pages13-23. 304KB Jun 04 2011 12:06:58 AM
Journal de Th´eorie des Nombres
de Bordeaux 17 (2005), 13–23
Binary quadratic forms and Eichler orders
par Montserrat ALSINA
´sume
´. Pour tout ordre d’Eichler O(D, N ) de niveau N dans
Re
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 quadratiques primitives de H(O(D, N )) pour rapport `a Γ(D, N ). En particulier nous retrouvons la classification des formes quadratiques
primitives et enti`eres de SL(2, Z). Un domaine fondamental explicite 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 indefinite 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 theory 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 =
i2
a, j 2
a,b
Q
be the quaternion Q-algebra of basis {1, i, j, ij}, satisfying
=
= 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.
The author was partially supported by MCYT BFM2000-0627.
Montserrat Alsina
14
be an indefinite quaternion algebra with a > 0.
Lemma 1.1. Let H = a,b
Q
An embedding Φ : H ֒→ M(2, R) is obtained by:
√
√
z + t √a
x + y √a
Φ(x + yi + zj + tij) =
.
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 = ac db ∈ GL(2, R) we denote by P(γ) the set of fixed points
in C of the transformation defined by γ(z) = az+b
cz+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 quotient 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 + p1 ), 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= ∅
Binary forms and Eichler orders
15
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 − DpF .
(ii) If p k N , then νp (D, N, d, m; O∗ ) is equal to 1 + DpF if p ∤ m, and
equal to 2 if p|m.
(iii) Assume N = pr u1 , with p ∤ u1 , r ≥ 2. Put m = pk u2 , 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
N,d, m; O ) is equal to 2ψp (m)
if
r = 2k + 1, then νp (D,
DF
DF
DF
k
= 1, equal to p if p = 0, and equal to 0 if p =
p
−1.
(c) If r = 2k, then νp (D, N, d, m; O∗ ) = pk−1 p + 1 + DpF .
(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 = ac db ∈ M(2, R), we put fα (x, y) := cx2 + (d − a)xy − by 2 . It
is called the binary quadratic form associated to α.
2
For a binary quadratic form f (x, y) :=Ax2 + Bxy
+ Cy = (A, B, C), we
A B/2
consider the associated matrix A(f ) = B/2 C , and the determinants
det1 (f ) = det A(f ) and det2 (f ) = 22 det A(f ) = −(B 2 − 4AC). Denote by
P(f ) the set of solutions in C of Az 2 + 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 −1 αγ ) = (det γ −1 )γ t A(fα )γ; in particular, if γ ∈ SL(2, R), z ∈ P(fα ) if and only if γ −1 (z) ∈ P(fγ −1 αγ ).
Definition 2.2. For a quaternion ω ∈ H ∗ , we define the binary quadratic
form associated to ω as the binary quadratic form fΦ(ω) .
16
Montserrat Alsina
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 mO of 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 proposition, 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}
√
= {(bβ ′ , α, −β) | α, β ∈ Q( a), tr(α) = 0} .
(iii) the binary quadratic forms of H(O) have coefficients in Z
Given a quaternion order O and a quadratic order Λ, put
h
1 √
mO , a
i
H(O, Λ) := {f ∈ H(O) : det1 (f ) = −DΛ }.
Remark that an imaginary quadratic order yields to consider definite binary quadratic forms, and a real quadratic order yields to indefinite binary
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−1
ω (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).
Binary forms and Eichler orders
17
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 + Cy 2 , denote by τ (f ) the solution of Az 2 + 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
p
mod 4, and we say it is of type B if DH = pq, q ≡ 1 mod 4 and q = −1.
It makes sense because of the following statement.
18
Proposition 4.2. For H =
statements holds:
Montserrat Alsina
p,q
Q
, p, q primes, exactly one of the following
(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
a b
O0 (1, N ) := 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
to work with
i
h case, it is also interesting
(−j+ij) 1−i
j+ij
the Eichler order O(1, N ) := Z 1, 2 , N 2 , 2 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 2−1
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 | > N1 for k ∈ Z, 0 < |k| ≤ N 2−1 .
Figure 4.1 shows the 46 points corresponding to reduced binary forms in
H∗ (Z + 2O0 (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).
Binary forms and Eichler orders
19
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 fundamental domain for X(1, 23).
(d, m)
τ (f )
n
(-7, 1)
2
(-7, 2)
2
(-11, 1)
2
(-14, 1)
8
(-19, 1)
2
(-19, 2)
6
(-22, 1)
4
(-23, 1)
3
(-23, 2)
3
(-43, 1)
2
(-67, 1)
2
(-83, 1)
6
(-91, 1)
4
√
√ o
7ι
τ1 = −19+
, τ2 = 19+46 7ι
46
n
√ o
√
4+ 7ι
7ι
,
τ
=
τ3 = −4+
4
23
23
n
o
√
√
τ5 = −9+46 11ι , τ6 = 9+4611ι
n
√
√
√
√
14ι
14ι
14ι
τ7 = −20+
, τ8 = −26+
, τ9 = −20+
, τ10 = −3+23 14ι ,
46
69
69
o
√
√
√
√
τ11 = 3+2314ι , τ12 = 20+69 14ι , τ13 = 26+69 14ι , τ14 = 20+46 14ι
n
o
√
√
19ι
τ15 = −21+
, τ16 = 21+46 19ι
46
n
√
√
√
19ι
19ι
, τ18 = −21+
, τ19 = −2+23 19ι ,
τ17 = −25+
92
92
o
√
√
√
τ20 = 2+2319ι , τ21 = 21+92 19ι , τ22 = 25+92 19ι
n
o
√
√
√
√
22ι
τ23 = −22+
, τ24 = −1+23 22ι , τ25 = 1+2322ι , τ26 = 22+46 22ι
46
n
o
√
√
√
√
23ι
23ι
τ27 = −23+
, τ28 = 23+92 23ι , τ29 = −23+
∼ 23+46 23ι
92
46
n
o
√
√
√
23+ 23ι
23ι
23ι
τ30 = −23+
,
τ
=
,
τ
=
31
32
69
23
69
n
o
√
√
τ33 = −7+46 43ι , τ34 = 7+4643ι
n
o
√
√
τ35 = −5+46 67ι , τ36 = 5+4667ι
n
√
√
√
83ι
83ι
, τ38 = −43+
, τ39 = −3+46 83ι ,
τ37 = −49+
138
138
o
√
√
√
τ40 = 3+4683ι , τ41 = 43+13883ι , τ42 = 49+13883ι
n
o
√
√
√
√
91ι
τ43 = −47+
, τ44 = −1+46 91ι , τ45 = 1+4691ι , τ46 = 47+23091ι
230
n
Montserrat Alsina
20
4.2. Small ramified case of type A. Let us consider HA (p) := p,−1
Q
i
h
p−1
1+i+j+ij
and the Eichler order OA (2p, N ) := Z 1, i, N j,
, for N | 2 , N
2
α β
1
square-free. The elements in the group ΓA (2p, N ) are γ = 2 −β ′ α′ such
′
√
√
√
that α, β ∈ Z[ p], α ≡ β ≡ α p mod 2, det γ = 1, N | tr(β) − β−β
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
a+b
,
,
b
=
1.
forms is gcd c+b
2
2N
(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 + 2OA (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.
and
4.3. Small ramified case of type B. Consider HB (p, q) := p,q
Q
i
h
i+ij
the Eichler order OB (pq, N ) := Z 1, N i, 1+j
, where N | q−1
2 , 2
4 , N
square-free and gcd(N, p) = 1. Then the group of quaternion transformations is
n
ΓB (pq, N ) = γ =
1
2
α β
qβ ′ α′
√
: α, β ∈ Z[ p], α ≡ β
mod 2,
o
′ −β+β ′
N | α−α2√
,
det
γ
=
1
.
p
We denote by XB (pq, N ) the corresponding Shimura curve of type B.
Binary forms and Eichler orders
21
Figure 4.2. Reduced binary forms in a fundamental domain for XA (6, 1).
det1 (f )
n
3
2
4
2
24
2
40
4
f
√ 2
√
√
(3 + 3)x + 2 3xy + (3 − 3)y 2
√ 2
√
√
(3 + 3)x − 2 3xy + (3 − 3)y 2
√
4x2 + 4 3xy + 4y 2
2x2 + 2y 2
√ 2
√
(6 + 2 3)x − (−6 + 2 3)y 2
√
6x2 − 4 3xy + 6y 2
√ 2
√
√
(10 + 2 3)x + 8 3xy − (−10 + 2 3)y 2
√ 2
√
√ 2
(8 + 2 3)x + 4 3xy − (−8 + 2 3)y
√
√
√
(8 + 2 3)x2 − 4 3xy − (−8 + 2 3)y 2
√ 2
√
√
(10 + 2 3)x − 8 3xy − (−10 + 2 3)y 2
τ (f )
v2
v4
v1
v6
τ1
τ2
τ3
τ4
τ5
τ6
√
1− 3
(1 − ι)
2 √
−1+ 3
(1 − ι)
√2
− 3+ι
∼ v3 ∼
2
=
=
=
= ι√
=
=
=
=
=
=
v5
√
6− 2)ι
√ 2√
− 3+ 6ι
∼√ τ2′ √
3
√
3−5 3
30
+ 5√10−
ι
11√
22 √
3−4 3
4 10− 30
+ √22 √ ι
13 √
30
−3+4 3
+ 4√10−
ι
13 √
22 √
−3+5 3
5 10− 30
+
ι
11
22
(
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
=
1.
forms in (i) is gcd a, b, c−b
2N
(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
elliptic points of order 3; thus they are the associated points to binary
Montserrat Alsina
22
Figure 4.3. Reduced binary forms in a fundamental domain for XB (10, 1).
det1 (f )
n
3
4
8
2
20
2
40
2
√
2
f
√
τ (f )
√
2
(−5 + 5 2)x + 2 2xy + (1 + 2)y
√
√
√
(5 + 5 2)x2 + 2 2xy + (−1 + 2)y 2
√ 2
√
√
(35 + 25 2)x − 2 2xy + (−7 + 5 2)y 2
√
√
√
(5 + 5 2)x2 + 2 2xy + (−1 + 2)y 2
√ 2
√
√ 2
5 √2x + 2 √2xy + √2y
(5 2x2 + 2 2xy + 2y 2
√
√
(10 + 10 2)x2 + (−2 + 2 2)y 2
√ 2
√
(−10 + 10 2)x + (2 + 2 2)y 2
√ 2
√ 2
(40 + 30 2)x + (−8 + 6 2)y
√
√
10 2x2 + 2 2y 2
√
√
− 2+ √3ι
∼ w3
w1 = 5(−1+
√ 2)
√
2+√ 3ι
w2 = −5(1+
√
√ 2)
2+ √3ι
w4 = 5(7+5
∼ w6
√ 2)
√
2+ √3ι
w5 = 5(1+ 2)
τ1 = −1+2ι
5
τ2 = 1+2ι
√5
√
5)ι
τ3 = (√10−
5√
5)ι
τ4 = ( √10+
5 √
(3 5−2 10)ι
τ5 = √
5
τ6 = 55ι
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.
Binary forms and Eichler orders
23
References
[AAB01] M. Alsina, A. Arenas, P. Bayer (eds.), Corbes de Shimura i aplicacions. STNB,
Barcelona, 2001.
[AB04] M. Alsina, P. Bayer, Quaternion orders, quadratic forms and Shimura curves. CRM
Monograph Series, vol. 22, American Mathematical Society, Providence, RI, 2004.
[Als00] M. Alsina, Dominios fundamentales modulares. Rev. R. Acad. Cienc. Exact. Fis. Nat.
94 (2000), no. 3, 309–322.
[Eic55] M. Eichler, Zur Zahlentheorie der Quaternionen-Algebren. J. reine angew. Math. 195
(1955), 127–151.
[Ogg83] A. P. Ogg, Real points on Shimura curves. Arithmetic and geometry, Vol. I, Progr.
Math., vol. 35, Birkh¨
auser Boston, Boston, MA, 1983, pp. 277–307.
[Shi67] G. Shimura, Construction of class fields and zeta functions of algebraic curves. Annals
of Math. 85 (1967), 58–159.
[Vig80] M.F. Vigneras, Arithm´
etique des alg`
ebres de quaternions. Lecture Notes in Math.,
no. 800, Springer, 1980.
Montserrat Alsina
Dept. Matem`
atica Aplicada III EUPM
Av. Bases de Manresa 61-73, Manresa-08240, Catalunya, Spain
E-mail : Montserrat.alsina@upc.edu
de Bordeaux 17 (2005), 13–23
Binary quadratic forms and Eichler orders
par Montserrat ALSINA
´sume
´. Pour tout ordre d’Eichler O(D, N ) de niveau N dans
Re
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 quadratiques primitives de H(O(D, N )) pour rapport `a Γ(D, N ). En particulier nous retrouvons la classification des formes quadratiques
primitives et enti`eres de SL(2, Z). Un domaine fondamental explicite 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 indefinite 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 theory 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 =
i2
a, j 2
a,b
Q
be the quaternion Q-algebra of basis {1, i, j, ij}, satisfying
=
= 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.
The author was partially supported by MCYT BFM2000-0627.
Montserrat Alsina
14
be an indefinite quaternion algebra with a > 0.
Lemma 1.1. Let H = a,b
Q
An embedding Φ : H ֒→ M(2, R) is obtained by:
√
√
z + t √a
x + y √a
Φ(x + yi + zj + tij) =
.
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 = ac db ∈ GL(2, R) we denote by P(γ) the set of fixed points
in C of the transformation defined by γ(z) = az+b
cz+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 quotient 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 + p1 ), 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= ∅
Binary forms and Eichler orders
15
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 − DpF .
(ii) If p k N , then νp (D, N, d, m; O∗ ) is equal to 1 + DpF if p ∤ m, and
equal to 2 if p|m.
(iii) Assume N = pr u1 , with p ∤ u1 , r ≥ 2. Put m = pk u2 , 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
N,d, m; O ) is equal to 2ψp (m)
if
r = 2k + 1, then νp (D,
DF
DF
DF
k
= 1, equal to p if p = 0, and equal to 0 if p =
p
−1.
(c) If r = 2k, then νp (D, N, d, m; O∗ ) = pk−1 p + 1 + DpF .
(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 = ac db ∈ M(2, R), we put fα (x, y) := cx2 + (d − a)xy − by 2 . It
is called the binary quadratic form associated to α.
2
For a binary quadratic form f (x, y) :=Ax2 + Bxy
+ Cy = (A, B, C), we
A B/2
consider the associated matrix A(f ) = B/2 C , and the determinants
det1 (f ) = det A(f ) and det2 (f ) = 22 det A(f ) = −(B 2 − 4AC). Denote by
P(f ) the set of solutions in C of Az 2 + 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 −1 αγ ) = (det γ −1 )γ t A(fα )γ; in particular, if γ ∈ SL(2, R), z ∈ P(fα ) if and only if γ −1 (z) ∈ P(fγ −1 αγ ).
Definition 2.2. For a quaternion ω ∈ H ∗ , we define the binary quadratic
form associated to ω as the binary quadratic form fΦ(ω) .
16
Montserrat Alsina
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 mO of 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 proposition, 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}
√
= {(bβ ′ , α, −β) | α, β ∈ Q( a), tr(α) = 0} .
(iii) the binary quadratic forms of H(O) have coefficients in Z
Given a quaternion order O and a quadratic order Λ, put
h
1 √
mO , a
i
H(O, Λ) := {f ∈ H(O) : det1 (f ) = −DΛ }.
Remark that an imaginary quadratic order yields to consider definite binary quadratic forms, and a real quadratic order yields to indefinite binary
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−1
ω (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).
Binary forms and Eichler orders
17
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 + Cy 2 , denote by τ (f ) the solution of Az 2 + 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
p
mod 4, and we say it is of type B if DH = pq, q ≡ 1 mod 4 and q = −1.
It makes sense because of the following statement.
18
Proposition 4.2. For H =
statements holds:
Montserrat Alsina
p,q
Q
, p, q primes, exactly one of the following
(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
a b
O0 (1, N ) := 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
to work with
i
h case, it is also interesting
(−j+ij) 1−i
j+ij
the Eichler order O(1, N ) := Z 1, 2 , N 2 , 2 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 2−1
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 | > N1 for k ∈ Z, 0 < |k| ≤ N 2−1 .
Figure 4.1 shows the 46 points corresponding to reduced binary forms in
H∗ (Z + 2O0 (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).
Binary forms and Eichler orders
19
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 fundamental domain for X(1, 23).
(d, m)
τ (f )
n
(-7, 1)
2
(-7, 2)
2
(-11, 1)
2
(-14, 1)
8
(-19, 1)
2
(-19, 2)
6
(-22, 1)
4
(-23, 1)
3
(-23, 2)
3
(-43, 1)
2
(-67, 1)
2
(-83, 1)
6
(-91, 1)
4
√
√ o
7ι
τ1 = −19+
, τ2 = 19+46 7ι
46
n
√ o
√
4+ 7ι
7ι
,
τ
=
τ3 = −4+
4
23
23
n
o
√
√
τ5 = −9+46 11ι , τ6 = 9+4611ι
n
√
√
√
√
14ι
14ι
14ι
τ7 = −20+
, τ8 = −26+
, τ9 = −20+
, τ10 = −3+23 14ι ,
46
69
69
o
√
√
√
√
τ11 = 3+2314ι , τ12 = 20+69 14ι , τ13 = 26+69 14ι , τ14 = 20+46 14ι
n
o
√
√
19ι
τ15 = −21+
, τ16 = 21+46 19ι
46
n
√
√
√
19ι
19ι
, τ18 = −21+
, τ19 = −2+23 19ι ,
τ17 = −25+
92
92
o
√
√
√
τ20 = 2+2319ι , τ21 = 21+92 19ι , τ22 = 25+92 19ι
n
o
√
√
√
√
22ι
τ23 = −22+
, τ24 = −1+23 22ι , τ25 = 1+2322ι , τ26 = 22+46 22ι
46
n
o
√
√
√
√
23ι
23ι
τ27 = −23+
, τ28 = 23+92 23ι , τ29 = −23+
∼ 23+46 23ι
92
46
n
o
√
√
√
23+ 23ι
23ι
23ι
τ30 = −23+
,
τ
=
,
τ
=
31
32
69
23
69
n
o
√
√
τ33 = −7+46 43ι , τ34 = 7+4643ι
n
o
√
√
τ35 = −5+46 67ι , τ36 = 5+4667ι
n
√
√
√
83ι
83ι
, τ38 = −43+
, τ39 = −3+46 83ι ,
τ37 = −49+
138
138
o
√
√
√
τ40 = 3+4683ι , τ41 = 43+13883ι , τ42 = 49+13883ι
n
o
√
√
√
√
91ι
τ43 = −47+
, τ44 = −1+46 91ι , τ45 = 1+4691ι , τ46 = 47+23091ι
230
n
Montserrat Alsina
20
4.2. Small ramified case of type A. Let us consider HA (p) := p,−1
Q
i
h
p−1
1+i+j+ij
and the Eichler order OA (2p, N ) := Z 1, i, N j,
, for N | 2 , N
2
α β
1
square-free. The elements in the group ΓA (2p, N ) are γ = 2 −β ′ α′ such
′
√
√
√
that α, β ∈ Z[ p], α ≡ β ≡ α p mod 2, det γ = 1, N | tr(β) − β−β
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
a+b
,
,
b
=
1.
forms is gcd c+b
2
2N
(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 + 2OA (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.
and
4.3. Small ramified case of type B. Consider HB (p, q) := p,q
Q
i
h
i+ij
the Eichler order OB (pq, N ) := Z 1, N i, 1+j
, where N | q−1
2 , 2
4 , N
square-free and gcd(N, p) = 1. Then the group of quaternion transformations is
n
ΓB (pq, N ) = γ =
1
2
α β
qβ ′ α′
√
: α, β ∈ Z[ p], α ≡ β
mod 2,
o
′ −β+β ′
N | α−α2√
,
det
γ
=
1
.
p
We denote by XB (pq, N ) the corresponding Shimura curve of type B.
Binary forms and Eichler orders
21
Figure 4.2. Reduced binary forms in a fundamental domain for XA (6, 1).
det1 (f )
n
3
2
4
2
24
2
40
4
f
√ 2
√
√
(3 + 3)x + 2 3xy + (3 − 3)y 2
√ 2
√
√
(3 + 3)x − 2 3xy + (3 − 3)y 2
√
4x2 + 4 3xy + 4y 2
2x2 + 2y 2
√ 2
√
(6 + 2 3)x − (−6 + 2 3)y 2
√
6x2 − 4 3xy + 6y 2
√ 2
√
√
(10 + 2 3)x + 8 3xy − (−10 + 2 3)y 2
√ 2
√
√ 2
(8 + 2 3)x + 4 3xy − (−8 + 2 3)y
√
√
√
(8 + 2 3)x2 − 4 3xy − (−8 + 2 3)y 2
√ 2
√
√
(10 + 2 3)x − 8 3xy − (−10 + 2 3)y 2
τ (f )
v2
v4
v1
v6
τ1
τ2
τ3
τ4
τ5
τ6
√
1− 3
(1 − ι)
2 √
−1+ 3
(1 − ι)
√2
− 3+ι
∼ v3 ∼
2
=
=
=
= ι√
=
=
=
=
=
=
v5
√
6− 2)ι
√ 2√
− 3+ 6ι
∼√ τ2′ √
3
√
3−5 3
30
+ 5√10−
ι
11√
22 √
3−4 3
4 10− 30
+ √22 √ ι
13 √
30
−3+4 3
+ 4√10−
ι
13 √
22 √
−3+5 3
5 10− 30
+
ι
11
22
(
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
=
1.
forms in (i) is gcd a, b, c−b
2N
(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
elliptic points of order 3; thus they are the associated points to binary
Montserrat Alsina
22
Figure 4.3. Reduced binary forms in a fundamental domain for XB (10, 1).
det1 (f )
n
3
4
8
2
20
2
40
2
√
2
f
√
τ (f )
√
2
(−5 + 5 2)x + 2 2xy + (1 + 2)y
√
√
√
(5 + 5 2)x2 + 2 2xy + (−1 + 2)y 2
√ 2
√
√
(35 + 25 2)x − 2 2xy + (−7 + 5 2)y 2
√
√
√
(5 + 5 2)x2 + 2 2xy + (−1 + 2)y 2
√ 2
√
√ 2
5 √2x + 2 √2xy + √2y
(5 2x2 + 2 2xy + 2y 2
√
√
(10 + 10 2)x2 + (−2 + 2 2)y 2
√ 2
√
(−10 + 10 2)x + (2 + 2 2)y 2
√ 2
√ 2
(40 + 30 2)x + (−8 + 6 2)y
√
√
10 2x2 + 2 2y 2
√
√
− 2+ √3ι
∼ w3
w1 = 5(−1+
√ 2)
√
2+√ 3ι
w2 = −5(1+
√
√ 2)
2+ √3ι
w4 = 5(7+5
∼ w6
√ 2)
√
2+ √3ι
w5 = 5(1+ 2)
τ1 = −1+2ι
5
τ2 = 1+2ι
√5
√
5)ι
τ3 = (√10−
5√
5)ι
τ4 = ( √10+
5 √
(3 5−2 10)ι
τ5 = √
5
τ6 = 55ι
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.
Binary forms and Eichler orders
23
References
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Monograph Series, vol. 22, American Mathematical Society, Providence, RI, 2004.
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[Eic55] M. Eichler, Zur Zahlentheorie der Quaternionen-Algebren. J. reine angew. Math. 195
(1955), 127–151.
[Ogg83] A. P. Ogg, Real points on Shimura curves. Arithmetic and geometry, Vol. I, Progr.
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auser Boston, Boston, MA, 1983, pp. 277–307.
[Shi67] G. Shimura, Construction of class fields and zeta functions of algebraic curves. Annals
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[Vig80] M.F. Vigneras, Arithm´
etique des alg`
ebres de quaternions. Lecture Notes in Math.,
no. 800, Springer, 1980.
Montserrat Alsina
Dept. Matem`
atica Aplicada III EUPM
Av. Bases de Manresa 61-73, Manresa-08240, Catalunya, Spain
E-mail : Montserrat.alsina@upc.edu