BIANCHI GROUPS AND INTEGRAL QUADRATIC FORMS

DONALD G. JAMES

1. Introduction

Let O_{d} be the ring of algebraic integers in K = ℚ(√−d), where d is a square-free positive integer. It was shown in Theorem 2.1 of James and Maclachlan [8] that the Bianchi group P SL(2, O_{d}), for d ≡ 1, 2 mod 4, is isomorphic to the spinor kernel O′(L) of an integral orthogonal group O(L). Here

L = ℤr ⊥ ℤs ⊥ (ℤt + ℤu)

is a lattice on a 4-dimensional quadratic space V with quadratic norm form n : V → ℚ and associated symmetric bilinear form f(x, y) = n(x + y) − n(x) − n(y), with n(r) = 1, n(s) = d, and t and u isotropic with f(t, u) = d. Also, the extended Bianchi group B_{d}, that is, the maximal discrete extension of P SL(2, O_{d}) in P SL(2, ℂ), is isomorphic to P SO(L). The lattice L must be replaced by L + ℤ2^{−1}(r − s) when d ≡ 3 mod 4. In this case 2 is either unramified or split in K, and locally 2 causes no difficulties. However, for d ≡ 1, 2 mod 4, 2 is ramified in K, and the local dyadic case can be complicated.

A Fuchsian subgroup F_{C} of P SL(2, O_{d}) stabilizes a circle or line C in the extended complex plane via its action as a linear fractional transformation. The main results in [8] gave a classification of the maximal non-elementary Fuchsian subgroups of a Bianchi group up to conjugacy. A key part of the argument used the strong approximation theorem on O′(L) to reduce a global integral representation problem involving the quadratic form to the corresponding local problems. The case d = 1 had previously been handled in [10] using instead Siegel’s analytical theorems on the representation of integers by indefinite quadratic forms.

Date: August, 1996.

1991 Mathematics Subject Classification. Primary 11F06, 20H10; Secondary 11E57, 20G30.

The author was partially supported by an NSF grant.

These results will now be extended, using ideas from [5] and [6], to obtain a classification of the maximal non-elementary Fuchsian subgroups of P SL(2, O_{d}) up to conjugacy in the extended Bianchi group B_{d} (see Theorems 4.5, 5.4 and 5.5). In general, there are now fewer conjugacy classes since the quotient group B_{d}/P SL(2, O_{d}) acts nontrivially. This action can be quite complicated, and depends on the values of various Legendre symbols involving the primes dividing d. Vulakh [12], [13] studied the maximal Fuchsian subgroups of B_{d} for some special values of d, including d = 2 and d = p ≡ 3 mod 4 a prime, by a different method involving hermitian forms. Theorems 3.2 and 3.3 give the connection between the maximal Fuchsian subgroup F_{C} and the unitary group P SU(H) where H is an integral 2 × 2 hermitian matrix determined by C. Our notation is similar to that used in [8]. See [3] for an algebraic treatment of Bianchi groups.

2. Bianchi groups and orthogonal groups

We summarize first the relationship between SL(2, K) and the orthogonal group O(V) of the quadratic norm form (see [7] and [8] for further details). The norm form comes from the determinant acting on the matrix ring 𝕄_{2}(K), viewed as a split quaternion algebra containing the ℚ-space V.

Let θ : SO(V) → ℚ^{∗}/ℚ^{∗2} denote the spinor norm, with spinor kernel O′(V). Then there exists an epimorphism Φ : A_{ℚ}^{∗} → SO(V), where A_{ℚ}^{∗} = {β ∈ GL(2, K) | detβ ∈ ℚ}, which restricts to an exact sequence

I → {±I} → SL(2, K) →^{Φ} O′(V) → I.

Put O′(L) = O′(V) ∩ O(L). Then Φ induces an isomorphism

Φ^{∗} : P SL(2, O_{d}) → O′(L)

which extends to an isomorphism from B_{d} onto P SO(L) (see Theorem 2.1 in [8], or Theorem 4.2 in [7] for a more general result on P SL(2, O_{S}) with ℚ replaced by a global field). Variations of these isomorphisms are also discussed in [2, §11], [4, §7.3B] and [11].

The index [SO(L) : O′(L)] = 2^{k+1}, where k is the number of distinct prime divisors of the discriminant of K. The explicit form of Φ given in equation (2) of [8] enables us to determine representatives for the k + 1 generators of the abelian 2-group SO(L)/O′(L). This group is essentially the 2-sylow subgroup of the ideal class group of K. Let Ψ(x) ∈ O(V), where x ∈ V with n(x) ≠ 0, denote the symmetry

Ψ(x) : v → v − n(x)^{−1}f(x, v)x, v ∈ V.

For a prime p dividing d (including p = 2 when d is even), put d = pq and choose a, b ∈ ℤ with aq + bp = 1. Then

σ(p) = −Ψ(t − bu)Ψ(bpr + at + bu)

is integral (check the effect of σ(p) on r, t and u) with spinor norm θ(σ(p)) = pℚ^{∗2}. When d ≡ 1 mod 4, put

σ(2) = Ψ(adr + as − 2abt − 2u)Ψ(at + u),

where now 2q = 1 + d and aq + 2b = 1. Again, θ(σ(2)) = 2ℚ^{∗2}. When d ≡ 1, 2 mod 4, the group SO(L) is generated over O′(L) by these σ(p), including σ(2), and −I. A similar result holds, without σ(2), for d ≡ 3 mod 4.

3. Fuchsian subgroups

A Fuchsian subgroup of P SL(2, O_{d}) stabilizing the circle C in the extended complex plane is called non-elementary if it contains at least two hyperbolic elements with distinct pairs of fixed points which lie on C. Put Z = X + iY ∈ ℂ. The circle C must then be of the form

AZZ̄ + BZ + B̄Z̄ + C = 0  (1)

with discriminant D = BB̄ − AC > 0, for some A, C ∈ ℤ and B = b_{1} + b_{2}ω ∈ O_{d} = ℤ[ω] where ω = √ − d for d ≡ 1, 2 mod 4 and ω = (1 + √ − d)/2 for d ≡ 3 mod 4. This follows since three fixed points on C will K-linearly determine the ratios of the values of the coefficients in (1), so that these ratios must lie in K. The coefficients can then be put into the above form. A more geometrical proof of this is given in [9, §2]. We will also assume (A, b_{1}, b_{2}, C) = 1.

Let C_{D} be the circle ZZ̄ = X^{2} + Y^{2} = D, and consider a linear fractional transformation

Z′ = (xZ + z)(yZ + w)^{−1}

induced by β = (\begin{matrix}x & y \\ z & w \end{matrix}) ∈ A_{ℚ}^{∗} which stabilizes C_{D}. Then Z′Z̄′ = D if

and only if xz̄ = Dyw̄ and Dxx̄ + zz̄ = D(Dyȳ + ww̄). Let Φ(β) = ϕ_{β} ∈ SO(V), with Φ as in §2, so that from equation (2) in [8],

(detβ)ϕ_{β}(t) = −dIm(xz̄)r + Re(xz̄)s + xx̄t − zz̄u

and

(detβ)ϕ_{β}(u) = dIm(yw̄)r − Re(yw̄)s − yȳt + ww̄u.

Hence the linear fractional transformation induced by β stabilizes C_{D} if and only if (detβ)ϕ_{β}(t + Du) = (xx̄ − Dyȳ)(t + Du). It follows, since ϕ_{β} ∈ SO(V), that xx̄ − Dyȳ = ±detβ and ϕ_{β} stabilizes ℤ(t + Du).

Moreover, using detβ = xw − yz, we can derive w = ±x̄ and z = ±Dȳ. Also,

Φ^{∗}(Stab(C_{D}, P SL(2, K))) = {ϕ ∈ O′(V) | ϕ(t + Du) = ±(t + Du)}.

For σ ∈ O(L) and Stab(w_{0}, O′(L)) = {ϕ ∈ O′(L) | ϕ(w_{0}) = w_{0}} where w_{0} ∈ L, we have

σStab(w_{0}, O′(L))σ^{−1} = Stab(σ(w_{0}), O′(L)).  (2)

Let F_{C} = Stab(C, P SL(2, O_{d})) be the maximal Fuchsian subgroup of P SL(2, O_{d}) stabilizing the circle C as in (1), and let F_{D} be the corresponding maximal subgroup that stabilizes C_{D}. Then, as above, but with β ∈ SL(2, O_{d}),

Φ^{∗}(F_{D}) = Stab(ℤ(t + Du), O′(L)).

The substitution Z′ = AZ + B̄, for A ≠ 0, sends the circle C_{D} to the

C in (1), and since γ_{A} = (\begin{matrix}A & 0 \\ B̄ & 1 \end{matrix}) ∈ A_{ℚ}^{∗}, we have Φ(γ_{A}) ∈ SO(V).

Computation then gives, for d ≡ 1, 2 mod 4,

Φ(γ_{A})(t + Du) = −db_{2}r + b_{1}s + At − Cu = w_{0} ∈ L  (3)

where n(w_{0}) = dD and (A, b_{1}, b_{2}, C) = 1. The form of w_{0} is slightly different for d ≡ 3 mod 4, see (7) in [8]. Hence, with γ_{A}^{∗} ∈ P SL(2, K),

γ_{A}^{∗}Stab(C_{D}, P SL(2, K))γ_{A}^{∗−1} = Stab(C, P SL(2, K))

and, therefore, as in (2),

Φ^{∗}(Stab(C, P SL(2, K))) = Stab(ℤw_{0}, O′(V)).

Theorem 3.1. Let C be as in (1), and w_{0} be as in (3). Then

Φ^{∗}(F_{C}) = Stab(ℤw_{0}, O′(L)).

Proof. When A ≠ 0, this follows from the above since

F_{C} = Stab(C, P SL(2, K)) ∩ P SL(2, O_{d}).

When C ≠ 0, the transformation Z′ = (BZ + C)Z^{−1} also sends the

circle C_{D} to C. Put γ_{C} = (\begin{matrix}B & 1 \\ C & 0 \end{matrix}) ∈ A_{ℚ}^{∗} so that Φ(γ_{C}) ∈ SO(V). Now

Φ(γ_{C})(t + Du) = −w_{0}, with w_{0} ∈ L as in (3), and the result follows as for A ≠ 0.

Finally, consider A = C = 0, so that C is the line BZ + B̄Z̄ = 0. The transformation Z′ = Z(BZ + 1)^{−1} sends this line to a circle with 2BB̄ = 2D > 0 as the new coefficient of ZZ̄, so that we can finish as

before after using (2) with σ = Φ((\begin{matrix}1 & B \\ 0 & 1 \end{matrix})).

A different proof of this theorem is given in [8] using a geometrical argument on hyperbolic 3-space instead of the more algebraic approach above.

The conjugacy classes of the special unitary groups

SU(H) = {β ∈ SL(2, O_{d}) | βHβ̄^{t} = H}

associated with integral hermitian matrices

H = (\begin{matrix}A & B \\ B̄ & C \end{matrix}) ,  (4)

with det H = −D, derived from the coefficients of the circles C, are studied in [7]. In fact, the Fushsian group F_{C} is essentially the unitary subgroup P SU(H) of P SL(2, O_{d}). The treatment of the groups SU(H) and their conjugacy classes in SL(2, O_{S}) given in [7] is more general and considers a quadratic extension of global fields K/F, with 2 ≠ 0,

and O_{S} a Dedekind ring of S-integers. Put H_{D} = (\begin{matrix}1 & 0 \\ 0 & −D \end{matrix}).

Theorem 3.2. Let H be as in (4), and w_{0} be as in (3). Then

Φ^{∗}(P SU(H)) = Stab(w_{0}, O′(L)).

Proof. First β = (\begin{matrix}x & y \\ z & w \end{matrix}) ∈ SL(2, O_{d}) lies in SU(H_{D}) if and only if

H_{D}β̄^{t} = β^{−1}H_{D}, and this is equivalent to w = x̄ and z = Dȳ. Then xx̄ − Dyȳ = detβ = 1. Hence Φ^{∗}(P SU(H_{D})) = Stab(t + Du, O′(L)). The remainder of the proof is similar to that of Theorem 3.1. For example, Ψ^{∗}(γ_{A}P SU(H_{D})γ_{A}^{−1}) = Stab(w_{0}, O′(L)) and γ_{A}H_{D}γ̄_{A}^{t} = AH when A ≠ 0, so that γ_{A}P SU(H_{D})γ_{A}^{−1} = P SU(H) and the result then follows. Similarly for γ_{C} when C ≠ 0.

Theorem 3.3. The maximal Fuchsian subgroup F_{C} for the circle C as in (1) satisfies [F_{C} : P SU(H)] ≤ 2 with H as in (4). Moreover,

F_{C} = P SU(H)

if and only if 1. d ≡ 1 mod 4 and D ≡ 0 mod 4, or

2. d ≡ 2 mod 8 and D ≡ 0 mod 8, or

3. there exists a prime p ≡ 3 mod 4 dividing both d and D.

Proof. The index inequality follows since Stab(w_{0}, O′(L)) has at most two cosets in Stab(ℤw_{0}, O′(L)). For each of the three conditions the equation xx̄ − Dyȳ = −1, which corresponds to ϕ(w_{0}) = −w_{0} for some ϕ ∈ O′(L), has no solution with x, y ∈ O_{d}. Therefore, F_{C} and P SU(H) have the same image under the isomorphism Φ^{∗}.

Now assume that none of the three conditions hold. The left hand side of the equation xx̄ − Dyȳ = −1 can be viewed as an indefinite quadratic form < 1, d, −D, −dD >, and this form locally represents −1 over the p-adic integers for all primes p (it suffices, by Hensel’s lemma, to check modulo 4p for all primes). Hence, by Theorem 1.5 in [1, p.131], the form represents −1 over ℤ. Then F_{C} ≠ P SU(H) since there now exists a ϕ ∈ O′(L) with ϕ(w_{0}) = −w_{0}.

4. Conjugacy classes of Fuchsian subgroups

The discriminant D is an invariant of the conjugacy class of the maximal Fuchsian subgroup F_{C} associated with the circle C in (1). Let L(D) be the set of representations of dD by w_{0} ∈ L as in (3) with (A, b_{1}, b_{2}, C) = 1 (with the w_{0} for d ≡ 3 mod 4 given in equation (7) of [8]). Put Ô′(L) = < O′(L), −I >.

Theorem 4.1. The number n_{d}(D) of conjugacy classes of maximal non-elementary Fuchsian subgroups F_{C} of P SL(2, O_{d}), which have discriminant D for C, is equal to the number of Ô′(L) equivalence classes of representations of dD from L(D).

It is the number of Ô′(L) equivalence classes of w_{0} ∈ L(D) that must be determined since this gives the number of distinct O′(L) orbits of the sublattices ℤw_{0}, that is, the number of conjugacy classes of F_{C}. The number n_{d}(D) is determined in [8]. Let N(L, D) be the number of O′(L) inequivalent representations of dD by L(D). Using the strong approximation theorem on O′(L),

N(L, D) = ∏_{p|2d} N(L_{p}, D) < ∞

where N(L_{p}, D) denotes the corresponding number of integral p-adic representations. The values of N(L_{p}, D) are computed in [8]. The number n_{d}(D) is then determined from N(L, D) by studying the effect of −I on the O′(L) orbits.

These conjugacy classes of the maximal non-elementary Fuchsian subgroups F_{C} of P SL(2, O_{d}) associated with circles C having discriminant D are usually not fixed under the action of the extended Bianchi group B_{d}, as already observed for d = 1 in [10], and in the final remark of [8]. Let b_{d}(D) be the number of distinct classes of these subgroups in P SL(2, O_{d}) under conjugacy in B_{d}, so that

1 ≤ b_{d}(D) ≤ n_{d}(D) < ∞.

Viewed in O′(L), a maximal non-elementary Fuchsian subgroup corresponds to the stabilizer of a sublattice ℤw_{0} of L. Hence, by extension to P SO(L), each maximal non-elementary Fuchsian subgroup of B_{d} corresponds to the stabilizer in P SO(L) of some ℤw_{0}. Therefore, b_{d}(D) also counts the number of conjugacy classes of maximal non-elementary Fuchsian subgroups of B_{d} with discriminant D. This number b_{d}(D) will now be determined when d ≡ 3 mod 4, where the difficulties with the prime 2 are avoided since then N(L_{2}, D) = 1. In the final section we consider the dyadic orbits when d ≡ 1, 2 mod 4. The method is a modification of the techniques used in [5] and [6]. Essentially, we study the action of −I, σ(p) ∈ SO(L) on the O′(L) orbits in L(D). Via the strong approximation theorem used to get the product formula above it suffices to consider the action on local orbits.

Let N(L_{p}, D) be the number of inequivalent local representations of dD by db_{2}r + b_{1}s + At + Cu ∈ L_{p}, for p odd and (A, b_{1}, b_{2}, C)_{p} = 1, under the action of the local spinor group O′(L_{p}) (for any d). Then N(L_{p}, D) = 1 when p|d and (p, D) = 1 (see Theorem 5.1 in [8]). Also, when p divides (d, D) and the Legendre symbol (\frac{d^{−1}D}{p}) ≠ 1, the value of N(L_{p}, D) is 2; representatives of the two orbits are t + Du and ϵ_{p}t + ϵ_{p}^{−1}Du, with ϵ_{p} a non-square unit. Our convention is (\frac{c}{p}) = 0 when p|c. When (\frac{d^{−1}D}{p}) = 1 there are four O′(L_{p}) orbits, with the two new orbits represented by ±dy_{p}, where y_{p} = e_{p}r + t + c_{p}u and n(y_{p}) = d^{−1}D = e_{p}^{2} + dc_{p}.

Lemma 4.2. Let p and l be two odd primes dividing (d, D). Then 1. σ(p) fixes the O′(L_{p}) orbit of t + Du if and only if (\frac{p^{−1}d}{p}) = 1.

2. σ(p) fixes the O′(L_{l}) orbit of t + Du if and only if (\frac{p}{l}) = 1.

3. When d ≡ 1, 2 mod 4, σ(2) fixes the O′(L_{l}) orbit of t+Du if and only if l ≡ ±1 mod 8.

Proof. A calculation shows that the coefficient of t in σ(p)(t + Du) is q −Dp, and the coefficient of u is a^{2}Dq −b^{2}p. It follows (as in Theorem 5.1 of [8]) that σ(p)(t + Du) and t + Du lie in the same O′(L_{p}) orbit if and only if q − Dp is a local square, that is, (\frac{q}{p}) = (\frac{p^{−1}d}{p}) = 1.

Viewed locally at l ≠ p, the above coefficient of u in σ(p)(t + Du) is a unit, but the coefficient of t is not, so that, after interchanging t and −u (using the isometry Ψ(t + u)Ψ(s) ∈ O′(L)), σ(p)(t + Du) and t + Du lie in the same O′(L_{l}) orbit if and only if b^{2}p − a^{2}Dq is a local square, that is, (\frac{p}{l}) = 1 since l|q. Similarly, for d even, σ(2) fixes the O′(L_{l}) orbit if and only if (\frac{2}{l}) = 1, that is, l ≡ ±1 mod 8. A separate calculation is needed for d ≡ 1 mod 4 where the coefficient of t in σ(2)(t + Du) is 2b^{2} − a^{2}Dq, and the coefficient of u is 2D − q.

The local information contained in this lemma will now be put into the form of a matrix G_{d}(D) over 𝔽_{2}. Let p_{1}, …, p_{h} be the odd primes dividing (d, D). Put G_{d}(D) = (0) if h = 0. Otherwise, let G_{d}(D) be the (h + 1) × h matrix over 𝔽_{2} with the (0, j)-entry 1 if and only if p_{j} ≡ 3 mod 4, and with the (i, j)-entry 1 if and only if (\frac{p_{i}}{p_{j}}) = −1 when

j ≠ i ≠ 0, and if and only if (\frac{p_{i}^{−1}d}{p_{i}}) = −1 when i = j ≠ 0. Thus the 0-row of G_{d}(D) determines which of the local O′(L_{p_{j}}) orbits of t + Du change under −I, and the i-th row determines which of the local orbits of t + Du change under σ(p_{i}). When d ≡ 1, 2 mod 4, let G_{d}(D) be further augmented by an another row with 1 in the j-th place if and only if p_{j} ≡ ±3 mod 8. Define

g = g(D) = rank G_{d}(D) ≤ h.

For example, let d = p_{1}p_{2} with p_{1} ≡ 1 mod 4 and p_{2} ≡ 3 mod 4 primes. Assume d|D so that h = 2. Using quadratic reciprocity, we find that g(D) = 1 when (\frac{p_{1}}{p_{2}} ) = 1, and g(D) = 2 when (\frac{p_{1}}{p_{2}} ) = −1.

Theorem 4.3. Assume d ≡ 3 mod 4, and (\frac{d^{−1}D}{p}) ≠ 1 for all primes p dividing (d, D). Then b_{d}(D) = 2^{h−g}.

Proof. Under the assumption on (\frac{d^{−1}D}{p}), N(L_{p}, D) = 2 for each p dividing (d, D). Since SO(L) is generated over O′(L) by σ(p_{i}), 1 ≤ i ≤ k, and −I, the result follows because g counts the number of independent elements of the elementary 2-group SO(L)/O′(L) that interchange distinct sets of local O′(L_{p}) orbits. The action of an element in SO(L) either halves the total number of O′(L) orbits, or leaves the number unchanged.

Lemma 4.4. Let p and l be odd primes dividing both d and D. Assume (\frac{d^{−1}D}{p}) = (\frac{d^{−1}D}{l}) = 1. Then

1. σ(p) interchanges the O′(L_{p}) orbits of dy_{p} and −dy_{p}.

2. σ(p) fixes the O′(L_{l}) orbit of dy_{l} when l ≠ p.

3. When d ≡ 1, 2 mod 4, σ(2) fixes the O′(L_{l}) orbit of dy_{l}.

Proof. The coefficient of r in σ(p)(y_{l}) is e′ = −e_{l} + 2e_{l}bp + bd + adc_{l}. When p ≠ l, so that l|q, we have e′ ≡ e_{l} mod l, and hence, as in Theorem 5.1 of [8], the orbit of dy_{l} is fixed by σ(p). This also includes the case p = 2 when d is even. However, e′ ≡ −e_{p} mod p when p = l, and the orbits of dy_{p} and −dy_{p} are interchanged by σ(p). A separate calculation is needed for σ(2) when d ≡ 1 mod 4.

We now generalize Theorem 4.3. Let p_{1}, …, p_{h} be the odd primes dividing (d, D), and let P be the subset of those with (\frac{d^{−1}D}{p}) = 1. For each subset S of P, let G_{d}(S) be the submatrix of G_{d}(D) with the rows and columns corresponding to p ∈ S omitted. Put

g(S) = rank G_{d}(S),

and let c(S) be the cardinality of S.

Theorem 4.5. Assume d ≡ 3 mod 4. Then

1. b_{d}(D) = 1 when (d, D) = 1.

2. b_{d}(D) = ∑_{S⊆P} 2^{h−c(S)−g(S)} when (d, D) > 1.

Proof. Assume h ≥ 1 since the case (d, D) = 1 is covered in [8]. For each S ⊆ P consider the set of O′(L_{p}) orbits corresponding to the pairs dy_{p} and −dy_{p} for p ∈ S, and the pairs t + Du and ϵ_{p}t + ϵ_{p}^{−1}Du for p ∉ S. There are 2^{h} global O′(L) orbits corresponding to these local orbits. Now consider the effect of SO(L)/O′(L) on these O′(L) orbits. Each σ(p), for p ∈ S, interchanges dy_{p} and −dy_{p}, and leaves fixed the other dy_{l}, for l ∈ S (and it is irrelevant what happens at p ∉ S). Hence each of the c(S) choices for σ(p) independently halves the number of orbits. As in Theorem 4.3, g(S) determines the number of σ(p), p ∉ S, along with −I, that independently further halve the number of orbits. Thus 2^{h−c(S)−g(S)} is the number of global inequivalent SO(L) orbits corresponding to this choice of S. The result follows by summing over all S, since then all possible choices of the local O′(L_{p}) orbits are covered.

Similar results hold for d ≡ 1, 2 mod 4 when N(L_{2}, D) = 1 and the dyadic orbits can be ignored.

Corollary 4.6. Let d = p ≡ 3 mod 4 be prime. Then

1. n_{p}(D) = b_{p}(D) = 2 if p|D and (\frac{d^{−1}D}{p}) = 1.

2. n_{p}(D) = b_{p}(D) = 1 otherwise.

This corollary was also obtained by Vulakh [12] using a different method involving hermitian forms. In this particular case the class number of K is odd and B_{p} = P GL(2, O_{p}).

The circle C corresponding to the Fuchsian group F_{C} can be obtained from the vector w_{0} ∈ L(D) representing dD. For example, ZZ̄ = D corresponds to w_{0} = t + Du. When p|D with (\frac{d^{−1}D}{p}) = 1, the circle dZZ̄ − eωZ − eω̄Z̄ = cd corresponds to w_{0} = dy_{p} = d(er + t + cu). The way that τ ∈ B_{d} changes the circle C in its conjugacy action on F_{C} corresponds to how Φ^{∗}(τ) ∈ P SO(L) changes the orbit of ℤw_{0}.

5. Dyadic orbits when 2 ramifies in K

The prime 2 ramifies in O_{d} when d ≡ 1, 2 mod 4, and it is necessary to also consider the dyadic orbits. There can be many complicated cases so we suppress some of the repetitive details of the proofs in this section. The next lemma is an extension of Theorem 5.3 in [8].

Lemma 5.1. Assume d ≡ 1 mod 4, and p is a prime dividing d.

1. When D ≡ 0 mod 8 and p ≡ 3 mod 4, the isometry σ(p) interchanges the O′(L_{2}) orbits of t + Du and −t − Du.

2. When D ≡ 0 mod 8 and d ≡ 5 mod 8, σ(2) interchanges the O′(L_{2}) orbits of t + Du and −t − Du.

3. When D ≡ 1 mod 4, the isometry σ(2) interchanges the O′(L_{2}) orbits of dr + 2t + \frac{D−d}{2}u and s + 2t + \frac{D−1}{2}u.

4. All other dyadic orbits are fixed by σ(2) and σ(p).

Proof. When D ≡ 0 mod 8, by Theorem 5.3 in [8], the O′(L_{2}) orbit of t + Du includes 5t + 5^{−1}Du, but not −t − Du. From the proof of Lemma 4.2, the coefficient of t in σ(p)(t+Du) is q−Dp, and hence σ(p) interchanges the two dyadic orbits if and only if q ≡ 3, 7 mod 8. That is, if and only if, p ≡ 3 mod 4 since d = pq ≡ 1 mod 4. However, for σ(2), after interchanging t and −u, the coefficient of t in σ(2)(t + Du) becomes q−2D. Since 2q = 1+d, σ(2) only interchanges the two dyadic orbits when d ≡ 5 mod 8. The remaining cases are similar.

Corollary 5.2. Let d = 1. Then 1. b_{1}(D) = 1 if D ≡ 0, 3 mod 4.

2. b_{1}(D) = 2 if D ≡ 1, 2 mod 4.

Proof. The first part follows from Theorem 5.7 in [8], or Theorem 5 in [10]. Also, n_{1}(D) = 3 for D ≡ 1 mod 4, but σ(2) interchanges two of the orbits so that b_{1}(D) = 2.

We next study the case d = p ≡ 1 mod 8 with p prime. The values for n_{p}(D) are 1, 2, 3, 4, 6, and 9, and depend on D mod 8p^{2}.

Theorem 5.3. Let d = p ≡ 1 mod 8 be prime. Then b_{p}(D) = 1 when (p, D) = 1, and for p|D

1. b_{p}(D) = 2 when (\frac{p^{−1}D}{p}) ≠ 1 and D ≡ 0, 3 mod 4.

2. b_{p}(D) = 3 when (\frac{p^{−1}D}{p}) = 1 and D ≡ 0, 3 mod 4.

3. b_{p}(D) = 4 when (\frac{p^{−1}D}{p}) ≠ 1 and D ≡ 1, 2 mod 4.

4. b_{p}(D) = 6 when (\frac{p^{−1}D}{p}) = 1 and D ≡ 1, 2 mod 4.

Proof. The case (p, D) = 1 is covered in [8], so assume p|D. By Lemmas 4.2, 4.4 and 5.1, σ(p) interchanges the O′(L_{p}) orbits of py_{p} and −py_{p} when (\frac{p^{−1}D}{p}) = 1, but σ(p) has no effect on the other local orbits. Also, σ(2) interchanges two of the O′(L_{2}) orbits when D ≡ 1 mod 4, but has no other effect on the local orbits. By Corollaries 5.2 and 5.4 in [8], −I interchanges two dyadic orbits when D ≡ 0 mod 8, and has the same effect as σ(p) on the O′(L_{p}) orbits. The result then follows from a case by case examination. For example, from Theorems 5.1 and 5.3 in [8], N(L, D) = 6 for D ≡ 1 mod 4 and (\frac{p^{−1}D}{p}) ≠ 1. But σ(2) interchanges two of the three dyadic orbits, while the two p-adic orbits are not effected by σ(p), σ(2), or −I. Hence b_{p}(D) = 4. Also, N(L, D) = 12 when D ≡ 1 mod 4 and (\frac{p^{−1}D}{p}) = 1. Now σ(p) interchanges two of the four O′(L_{p}) orbits, and σ(2) independently interchanges two of the three dyadic orbits, so that b_{d}(D) = 6.

The result is different when p ≡ 5 mod 8, since the effect of σ(2) changes. For example, b_{p}(D) = 1 when D ≡ 3 mod 4 and (\frac{p^{−1}D}{p}) ≠ 1.

Lemma 5.4. Assume d is even, and p is an odd prime dividing d.

1. When D ≡ 0 mod 8 and p ≡ 5, d + 5 mod 8, σ(p) interchanges the O′(L_{2}) orbits of t + Du and 5t + 5^{−1}Du.

2. When D ≡ 0 mod 8 and d ≡ ±6 mod 16, σ(2) interchanges the O′(L_{2}) orbits of t + Du and 5t + 5^{−1}Du.

3. When D ≡ d, d + 4 mod 16, σ(2) interchanges the O′(L_{2}) orbits of dy and −dy where y = r + γs + 2t + 2αu (with γ = 0, 1).

4. All other dyadic orbits are fixed by σ(2) and σ(p).

This follows from calculations similar to those in Lemma 5.1, using Theorem 5.5 in [8]. Finally, we consider d = 2p with p ≡ 3 mod 4 prime. By Theorem 5.7 in [8], the values for n_{d}(D) are 1, 2, 4 and 8.

Theorem 5.5. Let d = 2p with p ≡ 3 mod 4 prime. Then b_{d}(D) = 1 when (d, D) = 1, and for D odd with p|D,

1. b_{d}(D) = 2 when (\frac{d^{−1}D}{p}) ≠ 1.

2. b_{d}(D) = 4 when (\frac{d^{−1}D}{p}) = 1.

Proof. The case (d, D) = 1 is already covered in [8], so assume D is odd and p|D. The two dyadic orbits are fixed by σ(2), σ(p) and −I. Since −I interchanges the two or four O′(L_{p})-adic orbits in pairs, we have b_{d}(D) = 2 or 4.

The action of the group SO(L) on the local orbits is more complicated when D is even, with b_{d}(D) taking each of the values 1, 2, 3, 4 and 6.

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Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802

E-mail address: james@math.psu.edu