Arithmetic structures on noncommutative tori with real multiplication

0.37 MB·English

by  Jorge Plazas

ARITHMETIC STRUCTURES ON NONCOMMUTATIVE TORI
WITH REAL MULTIPLICATION
7
0
0 JORGEPLAZAS
2

n
a
J
3 Abstract. Westudythehomogeneouscoordinateringsofrealmultiplication
noncommutative tori as defined in [23]. Our aim is to understand how these
] rings give rise to an arithmetic structure on the noncommutative torus. We
A startbygivinganexplicitpresentationoftheseringsintermsoftheirnatural
generators. Thispresentationgivesusinformationabouttherationalityofthe
Q
ringsandisthestartingpointfortheanalysisofthecorrespondinggeometric
. data. Linear bases for these rings are also discussed. The basic role played
h
bythetafunctionsinthesecomputationsgivesrelationswithellipticcurvesat
t
a the level ofmodular curves. We usethe modularityof these coordinate rings
m to obtain algebras which do no depend on the choice of a complex structure
on the noncommutative torus. These algebras are obtained by an averaging
[
processover(limiting)modularsymbols.

2
v 0. Introduction
7
2 Noncommutative tori are standardprototypes of noncommutative spaces. Since
1
the early stages of noncommutative geometry these spaces have been central ex-
0
amples arising naturally in various context. In [14] Manin proposed the use of
1
6 noncommutative tori as a geometric framework for the study of abelian class field
0 theory of real quadratic fields. This is the so called “real multiplication program”.
/ Themainideaisthatnoncommutativetorimayplayaroleinthestudyofrealqua-
h
t draticfieldsanalogoustotheroleplayedbyellipticcurvesinthestudyofimaginary
a
quadraticfields. The relationbetweenthe endomorphismringsofnoncommutative
m
tori and orders in real quadratic fields is a good evidence supporting this point of
:
v view. If a noncommutative torus admits nontrivial Morita autoequivalences then
i the ring of such autoequivalences is an order in a real quadratic field. A noncom-
X
mutativetorus havingthis propertyis calledarealmultiplicationnoncommutative
r
torus.
a
The fact that noncommutative geometry may be relevant for addressing ques-
tions in number theory is also supported by variousresults and relationsthat have
emergedin the last years. In particular explicit class field theory of Q (Kronecker-
Weber theorem) and explicit class field theory of imaginary quadratic fields (com-
plex multiplication) can both be recovered from the dynamics of certain quantum
statistical mechanical systems ([2, 6, 7]). The existence of quantum statistical me-
chanical systems with rich arithmetical properties opens a new approach to the
studyofexplicitclassfieldtheoryusingthe toolsofquantumstatisticalmechanics.
Thefirstcaseforwhichthereisnotyetacompletesolutiontotheexplicitclassfield
2000 Mathematics Subject Classification. Primary81R60;Secondary16S37.
Key words and phrases. noncommutative tori with real multiplication, modular forms, theta
constants.
1
2 JORGEPLAZAS
theory problem is the case of real quadratic fields, K =Q(√D), where D N+ is
∈
a square free positive integer.
Noncommutative tori are, a priori, analytical objects. In order to achieve arith-
metical applications it is important to find appropriate algebraicstructures under-
lying these spaces. Rings admitting models algebraic over the corresponding base
fieldshaveprovedtobe essentialforthe analysisofquantumstatisticalmechanical
systems of arithmetic nature.
In [23] Polishchuk defined homogeneous coordinate rings for real multiplication
noncommutative tori endowed with a complex structure. These rings seem to be
goodcandidates for the applications described above. Our aim is to understand in
which sense these rings provide an arithmetic structure on noncommutative tori.
Starting from the explicit formulae defining these rings we study their rationality
properties and their dependence on the complex structure. Some examples related
to particular quadratic fields are treated.
Homogeneous coordinate rings for real multiplication noncommutative tori can
be viewed as a family of algebrasvarying with the parameter defining the complex
structure on the noncommutative torus. We give a presentation of these rings
in terms of modular forms and use this modularity to define rings which do not
depend on the choice of a complex structure. This is done by an averagingprocess
over (limiting) modular symbols. The fact that modular forms play an essential
role in our analysis points to deep relations with the quantum thermodynamical
systemintroducedbyConnesandMarcolliin[5]. Thisquantummechanicalsystem
recovers the class field theory of the modular field and provides a two dimensional
analog of the dynamical system corresponding to the Kronecker-Weber theorem.
Inthecaseofrealquadraticfieldsexplicitclassfieldtheoryisconjecturallygiven
in terms of special values of L-functions, this is the content of Stark’s famous con-
jectures[31]. Inordertoapplyourresultsonnoncommutativetoriinthisdirection
usingthetoolsofquantumstatisticalmechanicswestillneedtofindC -completions
∗
for the homogeneous coordinate rings of real multiplication noncommutative tori.
As a preliminary step in this direction we describe the geometric data associated
to the homogeneous coordinate rings of noncommutative tori. We expect that at
a future stage the use of the techniques developed in [4] will make it possible to
obtain suitable C -completions. In a different but related perspective arithmetic
∗
structures on noncommutative tori have been recently studied by Vlasenko in [32]
where the theory of rings of quantum theta functions is developed.
IwanttothankMatildeMarcolliforhercontinuedguidancethroughthisproject.
I am also grateful many people at MPI and UniBonn for very helpful discussions.
Special thanks go to NikolaiDurov,SnigdhayanMahanta and Eugene Ha. I thank
the Max-Planck-Institut fu¨r Mathematik for its support and hospitality.
1. Basic setting
In this section we recall the basic facts about noncommutative tori that will be
neededinthesequel. Also,werecallthedefinitionofhomogeneouscoordinaterings
of real multiplication noncommutative tori given in [23].
Noncommutativetoriaredefinedbytheiralgebrasofsmoothfunctionswhichare
noncommutative deformations of C (T), the algebra of smooth functions on the
∞
twodimensionaltorusT=S1 S1. Givenθ Rwedefine ,thealgebraofsmooth
θ
× ∈ A
functions on the noncommutative torus T ,as the algebraofformalpowerseriesin
θ
ARITHMETIC STRUCTURES ON RM NONCOMMUTATIVE TORI 3
twounitariesU andV withrapidlydecreasingcoefficientsandmultiplicationgiven
by the relation UV =e2πiθVU:
θ = C∞(Tθ)
A
= a= a UnVm a (Z2)
n,m n,m
{ |{ }∈S }
n,m Z
X∈
Let θ R be irrational. Following [3] we define, for any pair c,d Z, c > 0, a
∈ ∈
right -module E (θ) givenby the following actionof onthe Schwartzspace
θ d,c θ
A A
(R Z/cZ)= (R)c:
S × S
cθ+d
(fU)(x,α) = f(x ,α 1) (1.1)
− c −
αd
(fV)(x,α) = exp(2πi(x ))f(x,α) (1.2)
− c
Theright -moduleE (θ)isprojectiveandoffinitetype. Ifcanddarerelatively
θ d,c
A
prime we say that E (θ) is a basic -module. Being this the case the pair d,c
d,c θ
A
can be completed to a matrix
a b
g = SL (Z) (1.3)
c d ∈ 2
(cid:18) (cid:19)
In this case we write E (θ) for the module E (θ). By definition the degree of
g d,c
E (θ) is taken to be c. We also define the degree of a matrix g SL (Z), given
g 2
∈
as above, by deg(g) = c. The study of the differential geometry of T based on
θ
canonical derivations on and the corresponding compatible connections on the
θ
A
above modules carried out in [3] and further developed in [8] is the starting point
of the more recent algebro-geometricsetup we describe below.
Let SL (Z) actonR by fractionallineartransformations. Letg SL (Z) be as
2 2
∈
above and denote by U and V two generating unitaries of the algebra . We
′ ′ gθ
A
can define a left action of the algebra on E by:
gθ g
A
1
(U f)(x,α) = f x ,α a (1.4)
′
− c −
(cid:18) (cid:19)
x α
(V f)(x,α) = exp(2πi( ))f(x,α) (1.5)
′
cθ+d − c
This action gives an identification:
End (E (θ)) . (1.6)
Aθ g ≃Agθ
Note in particularthat ifgθ =θ then E (θ) has the structure ofa -bimodule.
g θ
A
Therefore in this case the algebra has nontrivial Morita auto equivalences. An
θ
A
irrational number θ R Q is a fixed point of a fractional linear transformation
∈ \
g SL (Z) if and only if it generates a quadratic extension of Q. In this case we
2
∈
say that the noncommutative torus T with algebra of smooth functions is a
θ θ
A
real multiplication noncommutative torus.
The tensor product of basic modules is again a basic module (c.f. [8, 25, 9]).
Given g ,g SL (Z), there is a well defined pairing of right -modules:
1 2 2 θ
∈ A
t :E (g θ) E (θ) E (θ) (1.7)
g1,g2 g1 2 ⊗C g2 → g1g2
This map gives rise to an isomorphism of bimodules:
Ag1g2θ−Aθ
E (g θ) E (θ) E (θ). (1.8)
g1 2 ⊗Ag2θ g2 → g1g2
4 JORGEPLAZAS
In particular, if gθ =θ one has an isomorphism
Eg(θ)⊗Aθ ···⊗Aθ Eg(θ)≃Egn(θ). (1.9)
n
Given τ C with Im(τ) < 0 we endow the noncommutative torus with a
∈ | {z }
complex structure given by the derivation δ : :
τ θ θ
A →A
δ : a UnVm (2πı) (nτ +m)a UnVm (1.10)
τ n,m n,m
7→
n,m Z n,m Z
X∈ X∈
This derivation should be viewed as an analog of the operator ∂¯ on a complex
elliptic curve. We will denote by T the noncommutative torus T equipped with
θ,τ θ
this complex structure.
A holomorphic structure on a right -module E is given by an operator ¯ :
θ
A ∇
E E which is compatible with the complex structure δ in the sense that it
τ
→
satisfies the following Leibniz rule:
¯(ea)= ¯(e)a+eδ (a), e E,a (1.11)
τ θ
∇ ∇ ∈ ∈A
Givena holomorphic structure ¯ ona right -module E the correspondingset of
θ
holomorphic sections is the spac∇e H0(Tθ,τ,EA¯):=Ker(¯).
On every basic module E one can defin∇e a family∇of holomorphic structures
d,c
¯ depending on a complex parameter z C:
z
{∇ } ∈
∂f dτ
¯ (f)= +2πi x+z f. (1.12)
z
∇ ∂x cθ+d
(cid:18) (cid:19)
By definition a standard holomorphic vector bundle on T is given by a basic
θ,τ
module E =E together with one of the holomorphic structures ¯ .
d,c g z
∇
Thespacesofholomorphicsectionsofastandardholomorphicvectorbundleson
T arefinitedimensional(c.f. [25],Section2). Ifcθ+d>0thendimH0(E , ¯ )=
θ,τ g 0
∇
c. Onwhatfollowswewill considerthe spacesofholomorphicsectionscorrespond-
ing to ¯ :
0
∇
Hg :=H0(Tθ,τ,Eg,∇¯0). (1.13)
A basis of is given by the Schwartz functions:
g
H
cτ x2
ϕ (x,β)=exp( )δβ α=1,...,c. (1.14)
α −cθ+d 2 α
The tensor product of holomorphic sections is again holomorphic. Using the
above basis the product can be written in terms of the corresponding structure
constants.
Theorem 1.1. ([25] Section2) Suppose g and g have positive degree. Then g g
1 2 1 2
has positive degree and t induces a well defined linear map
g1,g2
t : (g θ) (θ) (θ). (1.15)
g1,g2 Hg1 2 ⊗CHg2 →Hg1g2
Let g ,g and g g be given by
1 2 1 2
a b a b a b
g = 1 1 , g = 2 2 , g g = 12 12
1 c d 2 c d 1 2 c d
1 1 2 2 12 12
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
′
and let ϕ , ϕ and ψ be respectively the basis of (g θ), (θ) and
{ α} { β} { γ} Hg1 2 Hg2
(θ) as given in 1.14. Then
Hg1g2
t :ϕ ϕ′ Cγ ψ (1.16)
g1,g2 α⊗ β 7→ α,β γ
ARITHMETIC STRUCTURES ON RM NONCOMMUTATIVE TORI 5
where for α=1,...,c , β =1,...,c and γ =1,...,c we have:
1 2 12
τm2
Cγ = exp[πı − ] (1.17)
α,β 2c c c
1 2 12
m∈Ig1X,g2(α,β,γ)
with
I (α,β,γ)= n Z n c γ+c α mod c c ,
g1,g2 { ∈ | ≡− 1 12 12 1
n c d γ c d β mod c c .
2 12 12 2 12 2
≡ − }
Notation 1.2. Throughout we use the convention of summing over repeated in-
dexes.
Assume now that θ R is a quadratic irrationality. So there exist some g
∈ ∈
SL (Z) with gθ = θ and T has real multiplication. Fix a complex structure τ
2 θ
on T . In the case E = E (θ) we can extend a holomorphic structure on E to
θ g g
a holomorphic structure on the tensor powers E n. Following [23] we define a
g⊗
homogeneous coordinate ring for T by
θ,τ
Bg(θ,τ) = H0(Tθ,τ,E⊗¯n) (1.18)
n 0 ∇0
M≥
=
gn
H
n 0
M≥
One of the main results in [23] characterizes some structural properties of these
algebras in terms of the matrix elements of g:
Theorem1.3. ([23]Theorem3.5)Assumeg SL (Z)haspositiverealeigenvalues
2
∈
If c a+d then B (θ,τ) is generated over C by .
g g
• ≥ H
If c a+d+1 then B (θ,τ) is a quadratic algebra.
g
• ≥
If c a+d+2 then B (θ,τ) is a Koszul algebra.
g
• ≥
We will fix some notations and conventions for the rest of the paper. As above
g will denote a matrix
a b
g = SL (Z)
c d ∈ 2
(cid:18) (cid:19)
We will always assume that the following inequalities hold
Tr(g) =a+d>2 (g is hyperbolic) (1.19)
c Tr(g)+2 =a+d+2 (1.20)
≥
The first inequality implies that g has positive real eigenvalues. By Theorem 1.3
the secondinequality implies that B (θ,τ) is a quadraticKoszulalgebragenerated
g
in degree 1.
We denote by λ+ and λ the eigenvalues of g with 0 < λ+ < 1 and 1 < λ . It
− −
is important to note that λ is a fundamental unit for the quadratic extension it
−
generates. We also take
λ+ d λ d
−
θ = − , θ′ = − (1.21)
c c
These are the fixed points of g.
Proposition 1.4. Let α R be a quadratic irrationality. Then there exist g and
∈
θ satisfying 1.19 and 1.20 above and such that Q(α)=Q(θ).
6 JORGEPLAZAS
Proof. Suppose α R is a quadratic irrationality. Then there exist a hyperbolic
∈
element
a b
h= ′ ′ SL (Z)
c′ d′ ∈ 2
(cid:18) (cid:19)
having α as one of its two fixed points. Being a fixed point of h, α satisfies the
quadraticequationcα2+(d a)α b =0. Since ad bc =1 we canwrite the
′ ′ ′ ′ ′ ′ ′ ′
− − −
discriminant of this equation as D = (a +d)2 4 = Tr(h)2 4. The quadratic
′ ′
− −
irrationalities α and √D generate the same field extension of Q. Tr(h) > 2 and
| |
we may assume Tr(h)>2 since multiplying h by 1 does not change D. Define
−
Tr(h)+1 1
g = −
Tr(h)+2 1
(cid:18) − (cid:19)
Then g SL (Z) and Tr(g) = Tr(h) so the fixed points of g generate the same
2
extension∈of Q than α. By construction g satisfies 1.19 and 1.20. (cid:3)
2. B (θ,τ) in terms of generators and relations
g
We want to describe B (θ,τ) in terms of generatorsand relations. Let ϕ α=
g α
{ |
1,...,c be the basis for and ψ γ =1,...,c(a+d) the corresponding basis of
g γ
} H { | }
as given in 1.14.
g2
H
The multiplication map m: of the algebraB (θ,τ) is t . It is
g g g2 g g,g
H ⊗H →H
given in the above basis as in Theorem 1.1 by the structure constants:
m:ϕ ϕ Cγ ψ (2.1)
α⊗ β 7→ α,β γ
Our computation of the relations defining B (θ,τ) is based on the following
g
observation:
Lemma 2.1. Denote by the tensor algebra of . Then B (θ,τ) is isomorphic
g g
T H
to the quotient of by the homogeneous ideal generated by Ker(m) .
g g
T R ⊂H ⊗H
Sincem:vα,βϕ ϕ vα,βCγ ψ wehavethatvα,βϕ ϕ belongs
α⊗ β 7→ α,β γ α⊗ β ∈Hg⊗Hg
to Ker(m) if and only if vα,βCγ = 0 for all γ = 1,...,c(a+d). Using the bases
α,β
ϕ and ψ we identify and with Cc2 and Cc(a+d) respectively.
α γ g g g2
{ } { } H ⊗H H
Finding a set of defining relations of B (θ,τ) for the generators ϕ α = 1,...,c
g α
{ | }
amountstofindingabasisforthekernelofthelinearmapM :Cc2 Cc(a+d) given
by the Cγ . →
α,β
Lemma 2.2. The structure constant Cγ is different from zero if and only if
α,β
α d(γ β) mod c.
≡ −
Proof. The formula for the structure constants 1.17 in this case is:
τm2
Cγ = exp[πı − ] (2.2)
α,β c3(a+d)
m∈IgX,g(α,β,γ)
ARITHMETIC STRUCTURES ON RM NONCOMMUTATIVE TORI 7
The index set of the series is nonempty only when α d(γ β) mod c:
≡ −
I (α,β,γ)= cγ+c(a+d)α c(d2+bc)γ c(a+d)dβ mod c2(a+d)
g,g
6 ∅ ⇐⇒ − ≡ −
γ+(a+d)α (d2+bc)γ (a+d)dβ mod c(a+d)
⇐⇒ − ≡ −
(a+d)α (d2+bc+1)γ (a+d)dβ mod c(a+d)
⇐⇒ ≡ −
(a+d)α (d2+da)γ (a+d)dβ mod c(a+d)
⇐⇒ ≡ −
(a+d)α d(d+a)γ (a+d)dβ mod c(a+d)
⇐⇒ ≡ −
α d(γ β) mod c
⇐⇒ ≡ −
Thus Cγ =0 if α d(γ β) mod c.
α,β 6≡ −
Conversely if α d(γ β) mod c then
≡ −
I (α,β,γ) = n Zn cγ+c(a+d)α mod c2(a+d)
g,g
{ ∈ | ≡− }
= n Zn= cγ+c(a+d)α+mc2(a+d) for some m Z
{ ∈ | − ∈ }
And thus
( cγ+c(a+d)α+nc2(a+d))2
Cγ = exp[ πıτ − ]
α,β − c3(a+d)
n Z
X∈
( cγ+c(a+d)α)2 2n( cγ+c(a+d)α)
= exp[ πıτ( − + − +c(a+d)n2)]
− c3(a+d) c
n Z
X∈
((a+d)α γ)2
= exp[ πıτ − ] exp[ 2πıτ((a+d)α γ)n πıτc(a+d)n2]
− c(a+d) − − −
n Z
X∈
The last series is a theta series A.1:
[(a+d)α γ]2
Cγ = exp[ πıτ − ]ϑ( τ((a+d)α γ), τc(a+d))
α,β − c(a+d) − − −
And we can write it as a theta constant with rational coefficients by taking, τ =
′
τc(a+d) and l =c(a+d).
−
(a+d)α γ (a+d)α γ
Cγ = exp[ πıτc(a+d)[ − ]2]ϑ( τc(a+d) − , τc(a+d))
α,β − c(a+d) − c(a+d) −
(a+d)α γ (a+d)α γ
= exp[πıτ′[ − ]2]ϑ(τ′ − ,τ′)
l l
= ϑ(a+d)α−γ(τ′).
l
Now,byA.4 forr,s 1Zthe zeroesofϑ (z,τ )occuratthepointsofthe form
∈ l r,s ′
(r+p+1)τ +(s+q+1) forp,q Z. In particularthe zeroesof ϑ areat points
2 ′ 2 ∈ r,0
(r+p+ 1)τ +(q+ 1). Thus ϑ (τ ) = ϑ (0,τ ) = 0 for all r 1Z which proves
the lemm2a. ′ 2 r ′ r,0 ′ 6 ∈ l (cid:3)
The expression for the nonzero values of the structure constants in the proof of
Lemma 2.2 is crucial in all that follows. We state it as a corollary.
8 JORGEPLAZAS
Corollary 2.3. The nonzero values of the structure constants Cγ are theta con-
α,β
stants with rational characteristics depending on g. If α d(γ β) mod c then
≡ −
Cαγ,β = ϑ(a+d)α−γ(τ′)
l
= ϑ(a+d)d(γ−β)−γ(τ′)
l
where τ = τc(a+d) and l=c(a+d).
′
−
Proof. The first equality follows from the proof of Lemma 2.2. For the second one
note that the theta constant ϑ (τ ) only depend on the class of r in 1Z/Z so we
r ′ l
can replace α by d(γ β) mod c in the last formula for the nonzero structure
constants. − (cid:3)
WecanuseLemma2.2towritethelinearsystemM correspondingtomasasum
ofc independentsystems. We dothis by groupingthe nonzerostructureconstants.
Since c and d are relatively prime it follows that for γ in a fixed congruence class
mod c the value of β determines the unique value of α for which Cγ =0.
α,β 6
Notation 2.4. Given µ,β 1,...,c we denote by α(µ,β) the unique representa-
∈{ }
tive mod c of d(µ β) laying in 1,...,c .
− ∈{ }
Lemma 2.5. Let M :Cc2 Cc(a+d) be given by Cγ . Then M is equivalent to c
→ α,β
independent systems M(µ):Cc C(a+d), µ=1,...,c, each one of rank c (a+d).
→ −
Proof. Fix α and β in 1,2,...c . Given γ,γ 1,2,...c(a + d) we have by
′
{ } ∈ { }
Lemma 2.2
Cγ =0 and Cγ′ =0 α+dβ dγ and α+dβ dγ
α,β 6 α,β 6 ⇐⇒ ≡ ≡ ′
dγ dγ mod c
′
⇐⇒ ≡
γ γ′ mod c
⇐⇒ ≡
Therefore nonzero values of Cγ occur for values of γ in the same congruence
α,β
class mod candwecanarrangethesystemascindependentsystemsofdimension
c2 (a+d), each one corresponding to the values of of γ in the same congruence
×
class mod c. Again by Lemma 2.2 each one of these systems will have only c
nontrivial columns corresponding to the values of α and β satisfying α d(γ β)
≡ −
mod c. Leavingasidethezerostructureconstantsweareleftwiththecindependent
systems:
Cµ Cµ ... Cµ
α(µ,1),1 α(µ,2),2 α(µ,c),c
Cµ+c Cµ+c ... Cµ+c
 α(µ,1),1 α(µ,2),2 α(µ,c),c 
M(µ)= . (2.3)
 . 
 
 Cµ+(a+d−1)c Cµ+(a+d−1)c ... Cµ+(a+d−1)c 
 α(µ,1),1 α(µ,2),2 α(µ,c),c 
 
where µ 1,2,...,c . By Theorem 1.3 B (θ,τ) is generated by its degree 1 part
g
∈ { }
(rememberweassumedc a+d+2). Thisinparticularmeansthat ,itsdegree
g2
≥ H
two part, is generated by products of elements in so the multiplication map m
g
H
is surjective. At the level of the representing matrices this just means that M has
maximalrank. Also, since M is the directsum ofthe M(µ) eachone ofthese must
have maximal rank. (cid:3)
ARITHMETIC STRUCTURES ON RM NONCOMMUTATIVE TORI 9
Componentwise we have M(µ)i,j =Cαµ(+µ(,ij−),1j)c. By Lemma 2.2 the denominator
in the characteristic of the corresponding theta constant is (a+d)d(µ j) (µ+
− −
(i 1)c) so dividing out by l =c(a+d) one gets dµ dj µ i + 1 i.e.
− c − c − l − a+d a+d
M(µ) =ϑ (τ ) (2.4)
i,j q(µ) dj i ′
−c−a+d
Where q(µ) = dµ µ + 1 is the term not depending on i or j. It is important
c − l a+d
to note that the characteristics giving the theta constants which appear as the
coefficients of the M(µ) are all the same up to a shift. We can arrange these
characteristics in a matrix Λ = Λ(g) (1Z/Z) given by Λ = dj
i . The matrices M(µ) are then fun∈ctMionc×s(oaf+τd)anld µ determined biy,jg: −c −
(a+d)
(τ,µ) M(µ) =ϑ (τ ) (2.5)
7→ i,j q(µ)+Λi,j ′
Each one of the kernels of the matrices M(µ) gives us a set of relations for
B (θ,τ). Bythe abovediscussionweseethatthesesetsareindependent. We write
g
downa basis for the kernel of each M(µ) in terms of its minor determinants. First
we introduce some notation:
Notation 2.6. Let n > m and let L : Cn Cm be a surjective linear map.
→
Denote also by L (C) its matrix representation in the canonical basis.
m n
∈ M ×
Given i ,i ,...,i 1,2,...,n , i < i < ... < i we write L for the
1 2 m ∈ { } 1 2 m | |i1,i2,...,im
minor determinant corresponding to the columns i ,i ,...,i .
1 2 m
Lemma 2.7. Let n>m and let L (C). Assume the first m columns of L
n m
are linearly independent. For each k∈=M1,×...,n m let vk Cn be defined by:
− ∈
L if1 j m
1,2,...,j 1,m+k,j+1,...,n
| | − ≤ ≤
vk = L ifj =m+k (2.6)
j  1,2,...,m
−| |
0 otherwise

Then vk k =1,...,n m forms a basis for Ker(L)
{ | − }
Proof. Let L be as above. Denote by L1,...,Ln Cm its columns. Denote by
L˜ the matrix corresponding to the fir∈st m columns of L. For k
m m
1,∈...,Mn ×m let X˜k Cm be a solution of L˜Y = Lm+k we can complete X˜k t∈o
{ − } ∈ −
a vectorXk Cn by taking the remaining coordinatesto be 0 except for the n+k
∈
coordinate which we set to 1. Then LXk =0 for all k 1,...,n m . Also, it is
∈{ − }
clear from the construction that the Xk are linearly independent. In this way we
may construct a basis for the kernel of L. Now, we solve each one of the systems
L˜Y = Lm+k usingCramer’srule. Afterclearingdenominatorsinthesolutionand
comple−ting to a vector in Cn we get the vectors vk in 2.6. (cid:3)
It will be convenient to view the minor determinants of M(µ) as functions of τ:
10 JORGEPLAZAS
Definition 2.8. Let M(µ) be given as in 2.3 let i ,...,i 1,2,...,c with
1 a+d
∈ { }
i <i <...<i . Define
1 2 a+d
Fg,µ (τ) = M(µ)
i1,i2,...,ia+d | |i1,i2,...,ia+d
a+d
= sgn(σ) M(µ)
σ(k),ik
σ∈XSa+d kY=1
a+d
= sgn(σ) ϑq(µ) dik σ(k) (τ′)
σ∈XSa+d kY=1 − c −(a+d)
We will show later that for each g and µ these are modular functions on τ. By
applying Lemma 2.7 we can write now a explicit presentation of B (τ,θ):
g
Theorem 2.9. Given µ 1,2,...,c and k 1,...,c a d let vµ,k =vµ,k(τ)
∈{ } ∈{ − − } ∈
Cc be given by
Fg,µ (τ) if1 j a+d
1,2,...,j 1,a+d+k,j+1,...,a+d ≤ ≤
vµ,k =vµ,k(τ)= Fg,µ − (τ) ifj =m+k (2.7)
j j  1,2,...,a+d
0 otherwise
Then the algebra Bg(τ,θ) is generated by elements x1,...,xc of degree 1 subject to
relations fµ =0 where:
k
fµ =vµ,kx x +...+vµ,kx x (2.8)
k 1 α(µ,1) 1 c α(µ,c) c
Proof. Eachone ofthe matrices M(µ) has maximalrankequal to a+d. Therefore
there are a+d linearly independent columns and we can reorder them in order
to apply Lemma 2.7. We let x ,...,x be the generators of the tensor algebra
1 c
of corresponding to the basis ϕ ,...,ϕ . Thus = C x ,...,x . For each
g 1 c 1 c
T H T h i
µ 1,2,...,c Lemma2.7givesusabasisforthekernelofM(µ)whichcorresponds
by∈L{emma 2.1}to a set of defining relations for B (τ,θ). (cid:3)
g
Example 2.10. Let
4 1
g = −
5 1
(cid:18) − (cid:19)
The eigenvalues of g are
3 √5 3+√5
λ+ = − , λ− =
2 2
and the fixed points of g are
5 √5 5+√5
θ = − , θ′ = .
10 10
Fix now a complex structure τ on and consider the corresponding connection
θ
¯ on the -bimodule E (θ). SAince
0 θ 1,5
∇ A −
11 3
g2 = −
15 4
(cid:18) − (cid:19)
we have that C5 and C15. After choosing a basis the multiplication
g g2
H ≃ H ≃
map m : is represented by a matrix M (C). We take as
g g g2 15,25
H ⊗H → H ∈ M