A Boundary strata in genus three: Algorithms, examples, counting

# The Deligne-Mumford compactification of the real multiplication locus and Teichmüller curves in genus three

## 1 Introduction

Each Hilbert modular surface has a beautiful minimal smooth compactification due to Hirzebruch. Higher-dimensional Hilbert modular varieties instead admit many toroidal compactifications none of which is clearly the best. In this paper, we consider canonical compactifications of closely related varieties, namely the real multiplication locus in the moduli space of genus Riemann surfaces, as well as the locus of eigenforms in the bundle of holomorphic one-forms.

If is or , we give a complete description of the stable curves in the Deligne-Mumford compactification which are in the boundary of , and which stable curves equipped with holomorphic one-forms are in the boundary of the eigenform locus . If , we give strong restrictions on the stable curves in the boundary of . This allows one to reduce many difficult questions about Riemann surfaces with real multiplication to concrete problems in algebraic geometry and number theory by passing to the boundary of . In this paper, we apply our boundary classification to obtain finiteness results for Teichmüller curves in and noninvariance of the eigenform locus under the action of on .

### Boundary of the eigenform locus.

We now state a rough version of our calculation of the boundary of the eigenform locus. See Theorems 5.2, 8.1, and 8.5 for precise statements. Consider a totally real cubic field , and let be the ring of integers (we handle arbitrary orders , but stick to the ring of integers here for simplicity). The Jacobian of a Riemann surface has real multiplication by roughly if the endomorphism ring of contains a copy of (see §2 for details). We denote by the locus of Riemann surfaces whose Jacobians have real multiplication by . Real multiplication on determines an eigenspace decomposition of , the space of holomorphic one-forms on . The eigenform locus is the locus of pairs , where has real multiplication by , and is an eigenform.

The bundle extends to a bundle whose fiber over a stable curve is the space of stable forms on . A stable form over a stable curve is a form which is holomorphic, except for possibly simple poles at the nodes, such that the two residues at a single node are opposite (see §3 for details). We describe here the closure of in , which also determines the closure of in .

Consider the quadratic map , defined by

 Q(x)=NFQ(x)/x. (1.1)

We say that a finite subset satisfies the no-half-space condition if the interior of the convex hull of in the -span of in contains .

It is well known that every stable curve which is in the closure of the real multiplication locus has geometric genus or (we give a proof via complex analysis in §5). Our description of the closure of the eigenform locus is as follows.

###### Theorem 1.1.

A geometric genus stable form lies in the boundary of the eigenform locus if and only if:

• The set of residues of is a multiple of , for some subset , satisfying the no-half-plane condition and spanning an ideal , and for some embedding .

• If lies in a -subspace of , then an explicit additional equation, involving cross-ratios of the nodes of , is satisfied.

###### Remark.

The more precise version of this theorem, which we state in §5, gives a necessary condition which holds more generally in any genus. In §8, we show that this condition is sufficient in genus three. In fact, it is sufficient also in genus two, but we ignore this case as the boundary of the eigenform locus was previously calculated in the genus two case in [Bai07]. The higher genus cases are more difficult, as the Torelli map is no longer dominant.

The boundary of has a stratification into topological types, where two stable forms are of the same topological type if there is a homeomorphism between them which preserves residues up to constant multiple. We may encode a topological type by a directed graph with the edges weighted by elements of an ideal . Vertices represent irreducible components, edges represent nodes, and weights represent residues. The corresponding boundary stratum of is a product of moduli spaces , or a subvariety thereof. The possible topological types arising in the boundary of are shown in Figure 1. In Appendix  A, we give an algorithm for enumerating all boundary strata of associated to a given ideal . In Figure 4, we tabulate the number of two-dimensional boundary strata for many different fields.

An important special case is boundary strata parameterizing irreducible stable curves, otherwise known as trinodal curves. Consider a basis of an ideal . We say that is an admissible basis of if the satisfy the no-half-space condition. Let be the locus of trinodal forms having residues . Since a trinodal curve may be represented by points in identified in pairs, we may identify with the moduli space of such points. Suppose is admissible. As three points in whose convex hull contains must be contained in a subspace, we are in the second case of Theorem 1.1, so is cut out by a single polynomial equation on . We see in Theorem 8.5 that this equation is

 Ra11Ra22Ra33=1, (1.2)

where are certain cross-ratios of four points and the are integers determined explicitly by the .

### Intersecting flats in SL3(Z)∖SL3(R)/SO3(R).

In §7, we show that the notion of an admissible basis of a lattice in a totally real cubic number field is equivalent to a second condition on bases of totally real number fields, which we call rationality and positivity. Namely, a basis of is rational and positive if

 risi/rjsj∈Q+for all i≠j,

where is the dual basis of with respect to the trace pairing.

There is a classical correspondence between ideal classes in totally real degree number fields and compact flats in the locally symmetric space , the moduli space of lattices in . Given an lattice in a totally real number field , let be the group of totally positive units preserving , embedded in the group of positive diagonal matrices via the real embeddings of . There is an isometric immersion of the flat torus into arising from the right action of on . Let be the locus of lattices in which have an orthogonal basis. is a closed, but not compact, -dimensional flat. In §7, we show that rational and positive bases of lattices in number fields correspond to intersections of the corresponding compact flat with .

###### Theorem 1.2.

Given an lattice in a totally real number field, there is a natural bijection between the set and the set of rational and positive bases of up to multiplication by units, changing signs, and reordering.

Theorems 1.1 and 1.2 together imply that there is a natural bijection boundary strata of eigenform loci and intersection points of compact flats in with the distinguished flat . Note that is -dimensional, while each flat in is at most -dimensional, so one would not expect many intersections between these flats. Nevertheless, we show in §9 that the ring of integers in each totally real cubic field has some ideal which has an admissible basis. In fact, the computations described in Appendix A suggest that most lattices in cubic fields have many admissible bases, although there are also examples of lattices which have none. It would be an interesting problem to study the asymptotics of counting these bases.

### Algebraically primitive Teichmüller curves.

There is an important action of on , the study of which has many applications to the dynamics of billiards in polygons and translation flows. A major open problem is the classification of -orbit-closures. In genus two, this was solved by McMullen in [McM07], while next to nothing is known for higher genera.

Very rarely, a form has a -stabilizer which is a lattice in . In that case, the -orbit of projects to an algebraic curve in which is isometrically immersed with respect to the Teichmüller metric. Such a curve in is called a Teichmüller curve. A Teichmüller curve is uniformized by a Fuchsian group , called the Veech group of . The field generated by the traces of elements in is called the trace field of . The trace field is a totally real field of degree at most . See §10 for basic definitions around Teichmüller curves and the -action.

Our main motivation for this work was the problem of classifying algebraically primitive Teichmüller curves in , that is Teichmüller curves whose trace field has degree . Every algebraically primitive Teichmüller curve lies in for some order in its trace field by [Möl06b], and every Teichmüller curve has a cusp, so Theorem 1.1 allows one to approach the classification of Teichmüller curves by studying the possible stable curves which are limits of their cusps.

In , each eigenform locus is -invariant and contains one or two Teichmüller curves (see [McM03, McM05]). These Teichmüller curves lie in the stratum (where we write for the stratum of forms having zeros of order ). These Teichmüller curves were discovered independently by Calta in [Cal04].

A major obstacle to the existence of algebraically primitive Teichmüller curves in higher genus is that the eigenform loci are no longer -invariant. McMullen showed in [McM03] that is not -invariant for the ring of integers in . We prove in §11 the following stronger non-invariance statement

###### Theorem 1.3.

The eigenform locus is not invariant for the ring of integers in any totally real cubic field.

In contrast to the situation in , we give in this paper strong evidence for the following conjecture.

###### Conjecture 1.4.

There are only finitely many algebraically primitive Teichmüller curves in .

In §13, we prove the following instance of this conjecture.

###### Theorem 1.5.

There are only finitely many algebraically primitive Teichmüller curves generated by a form in the stratum .

The proof uses the cross-ratio equation (1.2) together with a torsion condition from [Möl06a] which gives strong restrictions on Teichmüller curves generated by forms with more than one zero. This torsion condition was used previously in [McM06b] to show that there is a unique primitive Teichmüller curve in and in [Möl08] to show finiteness of algebraically primitive Teichmüller curves in the hyperelliptic components of . Similar ideas should establish finiteness in the strata of with more than two zeros. More ideas are needed in the strata and the component of , as the torsion condition gives no information (in due to the presence of hyperelliptic curves).

While we cannot rule out infinitely many algebraically primitive Teichmüller curves in the stratum , Theorem 1.1 gives an efficient algorithm for searching any given eigenform locus for Teichmüller curves in this stratum. Given an order , first one lists all admissible bases of ideals in as described in Appendix A. For each admissible basis, there are a finite number of irreducible stable forms having these residues and a fourfold zero. One then lists these possible stable forms and then checks each to see if the cross-ratio equation (1.2) holds. If it never holds, then there are no possible cusps of Teichmüller curves in , so there are no Teichmüller curves.

Due to numerical difficulties with the odd component, we have only applied this algorithm to the hyperelliptic component . The algorithm recovers the one known example in this stratum, Veech’s -gon curve, contained in for the ring of integers in the unique cubic field of discriminant ; it has ruled out algebraically primitive Teichmüller curves in for every other eigenform locus it has considered.

###### Theorem 1.6.

Except for Veech’s -gon curve there are no algebraically primitive Teichmüller curves generated by a form in for the ring of integers in any of the totally real cubic fields of discriminant less than .

We discuss the algorithm on which this theorem is based in §14. We also give in this section some further evidence for Conjecture 1.4 in , that an infinite sequence of algebraically primitive Teichmüller curves in this stratum would have to satisfy some unlikely arithmetic restrictions on the widths of cylinders in periodic directions.

For completeness we mention that there is no hope of proving a finiteness theorem for algebraically primitive Teichmüller curves in without bounding . Already Veech’s fundamental paper [Vee89] and also [War98] and [BM] contain infinitely many algebraically primitive Teichmüller curves for growing genus .

### The eigenform locus is generic.

A rough dimension count leads one to expect Conjecture 1.4 to hold for the stratum , as the expected dimension of is , which is too small to contain a Teichmüller curve. On the other hand, if the eigenform locus is contained in some stratum besides the generic one , one would expect this intersection to be positive dimensional. This would be a source of possible Teichmüller curves. In §12, we prove that the eigenform locus is indeed generic.

###### Theorem 1.7.

For any order in a totally real cubic field, each component of the eigenform locus lies generically in .

The proof uses Theorem 1.1 to construct a stable curve in the boundary of where each irreducible component is a thrice-punctured sphere. A limiting eigenform on this curve must have a simple zero in each component.

### Primitive but not algebraically primitive Teichmüller curves.

From a Teichmüller curve in , one can construct many Teichmüller curves in higher genus moduli spaces by a branched covering construction. A Teichmüller curve is primitive if it does not arise from one in lower genus via this construction. Every algebraically primitive Teichmüller curve is primitive, but the converse does not hold. In , McMullen exhibited in [McM06a] infinitely many primitive Teichmüller curves with quadratic trace field. These curves lie in the intersection of with the locus of Prym eigenforms, that is, forms with an involution such that the part of is an Abelian surface with real multiplication having as an eigenform. It is not known whether all primitive Teichmüller curves in with quadratic trace fields arise from this Prym construction.

Our approach to classifying algebraically primitive Teichmüller curves could also be applied to the classification of (say) primitive Teichmüller curves in with quadratic trace field. Given a positive integer and an order in a real quadratic field , there is the locus of forms such that there exists a degree map of onto an elliptic curve with the kernel of the induced map having real multiplication by with as an eigenform. The locus is three-dimensional, and coincides with McMullen’s Prym eigenform locus. Teichmüller curves in having quadratic trace field must be generated by a form in some . There is a classification of the geometric genus zero forms in the boundary of , similar to that of Theorem 1.1, with the map replaced by a quadratic map

 Q:F⊕Q→F⊕Q.

Each boundary stratum of parameterizing trinodal curves is again a subvariety of cut out by an equation in cross-ratios similar to (1.2).

Since the cross-ratio equation (1.2) was responsible for ruling out algebraically primitive Teichmüller curves in , one might wonder why its analogue does not also rule out McMullen’s Teichmüller curves in . The difference is that the cross-ratio equation cutting out the trinodal boundary strata of no longer depends on the associated residues as in (1.2). Moreover, each such boundary stratum contains canonical forms having a fourfold zero, as opposed to the algebraically primitive case where these forms almost never exist. We hope to provide the details of this discussion in a future paper.

### Towards the proof of Theorem 1.1.

We conclude by summarizing the proof of Theorem 1.1. For simplicity, we continue to assume that is a maximal order. The reader may also wish to ignore the case of nonmaximal orders on a first reading.

The real multiplication locus (or more precisely, its lift to the Teichmüller space) is cut out by certain linear combinations of period matrices. To better understand the equations which cut out the real multiplication locus, in §4 we give a coordinate-free description of period matrices. Given an Abelian group , we define a cover , the space of Riemann surfaces equipped with a Lagrangian marking, that is, an isomorphism of onto a Lagrangian subspace of . We define a homomorphism

 Ψ:SZ(HomZ(L,Z))→Hol∗Mg(L),

where denotes the symmetric square, and is the group of nowhere vanishing holomorphic functions on . Each function is a product of exponentials of entries of period matrices. There is a Deligne-Mumford compactification of with a boundary divisor for each , consisting of stable curves where a curve homologous to has been pinched. In Theorem 4.1 we show that each is meromorphic on with order of vanishing

 ordDγΨ(a)=⟨a,γ⊗γ⟩

along .

Cusps of the real multiplication locus correspond to ideal classes in (or extensions of ideal classes if is nonmaximal). Given an ideal , we define in §5 a real multiplication locus , covering , of surfaces which have real multiplication in a way which is compatible with the Lagrangian marking by . The closure of in covers the closure of the cusp of corresponding to , so it suffices to compute the closure in . In §5, we construct a rank subgroup of (where is the inverse different of ) such that is cut out by the equations

 Ψ(a)=1 (1.3)

for all . The proof of Theorem 6.1 yields an identification of with a lattice in with the property that for each and , the order of vanishing of along the divisor is

 ordDtΨ(a)=⟨a,Q(t)⟩ (1.4)

with the pairing the trace pairing on and as in (1.1).

Now suppose that is a boundary stratum which is the intersection of the divisors for , and suppose that the do not satisfy the no-half-space condition. This means that we can find a vector such that for each with strict inequality for at least one. Multiplying by a sufficiently large integer, we may assume . From (1.3) we see that on , and from (1.4) we see that on . It follows that , from which we conclude the first part of Theorem 1.1.

If the lie in a subspace of , then we may choose to be orthogonal to each . By (1.4), the function is nonzero and holomorphic on . The equation restricted to cuts out a codimension-one subvariety of , which yields the second part of Theorem 1.1. In the case where parameterizes trinodal curves, the equation is exactly the cross-ratio equation (1.2). This concludes the necessity of the conditions of Theorem 1.1.

To obtain sufficiency of these conditions, in §8 we show that one can often define, using the functions , local coordinates from a neighborhood of a boundary stratum in into . In these coordinates, is , and the real multiplication locus is a subtorus of . The computation of the boundary of the real multiplication locus is thus reduced to the computation of the closure of an algebraic torus in , which is done in Theorem 8.14.

### Hilbert modular varities and the locus of real multiplication.

We conclude with a discussion of the relation between Hilbert modular varieties and the real multiplication locus. In several textbooks (e.g. [Fre90]) Hilbert modular varieties are defined as the quotients , where for some order , or even more restrictively for the ring of integers [Gor02]. There is a natural map from to the moduli space of Abelian varieties whose image is a component of the locus of Abelian varieties with real multiplication by . In Appendix B, we provide an example showing that the real multiplication locus need not be connected, so it is in general not the image of . This phenomenon is surely known to experts but is often swept under the rug. If one restricts to quadratic fields (as in [vdG88]) or to maximal orders (as in [Gor02]) this phenomenon disappears.

In this paper, we regard a Hilbert modular variety more generally as a quotient for any commensurable with . With this more general definition, the locus of Abelian varieties with real multiplication by is parametrized by a union of Hilbert modular varieties.

The eigenform loci which we compactify are closely related to the Hilbert modular varieties . In genus two, is isomorphic to , while in genus three, is a (degree-one) branched cover of . The real multiplication locus is a quotient of by an action of the Galois group. See §2 for details on Hilbert modular varieties and the various real multiplication loci.

### Acknowledgments.

The authors thank Gerd Faltings, Pascal Hubert, Curt McMullen and Don Zagier for providing useful ideas and arguments. The authors thank the MPIM Bonn for supporting the research of the second named author and providing both authors an excellent working atmosphere.

### Notation.

Throughout the paper, will denote a totally real number field, and order in , and a lattice whose coefficient ring contains .

Given an -module , we write for the submodule of fixed by the involution . We write for the quotient of by the submodule generated by the relations .

Given a bilinear pairing , we write and for the self-adjoint and anti-self-adjoint maps from to .

We write for the disk of radius about the origin in ; we write for the unit disk, and for the unit disk with the origin removed.

## 2 Orders, real multiplication, and Hilbert modular varieties

In this section, we discuss necessary background material on orders in number fields, Abelian varieties with real multiplication, and their various moduli spaces.

### Orders.

Consider a number field of degree . A lattice in (also called full module) is a subgroup of the additive group of isomorphic to a rank free Abelian group. An order in is a lattice which is also a subring of containing the identity element. The ring of integers in is the unique maximal order.

Given a lattice in , the coefficient ring of is the order

 OI={a∈F:ax∈M for all x∈M}.

We will sometimes write for the coefficient ring of .

Lattices in finite dimensional vector spaces over and their coefficient rings are defined similarly.

### Ideal classes.

Two lattices and in are similar if for some . An ideal class is an equivalence class of this relation. Given an order the set of ideal classes of lattices with coefficient ring is a finite set (see [BS66]). If is a maximal order, is the ideal class group of .

### Modules over orders.

Let be an order in a number field and a module over . The rank of is the dimension of as a vector space over . We say is proper if the -module structure on does not extend to a larger order in .

Every torsion-free, rank-one -module is isomorphic to a fractional ideal of , that is, a lattice in whose coefficient ring contains .

A symplectic -module is a torsion-free -module together with a unimodular symplectic form which is compatible with the -module structure in the sense that

 ⟨λx,y⟩=⟨x,λy⟩

for all and .

We equip with the symplectic pairing

 ⟨(α1,β1),(α2,β2)⟩=Tr(α1β2−α2β1). (2.1)

Every rank-two symplectic -module is isomorphic to a lattice in with coefficient ring contains such that the symplectic form on induces a unimodular symplectic paring .

### Inverse different.

Given a lattice with coefficient ring , the inverse different of is the lattice

 I∨={x∈F:Tr(xy)∈Z for all% y∈M}.

and have the same coefficient rings. The trace pairing induces an -module isomorphism .

The sum is a symplectic -module with the canonical symplectic form (2.1).

### Symplectic Extensions.

We now discuss the classification of certain extensions of lattices in number fields. This will be important in the discussion of cusps of Hilbert modular varieties below.

Let be a lattice in a number field with coefficient ring . An extension of by over an order is an exact sequence of -modules,

 0→I→M→I∨→0,

with a proper -module. Given such an extension, a -module splitting determines a -module isomorphism . The module inherits the symplectic form (2.1), which does not depend on the choice of the splitting . We say that this is a symplectic extension if the symplectic form is compatible with the -module structure of .

Let be the set of all symplectic extensions of by over any order up to isomorphisms of exact sequences which are the identity on and . We give the usual Abelian group structure: given two symplectic extensions,

 0→Iιi→Miπi−→I∨→0,

define by and by . The sum of the two extensions is

 0→I→Ker(π)/Im(ι)→I∨→0.

and the identity element is the trivial extension .

Let be the space of endomorphisms of that are self-adjoint with respect to the trace pairing. Note that . For , let denote the multiplication-by- endomorphism.

Given , let be the order

 {x∈F:[Mx,T](I∨)⊂I},

where is the commutator. That is a subring of follows from the formula

 Mλ[Mμ,T]+[Mλ,T]Mμ=[Mλμ,T].

Define a symplectic extension of by over by giving the -module structure

 λ⋅(α,β)=(λα+[Mλ,T](β),λβ).
###### Theorem 2.1.

The map induces an isomorphism

 Hom+Q(F,F)/(HomF(F,F)+Hom+Z(I∨,I))→E(I).
###### Proof.

To see that our map is a well-defined homomorphism is just a matter of working through the definitions, which we leave to the reader.

To show our map is a monomorphism, suppose is isomorphic to the trivial extension via . This isomorphism must be of the form for some self-adjoint . The condition that this is an -module isomorphism implies for all . Since is its own centralizer in , we must have , so .

Now consider the space . We write elements of as with for each . Let be those elements satisfying

 MxQy+QxMy=Qxy (2.2)

for all . We claim that every element of is of the form . To see this, let be a generator of over . The map sending to is injective by (2.2), so , where . The map sending to is injective so is an isomorphism because the domain also has dimension . Thus every element of has the desired form.

Now, every symplectic extension of by over an order is isomorphic as a symplectic -module to with the -module structure,

 λ⋅(α,β)=(λα+Qλ(β),λβ),

with . Since for some , our map is surjective. ∎

Given an order , let be the subgroup of extensions over some order such that , and let be the set of extensions over . From the above description of , we obtain:

###### Corollary 2.2.

is a torsion group with a finite subgroup.

If two lattices and are in the same ideal class, then the groups are canonically isomorphic.

### Real multiplication.

We now suppose is a totally real number field of degree .

Consider a principally polarized -dimensional Abelian variety . We let be the ring of endomorphisms of and the subring of endomorphisms such that the induced endomorphism of is self-adjoint with respect to the symplectic structure defined by the polarization.

Real multiplication by on is a monomorphism . The subring is an order in , and we say that has real multiplication by .

There can be many ways for a given Abelian variety to have real multiplication by . We write for the subgroup of the Galois group which preserves . If is real multiplication of on , then so is for any .

Let be the moduli space of -dimensional principally polarized Abelian varieties (where is the -dimensional Siegel upper half space). We denote by the locus of Abelian varieties with real multiplication by .

### Eigenforms.

Real multiplication induces a monomorphism , where is the vector space of holomorphic one-forms on . If is an embedding of , we say that is an -eigenform if

 λ⋅ω=ι(λ)ω

for all . Equivalently, is an -eigenform if

 ∫λ⋅γω=ι(λ)∫γω

for all and . If we do not wish to specify an embedding , we just call an eigenform.

Given an embedding and -eigenform , there is a unique choice of real multiplication which realizes as an -eigenform. Thus considering -eigenforms allows one to eliminate the ambiguity of the choice of real multiplication.

We denote by the one-dimensional space of -eigenforms. We obtain the eigenform decomposition,

 Ω(X)=⨁ι:F→RΩι(X), (2.3)

where the sum is over all field embeddings .

We denote by the moduli space of pairs where is a principally polarized Abelian variety and is a nonzero holomorphic one-form on . We write for the locus of eigenforms for real multiplication by and for the locus of -eigenforms. Note that for -conjugate embeddings and , the eigenform loci and coincide (as an -eigenform is simultaneously an -eigenform for a Galois conjugate real multiplication); however, each comes with a canonical choice or real multiplication which depends on .

### Hilbert modular varieties.

Choose an ordering of the real embeddings of . We use the notation . The group then acts on by , where acts on the upper-half plane by Möbius transformations in the usual way.

Given a lattice , we define to be the subgroup of which preserves . The Hilbert modular variety associated to is

 X(M)=Hg/SL(M).

Given an order , we define

 XO=∐MX(M),

where the union is over a set of representatives of all isomorphism classes of proper rank two symplectic -modules. If is a maximal order, then every rank two symplectic -module is isomorphic to (this also holds if ; see [McM07]), so in this case is connected. In general, is not connected, as there are nonisomorphic proper symplectic -modules; see Appendix B.

There are canonical maps and defined as follows. Given a lattice and , we define by

 ϕτ(x,y)=(x(i)+y(i)τi)gi=1.

The Abelian variety has real multiplication by defined by . The form is an -eigenform.

The map is an isomorphism, so we may regard as the moduli space of principally polarized Abelian varieties with a choice of real multiplication .

The Galois group acts on , and the map factors through to a generically one-to-one map .

### Cusps of Hilbert modular varieties.

The Baily-Borel-Satake compactification of is a projective variety obtained by adding finitely many points to which we call the cusps of . More precisely, we embed in by . We define with a certain topology whose precise definition is not needed for this discussion; see [BJ06]. The compactification of is . We define to be the union of the compactifications of its components.

###### Proposition 2.3.

There is a natural bijection between the set of cusps of and the set of isomorphism classes of symplectic extensions

 0→I→N→I∨→0 (2.4)

with a primitive rank-two symplectic -module and a torsion-free rank one -module. The cusps of correspond to the isomorphism classes of such extensions where as symplectic -modules.

###### Sketch of proof.

Fix a lattice . We must provide a -equivariant bijection between lines and extensions (up to isomorphism which is the identity on ). We assign to a line , the extension . The line is recovered from an extension by defining .

The bijection for cusps of follows immediately. ∎

Consider the set of all pairs , where is a lattice in whose coefficient ring contains , and . The multiplicative group of acts on such pairs by , where (using the identification of Theorem 2.1). We define a cusp packet for real multiplication by to be an equivalence class of a pair under this relation.

We denote by the finite set of cusp packets for real multiplication by . We have seen that there are canonical bijections between , the set of isomorphism classes of symplectic extensions of the form (2.4), the set of cusps of , and the set of cusps of . Moreover, there is a canonical bijection between the set of cusps of and .

## 3 Stable Riemann surfaces and their moduli

In this section, we discuss some background material on Riemann surfaces with nodal singularities, holomorphic one-forms, and their various moduli spaces.

### Stable Riemann surfaces.

A stable Riemann surface (or stable curve) is a connected, compact, one-dimensional, complex analytic variety with possibly finitely many nodal singularities – that is, singularities of the form  – such that each component of the complement of the singularities has negative Euler characteristic. In other terms, a stable Riemann surface can be regarded a disjoint union of finite volume hyperbolic Riemann surfaces with cusps, together with an identification of the cusps into pairs, each pair forming a node. We will refer to a pair of cusps facing a node as opposite cusps.

The arithmetic genus of a stable Riemann surface is the genus of the nonsingular surface obtained by thickening each node to an annulus; the geometric genus is the sum of the genera of its irreducible components.

### Homology.

Given a stable Riemann surface , let be the complement of the nodes. For each cusp of , let be the class of a positively oriented simple closed curve winding once around , and let be the subgroup generated by the expressions , where and are cusps joined to a node on .

We define . Defining to be the free Abelian subgroup (of rank equal to the number of nodes) generated by the , we have the canonical exact sequence

 0→C(X)→ˆH1(X;Z)→H1(˜X;Z)→0,

where is the normalization of .

### Markings.

Fix a genus surface , and let be a genus stable Riemann surface. A collapse is a map such that the inverse image of each node is a simple closed curve and is a homeomorphism on the complement of these curves.

A marked stable Riemann surface is a stable Riemann surface together with a collapse . Two marked stable Riemann surfaces and are equivalent if there is homeomorphism which is homotopic to the identity and a conformal isomorphism such that .

### Augmented Teichmüller space.

The Teichmüller space is the space of nonsingular marked Riemann surfaces of genus . It is contained in the augmented Teichmüller space , the space of marked stable Riemann surfaces of genus . We give the smallest topology such that the hyperbolic length of any simple closed curve is continuous as a function . Abikoff [Abi77] showed that this topology agrees with other natural topologies on defined via quasiconformal mappings or quasi-isometries.

### Deligne-Mumford compactification.

The mapping class group of orientation preserving homeomorphisms of defined up to isotopy acts on and by precomposition of markings. The moduli space of genus Riemann surfaces is the quotient . The Deligne-Mumford compactification of is , the moduli space of genus stable curves.

Over is the universal curve , a compact algebraic variety whose fiber over a point representing a stable curve is a curve isomorphic to (provided has no automorphisms).

Over