Generalized Humbert Schemes and Intersections of Humbert Surfaces

Ernst Kani

1 Introduction

In 1899 Humbert[Hu] attached to a principally polarized abelian surface (A, λ) which satisfies a certain “singular relation” an invariant Δ which is now called a Humbert invariant of (A, λ). This invariant gives rise to Humbert surfaces as follows.

Let A_{2} denote the moduli space of principally polarized abelian surfaces (A, λ). For a positive integer n > 0, let H_{n} ⊂ A_{2} denote the subset consisting of those surfaces (A, λ) ∈ A_{2} which have a singular relation with Humbert invariant Δ = n. It then follows from Humbert’s work that H_{n} is a closed analytic surface in A_{2}, provided that n ≡ 0, 1 (mod 4). The surfaces H_{n} are now called Humbert surfaces; cf. van der Geer[vdG], Ch. IX.

Here we want to study the components of the intersection H_{n} ∩ H_{m} of two Humbert surfaces. This problem was briefly addressed on p. 214 of [vdG], but the results obtained there (via analytic techniques) are somewhat incomplete; cf. Remark 11 below.

Here give a different approach to this problem by using the refined Humbert invariant which was introduced in [K1]. This approach is completely algebraic and hence has the advantage that it also works over an arbitrary algebraically closed field K instead of ℂ.

Using this invariant, we introduce in Section 2 (closed) subsets H(q) ⊂ A_{2}(K) which are indexed by positive integral quadratic forms q in r ≥ 1 variables. If r = 1, then these are precisely the Humbert surfaces, and so we will refer to the H(q)’s as generalized Humbert schemes.

From the definition of the H(q)’s it follows easily that if m ≠ n, then

(1)  H_{m} ∩ H_{n} = ⋃_{q} H(q),

where the union runs over all equivalence classes of positive definite binary quadratic forms q which primitively represent both m and n; cf. Proposition 12.

However, if we want to make use of this formula in order to determine the irreducible components of the intersection H_{m} ∩ H_{n}, then further work is necessary, for we need to know when H(q) is non-empty and what are its irreducible components.

Here we will solve this problem only in the special case that H(q) ⊂ H_{N^{2}}, for some N ≥ 1. In this case we can use the results from [K5] to prove in §2:

Theorem 1 Let q be a positive integral binary quadratic form which primitively represents N^{2}, for some N ≥ 1 with char(K) ∤ N. Then H(q) ≠ ∅ if and only if

(∗) every integer represented by q is congruent to 0 or 1 mod 4.

By combining this theorem with formula (1), it is not difficult to show:

Corollary 2 If char(K) ∤ N, then H_{N^{2}} ∩ H_{M} is non-empty, for any M ≡ 0, 1 (mod 4).

Let M_{2}(K) ⊂ A_{2}(K) denote the open subset consisting of the Jacobians of smooth genus 2 curves. Then H_{N^{2}} ∩M_{2} classifies the genus 2 curves which have an elliptic subcover of degree N; cf. [Hu], [K1]. In their paper, Accola and Previato[AP] raised the question of whether Lange’s moduli space[La] M_{2}(n, 1) = ∪_{N|n}H_{N^{2}} ∩M_{2} is connected. This follows from the following result which partially refines Corollary 2.

Corollary 3 If char(K) ∤ N ≥ 2, then H_{N^{2}} ∩ H_{m} ∩ M_{2} is non-empty, for any m ≡ 0, 1 (mod 4), m ≥ 4.

Theorem 1 answers the question of when H(q) is non-empty. To go further and to determine the irreducible components of H(q) is a much more difficult task. To state the result, it is useful to introduce the following terminology.

Definition. Let Q(N^{2}) denote the set of binary quadratic forms satisfying the conditions of Theorem 1. Thus, q ∈ Q(N^{2}) if and only if q is a positive integral binary quadratic form which satisfies condition (∗) and which primitively represents N^{2}.

Moreover, if N, m, d are positive integers, then we say that q is of type (N, m, d) if q ∈ Q(N^{2}), if m|N and if

(2)  disc(q) = −16m^{2}d, and gcd(N/m, d) = 1.

The following result gives a partial overview of the number of irreducible components of H(q).

Theorem 4 If q ∈ Q(N^{2}), then q is of type (N, m, d), for unique integers m|N and d ≥ 1. Put c_{m}(q) = gcd(cont(q), m), where cont(q) denotes the content of q, i.e., cont(q) = gcd(a, b, c), where a, b, c are the coefficients of q. Moreover, assume that char(K) ∤ Nd.

(a) H(q) has at most 2^{ω(c_{m}(q))} irreducible components, provided that 8 ∤ c_{m}(q). Here ω(n) = |{p|n}| denotes the number of distinct prime factors of n.

(b) If d > N^{4}/(4m^{2}) and if N is odd, then H(q) has precisely 2^{ω(c_{m}(q))} irreducible components, except when q is equivalent to the binary quadratic form N^{2}X^{2} + 4dY^{2}.

The above theorem is a special case of more general results which also deal with the case that 8|c_{m}(q) and analyze the exceptional cases; cf. Theorems 49 and 55.

Note that Theorem 4(a) implies the following irreducibility criterion, which generalizes that of the case m = 1 studied in [K3].

Corollary 5 If q has type (N, m, d) and if c_{m}(q) = 1, then H(q) is irreducible, provided that char(K) ∤ Nd.

It is useful to observe that the above corollary and (1) imply the following result.

Corollary 6 If char(K) = 0 and if gcd(N, M) = 1, then H(q) is irreducible whenever q ∈ Q(N^{2}) and H(q) ⊂ H_{N^{2}} ∩ H_{M}. Thus every irreducible component of H_{N^{2}} ∩ H_{M} is of the form H(q), for a suitable q ∈ Q(N^{2}).

Theorem 4 is a actually a consequence of several much finer results. The first of these identifies the irreducible components of H(q). For this, we will generalize the method of [K3] and show that the components of H(q) are the images \overline{T} _{α}^{N} ⊂ A_{2} of certain modular curves X_{α}^{N}. These are defined by primitive integral matrices α ∈ GL_{2}^{+}(ℚ) and give rise to the modular correspondences T_{α}^{N} on X(N) × X(N); cf. Section 4. To state the result, it is useful to introduce the following notation.

Notation. Let M_{d} denote the set of primitive 2 × 2 matrices of determinant d. If α ∈ M_{d} and if N ≥ 1, then we define the binary quadratic form q_{α}^{N} by

q_{α}^{N}(X, Y) = N^{2}X^{2} + 2mtXY + \frac{m^{2}}{N^{2}} (t^{2} + 4d)Y^{2}.

Here t = trace(βα), where β = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ), and \frac{N}{m} = gcd(x − w, y, z, N), if βα = ( \begin{matrix}x & y \\ z & w \end{matrix}).

Theorem 7 If q is a binary form of type (N, m, d), and if char(K) ∤ Nd, then

(3)  H(q) = ⋃_{α} \overline{T} _{α}^{N},

where the union is over all α ∈ M_{d} such that q_{α}^{N} is equivalent to q. Thus H(q) is an equidimensional curve which is a closed subscheme of A_{2}.

Note that if m > 1, then it can happen that gcd(N, d) > 1, so in this case the above union (3) consists of the images \overline{T} _{α}^{N} of the “bad” modular correspondences T_{α}^{N} whose determinant d = det(α) is not prime to the level N.

Theorem 7 has the following consequence.

Corollary 8 Let M ≡ 0, 1 (mod 4) be a positive integer. If char(K) = 0 or if char(K) > \frac{N^{2}M}{4} , then H_{N^{2}} ∩H_{M} is an equidimensional curve, provided that N^{2} ≠ M.

While Theorem 7 identifies the irreducible components of H(q), further work is necessary in order to determine which of the components are actually distinct. For this, we show:

Theorem 9 Let N, d be positive integers with char(K) ∤ Nd. If α_{1}, α_{2} ∈ M_{d}, then

(4)  gβα_{1}g^{−1} ≡ ±βα_{2} (mod N), for some g ∈ SL_{2}(ℤ) ⇒ \overline{T} _{α_{1}}^{N} = \overline{T} _{α_{2}}^{N}.

Furthermore, the converse implication holds if d > N^{4}/(4m^{2}), where m is determined by the (common) type (N, m, d) of q_{α_{i}}^{N}, for i = 1, 2.

From this result, it is clear that the SL_{2}(ℤ/Nℤ)-conjugacy classes of 2×2-matrices α ∈ M_{2}(ℤ/Nℤ) play an important role in the determination of the irreducible components of H(q). By extending the work of Nobs[No], it is possible to give a classification of these conjugacy classes; cf. §6.

It is much more difficult to analyze the precise number of components of H(q) when d ≤ N^{4}/(4m^{2}). Here we only do this in the cases that N = 2 and 4 (cf. Proposition 51 and Corollary 56), because already in the case of an odd prime N, the analysis is long and difficult and hence has been delegated to [K7]. There the following result is obtained. For this, recall from Gauss that a binary form q is called ambiguous if q is equivalent to aX^{2} + kaXY + cY^{2}, for some a, k, c ∈ ℤ.

Theorem 10 Let q ∈ Q(N^{2}), where N is a prime. If N = 2, then H(q) is irreducible, and when N > 2, then H(q) has at most 2 irreducible components. Furthermore, if N ≡ 1 (mod 4), then

(5)  H(q) is irreducible ⇔ q is primitive, i.e., cont(q) = 1,

whereas if N ≡ 3 (mod 4), then

(6)  H(q) is irreducible ⇔ q is primitive or ambiguous.

By using these results and the reduction theory of binary quadratic forms, it is possible to work out the irreducible components of H_{N^{2}} ∩ H_{M} explicitly for small values of N and M; cf. §8. Acknowledgment. This research was supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (NSERC).

2 The refined Humbert invariant and generalized Humbert schemes

Let A be an abelian surface over an algebraically closed field K of arbitrary characteristic, and assume that A has a principal polarization λ : A →^{∼} Â. Thus, λ = ϕ_{θ} for unique class θ = θ_{λ} in the Néron-Severi group NS(A) = Div(A)/≡, where ≡ denotes numerical equivalence.

As in [K1], [K3] and [K5], the refined Humbert invariant q̃_{(A,λ)} is defined by the formula

(7)  q̃_{(A,λ)}(D) = (D.θ_{λ})^{2} − 2(D.D), for D ∈ NS(A),

where (.) denotes the intersection number of divisors on A. From the Hodge index theorem it follows easily that q_{(A,λ)} defines a positive definite quadratic form q_{(A,λ)} on the quotient group NS(A, λ) = NS(A)/ℤθ_{λ}; cf. [K1], §3. It is clear that the isomorphism class of the quadratic module (NS(A, λ), q_{(A,λ)}) depends only the isomorphism class ⟨A, λ⟩ of the principally polarized abelian surface (A, λ).

As was explained in [K1], §5, q_{(A,λ)} is closely related to the classical Humbert invariant attached to an abelian surface A/ℂ. More precisely, any primitive element D ∈ NS(A, λ) gives rise to a “singular relation” in the sense of Humbert[Hu] (and conversely), and Δ = q_{(A,λ)}(D) is the Humbert invariant attached to this “singular relation” of the principally polarized abelian surface (A, λ) as defined by Humbert[Hu]. It thus follows from this that the subset

(8)  H_{Δ} = {⟨A, λ⟩ ∈ A_{2}(K) : q_{(A,λ)} primitively represents Δ}

of the moduli space A_{2} (which classifies isomorphism classes ⟨A, λ⟩ of principally polarized abelian surfaces over K) is precisely the Humbert surface of discriminant (or invariant) Δ; cf. [vdG], §IX.2. By Humbert, this defines (for K = ℂ) an irreducible surface in A_{2}(ℂ) whenever Δ ≡ 0, 1 (mod 4), and is empty otherwise.

The definition of a Humbert surface can be generalized as follows. Given any integral positive definite quadratic form q in r variables, let

H(q) = {⟨A, λ⟩ ∈ A_{2}(K) : q_{(A,λ)} primitively represents q}.

In other words, ⟨A, λ⟩ ∈ H(q) if and only if there exists an injective homomorphism f : ℤ^{r} ↪ NS(A, λ) with NS(A, λ)/f(ℤ^{r}) torsionfree such that q = q_{(A,λ)} ∘ f. Clearly, H(q) depends only on the GL_{r}(ℤ)-equivalence class of q.

Note that if q is a 1-variable quadratic form q_{Δ}(x) = Δx^{2}, then (8) shows that H(q_{Δ}) = H_{Δ} is the classical Humbert surface of discriminant Δ. Thus, the H(q)’s generalize Humbert surfaces and hence it is fitting to call them generalized Humbert sets. Below we will show (in certain cases) that they are closed subsets of A_{2}, so they inherit a scheme structure from A_{2} and hence can be called Humbert schemes.

Remark 11 When K = ℂ and q is a binary quadratic form, then H(q) is closely related to the (analytic) curves C_{U} defined on p. 214 of [vdG]. These curves are defined by positive definite 2-dimensional quadratic subspaces (U, ϕ) of a certain 5dimensional quadratic subspace (V_{ℚ}, Δ) of signature (3, 2). Unfortunately, van der Geer does not explain which (U, ϕ) can arise in this way, but only gives some necessary conditions. Moreover, he does not discuss the irreducible components of C_{U}.

As was mentioned in the introduction, the H(q)’s can be used to describe intersections of Humbert surfaces:

Proposition 12 If m and n are distinct positive integers, then

(9)  H_{m} ∩ H_{n} = ⋃_{q} H(q),

where the union runs over all equivalence classes of positive definite binary quadratic forms q which primitively represent both m and n.

Proof. Let q be such a form, and let ⟨A, λ⟩ ∈ H(q). Then q_{(A,λ)} primitively represents q. Since m is primitively represented q, it follows that m also primitively represented by q_{(A,λ)}, so ⟨A, λ⟩ ∈ H_{m} by (8). Thus H(q) ⊂ H_{m}, and similarly, H(q) ⊂ H_{n}, so H(q) ⊂ H_{n} ∩ H_{m}. This shows that the right side of (9) is contained in the left side.

Conversely, suppose that ⟨A, λ⟩ ∈ H_{m} ∩ H_{n}. Then there exist primitive vectors v, w ∈ M := NS(A, λ) such that q_{(A,λ)}(v) = m and q_{(A,λ)}(w) = n. If v and w were linearly dependent, then v = ±w and hence q_{(A,λ)}(v) = q_{(A,λ)}(w), contrary to the hypothesis. Thus, v and w are linearly independent and hence M_{0} := ℤv + ℤw has rank 2. Let M_{1} be the saturation of M_{0} in M. Then the restriction q of q_{(A,λ)} to M_{1} is a positive definite, binary quadratic form which is primitively represented by q_{(A,λ)}, and so ⟨A, λ⟩ ∈ H(q). Moreover, m = q(v) is primitively represented by q (because v is primitive in M, hence also in M_{1}. Similarly, n = q(w) is primitively represented by q. Thus q is one of the forms of the right side of (9), so ⟨A, λ⟩ ∈ ∪H(q).

Remark 13 (a) Note that there are only finitely many equivalence classes of forms q satisfying the conditions of Proposition 12 because their discriminants are bounded. Indeed, it follows from the above proof that we have |disc(q)| ≤ 4mn for any such q because |disc(q)| ≤ |disc(q_{|M_{0}|})| ≤ 4mn.

(b) The above proposition can be viewed as giving a partial answer to a question raised by McMullen[Mc], p. 96.

Proof of Theorem 1. Suppose first that H(q) ≠ ∅, so there exists ⟨A, λ⟩ ∈ A_{2}(K) such that q_{(A,λ)} primitively represents q. Now q_{(A,λ)} satisfies condition (∗) because (D.D) = 2χ(L(D)) ∈ 2ℤ by the Riemann-Roch Theorem and so (7) shows that q_{(A,λ)}(D) ≡ (D.θ_{λ})^{2} (mod 4), for all D ∈ Div(A). Thus, q also satisfies condition (∗).

Conversely, suppose that q satisfies condition (∗) and primitively represents N^{2}. Then by Proposition 7 of [K5] we know that q has type (N, m, d) for a unique pair m, d ≥ 1 with m|N and gcd(N/m, d) = 1. Since char(K) ∤ N, we have by Corollary 33 of [K5] that there exists a principally polarized abelian surface (A, λ) such that q_{(A,λ)} primitively represents q. Thus ⟨A, λ⟩ ∈ H(q), so H(q) ≠ ∅.

Proof of Corollary 2. We may assume that M ≠ N^{2} and that M > 1. (If M = 1, then interchange M and N^{2}.) Put ε = rem(M, 4) ∈ {0, 1} and consider the binary quadratic form q(X, Y) := N^{2}X^{2} + 2εNXY + MY^{2} which is positive definite because disc(q) = −4N^{2}(M − ε^{2}) < 0. Since q(1, 0) = N^{2} and q(0, 1) = M, we see that q primitively represents N^{2} and M, so H(q) ⊂ H_{N^{2}} ∩H_{M} by Proposition 12. Moreover, since q(X, Y) ≡ (NX + εY)^{2} (mod 4), for all X, Y ∈ ℤ, we see that property (∗) holds. Thus H(q) ≠ ∅ by Theorem 1, and so H_{N^{2}} ∩ H_{M} ≠ ∅, as claimed.

Proof of Corollary 3. Let q be as in the proof of Corollary 2, so q is a positive form which primitively represents N^{2} and M and satisfies (*). Thus, by Proposition 7 of [K5] there exists m|N and d ≥ 1 such that q has type (N, m, d). Moreover, since q(x, y) = (Nx + εy)^{2} + (M − ε^{2})y^{2} ≥ min(N^{2}, M − ε^{2}) ≥ 3, if (x, y) ≠ (0, 0), we see that q(x, y) ≠ 1, for all x, y ∈ ℤ.

Suppose first that K is not the algebraic closure of a finite field. Then we are in the situation of Corollary 34 of [K5], so there is a curve C/K such that q_{C} ∼ q, which means that ⟨J_{C}, θ_{C}⟩ ∈ H(q) ∩ M_{2} ⊂ H_{N^{2}} ∩ H_{M} ∩ M_{2}. This proves the assertion in this case.

Now suppose that K = \overline{𝔽}_{p} is the algebraic closure of a finite field. Choose an algebraically closed overfield K′ ⊃ K which is transcendental over K. Since H′(q) := H(q)∩M_{2} is a closed algebraic subset of M_{2} (cf. Theorem 7), and since H′(q)(K′) ≠ ∅ by what was proved before, it follows that also H′(q)(K) ≠ ∅, and so the assertion also holds in this case.

Remark 14 It follows from Corollary 3 that in particular H_{N^{2}} ∩ H_{M^{2}} ∩ M_{2} ≠ ∅ whenever N, M ≥ 2. This fact also follows from a result of [FPR]. Since H_{N^{2}} is irreducible by Humbert[Hu] and/or by Theorem 15 below, it thus follows that the moduli space

M_{2}(n, 1) = ⋃_{N|n,N>1} H_{N^{2}} ∩ M_{2}

consisting of all genus 2 curves admitting a surjective morphism of degree n to a curve of genus 1 is connected (provided that char(K) ∤ n). This answers a question raised (implicitly) by Accola and Previato[AP], §1.7. Note that the moduli space M_{2}(n, 1) was studied by Lange[La].

3 The morphism β_{N}

As a first step towards proving the main results of this paper, we recall from [K1], §3, and/or [FK], §4.4, that the Humbert surface H_{N^{2}} can be covered by the product surface X(N) × X(N), where X(N)/K is the (affine) modular curve of level N. In other words, there exists a finite morphism

β_{N} : X(N) × X(N) → A_{2}

whose image is the Humbert surface H_{N^{2}}. This morphism is, in fact, a variant of the “basic construction” of [FK], §2.3.

For our purposes, however, we need not only the existence (and finiteness) of this morphism, but also its modular description, which is a modification of that of [FK].

For this, fix an integer N ≥ 1 with char(K) ∤ N, where (as before) K is an algebraically closed field. Moreover, fix a primitive N-th root of unity ζ_{N} ∈ K^{×}. Recall from [DR], [KM] and/or the discussion and references of [K6], §4, that the (affine) modular curve X(N)/K (coarsely) represents the functor

X (N) = X (N)_{ζ_{N}} : (Sch/K) → (Sets)

which is defined by X (N)(S) = {⟨E/S, ϕ⟩}, where S is a K-scheme, E/S is an elliptic curve, ϕ : (ℤ/Nℤ)^{2} →^{∼} E[N] is a level N-structure of fixed determinant ζ_{N} ∈ K^{×}, and ⟨E/S, ϕ⟩ denotes the isomorphism class of the pair (E/S, ϕ).

Moreover, recall from [M1], Theorem 7.10, that the moduli space A_{2}/K coarsely represents the functor A_{2} = A_{2,1,1} : (Sch/K) → (Sets) which classifies principally polarized abelian surfaces, i.e., if S is a K-scheme, then A_{2}(S) is the set of isomorphism classes ⟨A, λ⟩ of pairs (A, λ) where A/S is an abelian scheme of relative dimension 2 and λ : A →^{∼} Â is a principal polarization.

Modifying the construction of [FK] slightly, we will define the morphism (of functors)

β̃_{N} : X (N) × X (N) → A_{2}

as follows. Let ⟨E_{i}/S, ϕ_{i}⟩ ∈ X (N)(S), where i = 1, 2, and put

(10)  ψ = ψ_{ϕ_{1},ϕ_{2}} := ϕ_{2} ∘ [\binom{1 0}{0 −1}]_{N} ∘ ϕ_{1}^{−1} : E_{1}[N] → E_{2}[N],

where, for an integral 2 × 2 matrix α, the symbol [α]_{N} denotes the endomorphism [α]_{N} ∈ End((ℤ/Nℤ)^{2}) whose matrix is α (mod N) with respect to the canonical basis ((1, 0), (0, 1)) of (ℤ/Nℤ)^{2}.

Let π_{ψ} : E_{1} × E_{2} → A_{ψ} := (E_{1} × E_{2})/G_{ψ} be the quotient map, where G_{ψ} = Graph(−ψ) ≤ E_{1}[N] × E_{2}[N] = (E_{1} × E_{2})[N]. Since ψ : E_{1}[N] → E_{2}[N] is an anti-isometry, there is a unique principal polarization λ_{ψ} : A_{ψ} → Â_{ψ} such that

(11)  π̂_{ψ} ∘ λ_{ψ} ∘ π_{ψ} = N(λ_{E_{1}} ⊗ λ_{E_{2}}),

where λ_{E_{1}} ⊗ λ_{E_{2}} is the product polarization on E_{1} × E_{2}; cf. [K2], Prop. 5.7. Thus ⟨A_{ψ}, λ_{ψ}⟩ ∈ A_{2}(S), and so the rule ⟨E_{1}/S, ϕ_{1}; E_{2}/S, ϕ_{2}⟩ ↦ ⟨A_{ψ}, λ_{ψ}⟩ defines a map

β̃_{N,S} : (X (N) × X (N))(S) → A_{2}(S).

Since the maps β̃_{N,S} are compatible with base-change, they define a morphism of functors β̃_{N} = {β̃_{N,S}}_{S}. By the universal property of the (coarse) moduli scheme X(N) × X(N), we thus have an induced K-morphism

β_{N} : X(N) × X(N) → A_{2}.

Theorem 15 The above morphism β_{N} : X(N) × X(N) → A_{2} is finite and its image is the Humbert surface H_{N^{2}}. In particular, H_{N^{2}} is an affine closed irreducible subscheme of A_{2} of dimension 2.

Proof. Although this follows from Corollary 3.10 of [K1], no proof is given there, so it is useful to present one here. (See also [FK], Proposition 4.11).

We first show that Im(β_{N}) = H_{N^{2}}. Indeed, if x = (x_{1}, x_{2}) ∈ (X(N) × X(N))(K), then each x_{i} corresponds to a tuple ⟨E_{i}/K, ϕ_{i}⟩ ∈ X (N)(K). By [K2], Corollary 5.9, the principally polarized surface (A_{ψ}, λ_{ψ}) contains an elliptic subgroup of degree N in the sense of [K1], and so by [K1], Corollary 3.9, we see that β_{N}(x) = ⟨A_{ψ}, λ_{ψ}⟩ ∈ H_{N^{2}}. Thus Im(β_{N}) ⊂ H_{N^{2}}. Conversely, if y = ⟨A, λ⟩ ∈ H_{N^{2}}, then by [K1], Corollary 3.9 and Theorem 3.1, (A, λ) has an elliptic subgroup E of degree N, and so by [K2], Propositions 5.2 and 5.5, there is another elliptic curve E′ and an anti-isometry ψ : E[N] → E′[N] such that (A_{ψ}, λ_{ψ}) ≃ (A, λ). Since K is algebraically closed and char(K) ∤ N, there exists a level N-structure ϕ such that ⟨E/K, ϕ⟩ ∈ X (N)(K). Since ψ is an anti-isometry, there is a unique level N-structure ϕ′ such that ⟨E′/K, ϕ′⟩ ∈ X (N)(K) and ψ_{ϕ,ϕ′} = ψ. We then have that β̃_{N,K}(⟨E/K, ϕ; E′/K, ϕ′⟩) = ⟨A, λ⟩, and so ⟨A, λ⟩ ∈ Im(β_{N}). This proves that Im(β_{N}) = H_{N^{2}}.

We next observe that β_{N} is quasi-finite. For this, let x′ ∈ β_{N}^{−1}(β_{N}(x)), where x = (x_{1}, x_{2}) is as above. Write x′ = (x′_{1}, x′_{2}), where x′_{i} = ⟨E′_{i}/K, ϕ′_{i}⟩ ∈ X (N)(K) and put ψ′ = ψ_{ϕ′_{1},ϕ′_{2}}. To show that β_{N}^{−1}(β_{N}(x′)) is finite, it suffices to show that the E′_{i}’s lie in finitely many K-isomorphism classes.

By hypothesis, ⟨A_{ψ}, λ_{ψ}⟩ = ⟨A_{ψ′}, λ_{ψ′}⟩, where ψ′ = ψ_{ϕ′_{1},ϕ′_{2}}, so there is an isomorphism α : A_{ψ} →^{∼} A_{ψ′} such that λ_{ψ} = α̂ ∘ λ_{ψ′} ∘ α. Thus, each elliptic subgroup of A_{ψ′} of degree N is isomorphic to an elliptic subgroup of A_{ψ} of degree N. In particular, since E_{ψ′,1} := π_{ψ′}(E′_{1} × {0}) ≃ E′_{1} and E_{ψ′,2} := π_{ψ′}({0} × E′_{2}) ≃ E′_{2} are both elliptic subgroups of degree N on A_{ψ′}, we see that E′_{1} and E′_{2} are each isomorphic to an elliptic subgroup of degree N on A_{ψ}. Thus the finiteness of β_{N}^{−1}(β_{N}(x′)) follows because A_{ψ} has only finitely many elliptic subgroups E ≤ A_{ψ} of degree N. (Indeed, since q_{(A,λ)} is positive-definite, there are only finitely many D̄ ∈ NS(A, λ) such that q_{(A,λ)}(D̄) = N^{2}, and so the assertion follows from the bijection constructed in [K1], Theorem 3.1.)

This, therefore, shows that β_{N} is quasi-finite at closed points. Since the quasifinite locus is open by [EGA], (IV, 13.1.4), it follows that β_{N} is quasi-finite because X(N) × X(N) is a Jacobson scheme.

To prove that β_{N} is actually finite, it suffices by [EGA], (III, 4.4.2) or (IV, 8.11.1), to verify that β_{N} is proper because β_{N} is quasi-finite. Moreover, since X(N) and A_{2} are separated K-schemes of finite type, it is enough to check that the functor β̃_{N} satisfies the valuative criterion of properness. Thus, let S = Spec(R), where R is a discrete valuation ring with quotient field F ⊃ K and let y = ⟨A, λ⟩ ∈ A_{2}(S) be such that there exists x_{F} = ⟨E_{1}, ϕ_{1}; E_{2}, ϕ_{2}⟩ ∈ (X (N) × X (N))(F) with β̃_{N,F} (x_{F}) = ⟨A_{F} , λ_{F} ⟩, where A_{F} = A ⊗ F and λ_{F} = λ ⊗ F. We want to show that x_{F} extends to x ∈ (X (N) × X (N))(S) and that β̃_{N,S}(x) = y. For this we observe that since A_{F} ≃ A_{ψ} is isogenous to E_{1} × E_{2} and A_{F} has good reduction over R by hypothesis, it follows that the same is true for E_{i} (cf. [ST], Cor. 2 on p. 493). Thus, there exist elliptic curves Ẽ_{i}/R with Ẽ_{i} ⊗ F = E_{i}. By [KM], Th. 3.7.1, it follows that ϕ_{i} extends uniquely to an isomorphism ϕ̃_{i} : (ℤ/Nℤ)^{2} →^{∼} Ẽ_{i}[N] (of determinant ζ_{N}) and so x̃ := ⟨Ẽ_{1}, ϕ̃_{1}; Ẽ_{2}, ϕ̃_{2}⟩ lies in (X (N) × X (N))(S) and satisfies x̃_{F} = x_{F}. Thus ỹ = β̃_{N,S}(x̃) satisfies ỹ_{F} = y_{F}, and so β_{N,S}(x̃) = y because A is the Neron model of A_{F}. Thus, β̃_{N} satisfies the valuative criterion of properness, and so β_{N} is proper and hence finite.

Since β_{N} is a closed map, it follows that H_{H^{2}} = β_{N}(X(N) × X(N)) is a closed subset of A_{2}. Moreover, since X(N) × X(N) is an affine and irreducible surface, it follows from Chevalley’s Theorem ([EGA] (II, 6.7.1)) that the same is true for H_{N^{2}}.

4 The modular correspondence T_{α}^{N}

From Theorem 15 we see that every curve on the Humbert surface H_{N^{2}} is the image of a curve on the product surface X(N)×X(N), and so the irreducible curve components of the intersection H_{N^{2}} ∩H_{M} can be obtained as images of curves lying on the product surface.

As is well-known (cf. [Sh]), the product surface X(N)×X(N) contains a multitude of interesting curves called modular correspondences. If K = ℂ, then each such curve is defined analytically as the image of the graph of a primitive integral matrix α ∈ GL_{2}^{+}(ℚ) acting on the upper half plane ℌ and hence is denoted by T_{α}^{N}.

As we shall see (cf. Theorem 7), the irreducible components of H_{N^{2}} ∩ H_{M} are all of the form β_{N}(T_{α}^{N}), for suitable matrices α. To prove this, we require a modular interpretation of the modular correspondences T_{α}^{N}. Such an interpretation was given in [K6]. However, in place of working with T_{α}^{N} directly, it is more convenient to work with the normalization τ_{α}^{N} : X_{α}^{N} → T_{α}^{N} of T_{α}^{N}.

In order to give the modular interpretation of the modular curves X_{α}^{N} and T_{α}^{N}, it is useful to introduce the following terminology.

Definition. Let α be an integral 2 × 2 matrix. If f : E → E′ is an S-isogeny of elliptic curves over a K-scheme S, and if ϕ and ϕ′ are level N-structures of E/S and E′/S respectively, then we say that the tuple (ϕ, f, ϕ′) is α-compatible if

(12)  f ∘ ϕ = ϕ′ ∘ [α]_{N}.

If α ∈ M_{d} is a primitive matrix of determinant d, then let X_{α}^{N} denote the moduli functor defined by the isomorphism classes of α-compatible tuples, i.e.,

X_{α}^{N}(S) = {⟨E/S, ϕ, E′/S, ϕ′, f⟩ : ⟨E/S, ϕ⟩, ⟨E′/S, ϕ′⟩ ∈ X (N)(S), f : E → E′ is a cyclic S-isogeny of degree d such that (ϕ, f, ϕ′) is α-compatible}.

Here, two such 5-tuples (E_{i}/S, ϕ_{i}, E′_{i}/S, ϕ′_{i}, f_{i}), for i = 1, 2, are isomorphic if there

exist S-isomorphisms g : E_{1} →^{∼} E_{2} and g′ : E′_{1} →^{∼} E′_{2} with f_{2} ∘ g = g′ ∘ f_{1}, ϕ_{2} = g ∘ ϕ_{1}

and ϕ′_{2} = g′ ∘ ϕ′_{1}.

The following result follows from the more general results of [K6].

Theorem 16 If char(K) ∤ Nd, then the moduli functor X_{α}^{N} is coarsely represented by a smooth, geometrically irreducible affine curve X_{α}^{N}/K which is a quotient of the modular curve X(Nd). If, moreover, N ≥ 3, then X_{α}^{N} is finely represented by X_{α}^{N}.

Proof. [K6], Theorems 2 and 3.

Remark 17 It was also shown in [K6] that the isomorphism class of the curves X_{α}^{N} only depends on d = det(α) and on N, and that when K = ℂ, then X_{α}^{N} is analytically isomorphic to the open Riemann surface Γ_{0}(N, d)\ℌ, where ℌ is the upper half plane and Γ_{0}(N, d) = Γ(N) ∩ Γ_{0}(Nd).

As was mentioned above, the curve X_{α}^{N} is the normalization of the modular correspondence T_{α}^{N} on X(N) × X(N). More precisely, the normalization morphism τ_{α}^{N} : X_{A}^{N} → T_{α}^{N} is induced by the forget morphism τ̃_{α}^{N} = {(τ_{α}^{N})_{S}}_{S} : X_{α}^{N} → X (N)×X (N), which is defined by the rule

(τ̃_{α}^{N})_{S}(⟨E_{1}/S, ϕ_{1}, E_{2}/S, ϕ_{2}, f⟩}) = ⟨E_{1}/S, ϕ_{1}, E_{2}/S, ϕ_{2}⟩.

Thus, by the universal property of the coarse moduli scheme X_{A}^{N}, the morphism τ̃_{A}^{N} induces a morphism of K-schemes τ_{α}^{N} : X_{α}^{N} → X(N) × X(N) and we have

Proposition 18 The morphism τ_{α}^{N} is finite and hence its image T_{α}^{N} is a closed irreducible curve on X(N) × X(N). Moreover, τ_{α}^{N} birational onto its image, and so X_{α}^{N} is the normalization of T_{α}^{N}.

Proof. [K6], Proposition 30.

Remark 19 It follows from the above that if x_{i} = ⟨E_{i}/K, ϕ_{i}⟩ ∈ X(N)(K), where i = 1, 2, then

(13) (x_{1}, x_{2}) ∈ T_{α}^{N}(K) ⇔ ∃ a cyclic isogeny f : E_{1} → E_{2} of degree d = det(α) such that (ϕ_{1}, f, ϕ_{2}) is α-compatible.

Corollary 20 If char(K) ∤ Nd, then for every α ∈ M_{d} there is a unique finite morphism

β_{α}^{N} : X_{α}^{N} → A_{2}

with the property that if x ∈ X_{α}^{N}(K) is defined by ⟨E_{1}/K, ϕ_{1}, E_{2}/K, ϕ_{2}, f⟩ ∈ X_{α}^{N}(K), then y = β_{α}^{N}(x) is given by ⟨A_{ψ}, λ_{ψ}⟩ ∈ A_{2}(K), where ψ = ψ_{ϕ_{1},ϕ_{2}} : E_{1}[N] → E_{2}[N] is defined by (10). Moreover, we have the relation

(14)  ϕ_{1}^{−1} ∘ ψ^{−1} ∘ f ∘ ϕ_{1} = [βα]_{N}, where β = \binom{1 0}{0 −1}.

Proof. Consider the morphism β_{α}^{N} := β_{N} ∘ τ_{α}^{N} : X_{α}^{N} → A_{2}, where τ_{α}^{N} is as in Proposition 18 and β_{N} as in Theorem 15. Since both τ_{α}^{N} and β_{N} are finite morphisms, so is β_{α}^{N}. It is clear from the definitions that β_{α}^{N} satisfies the asserted property. Since K is algebraically closed, this property determines β_{α}^{N} at all closed points of X_{α}^{N}, and hence also the morphism β_{α}^{N} itself because X_{α}^{N} is reduced (and Jacobson). Finally, property (14) follows directly from (12) and (10).

Remark 21 It follows from Corollary 20 and Theorem 16 that if char(K) ∤ Nd and if α ∈ M_{d}, then

\overline{T} _{α}^{N} := β_{α}^{N}(X_{α}^{N}) = β_{N}(T_{α}^{N})

is an irreducible curve lying on the modular three-fold A_{2}. Note that since the definition of X_{α}^{N} depends only on N, d and [α]_{N}, it follows that

(15)  α_{1} ≡ α_{2} (mod N) ⇒ \overline{T} _{α_{1}}^{N} = \overline{T} _{α_{2}}^{N}, for all α_{1}, α_{2} ∈ M_{d}.

5 The irreducible components of H(q)

By using the curves \overline{T} _{α}^{N} constructed in the previous section, we can now identify the irreducible components of the generalized Humbert schemes H(q), where q is a form of type (N, m, d). These results depend heavily on the results of [K5].

As a first step, we show that the curve \overline{T} _{α}^{N} is contained in H(q_{α}^{N}), where q_{α}^{N} is the binary quadratic form defined in §1.

Proposition 22 Let N and d be positive integers, and let α ∈ M_{d} be a primitive matrix of determinant d. Write βα = ( \begin{matrix}x & y \\ z & w \end{matrix}), where β = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ), and put g = gcd(x − w, y, z, N), m = \frac{N}{g}, and t = trace(βα) = x + w. Then the binary quadratic form q_{α}^{N} defined by q_{α}^{N} = [N^{2}, 2mt, \frac{m^{2}}{N^{2}} (t^{2} + 4d)], i.e., by

q_{α}^{N}(X, Y) = N^{2}X^{2} + 2mtXY + \frac{m^{2}}{N^{2}} (t^{2} + 4d)Y^{2},

has type (N, m, d), and we have that \overline{T} _{α}^{N} ⊂ H(q_{α}^{N}), provided that char(K) ∤ Nd.

Proof. Since βα is primitive and det(βα) = −d, it follows from Corollary 12 of [K5] that q_{α}^{N} has type (N, m, d). (Note that there is a typo in this corollary: the hypothesis “N-primitive” has to be replaced by “primitive”. Note also that the form given there is GL_{2}(ℤ)-equivalent to q_{α}^{N}.)

Assume now that char(K) ∤ Nd. To show that \overline{T} _{α}^{N} ⊂ H(q_{α}^{N}), let y ∈ \overline{T} _{α}^{N}(K). Since K is algebraically closed, there exists an x ∈ X_{α}^{N}(K) such that β_{α}^{N}(x) = y, and x is given by a 5-tuple ⟨E/K, ϕ, E′/K, ϕ′, f⟩ ∈ X_{α}^{N}(K). Then by Corollary 20 we have that y is given by ⟨A_{ψ}, λ_{ψ}⟩ ∈ A_{2}(K), where ψ = ψ_{ϕ,ϕ′} is as in (10). Thus, (E, E′, ψ, π_{ψ}) is an N-presentation of (A_{ψ}, λ_{ψ}) in the sense of [K5], §3.

By (14) we see that α′ := βα is the matrix of ψ^{−1} ∘ f_{|E[N]} with respect to the basis (P, Q) of E[N] defined by ϕ. Since α′ is primitive of determinant −d and f is cyclic of degree d, it follows from Proposition 28 and Remark 29 of [K5] that q_{(A_{ψ},λ_{ψ})} primitively represents q_{α}^{N}, and so y ∈ H(q_{α}^{N}). Thus \overline{T} _{α}^{N} ⊂ H(q_{α}^{N}), as claimed.

Proof of Theorem 7. By Proposition 22 we see that the right hand side of (3) is contained in the left hand side because H(q) = H(q_{α}^{N}), if q ∼ q_{α}^{N}.

To prove the opposite inclusion, let y ∈ H(q) be given by ⟨J, λ⟩ ∈ A_{2}(K). Then the definition of H(q) implies that q_{(J,λ)} primitively represents q, and so there exists a primitive submodule Ḡ of NS(B, λ) such that the restriction of q_{(J,λ)} to Ḡ is equivalent to q. Since q has type (N, m, d) by hypothesis, it follows from Theorem 31 of [K5] that there exists an N-presentation (E, E′, ψ, π) of (J, λ), and a cyclic isogeny h : E → E′ of degree d and an integer k with gcd(k, N/m) = 1 such that if M is any integral matrix with the property that M (mod N) is the matrix of ψ^{−1} ∘ h_{|E[N]} (with respect to some basis of E[N]), then q is GL_{2}(ℤ)-equivalent to the form q_{M,N,d,k} := [N^{2}, −2mT, \frac{m^{2}}{N^{2}} (T^{2} + 4d)], where T = −tr(M) + dk^{3}(det(M) + d).

Fix a level N-structure ϕ : (ℤ/Nℤ)^{2} →^{∼} E[N] of E, and put ϕ′ := ψ ∘ ϕ ∘ [β]_{N}. Note that ϕ′ is a level N-structure of E′, as is easy to check. Since ϕ^{−1} ∘ ψ^{−1} ∘ h ∘ ϕ ∈ End((ℤ/Nℤ)^{2}), it follows that there is an integral matrix M ∈ M_{2}(ℤ) such that

(16)  ϕ^{−1} ∘ ψ^{−1} ∘ h ∘ ϕ = [M]_{N}.

Note that this means that M (mod N) is the matrix of ψ^{−1} ∘ h_{|E[N]} with respect to the basis B = (ϕ(1, 0), ϕ(0, 1)) defined by ϕ, and so q is GL_{2}(ℤ)-equivalent to q_{M,N,d,k}.

Since h is a cyclic isogeny, it follows that M is N-primitive. Moreover, since ψ is an anti-isometry, we have that det(M) ≡ − deg(h) ≡ −d (mod N). Thus, by [K5], Lemma 30, there exists a primitive matrix α′ of determinant −d such that α′ ≡ M (mod N). Put α := βα′. Then α ∈ M_{d}, and [βα]_{N} = [M]_{N}, so q is GL_{2}(ℤ)- equivalent to q_{βα,N,d,k} = q_{α}^{N}, and it follows from (16) and the definition of ϕ′ that (ϕ, h, ϕ′) is α-compatible. Thus x = ⟨E/K, ϕ, E′/K, ϕ′, h⟩ ∈ X_{α}^{N} and we have by Corollary 20 that β_{α}^{N}(x) = y. This shows that y ∈ \overline{T} _{α}^{N}, for suitable α ∈ M_{d} with q_{α}^{N} ∼ q (where ∼ denotes GL_{2}(ℤ)-equivalence), and so (3) holds.

To prove the last assertion, note that it follows from (15) that the union on the right side of (3) consists of only finitely many distinct curves \overline{T} _{α}^{N}. Since each of these is a closed subset of A_{2} by Corollary 20, it follows that H(q) is a closed subset of A_{2}(K). Furthermore, since H(q) ≠ ∅ by Theorem 1 and since the \overline{T} _{α}^{N}’s are irreducible closed curves, it follows that H(q) is an equidimensional curve.

Remark 23 As the above proof shows, there exists a closed subscheme H(q)_{K} of A_{2}/K such that its set of K-rational points equals H(q) ⊂ A_{2}(K). We note that it follows from Theorem 7 that the scheme H(q)_{K} is compatible with base-change: if K′ is an algebraically closed extension field of K, then H(q)_{K} ⊗ K′ = H(q)_{K′} because the \overline{T} _{α}^{N}’s are compatible with base-change.

Proof of Corollary 8. From Corollary 2 it follows that H_{N^{2}} ∩ H_{M} ≠ ∅ and by Proposition 12 and Theorem 4 we know that H_{N^{2}} ∩ H_{M} is a union of H(q)’s, where q has type (N, m, d), for some m|N and some d ≥ 1. Since disc(q) = −16m^{2}d, we see that d ≤ m^{2}d = \frac{|disc(q)|}{16} ≤ \frac{N^{2}M}{4} by Remark 13. Thus, by the given hypothesis we see that char(K) ∤ Nd, and so by Theorem 7 we have that each such H(q) is an equidimensional curve, and hence so is H_{N^{2}} ∩ H_{N}.

For later use, we note the following partial refinement of Theorem 7.

Proposition 24 Let q be a form of type (N, m, d) and let α ∈ M_{d_{1}}. If char(K) ∤ Ndd_{1}, then \overline{T} _{α}^{N} ⊂ H(q) if and only if q is GL_{2}(ℤ)-equivalent to q_{α}^{N}, i.e., q ∼ q_{α}^{N}. If this is the case, then d_{1} = d.

The proof uses the following technical fact about non-CM points in X_{α}^{N} which will be used several times below. A point x = ⟨E/K, ϕ, E′/K, ϕ′, h⟩ ∈ X_{α}^{N}(K) is called a non-CM point if End(E) = ℤ.

Lemma 25 Let α ∈ M_{d} and N ≥ 1, and suppose that char(K) ∤ Nd and that K is not the algebraic closure of a finite field.

(a) There exist infinitely many non-CM points in X_{α}^{N}(K).

(b) If x ∈ X_{α}^{N}(K) is a non-CM point, and if β_{α}^{N}(x) = ⟨J, λ⟩, then q_{(J,λ)} ∼ q_{α}^{N}.

Proof. (a) The hypothesis on K implies that there exist infinitely many non-isomor-phic elliptic curves E_{i}/K such that End(E_{i}) = ℤ. Fix a level N-structure ϕ_{i} on E_{i}, and choose a cyclic subgroup C_{i} ≤ E_{i} of order Nd such that C_{i}[N] is generated by P_{i} = ϕ_{i1}(1, 0). Let h_{i} : E_{i} → E′_{i} := E_{i}/C_{i}[d] be the quotient isogeny. Then by [K6], Proposition 24, there exists a level N-structure ϕ′_{i1} on E′_{i} such that (ϕ_{i1}, h_{i}, ϕ′_{i1}) is α_{d}-compatible, where α_{d} = \binom{d 0}{0 1}. Now α = g_{1}α_{d}g_{2}, for some g_{1}, g_{2} ∈ SL_{2}(ℤ), so if we put ϕ_{i} = ϕ_{i1} ∘ [g_{2}]_{N} and ϕ′_{i} = ϕ′_{i1} ∘ [g_{1}^{−1}]_{N}, then (ϕ_{i}, h_{i}, ϕ′_{i}) is α-compatible, as is easy to check. Thus, x_{i} = ⟨E_{i}/K, ϕ_{i}, E′_{i}/K, ϕ′_{i}, h_{i}⟩ ∈ X_{α}^{N}(K) is a non-CM point. Moreover, x_{i} ≠ x_{j}, if i ≠ j (because E_{i} ≄ E_{j}), and so we obtain infinitely many non-CM points.

(b) By Proposition 22 we know that q_{(J,λ)} primitively represents q_{α}^{N}. But since Hom(E, E′) ≃ ℤ, we see that rank(q_{(J,λ)}) = 2, and so it follows that q_{(J,λ)} ∼ q_{α}^{N}.

Proof of Proposition 24. If q ∼ q_{α}^{N}, then by definition H(q) = H(q_{α}^{N}), and so \overline{T} _{α}^{N} ⊂ H(q_{α}^{N}) = H(q) by Proposition 22.

Now suppose conversely that \overline{T} _{α}^{N} ⊂ H(q). Then this also holds after a base-change of K (cf. Remark 23), so we may assume that K is not the algebraic closure of a finite field. Then by Lemma 25 there exists ⟨J, λ⟩ ∈ \overline{T} _{α}^{N}(K) such that q_{(J,λ)} ∼ q_{α}^{N}. On the other hand, since ⟨J, λ⟩ ∈ H(q), we have that q_{(J,λ)} primitively represents q, so q_{(J,λ)} ∼ q, and hence q_{α}^{N} ∼ q, as claimed.

To prove the last assertion, recall from [K5], Proposition 11, that q_{α}^{N} has type (N, m_{1}, d_{1}) for some unique m_{1}|N with gcd(d_{1}, \frac{N}{m_{1}} ) = 1. Now if q_{α}^{N} ∼ q, then q_{α}^{N} also has type (N, m, d), and so we have that m_{1} = m and d_{1} = d by [K5], Proposition 7.

We observe that the above Proposition 24 has the following interesting application.

Corollary 26 Let q_{i} ∈ Q(N, m_{i}, d_{i}), where i = 1, 2, be two binary forms which are not equivalent. If char(K) ∤ Nd_{1}d_{2}, then H(q_{1}) ∩ H(q_{2}) consists of finitely many CM-points.

Proof. If H(q_{1}) ∩ H(q_{2}) were infinite, then from Theorem 7 it follows that H(q_{1}) and H(q_{2}) would contain a common irreducible curve component T_{α}^{N}, where α ∈ M_{d_{1}}. By Proposition 24 we then have that q_{α}^{N} ∼ q_{i}, for i = 1, 2, and so q_{1} ∼ q_{2}, contrary to our hypothesis. Thus H(q_{1})∩H(q_{2}) is a finite set. Furthermore, if any of these intersection points ⟨J, λ⟩ were images of non-CM points, then by the proof of Proposition 24 we see that q_{(J,λ)} ∼ q_{i}, for i = 1, 2, so again we obtain the contradiction that q_{1} ∼ q_{2}.

We now turn to the proof of Theorem 9 of the introduction. The first assertion follows easily from the following lemma.

Lemma 27 Let α_{1}, α_{2} ∈ M_{d}, and suppose that

βα_{2} ≡ εg^{−1}βα_{1}g (mod N), for some g ∈ SL_{2}(ℤ) and ε ∈ {1, −1}.

If x_{1} := ⟨E/K, ϕ; E′/K, ϕ′; h⟩ ∈ X_{α_{1}}^{N}(K), then x_{2} := ⟨E/K, ϕ ∘ [g]_{N}; E′/K, ϕ′ ∘ [g′]_{N}; εh⟩ ∈ X_{α_{2}}^{N}(K), where g′ = β^{−1}gβ, and we have that β_{α_{1}}^{N}(x_{1}) = β_{α_{2}}^{N}(x_{2}).

Proof. Since g, g′ = β^{−1}gβ ∈ SL_{2}(ℤ), it is clear that ϕ ∘ [g]_{N} and ϕ′ ∘ [g′]_{N} are level N-structures of E/K and E′/K, respectively. Now since (ϕ, h, ϕ′) is α_{1}-compatible, it follows that (ϕ∘[g]_{N}, εh, ϕ∘[g′]_{N}) is α_{2}-compatible because εh∘ϕ∘[g]_{N} = εϕ′∘[α_{1}g]_{N} = ϕ′ ∘ [g′α_{2}]_{N} = ϕ′ ∘ [g′] ∘ [α_{2}]_{N}. This proves the first assertion.

To prove the second assertion, let ψ = ψ_{ϕ,ϕ′} : E[N] → E′[N] be defined as in (10), so β_{α_{1}}^{N}(x_{1}) = ⟨A_{ψ}, π_{ψ}⟩, where A = E × E′. Similarly, β_{α_{2}}^{N}(x_{2}) = ⟨A_{ψ′}, π_{ψ′}⟩, where ψ′ = ψ_{ϕ∘[g]_{N} ,ϕ′∘[g′]_{N}}. Now since gβ(g′)^{−1} = β, we see that

ψ′ = ϕ′ ∘ [g]_{N} ∘ [β]_{N} ∘ (ϕ ∘ [g′]_{N})^{−1} = ϕ′ ∘ [gβ(g′)^{−1}]_{N} ∘ ϕ^{−1} = ϕ′ ∘ [β]_{N} ∘ ϕ^{−1} = ψ,

and so it is clear that β_{α_{1}}^{N}(x_{1}) = β_{α_{2}}^{N}(x_{2}).

It is more difficult to prove the second assertion of Theorem 9. For this, we will prove the following more general assertion.

Theorem 28 Let q be a form of type (N, m, d) which satisfies the condition

(17)  |{(X, Y) ∈ ℤ^{2} : gcd(X, Y) = 1 and q(X, Y) = N^{2}}| = 2.

If char(K) ∤ Nd and if α_{1}, α_{2} ∈ M_{d} satisfy q_{α_{i}} ∼ q, for i = 1, 2, then

(18)  gβα_{1}g^{−1} ≡ ±βα_{2} (mod N), for some g ∈ SL_{2}(ℤ) ⇔ \overline{T} _{α_{1}}^{N} = \overline{T} _{α_{2}}^{N}.

We observe that the hypothesis (17) follows from the hypothesis on d and N which was made in Theorems 4 and 9.

Lemma 29 If q is a form of type (N, m, d) with d > N^{4}/(4m^{2}), then q satisfies condition (17).

Proof. Suppose first that q = [N^{2}, 2mt, m^{2}(t^{2} + 4d)/N^{2}], for some t ∈ ℤ, and assume that (X, Y) ∈ ℤ^{2} satisfies q(X, Y) = N^{2}. If Y ≠ 0, then N^{2} = q(X, Y) = (NX + \frac{m}{N}tY)^{2} + \frac{4m^{2}d}{N^{2}}Y^{2} ≥ \frac{4m^{2}d}{N^{2}}, which contradicts the hypothesis that N^{4}/(4m^{2}) < d. Thus Y = 0, and hence X = ±1 because gcd(X, Y) = 1. Thus q satisfies (17).

If q is any form of type (N, m, d), then q ∼ q′ := [N^{2}, 2mt, m^{2}(t^{2} + 4d)/N^{2}], for some t ∈ ℤ. By what was just proved, condition (17) holds for q′ and hence also for q because the left hand side of (17) is the same for all q ∼ q′.

Before proving Theorem 28, we observe that it and Lemma 27 imply Theorem 9.

Proof of Theorem 9 (using Theorem 28). The first assertion clearly follows from Lemma 27.

To prove the second assertion, suppose that \overline{T} _{α_{1}}^{N} = \overline{T} _{α_{2}}^{N}. Then \overline{T} _{α_{1}}^{N} ⊂ H(q_{α_{2}}^{N}) by Proposition 22, so q_{α_{1}}^{N} ∼ q_{α_{2}}^{N} by Proposition 24. (Recall that q_{α_{2}}^{N} has type (N, m_{2}, d), for some m_{i}|N, by [K5], Proposition 11.) Thus, q_{α_{1}}^{N} ∼ q_{α_{2}}^{N}, and so both have the same type (N, m, d).

By Lemma 29 we know that the hypothesis d > N^{4}/(4m^{2}) implies that q_{α_{i}}^{N} satisfies (17), and so the converse implication of (4) follows from (18).

To prove Theorem 28, we will use the following fact which will be proved in §6; cf. Corollary 37.

Lemma 30 Let α ∈ M_{d}, where d ≠ 0, and let α^{∗} = dA^{−1} be its adjoint. Then for any N ≥ 1, there exists a g ∈ SL_{2}(ℤ) such that βα^{∗}β^{−1} ≡ gαg^{−1} (mod N).

The following is the key result used in the proof of Theorem 28.

Proposition 31 Let α_{1}, α_{2} ∈ M_{d} and suppose that char(K) ∤ Nd. For i = 1, 2, let x_{i} = ⟨E_{i}, ϕ_{i}, E′_{i}, ϕ′, h_{i}⟩ ∈ X_{α_{i}}^{N}(K) be two non-CM points such that β_{α_{1}}^{N}(x_{1}) = β_{α_{2}}^{N}(x_{2}), so there exists an isomorphism φ : (A_{ψ_{2}}, λ_{ψ_{2}}) →^{∼} (A_{ψ_{1}}, λ_{ψ_{1}}), where ψ_{i} = ψ_{ϕ_{i},ϕ′_{i}}, for i = 1, 2. If

(19)  φπ_{2}(E_{2} × {0}) = π_{1}(E_{1} × {0}) or φπ_{2}({0} × E′_{2}) = π_{1}(E_{1} × {0}),

where π_{i} = π_{ψ_{i}} : E_{i} × E′_{i} → A_{ψ_{i}} is the quotient isogeny, for i = 1, 2, then there exists g ∈ SL_{2}(ℤ) such that gβα_{1}g^{−1} ≡ ±βα_{2} (mod N), and hence \overline{T} _{α_{1}}^{N} = \overline{T} _{α_{2}}^{N}.

Proof. Note first that (E_{i}, E′_{i}, ψ_{i}, π_{i}) is an N-presentation of (A_{ψ_{i}}, λ_{ψ_{i}}) in the sense of [K4], §2. Put (J, λ) := (A_{ψ_{1}}, λ_{ψ_{1}}); then (E_{2}, E′_{2}, ψ_{2}, φπ_{2}) is also an N-presentation of (J, λ). Put \overline{E} := π_{1}(E_{1} × \{0\}). We have to consider the following two cases: Case 1: \overline{E} = φπ_{2}(E_{2} × \{0\}). Here it follows from Proposition 10(e) of [K4] that there exist isomorphisms f : E_{1} →^{∼}

E_{2} and f′ : E′_{1} →^{∼} E′_{2} such that π_{1} = φ ∘ π_{2} ∘ (f × f′). Put h′_{1} = (f′)^{−1} ∘ h_{2} ∘ f ∈

Hom(E_{1}, E′_{1}). Since x_{1} is a non-CM point, we have that Hom(E_{1}, E′_{1}) = ℤh_{1}, and so h′_{1} = εh_{1} with ε ∈ {1, −1} because h′_{1} is cyclic. Moreover, since Graph(−ψ_{1}) = Ker(π_{1}) = Ker(π_{2}∘(f×f′)) = (f×f′)^{−1}(Graph(−ψ_{2})), we see that f′∘ψ_{1} = ψ_{2}∘f_{|E_{1}[N]}.

Recall from (14) that [βα_{i}]_{N} = ϕ_{i}^{−1} ∘ ψ_{i}^{−1} ∘ h_{i} ∘ ϕ_{i}, for i = 1, 2. Substituting h_{2} = f′∘(εh_{1})∘f^{−1} and ψ_{2}^{−1} = f ∘ψ^{−1}(f′_{E′_{1}[N]})^{−1} shows that [εβα_{2}]_{N} = ḡ∘[βα_{1}]_{N} ∘ḡ^{−1}, where ḡ := ϕ_{2}^{−1} ∘ f ∘ ϕ_{1} ∈ Aut(E[N]). Since f is an isomorphism of elliptic curves and the ϕ_{i} are level N-structures (of the same determinant), we see that det(ḡ) = 1. Thus, there exists g ∈ SL_{2}(ℤ) such that [g]_{N} = ḡ, and so εβα_{2} ≡ gβα_{1}g^{−1} (mod N). Thus \overline{T} _{α_{1}}^{N} = \overline{T} _{α_{2}}^{N} by Lemma 27. Case 2: \overline{E} = φπ_{2}(\{0\} × E′_{2}). Since (ϕ_{2}, h_{2}, ϕ′_{2}) is α_{2}-compatible, it follows that (ϕ′_{2}, h_{2}^{t}, ϕ_{2}) is α_{2}^{∗}-compatible, and so x_{2}^{∗} := ⟨E′_{2}, ϕ′_{2}, E_{2}, ϕ_{2}, h_{2}^{t}⟩ ∈ X_{α_{2}^{∗}}^{N}(K). Since β^{−1} = β, we see that ψ_{ϕ′_{2},ϕ_{2}} = ϕ_{2} ∘ [β]_{N} ∘ (ϕ′_{2})^{−1} = (ϕ′_{2} ∘ [β]_{N} ∘ (ϕ_{2})^{−1})^{−1} = ψ_{2}^{−1}. Let η : E′_{2} × E_{2} → E_{2} × E′_{2} be defined by η(x′, x) = (x, x′), and put π′_{2} = φ ∘ π_{2} ∘ η : E′_{2} × E_{2} → J. Then (E′_{2}, E_{2}, ψ_{2}^{−1}, π′_{2}) is an N-presentation of (J, λ), so β_{α_{2}^{∗}}^{N}(x_{2}^{∗}) = ⟨J, λ⟩.

Since π′_{2}(E′_{2} × \{0\}) = φπ_{2}(\{0\} × E′_{2}) = \overline{E}, we see that x_{1} and x_{2}^{∗} satisfy the hypotheses of the proposition and also the condition of Case 1. Thus, by what was proven above, it follows that there exists g_{1} ∈ SL_{2}(ℤ) and ε ∈ {±1} such that [εβα_{2}^{∗}]_{N} = [g_{1}βα_{1}g_{1}^{−1}]_{N}. On the other hand, by Lemma 30 there exists g_{2} ∈ SL_{2}(ℤ) such that [βα_{2}]_{N} = [g_{2}(β(βα_{2})^{∗}β^{−1})g_{2}^{−1}]_{N} = [g_{2}βα_{2}^{∗}g_{2}^{−1}]_{N}, where the latter equality follows because (βα_{2})^{∗} = α_{2}^{∗}β^{∗} = α_{2}^{∗}β. Thus, putting g = g_{2}g_{1} shows that [εβα_{2}]_{N} = [gβα_{1}g^{−1}]_{N}, and so \overline{T} _{α_{1}}^{N} = \overline{T} _{α_{2}}^{N} as before.

Corollary 32 Let α_{1}, α_{2} ∈ M_{d} and suppose that char(K) ∤ Nd and that q = q_{α_{1}}^{N} satisfies condition (17). If there exists a non-CM point x_{1} ∈ X_{α_{1}}^{N}(K) and a point x_{2} ∈ X_{α_{2}}^{N}(K) such that β_{α_{1}}^{N}(x_{1}) = β_{α_{2}}^{N}(x_{2}), then there exists g ∈ SL_{2}(ℤ) such that gβα_{1}g^{−1} ≡ ±βα_{2} (mod N), and hence \overline{T} _{α_{1}}^{N} = \overline{T} _{α_{2}}^{N}.

Proof. Write x_{i} = ⟨E_{i}/K, ϕ_{i}, E′_{i}/K, ϕ′_{i}, h_{i}⟩, and let ψ_{i}, π_{i}, φ and (J, λ) be as in the proof of Proposition 31. By Lemma 25 we know that q_{(J,λ)} ∼ q_{α_{1}}^{N} = q, so by Theorem 18 of [K4] we see that the condition (17) is equivalent to the condition |S_{N}(J, λ)| = 2, where S_{N}(J, λ) denotes the set of elliptic subgroups of J of λ-degree N.

By Proposition 10(d) of [K4] we know that \overline{E} := π_{1}(E_{1} × \{0\}) ∈ S_{N}(J, λ), and so S_{N}(J, λ) = \{\overline{E}, \overline{E}^{⊥}\} by Proposition 10(a) of [K4] and the fact that |S_{N}(J, λ)| = 2. Since also \overline{E}_{2} := φπ_{2}(E_{2} × \{0\}) ∈ S_{N}(J, λ) and \overline{E}′_{2} := φπ_{2}(\{0\} × E′_{2}) ∈ S_{N}(J, λ) = (\overline{E}_{2})^{⊥} ≠ \overline{E}_{2}, we thus have that either \overline{E} = \overline{E}_{2} or \overline{E} = \overline{E}′_{2}. This means that condition (19) holds and so the assertion follows from Proposition 31.

Proof of Theorem 28. The one implication of (18) follows from Lemma 27. To prove the converse implication, suppose that \overline{T} _{α_{1}}^{N} = \overline{T} _{α_{2}}^{N}. Then this equality of subschemes of A_{2}/K also holds over any extension field, and so we may assume that K is not the algebraic closure of a finite field. Then by Lemma 25 there is a non-CM point x_{1} ∈ X_{α_{1}}^{N}(K), and so there exists x_{2} ∈ X_{α_{2}}^{N}(K) such that β_{α_{1}}^{N}(x_{1}) = β_{α_{2}}^{N}(x_{2}) because \overline{T} _{α_{1}}^{N} = \overline{T} _{α_{2}}^{N}. We thus are in the situation of Corollary 32, and so (18) follows.

6 Conjugacy classes of matrices

In view of Theorems 9 and 28, it is important to understand the SL_{2}(ℤ/Nℤ)-conjugacy classes of matrices mod N. If N = p^{r} is a power of prime, then the conjugacy classes of matrices in SL_{2}(ℤ/Nℤ) were determined by Nobs[No]. His method can be generalized to obtain similar results for an arbitrary integer N and arbitrary matrices α mod N. For our purposes, however, it is more convenient to express this in terms of SL_{2}(ℤ)-conjugacy, using the fact that SL_{2}(ℤ) → SL_{2}(ℤ/Nℤ) is surjective.

The first step is to find a normal form for a representative of the conjugacy class of such matrices.

Notation. Let α = ( \begin{matrix}x & y \\ z & w \end{matrix}) ∈ M_{2}(ℤ) be an integral matrix, and let N ≥ 1 be an integer. Put

g_{α,N} := gcd(x − w, y, z, N), and x̄_{α,N} := rem(x, g_{α,N}).

Thus, if g = g_{α,N}, and x̄ = x̄_{α,N}, then 0 ≤ x̄ < g and x̄ ≡ x (mod g). In addition, put

(20)  τ_{α,N} := (tr(α) − 2x̄)/g and
δ_{α,N} := (det(α) + x̄^{2} − tr(α)x̄)/g^{2} = (det(α) − x̄^{2} − x̄gτ_{α,N})/g^{2}.

Lemma 33 If α ∈ M_{2}(ℤ) and N ≥ 1, then τ_{α,N} and δ_{α,N} are integers which satisfy the relation

(21)  tr(α)^{2} − 4 det(α) = (τ_{α,N}^{2} − 4δ_{α,N})g_{α,N}^{2}.

Proof. Put α′ = (α − x̄I)/g. From definitions of g = g_{α,N} and x̄ = x̄_{α,N} we see that α′ ∈ M_{2}(ℤ). We have that tr(α′) = (tr(α) − 2x̄)/g = τ_{α,N}, so τ_{α,N} = tr(α′) ∈ ℤ. Moreover, det(α) = det(x̄I + gα′) = x̄^{2} + x̄gtr(α′) + g^{2} det(α′), so we also have that δ_{α,N} = det(α′) ∈ ℤ.

Put τ = τ_{α,N} and δ = δ_{α,N}. From (20) we have that tr(α)^{2} − 4 det(α) = (2x̄ + gτ)^{2} − 4(x̄^{2} + x̄gτ + g^{2}δ) = g^{2}(τ^{2} − 4δ), as claimed.

The above quantities are invariants of the conjugacy class of α (mod N), as the following result shows.

Proposition 34 Let N ≥ 1, and let α_{i} ∈ M_{2}(ℤ), i = 1, 2 be two matrices such that

α_{1} ≡ γα_{2}γ^{−1} (mod N).

Then g_{α_{1},N} = g_{α_{2},N} and x̄_{α_{1},N} = x̄_{α_{2},N}. Moreover, if m := \frac{N}{g_{α_{1},N}}, then

(22)  τ_{α_{1},N} ≡ τ_{α_{2},N} (mod m) and δ_{α_{1},N} ≡ δ_{α_{2},N} (mod m).

Proof. Put α_{3} = γα_{2}γ^{−1}. Then g_{α_{1},N} = g_{α_{3},N} because g_{α,N} depends only on α (mod N). Moreover, since g_{α,N} is invariant under conjugation by [K5], Corollary 13 , it follows that g_{α_{3},N} = g_{α_{2},N}, and hence g_{α_{2},N} = g.

Put α′_{i} = (α_{i} − x̄_{i}I)/g, where x̄_{i} = x_{α_{i},N}. Then α′_{i} ∈ M_{2}(ℤ), and τ_{i} := τ_{α_{i},N} = tr(α′_{i}) and δ_{i} := δ_{α_{i},N} = det(α′_{i}), as we saw in the proof of Lemma 33. Now x̄_{1}I + gα′_{2} = α_{1} ≡ γα_{2}γ^{−1} ≡ x̄_{2}I + gγα′_{2}γ^{−1} (mod N), so x̄_{1}I ≡ x̄_{2}I (mod g), and hence x̄_{1} ≡ x̄_{2} (mod g), so x̄_{1} = x̄_{2} since 0 ≤ x̄_{i} < g. From this we see that the hypothesis implies that α′_{1} ≡ γα′_{2}γ^{−1} (mod m), so tr(α′_{1}) ≡ tr(α′_{2}) (mod m) and det(α′_{1}) ≡ det(α′_{2}) (mod m), and hence (22) follows.

We next observe that we can find a representative of the conjugacy class of α (mod N) of a special type.

Proposition 35 Let N ≥ 1 and α ∈ M_{2}(ℤ) and put g = g_{α,N}, m = N/g, x̄ = x̄_{α,N}, τ = τ_{α,N} and δ = δ_{α,N}. Then there exist z, z′ ∈ ℤ with zz′ ≡ 1 (mod m) such that

(23)  γαγ^{−1} ≡ x̄I + g ( \begin{matrix}0 & −δz′ \\ z & τ \end{matrix}) (mod N), for some γ ∈ SL_{2}(ℤ).

Proof. Put α′ = (α − x̄I)/g ∈ M_{2}(ℤ). By construction, g_{α′,m} = 1. Suppose first that m = p^{r} is a prime power. Then by Nobs[No], Lemma 4, there exists γ_{1} ∈ SL_{2}(ℤ) such that γ_{1}α′γ_{1}^{−1} ≡ α_{1} (mod m), where α_{1} = \binom{∗ ∗}{z ∗}, with gcd(z, m) = 1. Choose z′ such that zz′ ≡ 1 (mod m), and put α_{2} = ( \begin{matrix}0 & −δz′ \\ z & τ \end{matrix}). Then det(α_{2}) ≡ δ = det(α′) ≡ det(α_{1}) (mod m) and tr(α_{2}) = τ = tr(α′) ≡ tr(α_{1}) (mod m), so by Nobs[No], Lemma 5, there exists γ_{2} ∈ SL_{2}(ℤ) such that γ_{2}α_{1}γ_{2}^{−1} ≡ α_{2} (mod m). Put γ = γ_{2}γ_{1}. Then γα′γ^{−1} ≡ α_{2} (mod m), so γαγ^{−1} = x̄I + gγα′γ^{−1} ≡ x̄I + gα_{2} (mod gm), and hence (23) holds when m = p^{r}. If m is a product of prime powers, then the same result follows from the prime power cases by using the Chinese remainder theorem.

The above elements z, z′ ∈ ℤ are not uniquely determined mod m by the mod N conjugacy class of α. In order to classify the conjugacy classes we follow (and generalize) Nobs[No] and introduce the following set:

S_{N}(τ, δ) := {a ∈ (ℤ/Nℤ)^{×} : ∃(ξ, η) ∈ ℤ^{2} with ξ^{2} + τξη + δη^{2} ≡ a (mod N)}

It is easy to see (as in Nobs[No]) that S_{N}(t, d) is a subgroup of R_{N}^{×} := (ℤ/Nℤ)^{×}.

Proposition 36 Suppose that α_{1}, α_{2} ∈ M_{2}(ℤ) and N satisfy g_{α_{1},N} = g_{α_{2},N} = g, and that α_{i} = xI + gα′_{i}, i = 1, 2, where α′_{i} = \binom{∗ ∗}{z_{i} ∗} with gcd(z_{i}, m) = 1, for i = 1, 2, where m = N/g. Put τ = tr(α′_{1}) and δ = det(α′_{1}), and let z̄_{i} denote the image of z_{i} in R_{m}^{×} = (ℤ/mℤ)^{×}. Then the following conditions are equivalent:

(i) There exists γ ∈ SL_{2}(ℤ) such that γα_{1}γ^{−1} ≡ α_{2} (mod N).

(ii) tr(α′_{2}) ≡ τ (mod m), det(α′_{2}) ≡ δ (mod m) and z̄_{1}z̄_{2}^{−1} ∈ S_{m}(τ, δ).

Proof. It is clear that condition (i) is equivalent to the condition

(24)  There exists γ ∈ SL_{2}(ℤ) such that γα′_{1}γ^{−1} ≡ α′_{2} (mod m).

If m = p^{r} is a prime power, then the equivalence of (24) and (ii) follows from Nobs[No], Lemma 5, and hence the general case follows from this by using the Chinese remainder theorem.

Corollary 37 If α ∈ M_{2}(ℤ) and N ≥ 1, then there exists γ ∈ SL_{2}(ℤ) such that

(25)  βα^{∗}β ≡ γαγ^{−1} (mod N).

Proof. Note first that if α = \binom{x y}{z w}, then βα^{∗}β = \binom{w y}{z x}. Thus, if α ≡ α_{1} (mod N), then βα^{∗}β ≡ βα_{1}^{∗}β (mod N).

Put g = g_{α,N}, m = \frac{N}{g}, and let x̄, τ = τ(α), δ = δ(α) be defined as in (20). By Proposition 35 there exists α_{1} = \binom{0 δz′}{z τ} with zz′ ≡ 1 (mod m) such that if α′ = x̄I + gα_{1}, then α′ ≡ γ_{1}αγ_{1}^{−1} (mod N), for some γ_{1} ∈ SL_{2}(ℤ). Then β(α′)^{∗}β = x̄I+gβ(α_{1})^{∗}β. Since β(α_{1})^{∗}β = \binom{τ δz′}{z 0}, it follows from Proposition 36 that there exists γ_{2} ∈ SL_{2}(ℤ) such that β(α′)^{∗}β ≡ γ_{2}α′γ_{2}^{−1} (mod N). Now since α ≡ γ_{1}^{−1}α′γ_{1} (mod N), we have that βα^{∗}β ≡ β(γ_{1}^{−1}α′γ_{1})^{∗}β (mod N). Since γ_{1}^{∗} = γ_{1}^{−1}, we see that (γ_{1}^{−1}α′γ_{1})^{∗} = γ_{1}^{∗}(α′)^{∗}(γ_{1}^{−1})^{∗} = γ_{1}^{−1}(α′)^{∗}βγ_{1}. Thus, since β^{2} = 1, it follows that β(γ_{1}^{−1}α′γ_{1})^{∗}β = γ_{3}β(α′)^{∗}βγ_{3}^{−1}, where γ_{3} = β(γ_{1}^{−1})β ∈ SL_{2}(ℤ), and so βα^{∗}β ≡ γ_{3}β(α′)^{∗}βγ_{3}^{−1} ≡ γ_{3}γ_{2}α′γ_{2}^{−1}γ_{3}^{−1} ≡ γ_{3}γ_{2}γ_{1}αγ_{1}^{−1}γ_{2}^{−1}γ_{3}^{−1} (mod N), which shows that (25) holds with γ = γ_{3}γ_{2}γ_{1}.

In view of Proposition 36, it is of interest to determine the subgroup S_{m}(t, δ) in R_{m}^{×}. Nobs[No] computed S_{m}(t, d) for prime powers m = p^{r} and listed the result in Tabelle 1 of [No]. Using this, we see easily:

Proposition 38 Let τ, δ ∈ ℤ and N ≥ 1 be integers and put Δ_{N} = Δ_{N}(τ, δ) := gcd(τ^{2} − 4δ, N). Then

(26)  i_{N}(τ, δ) := [R_{N}^{×} : S_{N}(τ, δ)] ≤ 2^{ω(Δ_{N})},

except when 8|Δ_{N}, in which case i_{N}(t, d) ≤ 2^{ω(Δ_{N})+1}. Moreover, if 4 ∤ Δ_{N} and if ω′(Δ_{N}) := |{p|Δ_{N} : p is an odd prime}|, then

(27)  i_{N}(τ, δ) = 2^{ω′(Δ_{N})}.

Proof. If N = p_{1}^{e_{1}} ⋯ p_{r}^{e_{r}}, where the p_{i}’s are distinct primes, then by using the Chinese Remainder Theorem we see that

(28)  i_{N}(τ, δ) = ∏_{k=1}^{r} i_{p_{k}^{e_{k}}} (τ, δ).

From Tabelle 1 of Nobs[No] it follows that if p_{k} ∤ Δ_{N}, then i_{p_{k}^{e_{k}}} (τ, δ) = 1, and that if p_{k}|Δ_{N}, then

(29)  S_{p_{k}^{e_{k}}} (τ, δ) = (R_{p_{k}^{e_{k}}}^{×})^{2} and i_{p_{k}^{e_{k}}} (τ, δ) = 2, provided that p_{k} ≠ 2.

Moreover, by Nobs[No], loc. cit., we also have that i_{2}(τ, δ) = 1, that i_{4}(τ, δ)|2 and that i_{2^{e}}(τ, δ)|4, if e ≥ 3, and so the assertions follow.

If 4|Δ_{N}(τ, δ), then the exact formula for i_{N}(τ, δ) is much more complicated. To write it in a convenient form, we first introduce the the following notation.

Notation. If a ∈ ℤ, put

w_{4}(a) = \{ \begin{matrix}1 & if a ≡ 1, 2 (mod 4) \\ 2 & if a ≡ 0, 3 (mod 4) \end{matrix} and w_{8}(a) = \{ \begin{matrix}1 & if a ≡ 1, 5 (mod 8) \\ 2 & if a ≡ 2, 3, 4, 6, 7 (mod 8) \\ 4 & if a ≡ 0 (mod 8) \end{matrix}

From the table in Nobs[No] and (28) we deduce:

Proposition 39 Let τ, δ ∈ ℤ and suppose that 4|Δ_{N} = Δ_{N}(τ, δ), so Δ̃ := Δ/4 ∈ ℤ, where Δ = τ^{2} − 4δ. Then:

(30)  i_{N}(τ, δ) = \{ \begin{matrix}2^{ω′(Δ_{N})}w_{4}(Δ̃) & if 4||N, \\ 2^{ω′(Δ_{N})}w_{8}(Δ̃) & if 8|N, \end{matrix}

7 The number of components of H(q)

We now apply the above results on conjugacy classes to refine Theorem 7 as follows.

Theorem 40 Let q be a binary form of type (N, m, d), and put g = \frac{N}{m}. Then there exists a primitive matrix α ∈ M_{d} such that q ∼ q_{α}^{N}. Fix such a matrix α, and let z_{1}, …, z_{s} be integers such that their residues mod m is a system of representatives of R_{m}^{×}/S_{m}(τ, δ), where τ = τ_{βα,N} and δ = δ_{βα,N} are as in (20), and let z′_{i} ∈ ℤ be such that z_{i}z′_{i} ≡ 1 (mod m), for 1 ≤ i ≤ s. Then there exist primitive matrices α_{1}, …, α_{s} ∈ M_{d} such that

βα_{i} ≡ x̄I + g ( \begin{matrix}0 & −δz′_{i} \\ z_{i} & τ \end{matrix} ) (mod N), for 1 ≤ i ≤ s,

where x̄ = x̄_{βα,N}. Moreover, if char(K) ∤ Nd, then

(31)  H(q) = ⋃_{i=1}^{s} \overline{T} _{α_{i}}^{N}.

In particular, the number of irreducible components of H(q) satisfies the estimate

(32)  |Irr(H(q))| ≤ i_{m}(τ, δ) = [R_{m}^{×} : S_{m}(τ, δ)].

The key result for proving this is the following variant of Theorem 31 of [K5].

Proposition 41 Suppose that char(K) ∤ N and that ⟨J, λ⟩ ∈ H(q_{α}^{N}), where α ∈ M_{d}. Then there exists an N-presentation (E, E′, ψ, π) of (J, λ) and a cyclic iogeny h ∈ Hom(E, E′) of degree d such that if α′ ∈ M_{d} satisfies the relation

(33)  (ψ^{−1}h)_{|E[N]} = ϕ ∘ [βα′]_{N} ∘ ϕ^{−1},

for some level N-structure ϕ of E/K, then g_{βα′,N} = g_{βα,N} and x̄_{βα′,N} = x̄_{βα,N}, and

(34)  tr(βα′) ≡ tr(βα) (mod Ng),

where g = g_{βα,N}. Thus, if m = \frac{N}{g}, then

(35)  τ_{βα′,N} ≡ τ_{βα,N} (mod m) and δ_{βα′,N} ≡ β_{βα,N} (mod m).

In order to deduce this from the results of [K5], we need to recall some basic results which were (implicitly) proven there.

Lemma 42 Let N and d be positive integers and let α ∈ M_{d}. If g = g_{βα,N}, then there exists k ∈ ℤ such that kx̄_{βα,N} ≡ 1 (mod N), and for any such k we have that

(36)  r_{βα,k,N} := −(tr(βα) + kd(k^{2}d + 3))/g^{2} ∈ ℤ.

Proof. Since βα = \binom{x y}{z w} is primitive, and x̄ := x̄_{βα,N} ≡ x (mod g), we see that gcd(x̄, g) = gcd(x, g) = gcd(x, y, z, w, N) = 1, and so there exists a k ∈ ℤ such that kx̄ ≡ 1 (mod g). This proves the first assertion.

We observe that M = βα and k satisfy the hypotheses of Proposition 11 of [K5], so if r ∈ ℤ is defined by equation (9) of [K5], then q_{βα,N,d,k} = q_{N,m,d,k,r} by [K5], equation (7). This means that −2mtr(βα) = 2mT = 2m(kd(k^{2}d + 3) + rg^{2}), so r_{βα,,k,N} = r ∈ ℤ.

For the next result, recall from Lemma 21 of [K5] that if E/K and E′/K are elliptic curves, then every D ∈ NS(E × E′) has the unique representation D = D(a, b, h), where a, b ∈ ℤ and h ∈ Hom(E, E′).

Proposition 43 Let (E, E′, ψ, π) be an N-presentation of ⟨J, λ⟩ ∈ A_{2}(K), and let h ∈ Hom(E, E′) be a cyclic isogeny of degree d ≥ 1. Assume that char(K) ∤ N. Then there exists α ∈ M_{d} such that

(37)  (ψ^{−1}h)_{|E[N]} = ϕ ∘ [βα]_{N} ∘ ϕ^{−1},

holds for some level N-structure ϕ of E. Fix k such that kx̄_{βα,N} ≡ 1 (mod g), where g = g_{βα,N}, and put D_{1} = D(g^{2}, 0, 0) and D_{2} = D(kd(k^{2}d + 2), −kd, h). Then

(38)  m(rD_{1} + D_{2}) ∈ π^{∗}NS(J),

where m = \frac{n}{g} and r = r_{βα,k,N}. Moreover, r (mod m) is uniquely determined by (38).

Proof. Fix a level N-structure ϕ. As in the proof of Proposition 28 of [K5], we see that ϕ^{−1}∘ψ^{−1}∘h_{|E[N]}∘ϕ ∈ End(E[N]) is N-primitive and has determinant ≡ −d (mod N), so by Remark 29 of [K5] there exists α ∈ M_{d} such that (37) holds. Note that by Lemma 42 there exists a k such that kx̄_{βα,N} ≡ 1 (mod g).

By [K5], Propositions 28 and 25(c), we know that (38) holds for some r ∈ ℤ which is uniquely determined mod m by (38). Moreover, by equations (35), (42) and (9) of [K5] we have that r ≡ r_{βα,b,N} (mod m), and so the assertions follow.

Proof of Proposition 41. Put g = g_{βα,N} and m = \frac{N}{g}. Fix k ∈ ℤ such that kx̄_{βα,N} ≡ 1 (mod N) and put r := r_{βα,k,N} as in Lemma 42. Since q_{−α}^{N} = q_{βα,N,d,k} in the notation of Proposition 11 of [K5], we have by equation (7) of [K5] that q_{−α}^{N} = q_{N,n,d,k,r}. Thus q_{α}^{N} ∼ q_{−α}^{N} = q_{N,n,d,k,r}.

Since ⟨J, λ⟩ ∈ H(q_{α}^{N}), we are thus in the situation of Theorem 31 of [K5], and so we obtain an N-presentation (E, E′, ψ, π) of (J, λ) and a cyclic isogeny h : E → E′ of degree d such that if D_{1} = D(g^{2}, 0, 0) and D_{2} = D(kd(k^{2}d + 2), −kd, h), then (38) holds; cf. equation (44) of [K5].

Moreover, by Theorem 31(a) and Proposition 28 of [K5] we then have that g_{βα′,N} = g and that kx′ ≡ 1 (mod g), where βα′ = \binom{x′ y′}{z′ w′}, so x̄_{βα,N} = x̄_{βα′,N} because they are uniquely determined by k and g. Furthermore, both r and r_{βα′,k,N} satisfy (38), so by the uniqueness assertion of Proposition 43 we see that

r = r_{βα,k,N} ≡ r_{βα′,k,N} (mod m).

Thus g^{2}r_{β,k,N} ≡ g^{2}r_{βα′,k,N} (mod Ng), and so (34) follows. In view of Lemma 44 below, this implies that (35) also holds.

Lemma 44 Let α_{1}, α_{2} ∈ M_{2}(ℤ) be two integral matrices and let N ≥ 1 be an integer. Suppose that g_{α_{1},N} = g_{α_{2},N} and that x̄_{α_{1},N} = x̄_{α_{2},N}. If

(39)  tr(α_{1}) ≡ tr(α_{2}) (mod Ng), and det(α_{1}) ≡ det(α_{2}) (mod Ng),

where g = g_{α_{1},N}, then for m = \frac{N}{g} we have that

(40)  τ_{α_{1},N} ≡ τ_{α_{2},N} (mod N) and δ_{α_{1},N} ≡ δ_{α_{2},N} (mod m).

Proof. Put x̄ = x̄_{α_{i},N}, τ_{i} = τ_{α_{i},N} and δ_{i} = δ_{α_{i},N}, for i = 1, 2. Then g(τ_{1} − τ_{2}) = tr(α_{1}) − tr(α_{2}) ≡ 0 (mod Ng), so τ_{1} ≡ τ_{2} (mod N). Moreover, since g^{2}(δ_{1} − δ_{2}) = (det(α_{1}) − det(α_{2})) − x̄(τ_{1} − τ_{2})g ≡ 0 (mod g^{2}m), we see that δ_{1} ≡ δ_{2} (mod m).

Proof of Theorem 40. The existence of α ∈ M_{d} with q_{α}^{N} ∼ q follows from Theorem 16 of [K5]. Thus g_{βα,N} = g and gcd(x̄_{βα,N}, g) = 1; cf. the proof of Lemma 42.

Put ᾱ_{i} := x̄I + g ( \begin{matrix}0 & −δz′_{i} \\ z_{i} & τ \end{matrix} ). Then g_{ᾱ_{i},N} = g because gcd(z_{i}, m) = 1, and so ᾱ_{i} is N-primitive because gcd(x̄, g) = 1.

We next observe that if t := tr(α), then

(41)  tr(ᾱ_{i}) = t and det(ᾱ_{i}) ≡ −d (mod Ng), for 1 ≤ i ≤ s.

Indeed, tr(ᾱ_{i}) = 2x̄+gτ = t, and det(ᾱ_{i}) = x̄^{2}+ḡτ+g^{2}δz_{i}z_{i}^{∗} ≡ x̄^{2}+ḡτ+g^{2}δ (mod g^{2}m), so det(ᾱ_{i}) ≡ −d (mod g^{2}m) by the definition of δ. This proves (41).

It follows from (41) that det(ᾱ_{i}) ≡ −d (mod N), so by Lemma 30 of [K5] there exists a primitive matrix α′_{i} of determinant −d such that α′_{i} ≡ ᾱ_{i} (mod N). Put α_{i} := βα′_{i}. Then α_{i} ∈ M_{d}, and βα_{i} = α′_{i} ≡ ᾱ_{i} (mod N). This proves the existence of the matrices α_{1}, …, α_{s}.

To prove (31), let k ∈ ℤ be such that kx̄ ≡ 1 (mod g), and put t_{i} := −tr(α_{i}) − dk^{3}(det(ᾱ_{i}) + d). Then by (41) we see that t_{i} ≡ −t (mod Ng), for 1 ≤ i ≤ s. Thus, if we put q_{i} = [N^{2}, 2mt_{i}, (t_{i}^{2} + 4d)/g^{2}], then q_{i} ∼ q_{−α}^{N} = [N^{2}, −2mt, (t^{2} + 4d)/g^{2}]; cf. [K5], Lemma 9. Since q_{i} = q_{ᾱ_{i},N,d,k} in the notation of Proposition 11 of [K5], and since by construction βα_{i} ≡ ᾱ_{i} (mod N), it follows from Proposition 14 of [K5] that q_{−α_{i}}^{N} ∼ q_{−α}^{N}, for 1 ≤ i ≤ s. Thus q_{α_{i}}^{N} ∼ q because q_{−α}^{N} ∼ q_{α}^{N} ∼ q, and so it follows from Theorem 7 that \overline{T} _{α_{i}}^{N} ⊂ H(q). This proves that the right hand side of (31) is contained in H(q).

To prove the opposite inclusion, let ⟨J, λ⟩ ∈ H(q) = H(q_{α}^{N}). Then by Proposition 41 (and by Proposition 43) there exists an N-presentation (E, E′, ψ, π) of (J, λ), a cyclic isogeny h : E → E′ of degree d and a matrix α′ ∈ M_{d} such that (33) holds for some level N-structure ϕ of E/K. Then ϕ′ := ψ ∘ ϕ ∘ [β]_{N} is a level N-structure on E′/K and (33) shows that (ϕ, h, ϕ′) is α′-compatible in the sense of (12), and so x := ⟨E, ϕ, E′, ϕ′, h⟩ ∈ X_{α′}^{N}(K). Since ⟨J, λ⟩ = ⟨A_{ψ}, λ_{ψ}⟩ because (E, E′, ψ, π) is an N-presentation of (J, λ) and since ψ_{ϕ,ϕ′} = ψ, it follows from Corollary 20 that

(42)  ⟨J, λ⟩ = β_{α′}^{N}(x) ∈ β_{α′}^{N}(X_{α′}^{N}) = \overline{T} _{α′}^{N}.

By the second part of Proposition 41 we know that βα′ has the same invariants g, x̄, τ, δ as βα. Thus, by Proposition 35 there exists z ∈ R_{m}^{×} and γ_{1} ∈ SL_{2}(ℤ) such that γ_{1}βα′γ_{1}^{−1} ≡ α′′ := x̄I + g\binom{0 −δz′}{z τ} (mod N), where z′z ≡ 1 (mod m). Then z ≡ z_{k}σ (mod m), for some k with 1 ≤ k ≤ s and σ ∈ S_{m}(τ, δ) by our hypothesis on the z_{i}’s. Then by Lemma 33 and Proposition 36 there exists γ_{2} ∈ SL_{2}(ℤ) such that γ_{2}α′′γ_{2}^{−1} ≡ ᾱ_{k} ≡ βα_{k} (mod N), so if we put γ = γ_{2}γ_{1}, then γβα′γ^{−1} ≡ βα_{k} (mod N). Then by (4) we see that \overline{T} _{α′}^{N} = \overline{T} _{α_{k}}^{N}, and so ⟨J, λ⟩ ∈ \overline{T} _{α_{k}}^{N} by (42). This shows that opposite inclusion of (31) holds, and so concludes the proof of (31). From this (32) follows immediately since the \overline{T} _{α_{i}}^{N}’s are irreducible (but not necessarily distinct) components of H(q); cf. Remark 21.

Theorem 40 provides us with an upper bound for the number of irreducible components of H(q); cf. (32). To go further and to compute the number of components precisely in the situation of Theorem 4(b) (or of Theorem 28) requires further work.

Indeed, although the previous section analyzed the SL_{2}-conjugacy classes of matrices mod N, the conditions (4) and (18) involve SL_{2}-conjugacy classes up to to sign. Now in many cases this does not make a difference, but it does in some cases, which leads to certain exceptional forms. More precisely, we have:

Proposition 45 Let q be binary form of type (N, m, d) which satisfies the condition (17) of Theorem 28, and let τ, δ be as in Theorem 40. If char(K) ∤ Nd, then the number of irreducible components of H(q) is given by the formula

(43)  |Irr(H(q))| = i_{m}(τ, δ) = [R_{m}^{×} : S_{m}(τ, δ)],

except when −1 ∉ S_{m}(τ, δ) and q is equivalent to one of the following forms:

(44)  [N^{2}, 0, 4d], [N^{2}, N^{2}, (N^{2} + 16d)/4], [N^{2}, εN^{2}, (ε^{2}N^{2} + 4d)/4],

where 2|N, ε ∈ {0, 1} and d ≡ 1 + εN (mod 4) in the last case. For the exceptional cases we have

(45)  |Irr(H(q))| = i_{m}(t, δ)/2.

Proof. Suppose that the \overline{T} _{α_{i}}^{N} constructed in Theorem 40 are not all distinct. Then there exists i and j with 1 ≤ i < j ≤ s such that \overline{T} _{α_{i}}^{N} = \overline{T} _{α_{j}}^{N}, so by Theorem 18 we have that γβα_{i}γ^{−1} ≡ ±βα_{j} (mod N), for some γ ∈ SL_{2}(ℤ). We thus have that

(46)  γβα_{i}γ^{−1} ≡ −βα_{j} (mod N),

because the other case is ruled out by our choice of the α_{k}’s; cf. Proposition 36. Thus, since βα_{k} has the invariants x̄, τ (mod m) and δ (mod m), for all k, it follows from (46) and Proposition 34 that we have x̄_{−βα_{j},N} = x̄, τ_{−βα_{j}} ≡ τ (mod m) and δ_{−βα_{j}} ≡ δ (mod m). Moreover, if these conditions hold, then

−βα_{j} ≡ −x̄I + g ( \begin{matrix}0 & −δ(−z_{j}) \\ −z_{j} & −τ \end{matrix} ) ≡ x̄I + g ( \begin{matrix}∗ & δz_{j} \\ −z_{j} & ∗ \end{matrix} ) (mod N),

so by (46) and Proposition 36 we see that the residues of z_{i} and −z_{j} mod m lie in the same S_{m}(τ, δ)-coset of R_{m}^{×}. But this is only possible if −1 ∉ S_{m}(τ, δ).

We now analyze the above conditions on x̄, etc. Since x̄ = x̄_{βα_{i}} = x̄_{−βα_{j},N} ≡ −x̄ (mod g), we see that 2x̄ ≡ 0 (mod g). But since gcd(x̄, g) = 1, this implies that 2 ≡ 0 (mod g), which is only possible if g = 1 or g = 2. Case 1: g = 1. Here m = N and x̄ = 0, so it follows that tr(βα_{k}) = τ_{βα_{k},N} ≡ τ (mod N), for all k. But then τ ≡ tr(βα_{i}) ≡ −tr(βα_{j}) ≡ −τ (mod N), so 2τ ≡ 0 (mod N). Thus, either τ ≡ 0 (mod N) or 2|N and τ ≡ \frac{N}{2} (mod N). But since τ = tr(βα), this implies that q ∼ q_{α}^{N} ∼ [N^{2}, 0, 4d] or 2|N and q ∼ [N^{2}, N^{2}, (N/2)^{2} + 4d]; cf. [K5], Lemma 9. This gives us the first two cases of (44). Case 2: g = 2. Here m = \frac{N}{2} and x̄ = 1, so ᾱ_{k} = I + 2\binom{0 −δz′_{k}}{z_{k} τ}, for all k, where ᾱ_{k} is as in the proof of Theorem 40. (Thus, βα_{k} ≡ ᾱ_{k} (mod N), for all k.) We have x̄_{−ᾱ_{j},N} = x̄ = 1, so 2τ_{−ᾱ_{j},N} = tr(−ᾱ_{j})−2 = −tr(ᾱ_{j})−2 = −(2τ +2)−2 = −2τ −4, i.e., τ_{−ᾱ_{j},N} = −(τ +2). Thus 4δ_{−ᾱ_{j},N} = det(−ᾱ_{j})−1^{2} −2(−τ −2) = det(ᾱ_{j})−1^{2} −2τ +4τ +4 = 4δᾱ_{j} +4τ +4, and hence δ_{−ᾱ_{j},N} = δ_{ᾱ_{j},N} + τ + 1 ≡ δ + τ + 1 (mod m). Since δ_{βα_{k},N} ≡ δ (mod m), for all k, it follows from (46) and Proposition 34 that

δ ≡ δ_{βα_{i},N} ≡ δ_{−δα_{j},N} ≡ δ + τ + 1 (mod m),

so τ ≡ −1 mod m, and hence t := tr(βα) = 2τ +2 ≡ 0 (mod 2m). Since 2m = N, we thus see that t ≡ εN (mod 2N), where ε ∈ {0, 1}, and so q_{α}^{N} = [N^{2}, Nt, (t^{2} +4d)/4] ∼ q′ := [N^{2}, εN^{2}, (ε^{2}N^{2} +4d)/4]; cf. [K5], Lemma 9. Moreover, since −d+1^{2} −1·t = 4δ, we see that d ≡ 1 − t ≡ 1 − εN ≡ 1 + εN (mod 4) because 2|N. We thus obtain the last case of (44).

Now suppose that we are in the exceptional cases. Then −1 ∉ S_{m}(τ, δ), so 2|s = i_{m}(τ, δ). Choose z_{1}, …, z\frac{s}{2} in such a way that their residues mod m are a system of representatives of R_{m}^{×}/⟨−1, S_{m}(τ, δ)⟩ and put z_{k} = −z_{k− \frac{s}{2}} for k = \frac{s}{2} + 1, …, s. Then z_{1}, …, z_{s} satisfy the conditions of Theorem 40. Moreover, since x̄, g, τ and δ satisfy the above conditions, we see that βα_{k} ≡ −βα_{k+ \frac{s}{2}} (mod N), for 1 ≤ k ≤ \frac{s}{2}. Thus, \overline{T} _{α_{k}}^{N} = \overline{T} _{α_{k+ \frac{s}{2}}}^{N}, for these k’s. On the other hand, by construction (and by Proposition 36) we have that γβα_{i}γ^{−1} ≢ ±βα_{j}, for any γ ∈ SL_{2}(ℤ) and 1 ≤ i < j ≤ \frac{s}{2}, so by Theorem 28 it follows that \overline{T} _{α_{i}}^{N} ≠ \overline{T} _{α_{j}}^{N}, for 1 ≤ i < j ≤ \frac{s}{2}, and so (45) follows.

Remark 46 As the above proof shows, even if the hypothesis (17) does not hold for q, but if −1 ∉ S_{m}(τ, δ) and q is equivalent to one of the forms of (44), then we still have the estimate

(47)  |Irr(H(q))| ≤ i_{m}(τ, δ)/2.

To analyze the above results further, we first make the following observations.

Lemma 47 If q has type (N, m, d) and if α, τ, and δ are as Theorem 40, then q_{α}^{N} = [N^{2}, 2mt, Δ], where t = tr(βα) and Δ = Δ(τ, δ) = τ^{2} − 4δ. Thus, if cont(q) = gcd(a, b, c) denotes the content of q = [a, b, c], then we have

Δ_{m}(τ, δ) := gcd(Δ, m) = gcd(cont(q), m) =: c_{m}(q).

Proof. By definition, q_{α}^{N} = [N^{2}, 2mt, (t^{2} + 4d)/g^{2}], so the first assertion follows from (21) because det(βα) = −d. Thus gcd(m, cont(q_{α}^{N}) = gcd(m, N, 2mt, Δ) = gcd(m, Δ). Since q ∼ q_{α}^{N}, we have that cont(q) = cont(q_{α}^{N}), and so the assertion follows.

Lemma 48 Let τ, δ ∈ ℤ and let m ≥ 1. If 4 ∤ Δ_{m}(τ, δ), then −1 ∉ S_{m}(τ, δ) if and only if there exist a prime p ≡ 3 (mod 4) with p|Δ_{m}(τ, δ).

Proof. Write m = ∏_{i=1}^{r} p_{i}^{e_{i}}, where the p_{i}’s are distinct primes. Then the Chinese Remainder Theorem induces an isomorphism

(48)  S_{m}(τ, δ) ≃ ∏_{i=1}^{r} S_{p_{i}^{e_{i}}}(τ, δ),

so −1 ∈ S_{m}(τ, δ) ⇔ −1 ∈ S_{p_{i}^{e_{i}}}(τ, δ), for 1 ≤ i ≤ r. Since −1 ∈ S_{2}(τ, δ) = R_{2}^{×} = {1} and since for an odd prime p we have by (29) that −1 ∉ S_{p^{e}}(τ, δ) ⇔ p|Δ_{p^{e}}(τ, δ) and (\frac{−1}{p}) = −1 ⇔ p|Δ_{p^{e}}(τ, δ) and p ≡ 3 (mod 4), we see that the assertion follows.

We are now ready to prove the following refinement of Theorem 4.

Theorem 49 Let q be a binary form of type (N, m, d) and assume that 4 ∤ c_{m} = gcd(cont(q), m) and that char(K) ∤ Nd. Let ω′ = ω′(c_{m}) = |{p|c_{m} : p is an odd prime}|.

(a) We always have that |Irr(H(q))| ≤ 2^{ω′}.

(b) If there exists a prime p| gcd(N, d) with p ≡ 3 (mod 4) and if q is equivalent to one of the forms of (44), then then |Irr(H(q))| ≤ 2^{ω′−1}.

(c) Suppose that q satisfies condition (17). If q satisfies the conditions of part (b), then equality holds in part (b), otherwise equality holds in part (a).

Proof. By Lemma 47 we have that c_{m} = Δ_{m}(τ, δ), where τ and δ are as in Theorem 40, and so by (27) we have that i_{m}(τ, δ) = 2^{ω′} since 4 ∤ c_{m}. Thus, part (a) follows from (32). Moreover, in the situation of part (b) we have that −1 ∉ S_{m}(τ, δ) by Lemma 48 and the fact that p|c_{m} ⇔ p| gcd(N, d) for the forms of (44), so part (b) follows from Remark 46. Finally, part (c) follows from Proposition 45.

Corollary 50 If q is a binary form of type (N, m, d) with c_{m}(q) | 2, then H(q) is irreducible, provided that char(K) ∤ Nd.

Proof. Since 4 ∤ c_{m} = c_{m}(q) and ω′(c_{m}) = 0, it follows from Theorem 49(a) that |Irr(H(q)| ≤ 2^{0} = 1, so H(q) is irreducible.

We observe that the above corollary implies the following result for the case N = 2.

Proposition 51 If q has type (2, m, d), and if char(K) ∤ 2d, then H(q) is irreducible. Moreover, H(q) is uniquely determined by d if m = 1, whereas there are precisely two distinct H(q)’s for each d when m = 2.

Proof. Since c_{m}(q)|m|N = 2, it follows from Corollary 50 that H(q) is irreducible.

If m = 1, then gcd(d, 2/1) = 1, so d ≡ 1 (mod 2). Thus, it follows from [K5], Proposition 19, that q ∼ [4, 0, d], if d ≡ 1 (mod 4), whereas q ∼ [4, 4, d + 1], if d ≡ 3 (mod 4). Thus H(q) is uniquely determined by d in this case.

If m = 2, then by [K5], loc. cit., we know that either q ∼ q_{1} := [4, 0, 4d] or q ∼ q_{2} := [4, 4, 4d + 1]. Since q_{1} and q_{2} are distinct reduced forms, it follows that q_{1} ≁ q_{2}, and hence H(q_{1}) ≠ H(q_{2}) by Corollary 26.

If 4|c_{m}, then the situation is much more complicated because the formula for i_{m}(τ, δ) is more involved since it does not depend on gcd(m, τ^{2} − 4δ) alone, but involves the quantities w_{4}(Δ̃) and w_{8}(Δ̃), as we saw in Proposition 39. For this reason we introduce the following value w(q) for (special) binary quadratic forms q.

Notation. Let q = [N^{2}, 2mt, c] be a binary form of type (N, m, d) and suppose that 4|c_{m}(q) = gcd(m, cont(q)) = gcd(m, c). Put

(49)  w(q) = \{ \begin{matrix}w_{4} (\frac{c}{4}) & if 4||m, \\ w_{8} (\frac{c}{4}) & if 8|m \end{matrix}

We then have the following connection between w(q) and i_{m}(τ, δ).

Lemma 52 Let α ∈ M_{d} and N ≥ 1 and put g = g_{βα,N}, m = \frac{N}{g}, τ = τ_{βα,N}, and δ = δ_{βα,N}. Moreover, let q_{α}^{N} be the quadratic form associated to α. If 4|c_{m} = gcd(m, cont(q_{α}^{N})) = gcd(m, τ^{2} − 4δ), then

(50)  i_{m}(τ, δ) = 2^{ω′(c_{m})}w(q_{α}^{N}).

Proof. By definition and (21) we have q_{α}^{N} = [N^{2}, 2mt, Δ], where t = tr(βα) and Δ = τ^{2} − 4δ, and so c_{m} = gcd(m, cont(q_{α}^{N})) = gcd(m, Δ). Thus, if 4|c_{m}, then w(q_{α}^{N}) = w_{4}(Δ/4), when 4||m, and w(q_{α}^{N}) = w_{8}(Δ/4) otherwise, and so (50) is just a restatement of formula (30).

It is useful to observe that the value of w(q) only depends on the equivalence class of q and not on the particular form q of the form [N^{2}, 2mt, c]. Thus, we can extend the symbol w(q) to any form q of type (N, m, d) by setting w(q) = w([N^{2}, 2mt, c]), if q ∼ [N^{2}, 2mt, c]. This is justified by the following result.

Lemma 53 Let q_{i} := [N^{2}, 2mt_{i}, c_{i}], where i = 1, 2, be two forms of type (N, m, d). Suppose that q_{1} ∼ q_{2} and that 4|c_{m}(q_{i}). Then

(51)  \frac{c_{1}}{4} ≡ \frac{c_{2}}{4} (mod 2^{r}),

where r = min(3, v_{2}(m)). In particular, we have that w(q_{1}) = w(q_{2}),

Proof. Since 4|c_{m}(q_{i}) = gcd(m, c_{i}) we see that 4|m and hence 4m|m^{2}|N^{2}. Moreover, we have that 2|t_{i} because t_{i}^{2} + 4d = c_{i}g^{2} ≡ 0 (mod 4). By hypothesis, there exists γ = \binom{x y}{z w} ∈ GL_{2}(ℤ) such that γ^{t}M(q_{1})γ = M(q_{2}), where M(q_{i}) = \binom{N^{2} mt_{i}}{mt_{i} c_{i}}, for i = 1, 2. Thus

( \begin{matrix}N^{2} & mt_{2} \\ mt_{2} & c_{2} \end{matrix} ) = ( \begin{matrix}N^{2}x^{2} + 2mt_{1}xz + c_{1}z^{2} & N^{2}xy + mt_{1}(wx + yz) + c_{1}wz \\ N^{2}xy + mt_{1}(wx + yz) + c_{1}wz & N^{2}y^{2} + 2mt_{1}wy + c_{1}w^{2} \end{matrix} )

Since N^{2} ≡ 0 (mod 4m) and 2t_{1} ≡ 0 (mod 4), it follows by comparing the (1, 1)-entries and the (2, 2)-entries of this matrix identity that

(52)  c_{1}z^{2} ≡ 0 (mod 4m) and c_{1}w^{2} ≡ c_{2} (mod 4m).

Suppose first that 2 ∤ w, so w^{2} ≡ 1 (mod 8). Since \frac{c_{1}}{4} w^{2} ≡ \frac{c_{2}}{4} (mod m) by (52), it follows that \frac{c_{1}}{4} ≡ \frac{c_{1}}{4} w^{2} ≡ \frac{c_{2}}{4} (mod 2^{r}). Thus, (51) holds in this case.

Now suppose that 2|w. Then 2 ∤ z because gcd(z, w) = 1, so by the first equation of (52) we obtain that c_{1} ≡ c_{1}z^{2} ≡ 0 (mod 2^{s+2}), where s = v_{2}(m). Thus \frac{c_{1}}{4} ≡ 0 (mod 2^{s}), so by the second equation of (52) we obtain that \frac{c_{2}}{4} ≡ \frac{c_{1}}{4} w^{2} ≡ 0 (mod 2^{s}). This implies that \frac{c_{1}}{4} ≡ \frac{c_{2}}{4} ≡ 0 (mod 2^{r}), so (51) holds in all cases.

To prove the last assertion, suppose first that 4||m, so r = 2. Since the value of w_{4}(c_{i}/4) only depends on the residue class of c_{i}/4 mod 4, we have w_{4}(c_{1}/4) = w_{4}(c_{2}/4) by (51) and so w(q_{1}) = w(q_{2}). If 8|m, then r = 3, so w_{8}(c_{1}/4) = w_{8}(c_{2}/4) and w(q_{1}) = w(q_{2}).

We also require the following result from Nobs[No].

Lemma 54 Suppose that 4| gcd(m, Δ), where Δ = τ^{2} − 4δ, and put Δ̃ = \frac{Δ}{4}.

(a) If 4||m, then −1 ∉ S_{4}(τ, δ) if and only if Δ̃ ≢ 1, 2 (mod 4).

(b) If 8|2^{r}||m, then −1 ∉ S_{2^{r}}(τ, δ) if and only if Δ̃ ≢ 1, 2, 5 (mod 8).

Proof. This follows immediately from Tabelle 1 of Nobs[No].

We are now ready to study the number of irreducible components of H(q) in the case that 4|c_{m}(q). Here we have:

Theorem 55 Let q be a binary form of type (N, m, d) and assume that 4|c_{m} and that char(K) ∤ Nd.

(a) We always have that |Irr(H(q))| ≤ 2^{ω′(c_{m})}w(q).

(b) Suppose that q is equivalent to either [N^{2}, 0, 4d] or to [N^{2}, N^{2}, \frac{N^{2}}{4} + 4d], where 4|N. Put Δ̃ = d in the first case and Δ̃ = \frac{N^{2}}{16} + d in the second case. If either gcd(N, d) ≡ 0 (mod p) for some prime p ≡ 3 (mod 4) or if Δ̃ ≢ 1, 2 (mod 4) when 4||N or if Δ̃ ≢ 1, 2, 5 (mod 8) when 8|N, then |Irr(H(q))| ≤ 2^{ω′(c_{m})−1}w(q).

(c) Suppose that q satisfies condition (17). If q satisfies the conditions of part (b), then equality holds in part (b), otherwise equality holds in part (a).

Proof. (a) Let α ∈ M_{d} be such that q ∼ q_{α}^{N}, and let τ, δ be as in Theorem 40. Then the assertion follows directly from (32) and Lemma 52.

(b) Suppose first q ∼ q_{1} := [N^{2}, 0, 4d], so disc(q) = −16N^{2}d and hence m = N. Note that 4|c_{m} = (N, 4d) ⇔ 4|N. By Proposition 15 of [K5] we can choose α ∈ M_{d} such that q_{α}^{N} = q_{1}, and then τ^{2} − 4δ = 4d by (21). Thus by the proof of Lemma 48 we see that −1 ∉ S_{N}(τ, δ) if and only if either there exists p ≡ 3 (mod 4) with p|c_{m} = gcd(N, 4d) or if −1 ∉ S_{4}(τ, δ), when 4||m = N, or if −1 ∉ S_{2^{r}}(τ, δ), when 8|2^{r}||N. By Lemma 54, the latter two possibilities are equivalent to d = Δ̃ ≢ 1, 2 (mod 4), when 4||N, and to d = Δ̃ ≢ 1, 2, 5 (mod 8), when 8|N. If this is the case, then the assertion (for q ∼ q_{1}) follows from Remark 46 and Lemma 52.

Now suppose that q ∼ q_{2} := [N^{2}, N^{2}, \frac{N^{2}}{4} + 4d]. Here again disc(q) = −16N^{2}d, so m = N. Thus again 4|c_{m} = gcd(N, \frac{N^{2}}{4} + 4d) ⇔ 4|N, and so by a similar argument as in the case q ∼ q_{1} we see that the assertion holds for q ∼ q_{2} with Δ̃ = \frac{N^{2}}{16} + d.

(c) If q is as in part (b), then by the proof of part (b) we see that we are in the exceptional case of Proposition 45, and so by (45) and Lemma 50 we see that Irr(H(q)) = \frac{1}{2}i_{m}(τ, δ) = 2^{ω′(c_{m})−1}w(q).

Now assume that q is not as in part (b). We observe that the third case of (44) does not occur here because 4 ∤ c_{m}(q_{3}), where q_{3} := [N^{2}, εN^{2}, ε^{2} \frac{N^{2}}{4} + d] with 2|N, ε ∈ {0, 1}, and d ≡ 1 + εN (mod 4). Indeed, since disc(q_{3}) = −4N^{2}d, we see that m = \frac{N}{2} , so 4 ∤ c_{m} = gcd(\frac{N}{2} , d) because d ≡ 1 (mod 4) if 4|N.

From this (and the proof of part (b)) it follows that q does not satisfy the conditions of the exceptional case of Proposition 45 and so by (43) and Lemma 50 it follows that Irr(H(q)) = i_{m}(τ, δ) = 2^{ω′(c_{m})}w(q).

Corollary 56 If q has type (4, m, d) and if char(K) ∤ 2d, then H(q) is irreducible.

Proof. If c_{m} ≠ 4, then the assertion follows from Corollary 50 because the condition c_{m} ≠ 4 implies that c_{m}|2 (since c_{m}|N = 4). Thus, assume that c_{m} = 4, so we are in the situation of Theorem 55. Moreover, we have that m = 4 and 4|cont(q).

Since m = 4, we have that q ∼ [4^{2}, 8t, t^{2} + 4d] with 0 ≤ t ≤ \frac{4^{2}}{2·4} = 2, and since 4|cont(q), we see that 2|t. Thus, q ∼ [16, 0, 4d] or q ∼ [16, 16, 4 + 4d], which are precisely the two exceptional cases of Theorem 55(b) when N = 4. Put Δ̃ = d in the first case and Δ̃ = d + 1 in the second case, so w(q) = w_{4}(Δ̃) by (49).

If Δ̃ ≡ 1, 2 (mod 4), then w(q) = w_{4}(Δ̃) = 1 by definition (cf. §6), and thus |Irr(H(q))| ≤ 2^{0} = 1 by Theorem 55(a), so H(q) is irreducible in this case.

If Δ̃ ≢ 1, 2 (mod 4), then w(q) = w_{4}(Δ̃) = 2 by definition, and thus |Irr(H(q))| = \frac{1}{2}w(q) = 1 by Theorem 55(b), so H(q) is irreducible in this case as well. Thus, H(q) is irreducible in all cases.

Proof of Theorem 4. The first assertion follows from Proposition 7 of [K5].

(a) This follows from Theorem 40 and (26). Alternately, if 4 ∤ c_{m}(q), then this follows from Theorem 49(a) and if 4||c_{m}(q), then this follows from Theorem 55(a) because w(q)|2 when 4||m.

(b) From Lemma 29 we know that the given hypothesis on d implies that condition (17) of Theorem 28 holds for q. Since c_{m}(q)|N and 2 ∤ N, we are thus in the situation of Theorem 49(c), and hence the assertion follows from that theorem because the last two cases of (44) do not occur when 2 ∤ N.

Proof of Corollaries 5 and 6. The first corollary is the special case c_{m}(q) = 1 of Theorem 4(a). (Alternately, it is a special case of Corollary 50.)

To prove Corollary 6, let q ∈ Q(N^{2}). Then by Theorem 1 we know that q has type (N, m, d) for some m|N and d, and so by Corollary 34 of [K5] there exists (A, λ) ∈ A_{2}(K) such that q_{(A,λ)} ∼ q. Since (A, λ) ∈ H(q) ⊂ H_{M}, this implies that q → M and q → N^{2}. Thus, the content cont(q) of q divides N^{2} and M, so cont(q) = 1 (because gcd(N, M) = 1), and hence also c_{m}(q) = gcd(cont(q), m) = 1. Thus H(q) is irreducible by Corollary 5.

8 Examples

By using the above results (including Theorem 10, which is proved in [K7]), together with the reduction theory of binary quadratic forms, it is possible to work out explicitly the irreducible components of H_{N^{2}} ∩ H_{M} for small values of N and M. For example:

Proposition 57 (a) The nontrivial intersections of H_{1} with H_{M} for M ≤ 9 are:

H_{1} ∩ H_{4} = H_{1} ∩ H_{5} = H[1, 0, 4], H_{1} ∩ H_{8} = H[1, 0, 4] ∪ H[1, 0, 8], H_{1} ∩ H_{9} = H[1, 0, 8].

The listed H(q)’s are all irreducible if char(K) ≠ 2.

(b) We have the following intersections.

H_{4} ∩ H_{5} = H[1, 0, 4] ∪ H[4, 0, 5] ∪ H[4, 4, 5],
H_{4} ∩ H_{8} = H[1, 0, 4] ∪ H[4, 4, 8] ∪ H[4, 0, 4] ∪ H[4, 0, 8],
H_{4} ∩ H_{9} = H[4, 0, 5] ∪ H[4, 0, 9] ∪ H[4, 4, 9],
H_{4} ∩ H_{12} = H[4, 4, 4] ∪ H[4, 4, 12] ∪ H[4, 0, 8] ∪ H[4, 0, 12],
H_{4} ∩ H_{13} = H[4, 0, 1] ∪ [4, 0, 9] ∪ [4, 0, 13] ∪ [4, 4, 5] ∪ [4, 4, 13].

Moreover, the listed H(q)’s are irreducible if char(K) ≠ 2, 3, 5.

Proof. Let q be a form which appears on the right hand side of the formula (9) for H_{N^{2}} ∩ H_{M}. Then q primitively represents N^{2} and M. Assume that H(q) ≠ ∅. Then q ∈ Q(N^{2}) by Theorem 1, so q has type (N, m, d) for some m and d; cf. Theorem 4.

(a) If N = 1, then every form q of type (1, m, d) is equivalent to [1, 0, 4d]. Since such a form does not represent any numbers strictly between 1 and 4d, we must have 4d ≤ M. For each such form q, we make a table of the values n which are primitively represented by q (for n ≤ 9). It is then an easy check to see which of these forms represent a given M. Since these forms all have type (1, 1, d) with d|2, it follows from Corollary 5 that they are are irreducible if char(K) ≠ 2.

(b) Here N = 2. By the classification of type (2, m, d) (cf. the proof of Proposition 51) we know that q ∼ [4, 0, n] or q ∼ [4, 4, n] for some n ≥ 1. If n > 4, then these forms do not represent any numbers strictly between 4 and n, so if M ≥ 5, then n ≤ M. Thus, we need to look at only 2M forms and decide which of these has type (2, m, d) and primitively represents M, which is an easy exercise. From this list and Proposition 51 we see that H(q) is irreducible for these forms under the given hypotheses.

Note that the above forms are reduced (in the sense of reduction theory) when n ≥ 4, so if they are distinct, then they are not equivalent. We thus see that the associated H(q)’s are distinct by Corollary 26.

Remark 58 The method of analyzing the components of H_{N^{2}} ∩ H_{M} which was used in the proof of Proposition 57 can be extended to other cases, but the situation becomes more involved. On the one hand there are more forms to consider and one the other hand one has to apply the reduction algorithm to the forms to be sure to have distinct H(q)’s. For example, we have

H_{5} ∩ H_{9} = H[4, 0, 5] ∪ H[5, 2, 9] ∪ H[5, 4, 8],

where the first form is the reduction of the form [9, 8, 4] of type (3, 1, 5), the second is the reduction of the form [9, 2, 5] of type (3, 1, 11) and the third is the reduction of the form [9, 6, 5] of type (3, 3, 1).

Similarly, by using a computer we can use the results of the previous sections (together with Proposition 12 and reduction theory) to determine the number of irreducible components of the intersection H_{N^{2}} ∩ H_{m} for slightly larger values of N and M. Recall that M has to satisfy M ≡ 0, 1 (mod 4), for otherwise H_{M} = ∅.

Proposition 59 If char(K) = 0, then the number of irreducible components of the intersection H_{N^{2}} ∩ H_{M} of Humbert surfaces for N ≤ 5 and M ≤ 25 is given in the following table:

\begin{matrix}_{N^{2}}\^{m} & 1 & 4 & 5 & 8 & 9 & 12 & 13 & 16 & 17 & 20 & 21 & 24 & 25 \\ 1 & ∗ & 1 & 1 & 2 & 1 & 2 & 2 & 2 & 3 & 3 & 2 & 3 & 3 \\ 4 & 1 & ∗ & 3 & 4 & 3 & 4 & 5 & 5 & 5 & 6 & 5 & 6 & 6 \\ 9 & 1 & 3 & 3 & 5 & ∗ & 6 & 5 & 6 & 8 & 7 & 8 & 10 & 9 \\ 16 & 2 & 5 & 5 & 6 & 6 & 9 & 9 & ∗ & 9 & 12 & 10 & 11 & 12 \\ 25 & 3 & 6 & 7 & 8 & 9 & 9 & 10 & 12 & 15 & 16 & 11 & 13 & ∗ \end{matrix}

Moreover, all the H(q)’s appearing in these intersections are irreducible, with the exception of the cases (N, m) = (3, 21), (3, 24), (5, 5), and (5, 20). For these cases, the number of irreducible components is given in bold type in the above table.

Proof. (Sketch.) The cases N = 1 and N = 2 are handled by a similar method as in the proof of Proposition 57. For N ≥ 3 we use again Proposition 12 together with the fact that if H(q) appears in the intersection H_{N^{2}} ∩ H_{M}, then q has type (N, m, d) with m^{2}d ≤ \frac{N^{2}M}{4m^{2}} ; cf. Remark 13.

For N = 4 all the H(q)’s appearing in H_{16} ∩ H_{M} are irreducible by Corollary 56. For the cases N = 3 and N = 5 we apply Theorem 10. By that result (and

computer computations) we see that all the H(q)’s which appear in the intersection H_{N^{2}} ∩ H_{M} for M ≤ 25 are irreducible, with the following exceptions: 1) H[9, 6, 21], which appears in H_{9} ∩H_{21} and in H_{9} ∩H_{24}; 2) H[25, 10, 5], which appears in H_{25} ∩H_{5} and in H_{25} ∩ H_{20}; 3) H[25, 20, 20], which appears in H_{25} ∩ H_{20}; and 4) H[25, 0, 20], which also appears in H_{25}∩H_{20}. All these exceptions have two irreducible components by Theorem 10.

Note also that H[9, 6, 9], which appears in H_{9} ∩H_{12} and in H_{9} ∩H_{24}, is irreducible by Theorem 10 because [9, 6, 9] ∼ [24, 24, 9] is an ambiguous form.

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