An Analogue of Chern-Weil Theory for the Line Bundle of Weak Jacobi Forms on a Non-Compact Modular Surface

M. Jespers

An analogue of Chern-Weil theory for the line bundle of weak Jacobi forms on a non-compact modular surface

Master Thesis

Thesis supervisor: dr. R.S. de Jong

June 2016

Mathematisch Instituut, Universiteit Leiden Contents

Introduction 1

1 Background 3 1.1 The tower of compactiﬁcations of E0(Γ) ...... 4 1.2 Moduli spaces of elliptic curves ...... 5 1.3 Jacobi forms and theta divisors ...... 7 1.4 Metrics and Mumford-Lear extensions ...... 9

2 Tools 13 2.1 Method of test curves and locally generating sections ...... 13 2.2 Deligne pairing ...... 14 2.3 The Petersson metric near singular points of E(Γ) − E0(Γ) ...... 16

3 Calculations 18 3.1 Mumford-Lear extension of L4,4(Γ) to E(Γ) ...... 18 3.2 Comparison to Chern-Weil theory ...... 22

4 Concluding remarks 27

References 28 Introduction

Let X be a smooth compact algebraic surface over C. Chern-Weil theory implies that if (L, || · ||) is a line bundle on X equipped with a smooth metric, then we have Z ∧2 C · C = c1(L, || · ||) , X where C is the equivalence class of Weil divisors associated to L and c1(L, || · ||) is the first Chern form taken with respect to any connection on L compatible with || · ||. Let Γ ≤ SL2(Z) be a congruence subgroup which acts without fixed points on the upper half-plane H and let p: E0(Γ) → Y (Γ) be the universal elliptic curve over the modular curve Y (Γ) = Γ\H. We would like to use the aforementioned Chern-Weil theoretic result to compute the self-intersection of the theta divisor 0 on E (Γ) given by the image of the zero section, using a translation invariant metric || · ||Pet on the 0 line bundle of weak Jacobi forms L4,4(Γ) of weight 4 and index 4. But since E (Γ) is not compact, Chern-Weil theory can not be applied directly. Following the same line of reasoning as in [5], we try to remedy this by extending (L4,4(Γ), || · ||Pet) to a metrized line bundle (L4,4(Γ), || · ||) on the compactification p: E(Γ) → X(Γ) in the sense of Deligne-Rapoport. Let J be the divisor class associated to L4,4(Γ). It turns out not to be possible to extend ||·||Pet smoothly. The closest thing we can get to a smooth extension of ||·||Pet is a so-called Mumford-Lear extension. This extension picks up serious enough singularities along the 0 singular locus of E(Γ)−E (Γ) that the self-intersection of the associated divisor class [J, ||·||Pet]E(Γ) R ∧2 is not equal to E0(Γ) c1(L4,4(Γ), || · ||Pet) . The main result of this thesis, which is a generalization of [5, Theorem 5.2] to arbitrary congruence subgroups Γ ≤ SL2(Z) which act without fixed points on H, is the following. 0 Theorem. For every smooth compactification E of E (Γ), the Mumford-Lear extension of L4,4(Γ) to E exists. If we denote its associated divisor class by [J, || · ||Pet]E, then the following equality holds: Z 2 ∧2 lim [J, || · ||Pet]E = c1(L4,4(Γ), || · ||Pet) , E∈CPT(E0(Γ)) E0(Γ) where CPT(E0(Γ)) denotes the directed set of isomorphism classes of smooth compactifications of E0(Γ). Thus an analogue of Chern-Weil theory for the line bundle of weak Jacobi forms on E0(Γ) can be obtained by taking into account all smooth compactifications of E0(Γ).

We mention the following explicit results. If C(Γ) denotes the set of cusps of X(Γ) and hc the period of cusp c, the self-intersection of [J, || · ||Pet]E(Γ) is equal to 64 X 64 [J, || · || ]2 = d + Pet E(Γ) 12 Γ 12h2 c∈C(Γ) c where dΓ is the degree of the forgetful map X(Γ) → X(1). Furthermore, taking the limit of the self-intersection over all compactiﬁcations of E0(Γ), we obtain

2 64 lim [J, || · ||Pet]E = dΓ. E∈CPT(E0(Γ)) 12

1 Overview

Throughout this thesis, we fix a congruence subgroup Γ ≤ SL2(Z) which acts without fixed points on H. The material in this paper is organized as follows. In the first chapter, we introduce the main objects of study in this thesis, the elliptic surface E(Γ) and the line bundle Lk,m(Γ) of weak Jacobi forms of weight k and index m, and provide some background on what purpose they serve in the general body of mathematics. This chapter fulfills mainly an expository role and does not contain many proofs. In the second chapter, we introduce a set of tools to be used in the calculation of the Mumford-Lear 0 extension of || · ||Pet to the various smooth compactifications of E (Γ). We also give an explicit description of the Petersson metric near the singular locus of E(Γ) − E0(Γ).

Finally, in the third chapter, the Mumford-Lear extension of ||·||Pet to all smooth compactiﬁcations of E0(Γ) is computed and we show the associated system of divisors satisﬁes Chern-Weil theory and a Hilbert-Samuel type formula.

Acknowledgements

I would like to thank my supervisor Robin de Jong for his time and guidance the past year. His advice and feedback have been extremely helpful in the writing of this thesis, as well as in the research that went into it. I would also like to thank David Holmes and Peter Bruin for being in the reading committee. Finally, I would like to thank my family for the support they have given me the past year.

2 1 Background

This chapter serves to put the calculations in the next two chapters in context. We begin by introducing the central object of study of this thesis, the elliptic surface E(Γ) associated to a congruence subgroup Γ ≤ SL2(Z). As is well known (cf. [11, Proposition 2.1.1]), every Γ ≤ SL2(Z) acts properly discontinuously on the upper half plane H = {z ∈ C : =(z) > 0}, and this action J 2 extends to an action of Γ := Γ n Z on H × C given by a b aτ + b z + λτ + µ , (λ, µ) · (τ, z) = , . c d cτ + d cτ + d

0 J If Γ acts without ﬁxed points on H, the quotient E (Γ) = Γ \H × C comes equipped with the structure of non-compact complex manifold with the property that holomorphic functions on E0(Γ) J correspond to Γ -periodic holomorphic functions on H × C, and the purpose of section 1.1 is to describe the directed set

CPT(E0(Γ)) = {smooth compactiﬁcations of E0(Γ)}/=∼

0 of isomorphism classes of complete smooth algebraic surfaces X over C which contain E (Γ) as an open subset and which satisfy the property that X − E0(Γ) is a normal crossings divisor. We shall see that the minimal element of CPT(E0(Γ)) is E(Γ), and all other elements of CPT(E0(Γ)) can be obtained from E(Γ) by a finite series of blow-ups in points not contained in E0(Γ). Next, we give a relation between E(Γ) and certain moduli spaces of marked curves over a base scheme S. Here, by a curve over S (perhaps more aptly named a family of curves) we mean a 1- dimensional variety over S, i.e. a flat morphism X → S of finite type and of relative dimension 1 with geometrically connected fibers. If Γ ≤ SL2(Z) acts without fixed points on H and is equal to one of a b Γ(N) = ∈ SL ( ): a ≡ d ≡ 1 mod N, b ≡ c ≡ 0 mod N c d 2 Z a b Γ (N) = ∈ SL ( ): a ≡ d ≡ 1 mod N, c ≡ 0 mod N 1 c d 2 Z a b Γ (N) = ∈ SL ( ): c ≡ 0 mod N 0 c d 2 Z for some N > 0, then E0(Γ) can be interpreted as a moduli space of elliptic curves with extra structure. Here by an elliptic curve over S we mean a smooth proper curve over S whose geometric fibers are genus 1 curves, equipped with a section O : S → E. A similar interpretation can be given to E(Γ) in terms of moduli spaces of semi-stable curves, which we also introduce in section 1.2.

In section 1.3, we discuss the line bundle Lk,m(Γ) of weak Jacobi forms of weight k and index m associated to Γ. For certain k and m, there is a particularly nice interpretation of Lk,m(Γ) as the line bundle associated to a theta divisor. This gives a good motivation to study Jacobi forms and so we opt to develop the theory of theta divisors ﬁrst. Finally, in section 1.4 we introduce smooth metrics, Mumford-Lear extensions and the Petersson metric on the line bundle of weak Jacobi forms of weight k and index m.

3 1.1 The tower of compactiﬁcations of E0(Γ)

In this section we introduce the tower of smooth compactifications of E0(Γ). As we shall see shortly, all smooth compactifications of E0(Γ) are (by definition) elliptic surfaces. Roughly speaking, an elliptic surface is a family of curves parametrized by a base curve C, almost all of which are smooth curves of genus 1. Definition 1.1. Let k be a field. An elliptic surface E0 over a curve C/k is a smooth 2-dimensional variety over k, equipped with a proper morphism f : E0 → C with connected fibers such that the generic fiber f −1(η) of each irreducible component of C is a smooth genus 1 curve. For the remainder of this section, to avoid working with the abstract machinery of étalecoverings, we restrict our attention to varieties over C. There is a functor from the category of schemes of finite type over C to the category of complex analytic spaces, which sends a nonsingular variety over C to a complex manifold (cf. [12, Appendix B]). If X is a non-singular variety over C, we denote the image of X under this functor by Xan. Generally speaking, if E0 is an elliptic surface over a non-complete nonsingular curve C and C is the smooth completion of C, there is no way to define an elliptic surface π : Ef0 → C with the property that all fibers of π are smooth of genus 1. Instead we have to allow some of the fibers to be singular. As it turns out (cf. [2]) it is always possible to find an elliptic surface E → C extending E0 → C with the property that the complement E − E0 is a normal crossings divisor.

Definition 1.2. Let X be a complex manifold and write ∆ = {z ∈ C : |z| < 1}.A normal crossings divisor D of X is a complex submanifold which admits for every p ∈ D a coordinate ∼ n system z = (z1, . . . , zn): U −→ ∆ with the property that D∩U is given by the equation z1 . . . zk = 0. We say such a coordinate system z is adapted to D. Now if X is a connected non-compact complex manifold, we say Y is a smooth compactification of X if it is a connected compact complex manifold which contains X as an open subset and which satisfies the property that Y − X is a normal crossings divisor. Similarly, if Y is a complete non-singular an variety over C containing X as a Zariski open subset, we say Y is a compactification of X if Y is a compactification of Xan. We call D = Y − X the boundary divisor of the compactification Y/X. Given an elliptic surface E0 → C, there is a compactification E/C of E0, called the relatively minimal model, defined by the property that none of its fibers contain a −1-curve. Here, by a −1- curve we mean a rational curve on E0 with self-intersection −1. Recall that by Castelnuovo’s contractibility criterion, a curve on a surface can be blown down if and only if it is a −1-curve. Hence, since the Néron-Severi group of a surface is finitely generated and since a blow-down decreases the rank of the Néron-Severi group by one, a relatively minimal model of E0 can be obtained by starting with an arbitrary compactification Ef0/C and blowing down a finite number of −1-curves. This yields the following result: 0 Theorem 1.1. Let p: E → C be an elliptic surface over a non-compact smooth curve C over C with completion C. Consider the set CPT(E0) of isomorphism classes of pairs (E, π), where • π : E → C is an elliptic surface extending the map p: E0 → C, • E − E0 is a normal crossings divisor of E, 0 00 00 0 ordered by the relation that E E E if and only if E can be obtained from E as a tower

00 0 E = En → En−1 → ... → E0 = E ,

4 0 where for all k > 0, Ek is obtained from Ek−1 as a blow-up of Ek−1 in a point Ek−1 − E . Then CPT(E0) is a directed set, with minimum given by the relatively minimal model of E0. Equip Y (Γ) := Γ\H with the unique structure of Riemann surface satisfying the property that holomorphic functions Y (Γ) → C correspond to Γ-periodic holomorphic functions H → C. Associated to E0(Γ), we have a projection map

p: E0(Γ) → Y (Γ), [τ, z] 7→ [τ] and a zero section O : Y (Γ) → E0(Γ), [τ] 7→ [τ, 0], turning E0(Γ) into an elliptic surface over Y (Γ) in the strict sense that every fiber is an elliptic −1 curve: for every [τ] ∈ Y (Γ), the fiber p [τ] is equal to the complex torus Eτ := C/(Zτ ⊕ Z). The space Y (Γ) can be compactified by adding so-called cusps, cf. [11, section 2.4], and we denote the resulting curve by X(Γ). There is an interpretation of X(Γ) and E0(Γ) as complex manifolds coming from smooth algebraic varieties over C, and we define E(Γ)/X(Γ) to be the relatively minimal model of E0(Γ)/Y (Γ).

1.2 Moduli spaces of elliptic curves

The purpose of this section is to discuss certain moduli spaces and various canonical maps that exist between them, so let us begin by recalling the definition of a moduli space. If C is a category, X is an object in C and f : X → Y is a morphism in C, let hX = Hom(−,X) be the functor which sends Y to the set of morphisms from Y to X and denote by hf = Hom(−, f) the natural transformation given by post-composition with f. op Definition 1.3. Let S be a scheme and F : SchS → Sets a contravariant functor from SchS to Sets. A pair (X, τ) consisting of a scheme over S and a natural transformation τ : F → hX is said to be 1.A fine moduli space for F if τ is a natural isomorphism. 2.A coarse moduli space for F if τ satisfies the following properties:

• For every Y ∈ SchS and every natural transformation σ : F → hY , there is a unique morphism Y → X such that σ = hf ◦ τ. alg • For every one-point scheme pt = Spec(k) over S with k = k , the map τpt : F(pt) → X(pt) is a bijection. Note that if X is a fine moduli space for F, then there is a universal object over X associated to F, −1 given by UX = τX (idX ). No such object can in general be associated to a coarse moduli space. Roughly speaking, the difference between coarse and fine moduli spaces is that fine moduli spaces parametrize all “families of geometric objects of a certain type”, whereas coarse moduli spaces only parametrize “objects over points”. op We refer to a functor F : SchS → Sets as a moduli problem over S and to a fine moduli space (X, τ) for F as a solution of F.

Given a morphism X → S, a section e: S → X and a morphism f : T → S, write XT = X ×S T and eT : T → XT for the section corresponding to (e ◦ f, id) ∈ X(T ) ×S(T ) T (T ).

5 Let g and n be positive integers. A natural moduli problem to consider is the moduli problem Mg,n of n-pointed genus g curves. This is the moduli problem which associates to Y ∈ SchS the set Mg,n(Y ) of isomorphism classes of tuples (C,P1,...,Pn) where • C is a smooth, proper genus g curve over Y ,

• P1,...,Pn ∈ C(Y ) are disjoint sections of C → Y and to a morphism f : Y → Y 0 the function

0 Mg,n(Y ) → Mg,n(Y ):(C,P1,...,Pn) 7→ (CY ,P1Y ,...,PnY ).

Since the datum (C,P1,...,Pn) may have non-trivial automorphisms (cf. [1, Theorem XII.2.5]), there is in general no solution to Mg,n in the category of schemes. This type of issue is common in the theory of moduli spaces. If we want to have a scheme that represents a family of objects with automorphisms, then either we need to settle with coarse moduli spaces, or we need to consider a diﬀerent, rigidiﬁed moduli problem. In the special case that g = 1, we are dealing with elliptic curves over a base scheme and the N- torsion can be used to eliminate automorphisms.

Deﬁnition 1.4. Let n and N be positive integers and let Γ be one of Γ(N), Γ0(N) or Γ1(N). Let (E,O) be an elliptic curve over a scheme S over Spec(Z[1/N]), with N-torsion subgroup scheme E[N]. Let (Z/NZ)S be the scheme associated to the constant functor SchS → Sets with value Z/NZ. Then a level structure on (E,O) associated to Γ is: 2 • An isomorphism of group schemes (Z/NZ)S → E[N] if Γ = Γ(N).

• An embedding of group schemes (Z/NZ)S → E[N] if Γ = Γ1(N).

• A subgroup scheme of E[N] ´etalelocally isomorphic to Z/NZ if Γ = Γ0(N). The moduli problem of elliptic curves with level structure and n − 1 marked points is given on objects by the set M1,n(Γ)(Y ) of isomorphism classes of tuples (E,O,P2,...,Pn,T ), where • E is a smooth, proper genus 1 curve over Y ,

• O,P2,...,Pn ∈ E(Y ) are disjoint sections of E → Y , • T a level structure on (E,O) associated to Γ. There is a one-to-one correspondence between Y (Γ) and the set of isomorphism classes of elliptic curves over C with level structure associated to Γ (cf. [11, Theorem 1.5.1]). If N is such that Γ acts without fixed points on H, then M1,1(Γ) has a fine moduli space over Spec(Z[1/N]), and the analytification of its base change to C is isomorphic to Y (Γ) (cf. [1, chapter XVI]). Even if Γ does not act without fixed points, the space Y (Γ) is at least a coarse moduli space for M1,1(Γ) over Spec(C) (cf. [1, chapter XII]). If Γ = Γ(N), we use the abbreviation M1,n(Γ(N)) = M1,n(N).

Note that for n > 1, there is a projection map πn+1 : M1,n+1(Γ) → M1,n(Γ), induced by the natural transformation which forgets the last point:

πn+1(Y ): M1,n+1(Γ)(Y ) → M1,n(Γ)(Y ), (E,O,P2,...,Pn+1,T ) 7→ (E,O,P2,...,Pn,T )

0 The analytiﬁcation of the base change of M1,2(Γ) to C is isomorphic to E (Γ) − O (cf. [1, chapter XVI]).

6 As mentioned in the previous section, the spaces Y (Γ) and E0(Γ) are not compact, and neither are the M1,n(Γ) with n > 2. If we were to compactify these spaces, we would like the result to be the solution to some interesting moduli problem. The way to do this is to add degenerated curves to the moduli problem in the form of semi-stable curves. Definition 1.5. A nodal curve S is a curve over S with the property that every geometric fiber has only ordinary double points as singularities. We say a tuple (C,P1,...,Pn) consisting of a proper nodal curve over S and an n-tuple of disjoint S-valued smooth points is semi-stable if the automorphism group of (Cs,P1s,...,Pns) is reductive for all geometric points s of C. In particular, we see that every semi-stable curve C of genus 1 over a field k is a union of rational and smooth genus 0 and 1 curves, satisfying the following properties (cf. [1, chapter X], [9], [15]): • Every irreducible component of C that is smooth of genus 1 has at least 1 point that is marked or lies on another irreducible component. • Every irreducible component of C that is rational has at least 2 points that are marked or lie on another irreducible component. Now consider the moduli problem given on objects by the set of isomorphism classes of tuples (E,O,P2,...,Pn,T ) where • (E,O,T ) is a generalized elliptic curve with level structure in the sense of Deligne-Rapoport, cf. [10] sm • O,P2,...,Pn ∈ E (Y ) are disjoint sections of E → Y such that (E,O,P2,...,Pn) is semi- stable,

If Γ acts without fixed points on H then each M1,n(Γ) admits a solution over Spec(Z[1/N]) and we have natural projection maps M1,n(Γ) → M1,k(Γ) for all n > k. The analytification of the base change of M1,1(Γ) to Spec(C) is isomorphic to X(Γ), and the analytification of the base change of M1,2(Γ) to Spec(C) is isomorphic to E(Γ) − O (cf. [1, chapter XVI]).

1.3 Jacobi forms and theta divisors

Let E → S be an elliptic curve over a scheme, with zero section O : S → E. If S = Spec(k) for some ﬁeld k, we have the following bijection between the k-valued points of E and the degree zero divisors of E(k): 0 ϕO : E → Div (E),P 7→ [O] − [P ].

Note the role of the zero divisor O in this construction. There is a generalization of ϕO to arbitrary schemes S, where diﬀerent divisors (or rather, the line bundles associated to them) may take the place of O. This generalization makes use of the Picard scheme of E/S, which is the scheme op representing the functor PicE/S : SchS → Sets given by

∼ ∗ PicE/S(T ) = {(L, α): L ∈ Pic(ET ), α: OT −→ OT L}, where the datum α, called a rigidiﬁcation of L along OT , serves a similar purpose as the level structure on elliptic curves we saw earlier, in that it is there to eliminate automorphisms. For each T ∈ SchS, the set PicE/S(T ) forms a group under the operation

(L, α) · (L0, α0) = (L ⊗ L0, α ⊗ α0)

7 and this establishes a factorization of PicE/S through the forgetful functor Grp → Sets. Thus PicE/S 0 is in fact a group scheme. We refer to the connected component PicE/S ⊂ PicE/S of the unit element as the dual of E and denote it by E∨. The k-valued points of E∨ are precisely the isomorphism classes of degree zero line bundles on Ek, and the correct generalization of the map ϕO above is a ∨ polarization ϕL : E → E associated to a line bundle L ∈ Pic(E).

Deﬁnition 1.6. Let L be a line bundle on E. Consider the Mumford line bundle on E ×S E given by ∗ ∗ ⊗−1 ∗ ⊗−1 Λ(L) = m L ⊗ p1L ⊗ p2L .

This line bundle can be equipped with a rigidiﬁcation, and we have a map ϕL corresponding to Λ(L) via the universal property of PicE/S. Fiberwise, ignoring rigidiﬁcations, it is given by x 7→ ∗ ⊗−1 ∨ txL⊗L . Note that the relative degree of Λ(L) is zero. Hence ϕL factors through the map E → ∨ PicE/S. If ϕL is an isogeny E → E we call ϕL a polarization, and if ϕL is an isomorphism we call it a principal polarization.

Now a theta divisor on E is a divisor Θ on E with the property that the map ϕL with L = OE(Θ) is a principal polarization. Viewing E0(Γ) as a family of elliptic curves over Y (Γ), one can show that the polarization associated to the image of the zero section Y (Γ) → E0(Γ) is given ﬁberwise 0 by the map which sends P ∈ Eτ to OEτ (O − P ) ∈ Pic (Eτ ) for all τ ∈ H. Thus O is a theta divisor on E0(Γ). We describe the global sections of the line bundle associated to O. Because E0(Γ) is deﬁned as the quotient of a contractible space by a discontinuous group action, there is a fairly explicit description of Pic(E0(Γ)) in terms of systems of multipliers. Proposition 1.2. Let X and Y be complex manifolds. Suppose f : Y → X is a Galois cover with automorphism group Ψ and suppose the Picard group of Y is trivial. Then there is a one-to-one correspondence between Q 0 ∗ • Families of invertible holomorphic functions (eψ)ψ∈Ψ ∈ ψ∈Ψ H (Y, OY ) satisfying the property that 0 eψψ0 (z) = eψ(ψ · z)eψ0 (z) for all ψ, ψ0 ∈ Ψ and z ∈ Y . ∗ ∼ • Pairs (L, α), consisting of a line bundle L ∈ Pic(X) and a trivialization α: f L −→OY .

If (eψ)ψ∈Ψ is a system of multipliers, the vector space of global sections of the associated line bundle L is given by 0 0 H (X,L) = {θ(z) ∈ H (Y, OY ): θ(ψ · z) = eψ(z)θ(z) for all ψ ∈ Ψ}.

For details we refer the reader to [3, page 105]. Now to calculate the system of multipliers associated to O, we write down a holomorphic function f : H × C → C with div(f) = O and determine the quotient eγ(z) = f(γ · z)/f(z). A common choice for such a function is the Riemann theta function, which is deﬁned by the convergent power series 2 ! X 1 1 1 θ (τ, z) = exp πiτ n + + 2πi z + n + . 1,1 2 2 2 n∈Z Associated to the Riemann theta function we have the following system of multipliers: c(z + λτ + µ)2 c0(ψ; τ, z) = χ(γ) · (cτ + d)1/2 exp −πi λ2τ + 2λz − , cτ + d

8 a b J ∗ where ψ = ( c d , (λ, µ)) ∈ Γ and χ:Γ → C is a character which assigns to each γ ∈ Γ an eighth root of unity. We deﬁne the line bundle Lk,m(Γ) by a similar system of multipliers, c(z + λτ + µ)2 c(ψ; τ, z) = (cτ + d)k exp −2πim λ2τ + 2λz − , cτ + d 8 and call a global section of Lk,m(Γ) a weak Jacobi forms of weight k and index m. Thus θ1,1 is a weak Jacobi form of weight 4 and index 4, and the line bundle associated to 8O is isomorphic to L4,4(Γ). We end this section with a brief discussion of non-weak Jacobi forms. Consider the action of ΓJ on L4,4(Γ) given by (ψ · f)(τ, z) = f(ψ · (τ, z)). 1 m Note that weak Jacobi forms are invariant under ψ = (± ( 0 1 ) , (0, n)) for all m, n ∈ Z>0 with ψ ∈ J 1 0 J Γ . More generally, since (( 0 1 ) , (0, 1)) ∈ Γ for all congruence subgroups Γ ≤ SL2(Z), if c = α · ∞ is a cusp of X(Γ) and 1 m hc = min{m ∈ Z>0 : ((± 0 1 ) ∈ Γ} −1 is its period, then the function f(τ, z) · c(α ; τ, z) is hcZ-periodic in its ﬁrst and Z-periodic in its second argument. Therefore, the function admits a Fourier expansion

−1 X f(τ, z) · c(α ; τ, z) = ac(n, r) exp(2nπiτ/hc) exp(2πiz). n∈Z,r∈Z We say f is a Jacobi form of weight k and index m if it is a weak Jacobi form with the property 2 2 that ac(n, r) = 0 for all pairs (n, r) ∈ Z with n < 0 or 4mn − hcr < 0. If in addition this equality 2 holds for all (n, r) with n = 0 or 4mn − hcr = 0, we say f is a Jacobi cusp form. The Jacobi (cusp) cusp forms of weight k and index m form a C-vector space, which we denote by Jk,m(Γ) (Jk,m (Γ)), and as we will see in chapter 3, the quantity dim J4`,4`(Γ) grows like C · `2/2! + o(`2), as ` → ∞, where C is the limit of the self-intersection of the Mumford-Lear extension of L4,4(Γ) to all compactiﬁcations of E0(Γ).

1.4 Metrics and Mumford-Lear extensions

Suppose X is a complex manifold equipped with a holomorphic line bundle L. Let us begin by recalling the definition of a smooth metric on L and its associated first Chern form c1(L, || · ||). Definition 1.7. Let X be a complex manifold equipped with a holomorphic line bundle L.A smooth metric on L is a map ||·|| which associates to every local section s ∈ L(U) a function ||s||: U → R≥0 in such a manner that the following two conditions hold:

• For all f ∈ OX (U), we have ||f · s|| = |f| · ||s||. • For every section s ∈ L(U) generating L(U), the function ||s||2 is smooth and positive valued.

To a smooth metric || · ||, we associate a (1, 1)-form c1(L, || · ||), called the ﬁrst Chern form, by 1 choosing a connection ∇ on L compatible with || · || and taking c1(L, || · ||) = 2πi Tr(k∇), where k∇ is the curvature form associated to ∇. This Chern form can be shown to be equal to 1 c (L, || · ||) = ∂∂ log ||s||2, 1 2πi

9 where s is any locally generating section of L. Given smooth metrics ||·||1 and ||·||2 on line bundles L and M, the tensor product L ⊗ M comes equipped with a smooth metric || · || determined by the formula ||s ⊗ t|| = ||s||1 · ||t||2. The ﬁrst Chern form is additive on tensor products:

c1(L ⊗ M, || · ||) = c1(L, || · ||1) + c1(M, || · ||2). If f : Y → X is a morphism of complex manifolds, the pull-back line bundle f ∗L comes equipped with a smooth metric uniquely determined by the property that ||f ∗s|| = ||s|| ◦ f for all s ∈ L(U), ∗ ∗ and the ﬁrst Chern form is compatible with this pullback: c1(f L, || · ||) = f c1(L, || · ||). We have the following relation between the intersection pairing on line bundles on a complete smooth algebraic surface over C and integrals involving Chern forms of smooth metrics. Proposition 1.3. Let Xan be the complex manifold associated to a complete smooth algebraic an surface X over C. Suppose || · ||1 and || · ||2 are smooth metrics on line bundles L and M on X , and C and D are the equivalence classes of Weil divisors associated to L and M, respectively. Then Z C · D = c1(L, || · ||1) ∧ c1(M, || · ||2). X

Obviously it is of interest to extend this result as much as possible to metrized line bundles on non-compact algebraic surfaces. Let X be a non-compact algebraic surface over C equipped with a smoothly metrized line bundle (L, ||·||), let Y be a smooth compactification of X and write PicL(Y ) for the set of line bundles L ∈ Pic(Y ) such that L|X = L. We can try to find a smoothly metrized 0 line bundle (L, ||·|| ) on Y with the property that L|X is isometric to L. This is not always possible, but sometimes a singular metric on an extension L ∈ PicL(Y ) can be given which has mild enough singularities along Y − X that it can be used to determine intersection numbers. In particular this is the case if the extension can be equipped with a so-called pre-log metric (cf. [17]). Definition 1.8. Let Y be a smooth compactification of a non-compact complex manifold X and let D = Y − X be its boundary divisor. We say a holomorphic function f on X has log-log growth along D if for every coordinate system z = (z1, . . . , zn) adapted to D such that D is locally given by the equation z1 . . . zk = 0 there is a positive integer M and a positive real number C such that the following inequality holds:

k Y M |f(z1, . . . , zn)| < C · log(log(1/|zi|)) . i=1 Let j : X → Y be the inclusion map. The Dolbeault algebra of log-log forms is the sub-algebra of j∗EX which is closed under the operators ∧, ∂ and ∂ and which is generated by the holomorphic functions with log-log growth and the diﬀerential forms dz dz i , i for i = 1, . . . , k, zi log(1/|zi|) zi log(1/|zi|)

dzi, dzi for i > k.

We say a diﬀerential form η is pre-log-log if it is a log-log form with the property that ∂η, ∂η and ∂∂η are also log-log forms. Intuitively speaking, a function is pre-log-log if it has log-log growth and its “derivatives” also have log-log growth. We note that by [8, Proposition 7.4], the property of being a pre-log-log form is respected by pull-backs.

10 Deﬁnition 1.9. Let Xan be the complex manifold associated to a complete smooth algebraic an variety X over C of dimension n, equipped with a holomorphic line bundle L. Suppose D ⊂ X is a normal crossings divisor on Xan. Then a pre-log metric along D is a smooth metric ||·|| on X −D satisfying the following properties: • It has logarithmic growth along D: that is, for every x ∈ X, there is an adapted coordinate system z = (z1, . . . , zn), a positive integer M and a locally generating section s of L deﬁned on z such that

k k Y −M 0 Y M C · log(1/|zi|) < ||s(z1, . . . , zn)|| < C · log(1/|zi|) i=1 i=1

0 for all (z1, . . . , zn) ∈ X − D and some positive real numbers C,C . • For every rational section s of L, the function log ||s|| is a pre-log-log form along D − div(s) on X − div(s). Let E(Γ) be the elliptic surface associated to Γ and let Dsing be the singular locus of D = E(Γ) − 0 E (Γ). Write y = =(z) and η = =(τ). Then the line bundle Lk,m(Γ) of weak Jacobi forms can be equipped with the following so-called Petersson metric:

2 2 2 k ||f(τ, z)||Pet = |f(τ, z)| exp(−4πmy /η)η

The content of the next proposition is that || · ||Pet is translation invariant, meaning that we have J ||ψ · f||Pet = ||f||Pet for all local sections f of Lk,m(Γ) and all ψ ∈ Γ . a b J 0 Proposition 1.4. Let ψ = ( c d , (λ, µ)) ∈ Γ and let U be an open subset of E (Γ). For every 0 f ∈ H (U, Lk,m(Γ)) and (τ, z) ∈ H × C we have 2 aτ + b z + λτ + µ 2 f , = ||f(τ, z)||Pet. cτ + d cτ + d Pet

Proof. Tedious but straightforward computation.

As we will see in later chapters, the Petersson metric on the line bundle of weak Jacobi forms can be extended to a pre-log metric on E(Γ) − Dsing along D − Dsing, but not to a pre-log metric on the entirety of E(Γ). This phenomenon ﬁts in the framework of Mumford-Lear extensions, which was introduced in [5] and which we recall here.

Definition 1.10. Let Y be a smooth compactification of a smooth variety X over C and let (L, ||·||) be a smoothly metrized line bundle on X. Let D be the boundary divisor of Y/X.A Mumford-Lear extension of (L, || · ||) is a triple (L, || · ||0, e) consisting of a line bundle L ∈ Pic(Y ), a positive 0 ⊗e integer e and a metric || · || on L|X such that L|X is isometric to L and such that L|X is pre-log along D − S for some algebraic subset S ⊂ Y of codimension at least 2. Remark 1.11. Mumford-Lear extensions are so named because they are a common generalization of the extensions defined by Mumford (cf. [17]) and Lear (cf. [16]): a Mumford extension of a norm || · || on L to a line bundle L on Y is pre-log along the entirety of D, whereas a Lear extension of || · || to L is one that extends continuously, but only to a subset D − S with S of codimension at least 2 in Y .

11 0 1 Associated to a Mumford-Lear extension (L, || · || , e) we have the Q-line bundle [L, || · ||]Y = e L. Note that since the Picard group is not inﬂuenced by what happens on codimension two subsets, this line bundle is uniquely determined by (L, || · ||) (cf. [5, Proposition 3.9]). Note furthermore that Mumford-Lear extensions are compatible with tensor products. If C is the equivalence class of Weil divisors associated to L, we write [C, || · ||]Y for the divisor class associated to [L, || · ||]Y and call it the Mumford-Lear extension of C to Y . If it is clear from the context what metric is being used, we drop || · || from the notation.

12 2 Tools

In this chapter we introduce the tools necessary to compute the Mumford-Lear extension of L4,4(Γ) to the various compactifications of E0(Γ). We begin by discussing two techniques for calculating Mumford-Lear extensions that are somewhat similar in flavor, the method of test curves and the method of locally generating sections. The former allows us to reduce the calculation of the Mumford-Lear extension of a divisor to the case where the underlying manifold is ∆, while the latter provides a method to reason about Mumford-Lear extensions entirely by way of locally trivializing sections. In section 2.2, we introduce the Deligne pairing of line bundles, which is a relative version of the usual intersection pairing of divisors on algebraic surfaces. We collect some identities satisfied by this pairing, most notably an adjunction formula and a relation between Deligne pairings of degree zero divisors on one hand and the Poincarébundle on the Jacobian variety associated to a family of nodal curves on the other. In section 2.3 we describe the singularities of the Petersson metric in greater detail. As mentioned in the previous chapter, the smooth metric || · ||Pet can not be extended to a pre-log metric along the entirety of D = E(Γ) − E0(Γ). The degree to which the existence of such an extension fails can be quantified locally on a coordinate system (W, u, v) around a point p ∈ Dsing in the following manner. There is, associated to p, a rational number q ∈ Q>0 such that the metric || · || given by the formula log |u| log |v| log || · ||2 = log || · ||2 + q · . Pet log |u| + log |v| on E0(Γ) ∩ W does not only admit a Mumford-Lear extension, but in fact a Mumford extension along the entirety of D ∩ W . This property is the key ingredient in calculating the Mumford-Lear 0 extension of L4,4(Γ) to all compactifications of E (Γ).

2.1 Method of test curves and locally generating sections

Let (L, || · ||) be a metrized line bundle on a smooth algebraic variety X over C with associated divisor class C and let Y be a smooth compactiﬁcation of X. Write V = {D1,...,Dn} for the set of irreducible components of D = Y − X. Suppose a divisor C0 on Y is given with the property 0 ∼ that OY (C )|X = L. Suppose furthermore that the Mumford-Lear extension of L to Y exists. V 0 Pn Since Y is smooth, PicL(Y ) is a Z -torsor, so [C]Y is necessarily of the form [C]Y = C + i=1 aiDi for certain ai ∈ Q. The method of test curves allows us to determine these coeﬃcients ai in the following manner.

Proposition 2.1. Suppose we are given for each i = 1, . . . , n a curve fi : ∆ → Y which intersects Di ∗ transversally in a point pi ∈ Di − S and which restricts to a curve fi : ∆ → X. Then for each i = ∗ ∗ ∗ 0 1, . . . , n, the Mumford-Lear extension of fi L to ∆ exists, and we have [fi C]∆ = fi C + ai · [0], where [0] is the divisor on ∆ corresponding to the closed submanifold {0}.

Proof. Since pull-backs and Mumford-Lear extensions are compatible with tensor products, it suf- ﬁces to prove the proposition in the case that ai ∈ Z. Then [L]Y comes equipped with a smooth metric on [L]Y |X which is pre-log along D − S. Evidently this implies that the pull-back met- ∗ ric on fi [L]Y has logarithmic growth along {0}, so since pre-log-log forms are preserved under

13 ∗ ∗ pullbacks, we deduce that [fi C]∆ = fi [C]Y . Now we calculate:

 n  ∗ ∗ ∗ 0 X ∗ 0 [fi C]∆ = fi [C]Y = fi C + ajDj = fi C + ai · [0], j=1

∗ ∗ where in the last step we use that fi Di = [0] and fi Dj is the empty divisor for all i 6= j.

∗ Thus to find ai it suffices to compute the component of [0] for [fi C]∆. In view of Proposition 2.1, we say a curve fi is a test curve for the i-th component of D if it satisfies the properties in the statement of Proposition 2.1. Next we describe the method of locally generating sections. Evidently the property of being a Mumford extension of (L, || · ||) is a local one, so it suffices to characterize existence of Mumford extensions on compactifications of the form (∆∗)k × ∆n−k ⊂ ∆n. Then a generating section of an extension of L can be used to characterize existence of a Mumford extension in the following manner. 0 n 0 ∗ k Proposition 2.2. Let C be a divisor on ∆ with the property that O∆n (C ) restricts to L on (∆ ) × n−k 0 ∆ and let s be a generating section of O∆n (C ). Consider for each i = 1, . . . , k the divisor Di given by the equation zi = 0 and note that D1,...,Dk are the irreducible components of D = n ∗ k n−k ∆ − (∆ ) × ∆ . Let a1, . . . , ak ∈ Q>0. The following statements are equivalent: Pk n • The function log ||s|| − i=1 ai log |zi| on ∆ − D is a pre-log-log form along D 0 Pk • The Q-divisor C + i=1 ai · Di is the underlying Q-divisor of a Mumford extension of L to ∆n.

Proof. Again it suﬃces to prove this statement under the assumption that a1, . . . , ak ∈ Z. Then we −a1 −ak 0 Pk simply note that s·z1 . . . zk is a generating section of O∆n (C + i=1 ai ·Di), so all information about the growth of log ||·|| along D can be deduced from information about the growth of log ||s||− Pk i=1 ai log |zi|.

2.2 Deligne pairing

Throughout this section, by a nodal curve over C we mean a complete algebraic curve over C that has only ordinary double points as singularities. Definition 2.1. Let p: X → S be a family of nodal curves over a complex manifold S, that is, a flat proper holomorphic map p: X → S with fibers that are nodal curves over C, and let L and M be two line bundles on X. The Deligne pairing hL, Mi is defined as follows: • If S is a point, then X is a nodal curve. Consider the vector space V of pairs (`, m) of rational sections of L, M with the property that div(`) and div(m) have disjoint support and `, m are non-zero on the components of X and regular on the nodes of X. Then hL, Mi is the quotient of V by the relations (f`, m) ∼ f(div(m))(`, m) (`, gm) ∼ g(div(`))(`, m),

14 P where f, g are rational functions on X and where for a divisor D = P ∈X nP P on X we deﬁne Y f(D) = f(P )nP P ∈X Q • If S is not a point, we deﬁne hL, Mi(U) to be the set of tuples (us)s∈U ∈ s∈U hLs, Msi with the property that for all x ∈ U there are an open V 3 x and rational sections `, m on π−1(V ) with ut = h`t, mti for all t ∈ V . For a proof that hL, Mi is really a line bundle, we refer the reader to [1, section XIII.5]. From the “pointwise” construction of hL, Mi it is clear that the Deligne pairing is compatible with base change. Another property that is immediately apparent is the “bilinearity” and “symmetry” of the Deligne pairing: if L1, L2, M1 and M2 are line bundles on X, we have natural isomorphisms

hL1, M1i ⊗ hL1, M2i ⊗ hL2, M1i ⊗ hL2, M2i → hL1 ⊗ L2, M1 ⊗ M2i given by h`1, m1i ⊗ h`1, m2i ⊗ h`2, m1i ⊗ h`2, m2i 7→ h`1 ⊗ `2, m1 ⊗ m2i and

hL, Mi → hM, Li given by h`, mi 7→ hm, `i. Let P : S → X be a section of p such that the image of P is contained in the largest open subset U ⊂ X where p: X → S is smooth. Then there is a natural isomorphism between hL, O(P )i and P ∗L, cf. [1, XIII.5.18]. Thus we have the following “adjunction formula” involving the relative dualizing sheaf ωX/S: ∗ ⊗−1 Lemma 2.3. There is a natural isomorphism hO(P ), O(P )i → P ωX/S .

∗ ∼ Proof. By the usual adjunction formula, we have P (O(P ) ⊗ ωX ) = ωS. Note that as p is smooth ∗ ⊗−1 ∗ ∗ on U, we have ωU/S = ωU ⊗ p ωS . Hence, since P p ωS = ωS and since the image of P is contained in U, we have

∗ ∗ ⊗−1 ∗ ∗ ⊗−1 ∗ ∗ ⊗−1 ∗ ⊗−1 P O(P ) = P (ωX ⊗ p ωS) = P (ωU ⊗ p ωS) = P ωU/S = P ωX/S .

Now the result follows by applying the above natural isomorphism to L = O(P ).

We note that if X is a family of nodal curves over a compact Riemann surface S, then the Deligne pairing generalizes the usual intersection pairing in the sense that degShL, Mi = C · D, where C and D are the divisor classes associated to L and M. Next we give a relation between the Deligne pairing of two degree zero divisors and the Poincaré bundle on the Jacobian variety associated to a family of smooth curves p: X → S. To set this up, we need the fact that an analogue of the Picard scheme from section 1.3 exists in the category of complex analytic spaces. Once this is known, we infer by universality of PicX/S that the space X × PicX/S comes equipped with a line bundle P, called the Poincarébundle. Then on the Jacobian variety J(X/S) of X, which is by definition the connected component of the identity element of PicX/S, we get a line bundle on X×PicX/S by pulling back P along the inclusion map X×J(X/S) → X×PicX/S. By abuse of notation, we write P for this line bundle and also call it the Poincarébundle.

15 Proposition 2.4. Suppose E is a family of elliptic curves over a complex manifold S and suppose L and M are line bundles on E with the property that for all s ∈ S, Ls and Ms are degree zero line bundles on Es. Consider the map ν : S → J(E/S) ×S J(E/S) associated to

(L, M) ∈ Pic(E/S) ×S Pic(E/S) ∨ ∨∨ ∼ and let λ: E → E = E be the dual of the principal polarization associated to L = OE(O). Then there is a natural isomorphism hL, Mi⊗−1 −→∼ ((λ, id) ◦ ν)∗P

We refer the reader to [13, Corollary 4.2] for a proof of this proposition. We end this section by noting that if L and M are equipped with smooth metrics, the Deligne pairing hL, Mi comes equipped with a smooth metric as well. A smooth metric can also be defined on the Poincarébundle associated to X → S and the natural isomorphisms introduced in this section become isometries when the Deligne pairings and Poincarébundles are equipped with their natural smooth metrics.

2.3 The Petersson metric near singular points of E(Γ) − E0(Γ)

To introduce coordinates on E(Γ), one has no choice but to give an explicit construction of it. In [4], this is done using the theory of toric varieties. We characterize the resulting space in the following proposition. Proposition 2.5. Let c be a cusp of X(Γ). Then the ﬁber D = p−1{c} is an h-gon, where h is the period of c. Moreover, there is a numbering C0,...,Ch−1 of the irreducible components of D such that for each ν ∈ Z/hZ there is a coordinate system

(Wc,ν, uν, vν) centered around pν = Dν ∩ Dν+1 such that Wc,ν ∩ Cν is given by the equation uν = 0, Wc,ν ∩ Cν+1 is given by the equation vν = 0, and the following relations between (uν, vν) and the natural coordinate 0 system (τ, z) on Wc,ν ∩ E (Γ) hold: ν+1 ν uνvν = exp(2πiτ/h), uν vν = exp(2πiz).

Here, by an h-gon on a smooth algebraic surface X we mean the following. If h > 1, it is a 1 curve isomorphic to the one obtained by taking h copies {Cν}ν∈Z/hZ of P and gluing ∞ to 0 for 2 each ν ∈ Z/hZ, embedded in X in such a manner that Cν = −2 and Cν · Cν+1 = 1 for all ν ∈ Z/hZ. 2 2 If h = 1, it is a curve isomorphic to the projective closure of Spec(C[x, y]/(y − x (x + 1))).

With respect to the coordinate system (uν, vν), we have the following description of || · ||Pet:

Lemma 2.6. Let f be a local section of L4,4(Γ). Around pν = Cν ∩Cν+1, the metric ||·||Pet satisﬁes the following formula: 2 2 log||f(τ, z)||Pet|Wν = log(|f(τ, z)| )|Wν 8 2 2 log |uν| log |vν| + (ν + 1) log |uν| + ν log |vν| − h log |uν| + log |vν| 2h + 4 log − c (log |u| + log |v|) . 4π

16 1 Proof. The proof is identical to that of [5, Lemma 2.10], noting that log |uν| = 2 log(uνuν) and 1 log |vν| = 2 log(vνvν).

Now we have enough material to describe the singularities of || · ||Pet near the singular points of the boundary divisor of E(Γ)/E0(Γ). Proposition 2.7. Let p ∈ E(Γ) be a singular point of the boundary divisor of E(Γ)/E0(Γ), above a cusp of period h. Then there is a coordinate chart (W, u, v) around p such that the metric || · || determined by the formula

8 log |u| log |v| log || · ||2 = log || · ||2 + Pet h log |u| + log |v| admits a Mumford extension along {uv = 0}.

Proof. Assume without loss of generality that p = Cν ∩ Cν+1 and take u = uν, v = vν. By the method of locally generating sections, it suﬃces to check for a local generator s of L4,4(Γ) that there are a, b ∈ Q such that log ||s|| − a log |u| − b log |v| is pre-log-log along {uv = 0}. We shall do this 8 for θ1,1. Note that by Proposition 2.5, we can write θ1,1 as

X h/2(n+1/2)2+(ν+1)(n+1/2) h/2(n+1/2)2+ν(n+1/2) θ1,1(τ, z) = exp(π(n + 1)/2)u v . n∈Z Let c and d be the smallest powers of u respectively v occurring in the above power series. Then

8 θ1,1(τ, z) 2hc log log + 4 log − (log |u| + log |v|) u8cv8d 4π

m 2 m 2 8 is pre-log-log along {uv = 0}. Hence taking a = c + h (ν + 1) and b = d + h ν we see log ||θ1,1|| − a log |u| − b log |v| is pre-log-log along {uv = 0}, as was to be shown.

17 3 Calculations

This chapter contains the main calculations of this thesis. We first determine the Mumford-Lear extension of L4,4(Γ) to E(Γ). Then we compute the Mumford-Lear extension of L4,4(Γ) to all smooth compactifications of E0(Γ) and show the associated system of divisors satisfies Chern-Weil theory and a Hilbert-Samuel type formula.

3.1 Mumford-Lear extension of L4,4(Γ) to E(Γ)

Let C(Γ) = X(Γ) − Y (Γ) be the set of cusps of X(Γ) and consider the space

0 0 U(Γ) = E (Γ) ×Y (Γ) E (Γ), viewed as a family of elliptic curves over E0(Γ) via the projection on the first coordinate. Denote 0 by O,P : E (Γ) → U(Γ) the zero and diagonal section, respectively. Recall that L4,4(Γ) can be identified with OE0(Γ)(8O). Let ∗ Q = (id ×ϕO) P 0 0 ∨ be the pullback of the Poincarébundle P on E (Γ) ×Y (Γ) E (Γ) along the principal polarization ∗ associated to O. Consider the Hodge bundle λY (Γ) = O ωE0(Γ)/Y (Γ) on Y (Γ). This bundle comes equipped with a smooth metric || · ||Pet, with respect to which we have the following theorem: ∗ Theorem 3.1. Equip P Q with the pullback of the natural metric || · ||can on P along (id ×ϕO) ◦ P . Then there is an isometry

∼ ∗ ⊗4 ∗ ⊗4 (OE0(Γ)(8O), || · ||Pet) −→ (P Q, || · ||can) ⊗ (p λY (Γ), || · ||Pet) .

∗ Thus to find a Mumford-Lear extension of L4,4(Γ) it suffices to find Mumford-Lear extensions of P Q ∗ and p λY (Γ). Mumford’s original paper (cf. [17]) implies that λY (Γ) has a Mumford extension with ∗ ∗ underlying line bundle λX(Γ) = O ωE(Γ)/X(Γ). As for P Q, the existence of the Lear extension follows from Lear’s thesis (cf. [16]), and we will determine it using the method of test curves. Note that by Proposition 2.4, we have P ∗Q =∼ hP − O,P − Oi⊗−1. Hence, it suffices to determine ⊗−1 the Mumford-Lear extension of hP − O,P − Oi . Let c ∈ C(Γ) be a cusp of X(Γ) and let h = hc be its period. By Proposition 2.5, we know that the fiber above c is an h-gon D, say with irreducible components V = {Cc,0,...,Cc,h−1}, with the zero section O passing through Cc,0 and then ordered cyclically. Relabel the irreducible components by Ci = Cc,i for the moment and let f i : ∆ → E(Γ) be a test curve for the i-th component of D which does not intersect the zero section. Then by compatibility of the Deligne pairing and pullbacks, we have

∗ ⊗−1 ⊗−1 fi hP − O,P − Oi = hP∆∗ − O∆∗ ,P∆∗ − O∆∗ i ,

∗ where O∆∗ is the zero section of the ﬁbration p∆∗ : U(Γ)∆∗ → ∆ and P∆∗ is a section of p∆∗ which is disjoint from O∆∗ and which intersects the h-gon above 0 in the i-th component. Let U](Γ)∆∗ /∆ ∗ be a compactiﬁcation and desingularization of U∆∗ /∆ and let

O∆,P∆ : ∆ → U](Γ)∆∗

18 be the unique extensions of O∆∗ and P∆∗ to ∆. Since P∆∗ − O∆∗ is of relative degree 0, we have ⊗−1 that by [13, Theorem 2.2], the Mumford-Lear extension of hP∆∗ −O∆∗ ,P∆∗ −O∆∗ i to ∆ is equal to ⊗−1 ⊗−1 [hP∆∗ − O∆∗ ,P∆∗ − O∆∗ i ]∆ = hP∆ − O∆ + Φi,P∆ − O∆ + Φii , where Φi is a Q-divisor with support contained in D∆, uniquely determined by the properties that the coeﬃcient for C0∆ is zero and P∆ − O∆ + Φi has zero intersection with Cν∆ for all ν ∈ Z/hZ. Applying this second property to Φi to get the equality hP∆ − O∆ + Φi, Φii = 0, we see the Mumford-Lear extension can also be expressed as follows:

⊗−1 ⊗−1 [hP∆∗ − O∆∗ ,P∆∗ − O∆∗ i ]∆ = hP∆ − O∆ + Φi,P∆ − O∆ + Φii ⊗−1 = hP∆ − O∆ + Φi,P∆ − O∆ − Φii ⊗−1 = hP∆ − O∆,P∆ − O∆i ⊗ hΦi, Φii ⊗−1 2 = hP∆ − O∆,P∆ − O∆i ⊗ O∆(Φi · [0]),

2 where we use in the last equality that hΦi, Φii and O∆(Φi · [0]) are both uniquely characterized by 2 2 the fact that they are supported on [0] and have degree Φi . Thus the coeﬃcient for Ci is Φi . The 2 following lemma gives an explicit description of Φi . Lemma 3.2. There is exactly one -divisor Φ = P v C with v = 0 satisfying the Q i ν∈Z/hZ ν ν∆ 0 property that P∆ − O∆ + Φi has zero intersection with Cν∆ for all ν ∈ Z/hZ. Its coeﬃcients vν ∈ Q are given by ( ν · T0 if ν < i vν = , ν · T0 + i − ν else

h−i where T0 = h . The self-intersection of Φi is given by i(h − i) Φ2 = − . i h

Proof. Relabel the irreducible components of D∆ as Aν = Cν∆ for the moment. If Φi exists, it should satisfy the following relations:

0 = hP∆ − O∆ + Φi,Aii = 1 + hΦi,Aii

0 = hP∆ − O∆ + Φi,A0i = −1 + hΦi,A0i

0 = hP∆ − O∆ + Φi,Aνi = hΦ,Aνi for all ν 6= i

2 Thus, since Aν = −2 and Aν · Aν+1 = 1 for all ν ∈ Z/hZ, we see the vector v = (v0, . . . , vh−1) is a solution of the linear system Mv = b, where b = e0 − ei and

−2 1 0 ··· 1   1 −2 1 ··· 0     .. .  M =  0 1 −2 . .     ......   . . . . 1  1 0 ··· 1 −2

> Note that M is of rank h − 1, with kernel given by Q · (1, 1,..., 1) . Hence the solution set > h of Mv = b is of the form u + Q · (1, 1,..., 1) for some vector u ∈ Q . This implies that there

19 is exactly one v = (v0, . . . , vh−1) with v0 = 0 and Mv = b. Its coeﬃcients are determined by the following recurrence relations: ( 2vν−1 − vν−2 if ν 6= i + 1; vν = 2vi − vi−1 − 1 otherwise.

A small calculation shows that these relations are satisﬁed by the vν in the statement of this lemma, so we infer by unicity of Φi that the coeﬃcients of Φi are as claimed.

Next we calculate the self-intersection of Φi. Note that ( T0 if ν < i vν+1 − vν = T0 − 1 otherwise.

Hence,

h−1 h−1 h−1 2 X 2 X X Φi = −2 vν + 2 vνvν+1 = 2 (vν+1 − vν)vν ν=0 ν=0 ν=0 2 = T0 · i(i − 1) + (T0 − 1) · ((T0 − 1) · (h(h − 1) − i(i − 1) + 2(i(h − i))) i(h − i) = − , h as was to be shown.

Now consider for each c ∈ C(Γ) the Q-divisor

X i · (hc − i) Vc = − Cc,i. hc i∈Z/hcZ

∗ ⊗−1 N The previous discussion implies that [P Q, || · ||can]E(Γ) = hP − O,P − Oi ⊗ c∈C(Γ) OE(Γ)(Vc). ∗ ⊗−1 ⊗−1 Note that hP,Oi = P OU(Γ)(O) = OE(Γ)(O). Hence, since hP,P i and hO,Oi are isomorphic to λY (Γ) by Lemma 2.2, we ﬁnd using bilinearity of the Deligne pairing that

∗ ∗ ⊗2 O [P Q, || · ||can]E(Γ) = p λX(Γ) ⊗ OE(Γ)(2O) ⊗ OE(Γ)(Vc). c∈C(Γ)

Thus we arrive at the following result. ∗ Theorem 3.3. Let F be the divisor class associated to p λX(Γ) and let J be the divisor class P associated to L4,4(Γ). Then [J, || · ||Pet]E(Γ) = 12F + 8O + 4 c∈C(Γ) Vc.

To compute the self-intersection of [J, || · ||Pet]E(Γ), we need the following auxiliary results.

Lemma 3.4. For each c ∈ C(Γ), the self-intersection of Vc is given by

2 2 hc − 1 Vc = − 3hc

20 Proof. Fix a cusp c ∈ C(Γ) and relabel for the moment the period of c as h = hc and the irreducible components of D = Dc as Ci = Cc,i for all i ∈ Z/hZ. Then

h h X i2(h − i)2 X i(i + 1)(h − i)(h − i − 1) V 2 = −2 + 2 c h2 h2 i=0 i=0 h2(h + 1)2 − 1 (3h − 1)h(h + 1)(2h + 1) + h2(h + 1)(h − 1) = 3 h2 h2 − 1 = − 3h

Lemma 3.5. Let dΓ be the degree of the forgetful map π : X(Γ) → X(1). Then we have the following equality: X dΓ = hc. c∈C(Γ)

Proof. Choose N 0 such that Γ(N) ≤ Γ and let π1 : X(N) → X(1) and πΓ : X(N) → X(Γ) be the natural projection maps. Note that since Γ(N) is normal in SL2(Z), it is also normal in Γ. By [11, page 67] and the fact that Γ(N) is a normal subgroup of Γ, the ramification index ec of any cusp T ∈ C(Γ(N)) above c ∈ C(Γ) is equal to h/hc, where h is the period of T . As all cusps in X(N) have period N (cf. [11, page 101]), we deduce that hc = N/ec. ∗ Now consider the divisor of cusps δ0 := π [∞] of X(Γ). The pull-back of a point is supported on P the fiber above that point, so we may write δ0 = c∈C(Γ) nc ·c for certain nc ∈ Z. As π1 = π ◦πΓ, we ∗ ∗ deduce that πΓδ0 = π1[∞]. Therefore, since each cusp in X(N) has ramification index N above [∞], we find X X ∗ X X ncec · T = πΓ nc · c = N · T. c∈C(Γ) −1 c∈C(Γ) T ∈C(Γ(N)) T ∈πΓ (c) P We deduce that δ0 = c∈C(Γ) hc · c, so indeed X X hc = deg hc · c = deg δ0 = deg π · deg[∞] = dΓ · 1 = dΓ, c∈C(Γ) c∈C(Γ) as was to be shown.

We end this section with a calculation of the self-intersection of [J, || · ||Pet]E(Γ).

Theorem 3.6. Let dΓ be the degree of the forgetful map π : X(Γ) → X(1). Then the self-intersection of [J, || · ||Pet]E(Γ) is given by

2 64 X 64 [J, || · ||Pet]E(Γ) = dΓ + . 12 12hc c∈C(Γ)

Proof. Note that O·Vc = 0 for all c ∈ C(Γ) since O intersects Dc only in Cc,0 and since the coefficient for Cc,0 is zero. The self-intersection of F is also zero since F can be written as a sum of fibers, which have zero self-intersection since every fiber is algebraically equivalent to a fiber which doesn’t

21 intersect it. In a similar way we see that F has zero intersection with Vc for all c ∈ C(Γ). Thus we 2 obtain the following formula for [J, || · ||Pet]E(Γ):

2 2 2 X hc − 1 [J, || · ||Pet]E(Γ) = 192 · F · O + 64 · O − 64 12hc c∈C(Γ)

By the adjunction formula we have −O · O = F · O = dΓ (cf. [6, Proposition 3.2]). Substituting P 12 this in the above formula and using that dΓ = c∈C(Γ) hc we obtain 2 196 64 64 X 64 [J, || · ||Pet]E(Γ) = − − dΓ + 12 12 12 12hc c∈C(Γ) 64 X 64 = dΓ + , 12 12hc c∈C(Γ) as was to be shown.

3.2 Comparison to Chern-Weil theory

Let CPT0(E0(Γ)) be the subset of CPT(E0(Γ)) consisting of the isomorphism classes of compactifications Y/E0(Γ) that are obtained from E(Γ) by a finite number of blow-ups in singular points of the boundary divisor. By Lemma 3.7 below, to compute Mumford-Lear extensions to arbitrary 0 0 0 compactifications of E (Γ), it suffices to compute [L4,4(Γ), || · ||Pet]Y for all Y ∈ CPT (E (Γ)). Lemma 3.7. Let Y be a smooth compactification of a variety X over C with boundary divisor D and suppose (L, ||·||) is a smoothly metrized line bundle on X. Suppose furthermore that (L, ||·||, e) is a Mumford-Lear extension of L to Y and let S ⊂ D be the smallest subset for which

(L|X−S, || · ||) admits a Mumford extension along D − S. Let π : Y 0 → Y be the compactiﬁcation of X obtained by blowing up Y in a point p ∈ D − S. Then the Mumford-Lear extension of L to Y 0 exists and is equal to (π∗L, π∗|| · ||, e)

Proof. Just note that π∗|| · || is pre-log along π−1(D − S). Corollary 3.8. If E0/C is an elliptic surface equipped with a metrized line bundle (L, || · ||), the Mumford-Lear extensions of L to a smooth compactiﬁcation Y 0 of E0 can be computed in the follow- 0 0 0 ing manner: if Y is the greatest element of CPT (E ) with Y E Y and the Mumford-Lear extension of L to Y exists, then so does the Mumford-Lear extension of L to Y 0 and we have

∗ [L, || · ||]Y = π [L, || · ||]Y 0 , where π : Y 0 → Y is the projection map induced by the chain of blow-ups turning Y into Y 0. If we blow up a singular point p of the boundary divisor D of Y/E0(Γ), the exceptional divisor E intersects the strict transform of D in two points. Thus whenever we blow up a singular point, we get two singular points back, which again can be blown up to get four points and so on. This yields the following result.

22 Lemma 3.9. Take for each c ∈ C(Γ) and i = 0, . . . , hc − 1 a copy Tc,i of the full binary tree. Then CPT0(E0(Γ)) is order-isomorphic to

h −1 Y Yc Tc,i c∈C(Γ) i=0 equipped with the product order. Let Y be a compactification of E0(Γ). We say a singular point p ∈ Y − E0(Γ) has type (n, m) and multiplicity µ if there is an adapted coordinate system (V, u, v) of p with the property that the norm given by the formula 2µ log |u| log |v| log || · ||2 = log || · ||2 + Pet nm n log |u| + m log |v| admits a Mumford extension along D ∩ V . Note that by Proposition 2.12, a singular point p in the fiber above a cusp c ∈ C(Γ) has type (1, 1) and multiplicity 4/hc. As we will see in Theorem 3.11 below, if we blow up a singular point of type (n, m) and multiplicity µ we obtain two new points of multiplicity µ, one of type (n + m, m) and the other of type (n, n + m). Before we can prove this statement we need to study the expression on the right hand side of the above equation in more detail. Proposition 3.10. Let (n, m) be a pair of coprime positive integers and let D be the boundary 2 ∗ 2 ∗ 2 divisor of the compactification ∆ of (∆ ) . Let fn,m be the function (∆ ) → C defined by the formula 2 log |u| log |v| f (u, v) = . n,m nm n log |u| + m log |v| 2 2 Then fn,m is a pre-log-log function along D − {(0, 0)}. Moreover, if π is the function ∆ → ∆ given by (s, t) 7→ (st, t) then 2 f (π(s, t)) = log |t| + f (s, t). n,m nm(n + m) n,n+m

Similarly, if π0 is the function ∆2 → ∆2 given by (s, t) 7→ (s, st), then 2 f (π0(s, t)) = log |s| + f (s, t). n,m nm(n + m) n+m,n

Proof. For a proof that fn,m is pre-log-log along D − {(0, 0)}, we refer the reader to [5, Proposition 4.1.(i)]. As for the second statement, this follows from the following calculation: 2 log |t|(n log |s| + (n + m) log |t|) + m log |s| log |t| f (st, t) = n,m nm(n + m) n log |s| + (n + m) log |t| 2 = log |t| + f (s, t) nm(n + m) n,m The third statement is proved in an identical manner.

2 1 2 2 Let Bl(0,0)∆ ⊂ P × ∆ be the blow-up of ∆ in (0, 0). Note that the map π from the previous 2 2 2 proposition is the composition of the projection map Bl(0,0)∆ → ∆ with the coordinate map ∆ → 2 Bl(0,0)∆ (s, t) 7→ ((s : 1), st, t),

23 2 2 and that the exceptional divisor E of ψ : Bl(0,0)∆ → ∆ is given on this chart by the equation t = 0. Similarly, with respect to the coordinate map (s, t) 7→ ((1 : t), s, st) around p2 the projection map ψ is given by π0 and the exceptional divisor E by the equation t = 0. Theorem 3.11. Let Y be a compactiﬁcation of E0(Γ) with boundary divisor D and singular locus S. Suppose p ∈ D is a singular point of type (n, m) and multiplicity µ. Consider the blow-up ψ : Y 0 → Y 0 of Y in the point p. Then the singular locus of Y is equal to S − {p} ∪ {p1, p2}, where p1 and p2 are both of multiplicity µ, p1 is of type (n, n + m) and p2 is of type (n + m, n). Moreover, if the Mumford-Lear extension of L4,4(Γ) to Y exists then so does the Mumford-Lear extension of L4,4(Γ) to Y 0, and we have ∗ µ [J, || · || ] 0 = π [J, || · || ] − E, Pet Y Pet Y nm(n + m) where E is the exceptional divisor of the blow-up π : Y 0 → Y .

Proof. Let g be a generating section of L4,4|V , where (V, u, v) is a coordinate system for Y around p satisfying the property that the norm given by

2 2 log || · || = log || · ||Pet + µ · fn,m(u, v) admits a Mumford extension along D ∩ V . Denote by D1 = {u = 0} and D2 = {v = 0} the irreducible components of D ∩ V . Then by Proposition 2.2, there are c, d ∈ Q>0 such that log ||g|| − c · log |u| − d · log |v| is a pre-log-log function along D ∩ V . The points of intersection p1, p2 of the strict transform of D 2 with E correspond to the points ((0 : 1), 0, 0), ((1 : 0), 0, 0) on Bl(0,0)∆ and in the coordinate chart (s, t) around p1, the pull-back of || · || along ψ is given by µ log ||π∗g|| = log ||π∗g|| + log |t| + 1 µ · f (s, t). Pet nm(n + m) 2 n,n+m

∗ µ 1 Thus log ||π g||Pet − c log |s| + nm(n+m) log |t| + 2 µ · fn,n+m(s, t) is pre-log-log along D ∩ V , so p1 is a point of type (n, n + m) and multiplicity µ. In an entirely analogous manner, using that ψ is 0 given by π on the coordinate chart around p2, we see that p2 is a point of type (n + m, m) and multiplicity µ. sm Now since fk,`(u, v) is pre-log-log along D ∩V for all pairs of coprime integers (k, `), we deduce that 0 the Mumford-Lear extension of L4,4(Γ) to Y exists. As the pullback of D2 respectively D1 along ψ is given on the coordinate around p1 respectively p2 by the equation s = 0, we see the coeﬃcients for the irreducible components of ψ−1(D) and E of this Mumford-Lear extension coincide with those in the statement of this theorem, so we are done.

Now we have enough material to calculate the limit of the self-intersection of the Mumford-Lear 0 extension of L4,4(Γ) to all compactiﬁcations of E (Γ). 2 0 Theorem 3.12. The limit of the self-intersection [L4,4(Γ), || · ||Pet]Y over all Y ∈ CPT(E (Γ)) 64 exists and is equal to 12 dΓ.

Proof. Since the pull-back along the projection map associated to the blow-up in a non-singular point does not change the value of the self-intersection, we may as well calculate

2 lim [J, || · ||Pet]Y Y ∈CPT0(E0(Γ))

24 instead. For each c ∈ C(Γ) and i ∈ {0, . . . , hc − 1}, there is a one-to-one correspondence between the elements of Tc,i and the set N of pairs of coprime integers, defined recursively by the following conditions: • The element (1, 1) maps to the blow-up of E(Γ) in the i-th singular point above c. • If (n, m) maps to a compactification Y , then (n, n + m) maps to the blow-up of Y in the unique point of type (n, n + m) and (n + m, n) maps to the blow-up in the unique point of type (n + m, m). Recall that if π : Y → X is the blow-up of a smooth algebraic surface in a point p with exceptional ∗ 0 divisor E then Cl(X)⊕Z is isomorphic to Cl(Y ) via the map (D, n) 7→ π D +nE. Hence if Y → Y is the blow-up of a compactification Y/E0(Γ) in a point p of multiplicity µ and type (n, m) then

2 2 2 µ [J, || · || ] 0 = [J, || · || ] − . Pet Y Pet Y n2m2(n + m)2

µ2 Thus a point of multiplicity µ and type (n, m) contributes − n2m2(n+m)2 to the limit of self- intersections, so if the limit exists, it is equal to

2 2 2 X X 4 lim [J, || · ||Pet]Y = [J, || · ||Pet]E(Γ) − hc Y ∈CPT0(E0(Γ)) h2n2m2(n + m)2 c∈C(Γ) (n,m)∈N c   64 X 64 X 3 = dΓ + 1 − 2 2 2  . 12 12hc n m (n + m) c∈C(Γ) (n,m)∈N

P 1 As in the proof of [5, Theorem 4.11], we deduce from the results of [18] that the sum (n,m)∈N n2m2(n+m)2 converges to 1/3. Substituting this value in the above equation we obtain the desired result.

Next up is the promised analogue of Chern-Weil theory for the system of divisors associated to the tower of compactiﬁcations of E0(Γ). R ∧2 64 Theorem 3.13. The integral E0(Γ) c1(L4,4(Γ), || · ||Pet) exists and is equal to 12 dΓ.

0 Proof. Let ([J, || · ||Pet]E(Γ), 1, || · || ) be a Mumford extension of the norm || · || on L4,4(Γ) which is given locally around a double point p above a cusp of period h by the formula

8 log |u| log |v| log || · ||2 = log || · ||2 + . Pet h log |u| + log |v|

0 0 Write ω = c1(L4,4(Γ), || · ||), ω = c1(L4,4(Γ), || · || ) and

f = log ||s||2 − log ||s||02

0 1 for some meromorphic section s of L4,4(Γ). Then ω = ω + 2πi ∂∂f, so writing out the wedge product we see the following equality holds: Z Z Z ∧2 0∧2 1 ω = ω − d 2 ∂f ∧ ∂∂f . E0(Γ) E0(Γ) E0(Γ) (2πi)

25 Since an analogue of Chern-Weil theory holds for pre-log metrics and since pre-log-log forms have no residue (cf. [17]), we can rewrite the above integral as

hc−1 Z 2 X X [J, || · ||Pet] 0 − lim ∂f ∧ ∂∂f E (Γ) ε→0 c∈C(Γ) ν=0 ∂Vc,ν,ε where Vc,ν,ε = {(u, v) ∈ Wc,ν : |u| ≤ ε, |v| ≤ ε}. As in the proof of [5, Theorem 5.2], we compute Z 64 lim ∂f ∧ ∂∂f = . ε→0 2 ∂Vc,ν,ε 12hc Putting this all together, we ﬁnd Z ∧2 2 X 64 ω = [J, || · ||Pet]E(Γ) − hc · 2 0 12h E (Γ) c∈C(Γ) c 64 X 64 X 64 64 = dΓ + − = dΓ, 12 12hc 12hc 12 c∈C(Γ) c∈C(Γ) as was to be shown.

We end this section by showing that the system of divisors associated to the tower of compactifications of E0(Γ) satisfies a Hilbert-Samuel type formula, generalizing [5, Theorem 5.2] to arbitrary subgroups Γ ≤ SL2(Z) which act without fixed points on H. Proposition 3.14. The following equality holds:

2 dim J4`,4`(Γ) lim [J, || · ||Pet]X = lim . X∈CPT(E0(Γ)) `→∞ `2/2!

Proof. By [6, Proposition 3.8], the following equality holds for all ` 0 with hc|` for all c ∈ C(Γ):   16 2 X hc 16` hc X  dim J4`,4`(Γ) = dΓ · ` − dΓ · ` −  Q + H(∆) , 6  4 hc 2  c∈C(Γ) ∆|16`/hc,∆<0 16`/(hc∆) squarefree where Q(n) denotes the largest integer whose square divides n and H(∆) is the Hurwitz class number. As in the proof of [5, Proposition 2.6] and [5, Remark 2.7], we deduce 32 dim J (Γ) = d · `2 + o(`2). 4`,4` 12 Γ Now we calculate:

32 2 2 2 64 12 dΓ · ` + o(` ) dim J4`,4`(Γ) lim [J, || · ||Pet]X = dΓ = lim = lim . X∈CPT(E0(Γ)) 12 `→∞ `2/2! `→∞ `2/2!

26 4 Concluding remarks

We have succeeded in generalizing [5, Theorem 5.1] and [5, Theorem 5.2] to all congruence subgroups Γ ≤ SL2(Z) which act without ﬁxed points on H. The next question is whether these results can be generalized to arbitrary families of abelian varieties that come equipped with a theta divi- sor1. It goes beyond the scope of this thesis to even formulate an analogue of [5, Theorem 5.2] for families of abelian varieties. But we will mention the following higher-dimensional analogues of the 0 two main ingredients enabling the calculation of [J, || · ||Pet]E for all E ∈ CPT(E (Γ)): (i) Suppose p: A → Y is a family of abelian varieties and λ: A → A∨ is the principal polarization induced by a theta divisor Θ on A. Write P = (id, λ) and let P be the Poincar´ebundle ∨ ∗ 1 on A ×Y A . Let λY = det O ΩA/Y be the determinant of the Hodge bundle of A/Y . Then with respect to a translation invariant metric || · ||Pet on OA(2Θ) and certain natural metrics || · ||Pet on λY and || · ||can on P, we have an isometry

∼ ∗ ∗ (OA(2Θ), || · ||Pet) −→ P (P, || · ||can) ⊗ p (λY , || · ||Pet)

Thus, if p: A → Y is an extension of p, we can again use the original results of Mumford and Lear to deduce OA(2Θ) has a Mumford-Lear extension to A. (ii) In [7], it is proved that if A/Y is the Jacobian of a family of curves over Y satisfying certain properties, then the singularities of the Lear extension of P ∗P to A can be characterized on a coordinate neighborhood (U, z1, . . . , zn) adapted to D = A−A in the following manner. If D is given locally by the equation z1 . . . zk = 0 then there exists a positive integer d, a homogeneous k polynomial Q ∈ Z[x1, . . . , xk] of degree d with no zeroes on R>0 and for each locally generating ∗ section s of P P, a homogeneous polynomial Ps ∈ Z[x1, . . . , xk] of degree d + 1 such that for fs = Ps/Q we have that

− log ||s|| − fs(− log |z1|,..., − log |zk|)

is bounded on U ∩ A and extends continuously over U − Dsing. Although no analogue of relatively minimal models exists for higher-dimensional varieties, it may again be fruitful to study the behavior under blow-ups of rational functions that arise in this manner. 2 We also note that [J, || · ||Pet]E(Γ) can be expressed entirely in terms of invariants related to the action of Γ on H. This suggests that it may be possible to extend the results of this thesis to elliptic surfaces over curves that are deﬁned as the quotient of some space under a properly discontinuous group action.

1Dual abelian varieties and theta divisors on abelian varieties are deﬁned in exactly the same way as dual elliptic curves and theta divisors on elliptic curves.

27 References

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