On Quadratic Diophantine Equations in Four

Variables and Orders Associated with Lattices

Manabu Murata

Received: October 12, 2013 Revised: November 8, 2013

Communicated by Don Blasius

Abstract. This paper treats certain lattices in ternary quadratic spaces, which are obtained from the data of a non-zero element and a maximal lattice in a quaternary space. Each class in the genus of such a lattice with respect to the special orthogonal group corresponds to an isomorphism class in the genus of an order associated with the lattice in a quaternion algebra. Using this result together with the principle of Shimura, we show that the number of classes of the primitive solutions of a quadratic Diophantine equation in four variables coincides with the type number of the order under suitable conditions on the given element. We illustrate this result by explicit examples.

2010 Mathematics Subject Classification: 11E12, 11D09, 11E20

Introduction

Let (V, φ) be a nondegenerate quadratic space of dimension 4 over a number field F, that is, V is a four-dimensional vector space over F and φ is a nondegenerate symmetric F -bilinear form on V. For an element h of V such that φ[h](= φ(h, h)) ≠ 0, we put

W = (F h)^{⊥} = {x ∈ V | φ(x, h) = 0}

and consider a quadratic space (W, ψ) of dimension 3 over F with the restriction ψ of φ to W. In this paper the special orthogonal group SO^{ψ} of ψ is regarded as the subgroup {γ ∈ SO^{φ} | hγ = h} of the orthogonal group SO^{φ} of φ. The orthogonal group O^{ψ} of ψ is also identified with O^{φ} in a similar manner. For a maximal lattice L in (V, φ) we put

L[q, 𝔟] = {x ∈ V | φ[x] = q, φ(x, L) = 𝔟}, D(L) = {α ∈ O_{A}^{φ} | Lα = L}, where q = φ[h], 𝔟 = φ(h, L), and the subscript A means adelization. Since L[q, 𝔟] is stable under Γ^{·}(L) = O^{φ} ∩ D(L), the set L[q, 𝔟]/Γ^{·}(L) is meaningful in an obvious way. In the sense of Shimura [9, Introduction I], L[q, 𝔟] is the set of primitive solutions of the equation φ[x] = q. Our interest in this paper is basically the set

L[q, 𝔟]/Γ^{·}(L)

consisting of the classes of such solutions. By the principle of [9, Theorem 11.6], each class of solutions of L_{i}[q, 𝔟] modulo Γ^{·}(L_{i}) for i ∈ J corresponds to a class of O^{ψ} relative to an open subgroup O_{A}^{ψ} ∩ D(L) in O_{A}^{ψ}. Here {L_{i}}_{i∈J} is a set of representatives for the O^{φ}-classes in the O^{φ}-genus of L for which L_{i}[q, 𝔟] ≠ ∅ (see also (4.1) below). Now we consider the lattice L ∩ W in (W, ψ) and the even Clifford algebra A^{+}(W) of ψ, which is a quaternion algebra over F since the dimension of W is 3. Let A(N) be the order generated by 𝔤 and N in the Clifford algebra of ψ, and put A^{+}(N) = A^{+}(W) ∩ A(N) for an integral lattice N in (W, ψ). Here 𝔤 is the ring of all integers of F and the terms integral and maximal are given in §1.1. To L∩W we can associate an order 𝔒 in A^{+}(W), containing A^{+}(L∩W), whose discriminant is given by (3.22) below. The purpose of this paper is to define such an order 𝔒, to give a bijection of the SO^{ψ}-classes in the SO^{ψ}-genus of L ∩ W onto the isomorphism classes in the genus of 𝔒, and to prove

∑_{i∈J} # {L_{i}[q, 𝔟]/Γ^{·}(L_{i})} = t(𝔒)  (0.1)

Quadratic Diophantine Equations in Four Variables 249

through the above principle under suitable conditions on h ∈ L[q, 𝔟], where the genus of 𝔒 is defined by the set {α^{−1}𝔒α | α ∈ A^{+}(W)_{A}^{×}} and t(𝔒) is the type number of 𝔒. To obtain the order 𝔒, we proceed in a similar manner to [10, §4.6] under mild conditions on h ∈ L[q, 𝔟] (Proposition 3.3 (3)). As for the bijection, given a lattice N in the genus of L ∩ W, put N = (L ∩ W)τ(α) with α ∈ A^{+}(W)_{A}^{×}. Here τ is a surjective homomorphism of G^{+}(W)_{A} onto SO_{A}^{ψ} and G^{+}(W) is the even Clifford group of ψ, which is given by A^{+}(W)^{×}. Then our bijection is defined by N ⟼ α^{−1}𝔒α (Theorem 3.4 (2)). This result mainly relies on a description of L ∩ W in A^{+}(W) by means of A^{+}(L ∩ W) (cf. [4, Theorem 2.1]). Now in Proposition 4.3, we shall prove that O^{ψ}εD(L ∩ W) = O^{ψ}ε(O_{A}^{ψ} ∩ D(L)) for every ε ∈ O_{A}^{ψ} under several assumptions on h ∈ L[q, 𝔟]. Because W is odddimensional, the class number of the genus of L ∩ W with respect to O^{ψ} equals that with respect to SO^{ψ}. Hence by virtue of the principle mentioned above, the number ∑_{i∈J} # {L_{i}[q, 𝔟]/Γ^{·}(L_{i})} is equal to the number of SO^{ψ}-classes in the SO^{ψ}-genus of L ∩ W. Our main result (0.1) follows from this fact and the above bijection. It should be noted that the genus of 𝔒 is determined by the discriminant if 𝔒 has squarefree discriminant, for instance, if 𝔒 is maximal. When the quaternion algebra is totally definite, there are formulas for computing the type number of such an order; see Peters [5, Satz 14] or Pizer [6, Theorem A and Theorem B], etc.. In Section 5 we will take up totally-positive definite forms φ and employ their numerical tables [5, Tabelle 2] and [6, Table 1] for type numbers. It seems that there are few numerical examples for the type number of the genus of an order whose discriminant is not squarefree and for the class number of the genus of a lattice which is neither maximal nor unimodular with respect to a definite form. Here we assume that h satisfies the conditions in Proposition 4.3. Then (0.1) contains the case of non-maximal (and often non-unimodular) L ∩ W, more clearly, the case that 𝔒 has at most one higher-power prime 𝔭^{e} (e > 1) in its discriminant, where 𝔭 is a prime ideal of F (see also (4.10) below).

To see the existence of such an element h, as an application of Proposition 4.3, let B_{r, ∞} be a definite quaternion algebra over Q ramified exactly at a prime number r and take a prime number d prime to r so that d ≡ 1 (mod 4). In Theorem 5.1 we shall show: For every odd prime number p prime to dr and 0 ≤ n ∈ Z, except when n is odd and p remains prime in Q(\sqrt{d}), there exists a maximal lattice L in (V, φ) over Q of invariants \{4, Q(\sqrt{d}), B_{r, ∞}, 4\} (see (1.5) for the definitions) such that L[dp^{n}, 2^{−1}dZ] ≠ ∅. And moreover, formula (0.1) is valid for h ∈ L[dp^{n}, 2^{−1}dZ] with an order 𝔒 in B_{r, ∞} of discriminant rp^{n}Z. For example, let us take (V, φ) of invariants \{4, Q(\sqrt{29}), B_{2, ∞}, 4\}. By [9, Theorem 12.14 (vi)] the number of O^{φ}-classes in the O^{φ}-genus of maximal lattices in (V, φ) equals the number #{Λ[29, Z]/Γ^{·}(Λ)}, where Λ is a maximal lattice in a five-dimensional quadratic space over Q with respect to the quadratic form defined by the sum of five squares. In [9, §12.15], #{Λ[29, Z]/Γ^{·}(Λ)} was determined, and it is 3. Hence the genus of maximal lattices in (V, φ) consists of three O^{φ}-classes. For details, see the last part of Section 4.1 in the text. We can also see the representatives {L_{1}, L_{2}, L_{3}} of such classes by means of a bijection in Lemma 4.1 applied to the set Λ[29, Z]. Then we have the following numerical table:

\begin{matrix}p & N_{1}(29p) & N_{2}(29p) & N_{3}(29p) & t(2, p) & 𝔠(29p) \\ 1 & 2 & 0 & 0 & 1 & 1 \\ 5 & 0 & 2 & 0 & 1 & 1 \\ 7 & 0 & 0 & 2 & 1 & 1 \\ 13 & 0 & 2 & 2 & 2 & 2 \\ 23 & 0 & 0 & 6 & 1 & 1 \\ 53 & 0 & 10 & 6 & 3 & 3 \\ 59 & 24 & 8 & 6 & 3 & 3 \end{matrix}

Here we put N_{i}(29p) = #L_{i}[29p, 2^{−1} · 29Z], t(2, p) = t(𝔒) with an order 𝔒 in B_{2, ∞} of discriminant 2pZ, and 𝔠(29p) = ∑_{i=1}^{3} #{L_{i}[29p, 2^{−1} ·29Z]/Γ^{·}(L_{i})}; we quoted the type number t(2, p) in [6, Table 1]. Therefore we have

# {L_{i}[29 · 59, 2^{−1} · 29Z]/Γ^{·}(L_{i})} = 1 for i = 1, 2, 3,

for instance. It is noted that #Γ(L_{1}) = 48, #Γ(L_{2}) = 8, and #Γ(L_{3}) = 6, where Γ(L_{i}) = SO^{φ}∩D(L_{i}). In Section 5.3 we shall give a few numerical tables for r = 2 and d = 5, 13, 17, 29 including the above cases. As a special case of Theorem 5.1 we have Corollary 5.2, which states that for any d as in Theorem 5.1 only one O^{φ}-class in the genus satisfies L[d, 2^{−1}dZ] ≠ ∅ with a lattice L in the class and then (0.1) precisely gives #{L[d, 2^{−1}dZ]/Γ^{·}(L)} = 1, provided the type number of B_{r, ∞} is 1. The existence of a maximal lattice in Theorem 5.1 can be shown in a similar way to [3, Proposition 4.3] by means of two explicit formulas concerning #L[dp^{n}, 2^{−1}dZ] and #L[dp^{n}, 2^{−1}Z]. Those formulas will be given in (5.2) and (5.3) without detailed proofs because of the length of the paper. The author hopes to prove it in a subsequent paper.

Acknowledgements. I sincerely wish to thank Professor Koji Doi for his encouragement and for giving a remark on the discriminant of 𝔒 in (0.1). I am also thankful to Dr. Takashi Yoshinaga for supporting the computation of #L_{i}[dp^{n}, 2^{−1}dZ] in the numerical tables and for his much help to the determination of Λ[d, Z]/Γ(Λ) in Section 5.3 in November 2010. I would like to express my deep gratitude to the anonymous referee, who carefully read the manuscript and guided me to the connection of the classes of maximal lattices in a four-dimensional quadratic space with the sum of five squares.

Notation. We denote by Z, Q, and R the ring of rational integers, the field of rational numbers, and the field of real numbers, respectively. If R is an associative ring with identity element and if M is an R-module, then we write R^{×} for the group of all invertible elements of R and M_{n}^{m} the R-module of m × n-matrices with entries in M. We set R^{×2} = {a^{2} | a ∈ R^{×}}. For a finite set X, we denote by #X the number of elements in X. We set [a] = Max{n ∈ Z | n ≤ a}. Let V be a vector space over a field F of characteristic 0, and GL(V) the group of all F -linear automorphisms of V. We let GL(V) act on V on the right. Let F be an algebraic number field and 𝔤 the ring of all algebraic integers in F. For a fractional ideal of F we often call it a 𝔤-ideal. Let a, h, and r be the sets of archimedean primes, nonarchimedean primes, and real archimedean primes of F, respectively. We denote by F_{v} the completion of F at v ∈ a ∪ h, and by F_{A} and F_{A}^{×} the adele ring and the idele group of F, respectively. We often identify v with the prime ideal of F corresponding to v ∈ h, and write x_{v} for the image of x under the embedding of F into R over Q at v ∈ r. For v ∈ h we denote by 𝔤_{v}, 𝔭_{v}, and π_{v} the maximal order, the prime ideal, and a prime element of F_{v}, respectively. If K is a quadratic extension of F, we denote by D_{K/F} the relative discriminant of K over F, and put K_{v} = K ⊗_{F} F_{v} for v ∈ h.

By a 𝔤-lattice, or simply a lattice L in a vector space V over a number field or nonarchimedean local field F, we mean a finitely generated 𝔤-submodule in V containing a basis of V. By an order in a quaternion algebra B over F we mean a subring of B containing 𝔤 that is a 𝔤-lattice in B. For a symmetric F -bilinear form φ on V and two subspaces X and Y of V, we denote by X ⊕ Y the direct sum of X and Y if φ(x, y) = 0 for every x ∈ X and y ∈ Y; then we also denote by φ|_{X} the restriction of φ to X. When X is an object defined over a number field F, we often denote by X_{v} the localization at a prime v if it is meaningful. For given local objects X_{v} in the text with v ∈ a ∪ h, we put X_{a} = ∏_{v∈a} X_{v} and X_{h} = ∏_{v∈h} X_{v}.

1 Preliminaries for quadratic forms

1.1 Quadratic spaces and Clifford algebras

Let F be an algebraic number field or its completion at a prime. Throughout the paper we often call the former a global field and the latter a local field when it is nonarchimedean. Let (V, φ) be a quadratic space over F, that is, V is a vector space over F and φ is a symmetric F -bilinear form on V. In this paper we consider only a nondegenerate form φ. We put φ[x] = φ(x, x) for x ∈ V. We define the orthogonal group and the special orthogonal group of φ by

O^{φ}(V) = O^{φ} = {γ ∈ GL(V) | φ(xγ, yγ) = φ(x, y)},
SO^{φ}(V) = SO^{φ} = {γ ∈ O^{φ}(V) | det(γ) = 1}.

We denote by A(φ) = A(V) the Clifford algebra of φ and by A^{+}(φ) = A^{+}(V) the even Clifford algebra of φ. For x ∈ A(V) we mean x^{∗} the image of x under the canonical involution of A(V). We define the even Clifford group G^{+}(V) of (V, φ) by

G^{+}(V) = {α ∈ A^{+}(V)^{×} | α^{−1}V α = V }.  (1.1)

We denote by τ a homomorphism defined as follows:

τ : G^{+}(V) ⟶ SO^{φ}(V) via xτ(α) = α^{−1}xα for x ∈ V .  (1.2)

This is surjective and the kernel is F^{×}; see [9, Theorem 3.6], for example. For a 𝔤-lattice L in V we put

L̃ = L = {x ∈ V | 2φ(x, L) ⊂ 𝔤}.  (1.3)

We call L integral with respect to φ if φ[L] ⊂ 𝔤. We note that L ⊂ L̃ if L is integral. By a 𝔤-maximal, or simply a maximal, lattice L with respect to φ, we understand a 𝔤-lattice L in V which is maximal among 𝔤-lattices on which the values φ[x] are contained in 𝔤. For an integral lattice L in V with respect to φ, we denote by A(L) the subring of A(V) generated by 𝔤 and L. We also put

A^{+}(L) = A^{+}(V) ∩ A(L).  (1.4)

Then A(L) (resp. A^{+}(L)) is an order in A(V) (resp. A^{+}(V)) (cf. [9, §8.2]).

For a global field F and v ∈ a ∪ h, we put V_{v} = V ⊗_{F} F_{v} and denote by φ_{v} the F_{v}-bilinear extension of φ to V_{v}; we then put (V, φ)_{v} = (V_{v}, φ_{v}). For v ∈ h and a 𝔤_{v}-maximal lattice L_{v} in V_{v}, (V, φ)_{v} has a Witt decomposition as follows (cf. [9, Lemma 6.5]): V_{v} = Z_{v} ⊕ ∑_{i=1}^{r_{v}}(F_{v}e_{i} + F_{v}f_{i}) and L_{v} = N_{v} + ∑_{i=1}^{r_{v}}(𝔤_{v}e_{i} + 𝔤_{v}f_{i}) with some elements e_{i} and f_{i} (i = 1, ⋯, r_{v}) such that φ_{v}(e_{i}, e_{j}) = φ_{v}(f_{i}, f_{j}) = 0 and 2φ_{v}(e_{i}, f_{j}) = 1 or 0 according as i = j or i ≠ j. Here Z_{v} = {z ∈ V_{v} | φ_{v}(z, e_{i}) = φ_{v}(z, f_{i}) = 0 for every i}, on which φ_{v} is anisotropic; N_{v} = {x ∈ Z_{v} | φ_{v}[x] ∈ 𝔤_{v}}, which is a unique 𝔤_{v}-maximal lattice in Z_{v} with respect to φ_{v}. The dimension t_{v} of Z_{v} is uniquely determined by φ_{v} and 0 ≤ t_{v} ≤ 4 for v ∈ h (cf. [9, Theorem 7.6 (ii)]). We call Z_{v} a core subspace of (V, φ)_{v} and t_{v} the core dimension of (V, φ) at v. For convenience, we also call a subspace U_{v} of V_{v} anisotropic if φ_{v} is so on U_{v}. For 𝔤-lattices L and M in V over a global or local field F, we denote by [L/M] a 𝔤-ideal of F generated over 𝔤 by det(α) of all F -linear automorphisms α of V such that Lα ⊂ M. If F is a global field, then [L/M] = ∏_{v∈h}[L_{v}/M_{v}] with the localization [L/M]_{v} = [L_{v}/M_{v}] at each v. Following [11, §6.1], in both global and local F, we call [L̃/L] the discriminant ideal of (V, φ) if L is a 𝔤-maximal lattice in V with respect to φ. This is independent of the choice of L. If F is a local field, the discriminant ideal of φ coincides with that of a core subspace of φ.

By the invariants of (V, φ) over a number field F, we understand a set of data

\{n, F (\sqrt{δ}), Q(φ), \{s_{v}(φ)\}_{v∈r}\},  (1.5)

where n is the dimension of V, F (\sqrt{δ}) is the discriminant field of φ with δ = (−1)^{n(n−1)/2} det(φ), Q(φ) is the characteristic quaternion algebra of φ, and s_{v}(φ) is the index of φ at v ∈ r. For these definitions, the reader is referred to [11, §1.1, 3.1, and 4.1] (cf. also [4, (1.6)]). By virtue of [11, Theorem 4.2], the isomorphism class of (V, φ) is determined by \{n, F (\sqrt{δ}), Q(φ), \{s_{v}(φ)\}_{v∈r}\} and vice versa. The characteristic algebra Q(φ_{v}) is also defined for φ_{v} at v ∈ a ∪ h (cf. [11, §3.1]). By [11, Lemma 3.3] the isomorphism class of (V, φ)_{v} is determined by \{n, F_{v}(\sqrt{δ}), Q(φ_{v})\} if v ∈ h. As for v ∈ a, it is determined by \{n, s_{v}(φ)\} if v ∈ r, and by the dimension n if v ∉ r. If v ∈ r, then Q(φ_{v}) is given by [11, (4.2a) and (4.2b)], for example. If v ∉ r, then Q(φ_{v}) = M_{2}(C), where C is the field of complex numbers.

Let SO^{φ}(V)_{A} (resp. O^{φ}(V)_{A}) be the adelization of SO^{φ}(V) (resp. O^{φ}(V)) in the usual sense (cf. [9, §9.6]). For α ∈ SO^{φ}(V)_{A} and a 𝔤-lattice L in V, we denote by Lα the 𝔤-lattice in V whose localization at each v ∈ h is given by L_{v}α_{v}. We put

C(L) = {α ∈ SO^{φ}(V)_{A} | Lα = L}, C(L_{v}) = SO^{φ_{v}}(V_{v}) ∩ C(L), (v ∈ h) Γ(L) = SO^{φ}(V) ∩ C(L).

Then the map α ⟼ Lα^{−1} gives a bijection of SO^{φ}\SO_{A}^{φ}/C(L) onto {Lα | α ∈ SO_{A}^{φ}}/SO^{φ}. We call {Lα | α ∈ SO_{A}^{φ}} the SO^{φ}-genus of L, {Lγ | γ ∈ SO^{φ}} the SO^{φ}-class of L, and #{SO^{φ}\SO_{A}^{φ}/C(L)} the class number of SO^{φ} relative to C(L) or the class number of the genus of L with respect to SO^{φ}. It is known that all 𝔤-maximal lattices in V with respect to φ form a single SO^{φ}-genus. Let A^{+}(V)_{A}^{×} (resp. G^{+}(V)_{A}) be the adelization of A^{+}(V)^{×} (resp. G^{+}(V)). We can extend τ of (1.2) to a homomorphism of G^{+}(V)_{A} onto SO_{A}^{φ}. We denote it by the same symbol τ (cf. [9, §9.10]). For a 𝔤-lattice L in V, q ∈ F, and a 𝔤-ideal 𝔟 of F, we put

L[q] = {x ∈ L | φ[x] = q}, L[q, 𝔟] = {x ∈ V | φ[x] = q, φ(x, L) = 𝔟}.

Here φ(x, L) = {φ(x, y) | y ∈ L}, which becomes a 𝔤-ideal of F. Suppose F is a nonarchimedean local field. Let V have dimension n > 2 and L be a 𝔤-maximal lattice in V with respect to φ. Then [9, Theorem 10.5] due to Shimura shows that

L[q, 𝔟] = hC(L),  (1.6)

provided h ∈ L[q, 𝔟] (cf. also [12, Theorem 1.3]).

For a quaternion algebra B over F, we put 2β(x, y) = xy^{ι} + yx^{ι} for x, y ∈ B with the main involution ι of B. For an order 𝔬 in B it is known that [𝔬̃/𝔬] = d(𝔬)^{2} with an integral ideal d(𝔬) of F. Here 𝔬̃ is defined by (1.3) with β. The ideal d(𝔬) is called the discriminant of 𝔬. If F is a number field and 𝔬 is a maximal order, then d(𝔬) is the product of all prime ideals ramified in B, which is called the discriminant of B and denoted by D_{B}. We set

T (𝔬) = {α ∈ B_{A}^{×} | α𝔬 = 𝔬α}, T (𝔬_{v}) = B_{v}^{×} ∩ T (𝔬) (v ∈ a ∪ h),
Γ^{∗}(𝔬) = B^{×} ∩ T (𝔬),

where B_{A}^{×} is the adelization of B^{×}, B_{v} = B ⊗_{F} F_{v}, and 𝔬_{v} = 𝔬 ⊗_{𝔤} 𝔤_{v}. The number #{T (𝔬)\B_{A}^{×}/B^{×}} is called the type number of 𝔬. Let U = B_{a}^{×} ∏_{v∈h} 𝔬_{v}^{×} in B_{A}^{×}. Then the number of U \ B_{A}^{×}/B^{×} is called the class number of 𝔬.

Here we introduce two symbols below, which will be used throughout the paper. Let F be a nonarchimedean local field and 𝔭 the prime ideal of F. For b ∈ F^{×} we set

(F (\sqrt{b})/𝔭) = \begin{cases}1 & if F (\sqrt{b}) = F , \\ −1 & if F (\sqrt{b}) is an unramified quadratic extension of F , \\ 0 & if F (\sqrt{b}) is a ramified quadratic extension of F . \end{cases} For a quaternion algebra B over F we set

χ(B) = \begin{cases}1 & if B ≅ M_{2}(F), \\ −1 & if B is a division algebra. \end{cases}

1.2 Ternary quadratic spaces

We recall some basic facts on 3-dimensional quadratic spaces (W, ψ) over a number field or its completion F. The characteristic algebra Q(ψ) is given by A^{+}(W) by definition. The core dimension s_{v} of (W, ψ) at v ∈ h is determined by

s_{v} = \begin{cases}1 & if Q(ψ_{v}) = M_{2}(F_{v}), \\ 3 & if Q(ψ_{v}) is a division algebra. \end{cases}  (1.7)

This can be seen from [11, §3.2] and the proof of [11, Lemma 3.3]. There are isomorphisms of (W, ψ) onto (A^{+}(W)^{∘}, dν^{∘}) with d ∈ F^{×}. Here A^{+}(W)^{∘} = {x ∈ A^{+}(W) | x^{∗} = −x}, ν[x] = xx^{∗} for x ∈ A^{+}(W), and ν^{∘} is the restriction of ν to A^{+}(W)^{∘}. Let us explain such isomorphisms, following [9, §7.3]. Take an orthogonal basis {k_{1}, k_{2}, k_{3}} of W with respect to ψ, namely, an F - basis {k_{i}} of W such that ψ(k_{i}, k_{j}) = 0 for i ≠ j. Under the identification of W with the corresponding subspace in the Clifford algebra A(W), put ξ = k_{1}k_{2}k_{3} ∈ A(W)^{×}; then F + F ξ is the center of A(W). We see that A^{+}(W) = F +F k_{1}k_{2}+F k_{1}k_{3}+F k_{2}k_{3} and W ξ = F k_{1}k_{2}+F k_{1}k_{3}+F k_{2}k_{3}. By [9, Theorem 2.8 (ii)], A^{+}(W) is a quaternion algebra over F; the main involution coincides with the canonical involution ∗ restricted to A^{+}(W). Then the mapping x ⟼ xξ gives an F -linear isomorphism of W onto A^{+}(W)^{∘} such that (xξ)(xξ)^{∗} = ξξ^{∗}ψ[x] for x ∈ W. Putting ν[y] = yy^{∗} for y ∈ A^{+}(W), we have an isomorphism

(W, ψ) ≅ (A^{+}(W)^{∘}, (ξξ^{∗})^{−1}ν^{∘}) via x ⟼ xξ.  (1.8)

We note that ξξ^{∗} ∈ det(ψ)F^{×2}, since ξξ^{∗} = ψ[k_{1}]ψ[k_{2}]ψ[k_{3}] ∈ F^{×}. Let G^{+}(W) be the even Clifford group of (W, ψ) as in (1.1) and τ the homomorphism defined in (1.2). By the definition of A^{+}(W)^{∘}, α^{−1}A^{+}(W)^{∘}α = A^{+}(W)^{∘} for α ∈ A^{+}(W)^{×}. Hence we have G^{+}(W) = A^{+}(W)^{×}. Moreover, under the isomorphism (1.8) we can understand that

xτ(α) = α^{−1}xξαξ^{−1}

for x ∈ W and α ∈ A^{+}(W)^{×}. Now, the pair (A^{+}(W), ν) can be viewed as a quaternary quadratic space over F. We note that ν(x, y) = 2^{−1}T r_{A^{+}(W)/F} (xy^{∗}) for x, y ∈ A^{+}(W). For an integral lattice N in W with respect to ψ, we consider the order A^{+}(N) in A^{+}(W) defined by (1.4). Its discriminant d(A^{+}(N)) is given by [A^{+}(N) /A^{+}(N)] = d(A^{+}(N))^{2}, where A^{+}(N) is defined by (1.3) with ν. By [4, Lemma 1.1], d(A^{+}(N)) = 2^{−1}[Ñ/N]. It is noted that if the order A^{+}(N) is maximal in A^{+}(W) for an integral lattice N in (W, ψ), then N is 𝔤-maximal with respect to ψ. The converse is not true; namely, in general, A^{+}(N) is not maximal even if N is a maximal lattice.

2 Orthogonal complements in quaternary spaces

2.1 Invariants and discriminant ideals

Let (V, φ) be a 4-dimensional quadratic space over a number field F. The characteristic algebra Q(φ) is determined by A(φ) ≅ M_{2}(Q(φ)) by definition. Set B = Q(φ) and K = F (\sqrt{δ}) with δ = det(φ). The core dimension t_{v} of (V, φ) at v ∈ h is determined by

t_{v} = \begin{cases}0 & if F_{v}(\sqrt{δ}) = F_{v} and Q(φ_{v}) = M_{2}(F_{v}), \\ 4 & if F_{v}(\sqrt{δ}) = F_{v} and Q(φ_{v}) is a division algebra, \\ 2 & if F_{v}(\sqrt{δ}) ≠ F_{v}. \end{cases}  (2.1)

This can be seen from [11, §3.2] and the proof of [11, Lemma 3.3]. For h ∈ V such that φ[h] = q ≠ 0 we put

W = (F h)^{⊥} = {x ∈ V | φ(x, h) = 0}.  (2.2)

Then (W, ψ) is a nondegenerate ternary quadratic space over F with the restriction ψ of φ to W and (V, φ) = (W, ψ) ⊕ (F h, φ|_{F h}). The invariants of (W, ψ) are given by \{3, F (\sqrt{−δq}), Q(ψ), \{s_{v}(ψ)\}_{v∈r}\}, which are independent of the choice of h so that φ[h] = q. The characteristic algebra Q(ψ) = A^{+}(W) is determined by the local algebras Q(ψ_{v}) for all primes v of F. Then by [2, Theorem 1.1 (1)], Q(ψ_{v}) = M_{2}(F_{v}) holds exactly in the following cases:

δ ∈ F_{v}^{×2} and v ∤ D_{B},

δ ∉ F_{v}^{×2}, v ∤ D_{B}, and q ∈ κ_{v}[K_{v}^{×}], δ ∉ F_{v}^{×2}, v | D_{B}, and q ∉ κ_{v}[K_{v}^{×}], v ∈ r, q_{v} > 0, and s_{v}(φ) = 0, 2, v ∈ r, q_{v} < 0, and s_{v}(φ) = 0, −2, v ∈ a such that v ∉ r,

where κ_{v} is the norm form of K_{v}. It should be noted that

M_{2}(Q(φ)) ≅ Q(ψ) ⊗_{F} {K, q},  (2.3)

where F is a number field or its completion and {K, q} is the quaternion algebra over F defined in [4, (1.12)] if K ≠ F; we set {K, q} = M_{2}(F) if K = F (see also [9, §1.10]). This (2.3) can be seen from [11, Theorem 7.4 (i)]. The index at v ∈ r is given by s_{v}(ψ) = s_{v}(φ)−1 if q_{v} > 0 and s_{v}(ψ) = s_{v}(φ)+1 if q_{v} < 0. The core dimension of (W, ψ) at v ∈ h is determined by (1.7).

The Clifford algebra A(W) can be viewed as a subalgebra of A(V) with the restriction ψ. Then A^{+}(W) = {x ∈ A^{+}(V) | xh = hx} and G^{+}(W) = {α ∈ G^{+}(V) | αh = hα} by [9, Lemma 3.16]. The canonical involution of A(W) coincides with ∗ of A(V) restricted to A(W). In particular, such an involution ∗ gives the main involution of the quaternion algebra A^{+}(W).

Let L and M be 𝔤-maximal lattices in V and W with respect to φ and ψ, respectively. The discriminant ideals of φ and ψ are given as follows:

[L̃/L] = D_{K/F} 𝔢^{2},  (2.4)
[M̃/M] = 2𝔞^{−1}D_{Q(ψ)}^{2} ∩ 2𝔞,  (2.5)

where 𝔢 is the product of all the prime ideals which are ramified in B and which do not ramify in K; we understand D_{K/F} = 𝔤 if K = F; we put δq𝔤 = 𝔞𝔟^{2} with a squarefree integral ideal 𝔞 and a 𝔤-ideal 𝔟 of F. These (2.4) and (2.5) can be obtained by applying [11, Theorem 6.2] to (V, φ) and the complement (W, ψ). The intersection L ∩ W is an integral 𝔤-lattice in W with respect to ψ. It can be seen that [(L ∩ W) /L ∩ W] = [M/L ∩ W]^{2}[M̃/M] and [M/L ∩ W] is an integral ideal, which is independent of the choice of M; see [2, Lemma 2.2 (6)]. Moreover there is a 𝔤-ideal 𝔟(q) of F such that

[M/L ∩ W] = 𝔟(q)(2φ(h, L))^{−1}  (2.6)

by [2, Theorem 4.2]. We note that 2φ(h, L) must contain 𝔟(q) and that 2φ(h, L) ⊂ 𝔤 if h ∈ L. The ideal 𝔟(q) is determined by

2q[L̃/L] = 𝔟(q)^{2}[M̃/M]  (2.7)

(cf. [2, (4.1)]). Combining these, we obtain [(L ∩ W) /L ∩ W] = 2q[L̃/L](2φ(h, L))^{−2}. Now to L∩W we associate the order A^{+}(L∩W) defined by (1.4). Its discriminant is given by

d(A^{+}(L ∩ W)) = 2^{−1}[(L ∩ W) /L ∩ W] = q[L̃/L](2φ(h, L))^{−2}.  (2.8)

It is noted that the discriminant of A^{+}(W) divides q[L̃/L](2φ(h, L))^{−2}. We also note that if d(A^{+}(L ∩ W)) is squarefree, then 2φ(h, L) must be 𝔟(q) in (2.6), that is, L ∩ W is maximal in W.

For our later use, let us state a weak Witt decomposition of the local space (V, φ)_{v} whose core dimension t_{v} is 0 or 2. We fix a nonarchimedean prime v of F and drop the subscript v. Let L be a 𝔤-maximal lattice in V with respect to φ. We first note that φ is isotropic as t is 0 or 2. Let K be the discriminant algebra of φ defined by K = F × F if t = 0 and by K = F (\sqrt{det(φ)}) if t = 2; also let κ be the norm form defined by 2κ(x, y) = κ[x + y] − κ[x] − κ[y] and κ[(a, b)] = ab for x, y, (a, b) ∈ K if t = 0, and by 2κ(x, y) = xy^{ρ} + x^{ρ}y for x, y ∈ K with a nontrivial automorphism ρ of K over F if t = 2. Because K is embeddable in A^{+}(V), we identify K with the image of it. Then there is a weak Witt decomposition as follows (cf. [4, (1.19) and (1.20)]):

V = Kg ⊕ (F e + F f), L = 𝔯g + (𝔤e + 𝔤f),
(Kg, φ) ≅ (K, cκ) via xg ⟼ x  (2.9)

with some elements e and f of V such that φ[e] = φ[f] = 0 and 2φ(e, f) = 1, and g ∈ V such that g^{2} = c ∈ F^{×}. Here 𝔯 = 𝔤 × 𝔤 if t = 0 and 𝔯 is the maximal order of K if t = 2. We may assume that c = 1 if t = 0, c ∈ 𝔤^{×} if (K/𝔭) = −1 and χ(Q(φ)) = +1, c ∈ π𝔤^{×} if (K/𝔭) = −1 and χ(Q(φ)) = −1, and c ∈ 𝔤^{×} if (K/𝔭) = 0. We also note that A^{+}(Kg) = K and xg = gx^{∗} for x ∈ K, where (a, b)^{∗} = (b, a) for (a, b) ∈ K if t = 0 and the involution ∗ gives a nontrivial automorphism of K over F if t = 2.

2.2 The genus of L ∩ W

Let (V, φ) be a quaternary quadratic space over a number field F and (W, ψ) as in §2.1 with a fixed element h of V such that φ[h] ≠ 0.

Lemma 2.1. Let L be a 𝔤-maximal lattice in V with respect to φ. Then A^{+}(L∩ W) = A^{+}(L) ∩ A^{+}(W) for every h ∈ V such that φ[h] ≠ 0. The discriminant of A^{+}(L ∩ W) is given by (2.8).

This follows from the similar result [4, Lemma 3.2] on local orders A^{+}(L_{v} ∩W_{v}) by localization. We next restate [4, Corollary 2.2] which is a conclusion from the main result of [4]:

Theorem 2.2. Let (V, φ) be a quaternary quadratic space over a number field F and L a 𝔤-maximal lattice in V with respect to φ. For h ∈ V such that φ[h] ≠ 0 put W = (F h)^{⊥} and let ψ be the restriction of φ to W. Put 𝔬 = A^{+}(L ∩ W). Then C(L ∩ W) = τ(T (𝔬)) and Γ(L ∩ W) = τ(Γ^{∗}(𝔬)) hold. Consequently, the map N ⟼ A^{+}(N) gives a bijection of the SO^{ψ}(W)-classes in the SO^{ψ}(W)- genus of L ∩ W onto the conjugacy classes in the genus of 𝔬 which is the set {α^{−1}𝔬α | α ∈ A^{+}(W)_{A}^{×}}.

3 An order associated with L ∩ W

3.1 The local case

We first recall some general notation and results, following [9, §8 Part I]. For a quadratic space (V, φ) over a local field F, take a 𝔤-maximal lattice L in V with respect to φ. We define a subgroup J_{V} of G^{+}(V) by

J_{V} = {α ∈ G^{+}(V) | τ(α) ∈ C(L), αα^{∗} ∈ 𝔤^{×}}.  (3.1)

Put E_{V} = G^{1}(V) ∩ J_{V}, where G^{1}(V) = {α ∈ G^{+}(V) | αα^{∗} = 1} is the spin group of φ. If the dimension of V is even more than 2, then by virtue of [9, Theorem 8.9] specialized to this case,

[C(L) : τ(J_{V})] = \begin{cases}1 & if t = 0, or t = 2, (K/𝔭) = −1, and Q(φ) = M_{2}(F), \\ 2 & otherwise, \end{cases}

(3.2)

where t is the core dimension of (V, φ) and K = F (\sqrt{δ}) is the discriminant field of φ. If the dimension of V is odd more than 1, then by [9, Theorem 8.9] and [12, Theorem 1.8 (ii)],

[C(L) : τ(J_{V})] = \begin{cases}1 & if t = 1 and δ ∈ 𝔤^{×}F^{×2}, \\ 2 & otherwise. \end{cases}  (3.3)

Let (V, φ) be a quaternary quadratic space over F. For h ∈ V such that φ[h] = q ≠ 0, put W = (F h)^{⊥} and let ψ be the restriction of φ to W. Let K = F (\sqrt{δ}) be the discriminant field of φ. Also let L and M be 𝔤-maximal lattices in V and W with respect to φ and ψ, respectively. We define J_{V} in G^{+}(V) by (3.1) with L and J_{W} in G^{+}(W) with M. Let S_{V}^{+} (resp. S_{W}^{+} ) be the order in A^{+}(V) (resp. A^{+}(W)) generated by E_{V} and A^{+}(L) (resp. by E_{W} and A^{+}(M)) except the case where t = 2, (K/𝔭) = −1, and χ(Q(φ)) = −1 (resp. where t = 0 and q ∈ π𝔤^{×}F^{×2}, or t = 2, δq ∈ π𝔤^{×}F^{×2}, and Q(φ) = M_{2}(F)); in which cases we put

S_{V}^{+} = A^{+}(V) ∩ S_{V} if t = 2, (K/𝔭) = −1, and χ(Q(φ)) = −1,  (3.4)
S_{W}^{+} = A^{+}(W) ∩ S_{W} if \begin{cases}t = 0 and q ∈ π𝔤^{×}F^{×2}, \\ t = 2, δq ∈ π𝔤^{×}F^{×2}, and Q(ψ) = M_{2}(F), \end{cases} (3.5)

where S_{V} (resp. S_{W}) is a unique maximal order in A(V) (resp. A(W)) containing E_{V} and A(L) (resp. E_{W} and A(M)) given by [9, Theorem 8.6 (i)]. By [9, Theorem 8.6 (ii)] these S_{V}^{+} and S_{W}^{+} are maximal orders except in cases (3.4) and (3.5). It should be noted that we can prove this fact in a similar way to the proof of [9, Theorem 8.6 (ii)] even for the case which does not satisfy the assumption [9, (8.1)]. For the same reason we also see that S_{W} = A(M) in case (3.5). In all cases,

J_{V} = G^{+}(V) ∩ (S_{V}^{+})^{×},  (3.6)
J_{W} = G^{+}(W) ∩ (S_{W}^{+} )^{×} = (S_{W}^{+} )^{×}.  (3.7)

In fact, [9, Proposition 8.8 (ii)] together with G^{+}(W) = A^{+}(W)^{×} implies (3.7) except in case (3.5). As for (3.5), there is an order in A(W) containing J_{W} and M by [9, Lemma 8.4 (ii)]. In view of the uniqueness of S_{W} and E_{W} ⊂ J_{W}, S_{W} contains J_{W} and M, and hence [9, Proposition 8.8 (i)] is applicable to the case (3.5). This proves (3.7). Similarly we have (3.6). Now, A^{+}(W) = {x ∈ A^{+}(V) | xh = hx} and G^{+}(W) = {α ∈ G^{+}(V) | αh = hα} as mentioned in §2.1. It can be seen that

G^{+}(W) ∩ J_{V} = (A^{+}(W) ∩ S_{V}^{+})^{×}.  (3.8)

Thus G^{+}(W) ∩ J_{V} is the unit group of an order A^{+}(W) ∩ S_{V}^{+} in A^{+}(W).

Lemma 3.1. In the above setting the following assertions hold:

(1) [S_{W}^{+} /A^{+}(M)] is given by

\begin{cases}𝔭 & if t = 4 and q ∈ 𝔤^{×}F^{×2}, \\ & or t = 2, δq ∈ 𝔤^{×}F^{×2}, and χ(Q(ψ)) = −1, \\ 𝔤 & otherwise. \end{cases}  (3.9)

Here S_{W}^{+} may or may not be maximal when t = 0 and q ∈ π𝔤^{×}F^{×2} or when t = 2, δq ∈ π𝔤^{×}F^{×2}, and Q(ψ) = M_{2}(F).

(2) Assume that q ∈ 𝔤^{×}F^{×2} if t = 2, K/F is unramified, and Q(φ) is a division algebra. Then [A^{+}(W) ∩ S_{V}^{+}/A^{+}(L ∩ W)] is given by

\begin{cases}𝔭 & if t = 4 and q ∈ 𝔤^{×}F^{×2}, \\ & or t = 2, (K/𝔭) = −1, and χ(Q(φ)) = −1, \\ 𝔤 & otherwise. \end{cases}  (3.10)

Proof. Let s be the core dimension of (W, ψ). In view of (1.7), (2.3), and (2.1), we observe that

s = 1 and δq ∈ π𝔤^{×}F^{×2} ⟺ \begin{cases}t = 0 and q ∈ π𝔤^{×}F^{×2}, \\ t = 2, δq ∈ π𝔤^{×}F^{×2}, and Q(ψ) = M_{2}(F), \end{cases}

(3.11)

s = 3 and δq ∈ 𝔤^{×}F^{×2} ⟺ \begin{cases}t = 4 and q ∈ 𝔤^{×}F^{×2}, \\ t = 2, δq ∈ 𝔤^{×}F^{×2}, and χ(Q(ψ)) = −1. \end{cases}(3.12)

Then we can verify that

S_{V}^{+} = A^{+}(L) if \begin{cases}t = 0, \\ t = 2 except the case (K/𝔭) = −1 and χ(Q(φ)) = −1, \end{cases}

(3.13)

S_{W}^{+} = A^{+}(M) ⟺ \begin{cases}s = 1, \\ s = 3 and δq ∈ π𝔤^{×}F^{×2}. \end{cases}  (3.14)

In fact, if s = 1 and δq ∈ π𝔤^{×}F^{×2}, then the ‘if’-part of (3.14) follows from (3.11), (3.5), and S_{W} = A(M). If s = 3 and δq ∈ π𝔤^{×}F^{×2}, then S_{W}^{+} = A^{+}(M) because the discriminant of A^{+}(M) is 𝔭. If s = 3 and δq ∈ 𝔤^{×}F^{×2}, then since S_{W}^{+} is a maximal order in the division algebra A^{+}(W), it has discriminant 𝔭. Note that the discriminant of A^{+}(M) is 𝔭^{2}. Hence S_{W}^{+} ≠ A^{+}(M). Further, observing A^{+}(M) ⊂ S_{W}^{+} ⊂ (S_{W}^{+} ) ⊂ A^{+}(M) and applying [2, Lemma 2.2 (3)] with the norm form ν of A^{+}(W), we have [S_{W}^{+} /A^{+}(M)] = 𝔭. The remaining parts follow from [9, Theorem 8.6 (vi)]. From (3.14) and (3.12) we see that

S_{W}^{+} ≠ A^{+}(M) ⟺ \begin{cases}t = 4 and q ∈ 𝔤^{×}F^{×2}, \\ t = 2, δq ∈ 𝔤^{×}F^{×2}, and χ(Q(ψ)) = −1. \end{cases}

In this case [S_{W}^{+} /A^{+}(M)] = 𝔭, which proves (1). To prove (2), it is sufficient to observe the two cases that t = 4 or that t = 2, (K/𝔭) = −1, and χ(Q(φ)) = −1 by (3.13) and Lemma 2.1. If t = 4 and q ∈ π𝔤^{×}F^{×2}, then A^{+}(L ∩ W) ⊂ A^{+}(W) ∩ S_{V}^{+} ⊂ S_{W}^{+} = A^{+}(M). Thus A^{+}(W) ∩ S_{V}^{+} = A^{+}(L ∩ W) because L ∩ W is maximal. Suppose that t = 4 and q ∈ 𝔤^{×}F^{×2}. Then A^{+}(L ∩ W) has discriminant 𝔭^{2} and by Lemma 2.1, A^{+}(L ∩ W) ⊂ A^{+}(W) ∩ S_{V}^{+} ⊂ S_{W}^{+} in the division algebra A^{+}(W). We employ the setting and notation in the case where q_{0} ∈ 𝔤^{×} and (K_{1}/𝔭) = 0 in [4, §4.4]. In [4, (3.31)] observing (g_{2}g_{3})(g_{2}g_{3})^{∗} ∈ π^{2}𝔤^{×}, we set

𝔒 = 𝔤 + 𝔤g_{1}g_{2} + 𝔤g_{1}g_{3} + 𝔤π^{−1}g_{2}g_{3}.  (3.15)

This is an order in A^{+}(W) which contains but does not coincide with A^{+}(L ∩ W). Hence 𝔒 is a unique maximal order S_{W}^{+} in A^{+}(W). Now in the present setting, (V, φ) = (B, β) and L is a unique maximal order 𝔬 in B = Q(φ) with the norm form β. To see the order S_{V}^{+} in A^{+}(V), we here recall an F -linear mapping p defined in [9, §7.4 (B)]:

p : V ⟶ M_{2}(B) via p(x) = [ \begin{matrix}0 & x \\ x^{ι} & 0 \end{matrix} ] ,

where ι is the main involution of B. Then A^{+}(V) and S_{V}^{+} are given by

A^{+}(V) = \{[ \begin{matrix}x & 0 \\ 0 & y \end{matrix} ] | x, y ∈ B\} , S_{V}^{+} = \{[ \begin{matrix}x & 0 \\ 0 & y \end{matrix} ] | x, y ∈ 𝔬\} .

Under the identification of V with p(V) and of W with p(W), A^{+}(L ∩ W) and S_{W}^{+} are given by [4, (3.31)] and (3.15), respectively. Then we see that

π^{−1}g_{2}g_{3} = \frac{1}{π} [ \begin{matrix}0 & g_{2} \\ g_{2}^{ι} & 0 \end{matrix} ] [ \begin{matrix}0 & g_{3} \\ g_{3}^{ι} & 0 \end{matrix} ] = [ \begin{matrix}π^{−1}g_{2}g_{3}^{ι} & 0 \\ 0 & π^{−1}g_{2}^{ι}g_{3} \end{matrix} ] ,

β[π^{−1}g_{2}g_{3}^{ι}] = π^{−2} · πac · π^{2−2(κ−k)}(1 − c) ∈ 𝔤^{×}.

Thus both π^{−1}g_{2}g_{3}^{ι} and π^{−1}g_{2}^{ι}g_{3} belong to 𝔬, so that π^{−1}g_{2}g_{3} ∈ S_{V}^{+}. Therefore S_{V}^{+} contains S_{W}^{+} , which implies that A^{+}(W) ∩ S_{V}^{+} is the maximal order S_{W}^{+} . For the other cases S_{W}^{+} can be observed in a similar manner; we have then

S_{W}^{+} = \begin{cases}𝔤 + 𝔤1_{B}ω + 𝔤1_{B}(vω) + 𝔤π^{−1}ω(vω)  if (K_{1}/𝔭) = −1 and 𝔭 ∤ 2, \\ 𝔤 + 𝔤1_{B}ω + 𝔤1_{B}(uω) + 𝔤π^{−1}ω(uω)  if (K_{1}/𝔭) = −1 and 𝔭 | 2, \\ 𝔤 + 𝔤\sqrt{s}ω + 𝔤\sqrt{s}(vω) + 𝔤π^{−1}ω(vω)  if (K_{1}/𝔭) = 1. \end{cases}

Here the notation is the same as in each case of [4, §3.4]. Consequently A^{+}(W)∩ S_{V}^{+} = S_{W}^{+} in each case. This settles the case where t = 4 and q ∈ 𝔤^{×}F^{×2}. Suppose that t = 2, (K/𝔭) = −1, and χ(Q(φ)) = −1. In this case S_{V}^{+} is defined by (3.4) with the maximal order S_{V} in A(V). Let q ∈ π^{2ℓ}𝔤^{×} with ℓ ∈ Z. Then 𝔟(q) = 𝔭^{ℓ} as was seen in the case of q ∉ φ[Kg] in [4, §3.2] with g^{2} ∈ π𝔤^{×} in (2.9). Since A^{+}(W) is a division algebra, S_{W}^{+} is a unique maximal order in A^{+}(W) of discriminant 𝔭. We have by (2.8),

A^{+}(L ∩ W) ⊂ A^{+}(W) ∩ S_{V} ⊂ S_{W}^{+} ,
𝔭^{2ℓ+2}(2φ(h, L))^{−2} ⊂ d(A^{+}(W) ∩ S_{V}) ⊂ 𝔭.  (3.16)

Now put 2φ(h, L) = 𝔭^{m}, which satisfies m ≤ ℓ. We observe that qπ^{−m}e + π^{m}f ∈ L[q, 2^{−1}𝔭^{m}] = hC(L) by (1.6) with the same notation as in the proof of [4, Lemma 3.1]. Then identifying W with that in [4, (3.1)] and employing the isomorphism Ψ of A(V) in the proof of [4, Lemma 3.1], we can find the structure of A^{+}(W) ∩ S_{V} as follows:

Ψ(A^{+}(W) ∩ S_{V}) = 𝔯 + π^{−m}𝔯η,  (3.17)

where 𝔯 = 𝔤[ξ] is the maximal order of K and η is given by [4, (3.3)]. From this together with [4, (3.4)] we have [A^{+}(W) ∩ S_{V} /A^{+}(L ∩ W)] = [𝔯/𝔣] = 𝔭, where 𝔣 = 𝔤 + g^{2}𝔤ξ. To see (3.17), we recall by [9, Theorem 8.6 (iii)] that Ψ(S_{V}) = M_{2}(Q), where Q = 𝔯 + 𝔯g is a maximal order in the division algebra Q(φ) = A(Kg) = K + Kg. Then (3.17) can be seen from this and [4, (3.2)]. This completes the proof.

Lemma 3.2. Let the notation be the same as in Lemma 3.1 with h and L. Then the following assertions hold:

(1) Define an order 𝔒 in A^{+}(W) by

𝔒 = \begin{cases}S_{W}^{+} & if t = 4, \\ A^{+}(W) ∩ S_{V} & if t = 2, (K/𝔭) = −1, and χ(Q(φ)) = −1, \\ A^{+}(L ∩ W) & otherwise. \end{cases}

(3.18)

Then G^{+}(W) ∩ J_{V} = 𝔒^{×}.

(2) Assume that q = φ[h] ∈ 𝔤^{×}F^{×2} and 2φ(h, L) = 𝔟(q) if t = 2, K/F is unramified, and Q(φ) is a division algebra. Let 𝔒 be the order defined by (3.18). Then 𝔒 is a unique order in A^{+}(W), containing A^{+}(L ∩ W), of discriminant

𝔭  if t = 4,
or t = 2, (K/𝔭) = −1, and χ(Q(φ)) = −1,
q[L̃/L](2φ(h, L))^{−2} otherwise.

(3.19)

In particular, 𝔒 is a unique maximal order in the division algebra A^{+}(W) when t = 2, (K/𝔭) = −1, and χ(Q(φ)) = −1. Moreover, if L ∩ W ⊂ M, then 𝔒 ⊂ S_{W}^{+} .

Proof. To prove (1), let 𝔒 be the order given by (3.18). From Lemma 3.1 it can be seen that 𝔒 = A^{+}(W)∩S_{V}^{+}. Thus we have G^{+}(W)∩J_{V} = 𝔒^{×} by (3.8), which proves (1). To prove (2), by Lemma 3.1 (2) we see that A^{+}(W) ∩ S_{V}^{+} ≠ A^{+}(L ∩ W) if and only if t = 4 and q ∈ 𝔤^{×}F^{×2} or if t = 2, (K/𝔭) = −1, and χ(Q(φ)) = −1. If t = 4 and q ∈ 𝔤^{×}F^{×2}, then A^{+}(W) ∩ S_{V}^{+} = S_{W}^{+} as seen in the proof of Lemma 3.1 (2). If t = 2, (K/𝔭) = −1, and χ(Q(φ)) = −1, then, by our assumption, q ∈ π^{2ℓ}𝔤^{×} and 2φ(h, L) = 𝔟(q) = 𝔭^{ℓ} with ℓ ∈ Z. Thus applying (3.16) to m = ℓ, we have 𝔭^{2} ⊂ d(A^{+}(W)∩S_{V}) ⊂ 𝔭. Because A^{+}(W)∩S_{V} ≠ A^{+}(L∩W), A^{+}(W) ∩ S_{V} must be maximal in A^{+}(W). Consequently, if 𝔒 ≠ A^{+}(L ∩ W), it is a maximal order which is uniquely determined by discriminant 𝔭. As for the case of 𝔒 = A^{+}(L ∩ W), the discriminant is given by (2.8). Summing up these, we have the uniqueness of 𝔒. To prove the last assertion, suppose that L ∩ W ⊂ M. Then A^{+}(L ∩ W) ⊂ A^{+}(M) ⊂ S_{W}^{+} , which shows 𝔒 ⊂ S_{W}^{+} when 𝔒 = A^{+}(L ∩ W). If 𝔒 ≠ A^{+}(L ∩ W), then 𝔒 is maximal in A^{+}(W). Since S_{W}^{+} is also maximal, we have 𝔒 = S_{W}^{+} . Hence 𝔒 ⊂ S_{W}^{+} holds if L ∩ W ⊂ M. This proves (2).

3.2 The global case

Let (V, φ) and (W, ψ) be the quadratic spaces over a number field F in the setting of §2.2 with an element h of V such that φ[h] = q ∈ F^{×}. Let L and M be 𝔤-maximal lattices in V and W with respect to φ and ψ, respectively. Put

J_{V} = G^{+}(V)_{a} ∏_{v∈h} J_{V_{v}},  J_{W} = G^{+}(W)_{a} ∏_{v∈h} J_{W_{v}},  (3.20)

where J_{V_{v}} and J_{W_{v}} are given in §3.1. We have an order S_{W}^{+} in A^{+}(W) determined by S_{W_{v}}^{+} for all v ∈ h, where S_{W_{v}}^{+} is the order in A^{+}(W_{v}) given in §3.1; notice that S_{W_{v}}^{+} = A^{+}(M_{v}) for almost all v. Let us here insert a remark on the order in A^{+}(W) given in [12, Lemma 5.3 (ii)]. By applying that lemma to M, we have an order 𝔒_{0} containing A^{+}(M).

Then [(𝔒_{0})_{v}/A^{+}(M_{v})] is the same ideal as in (3.9) for each v ∈ h. This can be seen in the proof of [12, Lemma 5.3 (ii)]. Hence the order 𝔒_{0} coincides with S_{W}^{+} in the present situation.

Proposition 3.3. Let the notation be the same as above with h ∈ V and L. Also let K = F (\sqrt{δ}) be the discriminant field of φ. Then the following assertions hold:

(1) Let 𝔒 be the order in A^{+}(W) whose localization at v ∈ h is the local order defined by (3.18). Then

G^{+}(W)_{A} ∩ J_{V} = A^{+}(W)_{a}^{×}𝔒_{h}^{×}.  (3.21)

(2) T (A^{+}(M)) = T (S_{W}^{+} ) and J_{W} = A^{+}(W)_{a}^{×}(S_{W}^{+} )_{h}^{×}. Moreover J_{W} ⊂ T (A^{+}(M)) and G^{+}(W)_{A} ∩ J_{V} ⊂ T (A^{+}(L ∩ W)).

(3) Assume that q = φ[h] ∈ 𝔤_{v}^{×}F_{v}^{×2} and 2φ(h, L)_{v} = 𝔟(q)_{v} for every v ∈ h such that t_{v} = 2, K_{v}/F_{v} is unramified, and Q(φ)_{v} is a division algebra. Let 𝔒 be the order given in (1). Then 𝔒 is a unique order in A^{+}(W), containing A^{+}(L ∩ W), of discriminant

q[L̃/L](2φ(h, L))^{−2}𝔣^{−1}.  (3.22)

Here 𝔣 is the product of all the prime ideals 𝔭 of F such that t_{𝔭} = 4 and q ∈ 𝔤_{𝔭}^{×}F_{𝔭}^{×2}, or that t_{𝔭} = 2, K_{𝔭}/F_{𝔭} is unramified, and Q(φ)_{𝔭} is a division algebra.

(4) Under the assumptions of (3) suppose L ∩ W ⊂ M. Then 𝔒 ⊂ S_{W}^{+} and G^{+}(W)_{A} ∩ J_{V} ⊂ J_{W}.

Proof. To prove (1), we see that

G^{+}(W)_{A} ∩ J_{V} =

= G^{+}(W)_{A} ∩ (G^{+}(V)_{a} ∏_{v∈h}(J_{V})_{v}) = G^{+}(W)_{a} ∏_{v∈h}(G^{+}(W)_{v} ∩ (J_{V})_{v}).

Since G^{+}(W)_{v} ∩ (J_{V})_{v} = 𝔒_{v}^{×} by Lemma 3.2 (1), we have (3.21). From (3.14), (S_{W}^{+} )_{v} is generated by G^{1}(W)_{v} and A^{+}(M)_{v} if δ ∈ 𝔤^{×}F_{v}^{×2} and χ(Q(ψ)_{v}) = −1, and (S_{W}^{+} )_{v} = A^{+}(M)_{v} otherwise v ∈ h. Since

α^{−1}G^{1}(W)_{v}α = G^{1}(W)_{v} for every α ∈ A^{+}(W)_{v}^{×},

we have T (A^{+}(M)) ⊂ T (S_{W}^{+} ). Conversely, for α ∈ A^{+}(W)_{A}^{×}

α^{−1}S_{W}^{+} α = S_{W}^{+} ⟹ Mτ(α) = M ⟹ α^{−1}A^{+}(M)α = A^{+}(M).

This is because C(M) = τ(T (S_{W}^{+} )) by [12, Lemma 5.4]. Thus T (A^{+}(M)) = T (S_{W}^{+} ). Let x be an element of J_{W}. Since τ(x) ∈ τ(J_{W}) ⊂ C(M), together with C(M) = τ(T (A^{+}(M))), there is an element y of T (A^{+}(M)) such that τ(x) = τ(y). Hence x = ay with some a ∈ F_{A}^{×}. As F_{A}^{×} ⊂ T (A^{+}(M)), we have J_{W} ⊂ T (A^{+}(M)). Similarly let x ∈ G^{+}(W)_{A} ∩ J_{V}. Since τ(x) ∈ SO_{A}^{ψ} ∩ τ(J_{V}) ⊂ C(L ∩ W), together with C(L ∩ W) = τ(T (A^{+}(L ∩ W))) by Theorem 2.2, there is an element y of T (A^{+}(L ∩ W)) such that τ(x) = τ(y). From this, noticing F_{A}^{×} ⊂ T (A^{+}(L ∩ W)), we have G^{+}(W)_{A} ∩ J_{V} ⊂ T (A^{+}(L ∩ W)). This proves (2). To prove (3), we take the order 𝔒 of (1). Since Lemma 3.2 (2) is applicable to 𝔒_{v} for each v ∈ h under the assumption of (3), 𝔒_{v} contains A^{+}(L ∩ W)_{v} and has the discriminant given by (3.19). Also when 𝔒_{v} ≠ A^{+}(L ∩ W)_{v}, [𝔒_{v}/A^{+}(L ∩ W)_{v}] = 𝔭_{v} by Lemma 3.1 (2). Thus by applying [2, Lemma 2.2 (3)] to 𝔒 and A^{+}(L ∩ W) with the norm form ν of A^{+}(W), we have

[𝔒̃/𝔒] = [A^{+}(L ∩ W) /A^{+}(L ∩ W)][𝔒/A^{+}(L ∩ W)]^{−2}
= (q[L̃/L](2φ(h, L))^{−2})^{2} ∏_{𝔭|𝔣} 𝔭^{−2},

where 𝔣 is the ideal in the statement of (3). This gives (3.22). Now, let 𝔒′ be an order in A^{+}(W), containing A^{+}(L ∩ W), whose discriminant is given by (3.22). Then the localization 𝔒′_{v} at v ∈ h contains A^{+}(L ∩ W)_{v} and has the discriminant of (3.19). By Lemma 3.2 (2), 𝔒′_{v} = 𝔒_{v} for every v. Hence we have 𝔒′ = 𝔒, which shows the uniqueness of 𝔒. Keeping the assumptions of (3), let L ∩ W ⊂ M. Then applying Lemma 3.2 (2) with localization, we have 𝔒 ⊂ S_{W}^{+} . Thus G^{+}(W)_{A} ∩ J_{V} ⊂ J_{W} by (3.8) and (3.7). This proves (4).

Theorem 3.4. Let the notation and assumption be the same as in Proposition 3.3 (3) and 𝔒 the order in A^{+}(W) given in that proposition. Then the following assertions hold:

(1) C(L ∩ W) = τ(T (𝔒)) and Γ(L ∩ W) = τ(Γ^{∗}(𝔒)).

(2) The map (L ∩ W)τ(α) ⟼ α^{−1}𝔒α gives a bijection of the SO^{ψ}(W)- classes in the SO^{ψ}(W)-genus of L ∩ W onto the conjugacy classes in the genus of 𝔒 which is the set {α^{−1}𝔒α | α ∈ A^{+}(W)_{A}^{×}}.

(3) The type number of 𝔒 equals the type number of A^{+}(L ∩ W) and consequently is equal to the class number of the genus of L ∩ W with respect to SO^{ψ}(W).

Proof. In view of Lemma 3.1, 𝔒_{v} ≠ A^{+}(L ∩ W)_{v} if and only if t_{v} = 4 and q ∈ 𝔤_{v}^{×}F_{v}^{×2} or if t_{v} = 2, (K/v) = −1, and χ(Q(φ)_{v}) = −1 for v ∈ h. Since Lemma 3.2 (2) is applicable in our assumption, 𝔒_{v} is a unique maximal order in the division algebra A^{+}(W)_{v} in both cases. Furthermore, (L ∩ W)_{v} is a unique maximal lattice in the anisotropic space (W, ψ)_{v} becuase 2φ(h, L)_{v} = 𝔟(q)_{v}. Thus it can be found that

α^{−1}𝔒α = 𝔒 ⟺ α^{−1}A^{+}(L ∩ W)α = A^{+}(L ∩ W)

for α ∈ A^{+}(W)_{A}^{×}. This combined with Theorem 2.2 proves (1). To prove (2), let N be an arbitrary 𝔤-lattice in the genus of L ∩ W. Since N is integral we have the order A^{+}(N) in A^{+}(W). Taking α ∈ A^{+}(W)_{A}^{×} so that N = (L ∩ W)τ(α), we can put 𝔒(N) = α^{−1}𝔒α in A^{+}(W). In fact, if N = (L ∩ W)τ(α′) with some α′ ∈ A^{+}(W)_{A}^{×}, then (L ∩ W)τ(α(α′)^{−1}) = L ∩ W, whence α(α′)^{−1} belongs to T (A^{+}(L ∩ W)) = T (𝔒) by (1). This shows (α′)^{−1}𝔒α′ = α^{−1}𝔒α, namely, 𝔒(N) is independent of the choice of α. Moreover this is a unique order of discriminant q[L̃/L](2φ(h, L))^{−2}𝔣^{−1} containing A^{+}(N), where 𝔣 is the ideal in (3.22). Indeed, since 𝔒 contains A^{+}(L ∩ W) and has the discriminant given by (3.22), the order 𝔒(N) contains A^{+}(N) and has the same discriminant. The uniqueness of 𝔒(N) can be reduced to that of 𝔒. Our assertion (2) can be verified by using this fact and (1). Assertion (3) is a consequence from (2). This completes the proof.

4 Quadratic Diophantine equations in four variables

4.1 Quadratic Diophantine equations

Let (V, φ) be a quadratic space of dimension n over a number field F and L a 𝔤-lattice in V. We recall that

L[q, 𝔟] = {x ∈ V | φ[x] = q, φ(x, L) = 𝔟},

and this set is stable under Γ(L). For h ∈ V such that φ[h] = q ≠ 0 we set (W, ψ) as in (2.2). Assume that L is 𝔤-maximal with respect to φ and n > 2. Then

∑_{i∈I} # {L_{i}[q, 𝔟]/Γ(L_{i})} = # {SO^{ψ} \ SO_{A}^{ψ}/(SO_{A}^{ψ} ∩ C(L))} ,  (4.1)

where 𝔟 = φ(h, L), {L_{i}}_{i∈I} is a set of representatives for the SO^{φ}-classes in the SO^{φ}-genus of L for which L_{i}[q, 𝔟] ≠ ∅, and SO^{ψ} is regarded as the subgroup {γ ∈ SO^{φ} | hγ = h} of SO^{φ}. This is a consequence from the main theorem of quadratic Diophantine equations due to Shimura [9, Theorem 11.6] (cf. also [12, Theorem 2.2 and (2.7)]). For a 𝔤-lattice N in V we put

D(N) = {α ∈ O^{φ}(V)_{A} | Nα = N}, Γ^{·}(N) = O^{φ}(V) ∩ D(N)

as denoted in the Introduction. Then formula (4.1) is valid for (O^{φ}, O^{ψ}, D(L), Γ^{·}(L_{i}), J) in place of (SO^{φ}, SO^{ψ}, C(L), Γ(L_{i}), I) by [9, Theorem 11.6 (iii) and (v)], where {L_{i}}_{i∈J} is a set of representatives for the O^{φ}-classes in the O^{φ}-genus of L for which L_{i}[q, 𝔟] ≠ ∅ and O^{ψ} is regarded as the subgroup {γ ∈ O^{φ} | hγ = h} of O^{φ}. We note that the O^{φ}-genus of L coincides with the SO^{φ}-genus of L and that the class number of O^{φ} relative to D(L) equals the class number of SO^{φ} relative to C(L) when n is odd; see [9, Lemma 9.23 (i)], for example.

Now we pay attention to the following; if the number of the right-hand side of (4.1) coincides with #{SO^{ψ} \SO_{A}^{ψ}/C(L∩W)}, then the left-hand side of (4.1) is given by the class number of the genus of L ∩ W. Concerning this, there is a result [9, Proposition 11.13] for odd-dimensional spaces and also its analogue [2, Proposition 4.4] for even-dimensional spaces whose discriminant fields are the base fields. In Proposition 4.3 below we shall prove another analogue of [9, Proposition 11.13] to quaternary case. As for the representatives of classes in the genus of L ∩ W, by virtue of the principle in [9, Theorem 11.6 (i)], we have the following:

Lemma 4.1. Let the notation be as above. Fix an element h of L[q, 𝔟] (q ≠ 0) and set (W, ψ) as in (2.2). Then the map

kΓ(L_{i}) ⟼ (L_{i} ∩ (F k)^{⊥})γ^{−1}SO^{ψ}

defines a well-defined surjection of the union of the sets L_{i}[q, 𝔟]/Γ(L_{i}) for i ∈ I onto the SO^{ψ}-classes in the SO^{ψ}-genus of L ∩ W with γ ∈ SO^{φ} such that k = hγ for k ∈ L_{i}[q, 𝔟] and i ∈ I. In particular, if SO^{ψ}ε(SO_{A}^{ψ} ∩ C(L)) = SO^{ψ}εC(L ∩ W) for every ε ∈ SO_{A}^{ψ}, then the map is bijective. Moreover the assertions are true for (O^{ψ}, J, Γ^{·}(L_{i}), D(L), D(L ∩ W)) in place of (SO^{ψ}, I, Γ(L_{i}), C(L), C(L ∩ W)).

Proof. For k ∈ L_{i}[q, 𝔟] with i ∈ I there is γ ∈ SO^{φ} such that k = hγ as φ[k] = φ[h] by [9, Lemma 1.5 (ii)]. We set L = L_{i}α_{i} with α_{i} ∈ SO_{A}^{φ}. We may assume that (α_{i})_{v} = 1 for v ∈ a. Since h, k(α_{i})_{v} ∈ L_{v}[q, 𝔟_{v}] for v ∈ h, by (1.6), h = k(α_{i})_{v}α_{v} with some α_{v} ∈ C(L_{v}) for each v. Putting α_{v} = γ_{v}^{−1} for v ∈ a, we have α ∈ C(L) whose component is α_{v} for every prime v. Then by [9, Theorem 11.6 (i)] the map k ⟼ γα_{i}α induces a well-defined bijection of ⋃_{i∈I} L_{i}[q, 𝔟]/Γ(L_{i}) onto SO^{ψ} \ SO_{A}^{ψ}/(SO_{A}^{ψ} ∩ C(L)). Obviously γα_{i}α ⟼ (L ∩ W)(γα_{i}α)^{−1} gives a surjection of SO^{ψ} \ SO_{A}^{ψ}/(SO_{A}^{ψ} ∩ C(L)) onto the SO^{ψ}-classes in the genus of L ∩ W. On the other hand, we can consider a 𝔤-lattice L_{i} ∩ (F k)^{⊥} in the complement (F k)^{⊥}, which is isomorphic to (L_{i} ∩(F k)^{⊥})γ^{−1} in W under γ^{−1}. Then by localization (L∩W)_{v}(γα_{i}α)_{v}^{−1} = {L_{v}(α_{i})_{v}^{−1} ∩ (F_{v}h)^{⊥}(α_{i}α)_{v}^{−1}}γ^{−1} = (L_{i} ∩ (F k)^{⊥})_{v}γ^{−1} for every v ∈ h. This determines (L ∩ W)(γα_{i}α)^{−1} = (L_{i} ∩ (F k)^{⊥})γ^{−1}. We have thus the desired surjection. Clearly this map is bijective under the assumption in the statement. The assertions for O^{φ} can be handled in a similar way.

Here we apply Lemma 4.1 to the quadratic form defined by the sum of five squares; the result will be used in Section 5.3. Let X = Q_{5}^{1} and define Φ by Φ[x] = x ·^{t}x for x ∈ X. The pair (X, Φ) defines a quadratic space over Q whose invariants are {5, Q, B_{2, ∞}, 5}. These invariants can be determined by [11, (Q.5)] because of (X, Φ) ≅ (B_{2, ∞}, β) ⊕ (Qe, Φ|_{Qe}) with some e ∈ X so that Φ[e] = 1, where β is the norm form of B_{2, ∞}. Let Λ be a Z-maximal lattice in (X, Φ). It is known that #{O^{Φ} \ O_{A}^{Φ}/D(Λ)} = 1; see [9, §12.12], for example. By [9, Lemma 12.13 (i)], Λ[d, Z] ≠ ∅ for every squarefree positive integer d. Fixing k_{0} ∈ Λ[d, Z], we put V = (Qk_{0})^{⊥} and L = Λ ∩ V. Then by [9, Theorem 12.14 (ii)], L is a Z-maximal lattice in V with respect to the restriction φ of Φ to V. By virtue of (4.1) for O^{Φ}, #{Λ[d, Z]/Γ^{·}(Λ)} = #{O^{φ}\O_{A}^{φ}/(O_{A}^{φ}∩D(Λ))} holds. Suppose that d is an odd prime number. Then [9, Proposition 11.13 (iii)] is applicable to k_{0} ∈ Λ[d, Z]. We have thus

O^{φ}ε(O_{A}^{φ} ∩ D(Λ)) = O^{φ}εD(L)  (4.2)

for every ε ∈ O_{A}^{φ}. Therefore #{Λ[d, Z]/Γ^{·}(Λ)} equals the number of O^{φ}- classes in the O^{φ}-genus of Z-maximal lattices in (V, φ). This result can be found in [9, Theorem 12.14 (vi)]; the class number of SO^{φ} relative to C(L) equals #{Λ[d, Z]/Γ(Λ)} by the same theorem. Moreover (V, φ) has invariants \{4, Q(\sqrt{d}), B_{2, ∞}, 4\}, which can be seen by applying [2, Theorem 1.1 (2)] to (X, Φ) and d. In view of (4.2), by Lemma 4.1 we have a bijection

kΓ^{·}(Λ) ⟼ (Λ ∩ (Qk)^{⊥})γ^{−1}O^{φ}  (4.3)

of Λ[d, Z]/Γ^{·}(Λ) onto the O^{φ}-classes in the genus of L with some γ ∈ O^{Φ} so that k = k_{0}γ for every odd prime number d. A method of determining the set Λ[d, Z]/Γ^{·}(Λ) is explained in [9, §12.15]. In that explanation the case of d = 29 is treated and the result #{Λ[29, Z]/Γ^{·}(Λ)} = 3 is obtained with explicit representatives for Λ[29, Z]/Γ^{·}(Λ). Hence the class number of O^{φ} relative to D(L) is equal to 3, as mentioned in the Introduction. In Section 5.3 we shall list the representatives for Λ[d, Z]/Γ^{·}(Λ) and the corresponding lattices under the map (4.3) for d = 5, 13, 17, and 29.

4.2 Results for quaternary spaces

To apply our results in the previous section to quadratic Diophantine equations, let us assume n = 4 in the setting of §4.1 and take an element h of L[q, 𝔟]. Under suitable conditions on q and 𝔟, we have an order 𝔒 defined in Proposition 3.3 (3). The order satisfies inequalities

t(𝔒) ≤ #{SO^{ψ} \ SO_{A}^{ψ}/(SO_{A}^{ψ} ∩ C(L))} ≤ c(𝔒).  (4.4)

Here t(𝔒) (resp. c(𝔒)) is the type number (resp. the class number) of 𝔒. To show (4.4), we observe that

SO_{A}^{ψ} ∩ τ(J_{V}) ⊂ SO_{A}^{ψ} ∩ C(L) ⊂ C(L ∩ W).

Since the kernel of τ is F_{A}^{×}, the class number of 𝔒 is more than #{SO^{ψ} \ SO_{A}^{ψ}/(SO_{A}^{ψ} ∩ τ(J_{V}))} by (3.21). Further by Theorem 3.4 the type number of 𝔒 equals #{SO^{ψ} \ SO_{A}^{ψ}/C(L ∩ W)}. This proves (4.4).

Corollary 4.2. Let the notation and assumption be as in Proposition 3.3 (3) and 𝔒 the order in A^{+}(W) defined in that proposition with an element h of L[q, 𝔟]. Also let I (resp. J) be a set of representatives α for SO^{φ} \ SO_{A}^{φ}/C(L) (resp. O^{φ}\O_{A}^{φ}/D(L)) for which Lα^{−1}[q, 𝔟] ≠ ∅. Then the following inequalities hold:

t(𝔒) ≤ ∑_{α∈J} # {Lα^{−1}[q, 𝔟]/Γ^{·}(Lα^{−1})} ≤

≤ ∑_{α∈I} # {Lα^{−1}[q, 𝔟]/Γ(Lα^{−1})} ≤ c(𝔒).

Moreover, assume that 2φ(h, L)_{v} contains 𝔭_{v}^{[ν/2]} with q𝔤_{v} = 𝔭_{v}^{ν} for every v ∈ h such that t_{v} = 2 and K_{v}/F_{v} is ramified. Then the formula in [10, (1.9)] is applicable to h and it can be given as follows:

∑_{y} # {Lτ(y)^{−1}[q, 𝔟]/τ(G^{+}(V) ∩ yJ_{V} y^{−1})} = c(𝔒).  (4.5)

Here y runs over a set of all representatives for G^{+}(V)\G^{+}(V)_{A}/J_{V} such that G^{+}(W)_{A} ∩ G^{+}(V)yJ_{V} ≠ ∅.

Proof. To prove the first assertion, we recall that #{SO^{ψ} \ SO_{A}^{ψ}/C(L ∩ W)} = #{O^{ψ} \ O_{A}^{ψ}/D(L ∩ W)}, because W is odd-dimensional. Since O_{A}^{ψ} ∩ D(L) is contained in D(L ∩ W), by formula (4.1) for O^{φ}, we have the first inequality. Here we may assume that {Lα^{−1}}_{α∈J} ⊂ {Lα^{−1}}_{α∈I}. Clearly # {Lα^{−1}[q, 𝔟]/Γ^{·}(Lα^{−1})} ≤ # {Lα^{−1}[q, 𝔟]/Γ(Lα^{−1})} for every α ∈ J. Then the desired inequalities follow from these and (4.4) combined with (4.1). To prove (4.5), put q = q_{0}π_{v}^{2ℓ} and 2φ(h, L)_{v} = 𝔭_{v}^{m} with q_{0} ∈ 𝔤_{v}^{×} ∪ π_{v}𝔤_{v}^{×} and ℓ, m ∈ Z for v ∈ h. In order to apply [10, (1.9)], we have to verify that hC(L) = hτ(J_{V}) in V_{A} = V ⊗_{F} F_{A}. In view of (3.2) it is sufficient to observe the local cases where (i) t_{v} = 4, (ii) t_{v} = 2 and (K/v) = 0, (iii) t_{v} = 2, (K/v) = −1, and χ(Q(φ)_{v}) = −1. Our argument is basically the same as in [10, §4.3], and so we give only an outline of the proof to avoid a repetition of the same argument. Put C_{v} = C(L_{v}) and J_{v} = J_{V_{v}}.

(i) Through an isomomorphism of Q(φ)_{v} onto A^{+}(W)_{v} we have hC_{v} = hτ(J_{v}) in the same way as in §4.3 (i) of [10]. We note that C_{v} = SO_{v}^{φ} and C(L_{v}∩W_{v}) = SO_{v}^{ψ}. (ii) Assume that 2φ(h, L)_{v} ⊃ 𝔭_{v}^{ℓ}. In a Witt decomposition of φ in (2.9) with g^{2} ∈ 𝔤_{v}^{×}, take the same element ω_{v} ∈ K_{v}^{×} = G^{+}(K_{v}g) as in §4.3 (ii) of [10]. We take k_{v} = qπ_{v}^{−m}e + π_{v}^{m}f; then k_{v}τ(ω_{v}) = k_{v}. In a similar manner to [10, §4.3 (ii)] we have τ(ω_{v}) ∈ C_{v} and ω_{v} ∉ J_{v}, from which it follows that k_{v}C_{v} = k_{v}τ(J_{v}). Since, by our assumption, k_{v} ∈ L_{v}[q, 2^{−1}𝔭_{v}^{m}], we have k_{v} ∈ hC_{v} by (1.6). Thus the criterion [10, (1.10)] is applicable to k_{v}; we have hC_{v} = hτ(J_{v}). (iii) In a Witt decomposition of φ in (2.9) with g^{2} ∈ π_{v}𝔤_{v}^{×}, we take ω_{v} = g(qπ_{v}^{−m}e − π_{v}^{m}f) ∈ A^{+}(V_{v})^{×} and k_{v} = qπ_{v}^{−m}e + π_{v}^{m}f. Then it can be seen that

ν(ω_{v}) = ω_{v}ω_{v}^{∗} = −qg^{2} ∈ π_{v}q𝔤_{v}^{×}, k_{v}τ(ω_{v}) = k_{v},
L_{v}τ(ω_{v}) = 𝔯_{v}g + 𝔤_{v}q^{−1}π_{v}^{2m}f + 𝔤_{v}qπ_{v}^{−2m}e.

Because m = ℓ by the assumption on (iii), we have ν(ω_{v}) ∈ π_{v}^{2ℓ+1}𝔤_{v}^{×} and L_{v}τ(ω_{v}) = L_{v}. Hence k_{v}C_{v} = k_{v}τ(J_{v}) by the same way as in §4.3 (iii) of [10]. Since k_{v} ∈ L_{v}[q, 2^{−1}𝔭_{v}^{ℓ}] = hC_{v}, by [10, (1.10)], we have hC_{v} = hτ(J_{v}). Accordingly hC(L) = hτ(J_{V}) holds. Therefore [10, (1.9)] is applicable and the formula is given by

∑_{y} # {Lτ(y)^{−1}[q, 𝔟]/τ(G^{+}(V) ∩ yJ_{V} y^{−1})}

= # {G^{+}(W) \ G^{+}(W)_{A}/(G^{+}(W)_{A} ∩ J_{V})} ,  (4.6)

where y runs over all representatives for G^{+}(V) \ G^{+}(V)_{A}/J_{V} for which G^{+}(W)_{A}∩G^{+}(V)yJ_{V} ≠ ∅. Since G^{+}(W)_{A}∩J_{V} = A^{+}(W)_{a}^{×}𝔒_{h}^{×} by Proposition 3.3 (3), (4.6) equals the class number of 𝔒. Thus we obtain (4.5).

Let v be a prime of F in case (i), (ii), or (iii) of the proof of Corollary 4.2. As can be seen in the proof, there is an element ω_{v} of G^{+}(V_{v}) such that hτ(ω_{v}) = h, L_{v}τ(ω_{v}) = L_{v}, and ω_{v} ∉ J_{V_{v}}. This together with (3.2) shows that

[SO_{v}^{ψ} ∩ C(L_{v}) : SO_{v}^{ψ} ∩ τ(J_{V_{v}})] = [C(L_{v}) : τ(J_{V_{v}})]  (4.7)

for every v ∈ h under the two assumptions that 2φ(h, L)_{v} ⊃ 𝔭_{v}^{[ν/2]} if (K/v) = 0 and that ν ∈ 2Z and 2φ(h, L)_{v} = 𝔟(q)_{v} if (K/v) = −1 and χ(Q(φ)_{v}) = −1. Here K is the discriminant field of φ and q𝔤_{v} = 𝔭_{v}^{ν}. In the same assumptions we also see that

SO_{v}^{ψ} ∩ τ(J_{V_{v}}) = τ(𝔒_{v}^{×}).  (4.8)

These facts (4.7) and (4.8) are often useful in the application to quadratic Diophantine equations with four variables.

As for formula (4.1) for O^{φ}, we can state the following proposition:

Proposition 4.3. Let (V, φ) be a quadratic space of dimension 4 over a number field F and K = F (\sqrt{δ}) the discriminant field of φ. For an element h of V such that φ[h] = q ≠ 0 put W = (F h)^{⊥} and let ψ be the restriction of φ to W. Identify O^{ψ}(W) with {γ ∈ O^{φ}(V) | hγ = h}. Let L be a 𝔤-maximal lattice in V with respect to φ. Also let 𝔣_{1} be the product of all primes v ∈ h such that 2φ(h, L)_{v} ≠ 𝔟(q)_{v}. Suppose that for v ∈ h,

(1) v ∤ 2 and φ(h, L)_{v}^{2} = q𝔤_{v} if (K/v) = 0 and χ(Q(ψ)_{v}) = −1.

(2) q𝔤_{v} is a square ideal of F_{v} if (K/v) = −1 and χ(Q(φ)_{v}) = −1.

(3) 𝔣_{1} consists of the primes v such that t_{v} = 0 or that v ∤ 2, (K/v) = −1, χ(Q(φ)_{v}) = +1, and q𝔤_{v} is a square ideal of F_{v}.

Here t_{v} is the core dimension of φ at v. Let λ be the number of prime factors of 𝔣_{1}𝔣_{2}, where 𝔣_{2} is the product of all primes v ∈ h such that v ∤ 𝔣_{1}, t_{v} ≠ 4, (K/v) ≠ 0, and q𝔤_{v} is not a square ideal of F_{v}. Then [D(L∩W) : O_{A}^{ψ}∩D(L)] = [C(L ∩ W) : SO_{A}^{ψ} ∩ C(L)] = 2^{λ}. Moreover, if λ ≤ 1, then O^{ψ}εD(L ∩ W) = O^{ψ}ε(O_{A}^{ψ} ∩ D(L)) for every ε ∈ O_{A}^{ψ}.

Before stating the proof, we note a simple fact. Let G(V) be the Clifford group of φ. Then the homomorphism τ of (1.2) gives a surjection of G(V) onto O^{φ}(V), because V is even-dimensional.

Proof. In view of assumptions (2) and (3), we can take the order 𝔒 in Proposition 3.3 (3). Put q𝔤_{v} = 𝔭_{v}^{ν_{v}} with ν_{v} ∈ Z for v ∈ h. We note that v | 𝔣_{2} if and only if t_{v} = 0, v ∤ 𝔣_{1}, and ν_{v} is odd, or if (K/v) = −1, v ∤ 𝔣_{1}, and ν_{v} is odd. Suppose v ∤ 𝔣_{1}𝔣_{2}. Then (L ∩ W)_{v} is maximal in (W, ψ)_{v}. If t_{v} = 0, then ψ_{v} is isotropic and δq𝔤_{v} is square, which is because ν_{v} must be even by v ∤ 𝔣_{2}. Since C(L_{v}) = τ(J_{V_{v}}) by (3.2), we have SO_{v}^{ψ} ∩ C(L_{v}) = τ(𝔒_{v}^{×}) by (4.8). Clearly 𝔒_{v} = A^{+}(L ∩ W)_{v} by calculating the discriminant. Note that C(L_{v} ∩ W_{v}) = τ(A^{+}(L_{v} ∩ W_{v})^{×}) by [12, Lemma 5.4]. Hence we have C(L_{v} ∩ W_{v}) = SO_{v}^{ψ} ∩ C(L_{v}). If t_{v} = 4, then D(L_{v}) = O_{v}^{φ} and C(L_{v}) = SO_{v}^{φ}. Also D(L_{v} ∩W_{v}) = O_{v}^{ψ} and C(L_{v} ∩ W_{v}) = SO_{v}^{ψ} as ψ_{v} is anisotropic. Hence we have D(L_{v} ∩ W_{v}) = O_{v}^{ψ} ∩D(L_{v}) and C(L_{v} ∩W_{v}) = SO_{v}^{ψ} ∩C(L_{v}). Assume t_{v} = 2 and (K/v) = −1. Then δq𝔤_{v} must be square. If Q(φ)_{v} = M_{2}(F_{v}), then ψ_{v} is isotropic. In the same way as in the case t_{v} = 0 we see that C(L_{v}∩W_{v}) = SO_{v}^{ψ}∩C(L_{v}). If Q(φ)_{v} is a division algebra, then ψ_{v} is anisotropic and A^{+}(W)_{v} is a division algebra. Notice that C(L_{v} ∩ W_{v}) = τ(A^{+}(W)_{v}^{×}) as L_{v} ∩ W_{v} is maximal. Our order 𝔒_{v} has discriminant 𝔭_{v} by (3.22), whence it is maximal in A^{+}(W)_{v}. Observe that A^{+}(W)_{v}^{×} = F_{v}^{×}(𝔒_{v}^{×} ∪ 𝔒_{v}^{×}ω) with some ω ∈ A^{+}(W)_{v}^{×} so that ω^{2} is a prime element of F_{v}. Since SO_{v}^{ψ} ∩ τ(J_{V_{v}}) = τ(𝔒_{v}^{×}) by (4.8), we have [C(L_{v} ∩ W_{v}) : SO_{v}^{ψ} ∩ τ(J_{V_{v}})] = 2. In view of (4.7) together with [C(L_{v}) : τ(J_{V_{v}})] = 2 by (3.2), SO_{v}^{ψ} ∩ C(L_{v}) must coincide with C(L_{v} ∩ W_{v}). Assume (K/v) = 0. If Q(ψ)_{v} = M_{2}(F_{v}), we take a Witt decomposition of φ_{v} in (2.9) with g ∈ V_{v} so that g^{2} ∈ 𝔤_{v}^{×}. Then 𝔯_{v}g is a maximal lattice in the core subspace (K_{v}g, φ_{v}). Our assumption Q(ψ)_{v} = M_{2}(F_{v}) implies that there is an element k of K_{v}g such that φ_{v}[k] = q = φ[h]. Since the lattice 𝔯_{v}g ∩ (F_{v}k)^{⊥} is maximal in the complement (F_{v}k)^{⊥} in K_{v}g as (K_{v}g, φ_{v}) is anisotropic, we have 2φ_{v}(k, 𝔯_{v}g) = 𝔟(q)_{v} = 2φ(h, L)_{v}. Thus [9, Proposition 11.12 (iv) and (v)] are applicable to h. We have C(L_{v} ∩ W_{v}) = SO_{v}^{ψ} ∩ C(L_{v}) and D(L_{v} ∩ W_{v}) = O_{v}^{ψ} ∩ D(L_{v}). Similarly for the case where Q(ψ)_{v} is a division algebra, under the assumption (1), we have C(L_{v} ∩ W_{v}) = SO_{v}^{ψ} ∩ C(L_{v}) and D(L_{v} ∩ W_{v}) = O_{v}^{ψ} ∩ D(L_{v}). Suppose v | 𝔣_{1}. By assumption (3) such a prime satisfies either (i) t_{v} = 0 or (ii) v ∤ 2, (K/v) = −1, Q(φ)_{v} = M_{2}(F_{v}), and ν_{v} is even. In both cases (i) and (ii), 𝔒_{v} = A^{+}(L ∩ W)_{v} and SO_{v}^{ψ} ∩ C(L_{v}) = τ(𝔒_{v}^{×}). Moreover, we can prove that

T (𝔒_{v}) = F_{v}^{×}(𝔒_{v}^{×} ∪ 𝔒_{v}^{×}η)  (4.9)

with some element η of A^{+}(W)_{v}^{×} such that η𝔒_{v}^{×} = 𝔒_{v}^{×}η and ηη^{∗}𝔤_{v} = 𝔤_{v} or ηη^{∗}𝔤_{v} = 𝔭_{v} according as ν_{v} is even or odd. This can be handled in a similar way to the proof of [3, Theorem 3.1] for Case (i) and to [3, §3.4] for Case (ii). (We will determine the index [A^{+}(M_{v})^{×} : A^{+}(L_{v} ∩ W_{v})^{×}] in a subsequent paper, which may be used in the proof of (4.9).) We have therefore [C(L_{v} ∩ W_{v}) : τ(𝔒_{v}^{×})] = 2. Suppose v | 𝔣_{2}. Then (L∩W)_{v} is maximal and 𝔒_{v} = A^{+}(L∩W)_{v}. If t_{v} = 0, we have SO_{v}^{ψ} ∩C(L_{v}) = τ(𝔒_{v}^{×}). Since q𝔤_{v} is not square, by (3.5) and (3.7), J_{W_{v}} = A^{+}(L_{v} ∩ W_{v})^{×}. Hence [C(L_{v} ∩ W_{v}) : τ(𝔒_{v}^{×})] = 2 by (3.3). If (K/v) = −1, then Q(φ)_{v} must be M_{2}(F_{v}) under the assumption (2) as ν_{v} is odd. Applying (3.2) and (4.8), we have SO_{v}^{ψ} ∩ C(L_{v}) = τ(𝔒_{v}^{×}). Hence by the same way as in the case t_{v} = 0, [C(L_{v} ∩ W_{v}) : τ(𝔒_{v}^{×})] = 2. To prove [D(L_{v} ∩ W_{v}) : O_{v}^{ψ} ∩ D(L_{v})] = [C(L_{v} ∩ W_{v}) : SO_{v}^{ψ} ∩ C(L_{v})], we shall show that [O_{v}^{ψ} ∩ D(L_{v}) : SO_{v}^{ψ} ∩ C(L_{v})] = 2 because [D(L_{v} ∩ W_{v}) : C(L_{v} ∩ W_{v})] = 2 by [9, Lemma 6.8]. It is sufficient to investigate the following cases; (a) t_{v} = 0, (b) (K/v) = −1 and χ(Q(φ)_{v}) = +1, (c) (K/v) = −1 and χ(Q(φ)_{v}) = −1. In cases (a) and (b) we can verify the desired fact by the same technique as in the proof of [9, Proposition 11.12 (v)]; see the case L̃ = L and t ≠ 1 in that proof. As for case (c), we first note ν_{v} ∈ 2Z by our assumption (2); put ℓ = ν_{v}/2. Since (L ∩ W)_{v} must be maximal under assumption (3), 2φ(h, L)_{v} = 𝔟(q)_{v} = 𝔭_{v}^{ℓ} by (2.6) and (2.7). Now we take our setting and notation to be those in Case (iii) of the proof of Corollary 4.2. By (1.6), hα = k_{v} with some α ∈ C(L_{v}). Under such an α we may identify h, (W, ψ)_{v}, and (L ∩ W)_{v} with k_{v}, K_{v}g ⊕ F_{v}(qπ_{v}^{−ℓ}e − π_{v}^{ℓ}f), and 𝔯_{v}g + 𝔭_{v}^{−ℓ}(qπ_{v}^{−ℓ}e − π_{v}^{ℓ}f), respectively. Looking at the lattice 𝔯_{v}g in the subspace (K_{v}g, φ_{v}) of (V, φ)_{v}, we can find γ_{0} ∈ O(K_{v}g) such that det(γ_{0}) = −1 and (𝔯_{v}g)γ_{0} = 𝔯_{v}g by [9, Lemma 6.8]. Extend γ_{0} to an element γ of GL(V_{v}) by setting γ to be the identity map on (K_{v}g)^{⊥}. Then γ ∈ O_{v}^{φ}, hγ = h, det(γ) = −1, and L_{v}γ = L_{v}. This shows [O_{v}^{ψ} ∩ D(L_{v}) : SO_{v}^{ψ} ∩ C(L_{v})] = 2. Summing up all these results, we obtain the first assertion. To prove the second assertion, we borrow the idea of the proof of [9, Proposition 11.13 (ii)]. When there is no prime v dividing 𝔣_{1}𝔣_{2}, we have D(L ∩ W) = O_{A}^{ψ} ∩ D(L), and so our assertion is obvious. Hereafter we assume λ = 1. For ε ∈ O_{A}^{ψ} put Λ = Lε^{−1}, which is a 𝔤-maximal lattice in (V, φ). We consider τ(h) of O^{ψ}. Put 𝔞 = φ[h](2φ(h, L))^{−2}. Let c ∈ F_{A}^{×} so that 2cφ(h, L) = 𝔤; then 2φ_{v}(c_{v}h, Λ_{v}) = 𝔤_{v} and φ_{v}[c_{v}h]𝔤_{v} = φ_{v}[c_{v}h]φ_{v}(2c_{v}h, Λ_{v})^{−2} = 𝔞_{v} for every v ∈ h. Suppose 𝔞_{v} = 𝔤_{v}. Then φ_{v}[c_{v}h] ∈ 𝔤_{v}^{×} and 2φ_{v}(c_{v}h, Λ_{v}) = 𝔤_{v}. Hence c_{v}h belongs to Λ_{v} and also it is invertible in the order A(Λ)_{v}. Since this order contains Λ_{v} by definition, A(Λ)_{v} ∩ V_{v} = Λ_{v} by [9, Lemma 8.4 (iii)]. Thus we have Λ_{v}τ(h) = h^{−1}A(Λ)_{v}h ∩ V_{v} = Λ_{v}.

Let v ∤ 𝔣_{1}𝔣_{2}. If t_{v} = 0 or (K/v) = −1, then q𝔤_{v} = 𝔟(q)_{v}^{2} as ν_{v} is even. We have 𝔞_{v} = 𝔤_{v}, whence Λ_{v}τ(h) = Λ_{v}. If t_{v} = 4, then Λ_{v} is a unique maximal lattice in the anisotropic space (V, φ)_{v}. Hence Λ_{v}τ(h) = Λ_{v}. If (K/v) = 0 and Q(ψ)_{v} = M_{2}(F_{v}), we take a Witt decomposition of φ_{v} as in the same case of the proof of the first assertion with Λ_{v} in place of L_{v}. Because 2φ_{v}(h, Λ_{v}) = 2φ(h, L)_{v} = 𝔟(q)_{v}, by the same manner as in that proof, we can find α ∈ C(Λ_{v}) so that hα = k with some k ∈ K_{v}g. Then τ(h) = ατ(k)α^{−1} by [9, Lemma 3.8 (ii)]. We see that

Λ_{v}τ(h) = {(𝔯_{v}g)τ(k) + 𝔤_{v}eτ(k) + 𝔤_{v}fτ(k)}α^{−1} = Λ_{v}.

If (K/v) = 0 and Q(ψ)_{v} is a division algebra, then 𝔞_{v} = 𝔤_{v} by assumption (1), which leads Λ_{v}τ(h) = Λ_{v}. Let v | 𝔣_{1}𝔣_{2}. We take a weak Witt decomposition of φ_{v} as in (2.9) with Λ_{v} in place of L_{v}. Put q_{v} = φ_{v}[c_{v}h] and k = q_{v}e + f. We see that 𝔞_{v} = [M/L ∩ W]_{v}^{2} if v | 𝔣_{1} and ν_{v} ∈ 2Z, 𝔞_{v} = [M/L ∩ W]_{v}^{2}𝔭_{v} if t_{v} = 0 and ν_{v} ∉ 2Z, and 𝔞_{v} = 𝔭_{v} if v | 𝔣_{2} and (K/v) = −1, where M is a maximal lattice in (W, ψ). Since q_{v} ∈ 𝔞_{v}, it belongs to 𝔤_{v}. Hence we have k ∈ Λ_{v}[q_{v}, 2^{−1}𝔤_{v}]. By (1.6) there is α ∈ C(Λ_{v}) so that (c_{v}h)α = k. Moreover τ(k) = α^{−1}τ(h)α by [9, Lemma 3.8 (ii)]. Then α gives an isomorphism of W_{v} onto W′ = (F_{v}k)^{⊥} such that (Λ_{v} ∩ W_{v})α = Λ_{v} ∩ W′. Observe that Λ_{v} ∩ W′ = 𝔯_{v}g + 𝔤_{v}(q_{v}e − f). Employing [9, Lemma 3.10], we can find that

(Λ_{v} ∩ W′)τ(k) = {−x − a(q_{v}e − f) | x ∈ 𝔯_{v}g, a ∈ 𝔤_{v}} = Λ_{v} ∩ W′,
Λ_{v}τ(k) = {−x + q_{v}ae + q_{v}^{−1}bf | x ∈ 𝔯_{v}g, a, b ∈ 𝔤_{v}} ≠ Λ_{v},

because q_{v} ∈ 𝔞_{v} ⊂ 𝔭_{v} as seen above. Thus we have (Λ_{v}∩W_{v})τ(h) = Λ_{v}∩W_{v} but Λ_{v}τ(h) ≠ Λ_{v}. To sum up, τ(h) is an element of O^{ψ} such that (Λ ∩ W)τ(h) = Λ ∩ W and Λτ(h) ≠ Λ. Now, observe D(Λ ∩ W) = εD(L ∩ W)ε^{−1} and O_{A}^{ψ} ∩ D(Λ) = ε(O_{A}^{ψ} ∩ D(L))ε^{−1}. Since [D(L∩W) : O_{A}^{ψ} ∩D(L)] = 2 by λ = 1, we have [D(Λ∩W) : O_{A}^{ψ} ∩D(Λ)] = 2. By our result on τ(h) we obtain

D(Λ ∩ W) = (O_{A}^{ψ} ∩ D(Λ)) ∪ τ(h)(O_{A}^{ψ} ∩ D(Λ)).

Then our assertion follows from this and τ(h) ∈ O^{ψ}.

As a consequence, assuming that h ∈ L[q, 𝔟] satisfies all the assumptions with λ ≤ 1 in Proposition 4.3, by formula (4.1) for O^{φ} together with Proposition 4.3 and Theorem 3.4 (3), we obtain

∑_{α∈J} # {Lα^{−1}[q, 𝔟]/Γ^{·}(Lα^{−1})} = t(𝔒),  (4.10)

where J is a set of representatives α for O^{φ}\O_{A}^{φ}/D(L) for which Lα^{−1}[q, 𝔟] ≠ ∅ and 𝔒 is the order in A^{+}(W) defined in Proposition 3.3 (3) with h. It should be remarked that the discriminant of 𝔒 has at most one higher-power prime 𝔭^{e} (e > 1) if h satisfies λ ≤ 1. Note that formula (4.5) permits several such primes in the discriminant of 𝔒 if h satisfies the assumptions of Corollary 4.2. For example, the reader is referred to our notes after the proof of [3, Proposition 4.3], in which 𝔒 has discriminant 2 · 5^{2}g^{2}Z for a squarefree odd positive integer g prime to 5.

5 Applications and numerical examples

5.1 Applications to \{4, Q(\sqrt{d}), B_{r, ∞}, 4\}

Theorem 5.1. Let B_{r, ∞} be a definite quaternion algebra over Q ramified only at a prime number r. Take a quadratic space (V, φ) over Q whose invariants are \{4, Q(\sqrt{d}), B_{r, ∞}, 4\} with a prime number d prime to r such that d ≡ 1 (mod 4). Then for every odd prime number p prime to dr and 0 ≤ n ∈ Z there exist Z-maximal lattices L and L′ in (V, φ) such that

L[dp^{n}, 2^{−1}dZ] ≠ ∅,  L′[dp^{n}, 2^{−1}Z] ≠ ∅,  (5.1)

except when n ∉ 2Z and (\frac{d}{p}) = −1. Moreover the following formulas are valid:

∑_{α∈I} \frac{#Lα^{−1}[dp^{n}, 2^{−1}dZ]}{[Γ(Lα^{−1}) : 1]} = \frac{r − 1}{24} · \begin{cases}1 & if n = 0, \\ p^{n−1} (p + (\frac{d}{p})) & if n ≥ 1, \end{cases}  (5.2)

∑_{α∈I} \frac{#L′α^{−1}[dp^{n}, 2^{−1}Z]}{[Γ(L′α^{−1}) : 1]} = \frac{(r − 1)(d^{2} − 1)}{48} · \begin{cases}1 & if n = 0, \\ p^{n−1} (p + (\frac{d}{p})) & if n ≥ 1, \end{cases} (5.3)

∑_{α∈J} # {Lα^{−1}[dp^{n}, 2^{−1}dZ]/Γ^{·}(Lα^{−1})} = t(𝔒).  (5.4)

Here (\frac{d}{p}) is the quadratic residue symbol, I (resp. J) is a complete set of representatives for SO^{φ} \ SO_{A}^{φ}/C(L) (resp. O^{φ} \ O_{A}^{φ}/D(L)), 𝔒 is an order in the algebra A^{+}(W), which is isomorphic to B_{r, ∞}, of discriminant rp^{n}Z containing A^{+}(L ∩ W), and W = (Qh)^{⊥} with h ∈ L[dp^{n}, 2^{−1}dZ].

It is noted that Lα^{−1}[dp^{n}, 2^{−1}dZ] or L′α^{−1}[dp^{n}, 2^{−1}Z] may be empty for some α ∈ I or some α ∈ J.

Proof. First of all, under the assumption that L[dp^{n}, 2^{−1}dZ] ≠ ∅ and L′[dp^{n}, 2^{−1}Z] ≠ ∅ with some maximal L and L′ in V, we can derive formulas (5.2) and (5.3). We should mention that the proof will be given in a subsequent paper and that these formulas will be used in the present proof to show (5.1). By [8, Proposition 1.8], for any positive integer q there is a Z-maximal lattice L in (V, φ) such that L[q] ≠ ∅. Let h ∈ L[dp^{n}] with 0 ≤ n ∈ Z and take the complement (W, ψ) as in (2.2). Since d ≡ 1 (mod 4) and (\frac{d}{p}) = 1 if n

is odd, the quaternion algebra \{Q(\sqrt{d}), dp^{n}\} is M_{2}(Q). Hence Q(ψ) = B_{r, ∞} by (2.3) and so the invariants of ψ are \{3, Q(\sqrt{−p^{n}}), B_{r, ∞}, 3\}. We have then 𝔟(dp^{n}) = dp^{ℓ} with ℓ = [n/2]. Noticing 𝔟(dp^{n}) ⊂ 2φ(h, L) ⊂ Z as noted in §2.1, we see that

L[dp^{n}] = ⋃_{i=0}^{ℓ}{L[dp^{n}, 2^{−1}dp^{i}Z] ∪ L[dp^{n}, 2^{−1}p^{i}Z]}.  (5.5)

Applying the explicit formula of [8, Theorem 1.5 (II)] to L[dp^{n}], we can derive that

∑_{α∈I} \frac{#Lα^{−1}[dp^{n}]}{[Γ(Lα^{−1}) : 1]} = \frac{(r − 1) (d^{2} + (\frac{d}{p})^{n})}{48} ∑_{i=0}^{2ℓ} (\frac{d}{p})^{n+i} p^{i}.  (5.6)

We here recall our assumption that (\frac{d}{p}) = 1 if n is odd. We put

R[q] = ∑_{α∈I} \frac{#Lα^{−1}[q]}{#Γ(Lα^{−1})},  R[q, 𝔟] = ∑_{α∈I} \frac{#Lα^{−1}[q, 𝔟]}{#Γ(Lα^{−1})}  (5.7)

for q ∈ Z and a Z-ideal 𝔟 of Q. Suppose n = 2ℓ with 0 ≤ ℓ ∈ Z. We shall prove (5.1) by induction on ℓ. If ℓ = 0, then 𝔟(d) = dZ and L[d] = L[d, 2^{−1}dZ]∪L[d, 2^{−1}Z] by (5.5). Because L[d] ≠ ∅, either L[d, 2^{−1}dZ] or L[d, 2^{−1}Z] must be nonempty. If L[d, 2^{−1}dZ] ≠ ∅, then formula (5.2) is valid as mentioned above. Combining this with (5.6), we have R[d, 2^{−1}Z] = R[d]−R[d, 2^{−1}dZ] = 48^{−1}(r −1)(d^{2} −1). This implies that there is some α ∈ SO_{A}^{φ} so that Lα^{−1}[d, 2^{−1}Z] ≠ ∅. Conversely, if L[d, 2^{−1}Z] ≠ ∅, we have R[d, 2^{−1}dZ] = 24^{−1}(r − 1) in the same way, whence Lα^{−1}[d, 2^{−1}dZ] ≠ ∅ with some α ∈ SO_{A}^{φ}. As a consequence we can find maximal lattices L and L′ in (V, φ) such that L[d, 2^{−1}dZ] ≠ ∅ and L′[d, 2^{−1}Z] ≠ ∅. This settles the case ℓ = 0. Suppose ℓ > 0. In view of (5.5) we have

R[dp^{n}] = ∑_{i=0}^{ℓ} {R[dp^{n}, 2^{−1}dp^{i}Z] + R[dp^{n}, 2^{−1}p^{i}Z]} .  (5.8)

Observe that the mapping x ⟼ xp^{i} gives a bijection of Lα^{−1}[dp^{2(ℓ−i)}, 2^{−1}dZ] onto Lα^{−1}[dp^{n}, 2^{−1}dp^{i}Z] for i ≠ 0 and α ∈ SO_{A}^{φ} for which Lα^{−1}[dp^{n}, 2^{−1}dp^{i}Z] ≠ ∅. Similarly Lα^{−1}[dp^{2(ℓ−i)}, 2^{−1}Z] is mapped onto Lα^{−1}[dp^{n}, 2^{−1}p^{i}Z] under the above bijection if i ≠ 0 and Lα^{−1}[dp^{n}, 2^{−1}p^{i}Z] ≠ ∅. By our induction, (5.2) and (5.3) for 2(ℓ − i) in place of n are valid for i ≠ 0. Thus we see that

R[dp^{n}, 2^{−1}dZ] + R[dp^{n}, 2^{−1}Z]
= R[dp^{n}] − ∑_{i=0}^{ℓ} {R[dp^{2(ℓ−i)}, 2^{−1}dZ] + R[dp^{2(ℓ−i)}, 2^{−1}Z]}

= \frac{(r − 1)(d^{2} + 1)}{48} · p^{2ℓ−1} (p + (\frac{d}{p})) .  (5.9)

This shows that either L_{1}[dp^{n}, 2^{−1}dZ] or L_{1}[dp^{n}, 2^{−1}Z] is not empty with some maximal lattice L_{1} in V. Now, if L_{1}[dp^{n}, 2^{−1}dZ] ≠ ∅, then formula (5.2) is valid. Combining these results with (5.9), we have R[dp^{n}, 2^{−1}Z] ≠ 0, which implies that L_{1}α^{−1}[dp^{n}, 2^{−1}Z] ≠ ∅ with some α ∈ SO_{A}^{φ}. Conversely, if L_{1}[dp^{n}, 2^{−1}Z] ≠ ∅, we have L_{1}α^{−1}[dp^{n}, 2^{−1}dZ] ≠ ∅ with α ∈ SO_{A}^{φ} by the same way. Consequently we have maximal lattices L_{1} and L′_{1} in (V, φ) such that L_{1}[dp^{n}, 2^{−1}dZ] ≠ ∅ and L′_{1}[dp^{n}, 2^{−1}Z] ≠ ∅. This completes our induction on ℓ = n/2. The case of odd n can be proved similarly, which together with the case of even n shows (5.1) for every integer n ≥ 0. At the same time we obtain formulas (5.2) and (5.3). As for (5.4), observe first that the conditions of (1) and (2) in Proposition 4.3 are satisfied for h ∈ L[dp^{n}, 2^{−1}dZ] because r, d, and p are distinct prime numbers. Further (L ∩ W)_{v} is not maximal if and only if v = p as 𝔟(dp^{n}) = dp^{ℓ}Z, except when ℓ = [n/2] = 0, that is, when n = 0 or 1. Then we easily see that condition (3) of that proposition is satisfied; for instance, if p remains prime in Q(\sqrt{d}), then n must be even by our assumption, and so p satisfies (3). The ideal 𝔣_{2} of Proposition 4.3 in the present situation is Z, except when n = 1. If n = 0 or 1, then L ∩ W is maximal. Also 𝔣_{2} = Z or pZ according as n = 0 or 1. To sum up, Proposition 4.3 is applicable to h ∈ L[dp^{n}, 2^{−1}dZ] for every 0 ≤ n ∈ Z. Hence (5.4) follows from (4.10).

We note that when n ∉ 2Z and (\frac{d}{p}) = −1 in Theorem 5.1, L[dp^{n}, 2^{−1}dZ] = ∅

for any maximal lattice L and L′[dp^{n}, 2^{−1}Z] ≠ ∅ with some maximal lattice L′ in (V, φ). Formulas (5.2) and (5.3) can be derived by means of the mass formula due to Shimura [9, (13.18)], combined with a result in a subsequent paper as mentioned in the proof of Theorem 5.1. It should be remarked about (5.4) that the type number of 𝔒 is not determined by discriminant, but by the genus of 𝔒. (In other words, by Theorem 3.4 (2), the ideal [(L ∩ W) /L ∩ W] does not determine the genus of L ∩ W.) However, if L ∩ W is maximal, that is, if 𝔒 has squarefree discriminant, t(𝔒) is determined by the discriminant. In fact, 𝔒 is maximal or an order of squarefree discriminant rpZ according as n = 0 or 1. By a result due to Eichler [1, Satz 3], any order 𝔒′ of discriminant rpZ belongs to the genus of 𝔒 in the sense that 𝔒′ = y^{−1}𝔒y with some y ∈ A^{+}(W)_{A}^{×}. The similar fact is true for maximal orders, which have discriminant rZ. Accordingly in either case n = 0 or 1 the discriminant rp^{n}Z certainly determines the genus of 𝔒. Moreover the discriminant does not depend on d. In view of these together with (5.4), we can conclude

Corollary 5.2. Let the notation be as in Theorem 5.1. Then for n = 0 or 1 the number of the left-hand side of (5.4) is independent of the choice of d. Especially, if the type number of orders in B_{r, ∞} of discriminant rp^{n}Z is 1,

for any prime number d prime to rp^{n} such that d ≡ 1 (mod 4) and (\frac{d}{p})^{n} = 1 there exists only one O^{φ}-class in the genus of maximal lattices in (V, φ) of \{4, Q(\sqrt{d}), B_{r, ∞}, 4\} such that L_{1}[dp^{n}, 2^{−1}dZ] ≠ ∅ and

L_{1}[dp^{n}, 2^{−1}dZ] = hΓ^{·}(L_{1})

with a lattice L_{1} in the class and h ∈ L_{1}[dp^{n}, 2^{−1}dZ].

In Table 1 of Section 5.3 below we shall see a few numerical examples for r = 2 and n = 0 supporting this fact.

5.2 Examples for real quadratic fields

Let V be a totally definite quaternion algebra over F of discriminant 𝔤 and φ its norm form, where F is a totally real field of even degree. Taking a nonzero element h of V and a 𝔤-maximal lattice L in (V, φ), we have the complement (W, ψ) of F h and the lattice L ∩ W. We see that A^{+}(W) is isomorphic to the present V as quaternion algebras. Our order 𝔒 is then A^{+}(L ∩ W) and has discriminant q𝔟^{−2} with q = φ[h] and 𝔟 = 2φ(h, L). Let c(𝔒) denote the class number of 𝔒 as before.

Proposition 5.3. In the above setting with h ∈ L[q, 2^{−1}𝔟] assume that F has class number 1. Then there exists an order 𝔒 of discriminant q𝔟^{−2} in V such that ∑_{i∈I} # {L_{i}[q, 2^{−1}𝔟]/Γ(L_{i})} = c(𝔒), where {L_{i}}_{i∈I} is a set of representatives for the SO^{φ}-classes in the SO^{φ}-genus of L for which L_{i}[q, 2^{−1}𝔟] ≠ ∅.

We first note by [8, Proposition 1.8] that, for every totally positive integer q of F, there is a 𝔤-maximal lattice L in (V, φ) such that L[q] = {x ∈ L | φ[x] = q} ≠ ∅. Moreover if q𝔤 is squarefree, then L[q] = L[q, 2^{−1}𝔤] because of 𝔟(q) = 𝔤.

Proof. Clearly formula (4.5) is applicable to h ∈ L[q, 2^{−1}𝔟]. Since C(L) = τ(J_{V}) and F has class number 1, τ of (1.2) gives a bijection of G^{+}(V) \ G^{+}(V)_{A}/J_{V} onto SO^{φ} \ SO_{A}^{φ}/C(L). Furthermore we have τ(G^{+}(V) ∩ yJ_{V} y^{−1}) = Γ(Lτ(y)^{−1}) for every y ∈ G^{+}(V)_{A}. The assertion follows from these combined with (4.5).

For example, take (V, φ) as in Proposition 5.3 over F = Q(\sqrt{d}) with d = 5, 13, or 101. It is known that #{SO^{φ} \ SO_{A}^{φ}/C(L)} = 1 when d = 5, 13. As noted above, there is a maximal lattice L in (V, φ) such that L[q] = L[q, 2^{−1}𝔤] ≠ ∅ for a given totally-positive squarefree integer q of F. Applying Proposition 5.3 to h ∈ L[q], we have an order 𝔒 in V of discriminant q𝔤. Now suppose q ∈ 𝔤^{×}. Then 𝔒 is maximal as d(𝔒) = 𝔤. Its class number is 1 if d = 5, 13 and is 5 if d = 101. These results can be found in [5, Tabelle 2] due to Peters. Therefore by the same proposition, #{L[q, 2^{−1}𝔤]/Γ(L)} = 1 if d = 5, 13 and ∑_{i∈I} #{L_{i}[q, 2^{−1}𝔤]/Γ(L_{i})} = 5 if d = 101, where {L_{i}}_{i∈I} is a set of representatives of the SO^{φ}-classes in the genus of L for which L_{i}[q, 2^{−1}𝔤] ≠ ∅. We mention that there is a previous result [10, Theorem 1.11] concerning the application of [10, Theorem 1.6] to the norm forms of definite quaternion algebras over Q.

5.3 Numerical tables for \{4, Q(\sqrt{d}), B_{2, ∞}, 4\}

Let d be a prime number such that d ≡ 1 (mod 4). We take a quadratic space (V, φ) over Q of invariants \{4, Q(\sqrt{d}), B_{2, ∞}, 4\} and a complete set {L_{i}}_{i∈J} of representatives for the O^{φ}-classes in the O^{φ}-genus of maximal lattices in (V, φ). By (5.4) the number ∑_{i∈J} # {L_{i}[dp^{n}, 2^{−1}dZ]/Γ^{·}(L_{i})} is given by the type number t(𝔒) of some order 𝔒 in B_{2, ∞} of discriminant 2p^{n}Z for an odd prime number p prime to d and 0 ≤ n ∈ Z, where we assume (\frac{d}{p}) = 1 if n is odd and remark that L_{i}[dp^{n}, 2^{−1}dZ] may be empty

for some i ∈ J. We put 𝔠(dp^{n}) = ∑_{i∈J} # {L_{i}[dp^{n}, 2^{−1}dZ]/Γ^{·}(L_{i})} for convenience. We restrict ourselves to the case n = 0 or 1. In this section we shall not only give the numbers 𝔠(dp^{n}) by quoting t(𝔒), but also present #L_{i}[dp^{n}, 2^{−1}dZ] for i ∈ J by taking {L_{i}}_{i∈J} in the case of d = 5, 13, 17, or 29.

To obtain {L_{i}}_{i∈J} for these primes d, we proceed according to the viewpoint explained at the last part of §4.1. Let (X, Φ) be as in that section. We set

Λ = Ze_{1} + Ze_{2} + Ze_{3} + Zg + Ze_{5},

where {e_{i}} is the standard basis of Q_{5}^{1} and g = 2^{−1}(e_{1} + e_{2} + e_{3} + e_{4}). Then Λ is a Z-maximal lattice in (X, Φ). By (4.3) we have a bijection

k_{i}Γ^{·}(Λ) ⟼ (Λ ∩ (Qk_{i})^{⊥})γ_{i}^{−1}O^{φ}

of Λ[d, Z]/Γ^{·}(Λ) onto the O^{φ}-classes in the O^{φ}-genus of maximal lattices in (V, φ) with some γ_{i} ∈ O^{Φ} so that k_{i} = k_{0}γ for i ∈ J, where {k_{i}}_{i∈J} is a complete set of representatives for Λ[d, Z]/Γ^{·}(Λ) and k_{0} is an arbitrarily fixed element of Λ[d, Z]; we put V = (Qk_{0})^{⊥} and φ = Φ|_{V}. Hence the desired representatives {L_{i}}_{i∈J} can be obtained from explicit elements k_{i} for i ∈ J by taking (Λ ∩ (Qk_{i})^{⊥})γ_{i}^{−1} as L_{i}. A method of determining Λ[d, Z]/Γ^{·}(Λ) is explained in [9, §12.15]; in which {k_{i}}_{i∈J} was found for the case of d = 29. We employ that method for our purpose. Once such a set {k_{i}}_{i∈J} is obtained, using the lattice Λ ∩ (Qk_{i})^{⊥}, we can compute the number #L_{i}[dp^{n}, 2^{−1}dZ] for every i ∈ J.

Here is a list of the representatives k_{i} for Λ[d, Z]/Γ^{·}(Λ) and the corresponding lattices Λ ∩ (Qk_{i})^{⊥} for i ∈ J and d = 5, 13, 17, 29:

(1) d = 5.

k_{1} = 2e_{1} + e_{5}.
Λ ∩ (Qk_{1})^{⊥} = Ze_{2} + Ze_{3} + Ze_{4} + Z(g − e_{5}).

(2) d = 13.

k_{1} = 2e_{1} + 3e_{5}, k_{2} = 2(e_{2} + e_{3} + e_{4}) + e_{5}.
Λ ∩ (Qk_{1})^{⊥} = Z(e_{2} + e_{3} + e_{4}) + Ze_{2} + Ze_{3} + Z(3g − e_{5}),

Λ ∩ (Qk_{2})^{⊥} = Ze_{1} + Z(e_{2} − 2e_{5}) + Z(e_{3} − 2e_{5}) + Z(g − 3e_{5}).

(3) d = 17.

k_{1} = 4e_{4} + e_{5}, k_{2} = 2(e_{3} + e_{4}) + 3e_{5}.
Λ ∩ (Qk_{1})^{⊥} = Ze_{1} + Ze_{2} + Ze_{3} + Z(g − 2e_{5}),
Λ ∩ (Qk_{2})^{⊥} = Ze_{1} + Ze_{2} + Z(3e_{3} − 2e_{5}) + Z(g − e_{3}).

(4) d = 29.

k_{1} = 2e_{4} + 5e_{5}, k_{2} = 2(e_{3} + 2e_{4}) + 3e_{5},
k_{3} = 2(e_{1} + e_{2} + e_{3} + 2e_{4}) + e_{5}.
Λ ∩ (Qk_{1})^{⊥} = Ze_{1} + Ze_{2} + Ze_{3} + Z(5g − e_{5}),
Λ ∩ (Qk_{2})^{⊥} = Ze_{1} + Ze_{2} + Z(3e_{3} − 2e_{5}) + Z(g − e_{5}),
Λ ∩ (Qk_{3})^{⊥} = Z(e_{1} − 2e_{5}) + Z(e_{2} − 2e_{5}) +
+Z(e_{3} − 2e_{5}) + Z(g − 5e_{5}).

Here we note that the case d = 5 can be seen from [10, §4.4, (4.12c)]. It can also be verified that these k_{i} for i ∈ J form a complete set of representatives for Λ[d, Z]/Γ(Λ) for d = 5, 13, 17, 29. Since [9, Proposition 11.13 (ii)] is also applicable to k_{0} ∈ Λ[d, Z], by Lemma 4.1, {L_{i}}_{i∈J} gives a complete set of representatives for the SO^{φ}-classes in the SO^{φ}-genus of maximal lattices in (V, φ). We can further determine [Γ(L_{i}) : 1] for i ∈ J. In fact, by Theorem 5.1 we have an explicit formula (5.2) for R[dp^{n}, 2^{−1}dZ] with the notation of (5.7); then #Γ(L_{i}) is computable in an elementary way by using this formula combined with the numerical data of #L_{i}[dp^{n}, 2^{−1}dZ] in our tables. For example, if d = 29, then we have three maximal lattices {L_{1}, L_{2}, L_{3}} given above. Looking at Table 1 for d = 29 and at Table 3 for d = 29, p = 5 and 7, we have 2 · #Γ(L_{1})^{−1} = 24^{−1}, 2 · #Γ(L_{2})^{−1} = 4^{−1}, and 2 · #Γ(L_{3})^{−1} = 3^{−1} by (5.2). From these we get #Γ(L_{1}) = 48, #Γ(L_{2}) = 8, and #Γ(L_{3}) = 6. Moreover the mass of the genus with respect to SO^{φ} is 5/16, which indeed coincides with the mass derived from the exact formula of [7, Theorem 5.8]. Similarly for d = 5, 13, 17, we have #Γ(L_{1}) = 48 if d = 5; #Γ(L_{1}) = 48 and #Γ(L_{2}) = 12 if d = 13; #Γ(L_{1}) = #Γ(L_{2}) = 48 if d = 17.

In the numerical tables below, we put N_{i}(dp^{n}) = #L_{i}[dp^{n}, 2^{−1}dZ] and denote by t(2, p^{n}) (resp. c(2, p^{n})) the type number (resp. the class number) of 𝔒 in B_{2, ∞} of discriminant 2p^{n}Z. We quote t(2, p^{n}) and c(2, p^{n}) from [6, Table 1] due to Pizer. It is noted by Corollary 4.2 that the number ∑_{i∈J} # {L_{i}[dp^{n}, 2^{−1}dZ]/Γ(L_{i})} coincides with c(2, p^{n}) if t(2, p^{n}) = c(2, p^{n}).

\begin{matrix}d & N_{1}(d) & N_{2}(d) & N_{3}(d) & t(2, 1) & c(2, 1) & 𝔠(d) \\ 5 & 2 & ∗ & ∗ & 1 & 1 & 1 \\ 13 & 2 & 0 & ∗ & 1 & 1 & 1 \\ 17 & 2 & 0 & ∗ & 1 & 1 & 1 \\ 29 & 2 & 0 & 0 & 1 & 1 & 1 \end{matrix}

Table 1: 𝔠(d) for d = 5, 13, 17, 29

Let us verify our numerical results for 𝔠(dp) in a straightforward way by using the lattices listed above. As an example, we take up the case of d = 13 and p = 23. We begin with the 5-dimensional space (X, Φ) and Λ as above. Put k_{1} = 2e_{1} + 3e_{5} and k_{2} = 2(e_{2} + e_{3} + e_{4}) + e_{5}. In our list with d = 13, k_{1} and k_{2} form a complete set of representatives for Λ[13, Z]/Γ^{·}(Λ) (and it is true for Γ(Λ) in place of Γ^{·}(Λ)). Set V = (Qk_{2})^{⊥} and let φ be the restriction of Φ to V. Then (V, φ) has invariants \{4, Q(\sqrt{13}), B_{2, ∞}, 4\} and L_{2} = Λ ∩ V is Z-maximal in (V, φ). Since {e_{1}, e_{2} − 2e_{5}, e_{3} − 2e_{5}, g − 3e_{5}} is a Z-basis of L_{2}, representing φ by this basis, we may put V = Q_{4}^{1},

φ = [ \begin{matrix}1 & 0 & 0 & 1/2 \\ 0 & 5 & 4 & 13/2 \\ 0 & 4 & 5 & 13/2 \\ 1/2 & 13/2 & 13/2 & 10 \end{matrix} ] ,

and L_{2} = Z_{4}^{1}. Under this identification, Γ^{·}(L_{2}) is the subgroup {γ ∈ GL_{4}(Z) |

\begin{matrix}d & p & N_{1}(dp) & N_{2}(dp) & t(2, p) & c(2, p) & 𝔠(dp) \\ 5 & 11 & 24 & ∗ & 1 & 1 & 1 \\ 5 & 19 & 40 & ∗ & 2 & 3 & 2 \\ 5 & 29 & 60 & ∗ & 2 & 3 & 2 \\ 5 & 31 & 64 & ∗ & 2 & 4 & 2 \\ 5 & 41 & 84 & ∗ & 3 & 4 & 3 \\ 5 & 59 & 120 & ∗ & 3 & 5 & 3 \\ 5 & 61 & 124 & ∗ & 4 & 7 & 4 \\ 5 & 71 & 144 & ∗ & 2 & 6 & 2 \\ 5 & 79 & 160 & ∗ & 3 & 8 & 3 \\ 5 & 89 & 180 & ∗ & 5 & 8 & 5 \\ 5 & 101 & 204 & ∗ & 5 & 9 & 5 \\ 13 & 3 & 0 & 2 & 1 & 1 & 1 \\ 13 & 17 & 12 & 6 & 2 & 2 & 2 \\ 13 & 23 & 0 & 12 & 1 & 2 & 1 \\ 13 & 29 & 12 & 12 & 2 & 3 & 2 \\ 13 & 43 & 16 & 18 & 3 & 5 & 3 \\ 13 & 53 & 12 & 24 & 3 & 5 & 3 \\ 13 & 61 & 12 & 28 & 4 & 7 & 4 \\ 13 & 79 & 48 & 28 & 3 & 8 & 3 \\ 13 & 101 & 60 & 36 & 5 & 9 & 5 \end{matrix}

Table 2: 𝔠(dp) for d = 5, 13

Quadratic Diophantine Equations in Four Variables 281

\begin{matrix}d & p & N_{1}(dp) & N_{2}(dp) & N_{3}(dp) & t(2, p) & c(2, p) & 𝔠(dp) \\ 17 & 13 & 16 & 12 & ∗ & 2 & 3 & 2 \\ 17 & 19 & 16 & 24 & ∗ & 2 & 3 & 2 \\ 17 & 43 & 40 & 48 & ∗ & 3 & 5 & 3 \\ 17 & 47 & 48 & 48 & ∗ & 2 & 4 & 2 \\ 17 & 53 & 60 & 48 & ∗ & 3 & 5 & 3 \\ 17 & 59 & 72 & 48 & ∗ & 3 & 5 & 3 \\ 17 & 67 & 64 & 72 & ∗ & 4 & 7 & 4 \\ 17 & 83 & 72 & 96 & ∗ & 4 & 7 & 4 \\ 17 & 89 & 96 & 84 & ∗ & 5 & 8 & 5 \\ 17 & 101 & 96 & 108 & ∗ & 5 & 9 & 5 \\ 29 & 5 & 0 & 2 & 0 & 1 & 1 & 1 \\ 29 & 7 & 0 & 0 & 2 & 1 & 2 & 1 \\ 29 & 13 & 0 & 2 & 2 & 2 & 3 & 2 \\ 29 & 23 & 0 & 0 & 6 & 1 & 2 & 1 \\ 29 & 53 & 0 & 10 & 6 & 3 & 5 & 3 \\ 29 & 59 & 24 & 8 & 6 & 3 & 5 & 3 \\ 29 & 67 & 24 & 8 & 8 & 4 & 7 & 4 \\ 29 & 71 & 0 & 8 & 12 & 2 & 6 & 2 \\ 29 & 83 & 24 & 16 & 6 & 4 & 7 & 4 \\ 29 & 103 & 16 & 16 & 12 & 5 & 10 & 5 \end{matrix}

Table 3: 𝔠(dp) for d = 17, 29

Documenta Mathematica 19 (2014) 247–284

γφ ·^{t}γ = φ} of GL_{4}(Z). Then L_{2}[13p^{n}, 2^{−1} · 13Z] is given by

L_{2}[13p^{n}, 2^{−1} · 13Z] = {[x_{1} x_{2} x_{3} x_{4}] ∈ Z_{4}^{1} |
x_{1}^{2} + 5x_{2}^{2} + 5x_{3}^{2} + 10x_{4}^{2} + x_{1}x_{4} + 8x_{2}x_{3} + 13x_{2}x_{4} + 13x_{3}x_{4} = 13p^{n},
(2x_{1} + x_{4})Z + (10x_{2} + 8x_{3} + 13x_{4})Z + (8x_{2} + 10x_{3} + 13x_{4})Z
+(x_{1} + 13x_{2} + 13x_{3} + 20x_{4})Z = 13Z}.

Now for p = 23 and n = 1 we have all solutions in L_{2}[13 · 23, 2^{−1} · 13Z]:

[±5 ∓ 13 ∓ 13 ± 16], [±8 0 ∓ 13 ± 10], [±8 ∓ 13 0 ± 10],
[±18 0 ± 13 ∓ 10], [±18 ± 13 0 ∓ 10], [±21 ± 13 ± 13 ∓ 16].

We put

γ_{1} = [ \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & −1 & 0 & 0 \\ 1 & 0 & 0 & −1 \end{matrix} ] , γ_{2} = [ \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 1 & 1 & 1 & −2 \\ 1 & 0 & 0 & −1 \end{matrix} ] , γ_{3} = [ \begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 1 & 0 & 0 & −1 \end{matrix} ].

These matrices belong to Γ^{·}(L_{2}). Consider the subgroup U of Γ^{·}(L_{2}) generated by γ_{1}, γ_{2}, γ_{3}, and −1_{4}, where 1_{4} is the identity matrix of size 4. Put x = [5 − 13 − 13 16]. Then it can be seen that xU contains all elements of L_{2}[13 · 23, 2^{−1} · 13Z]. Thus we have L_{2}[13 · 23, 2^{−1} · 13Z] = xΓ^{·}(L_{2}). Similarly for k_{1}, we can consider a Z-lattice Λ ∩ (Qk_{1})^{⊥}. Denoting by φ_{1} the restriction of Φ to (Qk_{1})^{⊥}, we may put (Qk_{1})^{⊥} = Q_{4}^{1},

φ_{1} = [ \begin{matrix}3 & 1 & 1 & 9/2 \\ 1 & 1 & 0 & 3/2 \\ 1 & 0 & 1 & 3/2 \\ 9/2 & 3/2 & 3/2 & 10 \end{matrix} ] ,

Λ ∩ (Qk_{1})^{⊥} = Z_{4}^{1}, and Γ^{·}(Λ ∩ (Qk_{1})^{⊥}) = {γ ∈ GL_{4}(Z) | γφ_{1} ·^{t}γ = φ_{1}} under the identification with respect to a Z-basis {e_{2} + e_{3} + e_{4}, e_{2}, e_{3}, 3g − e_{5}} of Λ ∩ (Qk_{1})^{⊥}. Let L_{1} be the lattice in (V, φ) corresponding to Λ∩(Qk_{1})^{⊥} under some isomorphism of (V, φ) onto ((Qk_{1})^{⊥}, φ_{1}). Then the number #L_{1}[13p^{n}, 2^{−1} · 13Z] is equal to

#{[x_{1} x_{2} x_{3} x_{4}] ∈ Z_{4}^{1} |

3x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + 10x_{4}^{2} + 2x_{1}x_{2} + 2x_{1}x_{3} + 9x_{1}x_{4} + 3x_{2}x_{4} + 3x_{3}x_{4} = 13p^{n}, (6x_{1} + 2x_{2} + 2x_{3} + 9x_{4})Z + (2x_{1} + 2x_{2} + 3x_{4})Z + (2x_{1} + 2x_{3} + 3x_{4})Z

+(9x_{1} + 3x_{2} + 3x_{3} + 20x_{4})Z = 13Z}.

For p = 23 and n = 1 there is no elements of (Λ ∩ (Qk_{1})^{⊥})[13 · 23, 2^{−1} · 13Z]. Hence #L_{1}[13 · 23, 2^{−1} · 13Z] = 0. Because L_{1} and L_{2} are not in the same O^{φ}- class as k_{1}Γ^{·}(Λ) ≠ k_{2}Γ^{·}(Λ), we have therefore 𝔠(13 · 23) = #{L_{2}[13 · 23, 2^{−1} · 13Z]/Γ^{·}(L_{2})} = 1. This coincides with our result in the case of d = 13 and p = 23 in Table 2. We note that #U = 24, Γ^{·}(L_{2}) = U, and Γ(L_{2}) is generated by γ_{1}, γ_{2}, −1_{4}; furthermore we have xΓ(L_{2}) = L_{2}[13 · 23, 2^{−1} · 13Z], that is, #{L_{2}[13 · 23, 2^{−1} · 13Z]/Γ(L_{2})} = 1. As for Λ∩(Qk_{1})^{⊥}, four elements δ_{1}, ⋯, δ_{4} and −1_{4} generate Γ^{·}(Λ ∩ (Qk_{1})^{⊥}) and then Γ(Λ ∩ (Qk_{1})^{⊥}) is generated by δ_{1}δ_{2}, δ_{2}δ_{3}, δ_{4}, −1_{4}, where

δ_{1} = [ \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 3 & 0 & 0 & −1 \end{matrix} ] ,  δ_{2} = [ \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & −1 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ] ,

δ_{3} = [ \begin{matrix}1 & 0 & 0 & 0 \\ 1 & −1 & −1 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ] ,  δ_{4} = [ \begin{matrix}1 & −2 & 0 & 0 \\ 1 & −1 & −1 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & −3 & 0 & 1 \end{matrix} ] .

We shall show one more example for d = 13 and p = 79 obtained in the same manner:

# {L_{1}[13 · 79, 2^{−1} · 13Z]/Γ^{·}(L_{1})} = 1, # {L_{2}[13 · 79, 2^{−1} · 13Z]/Γ^{·}(L_{2})} = 2,

# {L_{1}[13 · 79, 2^{−1} · 13Z]/Γ(L_{1})} = 1, # {L_{2}[13 · 79, 2^{−1} · 13Z]/Γ(L_{2})} = 3.

Here L′_{1}[13 · 79, 2^{−1} · 13Z], with L′_{1} = Λ ∩ (Qk_{1})^{⊥} ≅ L_{1}, consists of 48 solutions

[±10 ∓ 39 ∓ 26 ± 2], ⋯, [±29 ∓ 13 ∓ 13 ∓ 2]

and L_{2}[13 · 79, 2^{−1} · 13Z] of 28 solutions

[±3 ± 13 ± 39 ∓ 32], ⋯, [±36 ± 39 ± 26 ∓ 46].

Accordingly ∑_{i=1}^{2} #{L_{i}[13·79, 2^{−1} ·13Z]/Γ(L_{i})} is a quantity that differs from both the type number and the class number of 𝔒.

References

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Manabu Murata

College of

Science and Engineering Ritsumeikan University Kusatsu, Shiga 525-8577 Japan murata31@pl.ritsumei.ac.jp