Vector-Valued Modular Forms from the Mumford Form, Schottky-Igusa Form, Product of Thetanullwerte and the Amazing Klein Formula

Vector-valued Siegel modular forms are the natural generalization of the classical elliptic modular forms as seen by studying the cohomology of the universal abelian variety. We show that for g>=4, a new class of vector-valued modular forms, defined on the Teichmuller space, naturally appears from the Mumford forms, a question directly related to the Schottky problem. In this framework we show that the discriminant of the quadric associated to the complex curves of genus 4 is proportional to the square root of the products of Thetanullwerte \chi_{68}, which is a proof of the recently rediscovered Klein `amazing formula'. Furthermore, it turns out that the coefficients of such a quadric are derivatives of the Schottky-Igusa form evaluated at the Jacobian locus, implying new theta relations involving the latter, \chi_{68} and the theta series corresponding to the even unimodular lattices E_8\oplus E_8 and D_{16}^+. We also find, for g=4, a functional relation between the singular component of the theta divisor and the Riemann period matrix.


Introduction
Here we shortly describe the results and introduce some simple (linear) algebraic facts that will be useful in the following and describe our notation for theta functions.
For each fixed positive integers g, n, define M n := g + n − 1 n , N n := (2n − 1)(g − 1) + δ n1 , K n := M n − N n , so that, for a complex curve C of genus g ≥ 2, M n and N n are the dimensions of Sym n H 0 (K C ) and H 0 (K n C ), respectively. Let be the Siegel upper half-space, i.e. the space of g × g complex symmetric matrices with positive definite imaginary part, and define the usual action of the symplectic group Γ g := Sp(2g, Z) on H g by Z → (AZ + B)(CZ + D) −1 , A B C D ∈ Γ g . Denote by M g the moduli stack of complex curves of genus g and by A g = Γ g \H g the moduli stack of principally polarized abelian g-folds.
According to Torelli's Theorem the morphism Jac : M g → A g , which on points takes the algebraic curves to its Jacobian, is injective. The question of characterizing the image of Jac is the Schottky problem. Let C g π −→ M g be the universal curve over M g and L n = Rπ * (K n Cg /Mg ) the vector bundle on M g of rank N n with fiber H 0 (K n C ) at the point of M g representing C. On A g it is the conormal bundle to the zero section in the universal abelian scheme; it pulls back via Jac to M g . A vector-valued Siegel modular form on A g is a global section of some tensor bundle of the Hodge vector bundle on A g , whose pull-back to M g via the Torelli map corresponds to L 1 . In spite of their relevance, e.g. in number theory, they have been studied essentially only in the case of genus two, where correspond to suitable commutators of Siegel modular forms. In Section 2 we will introduce a new set of vector-valued modular forms, defined on the Teichmüller space, strictly related to the Mumford forms, which are holomorphic global sections of the bundle F g = (det E g ) cn ⊗ (∧ Nn Sym n E g ) * , on M g , where E g ≡ L 1 and c n := 6n 2 − 6n + 1. Their weight is c n − g+n−1 n−1 . Such vector-valued Teichmüller modular forms provide a deep relation between Schottky's problem, quadrics describing C and θ-functions. The construction fits into the general problem of finding explicit formulae that relate theta nulls, which can be regarded as homogeneous coordinates on the moduli space of abelian varieties (with appropriate level structure), to projective invariants of curves, which are homogeneous coordinates on M g . Such formulae are also very interesting for physics, since they appear naturally in the computation of higher loop superstring amplitudes [1,2,3,4,5]. As a first application of the vector-valued Teichmüller modular forms, in Section 3 we consider the case g = 4. In particular, we will show that the discriminant of the quadric associated to C is proportional to the square root of the products of Thetanullwerte χ 68 , which is a proof of the recently rediscovered Klein "amazing formula". The relation with the Schottky's problem is due to the fact that the coefficients of such a quadric are derivatives with respect to the period matrix of the Schottky-Igusa form. This provides new theta relations involving the latter, χ 68 and the theta series corresponding to the even unimodular lattices E 8 ⊕ E 8 and D + 16 . Finally, we will find a functional relation between the singular component Θ s of the theta divisor and the Riemann period matrix and show that the hyperelliptic locus is a zero of order 10 for the Hessian of θ(e, Z), e ∈ Θ s . Given a basis v 1 , .
Mn the basis of the symmetrized tensor product Sym n V given by elements of the form with S n the group of permutations of n objects, taken with respect to an arbitrary ordering.
Proposition 1.1. Let V ∼ = C g be a g-dimensional complex vector space, and fix A ∈ GL(V ); then, the induced endomorphism on the 1-dimensional Consider the theta function with characteristics where z ∈ C g , Z ∈ H g and a, b ∈ R g . Let {α 1 , . . . , α g , β 1 , . . . , β g } be a symplectic basis of H 1 (C, Z). Denote by {ω i } 1≤i≤g the basis of H 0 (K C ) satisfying the normalization condition α i ω j = δ ij , and by τ ij := β i ω j ∈ H g the Riemann period matrix, i, j = 1, . . . , g. The basis of H 1 (C, Z) is determined up to the transformation which induces the following transformation on the period matrix For n ∈ Z, denote by J n (C) the principal homogeneous space of linear equivalence classes of divisors of degree n on C. The Jacobian J(C) := C g /L τ , L τ := Z g + τ Z g , is identified with J 0 (C): each point of J 0 (C) can be expressed as D 2 − D 1 , with D 1 and D 2 effective divisors of the same degree, which corresponds to D 2 D 1 ω ∈ J(C). Choose an arbitrary point p 0 ∈ C and let A(p) := (A 1 (p), . . . , A g (p)), A i (p) := p p 0 ω i , p ∈ C, be the Abel-Jacobi map. It embeds C into the Jacobian J 0 (C) and generalizes to a map from the space of divisors of C into J 0 (C) as A( i n i p i ) := i n i A(p i ), p i ∈ C, n i ∈ Z. By Jacobi Inversion Theorem the restriction of A to the space C g of divisors of degree g on C is a surjective map onto J 0 (C). Consider the vector of Riemann constants K p i : . . , g, for all p ∈ C. For any p ∈ C define the formal sum ∆ := (g − 1)p − K p so that, for any divisor ξ of degree g − 1 in C, ξ − ∆ is the point in C g given by ξ (g−1)p ω + K P . Under the projection C g → J 0 (C), ∆ becomes a distinguished point in J g−1 (C) depending only on the homological class of the marking (recall that a marking for C is given by fixing a canonical homotopy basis together with a basepoint p 0 ∈ C, see e.g. [6]). Furthemore, 2∆ = K C . We refer to [7] and [6] for further details.
We will also consider the prime form E(z, w) and the multi-valued g/2differential σ(z) on C with empty divisor, satisfying the property Such conditions fix σ(z) only up to a factor independent of z; the precise definition, to which we will refer, can be given, following [6], on the universal covering of C (see also [7]). Proposition 1.2. For each integer n, let φ n := {φ n i } 1≤i≤Nn be an arbitrary basis of H 0 (K n C ). Then for all p 1 , . . . , p g , y ∈ C, and for n ≥ 2, for all p 1 , . . . , p Nn ∈ C, depend only on the marking of C and on Proof. For each integer n, κ[φ n ] is a meromorphic function on C with empty divisor [7,8].

Vector-valued Teichmüller modular forms from the Mumford form
Let λ n := det L n be the determinant line bundle. According to Mumford [9] λ n ∼ = λ ⊗cn 1 , where c n := 6n 2 − 6n + 1. The Mumford form µ g,n is the unique, up to a constant, holomorphic section of λ n ⊗ λ −⊗cn 1 nowhere vanishing on M g . Comparing µ g,2 with the Polyakov measure for the bosonic string, Manin observed that c 2 = 13 in Mumford's formula coincides with the half of the string critical dimension. In a seminal paper [10] Belavin and Knizhnik proved that the Polyakov measure coincides with |µ g,2 | 2 . More generally −c n is the central charge of the chiral b − c system of conformal weight n [11].
Belavin and Knizhnik obtained µ g,2 from an expression for the curvature form of the determinant of Laplace operators. As observed in [12], this is a particular case of the similar formula for the determinants of Dirac operators on arbitrary compact manifolds, due to Bismut and Freed [13] (see also [14] and references therein). Such results lead to expressions in terms of complex geometry of the canonical curve C providing a link with the spectral invariants which appear using the formulae for the Laplace operator determinants by Ray and Singer [15] leading to sums over lengths of closed geodesics by means of the Selberg trace formula.
The expression of µ g,2 in terms of θ-functions has been derived in the context of string theory by Beilinson and Manin in [16].
The expression of µ g,n in terms of θ-functions, given by Verlinde and Verlinde [17] and Fay [6], follows immediately by (1.6) and (1.7). Nevertheless, it remains the hard question of expressing µ g,n without using points on C.
As we will see, there are exceptions for µ 2,2 and µ 3,2 . There is also a proof for the long-standing conjecture for µ 4,2 , here we shows that it follows immediately in the present context.
is its dependence on the basis {φ n i } 1≤i≤Nn : apparently, there is no natural choice for such a basis. In the case of µ 2,2 and µ 3,2 , since K 2 = 0 (i.e., Sym 2 H 0 (K C ) and H 0 (K 2 C ) have the same dimension), the natural choice is to use {ω i ω j } 1≤i,j≤g as basis of H 0 (K 2 C ). It turns out that such a choice reveals new interesting properties just in the case when K n > 0. Actually, the application of symmetric products of the ω i 's in this case introduces free vectorial indices and therefore leads to vector-valued modular forms defined on the Teichmüller space (see [18] for a very nice account on vector-valued Siegel modular forms) which are strictly related to the investigations in [8,19,20]. Remarkably, such a structure will also lead to a strict connection between Mumford forms, quadrics describing canonical curves, their discriminant and the Schottky problem. In particular, for g = 4 we will get some new results connecting the above structures to the Schottky-Igusa form, the theta series and the products of Thetanullwerte χ 68 , where Z ∈ H g , with k = 2 g−2 (2 g + 1). Let C be a Riemann surface of genus g ≥ 2 with a given symplectic basis for H 1 (C, Z). For each positive integer n, consider the basisω Mn of Sym n H 0 (K C ) whose elements, as in (1.1), are symmetrized tensor products of n-tuples of vectors of the basis ω 1 , . . . , ω g , taken with respect to an arbitrary ordering chosen once and for all. Denote by ω It is well known that such a map is surjective if and only if g = 2 or C is non-hyperelliptic of genus g > 2. Of course, when M n = N n the map is an isomorphism. In particular, for n = 2, this is the case for g = 2 and g = 3 non-hyperelliptic. It has been shown in [21,22,23] that for g = 2 κ[ω (2) ] = 1 π 12 χ 2 5 (τ ) . Furthermore, it has been conjectured in [21,22] and proved in [24,26] . . , g. Such a transformation property induces the Γ g -actionsρ (n) on Sym n H 0 (K C ) and ρ (n) := ψ •ρ (n) on H 0 (K n C ). Explicitly, Mn . .
[i 1 , . . . , i Kn |τ ] is independent of π and of x 1 , . . . , x Nn ∈ C; in fact, analogously to Proposition 1.2, one can check that it is a meromorphic function with zero divisor with respect to each x i (see [6,7]). This definition allows to express the generators of the kernel of the map ψ in (2.2) in terms of the basis ω Proof. If i 2 , . . . , i Kn are not pairwise distinct, this is obvious. Otherwise, the left hand side of (2.5) is proportional to where the sum is over the π in S Mn such that π(N n +2) = i 2 , . . . , π(M n ) = i Kn . Thus, (2.5) is equivalent to deti∈J If we identify a non-hyperelliptic smooth C with its canonical model in PH 0 (K C ), then the relations (2.5) generate the ideal of hypersurfaces of degree n containing the canonical curve.
Let H g be the closure of the locus of hyperelliptic Riemann period matrices in H g . Consider the basis vanishing precisely when ω (n) i Nn is not a basis of H 0 (K n C ). In particular, they have zeroes of order at least (n − 1)(g − 1) − 1 at τ ∈ H g .

Discriminant of the g = 4 quadric and the Schottky-Igusa modular form
Denotes by I g the closure of the locus of Riemann period matrices in H g .
The elements of such a locus can be naturally identified with the elements of the Torelli space T g . The Torelli space is the quotient of the Teichmüller space by the Torelli group, that is the normal subgroup of the mapping class group whose elements act trivially on H 1 (C, Z). Equivalently, the Torelli group is the kernel of the natural homomorphism of the mapping class group into the symplectic group Sp(2g, Z). Note that the hyperelliptic sublocus H g ⊂ H g can be naturally identified with the sublocus of T g corresponding to hyperelliptic Riemann surfaces. We denote byT g its complement in the Torelli space, whose elements correspond to non-hyperelliptic surfaces. The explicit expression of the coefficients of a quadric containing a canonical curve of genus g obviously depends on the choice of a coordinate basis of P g−1 or, equivalently, of a basis of H 0 (K C ). Therefore, it is natural to look for quantities characterizing such a curve that are invariant under the projective linear group PGL(g, C) of coordinate changes on P g−1 . We denote by I k an invariant of weight k, i.e. a function of the coefficients C ij of the quadric, transforming as where A is an element of GL(g, C), ρ : GL(g, C) → End(C g ) is the fundamental representation and A · C denotes the action of A ∈ GL(g, C) on the coefficients C ij .
If a symplectic basis of H 1 (C, Z) is fixed, such an invariant can be evaluated with respect to the basis of holomorphic abelian differentials canonically normalized with respect to the α-periods. It follows, by definition, that I k must transform as I k → I ′ k = det(Cτ + D) k I k , under a Γ g -transformation corresponding to a change of the symplectic basis of H 1 (C, Z).

4)
Z ∈ H g , is a modular form of weight 8. Up to normalization, it can be shown that F g is the difference of the theta series of the even unimodular lattices Proof. By (2.9), in the case n = 2 and g = 4 Lemma 3.2. Let C ∈ T 4 be a marked Riemann surface and let τ be its period matrix. Then

7)
i, j = 1, . . . , 4, with c ∈ C ⋆ independent of τ , so that Proof. Define some local coordinates t 1 , . . . , t 9 on T 4 centered at the point C, corresponding to the period matrix τ ∈ I 4 , and consider an arbitrary element ∂ t ∈ T C T 4 in the tangent space. Since F 4 vanishes identically on I 4 , we have Here, dτ ij is the element of the cotangent space T * C T 4 defined by dτ ij (∂ t ) := ∂ t τ ij , for all ∂ t ∈ T C T 4 . The Kodaira-Spencer map establishes an isomorphism between T * C T 4 and H 0 (K 2 C ). In particular, via the Rauch's variational formula [6], dτ ij corresponds to the quadratic differential ω i ω j . Since the identity (3.9) holds for an arbitrary ∂ t ∈ T C T 4 , it follows that i≤j ∂F 4 ∂Z ij Z=τ ω i ω j = as an element in H 0 (K 2 C ). Since the ideal of quadrics of a canonical curve is generated by Eq.(3.1), we have S 4ij (τ ) = c(τ ) 1+δ ij 2 [(ij)|τ ], for some holomorphic function c(τ ) on I 4 , independent of i, j = 1, . . . , 4. Let us prove that c(τ ) must be invariant under the action of Γ 4 on τ . Since F 4 (τ ) = 0 for all τ ∈ I 4 , it follows that on I 4 S 4 (γ · τ ) = det(Cτ + D) 8 t (Cτ + D)S 4 (τ )(Cτ + D) , (3.10) which is the same transformation property satisfied by [(ij)|τ ]. Thus, c is modular invariant, so it must be a constant. Finally, observe that c cannot vanish since it would imply ∂F 4 ∂Z ij (τ ) = 0 for all τ ∈ I 4 , which is impossible because F 4 is irreducible [28]. It is now useful to recall the following well known result. Proof. The modular form δ even θ[δ] vanishes if and only if C has an even singular spin structure δ. In this case, there are two holomorphic sections ξ 1 and ξ 2 of L δ , L 2 δ = K C , so that taking η 1 = ξ 2 1 , η 2 = ξ 2 2 and η 3 = ξ 1 ξ 2 one has η 2 3 = η 1 η 2 which is a quadric of rank 3 containing C.
Conversely, suppose that η 2 3 = η 1 η 2 for some η 1 , η 2 , η 3 ∈ H 0 (K C ). Set (η i ) = p∈C m i (p)p and consider the divisor D = p∈C min{m 1 (p), m 3 (p)}p. D has degree at most g−1, otherwise the ratio η 1 /η 3 would be a meromorphic function with at most g − 2 poles and the curve would be hyperelliptic for g = 4 or trigonal for g = 5. On the other hand, η 2 3 = η 1 η 2 implies that (η 1 ) ≤ 2(η 3 ) and since the supports of (η 1 ) − D and (η 3 ) − D are disjoint, the only possibility is that (η 1 ) = 2D. Therefore, η 1 is the square of a holomorphic section of the line bundle L δ , with L 2 δ = K C , corresponding to the divisor D. By the same reasoning it follows that η 2 is the square of a holomorphic section of the line bundle L δ ′ , with L 2 δ ′ = K C , corresponding to the divisor D ′ . Since (η 3 ) = D + D ′ is a canonical divisor it follows that δ ′ = δ. Then δ necessarily corresponds to an even singular spin structure, since a surface of genus g = 4, 5 admits no odd super-singular spin structures, that is spin structures with three or more holomorphic sections. where d ∈ C ⋆ is independent of τ . In particular, by Lemma 3.2, Furthermore, Proof. By (3.10) it follows that det S 4 (γ · τ ) = det(Cτ + D) 34 det S 4 (τ ) so that det S 4 is a modular form of weight 34 when restricted to I 4 . (Note that det S 4 is not a modular form on the whole H 4 , since the modular group action on S 4 is affine outside I 4 ). On the other hand, it has been proved in [29] that the square root of χ 68 in the RHS of (3.11) is well-defined when restricted to I 4 . Choose τ ∈ I 4 \ H 4 . By Lemma 3.2, the LHS of (3.11) is proportional to the discriminant. By Lemma 3.3, ∆ 4 (τ ) vanishes if and only if the Riemann surface has a singular even spin structure. The locus of Riemann surfaces with singular even spin structures in T 4 corresponds to the divisor of δ even θ[δ](0, τ ) in I 4 [29], so that the meromorphic function , has no poles on I 4 \ H 4 and therefore on I 4 , since H 4 has codimension 2.
Since it is a holomorphic modular invariant function not identically zero on I 4 , it must be a non-vanishing constant. Finally, (3.13) follows from (3.5) and (3.11).
When e ∈ Θ s is an even characteristic χ 68 (τ ) has at least a double zero, so that det S 4 (τ ) and by (3.14), det σ(e, τ ), has a zero of order at least one for all τ ∈ I 4 .
Remark 3.6. In [30] it has been re-obtained the Klein formula linking χ 18 to the square of the discriminant of plane quartics. The authors also studied possible generalizations to the case g > 3. In particular, in Eq.(3) of [30], it has been mentioned the following "amazing formula" by Klein in the footnote, p.462 in [31] χ 68 (τ ) = c ′∆ 4 (C) 2 T (C) 8 , with c ′ a constant. This formula relates χ 68 to the discriminant∆ 4 (C) of the Klein quadric and the tact invariant T (C) of the quadric and of the cubic (see pg.122 of [32]), whose intersection in P 3 determines C. Note that if 4 i,j=1 C ij ω i ω j = 0, denotes the Klein quadric, then withc ∈ C ⋆ independent of τ .
The following expression for µ 4,2 has been suggested in [33] and in [22] in the context of bosonic string theory. Its proof has been a long-standing problem and a more rigorous derivation has been provided in [34]. In the present approach it follows immediately.