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 31.1 The tower of compactifications of E0(Γ) . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Moduli spaces of elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Jacobi forms and theta divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Metrics and Mumford-Lear extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Tools 132.1 Method of test curves and locally generating sections . . . . . . . . . . . . . . . . . . 132.2 Deligne pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 The Petersson metric near singular points of E(Γ)− E0(Γ) . . . . . . . . . . . . . . 16

3 Calculations 183.1 Mumford-Lear extension of L4,4(Γ) to E(Γ) . . . . . . . . . . . . . . . . . . . . . . . 183.2 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, || · ||) isa line bundle on X equipped with a smooth metric, then we have

C · C =

∫Xc1(L, || · ||)∧2,

where C is the equivalence class of Weil divisors associated to L and c1(L, || · ||) is the first Chernform taken with respect to any connection on L compatible with || · ||. Let Γ ≤ SL2(Z) be acongruence 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 theaforementioned Chern-Weil theoretic result to compute the self-intersection of the theta divisoron E0(Γ) given by the image of the zero section, using a translation invariant metric || · ||Pet on theline bundle of weak Jacobi forms L4,4(Γ) of weight 4 and index 4. But since E0(Γ) is not compact,Chern-Weil theory can not be applied directly. Following the same line of reasoning as in [5], wetry to remedy this by extending (L4,4(Γ), || · ||Pet) to a metrized line bundle (L4,4(Γ), || · ||) on thecompactification

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 tobe possible to extend ||·||Pet smoothly. The closest thing we can get to a smooth extension of ||·||Pet isa so-called Mumford-Lear extension. This extension picks up serious enough singularities along thesingular locus of E(Γ)−E0(Γ) that the self-intersection of the associated divisor class [J, || · ||Pet]E(Γ)

is not equal to∫E0(Γ) c1(L4,4(Γ), || · ||Pet)

∧2. 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 pointson H, is the following.

Theorem. For every smooth compactification E of E0(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 equalityholds:

limE∈CPT(E0(Γ))

[J, || · ||Pet]2E =

∫E0(Γ)

c1(L4,4(Γ), || · ||Pet)∧2,

where CPT(E0(Γ)) denotes the directed set of isomorphism classes of smooth compactificationsof E0(Γ).

Thus an analogue of Chern-Weil theory for the line bundle of weak Jacobi forms on E0(Γ) can beobtained 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 periodof cusp c, the self-intersection of [J, || · ||Pet]E(Γ) is equal to

[J, || · ||Pet]2E(Γ) =

64

12dΓ +

∑c∈C(Γ)

64

12h2c

where dΓ is the degree of the forgetful map X(Γ) → X(1). Furthermore, taking the limit of theself-intersection over all compactifications of E0(Γ), we obtain

limE∈CPT(E0(Γ))

[J, || · ||Pet]2E =

64

12dΓ.

1

Overview

Throughout this thesis, we fix a congruence subgroup Γ ≤ SL2(Z) which acts without fixed pointson H.

The material in this paper is organized as follows. In the first chapter, we introduce the mainobjects of study in this thesis, the elliptic surface E(Γ) and the line bundle Lk,m(Γ) of weak Jacobiforms of weight k and index m, and provide some background on what purpose they serve in thegeneral body of mathematics. This chapter fulfills mainly an expository role and does not containmany proofs.

In the second chapter, we introduce a set of tools to be used in the calculation of the Mumford-Learextension of || · ||Pet to the various smooth compactifications of E0(Γ). We also give an explicitdescription 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 compactificationsof E0(Γ) is computed and we show the associated system of divisors satisfies Chern-Weil theoryand a Hilbert-Samuel type formula.

Acknowledgements

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

2

1 Background

This chapter serves to put the calculations in the next two chapters in context. We begin byintroducing the central object of study of this thesis, the elliptic surface E(Γ) associated to acongruence 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 actionextends to an action of ΓJ := Γ n Z2 on H× C given by((

a bc d

), (λ, µ)

)· (τ, z) =

(aτ + b

cτ + d,z + λτ + µ

cτ + d

).

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

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

of isomorphism classes of complete smooth algebraic surfaces X over C which contain E0(Γ) as anopen subset and which satisfy the property that X −E0(Γ) is a normal crossings divisor. We shallsee that the minimal element of CPT(E0(Γ)) is E(Γ), and all other elements of CPT(E0(Γ)) canbe 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 basescheme 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 1with geometrically connected fibers. If Γ ≤ SL2(Z) acts without fixed points on H and is equal toone of

Γ(N) =

(a bc d

)∈ SL2(Z) : a ≡ d ≡ 1 mod N, b ≡ c ≡ 0 mod N

Γ1(N) =

(a bc d

)∈ SL2(Z) : a ≡ d ≡ 1 mod N, c ≡ 0 mod N

Γ0(N) =

(a bc d

)∈ SL2(Z) : c ≡ 0 mod N

for some N > 0, then E0(Γ) can be interpreted as a moduli space of elliptic curves with extrastructure. Here by an elliptic curve over S we mean a smooth proper curve over S whose geometricfibers are genus 1 curves, equipped with a section O : S → E. A similar interpretation can be givento 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 massociated to Γ. For certain k and m, there is a particularly nice interpretation of Lk,m(Γ) as theline bundle associated to a theta divisor. This gives a good motivation to study Jacobi forms andso we opt to develop the theory of theta divisors first.

Finally, in section 1.4 we introduce smooth metrics, Mumford-Lear extensions and the Peterssonmetric 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, anelliptic surface is a family of curves parametrized by a base curve C, almost all of which are smoothcurves of genus 1.

Definition 1.1. Let k be a field. An elliptic surface E0 over a curve C/k is a smooth 2-dimensionalvariety over k, equipped with a proper morphism f : E0 → C with connected fibers such that thegeneric 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 etale coverings,we restrict our attention to varieties over C. There is a functor from the category of schemes offinite type over C to the category of complex analytic spaces, which sends a nonsingular varietyover C to a complex manifold (cf. [12, Appendix B]). If X is a non-singular variety over C, wedenote 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 : E0 → C with the propertythat 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 → Cwith 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 normalcrossings divisor D of X is a complex submanifold which admits for every p ∈ D a coordinatesystem z = (z1, . . . , zn) : U

∼−→ ∆n 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 Xif it is a connected compact complex manifold which contains X as an open subset and which satisfiesthe property that Y − X is a normal crossings divisor. Similarly, if Y is a complete non-singularvariety over C containing X as a Zariski open subset, we say Y is a compactification of X if Y an isa 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 relativelyminimal 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’scontractibility criterion, a curve on a surface can be blown down if and only if it is a −1-curve.Hence, since the Neron-Severi group of a surface is finitely generated and since a blow-down decreasesthe rank of the Neron-Severi group by one, a relatively minimal model of E0 can be obtained by

starting with an arbitrary compactification E0/C and blowing down a finite number of −1-curves.This yields the following result:

Theorem 1.1. Let p : E0 → C be an elliptic surface over a non-compact smooth curve C over Cwith 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,

ordered by the relation that E′ E E′′ if and only if E′′ can be obtained from E′ as a tower

E′′ = En → En−1 → . . .→ E0 = E′,

4

where for all k > 0, Ek is obtained from Ek−1 as a blow-up of Ek−1 in a point Ek−1 − E0.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 holo-morphic functions Y (Γ) → C correspond to Γ-periodic holomorphic functions H → C. Associatedto E0(Γ), we have a projection map

p : E0(Γ)→ Y (Γ), [τ, z] 7→ [τ ]

and a zero sectionO : 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 ellipticcurve: for every [τ ] ∈ Y (Γ), the fiber p−1[τ ] is equal to the complex torus Eτ := C/(Zτ ⊕ Z). Thespace Y (Γ) can be compactified by adding so-called cusps, cf. [11, section 2.4], and we denote theresulting curve by X(Γ). There is an interpretation of X(Γ) and E0(Γ) as complex manifolds comingfrom smooth algebraic varieties over C, and we define E(Γ)/X(Γ) to be the relatively minimal modelof 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 existbetween them, so let us begin by recalling the definition of a moduli space. If C is a category, X is anobject in C and f : X → Y is a morphism in C, let hX = Hom(−, X) be the functor which sends Yto the set of morphisms from Y to X and denote by hf = Hom(−, f) the natural transformationgiven by post-composition with f .

Definition 1.3. Let S be a scheme and F : SchopS → 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 uniquemorphism Y → X such that σ = hf τ .

• For every one-point scheme pt = Spec(k) over S with k = kalg, 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 ,given by UX = τ−1

X (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 spacesparametrize all “families of geometric objects of a certain type”, whereas coarse moduli spaces onlyparametrize “objects over points”.

We refer to a functor F : SchopS → 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 Tand 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 prob-lem Mg,n of n-pointed genus g curves. This is the moduli problem which associates to Y ∈ SchSthe 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 ′ the function

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 toMg,n in the category of schemes. This type of issue is common inthe theory of moduli spaces. If we want to have a scheme that represents a family of objects withautomorphisms, then either we need to settle with coarse moduli spaces, or we need to consider adifferent, 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 subgroupscheme E[N ]. Let (Z/NZ)S be the scheme associated to the constant functor SchS → Sets withvalue Z/NZ. Then a level structure on (E,O) associated to Γ is:

• An isomorphism of group schemes (Z/NZ)2S → 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 ] etale locally isomorphic to Z/NZ if Γ = Γ0(N).

The moduli problem of elliptic curves with level structure and n − 1 marked points is given onobjects 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 ellipticcurves over C with level structure associated to Γ (cf. [11, Theorem 1.5.1]). If N is such that Γacts without fixed points on H, then M1,1(Γ) has a fine moduli space over Spec(Z[1/N ]), and theanalytification of its base change to C is isomorphic to Y (Γ) (cf. [1, chapter XVI]). Even if Γdoes 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 abbreviationM1,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 naturaltransformation 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 )

The analytification of the base change of M1,2(Γ) to C is isomorphic to E0(Γ)−O (cf. [1, chapterXVI]).

6

As mentioned in the previous section, the spaces Y (Γ) and E0(Γ) are not compact, and neither arethe M1,n(Γ) with n > 2. If we were to compactify these spaces, we would like the result to be thesolution to some interesting moduli problem. The way to do this is to add degenerated curves tothe 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 fiberhas only ordinary double points as singularities. We say a tuple (C,P1, . . . , Pn) consisting of aproper nodal curve over S and an n-tuple of disjoint S-valued smooth points is semi-stable if theautomorphism 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 rationaland 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 markedor lies on another irreducible component.

• Every irreducible component of C that is rational has at least 2 points that are marked or lieon 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]

• O,P2, . . . , Pn ∈ Esm(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 eachM1,n(Γ) admits a solution over Spec(Z[1/N ]) and wehave natural projection maps M1,n(Γ) → M1,k(Γ) for all n > k. The analytification of the basechange of M1,1(Γ) to Spec(C) is isomorphic to X(Γ), and the analytification of the base changeof 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) forsome field k, we have the following bijection between the k-valued points of E and the degree zerodivisors of E(k):

ϕO : E → Div0(E), P 7→ [O]− [P ].

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

PicE/S(T ) = (L,α) : L ∈ Pic(ET ), α : OT∼−→ O∗TL,

where the datum α, called a rigidification of L along OT , serves a similar purpose as the levelstructure on elliptic curves we saw earlier, in that it is there to eliminate automorphisms. Foreach T ∈ SchS , the set PicE/S(T ) forms a group under the operation

(L,α) · (L′, α′) = (L⊗ L′, α⊗ α′)

7

and this establishes a factorization of PicE/S through the forgetful functor Grp→ Sets. Thus PicE/Sis in fact a group scheme. We refer to the connected component Pic0

E/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 isomorphismclasses of degree zero line bundles on Ek, and the correct generalization of the map ϕO above is apolarization 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 givenby

Λ(L) = m∗L⊗ p∗1L⊗−1 ⊗ p∗2L⊗−1.

This line bundle can be equipped with a rigidification, and we have a map ϕL correspondingto Λ(L) via the universal property of PicE/S . Fiberwise, ignoring rigidifications, it is given by x 7→t∗xL⊗L⊗−1. 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 callit 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 showthat the polarization associated to the image of the zero section Y (Γ) → E0(Γ) is given fiberwiseby the map which sends P ∈ Eτ to OEτ (O−P ) ∈ Pic0(Eτ ) for all τ ∈ H. Thus O is a theta divisoron E0(Γ).

We describe the global sections of the line bundle associated to O. Because E0(Γ) is defined as thequotient of a contractible space by a discontinuous group action, there is a fairly explicit descriptionof 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 withautomorphism group Ψ and suppose the Picard group of Y is trivial. Then there is a one-to-onecorrespondence between

• Families of invertible holomorphic functions (eψ)ψ∈Ψ ∈∏ψ∈ΨH

0(Y,O∗Y ) satisfying the prop-erty that

eψψ′(z) = eψ(ψ′ · z)eψ′(z)for all ψ,ψ′ ∈ 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 Lis given by

H0(X,L) = θ(z) ∈ H0(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 associatedto O, we write down a holomorphic function f : H × C → C with div(f) = O and determine thequotient 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

θ1,1(τ, z) =∑n∈Z

exp

(πiτ

(n+

1

2

)2

+ 2πi

(z +

1

2

)(n+

1

2

)).

Associated to the Riemann theta function we have the following system of multipliers:

c′(ψ; τ, z) = χ(γ) · (cτ + d)1/2 exp

(−πi

(λ2τ + 2λz − c(z + λτ + µ)2

cτ + d

)),

8

where ψ = ((a bc d

), (λ, µ)) ∈ ΓJ 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(ψ; τ, z) = (cτ + d)k exp

(−2πim

(λ2τ + 2λz − c(z + λτ + µ)2

cτ + d

)),

and call a global section of Lk,m(Γ) a weak Jacobi forms of weight k and index m. Thus θ81,1 is

a weak Jacobi form of weight 4 and index 4, and the line bundle associated to 8O is isomorphicto 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)).

Note that weak Jacobi forms are invariant under ψ = (± ( 1 m0 1 ) , (0, n)) for all m,n ∈ Z>0 with ψ ∈

ΓJ . More generally, since (( 1 00 1 ) , (0, 1)) ∈ ΓJ for all congruence subgroups Γ ≤ SL2(Z), if c = α ·∞

is a cusp of X(Γ) andhc = minm ∈ Z>0 : ((± 1 m

0 1 ) ∈ Γis its period, then the function f(τ, z) · c(α−1; τ, z) is hcZ-periodic in its first and Z-periodic in itssecond argument. Therefore, the function admits a Fourier expansion

f(τ, z) · c(α−1; τ, z) =∑

n∈Z,r∈Zac(n, r) exp(2nπiτ/hc) exp(2πiz).

We say f is a Jacobi form of weight k and index m if it is a weak Jacobi form with the propertythat ac(n, r) = 0 for all pairs (n, r) ∈ Z2 with n < 0 or 4mn− hcr2 < 0. If in addition this equalityholds for all (n, r) with n = 0 or 4mn−hcr2 = 0, we say f is a Jacobi cusp form. The Jacobi (cusp)forms of weight k and index m form a C-vector space, which we denote by Jk,m(Γ) (Jcuspk,m (Γ)), andas 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) toall 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 byrecalling the definition of a smooth metric on L and its associated first Chern form c1(L, || · ||).

Definition 1.7. LetX be a complex manifold equipped with a holomorphic line bundle L. A smoothmetric 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, bychoosing a connection ∇ on L compatible with || · || and taking c1(L, || · ||) = 1

2πi Tr(k∇), where k∇is the curvature form associated to ∇. This Chern form can be shown to be equal to

c1(L, || · ||) =1

2πi∂∂ log ||s||2,

9

where s is any locally generating section of L. Given smooth metrics ||·||1 and ||·||2 on line bundles Land M, the tensor product L ⊗M comes equipped with a smooth metric || · || determined by theformula ||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 equippedwith 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 smoothalgebraic 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 algebraicsurface X over C. Suppose || · ||1 and || · ||2 are smooth metrics on line bundles L and M on Xan,and C and D are the equivalence classes of Weil divisors associated to L andM, respectively. Then

C ·D =

∫Xc1(L, || · ||1) ∧ c1(M, || · ||2).

Obviously it is of interest to extend this result as much as possible to metrized line bundles onnon-compact algebraic surfaces. Let X be a non-compact algebraic surface over C equipped with asmoothly 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 metrizedline 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 enoughsingularities along Y −X that it can be used to determine intersection numbers. In particular thisis 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 andlet D = Y −X be its boundary divisor. We say a holomorphic function f on X has log-log growthalong D if for every coordinate system z = (z1, . . . , zn) adapted to D such that D is locally givenby the equation z1 . . . zk = 0 there is a positive integer M and a positive real number C such thatthe following inequality holds:

|f(z1, . . . , zn)| < C ·k∏i=1

log(log(1/|zi|))M .

Let j : X → Y be the inclusion map. The Dolbeault algebra of log-log forms is the sub-algebraof j∗EX which is closed under the operators ∧, ∂ and ∂ and which is generated by the holomorphicfunctions with log-log growth and the differential forms

dzizi log(1/|zi|)

,dzi

zi log(1/|zi|)for i = 1, . . . , k,

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 havelog-log growth. We note that by [8, Proposition 7.4], the property of being a pre-log-log form isrespected by pull-backs.

10

Definition 1.9. Let Xan be the complex manifold associated to a complete smooth algebraicvariety X over C of dimension n, equipped with a holomorphic line bundle L. Suppose D ⊂ Xan isa normal crossings divisor on Xan. Then a pre-log metric along D is a smooth metric || · || on X−Dsatisfying the following properties:

• It has logarithmic growth along D: that is, for every x ∈ X, there is an adapted coordinatesystem z = (z1, . . . , zn), a positive integer M and a locally generating section s of L definedon z such that

C ·k∏i=1

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

log(1/|zi|)M

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)−E0(Γ). Write y = =(z) and η = =(τ). Then the line bundle Lk,m(Γ) of weak Jacobi forms can beequipped with the following so-called Petersson metric:

||f(τ, z)||2Pet = |f(τ, z)|2 exp(−4πmy2/η)ηk

The content of the next proposition is that || · ||Pet is translation invariant, meaning that we have||ψ · f ||Pet = ||f ||Pet for all local sections f of Lk,m(Γ) and all ψ ∈ ΓJ .

Proposition 1.4. Let ψ = ((a bc d

), (λ, µ)) ∈ ΓJ and let U be an open subset of E0(Γ). For every

f ∈ H0(U,Lk,m(Γ)) and (τ, z) ∈ H × C we have∣∣∣∣∣∣∣∣f (aτ + b

cτ + d,z + λτ + µ

cτ + d

)∣∣∣∣∣∣∣∣2Pet

= ||f(τ, z)||2Pet.

Proof. Tedious but straightforward computation.

As we will see in later chapters, the Petersson metric on the line bundle of weak Jacobi forms canbe extended to a pre-log metric on E(Γ) −Dsing along D −Dsing, but not to a pre-log metric onthe entirety of E(Γ). This phenomenon fits in the framework of Mumford-Lear extensions, whichwas 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-Learextension of (L, || · ||) is a triple (L, || · ||′, e) consisting of a line bundle L ∈ Pic(Y ), a positiveinteger e and a metric || · ||′ on L|X such that L|X is isometric to L⊗e and such that L|X is pre-logalong 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 generalizationof the extensions defined by Mumford (cf. [17]) and Lear (cf. [16]): a Mumford extension of anorm || · || on L to a line bundle L on Y is pre-log along the entirety of D, whereas a Lear extensionof || · || to L is one that extends continuously, but only to a subset D− S with S of codimension atleast 2 in Y .

11

Associated to a Mumford-Lear extension (L, || · ||′, e) we have the Q-line bundle [L, || · ||]Y = 1eL.

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 furthermorethat Mumford-Lear extensions are compatible with tensor products. If C is the equivalence class ofWeil divisors associated to L, we write [C, || · ||]Y for the divisor class associated to [L, || · ||]Y andcall it the Mumford-Lear extension of C to Y . If it is clear from the context what metric is beingused, 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 calculat-ing Mumford-Lear extensions that are somewhat similar in flavor, the method of test curves andthe method of locally generating sections. The former allows us to reduce the calculation of theMumford-Lear extension of a divisor to the case where the underlying manifold is ∆, while the latterprovides a method to reason about Mumford-Lear extensions entirely by way of locally trivializingsections.

In section 2.2, we introduce the Deligne pairing of line bundles, which is a relative version of theusual intersection pairing of divisors on algebraic surfaces. We collect some identities satisfied bythis pairing, most notably an adjunction formula and a relation between Deligne pairings of degreezero divisors on one hand and the Poincare bundle on the Jacobian variety associated to a familyof nodal curves on the other.

In section 2.3 we describe the singularities of the Petersson metric in greater detail. As mentionedin the previous chapter, the smooth metric || · ||Pet can not be extended to a pre-log metric alongthe entirety of D = E(Γ) − E0(Γ). The degree to which the existence of such an extension failscan be quantified locally on a coordinate system (W,u, v) around a point p ∈ Dsing in the followingmanner. There is, associated to p, a rational number q ∈ Q>0 such that the metric || · || given bythe formula

log || · ||2 = log || · ||2Pet + q · log |u| log |v|log |u|+ log |v|

.

on E0(Γ) ∩W does not only admit a Mumford-Lear extension, but in fact a Mumford extensionalong the entirety of D ∩W . This property is the key ingredient in calculating the Mumford-Learextension of L4,4(Γ) to all compactifications of E0(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 associateddivisor class C and let Y be a smooth compactification of X. Write V = D1, . . . , Dn for the setof irreducible components of D = Y − X. Suppose a divisor C ′ on Y is given with the propertythat OY (C ′)|X ∼= L. Suppose furthermore that the Mumford-Lear extension of L to Y exists.Since Y is smooth, PicL(Y ) is a ZV -torsor, so [C]Y is necessarily of the form [C]Y = C ′+

∑ni=1 aiDi

for certain ai ∈ Q. The method of test curves allows us to determine these coefficients ai in thefollowing 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 =1, . . . , n, the Mumford-Lear extension of f∗i L to ∆ exists, and we have [f∗i 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 smoothmetric 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 [f∗i C]∆ = fi∗[C]Y . Now we calculate:

[f∗i C]∆ = fi∗[C]Y = fi

∗

C ′ + n∑j=1

ajDj

= fi∗C ′ + ai · [0],

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

Thus to find ai it suffices to compute the component of [0] for [f∗i 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 thestatement of Proposition 2.1.

Next we describe the method of locally generating sections. Evidently the property of being aMumford extension of (L, || · ||) is a local one, so it suffices to characterize existence of Mumfordextensions on compactifications of the form (∆∗)k × ∆n−k ⊂ ∆n. Then a generating section ofan extension of L can be used to characterize existence of a Mumford extension in the followingmanner.

Proposition 2.2. Let C ′ be a divisor on ∆n with the property that O∆n(C ′) restricts to L on (∆∗)k×∆n−k 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:

• The function log ||s|| −∑k

i=1 ai log |zi| on ∆n −D is a pre-log-log form along D

• The Q-divisor C ′ +∑k

i=1 ai · Di is the underlying Q-divisor of a Mumford extension of Lto ∆n.

Proof. Again it suffices to prove this statement under the assumption that a1, . . . , ak ∈ Z. Then wesimply note that s ·z−a1

1 . . . z−akk is a generating section of O∆n(C ′+∑k

i=1 ai ·Di), so all informationabout the growth of log || · || along D can be deduced from information about the growth of log ||s||−∑k

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 thathas 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, aflat proper holomorphic map p : X → S with fibers that are nodal curves over C, and let L and Mbe two line bundles on X. The Deligne pairing 〈L,M〉 is defined as follows:

• If S is a point, then X is a nodal curve. Consider the vector space V of pairs (`,m) of rationalsections of L,M with the property that div(`) and div(m) have disjoint support and `,m arenon-zero on the components of X and regular on the nodes of X. Then 〈L,M〉 is the quotientof V by the relations

(f`,m) ∼ f(div(m))(`,m)

(`, gm) ∼ g(div(`))(`,m),

14

where f, g are rational functions on X and where for a divisor D =∑

P∈X nPP on X wedefine

f(D) =∏P∈X

f(P )nP

• If S is not a point, we define 〈L,M〉(U) to be the set of tuples (us)s∈U ∈∏s∈U 〈Ls,Ms〉 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 = 〈`t,mt〉 for all t ∈ V .

For a proof that 〈L,M〉 is really a line bundle, we refer the reader to [1, section XIII.5].

From the “pointwise” construction of 〈L,M〉 it is clear that the Deligne pairing is compatible withbase change. Another property that is immediately apparent is the “bilinearity” and “symmetry”of the Deligne pairing: if L1,L2,M1 andM2 are line bundles on X, we have natural isomorphisms

〈L1,M1〉 ⊗ 〈L1,M2〉 ⊗ 〈L2,M1〉 ⊗ 〈L2,M2〉 → 〈L1 ⊗ L2,M1 ⊗M2〉

given by 〈`1,m1〉 ⊗ 〈`1,m2〉 ⊗ 〈`2,m1〉 ⊗ 〈`2,m2〉 7→ 〈`1 ⊗ `2,m1 ⊗m2〉 and

〈L,M〉 → 〈M,L〉

given by 〈`,m〉 7→ 〈m, `〉.

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 〈L,O(P )〉 and P ∗L,cf. [1, XIII.5.18]. Thus we have the following “adjunction formula” involving the relative dualizingsheaf ωX/S :

Lemma 2.3. There is a natural isomorphism 〈O(P ),O(P )〉 → P ∗ω⊗−1X/S .

Proof. By the usual adjunction formula, we have P ∗(O(P )⊗ ωX) ∼= ωS . Note that as p is smoothon U , we have ωU/S = ωU ⊗ p∗ω⊗−1

S . Hence, since P ∗p∗ωS = ωS and since the image of P iscontained in U , we have

P ∗O(P ) = P ∗(ω⊗−1X ⊗ p∗ωS) = P ∗(ω⊗−1

U ⊗ p∗ωS) = P ∗ω⊗−1U/S = P ∗ω⊗−1

X/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 Delignepairing generalizes the usual intersection pairing in the sense that degS〈L,M〉 = C · D, where Cand 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 Poincarebundle on the Jacobian variety associated to a family of smooth curves p : X → S. To set this up, weneed the fact that an analogue of the Picard scheme from section 1.3 exists in the category of complexanalytic spaces. Once this is known, we infer by universality of PicX/S that the space X × PicX/Scomes equipped with a line bundle P, called the Poincare bundle. Then on the Jacobian varietyJ(X/S) of X, which is by definition the connected component of the identity element of PicX/S , weget a line bundle onX×PicX/S by pulling back P along the inclusion mapX×J(X/S)→ X×PicX/S .By abuse of notation, we write P for this line bundle and also call it the Poincare bundle.

15

Proposition 2.4. Suppose E is a family of elliptic curves over a complex manifold S and suppose Land M are line bundles on E with the property that for all s ∈ S, Ls and Ms are degree zero linebundles 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). Thenthere is a natural isomorphism

〈L,M〉⊗−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 Delignepairing 〈L,M〉 comes equipped with a smooth metric as well. A smooth metric can also be definedon the Poincare bundle associated to X → S and the natural isomorphisms introduced in thissection become isometries when the Deligne pairings and Poincare bundles are equipped with theirnatural 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 followingproposition.

Proposition 2.5. Let c be a cusp of X(Γ). Then the fiber D = p−1c is an h-gon, where h is theperiod of c. Moreover, there is a numbering C0, . . . , Ch−1 of the irreducible components of D suchthat 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 isgiven by the equation vν = 0, and the following relations between (uν , vν) and the natural coordinatesystem (τ, z) on Wc,ν ∩ E0(Γ) hold:

uνvν = exp(2πiτ/h), uν+1ν vνν = exp(2πiz).

Here, by an h-gon on a smooth algebraic surface X we mean the following. If h > 1, it is acurve isomorphic to the one obtained by taking h copies Cνν∈Z/hZ of P1 and gluing ∞ to 0 foreach ν ∈ Z/hZ, embedded in X in such a manner that C2

ν = −2 and Cν ·Cν+1 = 1 for all ν ∈ Z/hZ.If h = 1, it is a curve isomorphic to the projective closure of Spec(C[x, y]/(y2 − x2(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 satisfiesthe following formula:

log||f(τ, z)||2Pet|Wν = log(|f(τ, z)|2)|Wν

+8

h

((ν + 1)2 log |uν |+ ν2 log |vν | −

log |uν | log |vν |log |uν |+ log |vν |

)+ 4 log

(−2hc

4π(log |u|+ log |v|)

).

16

Proof. The proof is identical to that of [5, Lemma 2.10], noting that log |uν | = 12 log(uνuν) and

log |vν | = 12 log(vνvν).

Now we have enough material to describe the singularities of || · ||Pet near the singular points of theboundary 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), abovea cusp of period h. Then there is a coordinate chart (W,u, v) around p such that the metric || · ||determined by the formula

log || · ||2 = log || · ||2Pet +8

h

log |u| log |v|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 themethod of locally generating sections, it suffices to check for a local generator s of L4,4(Γ) that thereare a, b ∈ Q such that log ||s|| − a log |u| − b log |v| is pre-log-log along uv = 0. We shall do thisfor θ8

1,1. Note that by Proposition 2.5, we can write θ1,1 as

θ1,1(τ, z) =∑n∈Z

exp(π(n+ 1)/2)uh/2(n+1/2)2+(ν+1)(n+1/2)vh/2(n+1/2)2+ν(n+1/2).

Let c and d be the smallest powers of u respectively v occurring in the above power series. Then

log

(log

∣∣∣∣θ1,1(τ, z)8

u8cv8d

∣∣∣∣)+ 4 log

(−2hc

4π(log |u|+ log |v|)

)is pre-log-log along uv = 0. Hence taking a = c+ m

h (ν + 1)2 and b = d+ mh ν

2 we see log ||θ81,1|| −

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-Learextension of L4,4(Γ) to E(Γ). Then we compute the Mumford-Lear extension of L4,4(Γ) to allsmooth compactifications of E0(Γ) and show the associated system of divisors satisfies Chern-Weiltheory 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

U(Γ) = E0(Γ)×Y (Γ) E0(Γ),

viewed as a family of elliptic curves over E0(Γ) via the projection on the first coordinate. Denoteby O,P : E0(Γ) → U(Γ) the zero and diagonal section, respectively. Recall that L4,4(Γ) can beidentified with OE0(Γ)(8O). Let

Q = (id×ϕO)∗P

be the pullback of the Poincare bundle P on E0(Γ) ×Y (Γ) E0(Γ)∨ along the principal polarization

associated to O. Consider the Hodge bundle λY (Γ) = O∗ωE0(Γ)/Y (Γ) on Y (Γ). This bundle comesequipped 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

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

⊗4.

Thus to find a Mumford-Lear extension of L4,4(Γ) it suffices to find Mumford-Lear extensions of P ∗Qand p∗λY (Γ). Mumford’s original paper (cf. [17]) implies that λY (Γ) has a Mumford extension withunderlying line bundle λX(Γ) = O∗ωE(Γ)/X(Γ). As for P ∗Q, the existence of the Lear extensionfollows 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 ∼= 〈P − O,P − O〉⊗−1. Hence, it suffices to determinethe Mumford-Lear extension of 〈P −O,P −O〉⊗−1. Let c ∈ C(Γ) be a cusp of X(Γ) and let h = hcbe its period. By Proposition 2.5, we know that the fiber above c is an h-gon D, say with irreduciblecomponents

V = Cc,0, . . . , Cc,h−1,

with the zero section O passing through Cc,0 and then ordered cyclically. Relabel the irreduciblecomponents by Ci = Cc,i for the moment and let f i : ∆ → E(Γ) be a test curve for the i-thcomponent of D which does not intersect the zero section. Then by compatibility of the Delignepairing and pullbacks, we have

f∗i 〈P −O,P −O〉⊗−1 = 〈P∆∗ −O∆∗ , P∆∗ −O∆∗〉⊗−1,

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 havethat by [13, Theorem 2.2], the Mumford-Lear extension of 〈P∆∗−O∆∗ , P∆∗−O∆∗〉⊗−1 to ∆ is equalto

[〈P∆∗ −O∆∗ , P∆∗ −O∆∗〉⊗−1]∆ = 〈P∆ −O∆ + Φi, P∆ −O∆ + Φi〉⊗−1,

where Φi is a Q-divisor with support contained in D∆, uniquely determined by the properties thatthe 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 〈P∆ − O∆ + Φi,Φi〉 = 0, we see theMumford-Lear extension can also be expressed as follows:

[〈P∆∗ −O∆∗ , P∆∗ −O∆∗〉⊗−1]∆ = 〈P∆ −O∆ + Φi, P∆ −O∆ + Φi〉⊗−1

= 〈P∆ −O∆ + Φi, P∆ −O∆ − Φi〉⊗−1

= 〈P∆ −O∆, P∆ −O∆〉⊗−1 ⊗ 〈Φi,Φi〉= 〈P∆ −O∆, P∆ −O∆〉⊗−1 ⊗O∆(Φ2

i · [0]),

where we use in the last equality that 〈Φi,Φi〉 and O∆(Φ2i · [0]) are both uniquely characterized by

the fact that they are supported on [0] and have degree Φ2i . Thus the coefficient for Ci is Φ2

i . Thefollowing lemma gives an explicit description of Φ2

i .

Lemma 3.2. There is exactly one Q-divisor Φi =∑

ν∈Z/hZ vνCν∆ with v0 = 0 satisfying theproperty that P∆−O∆ + Φi has zero intersection with Cν∆ for all ν ∈ Z/hZ. Its coefficients vν ∈ Qare given by

vν =

ν · T0 if ν < i

ν · T0 + i− ν else,

where T0 = h−ih . The self-intersection of Φi is given by

Φ2i = − i(h− i)

h.

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

0 = 〈P∆ −O∆ + Φi, Ai〉 = 1 + 〈Φi, Ai〉0 = 〈P∆ −O∆ + Φi, A0〉 = −1 + 〈Φi, A0〉0 = 〈P∆ −O∆ + Φi, Aν〉 = 〈Φ, Aν〉 for all ν 6= i

Thus, since A2ν = −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

M =

−2 1 0 · · · 11 −2 1 · · · 0

0 1 −2. . .

......

.... . .

. . . 11 0 · · · 1 −2

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

19

is exactly one v = (v0, . . . , vh−1) with v0 = 0 and Mv = b. Its coefficients are determined by thefollowing recurrence relations:

vν =

2vν−1 − vν−2 if ν 6= i+ 1;

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

vν+1 − vν =

T0 if ν < i

T0 − 1 otherwise.

Hence,

Φ2i = −2

h−1∑ν=0

v2ν + 2

h−1∑ν=0

vνvν+1 = 2h−1∑ν=0

(vν+1 − vν)vν

= T 20 · 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

Vc =∑

i∈Z/hcZ

− i · (hc − i)hc

Cc,i.

The previous discussion implies that [P ∗Q, || · ||can]E(Γ) = 〈P −O,P −O〉⊗−1 ⊗⊗

c∈C(Γ)OE(Γ)(Vc).

Note that 〈P,O〉 = P ∗OU(Γ)(O) = OE(Γ)(O). Hence, since 〈P, P 〉⊗−1 and 〈O,O〉⊗−1 are isomorphicto λY (Γ) by Lemma 2.2, we find using bilinearity of the Deligne pairing that

[P ∗Q, || · ||can]E(Γ) = p∗λ⊗2X(Γ) ⊗OE(Γ)(2O)⊗

⊗c∈C(Γ)

OE(Γ)(Vc).

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 classassociated 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

V 2c = −h

2c − 1

3hc

20

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

V 2c = −2

h∑i=0

i2(h− i)2

h2+ 2

h∑i=0

i(i+ 1)(h− i)(h− i− 1)

h2

=h2(h+ 1)2 − 1

3(3h− 1)h(h+ 1)(2h+ 1) + h2(h+ 1)(h− 1)

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 followingequality:

dΓ =∑

c∈C(Γ)

hc.

Proof. Choose N 0 such that Γ(N) ≤ Γ and let π1 : X(N) → X(1) and πΓ : X(N) → X(Γ) bethe 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 anycusp 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 onthe 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 π∗Γδ0 = π∗1[∞]. Therefore, since each cusp in X(N) has ramification index N above [∞],we find ∑

c∈C(Γ)

∑T∈π−1

Γ (c)

ncec · T = π∗Γ∑

c∈C(Γ)

nc · c =∑

T∈C(Γ(N))

N · T.

We deduce that δ0 =∑

c∈C(Γ) hc · c, so indeed∑c∈C(Γ)

hc = deg∑

c∈C(Γ)

hc · c = deg δ0 = deg π · deg[∞] = dΓ · 1 = dΓ,

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-intersectionof [J, || · ||Pet]E(Γ) is given by

[J, || · ||Pet]2E(Γ) =

64

12dΓ +

∑c∈C(Γ)

64

12hc.

Proof. Note that O ·Vc = 0 for all c ∈ C(Γ) since O intersects Dc only in Cc,0 and since the coefficientfor 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 weobtain the following formula for [J, || · ||Pet]

2E(Γ):

[J, || · ||Pet]2E(Γ) = 192 · F ·O + 64 ·O2 − 64

∑c∈C(Γ)

h2c − 1

12hc

By the adjunction formula we have −O · O = F · O = dΓ12 (cf. [6, Proposition 3.2]). Substituting

this in the above formula and using that dΓ =∑

c∈C(Γ) hc we obtain

[J, || · ||Pet]2E(Γ) =

(196

12− 64

12− 64

12

)dΓ +

∑c∈C(Γ)

64

12hc

=64

12dΓ +

∑c∈C(Γ)

64

12hc,

as was to be shown.

3.2 Comparison to Chern-Weil theory

Let CPT′(E0(Γ)) be the subset of CPT(E0(Γ)) consisting of the isomorphism classes of compact-ifications Y/E0(Γ) that are obtained from E(Γ) by a finite number of blow-ups in singular pointsof the boundary divisor. By Lemma 3.7 below, to compute Mumford-Lear extensions to arbitrarycompactifications of E0(Γ), it suffices to compute [L4,4(Γ), || · ||Pet]Y for all Y ∈ CPT′(E0(Γ)).

Lemma 3.7. Let Y be a smooth compactification of a variety X over C with boundary divisor Dand 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 ′ → Y be the compactification of X obtainedby blowing up Y in a point p ∈ D − S. Then the Mumford-Lear extension of L to Y ′ exists and isequal 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, || · ||), theMumford-Lear extensions of L to a smooth compactification Y ′ of E0 can be computed in the follow-ing manner: if Y is the greatest element of CPT′(E0) with Y EY ′ and the Mumford-Lear extensionof L to Y exists, then so does the Mumford-Lear extension of L to Y ′ and we have

[L, || · ||]Y = π∗[L, || · ||]Y ′ ,

where π : Y ′ → Y is the projection map induced by the chain of blow-ups turning Y into Y ′.

If we blow up a singular point p of the boundary divisor D of Y/E0(Γ), the exceptional divisor Eintersects the strict transform of D in two points. Thus whenever we blow up a singular point, weget two singular points back, which again can be blown up to get four points and so on. This yieldsthe 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 CPT′(E0(Γ)) is order-isomorphic to

∏c∈C(Γ)

hc−1∏i=0

Tc,i

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 thenorm given by the formula

log || · ||2 = log || · ||2Pet +2µ

nm

log |u| log |v|n log |u|+m log |v|

admits a Mumford extension along D ∩ V . Note that by Proposition 2.12, a singular point p in thefiber above a cusp c ∈ C(Γ) has type (1, 1) and multiplicity 4/hc. As we will see in Theorem 3.11below, if we blow up a singular point of type (n,m) and multiplicity µ we obtain two new points ofmultiplicity µ, one of type (n + m,m) and the other of type (n, n + m). Before we can prove thisstatement we need to study the expression on the right hand side of the above equation in moredetail.

Proposition 3.10. Let (n,m) be a pair of coprime positive integers and let D be the boundarydivisor of the compactification ∆2 of (∆∗)2. Let fn,m be the function (∆∗)2 → C defined by theformula

fn,m(u, v) =2

nm

log |u| log |v|n log |u|+m log |v|

.

Then fn,m is a pre-log-log function along D − (0, 0). Moreover, if π is the function ∆2 → ∆2

given by (s, t) 7→ (st, t) then

fn,m(π(s, t)) =2

nm(n+m)log |t|+ fn,n+m(s, t).

Similarly, if π′ is the function ∆2 → ∆2 given by (s, t) 7→ (s, st), then

fn,m(π′(s, t)) =2

nm(n+m)log |s|+ fn+m,n(s, t).

Proof. For a proof that fn,m is pre-log-log along D−(0, 0), we refer the reader to [5, Proposition4.1.(i)]. As for the second statement, this follows from the following calculation:

fn,m(st, t) =2

nm(n+m)

log |t|(n log |s|+ (n+m) log |t|) +m log |s| log |t|n log |s|+ (n+m) log |t|

=2

nm(n+m)log |t|+ fn,m(s, t)

The third statement is proved in an identical manner.

Let Bl(0,0)∆2 ⊂ P1 × ∆2 be the blow-up of ∆2 in (0, 0). Note that the map π from the previous

proposition is the composition of the projection map Bl(0,0)∆2 → ∆2 with the coordinate map ∆2 →

Bl(0,0)∆2

(s, t) 7→ ((s : 1), st, t),

23

and that the exceptional divisor E of ψ : Bl(0,0)∆2 → ∆2 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 ψis given by π′ 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 ′ → Yof 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 theMumford-Lear extension of L4,4(Γ) to Y exists then so does the Mumford-Lear extension of L4,4(Γ)to Y ′, and we have

[J, || · ||Pet]Y ′ = π∗[J, || · ||Pet]Y −µ

nm(n+m)E,

where E is the exceptional divisor of the blow-up π : Y ′ → Y .

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

log || · ||2 = log || · ||2Pet + µ · fn,m(u, v)

admits a Mumford extension along D ∩ V . Denote by D1 = u = 0 and D2 = v = 0 theirreducible 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 Dwith E correspond to the points ((0 : 1), 0, 0), ((1 : 0), 0, 0) on Bl(0,0)∆

2 and in the coordinatechart (s, t) around p1, the pull-back of || · || along ψ is given by

log ||π∗g|| = log ||π∗g||Pet +µ

nm(n+m)log |t|+ 1

2µ · fn,n+m(s, t).

Thus log ||π∗g||Pet − c log |s| + µnm(n+m) log |t| + 1

2µ · 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 isgiven by π′ on the coordinate chart around p2, we see that p2 is a point of type (n + m,m) andmultiplicity µ.

Now since fk,`(u, v) is pre-log-log alongDsm∩V for all pairs of coprime integers (k, `), we deduce thatthe Mumford-Lear extension of L4,4(Γ) to Y ′ exists. As the pullback of D2 respectively D1 along ψis given on the coordinate around p1 respectively p2 by the equation s = 0, we see the coefficients forthe irreducible components of ψ−1(D) and E of this Mumford-Lear extension coincide with thosein 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-Learextension of L4,4(Γ) to all compactifications of E0(Γ).

Theorem 3.12. The limit of the self-intersection [L4,4(Γ), || · ||Pet]2Y over all Y ∈ CPT(E0(Γ))

exists and is equal to 6412dΓ.

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

limY ∈CPT′(E0(Γ))

[J, || · ||Pet]2Y

24

instead. For each c ∈ C(Γ) and i ∈ 0, . . . , hc − 1, there is a one-to-one correspondence betweenthe elements of Tc,i and the set N of pairs of coprime integers, defined recursively by the followingconditions:

• 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 theunique point of type (n, n + m) and (n + m,n) maps to the blow-up in the unique point oftype (n+m,m).

Recall that if π : Y → X is the blow-up of a smooth algebraic surface in a point p with exceptionaldivisor E then Cl(X)⊕Z is isomorphic to Cl(Y ) via the map (D,n) 7→ π∗D+nE. Hence if Y ′ → Yis the blow-up of a compactification Y/E0(Γ) in a point p of multiplicity µ and type (n,m) then

[J, || · ||Pet]2Y ′ = [J, || · ||Pet]

2Y −

µ2

n2m2(n+m)2.

Thus a point of multiplicity µ and type (n,m) contributes − µ2

n2m2(n+m)2 to the limit of self-

intersections, so if the limit exists, it is equal to

limY ∈CPT′(E0(Γ))

[J, || · ||Pet]2Y = [J, || · ||Pet]

2E(Γ) −

∑c∈C(Γ)

hc∑

(n,m)∈N

42

h2cn

2m2(n+m)2

=64

12dΓ +

∑c∈C(Γ)

64

12hc

1−∑

(n,m)∈N

3

n2m2(n+m)2

.

As in the proof of [5, Theorem 4.11], we deduce from the results of [18] that the sum∑

(n,m)∈N1

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 thetower of compactifications of E0(Γ).

Theorem 3.13. The integral∫E0(Γ) c1(L4,4(Γ), || · ||Pet)

∧2 exists and is equal to 6412dΓ.

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

log || · ||2 = log || · ||2Pet +8

h

log |u| log |v|log |u|+ log |v|

.

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

f = log ||s||2 − log ||s||′2

for some meromorphic section s of L4,4(Γ). Then ω = ω′+ 12πi∂∂f , so writing out the wedge product

we see the following equality holds:∫E0(Γ)

ω∧2 =

∫E0(Γ)

ω′∧2 −∫E0(Γ)

d

(1

(2πi)2∂f ∧ ∂∂f

).

25

Since an analogue of Chern-Weil theory holds for pre-log metrics and since pre-log-log forms haveno residue (cf. [17]), we can rewrite the above integral as

[J, || · ||Pet]2E0(Γ) −

∑c∈C(Γ)

hc−1∑ν=0

limε→0

∫∂Vc,ν,ε

∂f ∧ ∂∂f

where Vc,ν,ε = (u, v) ∈Wc,ν : |u| ≤ ε, |v| ≤ ε. As in the proof of [5, Theorem 5.2], we compute

limε→0

∫∂Vc,ν,ε

∂f ∧ ∂∂f =64

12h2c

.

Putting this all together, we find∫E0(Γ)

ω∧2 = [J, || · ||Pet]2E(Γ) −

∑c∈C(Γ)

hc ·64

12h2c

=64

12dΓ +

∑c∈C(Γ)

64

12hc−∑

c∈C(Γ)

64

12hc=

64

12dΓ,

as was to be shown.

We end this section by showing that the system of divisors associated to the tower of compactifi-cations of E0(Γ) satisfies a Hilbert-Samuel type formula, generalizing [5, Theorem 5.2] to arbitrarysubgroups Γ ≤ SL2(Z) which act without fixed points on H.

Proposition 3.14. The following equality holds:

limX∈CPT(E0(Γ))

[J, || · ||Pet]2X = lim

`→∞

dim J4`,4`(Γ)

`2/2!.

Proof. By [6, Proposition 3.8], the following equality holds for all ` 0 with hc|` for all c ∈ C(Γ):

dim J4`,4`(Γ) =16

6dΓ · `2 − dΓ · `−

∑c∈C(Γ)

hc4 Q(

16`

hc

)+hc2

∑∆|16`/hc,∆<0

16`/(hc∆) squarefree

H(∆)

,

where Q(n) denotes the largest integer whose square divides n and H(∆) is the Hurwitz classnumber. As in the proof of [5, Proposition 2.6] and [5, Remark 2.7], we deduce

dim J4`,4`(Γ) =32

12dΓ · `2 + o(`2).

Now we calculate:

limX∈CPT(E0(Γ))

[J, || · ||Pet]2X =

64

12dΓ = lim

`→∞

3212dΓ · `2 + o(`2)

`2/2!= lim

`→∞

dim J4`,4`(Γ)

`2/2!.

26

4 Concluding remarks

We have succeeded in generalizing [5, Theorem 5.1] and [5, Theorem 5.2] to all congruence sub-groups Γ ≤ SL2(Z) which act without fixed points on H. The next question is whether these resultscan 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] forfamilies of abelian varieties. But we will mention the following higher-dimensional analogues of thetwo main ingredients enabling the calculation of [J, || · ||Pet]E for all E ∈ CPT(E0(Γ)):

(i) Suppose p : A → Y is a family of abelian varieties and λ : A → A∨ is the principal polarizationinduced by a theta divisor Θ on A. Write P = (id, λ) and let P be the Poincare bundleon A ×Y A∨. Let λY = detO∗Ω1

A/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 naturalmetrics || · ||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 andLear 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 certainproperties, then the singularities of the Lear extension of P ∗P to A can be characterized on acoordinate neighborhood (U, z1, . . . , zn) adapted to D = A−A in the following manner. If D isgiven locally by the equation z1 . . . zk = 0 then there exists a positive integer d, a homogeneouspolynomial Q ∈ Z[x1, . . . , xk] of degree d with no zeroes on Rk>0 and for each locally generatingsection s of P ∗P, a homogeneous polynomial Ps ∈ Z[x1, . . . , xk] of degree d + 1 such thatfor 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 ofrelatively minimal models exists for higher-dimensional varieties, it may again be fruitful tostudy the behavior under blow-ups of rational functions that arise in this manner.

We also note that [J, || · ||Pet]2E(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 ellipticsurfaces over curves that are defined as the quotient of some space under a properly discontinuousgroup action.

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

27

References

[1] E. Arbarello, M. Cornalba, P. Griffiths, Geometry of Algebraic Curves Volume II with a contri-bution by Joseph Daniel Harris, Grundlehren der mathematischen Wissenschaften, Volume 268,2011.

[2] W. Barth, C. Peters, A. van de Ven, Compact complex surfaces, A Series of Modern Surveys inMathematics, Volume 4, 2004.

[3] A. Beauville, Theta Functions, Old and New, Surveys of Modern Mathematics, Volume 6, 2013,p. 99-131.

[4] R. Berndt, Meromorphe Funktionen auf Mumford’s Kompaktifizierung der universellen elliptis-chen Kurve N -ter Stufe, J. reinde und angew. Math. Volume 326, 1981, p. 95-103.

[5] J. Burgos, U. Kuhn, J. Kramer, The singularities of the invariant metric on the line bundle ofJacobi forms, arXiv:1405.3075.

[6] J. Kramer, A geometrical approach to the theory of Jacobi forms, Compositio Math. Volume 79,1991, p. 1-19.

[7] J. Burgos, D. Holmes, R. de Jong, Singularities of the biextension metric for families of abelianvarieties, arxiv.org:1604.00686v1.

[8] J. Burgos, U. Kuhn, J. Kramer, Cohomological arithmetic Chow rings, J. Inst. Math. Jussieu 6(2007), no. 1, p. 1-172.

[9] P. Deligne, D. Mumford, The irreducibility of the space of curves of given genus, PublicationsMathmatiques de l’IHS, 36 (1969), p. 75-109.

[10] P. Deligne, M. Rapoport, Les schemas de modules de courbes elliptiques, Modular Functions ofOne Variable II (Proc. Internat. Summer School, Univ. Antwerp, 1972), 143316. Springer-Verlag,Berlin/Heidelberg, 1973.

[11] F. Diamond, J. Shurman, A first course in modular forms, volume 228 of Graduate Texts inMathematics. Springer, 2005.

[12] R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics. Springer, 1977

[13] D. Holmes, R. de Jong, Asymptotics of the Neron height pairing, arXiv:1304.4768v2.

[14] R. de Jong, Faltings delta-invariant and semistable degeneration, arXiv:1511.06567v1.

[15] F. Knudsen, The projectivity of the moduli space of stable curves. II. The stacks Mg,n, Math.Scand. 52 (1983), no. 2, p. 161-199.

[16] D. Lear, Extensions of normal functions and asymptotics of the height pairing, Ph.D. thesis,University of Washington, 1990.

[17] D. Mumford, Hirzebruch’s proportionality theorem in the noncompact case. Invent. Math. 42(1977), p. 239-272.

[18] L. Tornheim, Harmonic double series, Amer. J. Math 72 (1950), p. 303-314.

28