Pseudo-real multiplication and an application to Teichmüller curves

# Pseudo-real multiplication and an application to Teichmüller curves

Kolja Hept
###### Abstract

In this paper, we classify three-dimensional complex Abelian varieties isogenous to a product , where one of the factors admits real multiplication by a real quadratic order of discriminant . We show that the moduli space of these varieties essentially is the disjoint union of certain Hilbert modular varieties , each component depending on the choice of an ideal of . We give an explicit construction of these varieties.

We show that the boundary of the eigenform locus for pseudo-real multiplication by an order in over geometric genus zero stable curves is contained in the union of subvarieties defined by equations involving cross-ratios of projective coordinates. Moreover, restricted to certain topological types of stable curves relevant to the classification of primitive Teichmüller curves, we show that the boundary of the eigenform locus coincides with the subspace cut out by these cross-ratio equations. We compute these equations for the example of genus three Prym Teichmüller curves.

## 1 Introduction

Let  be the moduli space of Riemann surfaces of a fixed genus . Teichmüller curves in  arise naturally in the study of billiard tables and combine such various topics as algebraic geometry, ergodic theory, flat geometry and number theory. Much progress was made in classifying Teichmüller curves in the last two decades. We summarize the state of the art in Section 2. Teichmüller curves can roughly be distinguished by their trace field, a totally real number field of degree at most . Consider the bundle of non-zero holomorphic one-forms over . If any pair  generates a Teichmüller curve , then by [MartinVHS] the Jacobian  of  lies in the locus of principally polarized Abelian varieties isogenous to a product, where the one factor admits real multiplication by the trace field of  (and is consequently of dimension ). If the trace field of a Teichmüller curve is , then the curve comes from certain covering constructions of a torus.

Now consider the case . It is shown in [MartinMattPhilipp], that there are only finitely many Teichmüller curves with . Many partially results are known for the case . In particular for the Prym eigenform locus, there are several classification statements in [LanneauNguyen2] and [LanneauNguyen1] similar to those by McMullen in the genus two case. Furthermore, it is shown in [WrightOdd] and [WrightHyp], that there are only finitely many Teichmüller curves with in the minimal stratum outside of the Prym eigenform locus.

In this paper, we are dealing with the three-dimensional case where the trace field of has degree two over . Thus is isogenous to a product of an elliptic curve and a complex Abelian surface that admits real multiplication by a quadratic order  of some positive, non-square discriminant . The choice of such an Abelian subsurface together with the choice of real multiplication is equivalent to the choice of an endomorphism

 Q(√D)⊕Q↪End+(A)⊗Q,

what we will call pseudo-real multiplication.

Since we do not want to restrict ourselves to the Prym eigenform locus, the subvariety of admitting real multiplication can be of any type . Our first goal will be to classify all three-dimensional complex Abelian varieties admitting pseudo-real multiplication by a pseudo-cubic numberfield together with the choice of such an endomorphism. We will introduce in Section 5.2 a certain condition on depending on the discriminant , which we will call the prime factor condition. Let be relatively prime to the conductor of  and satisfying the prime factor condition. Then for each ideal of norm we will construct explicitly a Hilbert modular variety . Denoting by the moduli space of three-dimensional complex Abelian varieties together with some choice of pseudo-real multiplication by of type , we will prove the following theorem.

###### Theorem 1.1.

Let be a non-square discriminant and let be relatively prime to the conductor of . Then is non-empty if and only if satisfies the prime factor condition in . Moreover, in this case consists of the irreducible components , where is the number of splitting prime divisors of .

Let be the bundle of stable forms over the Deligne-Mumford compactification of . If is a lattice in a totally real number field of degree , then each boundary stratum of -weighted geometric genus zero curves can be embedded into the boundary of . Let be the locus of stable forms , where is some choice of a -eigenform for real multiplication on  by an order . In [MartinMatt], Bainbridge and Möller gave a certain condition for -weighted boundary strata called admissibility. They showed that each stable form in the boundary of lies in the image of some , a certain subvariety of an admissible boundary stratum  defined by some . Using the techniques developed in this paper, we will show that an analogous statement also holds in the pseudo-cubic case.

Furthermore, Bainbridge and Möller defined in [MartinMatt] for the genus three case the quadratic function mapping each to the number . They showed that an -weighted boundary stratum with weights is admissible if and only if the set is not contained in a closed half-space of its -span.

Using coordinates, we will also define a quadratic function

 Q: Q(√D)⊕Q→Q3.

After the preliminary work in the sections before, we can construct this function precisely in that way, that the analogous statement like in the cubic case holds. A boundary stratum of -weighted stable curves is isomorphic to a product of certain , the moduli spaces of genus zero curves with marked points. Thus each weighted stable curve in can be represented by projective coordinates. Using cross-ratios of these coordinates, it is shown in [MartinMatt], that each subvariety of an admissible stratum is cut ot by a single equation in these cross-ratios. We will show the analogous statement for the stratum of irreducible stable curves with cross-ratios  as follows.

###### Theorem 1.2.

Let be a -basis of and let be its dual basis of  with respect to the pseudo-trace pairing. The subvariety of the stratum of trinodal curves with weights is cut out by the equations

 pa123⋅pa213⋅pa312=exp(−2πi(a1b23+a2b13+a3b12)),

where the are certain rational numbers defined by and where runs over the integral solutions of

 a1s2s3+a2s1s3+a3s1s2=0. (1)

Contrary to the cubic case in [MartinMatt], where the subvariety is cut out by a single equation, note that the -module of solutions of Equation (1) may have rank two, see Example 8.19. This is a consequence of the fact, that in the pseudo-cubic case the trinodal boundary stratum is in general not a maximal dimensional boundary stratum. Furthermore, in [MartinMatt] it turns out, that for genus three, admissibility is also a sufficient condition, i.e. that the boundary of the real multiplication locus is precisely the union of all images in of the varieties , where  runs over all admissible boundary strata. We show that this statement holds in the pseudo-cubic case at least for those boundary strata relevant for classifying Teichmüller curves.

The quadratic function and the cross-ratio equations could be used in future work for constructing an algorithm to classify Teichmüller curves by their cusps in a similar spirit as the algorithm given in [MartinMatt]. Fixing the discriminant  and the type , one first uses the function  to list admissible bases for all lattices defining admissible -weighted boundary strata and then solve the cross-ratio equations. Presumably, one can use the techniques from [MartinMattPhilipp], to get a finiteness result for primitive (but not algebraically primitive) Teichmüller curves in .
Acknowledgements. This paper is my PhD-thesis at the Goethe Universität Frankfurt am Main. First of all I would like thank my adivsor Martin Möller for making all this work possible and for his excellent guidance. I am also grateful to my colleagues Matteo Constantini, Quentin Gendron, Thorsten Jörgens, Şevda Kurul, David Torres Teigell and Jonathan Zachhuber for proofreading this paper. Finally I would like to thank my family, especially my parents, and my friends for their continuous support.

## 2 Teichmüller curves: state of the art

The moduli space of genus compact Riemann surfaces carries a natural metric induced by the Teichmüller metric on its universal covering orbifold, the Teichmüller space . There is a natural action of on , the space of genus marked Riemann surfaces together with a non-zero quadratic differential. This action is induced by linear transformations on the quadratic differential. Fixing any point , post-composition of with the forgetful map factors through the quotient and yields an isometric embedding

 H=SO2(R)∖SL2(R)↪Tg.

Post-composing with the covering map , we obtain a complex geodesic

 f: H→Mg,

and we say that generates . Conversely, by a consequence of Teichmüller’s Theorem, every complex geodesic in is generated by a quadratic differential in this way. Of special interests are the complex geodesics generated by squares of a holomorphic one-form. One reason is, that by [Kra], Theorem 1, this happens if and only if the composed map

 H→Mg→Ag

is a geodesic (here denotes the Torelli map), where denotes the moduli space of -dimensional, principally polarized Abelian varieties. Another reason is, that any complex geodesic is commensurable to a complex geodesic generated by the square of a holomorphic one-form. There is also a natural action of (respectively ) on , the space of Riemann surfaces of genus together with a non-zero holomorphic one-forms. If for some non-zero , and setting for all , then the diagram

 SL2(R)F−−−−→ΩMg⏐⏐↓⏐⏐↓H=SO2(R)∖SL2(R)f−−−−→Mg

commutes. Thus one can look at holomorphic one-forms instead of quadratic differentials. Any Riemann surface with a holomorphic one-form can also be seen as a translation surface with singularities. The translation atlas is given by integrating , which allows a geometric approach to this topic. For this reason, any such pair is called a flat surface.
Teichmüller curves. A complex geodesic in , which is simultaneously an algebraic curve (which is usually not the case), is called a Teichmüller curve. It is well known that a flat surface generates a Teichmüller curve if and only if its -orbit is closed in . Furthermore, generates a Teichmüller curve if and only if or equivalently its Veech group (the image in of orientation-preserving affine diffeomorphisms under the derivative map) is a lattice in . Such a flat surface is called a Veech surface. In every stratum there are infinitely many Teichmüller curves, each of them generated by a torus covering ramified only over one point (also known as square-tiled surfaces).

In his famous work [Veech89], Veech constructed first examples of Teichmüller curves not of this type. He showed that the Veech group of any translation surface is a discrete subgroup of . Moreover, if it is a lattice, then it satisfies the Veech dichotomy: The geodesic flow in any direction is either periodic or uniquely ergodic. These surfaces with a lattice Veech group became an object of fruitful research, as they are combining such various topics as algebraic geometry, ergodic theory, flat geometry, number theory and Teichmüller theory.
Primitivity. A compact Riemann surface together with a quadratic differential  (respectively a flat surface ) is said to be geometrically primitive, if it is not the pullback of a quadratic differential (respectively a flat surface) of lower genus. A complex geodesic is said to be geometrically primitive, if any one (and thus each) generating differential is geometrically primitive. Möller showed in [MartinPeriodic], Theorem 2.6 that each flat surface is the pullback of a primitive flat surface . The primitive flat surface is also unique provided that the genus of  is at least two. Moreover, he showed that and are commensurable if generates a Teichmüller curve. Thus, for classification of Teichmüller curves it makes sense to classify the primitive ones.
Classification. During the last twelve years, much effort was made to classify all Teichmüller curves generated by flat surfaces. A complete classification for primitive Teichmüller curves in was achieved by McMullen in a series of papers in 2003 and 2004 ([McMBilliards], [McMSpin], [McMDecagon], [McMSines]). The central role here plays the Weierstrass locus , where is a positive discriminant (i.e. ). It consists of those genus two Riemann surfaces  such that the Jacobian  of  admits real multiplication by the real quadratic order  and such that there exists an eigenform  for some choice of real multiplication which has a double zero. The classification can be summarized as follows.

###### Theorem A (McMullen).

Let be a non-square discriminant.

1. If , then is a single primitive Teichmüller curve, generated by a certain -shaped translation surface.

2. If , then consists of two irreducible components, distinguished by their spin invariant. Each of the two components is a primitive Teichmüller curve generated by a certain -shaped translation surface.

3. The regular decagon is an eigenform in for real multiplication, and its orbit projects to a primitive Teichmüller curve in .

These curves are all primitive Teichmüller curves in , and they are pairwise different.

The curves in were indepently discovered by Calta in [Calta].
The Prym-eigenform locus. The next step was to look for primitive Teichmüller curves in higher genera, using a generalization of , namely the Prym-Weierstrass locus. This locus, denoted by , is defined to be the locus of Riemann surfaces , such that

1. there is a holomorphic involution , such that the Prym variety

 Jac(X)−:=(Ω(X)−)∗/H1(X;Z)−⊂Jac(X)

has dimension two (where denotes the eigenspace of to the eigenvalue ),

2. admits real multiplication by , and

3. there exists an eigenform for some choice of real multiplication which has only one zero.

By the Riemann-Hurwitz formula, is empty if . In genus two, this definition coincides with the old one, since each genus two Riemann surface is hyperelliptic. In [McMPrym], McMullen constructed infinitely primitive Teichmüller curves contained in for . He also showed that for not a square, the locus is a finite union of primitive Teichmüller curves.

Removing the ’only one zero’-condition in the definition of , we define  to be the locus of those , such that there is a holomorphic involution , with a two-dimensional subvariety of that admits real multiplication by the quadratic order , and such that  is in and an eigenform for some choice of real multiplication. For any partition of , there is the natural stratum in consisting of those flat surfaces , where  has exactly zeroes with of multiplicity . We set and finally we denote by respectively the image of respectively in  under the forgetful map. Note that and that the decagon form in Theorem 2 generates the only primitive Teichmüller curve in .
The trace field and splitting varieties. The trace field of a holomorphic form is defined to be the field extension of generated by the traces of all the elements in . By [McMBilliards], Theorem 5.1 we have

 [Q(SL(X,ω)):Q]≤g.

Flat surfaces with are called algebraically primitive, and the Teichmüller curve generated by such a form is also called algebraically primitive. Since the trace field is preserved by translation coverings, each algebraically primitive Teichmüller curve is also geometrically primitive. All the primitive Teichmüller curves mentioned above are generated by flat surfaces , such that is isogenous to a product of Abelian varieties, and where the one factor admits real multiplication by , in these cases a real quadratic number field.

This had to happen, as Möller has shown in [MartinVHS]:

###### Theorem B (Möller).

Let be a flat surface generating a Teichmüller curve  in . Then we have the following.

1. The trace field is totally real.

2. The image of under the Torelli map is contained in the locus of Abelian varieties isogenous to , where has dimension and real multiplication by . The generating differential is an eigenform for real multiplication by .

Thus, we have three possibilities for a flat surface generating a Teichmüller curve . The first one is, that the trace field is (these curves are called arithmetic). Then covers a torus, in particular is not primitive. The second one is, that is a totally real cubic number field, i.e. is algebraically primitive. In [MartinMattPhilipp], Bainbridge, Habegger and Möller have shown that there are only finitely many algebraically primitive Teichmüller curves in . The remaining case, where is a real quadratic number field, is what we are dealing with in this paper. Therefore, we are interested in three-dimensional principally polarized complex Abelian varieties isogenous to a product , where is a one-dimensional polarized complex Abelian variety of type  and is a two-dimensional polarized complex Abelian variety of type  admitting real multilplication by a real quadratic order .

In [LanneauNguyen1], Lanneau and Nguyen proved the following classification of the Prym-Weierstraß locus.

###### Theorem C (Lanneau, Nguyen).

For , is non empty if and only if or . All the loci are pairwise disjoint. Moreover, for the values ,, of discriminants, the following dichotomy holds. Either

1. is odd and then has exactly two connected components,

2. is even and is connected.

In addition, each component of corresponds to a closed -orbit. For , is non-empty if and only if and in this case it is connected.

In [LanneauNguyen2], they showed that the analogous satement holds for the loci and for . Note that the Prym variety for a genus three Riemann surface always carries a polarization of type . Beyond the curves of Prym loci, there are currently no classification results known for Teichmüller curves in genus three that are primitive but not algebraically primitive.

## 3 Complete statements of the results

As mentioned at the end of the last section, we are interested in principally polarized complex Abelian varieties isogenous to a product , where is a one-dimensional polarized complex Abelian variety of type  and is a two-dimensional polarized complex Abelian variety of type  admitting real multilplication by a real quadratic order . We will see in Section 7.1, that this is equivalent to the concept of pseudo-real multiplication.
Moduli-space of Abelian varieties with pseudo-real multiplication. Our first goal is to parameterize all these polarized Abelian varieties together with a choice of some real multiplication. With this goal in mind, we first consider the two-dimensional part. For a non-square discriminant and we denote by the space of all isomorphism classes , where is a polarized Abelian surface of type that admits real multiplication by and is some choice of real multiplication. If , it is well known that is isomorphic to the Hilbert modular surface

 XD=PSL2(OD⊕O∨D)∖H×H.

The reason that consists only of one irreducible component, is that every rank two symplectic -module (the lattice defining the Abelian surface) is isomorphic to , where is equipped with the natural trace pairing. This does not hold in the non-principal case. It turns out that, provided that the quotient is relatively prime to the conductor of , the lattice defining the surface is as a symplectic -module isomorphic to for some ideal of norm . The locus of such Abelian surfaces is denoted by

 Xa⊂XD,(d1,d2).

To count the irreducible components of , consider the prime factorization of . If is relatively prime to the conductor of , then we say that the pair satisfies the prime factor condition in , if the following two conditions hold.

1. No is inert over and

2. If is ramified over , then .

In Section 6.1, we use the prime ideal factorization for to prove the following.

###### Theorem 6.6.

Let be a non-square discriminant and let with and relatively prime to the conductor of . Then is non-empty if and only if satisfies the prime factor condition in . In this case consists of irreducible components , where is the number of splitting prime divisors of . For each component defined by the ideal , we have an isomorphism

 Φ: PSL2(OD⊕1√Da)∖H2≅Xa,[z]↦[(Az,Hz,ρz)]

between the corresponding Hilbert modular surface and .

Here we have seen why it makes sense to presuppose that  is relatively prime to the conductor of . Otherwise we would not have a unique prime ideal factorization for the ideal .

Now let us go back to dimension three. Similar to the lower dimensional case above, we denote by the space of all isomorphism classes , where  is a principally polarized Abelian variety of dimension three, is a two dimensional subvariety of  of type and is some choice of real multiplication by on . Again, we require that the degree  is relatively prime to the conductor of . We say that is of type for an ideal , if is in . We denote the locus of such classes by

 X(3)a⊂X(3)D,d.

Analogously to the dimension two case, in Section 6.3 we construct for each such a variety and get the following theorem.

###### Theorem 6.16.

Let be a non-square discriminant and let be relatively prime to the conductor of . Then is non-empty if and only if  satisfies the prime factor condition in . In this case consists of irreducible components , where  is the number of splitting prime divisors of . For each component defined by the ideal we have an isomorphism

 Φ: ΓD,d(η1,η2)∖H3≅X(3)a,[z]↦[(Az,Hz,sz,ρz)]

between the corresponding Hilbert modular variety and .

We will give an explicit description of the group , a finite-index subgroup of . Unfortunately, depends not only on the ideal , but also on the choice of a -basis of . Furthermore, does not have the structure of a direct product. We can construct a universal family

 ΓD,d(η1,η2)⋉Z6∖(H3×C3)→X(3)a

for those components, but for many other problems (like counting the cusps of ) the group is much too unwieldy. Nevertheless, we can give a lower and an upper bound for (with both inclusions of finite index), which do not dependent on any choice of basis and which are direct products. More precisely, we have the inclusion

 SL(1+a√DOD1√Da21+a)×Γ(d)⊆ΓD,d(η1,η2)⊆SL2(OD⊕1√Da)×SL2(Z)

and we interpret the quotients of by these groups as moduli spaces of polarized complex Abelian varieties with real multiplication and certain level structures.
Generalized period coordinates and cross-ratio equations. Before the final statement in [MartinMattPhilipp] mentioned in the previous section, Bainbridge and Möller have proven in [MartinMatt] the finiteness of algebraically primitive curves in  generated by flat surfaces in the stratum , using a generalization of the classical period coordinates.

Let be the space of arithmetic genus  stable curves together with a Lagrangian marking (weighting) by a rank  free Abelian group . There is a natural stratification of  defined by weight preserving homeomorphisms. If is a lattice in a totally real number field of degree , then each boundary stratum in  can be embedded into the boundary of , where  is the bundle of stable forms over the Deligne-Mumford compactification of . Bainbridge and Möller showed that a necessary condition for a boundary point  for lying in the closure of the real multiplication locus in is the following. There is some subvariety defined by of a so-called admissible -weighted boundary stratum , such that  lies in the image of under the forgetful map . They also showed, that this condition is sufficient in genus three.

The key was a description of the closure of the eigenform locus (where  is an order in a totally real number field  of degree ) in . For any free Abelian group of rank , they constructed a homomorphism

 Ψ: SZ(Hom(L,Z))→Hol∗(Mg(L)),

where denotes the symmetric square and the moduli space of Riemann surfaces together with a Lagrangian marking by . This homomorphism can be seen as a coordinate free version of exponentials of the classical period coordinates. They proved that for each , the function  extends to a meromorphic function on  and computed the orders of vanishing.

In genus three, they constructed a quadratic function and showed that a weighted boundary stratum is admissible if and only if the -images of the weights are not contained in a closed half-space of their -span. This geometric condition is called the no-half-space condition. For each , let be the moduli space of  labeled points on . Each -weighted boundary stratum is isomorphic to a product of certain , one factor for each irreducible component. Thus, any point of can be represented by an ordered tuple of elements in . They described the function  in terms of cross-ratios of these projective coordinates, and constructed single explicit equations cutting out the subvarieties . They used these so-called cross-ratio equations to prove the finiteness statement for algebraically primitive Teichmüller curves in the stratum . Moreover, they constructed an algorithm that searches for possible cusps (necessarily represented by irreducible stable forms) of algebraically primitive Teichmüller curves in the stratum . This algorithm works as follows. Fixing an order in a totally real cubic number field, first one lists all admissible bases of ideals in . For each admissible basis, there are just finitely many irreducible stable forms having these residues and a single zero. Then one checks, if the cross-ratio equation holds.

Our goal in Section 8 is, to construct analogues for the locus of those genus three Riemann surfaces , where admits multiplication by a pseudo-cubic number field . We first show that an isomorphism class of , where is such a choice of pseudo-real multiplication on a three-dimensional principally polarized Abelian variety, is the same as a class in . We formulate admissibility for the pseudo-cubic case and give the following necessary condition for lying in the boundary of the pseudo-real multiplication locus.

###### Corollary 8.13.

Let be a pseudo-cubic order. Each geometric genus zero stable curve lies in the image under the forgetful map of some admissible -weighted boundary stratum .

We construct a quadratic function and give a similar geometric reformulation of admissibility as in the cubic case. To be more precise, we define

 Q: F→Q3,(x,q)↦⎛⎜ ⎜⎝N(√Dx)tr(x)qtr(√Dx)q⎞⎟ ⎟⎠.

Then we show the following criterion.

###### Theorem 8.14.

An -weighted boundary stratum is admissible if and only if the set

 Q(S):={Q(w):w∈Weight(S)}

is not contained in a closed half-space of its -span.

Then we give also explicit equations in terms of cross-ratios that cut out the subvarieties . In the boundary stratum of irreducible stable curves of geometric genus zero, which we also call the stratum of trinodal type, this is the following.

###### Theorem 8.18.

Let be a -basis of , let be its dual basis of with respect to the pseudo-trace pairing and let be the -weighted boundary stratum of trinodal curves with weigths . Moreover, if is an element of , then we write

 h=3∑i,j=1bij(ri⊗rj)

with and . We identify with , the moduli space of six labeled points in , and denote the cross-ratios by

 pjk:=(pj,qj;qk,pk)∈C∖{0,1}

for with pairwise distinct entries. Then, the subvariety of is cut out by the equations

 pa123⋅pa213⋅pa312=exp(−2πi(a1b23+a2b13+a3b12)),

where runs over the integral solutions of

 a1s2s3+a2s1s3+a3s1s2=0.

As mentioned in the introduction, note that contrary to the cubic case, in general we do not get a single equation anymore. We get an analogous result for a second type of stratum, which we will call to be of nice non-trinodal type.

A cusp packet for a pseudo-cubic order  is a pair with a lattice  in  and a certain symplectic extension class . We will see that any boundary point of the pseudo-real multiplication locus corresponds to a cusp packet. Hence, at least for the strata relevant for Teichmüller curves, we can use the cross-ratio equations to get the following description of the boundary of the eigenform locus for pseudo-real multiplication.

###### Theorem 8.22.

Let be a pseudo-cubic order of degree relatively prime to the conductor of , and let be a cusp packet for . Furthermore, let be one of the two real quadratic pseudo-embeddings.

Then the intersection of the closure of the cusp of associated to with the union of the strata of trinodal and nice non-trinodal type is equal to the union of all , the -pseudo-embeddings of the subvarieties  of , where  runs over all admissible -weighted boundary strata of trinodal and nice non-trinodal type.

Finally, for doing a crosscheck, for every we take the Teichmüller curve generated by a specific -shaped surface (known as Thurston-Veech-construction) in the locus . We show that a cusp indeed lies in the image of some admissible -weighted boundary stratum and we compute the (in this case single) defining cross-ratio equation for the subvariety containing .

## 4 Complex Abelian varieties

Compact Riemann surfaces of a fixed genus can be parameterized via the Torelli map by their Jacobians, principally polarized complex Abelian varieties of dimension . Although this is our main motivation, arbitrary (not necessarily principally) polarized complex Abelian varieties are objects also worth for studying them in detail. In this paper non-principal polarized Abelian varieties appear as subvarieties of higher dimensional principally polarized ones. Here we summarize standard notations for complex Abelian varieties and a section with technical lemmata on symplectic forms on lattices.

### 4.1 Complex tori and Abelian varieties

We recall some definitions and properties concerning complex Abelian varieties. If not other stated, we follow [LangeBirke] and refer the reader there for more details.
Complex tori and period matrices. A lattice in a finite-dimensional complex vector space is a discrete subgroup of of maximal rank. Thus, if  has complex dimension , then is a free Abelian group of rank . The quotient , equipped with the Lie group structure inherited by , is called a complex torus, a connected compact complex Lie group of dimension . Conversely, each connected compact complex Lie group of dimension is isomorphic to a complex torus of dimension , since the universal cover of is a complex vector space of dimension  and the kernel of the universal covering map is a lattice in .

Any subset of a complex torus is a subtorus if and only if it is of the form , where is a subspace such that the intersection is a lattice in . A complex torus is said to be simple, if it does not have any non-trivial subtorus.

A homomorphism of complex tori is a holomorphic map compatible with the group structures. Each homomorphism of complex tori can be lifted to a unique -linear map . We call the analytic representation of and its restriction to the lattices the rational representation of . Conversely, a -linear map descends to a homomorphism of complex tori if and only if . A homomorphism of complex tori is called an isogeny, if it is surjective with finite kernel, and this holds if and only if its analytic representation is an isomorphism of vector spaces. Consequently, an isomomorphism is an isogeny with trivial kernel, respectively .

If is a -basis of and is a -basis of , then we can write and call the matrix

 Π=⎛⎜ ⎜⎝α1,1⋯⋯α1,2g⋮⋮αg,1⋯⋯αg,2g⎞⎟ ⎟⎠∈Mg,2g(C)

a period matrix for . Let and be period matrices for the tori and with respect to certain bases and let be a homomorphism. Then is represented by a matrix with respect to the chosen -bases and is represented by a matrix with respect to the chosen -bases and the relation

 MΠ=Π′R

holds.
Complex Abelian varieties and polarizations. From the viewpoint of algebraic geometry, a polarization on a complex torus is by definition the first Chern class of a positive definite holomorphic line bundle on . For our purposes it is more convenient to take an equivalent definition in terms of Hermitian forms.

###### Definition 4.1.

Let be a complex torus. A polarization on is a positive definite Hermitian form on , such that .

If there exists a polarization on a complex torus , then is called a complex Abelian variety and the pair is called a polarized complex Abelian Variety. A homomorphism of polarized complex Abelian varieties is a homomorphism of complex tori, such that the analytic representation preserves the Hermitian forms. For simplicity, we will ommit the term ’complex’ often.

Any positive definite Hermitian form on gives rise to a non-degenerated alternating -bilinearform

 EH: V×V→R,EH(v,w):=Im(H(v,w))

with the property that holds for all . Conversely, any non-degenerated alternating -bilinearform on with this property defines a positive definite Hermitian form

 HE: V×V→C,HE(v,w):=E(iv,w)+iE(v,w).

The assignment is a group isomorphism between the Néron-Severi group , the group of Hermitian forms on with , and the group of alternating -bilinearforms on with and for all . Its inverse is and it induces a bijection between the positive definite Hermitian forms and the non-degenerated alternating forms.

A symplectic form on a free -module is a non-degenerated alternating bilinearform with integer values. A symplectic module homomorphism of modules with symplectic forms is a module homomorphism preserving the forms. If the module with symplectic form is of finite rank, then it is of even rank and there exists a -basis of , such that is given by a matrix of the form

 (0D−D0)

with respect to this basis, where with and for all . Such a basis is called a symplectic basis and the tuple , as well as the diagonal matrix , is called the type of . The type does not depend on the choice of the basis, and so does the degree of , defined as . If is a polarization of an Abelian variety and if we speak of the type, respectively the degree of , we mean the type, respectively the degree of . We also refer to a symplectic basis for as a symplectic basis for . A polarization is said to be principal, if it is of type . That is, if and only if is unimodular.

Each subtorus of an Abelian variety is also an Abelian variety, because any polarization on induces the polarization on . Let be the orthogonal complement of in with respect to . Then is a lattice in and is called the complementary subvariety of  (with respect to ). In the case that is principal, we have the following relation between the types of the induced polarizations on complementary subvarieties.

###### Proposition 4.2.

Let be a principally polarized Abelian variety and let be complementary subvarieties with . If is of type , then is of type .

This is Corollary 12.1.5. in [LangeBirke]. In the next subsection, we will give a similar proof for the slightly more general situation, where we do not have a complex structure on our objects.
The moduli space of polarized Abelian varieties. Given any such type , polarized Abelian varieties of type can be constructed in the following way. We denote by

 Hg:={Z∈Mg(C):Zt=Z, Im(Z)>0}

the upper Siegel half space, i.e. the space of symmetric complex -matrices with positive definite imaginary part. For each we define to be the lattice in generated by the columns of . Then the bilinearform on defined by the matrix with respect to the standard basis is a polarization on the complex torus of type . A symplectic basis is given by the columns of .

An isomorphism between two triples and where is a polarized Abelian variety and a symplectic basis for , is an isomorphism between and such that the rational representation maps the ordered tuple onto . It is a consequence of the Riemann Relations (see [LangeBirke], Section 4.2) that any such triple is isomorphic to for some . Thus the upper Siegel half space , a complex manifold of dimension , is a moduli space for polarized Abelian varieties of type together with the choice of a symplectic basis. To get rid of the choice of a basis, let be the -module generated by the columns of the matrix and let be the subgroup of the symplectic group consisting of those matrices with . Then acts transitively, properly and discontinously on via

 (αβγδ).Z:=(αZ+β)(γZ+δ)−1.

Given any two , the polarized Abelian varieties and are isomorphic if and only if and are in the same -orbit.

###### Theorem 4.3 ([LangeBirke], Theorem 8.2.6.).

The normal complex analytic space is a moduli space for polarized Abelian varieties of type .

Endomorphism structures and real multiplication. In this paper, we are interested in Abelian varieties with a specific endomorphism structure. In general, the endomorphism -algebra of an Abelian variety  is determined by the endomorphism structure of its simple subvarieties. The reason is that for any two complementary subvarieties , there is an isogeny between and . Then, by induction, we get the following decomposition.

###### Theorem 4.4 (Poincaré’s Complete Reducibility Theorem).

Every complex Abelian variety is isogenous to a product

 An11×...×Anrr,

with pairwise not isogenous simple Abelian varieties . Moreover, the natural numbers are uniquely determined by and the Abelian varieties are uniquely determined by up to isogeny.

Given such an isogeny