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  • 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

    C · C = ∫ X c1(L, || · ||)∧2,

    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 on E0(Γ) given by the image of the zero section, using a translation invariant metric || · ||Pet on the line bundle of weak Jacobi forms L4,4(Γ) of weight 4 and index 4. But since E

    0(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 singular 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 points on 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 equality holds:

    lim E∈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 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

    [J, || · ||Pet]2E(Γ) = 64

    12 dΓ +

    ∑ c∈C(Γ)

    64

    12h2c

    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

    lim E∈CPT(E0(Γ))

    [J, || · ||Pet]2E = 64

    12 dΓ.

    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 extension of || · ||Pet to the various smooth compactifications of E0(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 extends to an action of ΓJ := Γ n Z2 on H× C given by((

    a b c 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 the structure 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 to describe 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 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

    Γ(N) =

    {( a b c d

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

    } Γ1(N) =

    {( a b c d

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

    } Γ0(N) =

    {( a b c 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 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 étale coverings, 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 nonsing