Arithmetic structures on noncommutative tori with real multiplication

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Arithmetic structures on

noncommutative tori with real multiplication

Dissertation

zur

Erlangung des Doktorgrades der

Mathematisch-Naturwissenschaftlichen Fakult¨at der

Rheinischen Friedrich-Wilhelms-Universit¨at Bonn

vorgelegt von Jorge Plazas

aus Bogot´a

Bonn 2006

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der Mathematisch-Naturwissenschaftlichen Fakult¨at der Rheinischen Friedrich-Wilhelms-Universit¨at Bonn

Erster Referent: Prof. Dr. Matilde Marcolli Zweiter Referent: Prof. Dr. Daniel Huybrechts

Tag der Promotion: 15. Januar 2007

Diese Dissertation ist auf dem Hochschulschriftenserver der ULB Bonn http://hss.ulb.uni-bonn.de/diss online elektronisch publiziert.

Erscheinungsjahr: 2007

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For Paula and Mateo

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Acknowledgments

First and foremost I want to thank Matilde Marcolli to whom I owe much more than I can adequately express. Her continued guidance during this three years made possible this project.

I also want to thank Daniel Huybrechts and Christian Kaiser whose comments and suggestions helped me to improve the present work.

Many thanks go to Armando Villamizar, Luis Fern´andez, Giovanni Landi and Lothar G¨ottsche. Their formative influence played a very important role in the earlier stages of my career.

I am grateful to my friends and colleges at MPI and UniBonn for making the fundamental interplay between mathematics and coffee so enjoyable.

Special thanks go to Nikolai Durov, Snigdhayan Mahanta and Eugene Ha for many helpful discussions.

I am happy to express my deep gratitude to my family, they are the greatest blessing of my life and I owe much to their love. Many thanks to Angela for making this period of my life so bright and colorful. My friends in Bonn made my stay here a great experience, I thank all of them. Also many thanks go to my friends in the high mountains in the other side of the big sea who always managed to be very close and encouraging.

I thank the Max Planck Institut f¨ur Mathematik for its support, its hospitality and its excellent conditions.

v

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Introduction

Noncommutative tori are standard prototypes of noncommutative spaces.

Since the early stages of noncommutative geometry these spaces have been central examples arising naturally in various context. In [16] Manin proposed the use of noncommutative tori as a geometric framework for the study of abelian class field theory of real quadratic fields. This is the so called “real multiplication program”. The main idea is that noncommutative tori may play a role in the study of real quadratic fields analogous to the role played by elliptic curves in the study of imaginary quadratic fields.

The relation between the endomorphism rings of noncommutative tori and orders in real quadratic fields is a good evidence supporting this point of view. If a noncommutative torus admits nontrivial Morita autoequivalences then the ring of such autoequivalences is an order in a real quadratic field.

A noncommutative torus having this property is called a real multiplication noncommutative torus.

The fact that noncommutative geometry may be relevant for addressing questions in number theory is also supported by various results and relations that have emerged in the last years. In particular explicit class field theory of Q (Kronecker-Weber theorem) and explicit class field theory of imaginary quadratic fields (complex multiplication) can both be recovered from the dynamics of certain quantum statistical mechanical systems ([2, 7, 8]). The existence of quantum statistical mechanical systems with rich arithmetical properties opens a new approach to the study of explicit class field theory using the tools of quantum statistical mechanics. The first case for which there is not yet a complete solution to the explicit class field 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 arithmetical applications it is important to find appropriate algebraic structures underlying these spaces. Rings admitting models algebraic over the corresponding base fields have proved to be essential for the analysis of quantum statistical mechanical systems of arithmetic nature.

In [26] Polishchuk defined homogeneous coordinate rings for real multiplication noncommutative tori endowed with a complex structure. These rings seem to be good candidates 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 formulas defining

1

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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 algebras varying 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 averaging process over (limiting) modular symbols.

The fact that modular forms play an essential role in our analysis points to deep relations with the quantum thermodynamical system introduced by Connes and Marcolli in [6]. This quantum mechanical system 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.

In the case of real quadratic fields explicit class field theory is conjec- turally given in terms of special values ofL-functions, this is the content of Stark’s famous conjectures [39]. In order to apply our results on noncommutative tori in this direction using the tools of quantum statistical mechanics we still need to find C^∗-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 [5] will make it possible to obtain suitableC^∗-completions.

In a different but related perspective arithmetic structures on noncommutative tori have been recently studied by Vlasenko in [40] where the theory of rings of quantum theta functions is developed.

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CHAPTER 1

Homogeneous coordinate rings on

noncommutative tori with real multiplication

In this chapter we introduce the main notions and notations used in the present work. In the first sections we introduce noncommutative tori with real multiplication and their homogeneous coordinate rings as defined by Polishchuk in [26]. In Section 5 we use the explicit formulas involved in the definition of these rings in order to obtain a presentation in terms of generators and relations, this section is the core of the chapter. We end with some particular examples in order to illustrate the behavior of the rings corresponding to different tori.

1. Noncommutative tori

On what follows we will denote byTthe two dimensional torusS¹×S¹. As a topological space T it is characterized by its algebra of continuous functions C(T). This algebra is a unital commutative C^∗-algebra. C(T) can be realized as the universal C^∗-algebra generated by two commuting unitaries U and V. Any element of C(T) admits a Fourier expansion in terms of powers of these unitaries and smooth functions are characterized as those functions whose coefficients in the corresponding Fourier expansion decay rapidly at infinity.

Noncommutative tori are defined by their function algebras which are noncommutative deformations of C(T) andC^∞(T).

Definition 1.1. Given θ ∈ R we define A_θ = C(Tθ), the algebra of continuous functions on the noncommutative torus Tθ, as the universalC^∗- algebra generated by two unitaries U and V subject to the relations:

U V =e^2πiθV U.

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Definition 1.2. Givenθ∈Rwe defineA_θ, thealgebra of smooth functions on the noncommutative torus Tθ, as the algebra of formal power series in two unitariesU and V with rapidly decreasing coefficients and multiplication given by the relation U V =e^2πiθV U:

A_θ = C^∞(Tθ)

= {a= X

n,m∈Z

an,mUⁿV^m| {a_n,m} ∈ S(Z²)}

3

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The compact group T acts on the algebras Aθ and A_θ. Much of the structure of these algebras is determined by this action. The action of T induces an action of its Lie algebraL=R² given by the derivations:

δ1(U) = 2πıU; δ1(U) = 0 (2)

δ2(U) = 0; δ2(V) = 2πıV (3)

The algebraA_θ is a pre-C^∗-algebra whose C^∗-completion is isomorphic toA_θ. The corresponding Frechet structure is determined by the derivations δ1 and δ2. Likewise one can obtain A_θ as the algebra of smooth elements ofA_θ determined by these derivations. The relation between these algebras parallels the classical situation which corresponds to the value θ = 0 for which one recovers C(T) and C^∞(T). We refer the reader to the seminal paper [3] and the survey [31] for the main results about the algebras Aθ= C(Tθ) and A_θ=C^∞(Tθ).

On what follows we will restrict to the case wereθis an irrational number. In this case the algebra A_θ is simple and admits a unique normalized traceχinvariant under the action ofT. In the algebraA_θ this trace is given by

χ(X

a_n,mUⁿV^m) =a_0,0. (4)

2. Morita equivalences and real multiplication

By the Serre-Swan theorem the theory of vector bundles overTis equivalent to to the theory of finite type projective modules over the algebraC(T).

To each complex vector bundle overTone associates theC(T)-module of its global sections. Smooth bundles correspond to finite type projective modules over C^∞(T) and every vector bundle over T is equivalent to a smooth one.

We consider projective finite type right A_θ-modules as vector bundles overTθ. If ˜E is a projective finite type right A_θ-module then there exists a projective finite type rightA_θ-moduleE such that one has an isomorphism of rightA_θ-modules:

E˜ 'E⊗A_θ Aθ.

Therefore, as in the commutative case, the categories of smooth and continuous vector bundles over Tθ are equivalent (c.f. [3]). On what follows we will restrict toA_θ-modules.

The traceχdefined in (4) can be extended to a traceT rχ on the matrix algebra M_n(A_θ) = End(Aⁿ_θ). A right A_θ-module E is projective of finite type if and only if there exists an idempotent e=e² =e^∗ inM_n(A_θ) such thatE 'eAⁿ_θ, thus we can define the rank ofE by

rk(E) =T rχ(e) (5)

Unless otherwise stated a right (resp. left)A_θ-module will always mean a right (resp. left) projective finite typeA_θ-module.

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2. MORITA EQUIVALENCES AND REAL MULTIPLICATION 5

Let θ ∈ R be irrational. Following [3] we define, for any pair c, d ∈ Z, c >0, a rightA_θ-moduleE_d,c(θ) given by the following action ofA_θ on the Schwartz spaceS(R×Z/cZ) =S(R)^c:

(f U)(x, α) = f(x−cθ+d

c , α−1) (6)

(f V)(x, α) = exp(2πi(x−αd

c ))f(x, α) (7)

The rank ofE_d,c(θ) is|cθ+d|and ifE is any rightA_θ-module withrk(E) =

|cθ+d|thenE'E_d,c(θ). The K0 group ofA_θ,K0(A_θ), is by definition the enveloping group of the abelian semigroup given by isomorphism classes of right A_θ-modules together with direct sum. The rank function rk extends to a injective morphism

rk :K0(A_θ)→R (8)

whose image is Z⊕θZ. Therefore one gets a ordered structure on K0(A_θ) given by the isomorphism

K0(A_θ)'Z⊕θZ⊂R (9)

The fact thatrkis injective is the content of the cancellation theorem due to Rieffel (c.f. [32]). From this theorem it follows that rightA_θ-modules are classified up to isomorphism by their rank and that any finite type projective right A_θ-module is either free or isomorphic to a right module of the form E_d,c(θ).

Ifcand dare relatively prime we say thatE_d,c(θ) is abasic A_θ-module.

Being this the case the pair d, ccan be completed to a matrix g=

a b c d

∈SL₂(Z) (10)

We writeE_g(θ) for the moduleE_d,c(θ). By definition the degree ofE_g(θ) is taken to bec. We also define the degree of a matrixg∈SL2(Z), given as above, by deg(g) =c.

Let SL2(Z) act on R by fractional linear transformations. Let g ∈ SL₂(Z) be as above and denote by U⁰ and V⁰ two generating unitaries of the algebra A_gθ. We can define a left action of the algebra A_gθ on E_g by:

(U⁰f)(x, α) = f

x−1 c, α−a

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(V⁰f)(x, α) = exp(2πi( x

cθ+d−α

c))f(x, α) (12)

This action gives an identification:

EndA_θ(E_g(θ))' A_gθ. (13)

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The tensor product of basic modules is again a basic module. More precisely, given g₁, g₂ ∈SL₂(Z), there is a well defined pairing of rightA_θ- modules:

tg1,g2 :Eg1(g2θ)⊗_CEg2(θ)→Eg1g2(θ) (14)

This map gives rise to an isomorphism ofA_g₁_g₂_θ− A_θ bimodules:

E_g₁(g₂θ)⊗_A_g

2θ E_g₂(θ)→E_g₁_g₂(θ).

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In particular, if gθ=θone has an isomorphism E_g(θ)⊗_A_θ · · · ⊗_A_θ E_g(θ)

| {z }

n

'E_gⁿ(θ).

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We say that two noncommutative tori Tθ⁰ and Tθ are Morita equivalent if there exist a A_θ⁰-A_θ-bimodule which is projective and of finite type both as a left A_θ⁰-module and as a right A_θ-module. We will consider the category whose objects are noncommutative tori and whose morphisms are given by isomorphism classes of finite type projective bimodules over the corresponding algebras of smooth functions. Composition is provided by tensor product over the corresponding algebra. A isomorphism in this category is called a Morita equivalence. From the discussion above we see that given a real number θ and a matrix g ∈ SL2(Z) the noncommutative tori Tgθ and Tθ are Morita equivalent. The inverse of the morphism represented by theA_gθ-A_θ-bimodule Eg(θ) is the morphism represented by theA_θ-A_gθ- bimoduleE_g⁻¹(gθ). By a result of Rieffel these are the only possible Morita equivalences in the category of noncommutative tori (c.f. [30]). More precisely, two noncommutative toriT⁰_θandTθ are Morita equivalent if and only if there exist g∈SL₂(Z) such that θ⁰ =gθ.

If gθ = θ then Eg(θ) has the structure of A_θ-bimodule. An irrational number θ ∈ R\ Q is a fixed point of a fractional linear transformation g∈SL2(Z) if and only if it generates a quadratic extension of Q.

Definition 2.1. The noncommutative torus Tθ with algebra of smooth functionsA_θ is areal multiplication noncommutative torus if the parameter θ generates a quadratic extension ofQ.

Thus Tθ is a real multiplication noncommutative torus if and only if it has nontrivial Morita autoequivalences.

In [16] Manin proposed the use of noncommutative tori as a geometric framework under which to attack the explicit class field theory problem for real quadratic extensions of Q. The explicit class field theory problem ask for explicit generators of the maximal abelian extension of a given field and the corresponding Galois action of the abelianization of the absolute Galois group on these generators. The only number fields for which a complete solution of this problem is known are the imaginary quadratic extensions of Q and Q itself. Elliptic curves whose endomorphism ring is nontrivial correspond to lattices generated by imaginary quadratic irrationalities and play a central role in the solution of the explicit class field theory problem

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3. COMPLEX STRUCTURES ON TORI AND HOLOMORPHIC CONNECTIONS 7

for the corresponding imaginary quadratic extensions. It is believed that noncommutative tori with real multiplication may play analogous role in the study of real quadratic extensions of Q. In order to achieve arithmetical applications it is important to realize algebraic structures underlying noncommutative tori. This is our main motivation for the study of the homogeneous coordinate rings described below.

3. Complex structures on tori and holomorphic connections A complex structure on the noncommutative torus Tθ is determined through the the derivations δ1 and δ2 by choosing a complex structure on the Lie algebra L = R² of T. For this we make a decomposition of the complexification of L into two complex conjugate subspaces. This can be done by choosing a complex parameter τ with nonzero imaginary part and taking{1, τ}as a basis for the holomorphic part of this decomposition. The resulting derivation δτ =τ δ1+δ2 is a complex structure Tθ. Explicitly we have:

δ_τ : X

n,m∈Z

a_n,mUⁿV^m7→(2πı) X

n,m∈Z

(nτ +m)a_n,mUⁿV^m (17)

This derivation should be viewed as an analog of the operator ¯∂on a complex elliptic curve. We will denote byTθ,τ the noncommutative torusTθequipped with this complex structure. In what follows we will assume thatIm(τ)<0.

We will also freely refer toτ as the complex structure onTθ,τ.

Complex structures on noncommutative tori were introduced by Connes in relation with the Yang Mills equation and positivity in Hochschild co- homology for noncommutative tori (c.f. [4]). The study of the structure of the space of connections associated to the above derivations was carried out in [9]. An approach through noncommutative analogs of theta functions was developed in [33, 10] were these are viewed as holomorphic sections on noncommutative tori. The resulting categories were studied throughly in [29]and [27].

A holomorphic structure on a rightA_θ-moduleEis given by an operator

∇¯ :E →E which is compatible with the complex structure δτ in the sense that it satisfies the following Leibniz rule:

∇(ea) = ¯¯ ∇(e)a+eδ_τ(a), e∈E, a∈ A_θ (18)

Given a holomorphic structure ¯∇ on a right A_θ-module E the corresponding set of holomorphic sections is the space

H⁰(Tθ,τ, E∇¯) :=Ker( ¯∇) (19)

On every basic moduleE_d,cone can define a family of holomorphic structures{∇¯_z} depending on a complex parameterz∈C:

∇¯_z(f) = ∂f

∂x + 2πi dτ

cθ+dx+z

f.

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By definition astandard holomorphic vector bundle on Tθ,τ is given by a basic module E_d,c =E_g together with one of the holomorphic structures

∇¯_z.

The spaces of holomorphic sections of a standard holomorphic vector bundles on Tθ,τ are finite dimensional (c.f. [29], Section 2). If cθ+d > 0 then dimH⁰(Eg,∇¯₀) = c. On what follows we will consider the spaces of holomorphic sections corresponding to ¯∇₀:

H_g :=H⁰(Tθ,τ, E_g,∇¯₀).

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A basis ofH_g is given by the Schwartz functions:

ϕα(x, β) = exp(− cτ cθ+d

x²

2 )δ_α^β α= 1, ..., c.

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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 3.1. ([29] Section 2) Suppose g₁ and g₂ have positive degree.

Then g1g2 has positive degree and tg1,g2 induces a well defined linear map t_g₁_,g₂ :H_g₁(g₂θ)⊗_CH_g₂(θ)→ H_g₁_g₂(θ).

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Let g₁, g₂ and g₁g₂ be given by g₁ =

a₁ b₁ c1 d1

, g₂ =

a₂ b₂ c2 d2

, g₁g₂=

a₁₂ b₁₂ c12 d12

and let {ϕ_α}, {ϕ⁰_β} and {ψ_γ} be respectively the basis of H_g₁(g₂θ), H_g₂(θ) and H_g₁_g₂(θ) as given in (22). Then

t_g₁_,g₂ :ϕ_α⊗ϕ⁰_β 7→C_α,β^γ ψ_γ (24)

Where for α= 1, ..., c₁, β= 1, ..., c₂ and γ = 1, ..., c₁₂ we have:

C_α,β^γ = X

m∈Ig1,g2(α,β,γ)

exp[πı −τ m² 2c1c2c12

] (25)

with

Ig1,g2(α, β, γ) ={n∈Z | n≡ −c₁γ+c12α mod c12c1, n≡c₂d₁₂γ−c₁₂d₂β mod c₁₂c₂} Notation 3.2. Throughout we use the convention of summing over re- peated indexes.

4. Homogeneous coordinate rings

Given a projective scheme Y over a field k together with an ample line bundle Lon Y one can construct the homogeneous coordinate ring

B=M

n≥0

H⁰(Y,L^⊗n).

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4. HOMOGENEOUS COORDINATE RINGS 9

This ring plays a prominent role in the study of the geometry ofY (c.f.[34]).

In [26] Polishchuk proposed an analogous definition of the homogeneous coordinate ring of a real multiplication noncommutative torus Tθ in terms of holomorphic sections of tensor powers of a standard holomorphic vector bundle on Tθ,τ. As mentioned above the real multiplication condition is fundamental in order to be able to perform the tensor power operation.

Assume thatθ∈Ris a quadratic irrationality. So there exist some g∈ SL2(Z) withgθ=θandTθhas real multiplication. Fix a complex structure τ on Tθ. In the case E =E_g(θ) we can extend a holomorphic structure on Eg to a holomorphic structure on the tensor powersE_g^⊗n. Following [26] we define a homogeneous coordinate ring for Tθ,τ by:

Bg(θ, τ) = M

n≥0

H⁰(Tθ,τ, E_∇^⊗n_¯

0) (26)

= M

n≥0

H_gⁿ

The category of holomorphic vector bundles onTθ,τ is equivalent to the heartC^θof a t-structure onD^b(Eτ), the derived category of the elliptic curve E_τ = C/(Z⊕τZ). In [26] Polishchuk exploits this equivalence in order to study the properties of the algebraBg(θ, τ) by studying the the corresponding image under this equivalence. The following result characterizes some structural properties ofBg(θ, τ) in terms of the matrix elements ofg:

Theorem4.1. ([26] Theorem 3.5)Assumeg∈SL₂(Z) has positive real eigenvalues

• If c≥a+d thenB_g(θ, τ) is generated over C by H_g.

• If c≥a+d+ 1 thenB_g(θ, τ) is a quadratic algebra.

• If c≥a+d+ 2 thenBg(θ, τ) is a Koszul algebra.

Let us briefly recall these definitions. If A = L

n≥0An is a connected graded algebra over a field k generated by its degree one piece A₁ then A is isomorphic to a quotient T(A1)/I where T(A1) = L

n≥0A^⊗n₁ is the tensor algebra of the vector space A₁ and I is a two sided ideal in T(A₁).

The algebra A is a quadratic algebra if the ideal I can be generated by homogeneous elements of degree two. Since Ais connected we can consider A0 =k as a left module overA. A quadratic A algebra is a Koszul algebra if the graded k-algebra L

n≥0Extⁿ_A(k, k) is generated byExt¹_A(k, k)'A^∗₁. We will fix some notations and conventions for the rest of the paper. As aboveg will denote a matrix

g=

a b c d

∈SL2(Z)

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We will always assume that the following inequalities hold T r(g) =a+d >2 (g is hyperbolic) (27)

c ≥T r(g) + 2 =a+d+ 2 (28)

The first inequality implies that g has positive real eigenvalues. By Theorem 4.1 the second inequality implies thatB_g(θ, τ) is a quadratic Koszul algebra generated in degree 1.

We denote byλ⁺ and λ⁻ the eigenvalues of gwith 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

θ= λ⁺−d

c , θ⁰ = λ⁻−d (29) c

These are the fixed points of g. By definition the n-graded part ofBg(θ, τ) isH_gⁿ. The dimension ofH_gⁿ is deg(gⁿ). Accordingly the Hilbert series for Bg(τ, θ) is given by (c.f [26]):

h_B_g_(τ,θ)(t) = 1 + (c−a−d) t+t² 1−(a+d)t+t² . (30)

Proposition 4.2. Let α ∈ R be a quadratic irrationality. Then there exist g and θ satisfying (27) and (28) such that Q(α) =Q(θ).

Proof. Supposeα ∈Ris a quadratic irrationality. Then there exist a hyperbolic element

h=

a⁰ b⁰ c⁰ d⁰

∈SL2(Z)

having α as one of its two fixed points. Being a fixed point of h,α satisfies the quadratic equationc⁰α²+ (d⁰−a⁰)α−b⁰ = 0. Sincea⁰d⁰−b⁰c⁰ = 1 we can write the discriminant of this equation as D= (a⁰+d⁰)²−4 =T r(h)²−4.

α and √

D generate the same field extension ofQ. |T r(h)|>2 and we may assume T r(h)>2 since multiplyingh by −1 does not changeD. Define

g=

T r(h) + 1 −1 T r(h) + 2 −1

Then g ∈SL₂(Z) and T r(g) = T r(h) so the fixed points of g generate the same extension ofQthanα. By constructiong satisfies (27) and (28).

5. A presentation in terms of generators and relations We want to describe Bg(θ, τ) in terms of generators and relations. Let {ϕ_α|α = 1, ..., c} be the basis forH_g and {ψ_γ|γ = 1, ..., c(a+d)} the corresponding basis ofH_g2 as given in (22).

The multiplication map m : H_g ⊗ H_g → H_g2 of the algebra B_g(θ, τ) is t_g,g. It is given in the above basis as in Theorem 3.1 by the structure

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5. A PRESENTATION IN TERMS OF GENERATORS AND RELATIONS 11

constants:

m:ϕα⊗ϕ_β 7→C_α,β^γ ψγ

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Our computation of the relations defining Bg(θ, τ) is based on the following observation:

Lemma 5.1. Denote by T the tensor algebra of H_g. Then B_g(θ, τ) is isomorphic to the quotient of T by the homogeneous ideal R generated by Ker(m)⊂ H_g⊗ H_g.

Proof. The algebra gradedB_g(θ, τ) is generated by its degree one part H_g therefore it is a quotient of T. Being quadratic the ideal of relations R=Ker(m) consist of degree two elements inT. Let v^α,βϕ_α ⊗ϕ_β ∈ H_g ⊗ H_g, v^α,β ∈ C be an arbitrary homogeneous element of T of degree 2. Since m : v^α,βϕα ⊗ϕ_β 7→ v^α,βC_α,β^γ ψγ we have that v^α,βϕ_α ⊗ϕ_β belongs to Ker(m) if and only if v^α,βC_α,β^γ = 0 for all γ = 1, ..., c(a+d). Using the bases{ϕ_α} and{ψ_γ}we identifyH_g⊗ H_g and H_g2 withC^c² andC^c(a+d) respectively. Finding a set of defining relations of B_g(θ, τ) for the generators {ϕ_α|α = 1, ..., c} amounts to finding a basis for the kernel of the linear mapM :C^c² →C^c(a+d) given by theC_α,β^γ .

Lemma 5.2. The structure constant C_α,β^γ is different from zero if and only ifα ≡d(γ−β) modc.

Proof. The formula for the structure constants (25) in this case is:

C_α,β^γ = X

m∈Ig,g(α,β,γ)

exp[πı −τ m² c³(a+d)] (32)

The index set of the series is nonempty only whenα≡d(γ−β) modc.

Ig,g(α, β, γ)6=∅ ⇐⇒ −cγ+c(a+d)α≡c(d²+bc)γ−c(a+d)dβ mod c²(a+d)

⇐⇒ −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 modc(a+d)

⇐⇒ α≡d(γ−β) mod c ThusC_α,β^γ = 0 if α6≡d(γ−β) modc.

Conversely if α≡d(γ−β) mod cthen

Ig,g(α, β, γ) = {n∈Z|n≡ −cγ+c(a+d)α mod c²(a+d)}

= {n∈Z|n=−cγ+c(a+d)α+mc²(a+d) for some m∈Z}

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And thus C_α,β^γ = X

n∈Z

exp[−πıτ(−cγ+c(a+d)α+nc²(a+d))² c³(a+d) ]

= X

n∈Z

exp[−πıτ((−cγ+c(a+d)α)²

c³(a+d) +2n(−cγ+c(a+d)α)

c +c(a+d)n²)]

= exp[−πıτ((a+d)α−γ)² c(a+d) ]X

n∈Z

exp[−2πıτ((a+d)α−γ)n−πıτ c(a+d)n²]

The last series is a theta series (73):

C_α,β^γ = exp[−πıτ[(a+d)α−γ]²

c(a+d) ]ϑ(−τ((a+d)α−γ),−τ c(a+d)) And we can write it as a theta constant with rational coefficients by taking, τ⁰ =−τ c(a+d) and l=c(a+d).

C_α,β^γ = exp[−πıτ c(a+d)[(a+d)α−γ

c(a+d) ]²]ϑ(−τ c(a+d)(a+d)α−γ

c(a+d) ,−τ c(a+d))

= exp[πıτ⁰[(a+d)α−γ

l ]²]ϑ(τ⁰(a+d)α−γ l , τ⁰)

= ϑ(a+d)α−γ l

(τ⁰).

Now, by Lemma 0.5 forr, s∈ ¹_lZthe zeroes ofϑ_r,s(z, τ⁰) occur at the points of the form (r+p+¹₂)τ⁰+ (s+q+¹₂) forp, q∈Z. In particular the zeroes of ϑr,0 are at points (r+p+¹₂)τ⁰+ (q+¹₂). Thus ϑr,0(0, τ⁰)6= 0 for allr ∈ ¹_lZ

which proves the lemma.

The expression for the nonzero values of the structure constants in the proof of Lemma 5.2 is crucial in all that follows. We state it as a corollary.

Corollary5.3. The nonzero values of the structure constantsC_α,β^γ are theta constants with rational characteristics depending ong. Ifα ≡d(γ−β) mod c then

C_α,β^γ = ϑ(a+d)α−γ l

(τ⁰)

= ϑ(a+d)d(γ−β)−γ l

(τ⁰) where τ⁰ =−τ c(a+d) and l=c(a+d).

Proof. The first equality follows from the proof of Lemma 5.2. For the second one note that the theta constantϑ_r(τ⁰) only depend on the class of

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5. A PRESENTATION IN TERMS OF GENERATORS AND RELATIONS 13

r in ¹_lZ/Z so we can replace α by d(γ−β) modc in the last formula for

the nonzero structure constants.

We can use Lemma 5.2 to write the linear system M corresponding to m as a sum ofc independent systems. We do this by grouping the nonzero structure constants. 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 C_α,β^γ 6= 0.

Notation 5.4. Given µ, β ∈ {1, ..., c} we denote by α(µ, β) the unique representative mod c of d(µ−β) laying in ∈ {1, ..., c}.

Lemma 5.5. Let M : C^c

2 → C^c(a+d) be given by C_α,β^γ . Then M is equivalent to c independent systems M(µ) :C^c → C^(a+d), µ= 1, ..., c, each one of rank c−(a+d).

Proof. Fix α and β in {1,2, ...c}. Given γ, γ⁰ ∈ {1,2, ...c(a+d)} we have by Lemma 5.2

C_α,β^γ 6= 0 and C_α,β^γ⁰ 6= 0 ⇐⇒ α+dβ≡dγ and α+dβ≡dγ⁰

⇐⇒ dγ ≡dγ⁰ modc

⇐⇒ γ ≡γ⁰ modc

Therefore nonzero values ofC_α,β^γ occur for values ofγin the same congruence class mod c and we can arrange the system as c independent systems of dimension c² ×(a+d), each one corresponding to the values of of γ in the same congruence class modc. Again by Lemma 5.2 each one of these systems will have only c nontrivial columns corresponding to the values of α and β satisfying α ≡d(γ −β) modc. Leaving aside the zero structure constants we are left with the cindependent systems:

M(µ) =







C_α(µ,1),1^µ C_α(µ,2),2^µ ... C_α(µ,c),c^µ C_α(µ,1),1^µ+c C_α(µ,2),2^µ+c ... C_α(µ,c),c^µ+c

. .

C_α(µ,1),1^{µ+(a+d−1)c} C_α(µ,2),2^{µ+(a+d−1)c} ... C_α(µ,c),c^{µ+(a+d−1)c}





 (33)

whereµ∈ {1,2, ..., c}.

By Theorem 4.1B_g(θ, τ) is generated by its degree 1 part (remember we assumed c≥a+d+ 2). This in particular means that H_g2, its degree two part, is generated by products of elements in H_g so the multiplication map m is surjective. At the level of the representing matrices this just means that M has maximal rank. Also, since M is the direct sum of the M(µ)

each one of these must have maximal rank.

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Componentwise we have

M(µ)i,j =C_α(µ,j),j^µ+(i−1)c.

By Lemma 5.2 the denominator in the characteristic of the corresponding theta constant is (a+d)d(µ−j)−(µ+(i−1)c) so dividing out byl=c(a+d) one gets ^dµ_c −^dj_c −^µ_l −_a+dⁱ +_a+d¹ i.e.

M(µ)i,j =ϑ_q(µ)−^dj

c− ⁱ

a+d

(τ⁰) (34)

Whereq(µ) = ^dµ_c −^µ_l +_a+d¹ is the term not depending oniorj.

It is important 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) ∈ M_c×(a+d)(¹_lZ/Z) given by Λi,j =−^dj_c −_(a+d)ⁱ . The matricesM(µ) are then functions ofτ andµ determined byg:

(τ, µ)7→M(µ)_i,j =ϑ_q(µ)+Λ_i,j(τ⁰) (35)

Each one of the kernels of the matricesM(µ) gives us a set of relations forB_g(θ, τ). By the above discussion we see that these sets are independent.

We write down a basis for the kernel of each M(µ) in terms of its minor determinants. First we introduce some notation:

Notation 5.6. Let n > m and let L : Cⁿ → C^m be a surjective linear map. Denote also by L ∈ M_m×n(C) its matrix representation in the canonical basis. Given i₁, i₂, ..., i_m ∈ {1,2, ..., n}, i₁ < i₂ < ... < i_m we write |L|_i₁_,i₂_,...,i_m for the minor determinant corresponding to the columns i₁, i₂, ..., i_m.

Lemma 5.7. Let n > m and let L ∈ M_n×m(C). Assume the first m columns ofLare linearly independent. For eachk= 1, ..., n−mlet v^k∈Cⁿ be defined by:

v_j^k =







|L|1,2,...,j−1,m+k,j+1,...,n if1≤j ≤m

−|L|_1,2,...,m if j =m+k

0 otherwise

(36)

Then {v^k|k= 1, ..., n−m} forms a basis for Ker(L).

Proof. Let L be as above. Denote by L¹, ..., Lⁿ ∈ C^m its columns.

Denote by ˜L∈ M_m×m the matrix corresponding to the first m columns of L. Fork∈ {1, ..., n−m}let ˜X^k∈C^m be a solution of ˜LY =−L^m+kwe can complete ˜X^k to a vector X^k ∈ Cⁿ by taking the remaining coordinates to be 0 except for the n+k coordinate which we set to 1. ThenLX^k = 0 for allk∈ {1, ..., n−m}. Also, it is clear from the construction that theX^kare linearly independent. In this way we may construct a basis for the kernel of L. Now, we solve each one of the systems ˜LY =−L^m+k using Cramer’s rule. After clearing denominators in the solution and completing to a vector

inCⁿ we get the vectorsv^k in (36).

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6. FIRST EXAMPLES 15

It will be convenient to view the minor determinants of M(µ) as functions of τ.

Definition 5.8. LetM(µ) be given as in 33 leti1, ..., ia+d∈ {1,2, ..., c}

withi₁ < i₂ < ... < i_a+d. Define F_i^g,µ

1,i2,...,ia+d(τ) = |M(µ)|_i₁_,i₂_,...,i_a+d

= X

σ∈S_a+d

sgn(σ)

a+d

Y

k=1

M(µ)_σ(k),i_k

= X

σ∈S_a+d

sgn(σ)

a+d

Y

k=1

ϑ_q(µ)−dik

c −_(a+d)^σ(k) (τ⁰)

We will show later that for eachgandµthese are modular functions on τ.

By applying Lemma 5.7 we can write now a explicit presentation of Bg(τ, θ):

Theorem 5.9. Given µ ∈ {1,2, ..., c} and k ∈ {1, ..., c −a−d} let v^µ,k=v^µ,k(τ)∈C^c be given by

v^µ,k_j = v^µ,k_j (τ) (37)

=







F1,2,...,j−1,a+d+k,j+1,...,a+d^g,µ (τ) if1≤j≤a+d F1,2,...,a+d^g,µ (τ) if j=m+k

0 otherwise

Then the algebraB_g(τ, θ)is generated by elements x₁, ..., x_c of degree1 subject to relations f_k^µ= 0 where:

f_k^µ=v^µ,k₁ x_α(µ,1)x₁+...+v_c^µ,kx_α(µ,c)x_c (38)

Proof. Each one of the matricesM(µ) has maximal rank equal toa+d.

Therefore there area+dlinearly independent columns and we can reorder them in order to apply Lemma 5.7. We let x₁, ..., x_c be the generators of the tensor algebra T of H_g corresponding to the basis ϕ1, ..., ϕc. Thus T =Chx₁, ..., x_ci. For eachµ ∈ {1,2, ..., c} Lemma 5.7 gives us a basis for the kernel of M(µ) which corresponds by Lemma 5.1 to a set of defining

relations for B_g(τ, θ).

6. First examples

In this section we look at the behavior of B_g(θ, τ) for some particular values of the matrix g.

Example 6.1. Let

g=

4 −1 5 −1

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The eigenvalues of g are:

λ⁺= 3−√ 5

2 , λ⁻= 3 +√ 5 2 and the fixed points ofg are:

θ= 5−√ 5

10 , θ⁰ = 5 +√ 5 10

Fix now a complex structureτ on A_θ and consider the corresponding connection ¯∇₀ on the A_θ-bimodule E_g(θ) = E−1,5(θ). The Hilbert series for Bg(τ, θ) is given by (30):

h_B_g_(τ,θ)(t) = 1 + 2t+t² 1−3t+t²

= 1 + 5t+ 15t²+ 40t³+ 105t⁴+ 275t⁵+. . .

In particularH_g 'C⁵ and H_g2 'C¹⁵. After choosing a basis the multiplication map m : H_g⊗ H_g → H_g2 is represented by a matrix M ∈ M_15,25. We take as above {ϕ_α ⊗ϕ_β|α, β = 1, ...,5} as basis for H_g ⊗ H_g and {ψ_γ|γ = 1, ...,15} as basis forH_g2 so thatM is the matrix corresponding to the structure constants

C_α,β^γ =

( ϑ^3β−4γ

15

(−15τ) if α≡d(γ−β) modc 0 otherwise

We write it asM 'M(1)⊕M(2)⊕M(3)⊕M(4)⊕M(5) where the elements of M(µ)∈ M_3,5(C) are given by

M(µ)i,j =ϑ_q(µ)+Λ_i,j(−15τ) Withq(µ) = ^5−4µ₁₅ and

Λ = 1 15





2 14 11 8 5

7 4 1 13 10

12 9 6 3 0





Each µ ∈ {1, ...,5} gives us a set of 2 relations corresponding to a basis for the kernel of M(µ). In this case Bg(θ, τ) is a quadratic algebra with 5 generators of degree 1 and 10 quadratic relations.

The minors of M(µ) give the functions of τ appearing as coefficients of the defining relations of B_g(θ, τ). For each ordered triple i₁, i₂, i₃ ∈ {1,2,3,4,5} we have:

F_i^g,µ₁_,i₂_,i₃(τ) = |M(µ)|_i₁_,i₂_,i₃

= X

σ∈S₃

sgn(σ)

3

Y

k=1

ϑ5−4µ

15 −^dik₅ −^σ(k)₃ (−15τ)

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Thus each one of the coefficients of the defining relations is a sum of triple products of theta constants. For example, taking µ = 1 we get the two relations:

f₁¹=F_4,2,3^g,1 (τ)x5x1+F_1,4,3^g,1 (τ)x1x2+F_1,2,4^g,1 (τ)x2x3−F_1,2,3^g,1 (τ)x3x4

and

f₂¹=F_5,2,3^g,1 (τ)x₄x₁+F_1,5,3^g,1 (τ)x₅x₂+F_1,2,5^g,1 (τ)x₁x₃−F_1,2,3^g,1 (τ)x₃x₅. Example 6.2. Let

g=

5 −1 6 −1

The eigenvalues of g are

λ⁺= 2−√

3, λ⁻ = 2 +√ 3 and the fixed points ofg are

θ= 3−√ 3

6 , θ⁰= 3 +√ 3 6 .

Fix now a complex structure τ on A_θ and consider the corresponding connection ¯∇₀ on theA_θ-bimoduleEg(θ) =E−1,6(θ). The Hilbert series for B_g(τ, θ) is given by (30):

h_B_g_(τ,θ)(t) = 1 + 2t+t² 1−4t+t²

= 1 + 6t+ 24t²+ 90t³+ 336t⁴+ 1254t⁵+. . .

In particularH_g 'C⁶ and H_g2 'C²⁴. After choosing a basis the multiplication map m : H_g⊗ H_g → H_g2 is represented by a matrix M ∈ M_24,36. We take as above {ϕ_α ⊗ϕ_β|α, β = 1, ...,6} as basis for H_g ⊗ H_g and {ψ_γ|γ = 1, ...,24} as basis forH_g2 so thatM is the matrix corresponding to the structure constants

C_α,β^γ =

( ϑ^4β−5γ

24

(−24τ) if α≡d(γ−β) mod 6 0 otherwise

We write it asM 'M(1)⊕M(2)⊕M(3)⊕M(4)⊕M(5)⊕M(6) where the elements of M(µ)∈ M_4,6(C) are given by

M(µ)i,j =ϑ_q(µ)+Λ_i,j(−24τ) Withq(µ) = ^6−5µ₂₄ and

Λ = 1 24







2 22 18 14 10 6

8 4 0 20 16 12

14 10 6 2 22 18

20 16 12 8 4 0







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Each µ ∈ {1, ...,6} gives us a set of 2 relations corresponding to a basis for the kernel of M(µ). In this case B_g(θ, τ) is a quadratic algebra with 6 generators of degree 1 and 12 quadratic relations.

Example 6.3. Let g=

3 1 8 3

, θ=

√ 2 4

Fix now a complex structure τ on A_θ and consider the corresponding connection ¯∇₀ on theA_θ-bimodule Eg(θ) = E3,8(θ). The Hilbert series for B_g(τ, θ) is given by (30):

h_B_g_(τ,θ)(t) = 1 + 2t+t² 1−6t+t²

= 1 + 8t+ 48t²+ 280t³+ 1632t⁴+ 9512t⁵+ +. . . In particularH_g 'C⁸ and H_g2 'C⁴⁸. After choosing a basis the multiplication map m : H_g⊗ H_g → H_g2 is represented by a matrix M ∈ M_48,64. We take as above {ϕ_α ⊗ϕβ|α, β = 1, ...,8} as basis for H_g ⊗ H_g and {ψ_γ|γ = 1, ...,48} as basis for H_g2. We write the matrix corresponding to the structure constants as M ' M(1)⊕ · · · ⊕M(8) where the elements of M(µ)∈ M_6,8(C) are given by

M(µ)i,j =ϑ_q(µ)+Λ_i,j(−48τ) Withq(µ) = ^17µ₄₈ +¹₆ and

Λ = 1 48







26 44 14 32 2 20 38 8 34 4 22 40 10 28 46 16 42 12 30 0 18 36 6 24 2 20 38 8 26 44 14 32 10 28 46 16 34 4 22 40 18 36 6 24 42 12 30 0







Example 6.4. Let g=

−1 −1

7 6

, θ=

√ 21−7

14

Fix now a complex structure τ on A_θ and consider the corresponding connection ¯∇₀ on theA_θ-bimodule E_g(θ) = E_6,7(θ). The Hilbert series for B_g(τ, θ) is given by (30):

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h_B_g_(τ,θ)(t) = 1 + 2t+t² 1−5t+t²

= 1 + 7t+ 35t²+ 168t³+ 805t⁴+ 3857t⁵+. . .

In particularH_g 'C⁷ and H_g2 'C³⁵. After choosing a basis the multiplication map m : H_g⊗ H_g → H_g2 is represented by a matrix M ∈ M_35,49. We take as above {ϕ_α ⊗ϕ_β|α, β = 1, ...,7} as basis for H_g ⊗ H_g and {ψ_γ|γ = 1, ...,35} as basis for H_g2. We write the matrix corresponding to the structure constants as M ' M(1)⊕ · · · ⊕M(7) where the elements of M(µ)∈ M_5,7(C) are given by

M(µ)_i,j =ϑ_q(µ)+Λ_i,j(−48τ) Withq(µ) = ^17µ₄₈ +¹₆ and

Λ = 1 35







2 32 27 22 17 12 7 9 4 34 29 24 19 14 16 11 6 1 31 26 21 23 18 13 8 3 33 28 30 25 20 15 10 5 0







Example 6.5. Let g=

4 1 15 4

, θ=−

√ 15 15

Fix now a complex structure τ on A_θ and consider the corresponding connection ¯∇₀ on theA_θ-bimodule E_g(θ) =E_4,15(θ). The Hilbert series for Bg(τ, θ) is given by (30):

h_B_g_(τ,θ)(t) = 1 + 7t+t² 1−8t+t²

= 1 + 15t+ 120t²+ 945t³+ 7440t⁴+ 58575t⁵+. . . In particular H_g ' C¹⁵ and H_g2 ' C¹²⁰. After choosing a basis the multiplication map m : H_g ⊗ H_g → H_g2 is represented by a matrix M ∈ M_120,225. We take as above {ϕ_α⊗ϕ_β|α, β= 1, ...,15} as basis forH_g⊗ H_g and{ψ_γ|γ= 1, ...,120}as basis forH_g2. We write the matrix corresponding to the structure constants asM 'M(1)⊕ · · · ⊕M(15) where the elements of M(µ)∈ M_8,15(C) are given by

M(µ)i,j =ϑ_q(µ)+Λ_i,j(−120τ)

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Withq(µ) = ^31µ₁₂₀ +¹₈ and Λ∈ M_15×8(₁₂₀¹ Z/Z) is given by Λi,j =−4j

15− i

8; i= 1, . . . ,8; j= 1, . . . ,15

Example 6.6. Let g=

−1 −1 11 10

, θ=−

√77−11 22

Fix now a complex structure τ on A_θ and consider the corresponding connection ¯∇₀ on the A_θ-bimodule E_g(θ) = E_11,10(θ). The Hilbert series forBg(τ, θ) is given by (30):

h_B_g_(τ,θ)(t) = 1 + 2t+t² 1−9t+t²

= 1 + 11t+ 99t²+ 880t³+ 7821t⁴+ 69509t⁵+. . . In particular H_g ' C¹¹ and H_g2 ' C⁹⁹. After choosing a basis the multiplication map m : H_g ⊗ H_g → H_g2 is represented by a matrix M ∈ M_99,121. We take as above {ϕ_α⊗ϕβ|α, β = 1, ...,11} as basis for H_g⊗ H_g and {ψ_γ|γ = 1, ...,99}as basis for H_g2. We write the matrix corresponding to the structure constants asM 'M(1)⊕ · · · ⊕M(11) where the elements of M(µ)∈ M_9,11(C) are given by

M(µ)i,j =ϑ_q(µ)+Λ_i,j(−99τ)

Withq(µ) = ^89µ₉₉ +¹₉ and Λ∈ M_11×9(₉₉¹Z/Z) is given by Λi,j =−10j

11 − i

9; i= 1, . . . ,9; j= 1, . . . ,11

Example 6.7. Let g=

6 1 35 6

, θ=−

√35 35

Fix now a complex structure τ on A_θ and consider the corresponding connection ¯∇₀ on theA_θ-bimodule E_g(θ) =E_6,35(θ). The Hilbert series for B_g(τ, θ) is given by (30):

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h_B_g_(τ,θ)(t) = 1 + 23t+t² 1−12t+t²

= 1 + 35t+ 420t²+ 5005t³+ 59640t⁴+ 710675t⁵+. . . In particular H_g ' C³⁵ and H_g2 ' C⁴²⁰. After choosing a basis the multiplication map m : H_g ⊗ H_g → H_g2 is represented by a matrix M ∈ M_420,1225. We take as above{ϕ_α⊗ϕ_β|α, β = 1, ...,35}as basis forH_g⊗ H_g and{ψ_γ|γ= 1, ...,420}as basis forH_g2. We write the matrix corresponding to the structure constants asM 'M(1)⊕ · · · ⊕M(35) where the elements of M(µ)∈ M_12,35(C) are given by

M(µ)_i,j =ϑ_q(µ)+Λ_i,j(−120τ)

Withq(µ) = ^71µ₄₂₀ +₁₂¹ and Λ∈ M_35×12(₄₂₀¹ Z/Z) is given by Λ_i,j =−6j

35 − i

12; i= 1, . . . ,12; j= 1, . . . ,35.

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Arithmetic structures on noncommutative tori with real multiplication