases when 𝔫 is square-free, J_{0}(𝔫) is non-trivial, and J_{0}(𝔫)^{new} = J_{0}(𝔫). The first case is 𝔫 = xy; Conjecture 9.3 then specializes

\begin{matrix}v_{+} & ⋮ & v_{−} & e_{u} \\ Figure 6. Γ^{xy} \ 𝒯 \end{matrix}

to the conjecture in [41]. We will prove this conjecture in §9.4 under certain assumptions, and unconditionally for some small q. The second case is 𝔫 = 𝔭𝔮, where 𝔭 ≠ 𝔮 are primes of degree 2. In this case the conjecture predicts that there is a Jacquet-Langlands isogeny whose kernel is isomorphic to ℤ/(q^{2} + 1)ℤ × ℤ/(q^{2} + 1)ℤ; cf. Example 6.13. The method of this paper should be possible to adapt to this latter case, and prove the conjecture for some small q.

9.4. Special case. Assume 𝔫 = xy. To simplify the notation we put

J := J_{0}(xy), J′ := J^{xy}, H := H_{0}(xy, ℤ), H′ := H(xy, ℤ)′, 𝕋 := 𝕋(xy).

First, we prove the analogue of Lemma 8.16 for J′.

Lemma 9.4. Let Φ′_{∞} be the component group of J′ at ∞. Let 𝔭 ⊲ A be any prime not dividing xy. Then T_{𝔭} − |𝔭| − 1 annihilates Φ′_{∞} ≅ ℤ/(q + 1)ℤ.

Proof. The quotient graph Γ^{xy} \ 𝒯 has two vertices joined by q + 1 edges; see [41, Prop. 6.5]. We label the vertices by v_{+} and v_{−}, and label the edges by the elements of ℙ^{1}(𝔽_{q}); see Figure 6. Let γ ∈ GL_{2}(F_{∞}) be an arbitrary element. Then γv_{±} ≡ v_{±} mod Γ^{xy} if ord_{∞}(det γ) is even, and γv_{±} ≡ v_{∓} mod Γ^{xy} if ord_{∞}(det γ) is odd. Consider the free ℤ-module with generators \{e_{u}, \overline{e_{u}} | u ∈ ℙ^{1}(𝔽_{q})\}, modulo the relations \overline{e_{u}} = −e_{u}. The action of T_{𝔭} on this module satisfies

T_{𝔭} ∑_{u∈ℙ^{1}(𝔽_{q})} e_{u} = (|𝔭| + 1) \begin{cases}∑_{u∈ℙ^{1}(𝔽_{q})} e_{u} & if deg(𝔭) is even, \\ − ∑_{u∈ℙ^{1}(𝔽_{q})} e_{u} & if deg(𝔭) is odd. \end{cases}

H′ is generated by the cycles φ_{u} = e_{u} − e_{∞}, u ∈ 𝔽_{q}. Let φ_{u}^{∗} be the dual basis of Hom(H′, ℤ). The map in (9.5) sends φ_{u} to φ_{u}^{∗} + ∑_{w∈𝔽_{q}} φ_{w}^{∗}. It is easy to see from this that Φ′_{∞} is cyclic of order q + 1 and is generated by ∑_{w∈𝔽_{q}} φ_{w}^{∗}. Note that |𝔭| + 1 ≡ 0 mod (q + 1) if deg(𝔭) is odd. Hence

T_{𝔭} ∑_{w∈𝔽_{q}} φ_{w}^{∗} = T_{𝔭} ( ∑_{u∈ℙ^{1}(𝔽_{q})} e_{u}^{∗} − (q + 1)e_{∞}^{∗}) = ±(|𝔭|+1) ∑_{u∈ℙ^{1}(𝔽_{q})} e_{u}^{∗}−(q+1)T_{𝔭}e_{∞}^{∗}

≡ (|𝔭| + 1) ∑_{u∈ℙ^{1}(𝔽_{q})} e_{u}^{∗} − (q + 1)(|𝔭| + 1)e_{∞}^{∗} = (|𝔭| + 1) ∑_{w∈𝔽_{q}} φ_{w}^{∗} mod (q + 1).

This implies that T_{𝔭} acts by multiplication by |𝔭| + 1 on Φ′_{∞}. □

Theorem 9.5. Assume H ≅ H′ as 𝕋-modules. There is an isogeny J → J′ defined over F whose kernel is cyclic of order q^{2} + 1 and annihilated by the Eisenstein ideal.

Proof. By (4.3) and (4.4), the rigid-analytic uniformization of J over F_{∞} is given by the 𝕋^{0}-equivariant sequence

0 → H → Hom(H, ℂ_{∞}^{×}) → J(ℂ_{∞}) → 0.

By Proposition 2.21, Hom(H, ℤ) ≅ 𝕋 = 𝕋^{0}, so we can write this sequence as the 𝕋-equivariant sequence

0 → H → 𝕋 ⊗ ℂ_{∞}^{×} → J(ℂ_{∞}) → 0.

By Theorem 5.5, the sequence derived from this using the valuation homomorphism ord_{∞} is

0 → H → 𝕋 → Φ_{∞} → 0,

where Φ_{∞} is the component group of the Néron model of J at ∞. Now we can consider H as ideal in 𝕋. We know from Lemma 8.16 that the Eisenstein ideal 𝔈 annihilates Φ_{∞} ≅ ℤ/(q^{2} + 1)(q + 1)ℤ. Hence Φ_{∞} is a quotient of 𝕋/𝔈. On the other hand, by Corollary 3.18, 𝕋/𝔈 ≅ ℤ/(q^{2} + 1)(q + 1)ℤ. Comparing the orders of Φ_{∞} and 𝕋/𝔈, we conclude that Φ_{∞} ≅ 𝕋/𝔈 and H ≅ 𝔈. From the discussion in §9.2, if we assume H ≅ H′ as 𝕋-modules, the rigidanalytic uniformization of J′ over F_{∞} is given by the 𝕋-equivariant sequence

0 → H′ → 𝕋 ⊗ ℂ_{∞}^{×} → J′(ℂ_{∞}) → 0.

The argument in the previous paragraph allows us to identify H′ in the above sequence with the annihilator 𝔈′⊲𝕋 of Φ′_{∞} in 𝕋. On the other hand, by Lemma 9.4, T_{𝔭} − |𝔭| − 1 ∈ 𝔈′ for any 𝔭 ∤ xy. Therefore, 𝔈 ⊂ 𝔈′ and

𝕋/𝔈′ ≅ Φ′_{∞} ≅ ℤ/(q + 1)ℤ.

Hence 𝔈′/𝔈 ≅ ℤ/(q^{2} + 1)ℤ. We have identified the uniformizing tori of J and J′ with 𝕋 ⊗ ℂ_{∞}^{×}, and the uniformizing lattices H and H′ with 𝔈 and 𝔈′, respectively. Now specializing a theorem of Gerritzen [19] to this situation, we get a natural bijection

Hom_{𝕋}(𝕋 ⊗ ℂ_{∞}^{×}, 𝔈; 𝕋 ⊗ ℂ_{∞}^{×}, 𝔈′) ⟶^{∼} Hom_{𝕋}(J_{F_{∞}}, J′_{F_{∞}}),

where on the left hand-side is the group of homomorphisms 𝕋 ⊗ ℂ_{∞}^{×} → 𝕋 ⊗ ℂ_{∞}^{×} which map 𝔈 to 𝔈′ and are compatible with the action of 𝕋. It is clear that identity map on 𝕋⊗ℂ_{∞}^{×} is in this set. The snake lemma applied to the resulting diagram

[figure: 0; 𝔈; 𝕋 ⊗ ℂ_{∞}^{×}; J(ℂ_{∞}); π; 0; 0; 𝔈′; 𝕋 ⊗ ℂ_{∞}^{×}; J′(ℂ_{∞}); 0]

shows that there is an isogeny π : J → J′ with ker(π) ≅ 𝔈′/𝔈. Moreover, since Hom_{𝕋}(𝕋, 𝕋) ≅ 𝕋, every 𝕋-equivariant homomorphism J → J′ can be obtained as a composition of π with an element of 𝕋. We know that there is an isogeny J → J′ defined over F. Since the endomorphisms of J induced by the Hecke operators are also defined over F, we conclude that π is defined over F. □

Theorem 9.6. In addition to the assumption in Theorem 9.5, assume 𝕋_{𝔐} is Gorenstein for all maximal Eisenstein ideals 𝔐 of residue characteristic dividing q^{2} + 1. Then there is an isogeny J → J′ whose kernel is ⟨c_{y}⟩.

Proof. By Theorem 9.5, there is an isogeny J → J′ defined over F whose kernel H ⊂ J[𝔈] is cyclic of order q^{2} + 1. Assume 𝕋_{𝔐} is Gorenstein for all maximal Eisenstein ideals 𝔐 of residue characteristic dividing q^{2} +1. First, by Theorem 7.13 and Proposition 8.18, this implies J[2, 𝔈] = C[2]. Next, let ℓ|(q^{2} + 1) be an odd prime. From the discussion after Proposition 8.18 we know that the action of inertia at y on J_{ℓ}[𝔈] is unipotent and the I_{y}-invariant subgroup of J_{ℓ}[𝔈] is C_{ℓ} = ⟨c_{y}⟩_{ℓ}. Since 4 does not divide q^{2} +1, these two observations imply that there is an isogeny J → J′ whose kernel is cyclic of order q^{2} + 1 and is contained in C[2] + ⟨c_{y}⟩. If q is even, then C[2]+⟨c_{y}⟩ = ⟨c_{y}⟩, and we are done. If q is odd, then the kernel is generated by c_{y}, or c_{y} + \frac{q+1}{2}c_{x}, or 2c_{y} + \frac{q+1}{2}c_{x}. We know that ℘_{y}(c_{y}) = 0 and ℘_{y}(c_{x}) = 1 (cf. Proposition 6.7). If H is generated by c_{y} + \frac{q+1}{2}c_{x} or 2c_{y} + \frac{q+1}{2}c_{x}, then the specialization map ℘_{y} gives the exact sequence

0 → \frac{ℤ}{\frac{q^{2}+1}{2}ℤ} → H ⟶^{℘_{y}} ℤ/2ℤ → 0.

It is a consequence of the uniformization theorem in [23] that X_{F_{𝔭}}^{𝔫} is a twisted Mumford curve for any 𝔭|𝔫 (here 𝔫 is arbitrary). This implies that J^{𝔫} has toric reduction at 𝔭. In particular, J′ has toric reduction at y. Now we can apply Theorem 4.3 in [41] to get an exact sequence

0 → ℤ/2ℤ → Φ_{y} → Φ′_{y} → \frac{ℤ}{\frac{q^{2}+1}{2}ℤ} → 0,

where Φ′_{y} is the component group of J′ at y. This implies that the order of Φ′_{y} is (q + 1)(q^{2} + 1)/4. But according to [41, Thm. 6.4], Φ′_{y} ≅ ℤ/(q^{2} + 1)(q + 1)ℤ, which leads to a contradiction. □

To be able to verify the assumptions in Theorem 9.6 computationally, it is crucial to be able to compute the action of 𝕋 on H and H′. The methods for doing this will be discussed in Section 10. Our calculations lead to the following:

Proposition 9.7. The assumptions of Theorem 9.6 hold for q = 2, and for the 12 cases listed in Table 4. In particular, in these cases there is an isogeny J → J′ whose kernel is ⟨c_{y}⟩.

Remark 9.8. We believe that the assumptions in Theorem 9.6 hold in general. Our method for verifying that H and H′ are isomorphic 𝕋-modules relies on finding a perfect 𝕋-equivariant pairing 𝕋 × H′ → ℤ. Unlike the case of H,

\begin{matrix}q & y & disc(𝕋) \\ 2 & T^{2} + T + 1 & 4 \\ 3 & T^{2} + 1 & 80 \\ 3 & T^{2} + T + 2 & 68 \\ 3 & T^{2} + 2T + 2 & 68 \\ 5 & T^{2} + T + 1 & 265216 \\ 5 & T^{2} + T + 2 & 278800 \\ 7 & T^{2} + 1 & 7372800000 \\ 7 & T^{2} + T + 4 & 6567981056 \end{matrix}

Table 3. Discriminant of 𝕋(xy)

there is no natural pairing between these modules, so our method is by trialand-error. We essentially construct some 𝕋-equivariant pairings, and check if one of those is perfect (see the discussion after (10.2)). This method is very inefficient, and our computer calculations terminated in a reasonable time only in the cases listed in Table 4.

10. Computing the action of Hecke operators

10.1. Action on H. Assume 𝔫 = xy. To simplify the notation, we denote H = H_{0}(xy, ℤ), H′ = H(xy, ℤ)′, 𝕋 = 𝕋(xy). Assume x = T. Theorem 6.8 in [9] gives a recipe for computing a matrix by which T_{x−s} acts on H for any s ∈ 𝔽_{q}^{×}. Since by Proposition 2.21 the operators {1, T_{x−s} | s ∈ 𝔽_{q}^{×}} form a ℤbasis of 𝕋, this essentially gives a complete description of the action of 𝕋 on H. This also allows us to compute the discriminant of 𝕋, an interesting invariant measuring the congruences between Hecke eigenforms. (Recall that disc(𝕋) is the determinant of the q × q matrix (Trace(t_{i}t_{j}))_{1≤i,j≤q}, where {t_{1}, …, t_{q}} is a ℤ-basis of 𝕋.) Table 3 lists the values of disc(𝕋) in some cases.

Remark 10.1. The algebra 𝕋(𝔫)⊗ℂ_{∞} is isomorphic to the Hecke algebra acting on doubly cuspidal Drinfeld modular forms of weight 2 and type 1 on Γ_{0}(𝔫); see [18, (6.5)]. The algebra 𝕋(𝔫) ⊗ ℂ_{∞} has no nilpotent elements if and only if p ∤ disc(𝕋). Table 3 indicates that p ∤ disc(𝕋) for q = 3 and arbitrary y, but for q = 5 or 7, there exist y_{1} and y_{2} such that p | disc(𝕋(xy_{1})) and p ∤ disc(𝕋(xy_{2})). It seems like an interesting problem to investigate the frequency with which p divides disc(𝕋).

Remark 10.2. For the sake of completeness, and also because [9] is in German, we give Gekeler’s method for computing a matrix G(x − s) ∈ Mat_{q}(ℤ), s ∈ 𝔽_{q}^{×}, representing the action of T_{x−s} on H. Let y = T^{2} + aT + b. Label the rows and columns of G(x − s) by u, w ∈ 𝔽_{q}. Then the (u, w) entry of G(x − s) is equal to

2 − Q(u, w) − (q + 1)δ_{w,s} + qδ_{u,0}δ_{w,b/s}, where δ is Kroneker’s delta, and Q(u, w) is the number of solutions β ∈ 𝔽_{q} (without multiplicities) of the equation

(u − β)(w − β)(s − β) + β(β^{2} + aβ + b) = 0

plus 1 if u + w + s + a = 0.

By comparing the discriminant of the characteristic polynomial of T_{x−s} with disc(𝕋), one can deduce that in some cases 𝕋 is monogenic, i.e., is generated by a single element as a ℤ-algebra:

Example 10.3. For q = 2 and y = T^{2} + T + 1

𝕋 ≅ ℤ[T_{x−1}] ≅ ℤ[X]/X(X + 2).

For q = 3 and y = T^{2} + T + 2

𝕋 ≅ ℤ[T_{x−2}] ≅ ℤ[X]/(X + 1)(X^{2} − X − 4).

If 𝕋 is monogenic, then its localization at any maximal ideal is Gorenstein; see [32, Thm. 23.5]. This is stronger than what we need for Theorem 9.6, but it is also computationally harder to establish. A simpler test is based on the following lemma:

Lemma 10.4. Let ℓ be a prime number dividing (q + 1)(q^{2} + 1). Suppose there is an element η in 𝔈 such that

dim_{𝔽_{ℓ}} H_{00}(xy, 𝔽_{ℓ})[η] = 1.

Then 𝕋_{𝔐} is Gorenstein, where 𝔐 = (𝔈, ℓ).

Proof. Note that the dimension of H_{00}(xy, 𝔽_{ℓ})[η] is at least 1 since E_{00}(xy, 𝔽_{ℓ}) ≅ 𝔽_{ℓ} is a subspace. Now consider the ideal ℑ = (ℓ, η)𝕋_{ℓ}. We have H_{00}(xy, 𝔽_{ℓ})[ℑ] = H_{00}(xy, 𝔽_{ℓ})[η]. On the other hand, H_{00}(xy, 𝔽_{ℓ})[ℑ] is 𝔽_{ℓ}-dual to 𝕋_{ℓ}/ℑ, cf. Proposition 2.21. Hence, if dim_{𝔽_{ℓ}} H_{00}(xy, 𝔽_{ℓ})[η] = 1, then 𝕋_{ℓ}/ℑ ≅ 𝔽_{ℓ}. This implies 𝕋_{ℓ}/𝔐 ≅ 𝕋_{ℓ}/ℑ. Since ℑ ⊂ 𝔐, we get 𝔐 = (ℓ, η), which implies that 𝕋_{𝔐} is Gorenstein by Proposition 15.3 in [33]. □

Any ℤ-linear combination η of the operators {T_{x−s}−(q+1)}_{s∈𝔽_{q}^{×}} is in 𝔈. We can compute the characteristic polynomial of such η acting on H using Gekeler’s method. Fix ℓ dividing (q + 1)(q^{2} + 1). If we find η whose characteristic polynomial modulo ℓ does not have 0 as a multiple root, then we can apply Lemma 10.4 to conclude that 𝕋_{𝔐} is Gorenstein for Eisenstein 𝔐 of residue characteristic ℓ. Using this strategy, for each prime q ≤ 7 we found by computer calculations an appropriate η for any ℓ dividing (q + 1)(q^{2} + 1).

10.2. Action on H′. Suppose y = T^{2} + aT + b. We denote the place x by ∞′. Let T′ = T^{−1}, A′ = 𝔽_{q}[T′], and y′ = T^{′2} + ab^{−1}T′ + b^{−1}. We have the correspondence of places of F:

\begin{aligned}∞ & ⟷ & T′ \\ T & ⟷ & ∞′ \\ y & ⟷ & y′ \end{aligned}

Let D be the quaternion algebra over F ramified precisely at ∞′ and y′ (i.e. ramified at x and y). Take an Eichler A′-order D′ in D of level T′. More precisely, D′_{𝔭′} := D′ ⊗_{A′} O_{𝔭′} is a maximal O_{𝔭′}-order in D_{𝔭′} := D ⊗_{F} F_{𝔭′} for each prime 𝔭′ of A′ with 𝔭′ ≠ T′, and there exists an isomorphism ι_{T′} : D_{T′} := D ⊗_{F} F_{T′} ≅ Mat_{2}(F_{T′}) such that

ι_{T′}(D′_{T′}) = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Mat_{2}(O_{T′}) | c ≡ 0 mod T′\} .

Let O_{D_{∞′}} be the maximal O_{∞′}-order in D_{∞′}. Consider the double coset spaces

G′_{xy} := D^{×}\D^{×}(𝔸_{F})/ (D̂′^{×} · (O_{D_{∞′}}^{×} F_{∞′}^{×})) and Cl(D′) := D^{×}\D^{×}(𝔸_{F}^{∞′})/D̂′^{×},

where:

• 𝔸_{F} is the adele ring of F, i.e. 𝔸_{F} is the restricted direct product ∏′_{v} F_{v};

• 𝔸_{F}^{∞′} is the finite (with respect to ∞′) adele ring of F, i.e. 𝔸_{F}^{∞′} = ∏_{v≠∞′} F_{v};

• D^{×}(𝔸_{F}) (resp. D^{×}(𝔸_{F}^{∞′})) denotes (D ⊗_{F} 𝔸_{F})^{×} (resp. (D ⊗_{F} 𝔸_{F}^{∞′})^{×});

• D̂′ = ∏_{𝔭′⊲A′} D′_{𝔭′}. Then the strong approximation theorem (with respect to {∞}) shows that

Lemma 10.5. The double coset space G′_{xy} can be identified with the set of the oriented edges of the quotient graph Γ^{xy}\𝒯 in Figure 6.

Note that Cl(D′) can be identified with the locally-principal right ideals of D′ in D, and #Cl(D′) = q + 1. Moreover, if we take i_{u} ∈ GL_{2}(F_{T′}), u ∈ ℙ^{1}(𝔽_{q}), to be

i_{u} = \begin{cases}(\begin{matrix}u & 1 \\ 1 & 0 \end{matrix}) , & if u ∈ 𝔽_{q}, \\ (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) , & if u = ∞, \end{cases}

then, via the natural embedding GL_{2}(F_{T′}) ≅ D_{T′}^{×} ↪ D^{×}(𝔸_{F}^{∞′}), {i_{u} | u ∈ ℙ^{1}(𝔽_{q})} is a set of representatives of Cl(D′). Take an element ϖ_{∞′} ∈ D_{∞′} such that its reduced norm Nr(ϖ_{∞′}) = T. From the natural surjection from G′_{xy} to Cl(D′), one observes that

{(i_{u}, ϖ_{∞′}^{c}) ∈ D^{×}(𝔸_{F}^{∞′}) × D_{∞′}^{×} = D^{×}(𝔸_{F}) | u ∈ ℙ^{1}(𝔽_{q}), c = 0, 1}

is a set of representatives of G′_{xy}. We may take e_{u} := [i_{u}, 1] ∈ G′_{xy}. Then there exists a unique permutation γ : ℙ^{1}(𝔽_{q}) → ℙ^{1}(𝔽_{q}) of order 2 so that

\overline{e_{u}} := [i_{u} (\begin{matrix}0 & 1 \\ T′ & 0 \end{matrix}) , 1] = [i_{γ(u)}, ϖ_{∞′}].

Moreover, H′ can be viewed as the set of ℤ-valued functions f on G′_{x,y} satisfying

f(e_{u}) + f(\overline{e_{u}}) = 0, ∀u ∈ ℙ^{1}(𝔽_{q}) and ∑_{u∈ℙ^{1}(𝔽_{q})} f(e_{u}) = 0.

Let I_{u} be the right ideal of D′ in D corresponding to i_{u}, i.e.,

I_{u} := D ∩ i_{u} · D̂′.

Then the reduced norm of I_{u} is trivial, i.e., the ideal of A′ generated by the reduced norms of all elements in I_{u} is A′. For each ideal 𝔪′ ⊲ A′, the 𝔪′-th Brandt matrix B(𝔪′) = (B_{u,u′}(𝔪′))_{u,u′} ∈ Mat_{q+1}(ℤ) is defined by

B_{u,u′}(𝔪′) := \frac{#\{b ∈ I_{u}I_{u′}^{−1} : Nr(b) · A′ = 𝔪′\}}{q − 1}.

In Section 10.3, we give an explicit recipe of computing B_{u,u′}(𝔪′) for deg(𝔪′) = 1 when q is odd and the constant term b of y is not a square in 𝔽_{q}^{×}, and also when q = 2. By an analogue of Hecke’s theory (cf. [5]) we obtain the following result:

Proposition 10.6.

(1) Viewing γ as a permutation matrix in Mat_{q+1}(ℤ), one has

B(T′) = 2 · J − γ,

where every entry of J is 1.

(2) Identifying the place x − s and T′ − s^{−1} of F, one has that for s ∈ 𝔽_{q}^{×},

T_{x−s}e_{u} = ∑_{u′∈ℙ^{1}(𝔽_{q})} B_{u,γ(u′)}(T′ − s^{−1})\overline{e_{u′}}.

Proof. By Lemma II.5 in [5], we observe that for every u ∈ 𝔽_{q},

[i_{γ(u)}] + ∑_{u′∈𝔽_{q}} B_{u,u′}(T′)[i_{u′}] = 2 ∑_{u′′∈𝔽_{q}}[i_{u′′}] ∈ ℤ[Cl(D′)].

This shows (1). To prove (2), notice that for every g_{v} ∈ GL_{2}(F_{v}) with v ≠ ∞′ and u ∈ 𝔽_{q}, there exists u′ ∈ 𝔽_{q} such that

[i_{u}g_{v}, 1] = [i_{u′}, ϖ_{∞′}^{c}],

where c = ord_{v}(det g_{v}) · deg v. By Proposition II.4 in [5] we have that for s ∈ 𝔽_{q}^{×},

T_{x−s}e_{u} = ∑_{u′∈ℙ^{1}(𝔽_{q})} B_{u,u′}(T′ − s^{−1})[i_{u′}, ϖ_{∞′}]
= ∑_{u′∈ℙ^{1}(𝔽_{q})} B_{u,u′}(T′ − s^{−1})\overline{e_{γ(u′)}}
= ∑_{u′∈ℙ^{1}(𝔽_{q})} B_{u,γ(u′)}(T′ − s^{−1})\overline{e_{u′}}.
□

For u ∈ 𝔽_{q}, let f_{u} ∈ H′ such that f_{u}(e′_{u}) = δ_{u,u′} for u′ ∈ 𝔽_{q} and f_{u}(e_{∞}) = −1. We immediately get the following.

Corollary 10.7. For u, u′ ∈ 𝔽_{q} and s ∈ 𝔽_{q}^{×}, set

B′_{u′,u}(x − s) := B_{u′,γ(∞)}(T′ − s^{−1}) − B_{u′,γ(u)}(T′ − s^{−1}).

Then

T_{x−s}f_{u} = ∑_{u′∈𝔽_{q}} B′_{u′,u}(x − s)f_{u′}.

In other words, B′(x − s) := (B′_{u′,u}(x − s))_{u′,u} ∈ Mat_{q}(ℤ) is a matrix representation of T_{x−s} acting on H′ with respect to the basis {f_{u} | u ∈ 𝔽_{q}}.

Remark 10.8. From (2.7) we know that ∑_{s∈𝔽_{q}} T_{x−s} = −1, as endomorphisms of H. On the other hand, by Remark 9.2, JL : H⊗ℚ ≅ H′⊗ℚ is T_{x}-equivariant. Hence the previous corollary allows us to obtain also the matrix representation of T_{x} acting on H′.

Remark 10.2 and Corollary 10.7 give the matrices by which T_{x−s} (s ∈ 𝔽_{q}^{×}) acts on H and H′. Since T_{x−s} generate 𝕋, the condition that the 𝕋-modules H and H′ are isomorphic in Theorem 9.5 is equivalent to the matrices B′(x − s) being simultaneously ℤ-conjugate to Gekeler’s matrices G(x−s), i.e., to the existence of a single matrix C ∈ GL_{q}(ℤ) such that

(10.1)  C^{−1} · B′(x − s) · C = G(x − s)  ∀s ∈ 𝔽_{q}^{×}.

Remark 10.9. Due to Jacquet-Langlands correspondence, there does exist a matrix C ∈ GL_{q}(ℚ) satisfying (10.1), but the existence of an integral matrix is more subtle; cf. [27].

Example 10.10. Let q = 2 and y = T^{2} + T + 1. By Remark 10.2

G(x − 1) = (\begin{matrix}0 & 0 \\ 1 & −2 \end{matrix}) .

On the other hand, with respect to the basis {i_{∞}, i_{0}, i_{1}} of ℤ[Cl(D′)] we calculate that (see Remark 10.12)

γ = (\begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}) and B(T′ − 1) = (\begin{matrix}0 & 2 & 1 \\ 2 & 0 & 1 \\ 1 & 1 & 1 \end{matrix}) .

Therefore

B′(x − 1) = (\begin{matrix}−2 & −1 \\ 0 & 0 \end{matrix}) .

We can take C = (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}).

Note that the matrix (\begin{matrix}0 & 0 \\ 0 & −2 \end{matrix}) is conjugate to G(x − 1) over ℚ, but not over

ℤ; in fact, there are exactly two conjugacy classes of matrices in Mat_{2}(ℤ) with characteristic polynomial X(X + 2).

\begin{matrix}q & y & α \\ 3 & T^{2} + T + 2 & [0, 1, 0] \\ 3 & T^{2} + 2T + 2 & [0, 0, 1] \\ 5 & T^{2} + T + 2 & [−1, 1, 4, 5, 2] \\ 5 & T^{2} + 2T + 3 & [−1, −3, −6, −5, −2] \\ 5 & T^{2} + 3T + 3 & [−1, −6, −3, −2, −5] \\ 5 & T^{2} + 4T + 2 & [−1, 4, 1, 2, 5] \\ 7 & T^{2} + T + 6 & [−8, 0, −6, −5, −8, −7, 5] \\ 7 & T^{2} + 2T + 3 & [−8, −7, −7, 2, 3, −6, −6] \\ 7 & T^{2} + 3T + 5 & [−5, −6, −6, −4, 2, 3, −5] \\ 7 & T^{2} + 4T + 5 & [−8, −8, 0, −7, −6, 5, −5] \\ 7 & T^{2} + 5T + 3 & [−5, −4, −5, −6, 3, −6, 2] \\ 7 & T^{2} + 6T + 6 & [−5, −6, 2, −5, −6, −4, 3] \end{matrix}

Table 4. Available choice of α

There exists an algorithm for deciding whether, for two collections of integral matrices {X_{1}, …, X_{m}} and {Y_{1}, …, Y_{m}}, there exists an integral and integrally invertible matrix C relating them via conjugation, i.e., such that C^{−1}X_{i}C = Y_{i} for all i; see [48]. Unfortunately, this algorithm is complicated and does not seem to have been implemented into the standard computational programs, such as Magma. Instead of trying to find C, we take a different approach to proving that the 𝕋-modules H and H′ are isomorphic. Note that the pairing in Proposition 2.21 gives an isomorphism H ≅ Hom(𝕋, ℤ) of 𝕋-modules. If one constructs a perfect 𝕋-equivariant pairing

(10.2)  H′ × 𝕋 → ℤ,

then the desired isomorphism H′ ≅ Hom(𝕋, ℤ) ≅ H follows. The absence of Fourier expansion in the quaternionic setting makes the construction of such a pairing ad hoc. Note that Hom(H′, ℤ) = ⨁_{u∈𝔽_{q}} ℤe_{u}^{∗}, where ⟨f_{u}, e_{u′}^{∗}⟩ := f_{u}(e_{u′}) = δ_{u,u′}. One way to construct (10.2) is to find a ℤ-linear combination α = ∑_{u} a_{u}e_{u}^{∗} such that

det (⟨f_{u}, T_{x−s}α⟩)_{u,s∈𝔽_{q}} = ±1.

We were able to find such α in several cases; see Table 4 where α is given as [a_{0}, a_{1}, …, a_{p−1}].

10.3. Computation of Brandt matrices. Recall that we denote y = T^{2} + aT + b ∈ A and y′ = T^{′2} + ab^{−1}T + b^{−1}, where T′ = 1/T. Assume q is odd and b is not a square in 𝔽_{q}^{×}. Set y_{0} := bT^{′2} + aT′ + 1. Let

D := F + F i + F j + F ij

where i^{2} = T′, j^{2} = y_{0}, ij = −ji. Since we have the Hilbert quadratic symbols (T′, y_{0})_{T} = (T′, y_{0})_{y} = −1 and (T′, y_{0})_{v} = 1 for every places v ≠ T or y, the quaternion algebra D is ramified precisely at T and y.

Let D′ = A′ + A′i + A′j + A′ij, which is an Eichler A′-order in D of level T′y′. We choose the representatives of locally principal right ideal classes of D′ as follows.

I_{∞} := D′ = A′ + A′i + A′j + A′ij,
I_{u} := A′(1 − j) + A′(\frac{i + ij}{T′} + 2uj) + A′T′j + A′ij, u ∈ 𝔽_{q}.

Then the reduced norm of I_{u} is trivial for every u ∈ ℙ^{1}(𝔽_{q}). Moreover, for u, u′ ∈ 𝔽_{q}, I_{u}I_{u′}^{−1} is equal to

\begin{cases}A′ + A′T′i + A′(j − 2ui) + A (\frac{i(1+j)^{2}}{T′} + 4u^{2}i) , & if u = u′, \\ A′T′ + A(i − (u′ − u)^{−1}) + A (j − \frac{u′+u}{u′−u}) + A (\frac{i(1+j)^{2}}{T′} + \frac{4uu′}{u′−u}) , & if u ≠ u′. \end{cases}

After tedious calculations, we obtain the following:

Lemma 10.11.

(1) For s ∈ 𝔽_{q},

\frac{#\{z ∈ I_{∞} | Nr(z)A′ = (x − s)A′\}}{q − 1} = \begin{cases}2 & if s ∈ (𝔽_{q}^{×})^{2}, \\ 1 & if s = 0, \\ 0 & otherwise. \end{cases}

(2) For s, u ∈ 𝔽_{q}, set

α_{y}(s, u) := 1 + s(4u^{2} + a + sb) and β_{y}(s, u) = ((a + 4u^{2})^{2} − 4b)s + 16u^{2}.

Then

\frac{#\{z ∈ I_{u} | Nr(z)A′ = (x − s)A′\}}{q − 1} = \begin{cases}2 & if α_{y}(s, u) ∈ (𝔽_{q}^{×})^{2}, \\ 1 & if α_{y}(s, u) = 0, \\ 0 & otherwise. \end{cases}

\frac{#\{z ∈ I_{u}I_{u}^{−1} | Nr(z)A′ = (x − s)A′\}}{q − 1} = \begin{cases}2 & if β_{y}(s, u) ∈ (𝔽_{q}^{×})^{2}, \\ 1 & if β_{y}(s, u) = 0, \\ 0 & otherwise. \end{cases}

(3) For s, u, u′ ∈ 𝔽_{q} with u ≠ u′, set

ξ_{y}(s, u, u′) :=(2u^{2} + 2u^{′2} + s(u′ − u)a)^{2}
− (1 − s(u′ − u)^{2})(16u^{2}u^{′2} − s(u′ − u)^{2}(a^{2} − 4b)).

Then

\frac{#\{z ∈ I_{u}I_{u′}^{−1} | Nr(z)A′ = (x − s)A′\}}{q − 1} = \begin{cases}2 & if ξ_{y}(s, u) ∈ (𝔽_{q}^{×})^{2}, \\ 1 & if ξ_{y}(s, u) = 0, \\ 0 & otherwise; \end{cases}

Documenta Mathematica 20 (2015) 551–629

Since the Brandt matrices are symmetric under our settings, the above lemma gives us the recipe of computing the Brandt matrices B(T′ − s) for s ∈ 𝔽_{q}. In particular, we can get

γ(∞) = ∞, γ(0) = 0, and γ(u) = −u ∀u ∈ 𝔽_{q}^{×}.

Remark 10.12. In characteristic 2 the presentation of quaternion algebras has to be modified. Consider the case when q = 2 and y = T^{2} + T + 1. In this case, we let D = F + F i + F j + F ij, where

i^{2} + i = T′, j^{2} = y_{0} = T^{′2} + T′ + 1, ji = (i + 1)j.

Let D′_{max} = A + Ai + Aj + Aij, which is a maximal A′-order in D. We take the Eichler A′-order D′ = A′ + A′i + A′T′j + A′ij and the representatives of ideal classes of D′ in the following:

I_{∞} = D′ = A′ + A′i + A′T′j + A′ij,
I_{0} = A′T′ + A′(1 + i) + A′j + A′ij,
I_{1} = A′T′ + A′(1 + i) + A′(1 + j) + A′ij.

Then

I_{0}I_{0}^{−1} = A′ + A′i + A′xj + A′(i + 1)j,
I_{0}I_{1}^{−1} = A′T′ + A′i + A′(1 + j) + A′(1 + ij),
I_{1}I_{1}^{−1} = A′ + A′T′i + A′j + A′(i + ij).

Since

{z ∈ D′_{max} | Nr(z) = T′} = {i, 1 + i, 1 + T′ + j}

and

{z ∈ D′_{max} | Nr(z) = 1 + T′} = {T′ + j, T′ + i + j, 1 + T′ + i + j},

we can get

B(T′) = (\begin{matrix}2 & 1 & 2 \\ 1 & 2 & 2 \\ 2 & 2 & 1 \end{matrix}) and B(T′ − 1) = (\begin{matrix}0 & 2 & 1 \\ 2 & 0 & 1 \\ 1 & 1 & 1 \end{matrix}) .

Acknowledgements. Part of this work was carried out while the first author was visiting Taida Institute for Mathematical Sciences in Taipei and National Center for Theoretical Sciences in Hsinchu. He heartily thanks Professor Winnie Li and Professor Jing Yu for their invitation, and the members at these institutions for creating a welcoming and productive atmosphere. This research was completed while the second author was visiting Max Planck Institute for Mathematics. He wishes to thank the institute for kind hospitality and good working conditions.

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Mihran Papikian

Department of Mathematics Pennsylvania State Univ. University Park PA 16802 U.S.A. papikian@psu.edu Fu-Tsun Wei Institute of Mathematics Academia Sinica 6F Astronomy-Mathematics Building No. 1 Sec. 4 Roosevelt Road Taipei 10617 Taiwan ftwei@math.sinica.edu.tw 630

Documenta Mathematica 20 (2015)