Montserrat Alsina
Binary quadratic forms and Eichler orders Tome 17, no1 (2005), p. 13-23.
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Binary quadratic forms and Eichler orders
par Montserrat ALSINAR´esum´e. Pour tout ordre d’Eichler O(D, N ) de niveau N dans une alg`ebre de quaternions ind´efinie de discriminant D, il existe un groupe Fuchsien Γ(D, N ) ⊆ SL(2, R) et une courbe de Shimura X(D, N ). Nous associons `a O(D, N ) un ensemble H(O(D, N )) de formes quadratiques binaires ayant des coefficients semi-entiers quadratiques et developpons une classification des formes quadra-tiques primitives de H(O(D, N )) pour rapport `a Γ(D, N ). En par-ticulier nous retrouvons la classification des formes quadratiques primitives et enti`eres de SL(2, Z). Un domaine fondamental ex-plicite pour Γ(D, N ) permet de caract´eriser les Γ(D, N ) formes r´eduites.
Abstract. For any Eichler order O(D, N ) of level N in an in-definite quaternion algebra of discriminant D there is a Fuchsian group Γ(D, N ) ⊆ SL(2, R) and a Shimura curve X(D, N ). We associate to O(D, N ) a set H(O(D, N )) of binary quadratic forms which have semi-integer quadratic coefficients, and we develop a classification theory, with respect to Γ(D, N ), for primitive forms contained in H(O(D, N )). In particular, the classification the-ory of primitive integral binary quadratic forms by SL(2, Z) is recovered. Explicit fundamental domains for Γ(D, N ) allow the characterization of the Γ(D, N )-reduced forms.
1. Preliminars Let H = a,b Q
be the quaternion Q-algebra of basis {1, i, j, ij}, satisfying i2 = a, j2 = b, ji = −ij, a, b ∈ Q∗. Assume H is an indefinite quaternion algebra, that is, H ⊗Q R ' M(2, R). Then the discriminant DH of H is
the product of an even number of different primes DH = p1· · · p2r ≥ 1 and
we can assume a > 0. Actually, a discriminant D determines a quaternion algebra H such that DH = D up to isomorphism. Let us denote by n(ω)
the reduced norm of ω ∈ H.
Fix any embedding Φ : H ,→ M(2, R). For simplicity we can keep in mind the embedding given at the following lemma.
Lemma 1.1. Let H =a,b
Q
be an indefinite quaternion algebra with a > 0. An embedding Φ : H ,→ M(2, R) is obtained by:
Φ(x + yi + zj + tij) = x + y√a z + t√a b(z − t√a) x − y√a .
Given N ≥ 1, gcd(D, N ) = 1, let us consider an Eichler order of level N , that is a Z-module of rank 4, subring of H, intersection of two maximal orders. By Eichler’s results it is unique up to conjugation and we denote it by O(D, N ).
Consider Γ(D, N ) := Φ({ω ∈ O(D, N )∗ | n(ω) > 0}) ⊆ SL(2, R) a group of quaternion transformations. This group acts on the upper complex half plane H = {x + ιy ∈ C | y > 0}. We denote by X(D, N ) the canonical model of the Shimura curve defined by the quotient Γ(D, N )\H, cf. [Shi67], [AAB01].
For any γ = a b
c d ∈ GL(2, R) we denote by P(γ) the set of fixed points
in C of the transformation defined by γ(z) = az+bcz+d.
Let us denote by E (H, F ) the set of embeddings of a quadratic field F into the quaternion algebra H. Assume there is an embedding ϕ ∈ E (H, F ). Then, all the quaternion transformations in Φ(ϕ(F∗)) ⊂ GL(2, R) have the same set of fixed points, which we denote by P(ϕ). In the case that F is an imaginary quadratic field it yields to complex multiplication points, since P(ϕ) ∩ H is just a point, z(ϕ).
Now, we take in account the arithmetic of the orders. Let us consider the set of optimal embeddings of quadratic orders Λ into quaternion orders O,
E∗(O, Λ) := {ϕ | ϕ : Λ ,→ O, ϕ(F ) ∩ O = ϕ(Λ)}.
Any group G ≤ Nor(O) acts on E∗(O, Λ), and we can consider the quo-tient E∗(O, Λ)/G. Put ν(O, Λ; G) := ] E∗(O, Λ)/G. We will also use the notation ν(D, N, d, m; G) for an Eichler order O(D, N ) ⊆ H of level N and the quadratic order of conductor m in F = Q(√d), which we denote Λ(d, m).
Since further class numbers in this paper will be related to this one, we include next theorem (cf. [Eic55]). It provides the well-known relation between the class numbers of local and global embeddings, and collects the formulas for the class number of local embeddings given in [Ogg83] and [Vig80] in the case G = O∗. Consider ψp the multiplicative function given
by ψp(pk) = pk(1 + 1p), ψp(a) = 1 if p - a. Put h(d, m) the ideal class
number of the quadratic order Λ(d, m).
Theorem 1.2. Let O = O(D, N ) be an Eichler order of level N in an indefinite quaternion Q-algebra H of discriminant D. Let Λ(d, m) be the quadratic order of conductor m in Q(√d). Assume that E (H, Q(√d)) 6= ∅
and gcd(m, D) = 1. Then,
ν(D, N, d, m; O∗) = h(d, m) Y
p|DN
νp(D, N, d, m; O∗).
The local class numbers of embeddings νp(D, N, d, m; O∗), for the primes
p|DN , are given by (i) If p|D, then νp(D, N, d, m; O∗) = 1 − DF p . (ii) If p k N , then νp(D, N, d, m; O∗) is equal to 1 +
DF p if p - m, and equal to 2 if p|m.
(iii) Assume N = pru1, with p - u1, r ≥ 2. Put m = pku2, p - u2.
(a) If r ≥ 2k + 2, then νp(D, N, d, m; O∗) is equal to 2ψp(m) if
DF
p
= 1, and equal to 0 otherwise.
(b) If r = 2k + 1, then νp(D, N, d, m; O∗) is equal to 2ψp(m) if DF p = 1, equal to pk if DF p = 0, and equal to 0 if DF p = −1. (c) If r = 2k, then νp(D, N, d, m; O∗) = pk−1 p + 1 +DF p . (d) If r ≤ 2k − 1, then νp(D, N, d, m; O∗) is equal to pk/2+ pk/2−1
if k is even, and equal to 2pk−1/2 if k is odd.
2. Classification theory of binary forms associated to quaternions
Given α = a bc d ∈ M(2, R), we put fα(x, y) := cx2+ (d − a)xy − by2. It
is called the binary quadratic form associated to α.
For a binary quadratic form f (x, y) := Ax2+ Bxy + Cy2 = (A, B, C), we
consider the associated matrix A(f ) =
A B/2 B/2 C
, and the determinants det1(f ) = det A(f ) and det2(f ) = 22det A(f ) = −(B2− 4AC). Denote by
P(f ) the set of solutions in C of Az2+ Bz + C = 0. If f is (positive or
negative) definite, then P(f ) ∩ H is just a point which we denote by τ (f ). The proof of the following lemma is straightforward.
Lemma 2.1. Let α ∈ M(2, R).
(i) For all λ, µ ∈ Q, we have fλα= λfα and fα+µ Id= fα; in particular,
P(fλα+µ Id) = P(fα).
(ii) z ∈ C is a fixed point of α if and only if z ∈ P(fα), that is, P(fα) =
P(α).
(iii) Let γ ∈ GL(2, R). Then A(fγ−1αγ) = (det γ−1)γtA(fα)γ; in
particu-lar, if γ ∈ SL(2, R), z ∈ P(fα) if and only if γ−1(z) ∈ P(fγ−1αγ).
Definition 2.2. For a quaternion ω ∈ H∗, we define the binary quadratic form associated to ω as the binary quadratic form fΦ(ω).
Given a quaternion algebra H denote by H0 the pure quaternions. By
using lemma 2.1 it is enough to consider the binary forms associated to pure quaternions:
H(a, b) = {fΦ(ω) : ω ∈ H0}, H(O) = {fΦ(ω): ω ∈ O ∩ H0}.
Definition 2.3. Let O be an order in a quaternion algebra H. We define the denominator mOof O as the minimal positive integer such that mO·O ⊆
Z[1, i, j, ij]. Then the ideal (mO) is the conductor of O in Z[1, i, j, ij].
Properties for these binary forms are collected in the following proposi-tion, easy to be verified.
Proposition 2.4. Consider an indefinite quaternion algebra H = a,b
Q
, and an order O ⊆ H. Fix the embedding Φ as in lemma 1.1. Then:
(i) There is a bijective mapping H0 → H(a, b) defined by ω 7→ fΦ(ω).
Moreover det1(fΦ(ω)) = n(ω). (ii) H(a, b) = {(b(λ2+ λ3 √ a), λ1 √ a, −λ2+ λ3 √ a) | λ1, λ2, λ3 ∈ Q} = {(bβ0, α, −β) | α, β ∈ Q(√a), tr(α) = 0} . (iii) the binary quadratic forms of H(O) have coefficients in Z
h 1 mO, √ a i Given a quaternion order O and a quadratic order Λ, put
H(O, Λ) := {f ∈ H(O) : det1(f ) = −DΛ}.
Remark that an imaginary quadratic order yields to consider definite bi-nary quadratic forms, and a real quadratic order yields to indefinite bibi-nary forms. Given ω ∈ O ∩H0, consider Fω = Q( √ d), d = − n(ω). Then ϕω( √ d) = ω defines an embedding ϕω ∈ E(H, Fω). By considering Λω := ϕ−1ω (O) ∩
Fω, we have ϕω ∈ E∗(O, Λω). Therefore, by construction, it is clear that
P(fΦ(ω)) = P(Φ(ω)) = P(ϕω). In particular, if we deal with quaternions of
positive norm, we obtain definite binary forms, imaginary quadratic fields and a unique solution τ (fΦ(ω)) = z(ϕω) ∈ H. The points corresponding to
these binary quadratic forms are in fact the complex multiplication points. Theorem 4.53 in [AB04] states a bijective mapping f from the set E (O, Λ) of embeddings of a quadratic order Λ into a quaternion order O onto the set H(Z + 2O, Λ) of binary quadratic forms associated to the orders Z + 2O and Λ. By using optimal embeddings, a definition of primitivity for the forms in H(Z + 2O, Λ) was introduced. We denote by H∗(Z + 2O, Λ) the corresponding subset of (O, Λ)-primitive binary forms. Then equivalence of embeddings yields to equivalence of forms.
Corollary 2.5. Given orders O and Λ as above, for any G ⊆ O∗ consider Φ(G) ⊆ GL(2, R). There is a bijective mapping between E∗(O, Λ)/G and H∗(Z + 2O, Λ)/Φ(G).
Fix O = O(D, N ), Λ = Λ(d, m) and G = O∗. We use the notation h(D, N, d, m) := ]H∗(Z + 2O(D, N ), Λ(d, m))/ΓO∗. Thus, h(D, N, d, m) =
ν(D, N, d, m; O∗), which can be computed explicitly by Eichler results (cf. theorem 1.2).
3. Generalized reduced binary forms
Fix an Eichler order O(D, N ) in an indefinite quaternion algebra H. Consider the associated group Γ(D, N ) and the Shimura curve X(D, N ).
For a quadratic order Λ(d, m), consider the set H∗(Z + 2O(2p, N ), Λ) of binary quadratic forms. As above, for a definite binary quadratic form f = Ax2+ Bxy + Cy2, denote by τ (f ) the solution of Az2+ Bz + C = 0 in H.
Definition 3.1. Fix a fundamental domain D(D, N ) for Γ(D, N ) in H. Make a choice about the boundary in such a way that every point in H is equivalent to a unique point of D(D, N ). A binary form f ∈ H∗(Z + 2O(D, N ), Λ) is called Γ(D, N )-reduced form if τ (f ) ∈ D(D, N ). Theorem 3.2. The number of positive definite Γ(D, N )-reduced forms in H∗(Z + 2O(D, N ), Λ(d, m)) is finite and equal to h(D, N, d, m).
Proof. We can assume d < 0, in order H∗(Z + 2O(D, N ), Λ(d, m)) consists on definite binary forms. By lemma 2.1 (iii), we have that Γ(D, N )-equivalence of forms yields to Γ(D, N )-equivalence of points. Note that τ (f ) = τ (−f ), but f is not Γ(D, N )-equivalent to −f . Thus, in each class of Γ(D, N )-equivalence of forms there is a unique reduced binary form. Consider G = {ω ∈ O∗| n(ω) > 0} in order to get Φ(G) = Γ(D, N ). The group G has index 2 in O∗and the number of classes of Γ(D, N )-equivalence in H∗(Z + 2O(D, N ), Λ(d, m)) is 2 h(D, N, d, m). In that set, positive and negative definite forms were included, thus the number of classes of positive
definite forms is exactly h(D, N, d, m).
4. Non-ramified and small ramified cases
Definition 4.1. Let H be a quaternion algebra of discriminant D. We say that H is nonramified if D = 1, that is H ' M(2, Q). We say H is small ramified if D = pq; in this case, we say it is of type A if D = 2p, p ≡ 3 mod 4, and we say it is of type B if DH = pq, q ≡ 1 mod 4 and
p q
= −1. It makes sense because of the following statement.
Proposition 4.2. For H =p,q
Q
, p, q primes, exactly one of the following statements holds:
(i) H is nonramified.
(ii) H is small ramified of type A. (iii) H is small ramified of type B.
We are going to specialize above results for reduced binary forms for each one of these cases.
4.1. Nonramified case. Consider H = M(2, Q) and take the Eichler order
O0(1, N ) :=
a b
cN d | a, b, c, d ∈ Z .
Then Γ(1, N ) = Γ0(N ) and the curve X(1, N ) is the modular curve X0(N ).
To unify results with the ramified case, it is also interesting to work with the Eichler order O(1, N ) := Zh1,j+ij2 , N(−j+ij)2 ,1−i2 i in the nonramified quaternion algebra 1,−1
Q
.
Proposition 4.3. Consider the Eichler order O = O0(1, N ) ⊆ M(2, Q)
and the quadratic order Λ = Λ(d, m). Then:
(i) H∗(Z + 2O, Λ) ' {f = (N a, b, c) | a, b, c ∈ Z, det2(f ) = −DΛ}.
(ii) The (O, Λ)-primitivity condition is gcd(a, b, c) = 1.
(iii) If d < 0, the number of Γ0(N )-reduced positive definite primitive
binary quadratic forms in H∗(Z + 2O, Λ) is equal to h(1, N, d, m). For N = 1, the well-known theory on reduced integer binary quadratic forms is recovered. In particular, the class number of SL(2, Z)-equivalence is h(d, m).
For N > 1, a general theory of reduced binary forms is obtained. For N equal to a prime, let us fix the symmetrical fundamental domain
D(1, N ) =z ∈ H | | Re(z)| ≤ 1/2,
z − Nk > N1, k ∈ Z, 0 < |k| ≤ N −12 given at [AB04]; a detailed construction can be found in [Als00]. Then a positive definite binary form f = (N a, b, c), a > 0, is Γ0(N )-reduced if
and only if |b| ≤ N a and |τ (f ) − Nk| > 1
N for k ∈ Z, 0 < |k| ≤ N −1
2 .
Figure 4.1 shows the 46 points corresponding to reduced binary forms in H∗(Z + 2O
0(1, 23), Λ) for DΛ = 7, 11, 19, 23, 28, 43, 56, 67, 76, 83, 88, 91, 92,
which occurs in an special graphical position. In fact these points are exactly the special complex multiplication points of X(1, 23), characterized by the existence of elements α ∈ Λ(d, m) of norm DN (cf. [AB04]). The table describes the n = h(1, 23, d, m) inequivalent points for each quadratic order Λ(d, m).
Note that for these symmetrical domains it is easy to implement an algorithm to decide if a form in this set is reduced or not, by using isometric circles.
Figure 4.1. The points τ (f ) for some f reduced binary forms corresponding to quadratic orders Λ(d, m) in a fun-damental domain for X(1, 23).
(d, m) n τ (f ) (-7, 1) 2 nτ1=−19+ √ 7ι 46 , τ2= 19+√7ι 46 o (-7, 2) 2 nτ3=−4+ √ 7ι 23 , τ4= 4+√7ι 23 o (-11, 1) 2 nτ5=−9+ √ 11ι 46 , τ6= 9+√11ι 46 o (-14, 1) 8 n τ7=−20+ √ 14ι 46 , τ8= −26+√14ι 69 , τ9= −20+√14ι 69 , τ10= −3+√14ι 23 , τ11=3+ √ 14ι 23 , τ12= 20+√14ι 69 , τ13= 26+√14ι 69 , τ14= 20+√14ι 46 o (-19, 1) 2 nτ15= −21+ √ 19ι 46 , τ16= 21+√19ι 46 o (-19, 2) 6 n τ17= −25+ √ 19ι 92 , τ18= −21+√19ι 92 , τ19= −2+√19ι 23 , τ20=2+ √ 19ι 23 , τ21= 21+√19ι 92 , τ22= 25+√19ι 92 o (-22, 1) 4 nτ23= −22+ √ 22ι 46 , τ24= −1+√22ι 23 , τ25= 1+√22ι 23 , τ26= 22+√22ι 46 o (-23, 1) 3 n τ27= −23+ √ 23ι 92 , τ28= 23+√23ι 92 , τ29= −23+√23ι 46 ∼ 23+√23ι 46 o (-23, 2) 3 nτ30= −23+ √ 23ι 69 , τ31= √ 23ι 23 , τ32= 23+√23ι 69 o (-43, 1) 2 nτ33= −7+ √ 43ι 46 , τ34= 7+√43ι 46 o (-67, 1) 2 nτ35= −5+ √ 67ι 46 , τ36= 5+√67ι 46 o (-83, 1) 6 n τ37= −49+ √ 83ι 138 , τ38= −43+√83ι 138 , τ39= −3+√83ι 46 , τ40=3+ √ 83ι 46 , τ41= 43+√83ι 138 , τ42= 49+√83ι 138 o (-91, 1) 4 nτ43= −47+ √ 91ι 230 , τ44= −1+√91ι 46 , τ45= 1+√91ι 46 , τ46= 47+√91ι 230 o
4.2. Small ramified case of type A. Let us consider HA(p) :=
p,−1 Q
and the Eichler order OA(2p, N ) := Z
h
1, i, N j,1+i+j+ij2 i
, for N | p−12 , N square-free. The elements in the group ΓA(2p, N ) are γ = 12
α β −β0α0
such that α, β ∈ Z[√p], α ≡ β ≡ α√p mod 2, det γ = 1, N |
tr(β) −β−β√ 0 p
. We denote by XA(2p, N ) the Shimura curve of type A defined by ΓA(2p, N ).
Proposition 4.4. Consider the Eichler order OA(2p, N ) and the quadratic
order Λ = Λ(d, m).
(i) The set H(Z + 2OA(2p, N ), Λ) of binary forms is equal to
{f = (a + b√p, 2c√p, a − b√p) : a, b, c ∈ Z, a ≡ b ≡ c mod 2, N | (a + b), det1(f ) = −DΛ}.
(ii) The (OA(2p, N ), Λ)-primitivity condition for these binary quadratic
forms is gcd c+b2 ,a+b2N , b = 1.
(iii) If d < 0, the number of ΓA(2p, N )-reduced positive definite primitive
binary forms in H∗(Z + 2OA(2p, N ), Λ) is equal to h(2p, N, d, m).
For example, consider the fundamental domain D(6, 1) for the Shimura curve XA(6, 1) in the Poincar´e half plane defined by the hyperbolic
polygon of vertices {v1, v2, v3, v4, v5, v6} at figure 4.2 (cf. [AB04]). The
table contains the corresponding reduced binary quadratic forms f ∈ H∗(Z + 2O
A(6, 1), Λ(d, 1)) and the associated points τ (f ) for det1(f ) =
4, 3, 24, 40, that is d = −1, −3, −6, −10. Since the vertices are elliptic points of order 2 or 3, they are the associated points to forms of determinant 4 or 3, respectively. We put n = h(6, 1, d, 1) the number of such reduced forms for each determinant.
4.3. Small ramified case of type B. Consider HB(p, q) :=
p,q Q
and the Eichler order OB(pq, N ) := Z
h
1, N i,1+j2 ,i+ij2 i, where N |q−14 , N square-free and gcd(N, p) = 1. Then the group of quaternion transfor-mations is ΓB(pq, N ) = n γ = 12 α β qβ0 α0 : α, β ∈ Z[√p], α ≡ β mod 2, N | α−α20√−β+β0 p , det γ = 1 o . We denote by XB(pq, N ) the corresponding Shimura curve of type B.
Figure 4.2. Reduced binary forms in a fundamental do-main for XA(6, 1). det1(f ) n f τ (f ) 3 2 (3 +√3)x2+ 2√3xy + (3 −√3)y2 v2= 1− √ 3 2 (1 − ι) (3 +√3)x2− 2√3xy + (3 −√3)y2 v4= −1+ √ 3 2 (1 − ι) 4 2 4x2+ 4√3xy + 4y2 v 1= − √ 3+ι 2 ∼ v3∼ v5 2x2+ 2y2 v6= ι 24 2 (6 + 2√3)x2− (−6 + 2√3)y2 τ1=( √ 6−√2)ι 2 6x2− 4√3xy + 6y2 τ2=− √ 3+√6ι 3 ∼ τ 0 2 40 4 (10 + 2√3)x2+ 8√3xy − (−10 + 2√3)y2 τ 3=3−5 √ 3 11 + 5√10−√30 22 ι (8 + 2√3)x2+ 4√3xy − (−8 + 2√3)y2 τ4=3−4 √ 3 13 + 4√10−√30 22 ι (8 + 2√3)x2− 4√3xy − (−8 + 2√3)y2 τ5=−3+4 √ 3 13 + 4√10−√30 22 ι (10 + 2√3)x2− 8√3xy − (−10 + 2√3)y2 τ6=−3+5 √ 3 11 + 5√10−√30 22 ι
Proposition 4.5. Consider the Eichler order OB(pq, N ) in HB(p, q) and
the quadratic order Λ = Λ(d, m).
(i) The set H(Z + 2OB(pq, N ), Λ) of binary forms contains precisely the
forms f = (q(a + b√p), 2c√p, −a + b√p) where a, b, c ∈ Z, 2N |(c − b) and det1(f ) = −DΛ.
(ii) The (OB(pq, N ), Λ)-primitivity condition for these binary quadratic
forms in (i) is gcd a, b,c−b2N = 1.
(iii) If d < 0, the number of ΓB(pq, N )-reduced positive definite primitive
binary forms in H∗(Z + 2OB(pq, N ), Λ) is equal to h(pq, N, d, m).
In figure 4.3 we show a fundamental domain for ΓB(10, 1) given by the
hyperbolic polygon of vertices {w1, w2, w3, w4, w5, w6}. All the vertices are
Figure 4.3. Reduced binary forms in a fundamental do-main for XB(10, 1). det1(f ) n f τ (f ) 3 4 (−5 + 5√2)x2+ 2√2xy + (1 +√2)y2 w1= − √ 2+√3ι 5(−1+√2) ∼ w3 (5 + 5√2)x2+ 2√2xy + (−1 +√2)y2 w2=− √ 2+√3ι 5(1+√2) (35 + 25√2)x2− 2√2xy + (−7 + 5√2)y2 w 4= √ 2+√3ι 5(7+5√2) ∼ w6 (5 + 5√2)x2+ 2√2xy + (−1 +√2)y2 w 5= √ 2+√3ι 5(1+√2) 8 2 5√2x2+ 2√2xy +√2y2 τ 1=−1+2ι5 (5√2x2+ 2√2xy +√2y2 τ2=1+2ι5 20 2 (10 + 10√2)x2+ (−2 + 2√2)y2 τ3=( √ 10−√5)ι 5 (−10 + 10√2)x2+ (2 + 2√2)y2 τ4=( √ 10+√5)ι 5 40 2 (40 + 30√2)x2+ (−8 + 6√2)y2 τ5=(3 √ 5−2√10)ι 5 10√2x2+ 2√2y2 τ6= √ 5ι 5
forms of determinant 3. We also represent the points corresponding to reduced binary quadratic forms f with det1(f ) = 40, which correspond to
special complex points. The table also contains the explicit reduced definite positive binary forms and the corresponding points for determinants 8 and 20.
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Montserrat Alsina
Dept. Matem`atica Aplicada III EUPM
Av. Bases de Manresa 61-73, Manresa-08240, Catalunya, Spain E-mail : Montserrat.alsina@upc.edu
