Hyperelliptic Schottky Problem

and Stable Modular Forms

Giulio Codogni

Received: August 26, 2014 Revised: February 15, 2016

Communicated by Gavril Farkas

Abstract. It is well known that, fixed an even, unimodular, positive definite quadratic form, one can construct a modular form in each genus; this form is called the theta series associated to the quadratic form. Varying the quadratic form, one obtains the ring of stable modular forms. We show that the differences of theta series associated to specific pairs of quadratic forms vanish on the locus of hyperelliptic Jacobians in each genus. In our examples, the quadratic forms have rank 24, 32 and 48. The proof relies on a geometric result about the boundary of the Satake compactification of the hyperelliptic locus. We also study the monoid formed by the moduli space of all principally polarised abelian varieties, the operation being the product of abelian varieties. We use this construction to show that the ideal of stable modular forms vanishing on the hyperelliptic locus in each genus is generated by differences of theta series.

2010 Mathematics Subject Classification: 14H42 , 32G20 Keywords and Phrases: Schottky problem, Satake compactification, theta series

1. Introduction

The hyperelliptic Schottky problem is to characterise the locus of Jacobians of hyperelliptic curves inside the moduli space of principally polarised abelian varieties. A classical approach is to look for modular forms vanishing along the hyperelliptic locus; in other words, one looks for the equations of the hyperelliptic locus inside the moduli space of principally polarised abelian varieties. A special kind of modular forms are the theta constants; these were used by Mumford to give a solution to the hyperelliptic Schottky problem, as reviewed in Theorem 4.4.

In this paper, we deal with stable modular forms. One nice feature of these forms is that they relate the theory of moduli spaces with the theory of quadratic forms. To start with, let us explain what we mean by stable modular forms. Let A_{g} be the moduli space of principally polarised abelian g-fold defined over the field of complex numbers. We consider the Satake compactification A_{g}^{S} of A_{g}. This comes with a stratification

A_{g}^{S} = A_{g} ⊔ A_{g−1} ⋯ A_{1} ⊔ A_{0}

In particular, we have a closed embedding

ι_{g}: A_{g−1}^{S} ↪ A_{g}^{S}

The collection of the moduli spaces A_{g}^{S} and these maps form a direct system of varieties; we can thus consider the ind-scheme

A_{∞} := lim_{g} A_{g}^{S}

The basic definitions about ind-schemes are recalled in Section 2 Stable modular forms are naturally defined on A_{∞}. A stable modular form F is a collection of modular forms (F_{g})_{g≥0}: each F_{g} is modular form on A_{g}^{S} and ι_{g}^{∗}F_{g} = F_{g−1}. We will recall the theory in Section 3, in particular see Definition 3.3. A classical and surprising fact is that we can construct a stable modular form out of an even, unimodular, positive definite quadratic form Q. This stable modular form is called theta series associated to Q, and is denoted by Θ_{Q}; see Definition 3.6. In particular, for any g, Θ_{Q,g} is a modular form on A_{g}^{S}. In [Fre77], Freitag showed that all stable modular forms are linear combinations of theta series. In this set up, we can consider the ideal of stable modular forms vanishing on the locus Hyp_{g} of hyperelliptic Jacobians in every genus. Let us formalise this with definition.

Definition 1.1 (Stable Equation). A stable equation for the hyperelliptic locus is a stable modular form (F_{g})_{g≥0} such that F_{g} vanishes along the hyperelliptic locus Hyp_{g} for every g.

Our first result is the following:

Theorem 1.2 (= Theorem 4.2). The ideal of stable equations of the hyperelliptic locus is generated by differences of theta series

Θ_{P} − Θ_{Q}

where P and Q are even, unimodular, positive definite quadratic forms of the same rank.

A key ingredient in the proof of this results is a natural monoidal structure that one can put on A_{∞}. Given two principally polarised abelian varieties, their product is still a principally polarised abelian variety but of higher dimension. This defines an operation

m: A_{∞} × A_{∞} → A_{∞}

The pull-back m^{∗} gives to the ring of stable modular forms the structure of a co-commutative co-algebra. Because of this, we can run the general machinery explained in Section 2 to prove Theorem 1.2. So far, the ideal of stable equations for the hyperelliptic locus could be trivial. Indeed, this is the case for the moduli space of curves: in [CSB14], it is shown that the ideal of stable modular form vanishing on the Jacobian locus in any genus is trivial. In other words, given a non-zero stable modular form F, there exists a g such that F_{g} does not vanish on the moduli space of genus g curves. In [SB13], it is similarly shown that the ideal of stable equation for the n-gonal locus, with n ≥ 3, is trivial. However, as we are going to see, the ideal of stable equations for the locus of Jacobians of hyperelliptic curves is far from being trivial. The first stable equation for the hyperelliptic locus was discovered by C. Poor ([Poo96]): it is the difference of the theta series associated to the quadratic forms D_{16}^{+} and E_{8} ⊕ E_{8}; this modular form is also called the called Schottky form. To construct new stable equations, we need to know more about the geometry of the Satake compactification. The Satake compactification Hyp_{g}^{S} of Hyp_{g} will be defined in Section 4. We denote by A_{g}^{ind} the moduli space of indecomposable principally polarised abelian g-fold.

Theorem 1.3 (= Theorem 4.3; Transversality). The intersection of the Stake compactification Hyp_{g+1}^{S} and A_{g}^{ind} inside A_{g+1}^{S} is scheme theoretically equal to Hyp_{g}.

The statement was well-known at the level of sets; we call it a transversality result because it states that the scheme structure of the intersection is the reduced one. The analogue result does not hold for the moduli of curves [CSB14, Theorem 1.1] and for the moduli of n-gonal curves [SB13], with n ≥ 3. In those cases, the failure of the transversality implies that there are no stable equations; in the hyperelliptic case, this transversality result is key in the construction of stable equations. Combining Theorem 1.3 and Criterion 5.2 we can prove the following

Theorem 1.4 (= Corollary 5.7 and Theorem 6.1). The difference of theta series

Θ_{P} − Θ_{Q}

is a stable equation for the hyperelliptic locus when one of the following hold:

(1) rk(P) = rk(Q) = 24 and the two quadratic forms have same number of vectors of norm 2;

(2) rk(P) = rk(Q) = 32 and the two quadratic forms do not have any vector of norm 2;

(3) rk(P) = rk(Q) = 48 and the two quadratic forms do not have any vector of norm 2 or 4;

Each item of Corollary 1.4 concerns a finite positive number of pairs of quadratic forms. In [Kin03], it is shown that there are more than ten millions of quadratic forms meeting the hypothesis of the second item. In the first item, the "slope" of the quadratic form, i.e. the ratio between the rank and the norm of the shortest non-zero vector, is strictly bigger than the slope of the hyperelliptic locus; because of this, in the proof we need to use some non-trivial arithmetic properties of the quadratic forms: namely we use Theorem 6.2 via Corollary 6.3. We think that the ideal of stable equations defines scheme theoretically the hyperelliptic locus inside the moduli space of indecomposable principally polarised abelian varieties in any genus. This, in particular, would imply that there are infinitely many pairs of quadratic forms which give stable equations for the hyperelliptic locus. In order to give a characterisation of these pairs, we think one should relate theta series to partition functions, as partially suggested in [GV09], [GKV10] and [Mat15].

Acknowledgement. This work is part of my Ph.D. thesis [Cod14], I would like to thank my advisor Nick Shepherd-Barron, without whom this work would not have been possible. It is a pleasure to thank Riccardo Salvati Manni for many useful advices. I also had the benefit of conversations with Enrico Arbarello, Gavril Farkas, Sam Grushevsky, Marco Matone, Sara Perna, Filippo Viviani and Roberto Volpato on the topics of this paper. I would like to thank the refree for useful comments. My Ph.D. was funded by the DPMMS, the EPSRC and Selwyn College. I have been also supported by the Physics department of the University of Padova during two visits. Now, I am funded by the grant FIRB 2012 "Moduli Spaces and their applications" and by the ERC StG 307119 - StabAGDG.

2. Ind-varieties and commutative Monoids

In this section, we recall some general definitions and results about ind-varieties and monoid. A reference about ind-variety is [Kum02, Chapter IV]. An indvariety X is a collection (X_{n})_{n≥0} of algebraic varieties and a collection of closed embeddings

ι_{n}: X_{n−1} ↪ X_{n}

We write

X = lim_{n} X_{n}

This limit exists in the category of locally ringed spaces; however, we prefer to enlarge the category of schemes including all direct systems. This means that for us an ind-variety is a direct system of algebraic varieties. A line bundle L on X is the data of a line bundle L_{n} on each X_{n} such that ι_{n}^{∗}L_{n} = L_{n−1}. A section s of L is a collection of sections (s_{n})_{n≥0} such that s_{n} is a section of L_{n} on X_{n} and the restriction of s_{n} to X_{n−1} is s_{n−1}. We assume that the vector space H^{0}(X, L^{k}) is finite dimensional for every k. The ring of sections of L is thus defined as a projective limit in the category of graded rings

R(X, L) := \underleftarrow{lim} R(X_{n}, L_{n}) .

We do not have to worry about the topology of this ring because of graded pieces H^{0}(X, L^{k}) are finite dimensional. In other words, elements on R(X, L) are not formal power series.

Remark 2.1 (Ampleness on ind-varieties). The concept of ampleness for a line bundle L on an ind-variety X is subtle and, to the best of our knowledge, there is not a standard definition. A first definition could be that there exists a k such that L_{n}^{k} is very ample on X_{n} for every n. Remark that k does not depend on n. If H^{0}(X, L^{k}) is finite dimensional for every k but the dimension of X_{n} tends to infinity when n grows, L can not be ample. A weaker definition is to ask that for every n there exists a k = k(n) such that L_{n}^{k} is very ample on X_{n}. The example that we will study in this paper is ample just in the sense of the second definition. This second definition does not imply the classical consequences of ampleness: for instance, in this generality, it is not even clear that an ample line bundle is effective.

An ind-monoid is an ind-variety M with an associative multiplication and an identity element 1_{M}. A multiplication m is a family of maps

m_{g,h}: M_{g} × M_{h} → M_{g+h}

compatible with the restrictions. M is commutative if the multiplication is.

Definition 2.2 (Split monoid). Let M be a commutative ind-monoid and L a line bundle on M. We say that M is split with respect to L if the following two conditions hold:

(1) For every g and h

m_{g,h}^{∗}L_{g+h} ≅ pr_{1}^{∗}L_{g} ⊗ pr_{2}^{∗}L_{h} =: L_{g} ⊠ L_{h}

where pr_{i} are the projections;

(2) for every k, the vector space H^{0}(M, L^{k}) is finite dimensional and spanned by characters, where a section χ of L is a character if

m_{g,h}^{∗}χ_{g+h} = χ_{g} ⊠ χ_{h} ∀ g, h .

In the language of Hopf algebras, condition (1) means that the pull-back m^{∗} is a co-commutative co-multiplication for R(M, L). The definition of character makes sense only if condition (1) holds. With a slight abuse of notations, we will speak about characters of M rather than characters of the co-algebra R(M, L), and we will write χ(αβ) = χ(α)χ(β) instead of m_{g,h}^{∗}χ_{g+h}(α × β) = χ_{g}(α) ⊠ χ_{h}(β).

Lemma 2.3. Let M be a commutative monoid, suppose it is split with respect to a line bundle L, then, any set of characters is linearly independent.

Proof. This proof is standard. We argue by contradiction. Take n minimal such that there exist n linearly dependent characters χ_{1}, …, χ_{n}. We can write

χ_{n} = ∑_{i=1}^{n−1} λ_{i}χ_{i} λ_{i} ∈ ℂ .

Pick α ∈ M such that χ_{1}(α) ≠ χ_{n}(α). For any β ∈ M we have

∑_{i=1}^{n−1} λ_{i}χ_{i}(α)χ_{i}(β) = χ_{n}(α)χ_{n}(β) = χ_{n}(α) (∑_{i=1}^{n−1} λ_{i}χ_{i}(β))

Since β is arbitrary we get

∑_{i=i}^{n−1} λ_{i}(χ_{i}(α) − χ_{n}(α))χ_{i} = 0 .

The coefficient χ_{1}(α) − χ_{n}(α) is non-zero, so we have written a non-trivial linear relation among fewer than n characters. This contradicts the minimality of n. □

Proposition 2.4. Let M be a commutative ind-monoid and N a submonoid. Suppose that M is split with respect to a line bundle L. Then the ideal I_{N} in R(M, L) of sections vanishing on N is generated by differences of characters

χ_{i} − χ_{j}

Proof. Take s in I_{N}. We can assume that s is homogeneous and write it as a linear combination

s = λ_{1}χ_{1} + ⋯ + λ_{n}χ_{n}

where χ_{i} are characters and λ_{i} are constants. Restricting χ_{i} to N some of them might become equal. Up to relabelling the χ_{i}, we can fix integers 0 = m_{0} < m_{1} < ⋯ < m_{k} = n and distinct characters θ_{1}, …, θ_{k} of N such that

χ_{i} |_{N}= θ_{j} ⟺ m_{j−1} < i ≤ m_{j}

For j = 1, …, k, let us define

μ_{j} := ∑_{i=m_{j−1}+1}^{m_{j}} λ_{i} .

By hypothesis we know that

0 = s |_{N} = ∑_{j=1}^{k} μ_{j}θ_{j} .

By Lemma 2.3 we have μ_{j} = 0 for every j, so

s = s − ∑_{j=1}^{k} μ_{j}χ_{m_{j}} = ∑_{j=1}^{k} ∑_{i=m_{j−1}+1}^{m_{j}} λ_{i}(χ_{i} − χ_{m_{j}})

The differences χ_{i} − χ_{m_{j}} vanish on N for m_{j−1} < i ≤ m_{j}, so we have just expressed s as linear combination of differences of characters vanishing on N.

□

The previous argument actually shows that every element of the ideal can be written as a linear combination of differences of characters. These results are special cases of a more general theory of Milnor and Moore. They have many applications in the study of moduli spaces, e.g. [GHT14].

3. Satake compactification, modular forms and theta series

We recall some facts about modular forms and the Satake compactification of A_{g}. General references about modular forms are [BvdGHZ08] and [Mum07]. The Satake compactification was first defined in [Sat56]; a comprehensive reference is [Fre83]. The line bundle L_{g} of weight one modular forms on A_{g} is defined as the determinant of the Hodge bundle; it is ample and it generates the rational Picard group.

Definition 3.1 (Siegel modular form). A weight k and degree g Siegel modular form is a section of L_{g}^{k} on A_{g}.

The universal cover of A_{g} is the Siegel upper half space H_{g}; the symplectic group Sp(2g, ℤ) acts on H_{g} and

A_{g} = H_{g}/Sp(2g, ℤ)

The line bundle L_{g} is trivial when it s pulled back to H_{g}; therefore a modular form can be also defined as a holomorphic function on H_{g} which transforms appropriately under the action of Sp(2g, ℤ). The Satake compactification A_{g}^{S} is a normal projective variety defined as follows

A_{g}^{S} := Proj(⨁_{n≥0} H^{0}(A_{g}, L_{g}^{n}))

This is the compactification "seen" by modular forms. The line bundle L_{g} extends naturally to A_{g}^{S} because it is the O(1) of this Proj. For the same reason, all modular forms extend to A_{g}^{S}

Definition 3.2 (The Siegel operator). The Siegel operator Φ is a map of graded rings

Φ: ⨁_{n≥0} H^{0}(A_{g}, L_{g}^{n}) → ⨁_{n≥0} H^{0}(A_{g−1}, L_{g−1}^{n})

defined as

Φ(F)(τ) := lim_{t→+∞} F (τ ⊕ it) ,

where τ is an element of H_{g−1} and t ∈ ℝ. Here, we are thinking at F as a holomorphic function on H_{g}.

Clearly, there is some work to do to show that Φ(F) is a well defined element of H^{0}(A_{g−1}, L_{g}^{n}); the interested reader cal look at [Fre83]. The Siegel operator is surjective for n even and larger than 2g ([Fre83] page 64); this means that the Siegel operator defines a closed embedding of ι_{g}: A_{g−1}^{S} ↪ A_{g}^{S}. One can check that the image of A_{g−1}^{S} is the boundary ∂A_{g}^{S} of A_{g}^{S}, so we obtain a stratification

A_{g}^{S} = A_{g} ⊔ A_{g−1}^{S} = A_{g} ⊔ A_{g−1} ⋯ A_{1} ⊔ A_{0} .

By construction, the pull-back ι_{g}^{∗}L_{g} is isomorphic to L_{g−1}, and the pull-back ι_{g}^{∗}: H^{0}(A_{g}, L_{g}^{n}) → H^{0}(A_{g−1}, L_{g−1}^{n}) is the Siegel operator. Again, a reference is [Fre83].

The system of varieties A_{g}^{S} together with the closed embeddings ι_{g} induced by the Siegel operators forms a direct system, so we can define the ind-variety

A_{∞} := lim_{g} A_{g}^{S}

We follow the notations of Section 2. The line bundles L_{g} define a line bundle

L_{∞} := lim_{g} L_{g}

on A_{∞}. This line bundle is called the line bundle of weight one stable stable modular forms.

Definition 3.3 (Stable modular forms). A weight k stable modular form F is a section of L_{∞}^{k}. More concretely, it is a collection

F = (F_{g})_{g≥0}

where F_{g} is a modular form of weight k on A_{g} and

Φ(F_{g+1}) = F_{g}

Recall that each line bundle L_{g} is ample on A_{g}; however, the same assertion is problematic for L_{∞}, as explained in Remark 2.1. We now define a structure of commutative monoid on A_{∞}. Given two principally polarised abelian varieties of dimension respectively g and h, their product is still a principally polarised abelian variety of dimension g + h. This gives a commutative operation

\begin{aligned}m: & A_{∞} × A_{∞} & → & A_{∞} \\ & ([X], [Y]) & ↦ & [X × Y] \end{aligned}

The identity element is A_{0}.

Lemma 3.4. On A_{∞} × A_{∞}, we have

m^{∗}L_{∞} = L_{∞} ⊠ L_{∞} .

Proof. For every pair of integers g and h one looks at the morphism

m: A_{g} × A_{h} → A_{g+h}

The fibre of the Hodge bundle E_{g} at a point [X] of A_{g} is the tangent space at the identity of X. This implies that m^{∗}E_{g+h} is E_{g} ⊠ E_{h}. The statement on L_{g} follows by taking the determinant. □

We have the following useful formal consequence

Proposition 3.5. The pull-back m^{∗} defines a co-commutative comultiplication on the algebra of stable modular forms R(A_{∞}, L_{∞}).

The ring of stable modular form, so far, could be trivial. However, there is a classical and surprising way to produce plenty of stable modular forms out of quadratic forms. Let us go trough all definitions. A quadratic form is a pair (Λ, Q), where Λ is a finitely generated free group and Q is a ℤ-valued bilinear form on Λ. The rank of the quadratic form is defined as the rank of Λ. The elements of Λ are called vectors, and the norm of a vector v is Q(v, v). We always assume Q to be even (i.e. Q(v, v) is even for every v), unimodular (i.e. det Q = 1) and positive definite. Often, we will denote a quadratic form just by Q.

Definition 3.6 (Theta series). Let (Λ, Q) be an even unimodular positive definite quadratic form and g a positive integer, the associated theta series is

Θ_{Q,g}(τ) := ∑_{x_{1},...,x_{g}∈Λ} exp(πi ∑_{i,j} Q(x_{i}, x_{j})τ_{ij})

where τ belongs to H_{g}.

This is a weight \frac{1}{2} rk(Λ) and degree g modular form. By explicit computation, one sees that the Siegel operator 3.2 acts as follows

Φ(Θ_{Q,g+1}) = Θ_{Q,g} ,

so the collection of all theta series

Θ_{Q} := (Θ_{Q,g})_{g≥0}

is a stable modular form. Given X ∈ A_{g} and Y ∈ A_{h}, we have the factorisation property

Θ_{Q,g+h}([X × Y]) = Θ_{Q,g}(X)Θ_{Q,h}(Y) ,

which means that the theta series are characters for the monoid A_{∞}.

Example 3.7 (Quadratic forms and theta constants). In some cases, theta series can be written in term of theta constants, let us give some examples following [Igu81]. Let E_{8} be the quadratic form associate to the Dynkin diagram E_{8}. Using a similar definition, for every integer k one can define the Witt quadratic forms W_{8k}. The quadratic form W_{8k} has rank 8k, it is equal to E_{8} for k = 1 and to D_{16}^{+} for k = 2. Up to a constant, we have the following expansion

Θ_{W_{8k},g}(τ) = ∑_{ϵ even} θ[ϵ]^{8k}(τ) ,

where the sum runs over all the even theta characteristics. In particular, the well-known Schottky form can be written as

Θ_{D_{16}^{+}} − Θ_{E_{8}⊕E_{8}} = 2^{−g} ∑_{ϵ even} θ[ϵ]^{16}(τ) − 2^{−2g}( ∑_{ϵ even} θ[ϵ]^{8}(τ))^{2}

In general, a theta series will not have such a simple expression in term of theta constants. In [SM89, Section 3], there is a systematic analysis of the theta series which can be expanded in this way; the results of that paper relies upon [Mum07, Theorem 6.3].

The ring of stable modular forms is described by the following result of Freitag:

Theorem 3.8 (Theorem 2.5 of [Fre77]). The ring of stable modular forms R(A_{∞}, L_{∞}) is the polynomial ring in the theta series associated to irreducible quadratic forms.

Freitag’s main contribution was to show that H^{0}(A_{∞}, L_{∞}^{k}) is spanned by theta series for every k. There are finitely many quadratic forms of a given rank, so we already learn that H^{0}(A_{∞}, L_{∞}^{k}) is finite dimensional. This result, together with Lemma 3.4 and the fact that theta series are characters, means that A_{∞} equipped with the line bundle L_{∞} is a split monoid in the sense of Definition 2.2. Freitag’s claim about the polynomial structure now follows easily from Proposition 2.4.

4. Satake compactification of the hyperelliptic locus

In this section we define the Satake compactification of the hyperelliptic locus and we prove Theorems 1.2 and 1.3. Consider the Jacobian morphism

j: Hyp_{g} → A_{g}

mapping a curve to its Jacobian.

Definition 4.1 (Satake compactification). The Satake compactification Hyp_{g}^{S} of the hyperelliptic locus Hyp_{g} is the scheme-theoretic closure of j(Hyp_{g}) inside A_{g}^{S}.

A degeneration of a hyperelliptic Jacobian is still the Jacobian of a hyperelliptic curve ([Hoy63]), so we have a stratification

Hyp_{g}^{S} = ⨆ ; ∑ g_{i}≤g Hyp_{g_{1}} × ⋯ × Hyp_{g_{k}}

Equivalently, the Satake compactification is the image of the Deligne-Mumford compactification of Hyp_{g} under the morphism which maps a curve to the Jacobian of its normalisation. In particular, Hyp_{g+1}^{S} contains Hyp_{g}^{S} as a scheme, so we can define the commutative ind-monoid

Hyp_{∞} := lim_{g} Hyp_{g}^{S}

Using the monoid structure we can show the following

Theorem 4.2. The ideal of stable modular forms vanishing on Hyp_{∞} is generated by differences of theta series

Θ_{P} − Θ_{Q}

where P and Q are even, unimodular, positive definite quadratic forms of the same rank.

Proof. We know that Hyp_{∞} is a commutative sub-monoid of A_{∞} and A_{∞} satisfies the hypotheses of Definition 2.2. The theta series are the characters of co-algebra R(A_{∞}, L_{∞}), so the result is a direct consequence of Proposition 2.4. □

So far, the ideal studied in the theorem could be trivial. A key tool to show that a modular form is a stable equation for the hyperelliptic locus is the following geometric result. Let A_{g}^{ind} be the moduli space of indecomposable principally polarised abelian g-fold.

Theorem 4.3 (Transversality). Inside A_{g+1}^{S} , the intersection of Hyp_{g+1}^{S} and A_{g}^{ind} is scheme theoretically equal to Hyp_{g}.

The statement was well-known at the level of sets (cf [Hoy63] or [ACG11, Lemma 11.6.14]); we call it a transversality result because it states that the scheme structure of the intersection is the reduced one.

Proof. We work with level structure (4, 8), and we denote by A_{g}^{S}(4, 8) the Satake compactification of the moduli space A_{g}(4, 8) of principally polarised abelian g-fold with level structure (4, 8). This amounts to take a finite Galois cover of A_{g}^{S}. Now, A_{g+1}^{S}(4, 8) has several boundary components, all isomorphic to A_{g}^{S}(4, 8). We will fix one of them, let us call it V. The hyperelliptic locus Hyp_{g+1}^{S}(4, 8) breaks into several irreducible components (cf. [Tsu91]); an irreducible component is identified by the choice of a fundamental system 𝔪 of theta characteristic, we will fix such an 𝔪 and denote by Y = Y_{𝔪} the corresponding irreducible component. Since the cover is Galois, locally the intersection of Hyp_{g+1}^{S} and A_{g} is isomorphic to the intersection of Y and V. Because of this, it is enough to show that the scheme-theoretic intersection of V^{ind} and Y is reduced. We need to work at level (4, 8) to apply the following result, which is due set-theoretically to Mumford and scheme theoretically to Salvati Manni.

Theorem 4.4 ([SM03]). Fix a fundamental system of theta characteristic 𝔪 = (m_{1}, …, m_{2g+1}), and let b be the sum of the odd m_{i}. Then, the corresponding irreducible component Y = Y_{𝔪} of Hyp_{g}(4, 8) is scheme theoretically defined by the vanishing of the theta constants θ_{m+b} such that m = m_{i_{1}} + ⋯ + m_{i_{k}} for k < g and k ≡ g, g +1, and the non-vanishing of the remaining theta constants.

In the statement, the non-vanishing of the remaining theta constants is needed to rule out the loci of decomposable abelian varieties; our statement is about Jacobians of smooth curves, so we are already working outside these loci. Let 𝔪 be a g + 1 dimensional system of fundamental theta characteristic, e.g. the one defined in equation 7 of [SM03]. Let Y = Y_{𝔪} be the corresponding irreducible component of Hyp_{g+1}^{S}(4, 8). Let I_{g+1}(Y_{𝔪}) be the ideal of modular forms generated by the theta constants vanishing along Y_{𝔪}. Because of Salvati Manni’s result, this ideal defines scheme-theoretically Y_{𝔪}. By direct computation one sees that

Φ(θ [ \begin{matrix}ϵ & 0 \\ ϵ′ & δ \end{matrix} ]) = θ [ \begin{matrix}ϵ \\ ϵ′ \end{matrix} ]

for δ equal either to 0 or 1. In the previous formula, Φ is the Siegel operator, i.e. the restriction operator from A_{g+1}^{S}(4, 8) to one of the boundary component, say to V. Because of the theorem quoted above, this means that, scheme theoretically, the intersection of Y_{𝔪} and V ≅ A_{g}(4, 8)^{S} away from decomposable abelian varieties is isomorphic to Y_{𝔫}; where 𝔫 is the g dimensional system of fundamental theta characteristic defined in equation 7 of [SM03]. □

We now complete the description of the tangent space of Hyp_{g+1}^{S} along Hyp_{g}. Let C be a smooth genus g hyperelliptic curve and X its Jacobian. To start with, let us describe the normal bundle exact sequence of A_{g} in A_{g+1}^{S} at X. This sequence reads

0 → T_{X}A_{g} → T_{X}A_{g+1}^{S} → H^{0}(X, 2Θ)^{∨} → 0

We will need the following explicit description of the action of these derivations. Let F_{g+1} be a modular form on A_{g+1} and F_{g} its restriction to A_{g}. For any element T in the Siegel upper half space H_{g+1} write

T = ( \begin{matrix}τ & z \\ ^{t}z & t \end{matrix} ) ,

with t in H_{1} and τ in H_{g}. Let q := exp(2πit); then the Fourier-Jacobi expansion of F_{g+1} is

(1)  F_{g+1}(T) = F_{g}(τ) + ∑_{n≥1} f_{n}(τ, z)q^{n}

where f_{n} is a section of H^{0}(X_{τ} , 2nΘ), X_{τ} is the principally polarised abelian variety defined by τ, and z is a system of co-ordinates on X_{τ}. A derivation D ∈ T_{X}A_{g} acts as D(F_{g+1}) = D(F_{g}); a derivation in D ∈ H^{0}(X, 2Θ)^{∨} acts as D(F_{g+1}) = D(f_{1}). The normal bundle exact sequence for Hyp_{g} in Hyp_{g+1}^{S} at C is a subsequence of the normal bundle exact sequence of A_{g} in A_{g+1}^{S} at X. To describe it, we need to introduce the following morphism

(2)  \begin{aligned}Ψ : & C & ⟶^{f} & C × C & ⟶^{δ} & X \\ & p & ↦ & (p, ι(p)) \\ & & & (a, b) & ↦ & AJ(a) − AJ(b) \end{aligned}

where ι is the hyperelliptic involution and AJ is the Abel-Jacobi map.

Lemma 4.5. The pull-back Ψ^{∗}2Θ is isomorphic to 2(K_{C} + W), where W is the divisor of Weierstrass points on C.

Proof. The pull-back δ^{∗}2Θ is K_{C} ⊠ K_{C}(2Δ), where Δ is the diagonal; this is well-known, e.g. [Wel86, Equation 4.4]. Now, we pull-back K_{C} ⊠ K_{C}(2Δ) via f. The pull-back of Δ is W; the pull-back of K_{C} ⊠ K_{C} is K_{C} + ι^{∗}K_{C} = 2K_{C}.

□

Theorem 4.6. Keep notation as above, the normal bundle exact sequence for Hyp_{g} in Hyp_{g+1} at C is

0 → T_{C}Hyp_{g} → T_{C}Hyp_{g+1}^{S} → P_{C} → 0

where P_{C} is the image of the map

Ψ^{∗}: H^{0}(X, 2Θ) → H^{0}(C, 2(K_{C} + W)) .

In other words, the normal tangent cone at C is the cone over (Ψ(C), 2Θ).

Proof. There are two things we have to prove, first

T_{C}Hyp_{g+1}^{S} ∩ T_{C}A_{g} = T_{C}Hyp_{g};

but this is equivalent to Theorem 4.3. To describe the co-kernel of the inclusion

T_{C}Hyp_{g} ↪ T_{C}Hyp_{g+1}^{S}

we need to know that, after blowing up A_{g}^{S} in A_{g+1}^{S}, the proper transform of Hyp_{g+1}^{S} meets the Kummer variety of X in Ψ(C). This is proved in [Nam73, Theorem 6], just remark that to obtain a generic irreducible nodal hyperelliptic curve we need to glue two points conjugated under the hyperelliptic involution.

□

In [Cod14, Lemma 3.6], it is shown that Ψ^{∗} is not surjective and P_{C} has rank 2g; however, we do not need this result here.

Remark 4.7 (Failure of Theorem 4.3 at the locus of decomposable abelian varieties). For the sake of completeness, let us sketch a proof of the failure of Theorem 4.3 at the locus of decomposable abelian varieties. This result is not needed in this paper, but we think that the study of this intersection is interesting on its own. Pick two integers such that g_{1} +g_{2} = g; fix a hyperelliptic curve C of genus g_{1} and a hyperelliptic curve D of genus g_{2}. Call ι the hyperelliptic involution. The point (C, D) in Hyp_{g+1}^{S} represents all the hyperelliptic curves of the form C ⊔ D/(p ∼ q, ι(p) ∼ ι(q)), where p is a point varying in C and q is varying in D. Recall that we have an identification

Sym^{2}(H^{0}(C, K_{C})^{∨} ⊕ H^{0}(D, K_{D})^{∨}) = T_{(J(C)×J(D))}A_{g}

Arguing as in [CSB14], we can show that the tangent space of Hyp_{g+1}^{S} at (C, D) contains the image of the map

\begin{aligned}ψ: & C/ι × D/ι & → & ℙ Sym^{2}(H^{0}(C, K_{C})^{∨} ⊕ H^{0}(D, K_{D})^{∨}) \\ & (p, q) & ↦ & ω_{i}(p)ψ_{j}(q) + ω_{j}(p)ψ_{i}(q) \end{aligned}

where ω_{i} are a basis of H^{0}(C, K_{C}) and ψ_{i} are a basis of H^{0}(D, K_{D}). This is the same tangent direction we get when we consider an appropriate smoothing of the genus g nodal curve C ⊔ D/p ∼ q; this smoothing is not hyperelliptic, so the intersection of T_{(C,D)}Hyp_{g+1}^{S} with T_{(J(C)×J(D))}A_{g} is strictly bigger than T_{(C,D)}Hyp_{g}^{S}.

5. Projective invariants of hyperelliptic curves

In this section, we review the theory of projective invariants, and we use it to show that certain modular forms vanish on the hyperelliptic locus. To start with, let us introduce the auxiliary space B_{g}. This is the moduli space of 2g + 2 points on ℙ^{1}, up to permutation and projectivity. The points are counted with multiplicity, points are not allowed to have multiplicity bigger that g + 1. This space is classically constructed as a GIT quotient; it is irreducible, it has an open dense subset B_{g}^{∘} where the 2g + 2 points are all distinct, and a boundary D where at least two points coincide. Let C be a smooth genus g hyperelliptic curve, fix a two to one map π: C → ℙ^{1}. This morphism is unique up to projective transformations of ℙ^{1}, it ramifies at 2g +2 points. A point p is called a Weierstrass point if it is a ramification point for π.

Definition 5.1 (Projective invariants). The projective invariants of C are the image of the Weierstrass points under π, considered up to permutations and projective automorphisms of ℙ^{1}.

Equivalently, the projective invariants of C are the points of the branch divisor of π, considered up to projectivity. The projective invariants of a smooth hyperelliptic curve C are naturally a point of B_{g}^{∘}, so we have a morphism

f_{g}: Hyp_{g} → B_{g}^{∘}

One can reconstruct a hyperelliptic curve out of its projective invariants, and given 2g + 2 points on ℙ^{1} there is a hyperelliptic curve with that projective invariants; this means that f_{g} is an isomorphism. This construction has been extensively used to study the moduli space Hyp_{g}; references are [Igu67], [AL02] and [Pas11, Chapter 2]. As an aside, let us recall that the Thomæ’s formula permits to write the crossratios of the projective invariants in term of second order theta functions evaluated at the period matrix of C. Following [AL02], the map f_{g} extends to an isomorphism

f_{g}: Hyp_{g} ⊔ η_{0}^{∗} → B_{g}^{∘} ⊔ D^{∗}

where η_{0}^{∗} parametrises irreducible singular hyperelliptic curves with just one node, i.e. curves of the form C/p ∼ ι(p), and D^{∗} parametrises 2g + 2 points on ℙ^{1} such that exactly 2 points coincide. The image of a curve in η_{0}^{∗} is a set of 2g + 2 points of the form {p_{1}, …, p_{2g}, p, p}, where {p_{1}, …, p_{2g}} are the projective invariants of the normalisation and the glued points are the preimages of p under π. Always following [AL02], we can extend f_{g} further to a morphism

f_{g}: \overline{Hyp}_{g} → B_{g}

which contracts the boundary divisors of the Deligne-Mumford compactification different from the closure of η_{0}^{∗} to high co-dimension loci. We need some more information about B_{g}, again references are [Igu67], [AL02] and [Pas11, Chapter 2]. From the GIT point of view, the moduli space B_{g} can be constructed as the Proj of the ring S(2, 2g + 2), which is the ring of co-invariant of binary forms of degree 2g + 2. This ring is formally constructed as follows: let Z be the cartesian product of 2g + 2 copies of ℙ^{1}, on this variety we have a diagonal action of SL(2, ℂ) and an action of the symmetric group; this action linearise to an action on the line bundle M := O(1, …, 1), the ring S(2, 2g + 2) is the ring of invariant element of R(Z, M). More concretely, S(2, 2g + 2) is the ring of symmetric functions in 2g + 2 variables, which are co-invariant under the natural action of SL(2, ℂ). The discriminant Δ is an element of S(2, 2g + 2) of degree 4g + 2, it cuts out the boundary divisor D. We now consider also the Jacobian morphism

j: \overline{Hyp}_{g} → Hyp_{g}^{S} ↪ A_{g}

Take the composition

j ∘ f_{g}^{−1} : B_{g} −−→ A_{g}

Under this map, D^{∗} dominates Hyp_{g−1}; explicitly, the set {p_{1}, …, p_{2g}, p, p} is mapped to the Jacobian of the smooth hyperelliptic curve defined by the projective invariants {p_{1}, …, p_{2g}}. Taking the pull-back we obtain a morphism of graded ring

ρ: R(A_{g}, L_{g}) → S(2, 2g + 2)

whose kernel is exactly the ideal of modular forms vanishing on the hyperelliptic locus. This map is sometime called Igusa morphism of projective invariants, it was introduced in [Igu67], where Igusa proved that its degree is \frac{1}{2}g ^{1}. Using this construction we can prove the following criterion.

Criterion 5.2 (Weissauer - unpublished). Let F_{g} be a weight n and degree g modular form. Restrict it to Hyp_{g}^{S} and say it vanishes along Hyp_{g−1} with multiplicity at least k. If

\frac{n}{k} < 8 + \frac{4}{g}

then F_{g} vanishes on Hyp_{g}.

Proof. Suppose F_{g} vanishes with multiplicity at least k on Hyp_{g−1}. This means that (j ∘ f_{g}^{−1})^{∗}F_{g} vanishes with multiplicity at least k on D. In other words, Δ^{k} divides ρ(F_{g}). The degree of the discriminant in S(2, 2g + 2) is 4g + 2, the degree of ρ(F_{g}) is \frac{1}{2}gn. Since, by hypothesis,

k(4g + 2) > \frac{1}{2}gn

we obtain that ρ(F_{g}) is equal to zero, so the claim. □

Remark 5.3 (Relation with Theorem 4.6). To show that F_{g} vanishes along Hyp_{g−1} with multiplicity at least 2 one needs to know the tangent space of Hyp_{g}^{S} along Hyp_{g−1}; in the applications, especially in Theorem 6.1, we will use the description given in Theorem 4.6.

Remark 5.4 (Other versions of Criterion 5.2). A weaker version of Criterion 5.2 can be found in [Poo96]. An alternative proof is in [Pas11]. In [SM00], Salvati Manni attributed this criterion to Weissauer, in an unpublished preprint, and showed that the inequality is sharp. This Criterion is also related to the slope of the hyperelliptic locus, in the sense of slope of the cone of effective divisors (cf. [CH88]).

1If g is odd, all non-trivial modular forms have odd degree, so the factor \frac{1}{2} should not worry the reader.

Combining Theorem 1.3 and Criterion 5.2 we can find a first group of stable equations for the hyperelliptic locus. We will need the following basic invariant of a quadratic form (Q, Λ)

μ(Q) := min{Q(v, v) | v ∈ Λ; v ≠ 0} = min{n | R_{n}(Λ) ≠ ∅} ,

where R_{n}(Λ) is the set of vectors of Λ of norm 2n.

Theorem 5.5. Let (Q, Λ) and (P, Γ) be two even positive definite unimodular quadratic forms of rank N and let μ := min{μ(Q), μ(P)}. If

\frac{N}{μ} ≤ 8 ,

then

F := Θ_{Q} − Θ_{P}

is a stable equation for the hyperelliptic locus. In other words, F_{g} vanishes on Hyp_{g} for every g.

Proof. The proof is by induction on g. The difference of two theta series vanishes on A_{0}. Suppose the statement true for g, we want to apply Criterion 5.2 to F_{g+1}. Call k := \frac{1}{2}μ, we need to prove that F_{g+1} vanishes at the boundary component Hyp_{g} with multiplicity at least k. We first compute the multiplicity along tangent direction parallel to the boundary, namely along T_{C}Hyp_{g+1}^{S}∩T_{C}A_{g}, where C is a smooth genus g hyperelliptic curve. This intersection is, by Theorem 1.3, equal to T_{C}Hyp_{g}. By induction, F_{g} vanishes along Hyp_{g}, so F_{g+1} is annihilated by the derivations contained in T_{C}Hyp_{g+1}^{S} ∩ T_{C}A_{g}. Let us now look at the normal direction to A_{g}; we will use the Fourier-Jacobi expansion introduced in the equation (1). Writing out the Fourier-Jacobi expansion of F_{g+1}, the hypothesis on μ implies that the first k terms vanish. This means that F_{g+1} vanishes with order at least k along the normal direction to A_{g} in A_{g+1}^{S}; in particular, we obtain that it vanishes along Hyp_{g} with multiplicity at least k and we can apply Criterion 5.2. □

The hypotheses of Theorem 5.5 are quite restrictive; let us describe the cases where the Theorem can be applied.

Proposition 5.6. Let (Q, Λ) be an even, positive definite, unimodular quadratic form, then

\frac{rk(Q)}{μ(Q)} ≤ 8

if and only if the pair (rk(Q), μ(Q)) is equal to one of the following pairs: (8, 2), (16, 2), (32, 4) or (48, 6).

Proof. Given any even unimodular quadratic form (Q, Λ), there is an upper bound

μ(Q) ≤ 2⌊\frac{rk(Q)}{24}⌋ + 2

where “⌊ ⌋” is the round down (see [CS99, Section 7.7 Corollary 21] ). This bound, combined with the fact that the rank is divisible by 8, gives the Proposition.

□

Let us call the type of quadratic form the pair (rk(Q), μ(Q)). There is just one quadratic form of type (8, 2) and one of type (24, 4), so we do not get any stable equation in these cases. The rank 16 case was considered by Poor in [Poo96]: there are two quadratic forms of type (16, 2), so one gets one equation. In [Kin03, Corollary 5], using a generalization of the mass formula, it is shown that there exist at least ten millions of quadratic forms of rank 32 and μ = 4 ; however, just 15 of them are known explicitly. The situation for quadratic forms of type (48, 6) is not clear: believably, there exist many of them, see [Kin03, Page 15], but there is not any lower bound and just 3 of them are known explicitly. (King adopts a slightly different notation: every quadratic form is tacitly assumed to be positive definite.) To summarise, Theorem 5.5 can be applied to the following cases

Corollary 5.7. If P and Q are two even, unimodular, positive definite quadratic forms meeting one of the following two hypotheses:

(1) rk(P) = rk(Q) = 32 and μ(P) = μ(Q) = 4; that is, the quadratic forms do not have any vector of norm 2;

(2) rk(P) = rk(Q) = 48 and μ(P) = μ(Q) = 6; that is, the quadratic forms

do not have any vector of norm 2 and 4; then, the difference

Θ_{P} − Θ_{Q}

is a stable equation for the hyperelliptic locus.

6. Niemeier quadratic forms

Niemeier quadratic forms are rank 24 quadratic forms. In this section we prove the following:

Theorem 6.1. Let (P, Γ) and (Q, Λ) be two rank 24 quadratic forms with the same number of vectors of norm 2, then the difference

Θ_{P} − Θ_{Q}

is a stable equation for the hyperelliptic locus.

Vectors of norm 2 are usually called roots. This result concerns the following 5 pairs of quadratic forms

quadratic forms A_{5}^{4}D_{4} , D_{4}^{6} A_{9}^{2}D_{6} , D_{6}^{4} A_{11}D_{7}E_{6}, E_{6}^{4} ; # roots  72  120  144

\begin{matrix}A_{17}E_{7} , D_{10}E_{7}^{2} & E_{8}D_{16}, E_{8}^{3} \\ 216 & 360 \end{matrix}

Documenta Mathematica 21 (2016) 445–466

where a quadratic form of rank 24 is labelled by its root system (see e.g. [Ebe13, Section 3] for more details). The pair E_{8}^{3}, D_{16}E_{8} corresponds to the modular form Θ_{E_{8}}(Θ_{E_{8}⊕E_{8}} − Θ_{D_{16}^{+}}), so its behaviour was well-understood. The others cases can not be expressed as a product of lower weight stable modular forms and they are not covered by previous results. This result is surprising because the slope of these quadratic forms, i.e. the ratio between the rank and the norm of the shortest vector, is strictly bigger than 8, so they were not expected to vanish on the hyperelliptic locus in every genus. Before proving our theorem we need two preliminary results.

6.1. A formula for sections of 2Θ. Let s be a section of 2Θ on the Jacobian of a curve C with period matrix τ. For every couple of points a and b of C the following classical formula holds:

(3) s(τ, a − b) = E(a, b)^{2}[s(τ, 0)ω(a, b) + ∑_{i,j} \frac{∂^{2}s}{∂z_{i}∂z_{j}} (τ, 0)ω_{i}(a)ω_{j}(b)]

where E is the Prime form, {ω_{i}} is the basis of the holomorphic differentials on C corresponding to the basis \{\frac{∂}{∂z_{i}} \} of the tangent space at the origin of the Jacobian, ω(a, b) is the fundamental normalised bi-differential, and everything is trivialised with respect to a choice of local co-ordinates z_{a} and z_{b}. This formula is well known, see e.g. [MV10, Appendix A].

6.2. The “heat equation” for Niemeier quadratic forms. The classification of rank 24 quadratic forms is due to Niemeier, but it has been simplified by Venkov proving and using the following identity

Theorem 6.2 (Venkov, cf. [Ebe13] Section 3). Let (Λ, Q) be a rank 24 quadratic form, then

r_{2}(Λ)Q(v, w) = 8 ∑_{y∈R_{2}(Λ)} Q(y, w)Q(y, v) ∀ v, w ∈ Λ

where r_{2}(Λ) is the number of roots and R_{2}(Λ) is the set of roots.

The proof relies upon the theory of degree 1 modular forms with harmonic coefficients. Let us draw a consequence of this result about the Fourier-Jacobi expansion of theta series (cf. equation (1)).

Corollary 6.3 (Heat equation). The first Fourier-Jacobi coefficient f_{1} of a theta series associated to a rank 24 quadratic form Λ satisfies the following “heat equation”

r_{2}(Λ)πi\frac{∂f_{1}}{∂τ_{ij}} (τ, 0) = 3(1 + δ_{ij})\frac{∂^{2}f_{1}}{∂z_{i}∂z_{j}} (τ, 0) ,

where r_{2}(Λ) is the number of roots of Λ.

Proof. By explicit computation, we can write out the first Fourier-Jacobi coefficient of a theta series:

f_{1}(τ, z) = ∑_{x_{1},...,x_{g}∈Λ} ∑_{y∈R_{2}(Λ)} exp(πi ∑_{i,j} Q(x_{i}, x_{j})τ_{ij} + 2πi ∑_{i} Q(y, x_{i})z_{i}) .

Fix two indexes i and j, by explicit computation we have

\frac{∂^{2}f_{1}}{∂z_{i}∂z_{j}} (τ, 0) = (2πi)^{2} ∑_{x_{1},...,x_{g}∈Λ} ∑_{y∈R_{2}(Λ)}Q(y, x_{i})Q(y, x_{j}) exp(πi ∑_{i,j} Q(x_{i}, x_{j})τ_{ij}) ,

On the other hand

(1 + δ_{ij})\frac{∂f_{1}}{∂τ_{ij}} (τ, 0) = 2πi ∑_{x_{1},...,x_{g}∈Λ} Q(x_{i}, x_{j}) exp(πi ∑_{i,j} Q(x_{i}, x_{j})τ_{ij}) ,

the coefficient (1 + δ_{ij}) is because the variables on H_{g} are τ_{ij} with i ≤ j, so when we compute the derivative with respect to τ_{ij} we need to derive both τ_{ij} and τ_{ji}. Applying Theorem 6.2 we obtain the result. □

This formula is also discussed in [MV10, page 16]. This result is generalised to higher order Fourier-Jacobi coefficients and higher rank quadratic forms in [Cod14, Theorem 10.3].

6.3. Proof of Theorem 6.1. We want to show that F_{g} = Θ_{P,g} − Θ_{Q,g} is zero on Hyp_{g} for every g; we argue by induction on g. The case g = 0 is easy. To prove the inductive step we use Criterion 5.2: we need to show that F_{g+1} vanishes along Hyp_{g} with multiplicity at least 2. As in Theorem 5.5, the derivative along directions tangent to A_{g} vanishes because of Theorem 1.3 and the inductive hypothesis. The normal direction is quite different: now the first Fourier-Jacobi coefficient is not trivial, so there is some work to do. Because of Theorem 4.6, it is enough to check that f_{1} vanishes when restricted to points of the form (τ, p − ι(p)), where τ is the period matrix of a smooth hyperelliptic curve C, p is a point of C and ι is the hyperelliptic involution. To show this we argue as follows. First remark that

f_{1}(τ, 0) = F_{g}(τ) = 0

Then we apply the formula (3), trivialising everything with respect to coordinates z_{p} and ι^{∗}z_{p} and recalling that

\frac{ω}{dz_{p}} (p) = \frac{ω}{ι^{∗}dz_{p}} (ι(p))

we get

f_{1}(τ, p − ι(p)) = E(p, ι(p))^{2} ∑_{i,j} \frac{∂^{2}f_{1}}{∂z_{i}∂z_{j}} (τ, 0)ω_{i}(p)ω_{j}(p)

Now the heat equation 6.3 and the hypothesis on the number of roots come into the game: since r_{2}(Λ) = r_{2}(Γ) =: r, we have

6 ∑_{i,j} \frac{∂^{2}f_{1}}{∂z_{i}∂z_{j}} (τ, 0)ω_{i}(p)ω_{j}(p) = rπi ∑_{i≥j} \frac{∂F_{g}}{∂τ_{ij}} (τ)ω_{i}(p)ω_{j}(p) = (rπi)dF_{g}(τ)(p)

Let us explain the last equality: the fibre of the cotangent bundle of A_{g} at Jac(C) is isomorphic to Sym^{2} H^{0}(C, K_{C}), so dF_{g}(τ) is a quadric in ℙH^{0}(C, K_{C})^{∨} and we can evaluate it on the image of p under the canonical map. The co-normal bundle of Hyp_{g} in M_{g} is given by the −1 eigenspace of H^{0}(C, 2K_{C}); the image of the co-differential m: Sym^{2} H^{0}(C, K_{C}) → H^{0}(C, 2K_{C}) is the +1 eigenspace; we conclude that the quadric in the conormal bundle of Hyp_{g} in A_{g} vanishes along the canonical image of C. Since F_{g} vanishes along Hyp_{g}, dF_{g} is a quadric containing the canonical image of C; in other words, it has to vanish when evaluated at any point p of C. This concludes the proof of Theorem 6.1.

6.4. Other results about Niemeir quadratic forms. With similar tools, we can prove other results about the behaviour of these modular forms on the moduli space of curves and abelian varieties.

Theorem 6.4 ([Cod14] Corollary 11.2). Let P and Q be two even positive definite unimodular quadratic forms of rank 24 with the same number of roots, then the stable modular form

F := Θ_{P} − Θ_{Q}

is zero on M_{g} for g ≤ 4, and it cuts a divisor of slope 12 on M_{5}.

Theorem 6.5 ([Cod14] Theorem 11.3). The following degree 5 modular forms are non-trivial cusp forms

Θ(D_{16}E_{8}) − Θ(E_{8}^{3}) − \frac{21504}{24}(Θ(A_{5}^{4}D_{5}) − Θ(D_{4}^{6}))

Θ(D_{16}E_{8}) − Θ(E_{8}^{3}) − \frac{21504}{216}(Θ(A_{9}^{2}D_{6}) − Θ(D_{6}^{4}))

Θ(D_{16}E_{8}) − Θ(E_{8}^{3}) − \frac{21504}{480}(Θ(A_{11}D_{7}E_{6}) − Θ(E_{6}^{4}))

Θ(D_{16}E_{8}) − Θ(E_{8}^{3}) − \frac{21504}{−2520}(Θ(A_{17}E_{7}) − Θ(D_{10}E_{7}^{2}))

where, for typographical reasons, we write Θ(Q) rather than Θ_{Q,5}.

References

[ACG11] E. Arbarello, M. Cornalba, and P. A. Griffiths. Geometry of algebraic curves. Volume II, volume 268 of Grundlehren der Mathematischen Wissenschaften. Springer, Heidelberg, 2011. With a contribution by J. D. Harris.

[AL02] D. Avritzer and H. Lange. The moduli spaces of hyperelliptic curves and binary forms. Mathematische Zeitschrift, 242(4):615–632, 2002.

[BvdGHZ08] J. H. Bruinier, G. van der Geer, G. Harder, and D. Zagier. The 1-2-3 of modular forms. Universitext. Springer-Verlag, Berlin, 2008.

[CH88] M. Cornalba and J. Harris. Divisor classes associated to families of stable varieties, with applications to the moduli space of curves. Ann. Sci. École Norm. Sup. (4), 21(3):455–475, 1988.

[Cod14] G. Codogni. Satake compactifications, Lattices and Schottky problem. PhD thesis, University of Cambridge, 2014.

[CS99] J. H. Conway and N. J. A. Sloane. Sphere packings, lattices and groups, volume 290 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, New York, 1999.

[CSB14] G. Codogni and N. I. Shepherd-Barron. The non-existence of stable Schottky forms. Compos. Math., 150(4):679–690, 2014.

[Ebe13] W. Ebeling. Lattices and codes. Advanced Lectures in Mathematics. Springer Spektrum, Wiesbaden, third edition, 2013. A course partially based on lectures by Friedrich Hirzebruch.

[Fre77] E. Freitag. Stabile Modulformen. Math. Ann., 230(3):197–211, 1977.

[Fre83] E. Freitag. Siegelsche Modulfunktionen, volume 254 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, 1983.

[GHT14] S. Grushevsky, K. Hulek, and O. Tommasi. Stable cohomology of the perfect cone toroidal compactification of the moduli space of abelian varieties. 2014. To appear in Journal für die reine und angewandte Mathematik.

[GKV10] M. R. Gaberdiel, C. A. Keller, and R. Volpato. Genus two partition functions of chiral conformal field theories. Commun.Num.Theor.Phys, (4):295–364, 2010.

[GV09] M. R. Gaberdiel and R. Volpato. Higher genus partition functions of meromorphic conformal field theories. J. High Energy Phys., (6):048, 43, 2009.

[Hoy63] W. L. Hoyt. On products of algebraic families of jacobian varieties. Annals of Mathematics, 77(3):415–423, 1963.

[Igu67] J. Igusa. Modular forms and projective invariants. Amer. J. Math., 89:817–855, 1967.

[Igu81] J. Igusa. Schottky’s invariant and quadratic forms. In E. B. Christoffel (Aachen/Monschau, 1979), pages 352–362. Birkhäuser, Basel, 1981.

[Kin03] O. D. King. A mass formula for unimodular lattices with no roots. Mathematics of Computation, 72(242):839–863 (electronic), 2003.

[Kum02] S. Kumar. Kac-Moody groups, their flag varieties and representation theory, volume 204 of Progress in Mathematics. Birkhäuser Boston Inc., 2002.

[Mat15] M. Matone. A surprising relation for the effective coupling constants of N=2 super yang-mills theories. J. High Energy Phys., 1506 (2015), 109.

[Mum07] D. Mumford. Tata lectures on theta. I. Modern Birkhäuser Classics. Birkhäuser Boston Inc., Boston, MA, 2007. With the collaboration of C. Musili, M. Nori, E. Previato and M. Stillman, Reprint of the 1983 edition.

[MV10] M. Matone and R. Volpato. Getting superstring amplitudes by degenerating Riemann surfaces. Nuclear Phys. B, 839(1-2):21–51, 2010.

[Nam73] Y. Namikawa. On the canonical holomorphic map from the moduli space of stable curves to the Igusa monoidal transform. Nagoya Math. J., 52:197–259, 1973.

[Pas11] S. Pascolutti. Boundaries of Moduli Space of Hyperelliptic Curves with a Level 2 Structure. PhD thesis, La Sapienza, Università di Roma, 2011.

[Poo96] C. Poor. Schottky’s form and the hyperelliptic locus. Proc. Amer. Math. Soc., 124(7):1987–1991, 1996.

[Sat56] I. Satake. On the compactification of the Siegel space. J. Indian Math. Soc. (N.S.), 20:259–281, 1956.

[SB13] N. Shepherd-Barron. Siegel modular forms and the gonality of curves. arXiv:1306.6253, 2013.

[SM89] R. Salvati Manni. Thetanullwerte and stable modular forms. Amer. J. Math., 111 (3) 435–455, 1989

[SM00] R. Salvati Manni. Slope of cusp forms and theta series. J. Number Theory, 83 (2) 282–296, 2000

[SM03] R. Salvati Manni. Modular forms vanishing on the hyperelliptic locus. Japan. J. Math. (N.S.), 29 (1) 135–142, 2003

[Tsu91] S. Tsuyumine. Thetanullwerte on a moduli space of curves and hyperelliptic loci. Mathematische Zeitschrift, 207 (4) :539–568, 1991.

[Wel86] G. E. Welters. The surface C − C on Jacobi varieties and 2nd order theta functions. Acta Math., 157(1-2):1–22, 1986.

Giulio Codogni Dipartimento di Matematica e Fisica Università degli Studi Roma Tre Largo S. Leonardo Murialdo 1 00146 Rome (Italy) codogni@mat.uniroma3.it