Journal de Théorie des Nombres
de Bordeaux 17 (2005), 13–23
Binary quadratic forms and Eichler orders
par Montserrat ALSINA
Résumé. Pour tout ordre d’Eichler O(D, N ) de niveau N dans
une algèbre de quaternions indéfinie de discriminant D, il existe
un groupe Fuchsien Γ(D, N ) ⊆ SL(2, R) et une courbe de Shimura
X(D, N ). Nous associons à 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 à Γ(D, N ). En particulier nous retrouvons la classification des formes quadratiques
primitives et entières de SL(2, Z). Un domaine fondamental explicite pour Γ(D, N ) permet de caractériser les Γ(D, N ) formes
réduites.
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 =
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.
i2
a, j 2
The author was partially supported by MCYT BFM2000-0627.
Montserrat Alsina
14
Lemma 1.1. Let H = a,b
be an indefinite quaternion algebra with a > 0.
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
νp (D, N, d, m; O∗ ).
p|DN
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 r = 2k + 1, then νp (D,
N,d, m; O ) is equal to 2ψp (m)
if
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 0 , 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 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)
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
τ (f )
√
√ o
7ι
τ1 = −19+
, τ2 = 19+46 7ι
46
n
√
√ o
7ι
4+ 7ι
τ3 = −4+
,
τ
=
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
τ17 = −25+
, τ18 = −21+
, τ19 = −2+23 19ι ,
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
√
√
√
23ι
23ι
23+ 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
τ37 = −49+
, τ38 = −43+
, τ39 = −3+46 83ι ,
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
h
i
1+i+j+ij
p−1
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 0 α0 such
0
√
√
√
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
forms is gcd c+b
,
,
b
=
1.
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 Poincaré 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
h
i
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 0 α0
√
: α, β ∈ Z[ p], α ≡ β
mod 2,
o
0 −β+β 0
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 )
√
1− 3
(1 − ι)
2 √
−1+ 3
(1 − ι)
√2
− 3+ι
∼ v3 ∼
2
v2
v4
v1
v6
=
=
=
= ι√
τ1
τ2
τ3
τ4
τ5
τ6
=
=
=
=
=
=
v5
√
6− 2)ι
√ 2√
− 3+ 6ι
∼√ τ20 √
3
√
3−5 3
30
+ 5√10−
ι
11√
22 √
4 10− 30
3−4 3
+ √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
forms in (i) is gcd a, b, c−b
=
1.
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
f
√ 2
√
√
(−5 + 5 2)x + 2 2xy + (1 + 2)y 2
√
√
√
(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
√ 2
(−10 + 10 2)x + (2 + 2 2)y
√
√
(40 + 30 2)x2 + (−8 + 6 2)y 2
√ 2
√ 2
10 2x + 2 2y
τ (f )
√
√
− 2+ √3ι
w1 = 5(−1+
∼ w3
√
√ 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, Birkhäuser 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, Arithmétique des algèbres de quaternions. Lecture Notes in Math.,
no. 800, Springer, 1980.
Montserrat Alsina
Dept. Matemàtica Aplicada III EUPM
Av. Bases de Manresa 61-73, Manresa-08240, Catalunya, Spain
E-mail : Montserrat.alsina@upc.edu