Hurwitz quaternion order and arithmetic Riemann surfaces

0.16 MB·English

by  Mikhail G. Katz

HURWITZ QUATERNION ORDER AND ARITHMETIC
0 RIEMANN SURFACES
1
0
2 MIKHAIL G. KATZ1, MARY SCHAPS, AND UZI VISHNE2

r
a
M Abstract. WeclarifytheexplicitstructureoftheHurwitzquater-
nionorder,whichisoffundamentalimportanceinRiemannsurface

1
theory and systolic geometry.
1

]
A
R
h. Contents
t
a
1. Congruence towers and the 4/3 bound 1
m
2. The Hurwitz order 3
[ Hur
Q
3. Maximality of the order 6
4 QHur
v 4. The (2,3,7) group inside 7
Hur
7 Q
5. Azumaya algebras 8
3
1 6. Quotients of 9
Hur
1 Q
References 12
0
7
0
/
h
t
a
1. Congruence towers and the 4/3 bound
m
: Hurwitz surfaces are an important and famous family of Riemann
v
i surfaces.
X
r We clarify the explicit structure of the Hurwitz quaternion order,
a
which is of fundamental importance in Riemann surface theory and
Date: March 12, 2010.
2000 Mathematics Subject Classification. 11R52; 53C23, 16K20.
Key words and phrases. arithmetic lattice, Azumaya algebras, Fuchsian group,
Hurwitz group, Hurwitz order, hyperbolic surface, hyperbolic reflection group,
quaternion algebra, subgroup growth, systole.
1SupportedbytheIsraelScienceFoundation(grantsno.84/03and1294/06)and
the Binational Science Foundation (grant 2006393).
2Supported by the EU research and training network HPRN-CT-2002-00287,
ISF Center of Excellence grant 1405/05,and BSF grant no. 2004-083.
1
2 M. KATZ,M. SCHAPS,AND U.VISHNE
systolic geometry.1 A Hurwitz surface X by definition attains the up-
per bound of 84(g 1) for the order Aut(X) of the holomorphic
X − | |
automorphism group of X, where g is its genus. In [11], we proved a
X
systolic bound
sysπ (X) 4 log(g ) (1.1)
1 ≥ 3 X
for Hurwitz surfaces in a principal congruence tower (see below). Here
the systole sysπ is the least length of a noncontractible loop in X.
1
The question of the existence of other congruence towers of Riemann
surfaces satisfying the bound (1.1), remains open.
Marcel Berger’s monograph [2, pp. 325–353] contains a detailed
exposition of the state of systolic affairs up to ’03. More recent devel-
opments are covered in [10].
While (1.1) can be thought of as a differential-geometric application
ofquaternion algebras,suchanapplicationofLie algebras maybefound
in [1].
We will give a detailed description of a specific quaternion algebra
order, which constitutes the arithmetic backbone of Hurwitz surfaces.
The existence of a quaternion algebra presentation for Hurwitz surfaces
is due to G. Shimura [16, p. 83]. An explicit order was briefly described
by N. Elkies in [6] and in [7], with a slight discrepancy between the two
descriptions, see Remark 2.3 below. We have been unable to locate a
more detailed account of this important order in the literature. The
purpose of this note is to provide such an account.
A Hurwitz group is by definition a (finite) group occurring as the
(holomorphic) automorphism groupofa Hurwitz surface. Such a group
is the quotient ∆ /Γ of a pair of Fuchsian groups. Here ∆ is
2,3,7 2,3,7
the (2,3,7) triangle group, while Γ(cid:1)∆ is a normal subgroup.
2,3,7
Let η = 2cos 2π and K = Q[η], a cubic extension of Q. A class of
7
Hurwitz groups arise from ideals in the Hurwitz quaternion order
(η,η) ,
Hur K
Q ⊂
see Definition 2.1 for details.
Recall that for a division algebra D over a number field k, the dis-
criminant disc(D) is the product of the finite ramification places of D.
Let be an order of D, and let O be the ring of algebraic integers
Q K
1In the literature, the term “Hurwitz quaternion order” has been used both in
the sense used in the presenttext, andin the sense of the unique maximalorder of
Hamilton’s rational quaternions.
HURWITZ QUATERNION ORDER AND ARITHMETIC RIEMANN SURFACES3
in K. By definition 1 is the group of elements of norm 1 in , and a
Q Q
principal congruence subgroup of 1 is a subgroup of the form
Q
1(I) = x 1: x 1 (mod I ) (1.2)
Q { ∈ Q ≡ Q }
where I(cid:1)O . Any subgroup containing such a subgroup is called a
K
congruence subgroup.
TheHurwitzorderisdescribedinSection2. InSection3weverifyits
maximality. Thepreciserelationshipbetween theorderandthe(2,3,7)
groupisgiven inSection 4. In Section 5 we notethat the Hurwitz order
is Azumaya, which implies that every ideal of the order is generated by
acentral element, andevery automorphismisinner. It alsofollows that
all quotient rings are matrix rings, and in Section 6 we present some
explicit examples of these quotients and their associated congruence
subgroups.
2. The Hurwitz order
Hur
Q
Let 2 denote the hyperbolic plane. Let Aut( 2) = PSL (R) be its
2
H H
groupoforientation-preservingisometries. Considerthelattice∆
2,3,7
⊂
Aut( 2), defined as the even part of the group of reflections in the
H
sides of the (2,3,7) hyperbolic triangle, i.e. geodesic triangle with an-
gles π, π, and π. We follow the concrete realization of ∆ in terms of
2 3 7 2,3,7
the group of elements of norm one in an order of a quaternion algebra,
given by N. Elkies in [6, p. 39] and in [7, Subsection 4.4].
Let K denote the real subfield of Q[ρ], where ρ is a primitive 7th root
of unity. Thus K = Q[η], where the element η = ρ+ρ−1 satisfies the
relation
η3 +η2 2η 1 = 0. (2.1)
− −
Note the resulting identity
(2 η)3 = 7(η 1)2. (2.2)
− −
There are three embeddings of K into R, defined by sending η to any
of the three real roots of (2.1), namely
2cos 2π ,2cos 4π ,2cos 6π .
7 7 7
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
We view the first embedding as the ‘natural’ one K֒ R, and denote
→
the others by σ ,σ :K R. Notice that 2cos(2π/7) is a positive root,
1 2
→
while the other two are negative.
4 M. KATZ,M. SCHAPS,AND U.VISHNE
From the minimal polynomial we have Tr (η) = 1. Multiplying
K/F −
(2.1) by the ‘conjugate’ η3 η2 2η+1 gives
− −
(η2)3 5(η2)2 +6(η2) 1 = 0,
− −
so similarly Tr (η2) = 5. The recursion relation
K/F
Tr(η3+i) = Tr(η2+i)+2Tr(η1+i)+Tr(ηi)
−
provides Tr(η3) = 4 and Tr(η4) = 13. Traces of multiples in the
−
integral basis 1,η,η2 then give the discriminant
3 1 5
−
disc(K/Q) = (cid:12) 1 5 4 (cid:12) = 49.
(cid:12) − − (cid:12)
(cid:12) 5 4 13 (cid:12)
(cid:12) − (cid:12)
(cid:12) (cid:12)
By Minkowski’s bound [9, Su(cid:12)bsection 30.3(cid:12).3], it follows that every
ideal class contains an ideal of norm < 3!√49 < 2, which proves
33
that O = Z[η] is a principal ideal domain. The only ramified prime
K
is7O = 2 η 3, cf. (2.2)andusing thefactthatη 1 = (η2+2η)−1 is
K h − i −
invertibleinO . Notethattheminimalpolynomialf(t) = t3+t2 2t 1
K − −
remains irreducible modulo 2, so that
O / 2 = F [η¯] = F , (2.3)
K h i ∼ 2 8
the field with 8 elements.
Definition 2.1. We let D be the quaternion K-algebra
(η,η) = K[i,j i2 = j2 = η, ji = ij]. (2.4)
K
| −
As mentioned above, the root η > 0 defines the natural imbedding
of K into R, and so D R = M (R), and thus the imbedding is
2
⊗
unramified. On the other hand, we have σ (η),σ (η) < 0, so the alge-
1 2
bras D R and D R are isomorphic to the standard Hamilton
⊗σ1 ⊗σ2
quaternion algebra over R.
Moreover,DisunramifiedoverallthefiniteplacesofK [11,Prop.7.1].
Remark 2.2. By the Albert-Brauer-Hasse-Noether theorem [14, The-
orem 32.11], D is the only quaternion algebra over K with this ramifi-
cation data.
Let D be the order defined by
O ⊆
= O [i,j].
O K
HURWITZ QUATERNION ORDER AND ARITHMETIC RIEMANN SURFACES5
Clearly, the defining relations of D serve as defining relations for as
O
well:
= Z[η][i,j i2 = j2 = η, ji = ij]. (2.5)
∼
O | −
Fix the element τ = 1 + η + η2, and define an element j′ D by
∈
setting
1
′
j = (1+ηi+τj).
2
′
Notice that j is an algebraic integer of D, since the reduced trace
is 1, while the reduced norm is
1
(1 η η2 η τ2 +η2 0) = 1 3η,
4 − · − · · − −
so that both are in O . In particular, we have the relation
K
′2 ′
j = j +(1+3η). (2.6)
We define an order D by setting
Elk
Q ⊂
′
= Z[η][i,j ]. (2.7)
Elk
Q
Finally, we define a new order D by setting
Hur
Q ⊂
′
= Z[η][i,j,j ]. (2.8)
Hur
Q
Remark 2.3. There is a discrepancy between the descriptions of a
maximal order of D in [6, p. 39] and in [7, Subsection 4.4]. According
to [6, p. 39], Z[η][i,j,j′] is a maximal order. Meanwhile, in [7, Subsec-
tion4.4], themaximalorderisclaimedtobetheorder = Z[η][i,j′],
Elk
Q
described as Z[η]-linear combinations of the elements 1,i,j′, and ij′,
on the last line of [7, p. 94]. The correct answer is the former, i.e. the
order (2.8).
We correct this minor error in [7], as follows.
Lemma 2.4. The order strictly contains = Z[η][i,j′].
Hur Elk
Q Q
Proof. The identities (2.6) and
j′i = η2 +i ij′ (2.9)
−
show that the module
′ ′ ′
= Z[η]+Z[η]i+Z[η]j +Z[η]ij , (2.10)
Elk
Q
6 M. KATZ,M. SCHAPS,AND U.VISHNE
which is clearly contained in , is closed under multiplication, and
Elk
Q
′ ′
thusequalto . Moreover, theset 1,i,j ,ij isabasisofD overK,
Elk
Q { }
and a computation shows that
9+2η +3η2 3 3η η2 18 4η 6η2
′
j = − + − − i+ − − j ,
7 7 7
with non-integral coefficients. Therefore j . (cid:3)
Elk
6∈ Q
3. Maximality of the order
Hur
Q
Proposition 3.1. The order is a maximal order of D.
Hur
Q
Proof. By Lemma 2.4 it is enough to show that every order contain-
ing is contained in . Let M be an order, namely a
Elk Hur Elk
Q Q ⊇ Q
ring which is finite as a O -module, and let x M. Since 1,i,j,ij
K ∈ { }
is a K-basis for the algebra D, we can write
x = 1(a+bi+cj +dij)
2
forsuitablea,b,c,d K. Recall thatevery element ofanorder satisfies
∈
amonicpolynomialover O , soinparticularithasintegraltrace. Since
K
we have x,ix,jx,ijx M, with traces a,ηb,ηc, η2d, respectively,
∈ −
while the element η = (η2 +η 2)−1 is invertible in O , we conclude
− K
that, in fact, a,b,c,d O . Now,
∈ K
1 1
tr(xj′) = tr((a+bi+cj +dij)(1+ηi+τj)) = (a+η2b+ητc)
4 2
and
1 1
tr(xij′) = tr((a+bi+cj+dij)i(1+ηi+τj)) = (η2a+ηb η2τd).
4 2 −
Since these are integers, and since ητ η + 1 and η3τ 1 modulo
≡ ≡
2O , we have that a η2b+(η +1)c, and d η3a+η2b τb+ηc.
K ≡ ≡ ≡
It then follows that
x (η2 +2η +1)cj′ ((η2 +3η+1)c+b)ij′ O [i,j],
− − ∈ K
so that x . (cid:3)
Hur
∈ Q
Remark 3.2. Since K = D, the center of is
Hur Hur
Q Q
Cent( ) = Cent(D) = K = O .
QHur QHur ∩ QHur ∩ K
While admits the presentation (2.5), typical of symbol algebras,
O
it should be remarked that cannot have such a presentation.
Hur
Q
HURWITZ QUATERNION ORDER AND ARITHMETIC RIEMANN SURFACES7
Remark 3.3. There is no pair of anticommuting generators of
Hur
Q
over Z[η].
Proof. One can compute that /2 is a 2 2 matrix algebra [11,
Hur Hur
Q Q ×
Lemma 4.3], and in particular non-commutative; however anticommut-
(cid:3)
ing generators will commute modulo 2.
The prime 2 poses the only obstruction to the existence of an anti-
commuting pair of generators. Indeed, adjoining the fraction 1, we
2
clearly have
[1] = [1] = O [1][i,j i2 = j2 = η, ji = ij],
QHur 2 O 2 K 2 | −
and this is an Azumaya algebra over O [1], see Definition 5.1. A pre-
K 2
sentation of is given in Lemma 4.2.
Hur
Q
4. The (2,3,7) group inside
Hur
Q
The group of elements of norm 1 in the order , modulo the
Hur
Q
center 1 , is isomorphic to the (2,3,7) group [7, p. 95]. Indeed,
{± }
Elkies gives the elements
1
g = ij,
2
η
1
g = (1+(η2 2)j +(3 η2)ij),
3
2 − −
1
g = ((τ 2)+(2 η2)i+(τ 3)ij),
7
2 − − −
satisfying the relations g2 = g3 = g7 = 1 and g = g g , which
2 3 7 − 2 7 3
therefore project to generators of ∆ PSL (R).
2,3,7 2
⊂
Lemma 4.1. The Hurwitz order is generated, as an order, by the ele-
ments g and g , so that we can write = O [g ,g ].
2 3 QHur K 2 3
Proof. We have g ,g ,g by the invertibility of η in O and the
2 3 7 ∈ QHur K
equalities
g = (3+6η η2)+(1+3η)i (2+η2)jj′ 2(ijj′ (1 η)ij),
3
− − − − −
g = (τ +3η)+2(1+η)i (η +η2)jj′ +(2 τ)(ijj′ (1 η)ij).
7
− − − −
Conversely, we have the relations
i = (1+η)(g g g g ),
3 2 2 3
−
j = (1+η)(1+(η2 +η 1)g 2g ),
2 3
− −
j′ = (1+ηi)g +(η2 2)ij +j,
3
−
8 M. KATZ,M. SCHAPS,AND U.VISHNE
(cid:3)
proving the lemma.
Lemma 4.2. A basis for the order as a free module over Z[η] is
Hur
Q
given by the four elements 1, g , g , and g g .
2 3 2 3
The defining relations g2 = 1, g2 = g 1 and g g + g g =
2 3 3 2 3 3 2
− −
g (η2 +η 1) provide a presentation of as an O -order.
2 − − QHur K
Proof. The module spanned by 1,g ,g ,g g is closed under multiplica-
2 3 2 3
tion by the relations given in the statement (which are easily verified);
thus O [g ,g ] = span 1,g ,g ,g g . The relations suffice since
K 2 3 OK{ 2 3 2 3}
the ring they define is a free module of rank 4, which clearly project
(cid:3)
onto .
Hur
Q
Remark 4.3. An alternative basis for the order as a free module
Hur
Q
over Z[η] is given by the four elements 1, i, jj′, and ℓ = ijj′ (1 η)ij.
− −
Remark 4.4. Since is a free module of rank 4 over O , so is 1 ,
O K 2O
and 1 / is a 4-dimensional vector space over O /2O , which is the
2O O K K
field of order 8. Furthermore, one can check that / is a two-
Hur
Q O
dimensional subspace, namely [1 : ] = [ : ] = 26 where we
2O QHur QHur O
are calculating the indices of the orders as abelian groups .
5. Azumaya algebras
We briefly describe a useful generalization of the class of central
simple algebras over fields, to algebras over commutative rings.
Definition 5.1 (e.g. [15, Chapter 2]). Let R be a commutative ring.
Let A be an R-algebra which is a faithful finitely generated projective R-
module. If the natural map A Aop End (A) (action by left and
R R
⊗ →
right multiplication) is an isomorphism, then A is an Azumaya algebra
over R.
Suppose every non-zero prime ideal of R is maximal (which is the
casewitheveryDedekinddomain, suchasO = Z[η]), andletF denote
K
the ring of fractions of R. It is known that if A is an R-algebra, which
is a finite module, such that
(1) for every maximal ideal M(cid:1)R, A/MA is a central simple alge-
bra, of fixed degree, over R/MR; and
(2) A F is central simple, of the same degree, over F,
R
⊗
then A is Azumaya over R [15, Theorem 2.2.a]. The second condition
clearly holds for over O since K = D.
QHur K QHur⊗OK ∼
HURWITZ QUATERNION ORDER AND ARITHMETIC RIEMANN SURFACES9
In [11, Lemma 4.3] we proved the following theorem.
Theorem 5.2. For every ideal I(cid:1)O , we have an isomorphism
K
/I = M (O /I).
QHur QHur ∼ 2 K
This was proved in [11] for an arbitrary maximal order in a division
algebra with no finite ramification places, by decomposing I as a prod-
uct of prime power ideals, applying the isomorphism /pt = /pt
p p
Q Q Q
[14, Section 5] for = ( being the completion with respect to
Hur p
Q Q Q
the p-adic valuation), and using the structure of maximal orders over
a local ring [14, Section 17].
We therefore obtain the following corollary.
Corollary 5.3. The order is an Azumaya algebra.
Hur
Q
This fact has various ring-theoretic consequences. In particular,
there is a one-to-one correspondence between two-sided ideals of
Hur
Q
and ideals of its center, O [15, Proposition 2.5.b]. Since O is a
K K
principal ideal domain, if follows that every two-sided ideal of is
Hur
Q
generated by a single central element.
Another property of Azumaya algebras is that every automorphism
is inner (namely, induced by conjugation by an invertible element) [15,
Theorem 2.10], in the spirit of the Skolem-Noether theorem, cf. [13,
p. 107].
6. Quotients of
Hur
Q
In order to make Theorem 5.2 explicit, suppose I(cid:1)O is an odd
K
ideal (namely I +2O = O ). By the inclusion 2 , we have
K K QHur ⊆ O
that +I = and I = I . Therefore
Hur Hur Hur
O Q Q O∩ Q O
/I = /( I ) = (I + )/I = /I ,
Hur ∼ Hur Hur Hur Hur
O O O O∩ Q Q O Q Q Q
and so /I = M (L) for L = O /I. From the presentation of , see
O O ∼ 2 K O
(2.5), it follows that
/I = L[i,j i2 = j2 = η,ji = ij],
∼
O O | −
which allows for an explicit isomorphism /I M (L).
2
O O→
Example 6.1 (First Hurwitz triplet). The quotient /13 can
Hur Hur
Q Q
be analyzed as follows. Since the minimal polynomial λ3+λ2 2λ 1
− −
10 M. KATZ,M. SCHAPS,AND U.VISHNE
factors over F as (λ 7)(λ 8)(λ 10), we obtain the ideal decom-
13
− − −
position
13O = 13,η 7 13,η 8 13,η 10 , (6.1)
K h − ih − ih − i
andtheisomorphismO / 13 F F F ,definedbyη (7,8,10).
K h i→ 13× 13× 13 7→
In fact, one has
13 = η(η+2)(2η 1)(3 2η)(η+3),
− −
where η(η + 2) is invertible, and the other factors generate the ideals
given above, in the respective order; therefore, (6.1) can be rewritten
as
13O = (2η 1)O (3 2η)O (η+3)O .
K − K · − K · K
An embedding O [i]/ 13 ֒ M (F ) M (F ) M (F ) is obtained
K h i → 2 13 × 2 13 × 2 13
by mapping the generator i via
0 1 0 1 0 1
i , , ,
7→ (cid:18)(cid:18)7 0(cid:19) (cid:18)8 0(cid:19) (cid:18)10 0(cid:19)(cid:19)
satisfying the defining relation i2 = η. In order to extend this em-
bedding to /13 , we need to find in each case a matrix which
Hur Hur
Q Q
anti-commutes with i, andwhose square isη. Namely, we seek a matrix
a b
,
(cid:18) ηb a(cid:19)
− −
such that a2 ηb2 = η (η stands for 7,8 or 10, respectively). Solving
−
this equation in each case, the map
/13 M (F ) M (F ) M (F )
Hur Hur 2 13 2 13 2 13
Q Q → × ×
may then be defined as follows:
1 1 4 1 6 0
j , , .
7→ (cid:18)(cid:18)6 12(cid:19) (cid:18)5 9(cid:19) (cid:18)0 7(cid:19)(cid:19)
The map is obviously onto each of the components, and thus, by the
Chinese remainder theorem, onto on the product.
The three prime ideals define a triplet of principal congruence sub-
groups, as in (1.2). One therefore obtains a triplet of distinct Hurwitz
surfaces of genus 14. All three differ both in the value of the systole
and in the number of systolic loops [17].