Concomitants of Ternary Quartics and Vector-valued Siegel and Teichm\"uller Modular Forms of Genus Three

We show how one can use the representation theory of ternary quartics to construct all vector-valued Siegel modular forms and Teichm\"uller modular forms of degree 3. The relation between the order of vanishing of a concomitant on the locus of double conics and the order of vanishing of the corresponding modular form on the hyperelliptic locus plays an important role. We also determine the connection between Teichm\"uller cusp forms on \overline{M}_g and the middle cohomology of symplectic local systems on M_g. In genus 3, we make this explicit in a large number of cases.


Introduction
This paper contains two main results. Firstly, we show how the representation theory associated to ternary quartics can be used to describe and construct all vector-valued Siegel and Teichmüller modular forms of degree 3 (Theorem 11.6). This uses the classical notion of concomitants, of which invariants, covariants, and contravariants are special cases. Secondly, we describe for arbitrary g the precise relation between certain spaces of Teichmüller cusp forms on M g and the middle cohomology of the standard symplectic local systems on M g (Theorem 13.1). We illustrate the main results by a substantial number of examples, focusing on genus 3 for the second theorem, and obtain several other results of independent interest. Let A 3 be the moduli space of principally polarized abelian varieties of dimension 3. Over the complex numbers the orbifold A 3 (C), associated to the moduli space A 3 , can be written as an arithmetic quotient Γ 3 \H 3 , where the Siegel modular group Γ 3 = Sp(6, Z) of degree 3 acts on the Siegel upper half space H 3 of degree 3 in the usual way. The moduli stack A 3 carries a natural vector bundle E of rank 3, the Hodge bundle with fibre H 0 (X, Ω 1 X ) over the point [X] of A 3 . Over C it can be given as a quotient Γ 3 \(H 3 × C 3 ) where the action on C 3 corresponds to the standard representation of GL (3). In a similar way, for each irreducible representation ρ of GL (3) there is an associated vector bundle E ρ that can be constructed from E by using a Schur functor. Sections of powers of the determinant bundle det(E) on A 3 can be identified with scalar-valued Siegel modular forms, while sections of E ρ can be identified with vector-valued Siegel modular forms. The vector bundle E, and more generally all E ρ , extend in a natural way over the standard smooth compactification of A 3 . By the Koecher principle holomorphic sections of E ρ extend over this compactification.
Let M 3 denote the moduli space of curves of genus 3. The Torelli morphism t : M 3 → A 3 is a morphism of algebraic stacks of degree 2 ramified along the hyperelliptic locus. By pullback under t we obtain the Hodge bundle E ′ on M 3 and for each irreducible representation ρ of GL(3) a vector bundle E ′ ρ on M 3 . Sections of such a bundle E ′ ρ are called Teichmüller modular forms of degree 3. The vector bundle E ′ and hence all the E ′ ρ can be extended in a natural way over the Deligne-Mumford compactification M 3 and we show that a holomorphic section of E ′ ρ automatically extends to a holomorphic section of the extended bundle.
The first Teichmüller modular form that is not (a pullback under the Torelli map of) a Siegel modular form is the scalar-valued form χ 9 of weight 9 whose existence was proven by Ichikawa [24,25]. There is an involution on the space of Teichmüller forms such that a Teichmüller modular form F that is invariant is the pullback of a Siegel modular form, while an anti-invariant form is divisible by χ 9 , with quotient the pullback of a Siegel modular form. The study of Teichmüller modular forms of degree 3 reduces therefore to that of Siegel modular forms ( §7).
A nonhyperelliptic curve of genus 3 has as canonical image a quartic curve in P 2 and thus the open part M nh 3 of M 3 that corresponds to nonhyperelliptic curves has a description as the quotient of an open part of the space of ternary quartics under the action of GL (3). Thus the representation theory of ternary quartics enters, i.e., the decomposition of Sym d (Sym 4 (C 3 )) into irreducible representations of GL (3). The classical notion of concomitants of ternary quartics makes this decomposition explicit (see [9]). Our first main result is a complete description of Siegel (and Teichmüller) modular forms of degree 3 in terms of concomitants. We associate to a concomitant a meromorphic Siegel or Teichmüller modular form which is holomorphic outside the hyperelliptic locus on M 3 or A 3 (see §10). This is analogous to our description of Siegel modular forms of degree 2 in terms of covariants of the action of GL(2) on the space of binary sextics, see [10].
The fact that in general concomitants define meromorphic modular forms that become holomorphic after multiplication with a suitable power of χ 9 forces us to analyze the order of vanishing of the modular form associated to a concomitant along the hyperelliptic locus. We express this order of vanishing in terms of the order of vanishing of the concomitant along the locus of double conics in the space of ternary quartics. Remarkably, this enables us to identify the spaces of concomitants of ternary quartics with given order of vanishing along the locus of double conics with the spaces of Siegel modular forms with given order of vanishing along the divisor at infinity (Theorem 11.6).
Instead of working with χ 4,0,−1 and multiplying with a power of χ 9 , we can also work with χ 4,0,8 and obtain from a concomitant a holomorphic Siegel modular form, which may be divisible by a power of χ 9 . In order to use this in an efficient way, we need to know the Fourier expansion of χ 4,0,8 rather well. We obtain it by analyzing the Schottky form, a scalar-valued Siegel cusp form of degree 4 and weight 8, along the 'diagonally embedded' H 3 × H 1 ⊂ H 4 in the Siegel upper half space of degree 4 (see § §4-5).
To demonstrate our approach, we construct a substantial number of Siegel cusp forms and we compute some of their Hecke eigenvalues, finding agreement with [5].
In [4], Bergström and two of the present authors studied the cohomology of symplectic local systems on A 3 . The same method can also be used to study the cohomology of the corresponding local systems on M 3 . (As is well-known, this cohomology is very closely related to the cohomology of the moduli spaces M 3,n of n-pointed curves of genus 3.) If the local system is even, its cohomology on M 3 equals that on t(M 3 ) = A 3 − A 2,1 , the moduli space of indecomposable principally polarized abelian threefolds. However, in the odd case, the cohomology can not be explained in terms of A 3 . It is here that Teichmüller modular forms enter. Our second main result (Theorem 13.1) gives the precise relationship between certain spaces of Teichmüller cusp forms and the middle cohomology of symplectic local systems on M g . This is a partial analogue of the results of Faltings and Chai [17] for A g . We then specialize to genus 3, where we have determined these spaces of Teichmüller cusp forms in a substantial number of cases, via computations with concomitants. Finally, we discuss how these results match perfectly with conjectural formulas for the 'motivic' Euler characteristics of the symplectic local systems of weight at most 20 on M 3 , obtained from counts of curves over finite fields, as in [4]. of degree g in the usual way: The moduli space A g carries a rank g vector bundle, the Hodge bundle E. The induced bundle on A g (C) corresponds to the factor of automorphy For an irreducible representation ρ of GL(g, C) of highest weight (ρ 1 , . . . , ρ g ) with ρ 1 ≥ ρ 2 ≥ . . . ≥ ρ g , we have a corresponding vector bundle E ρ on A g ; the Hodge bundle corresponds to the standard representation with highest weight (1, 0, . . . , 0) and its determinant has highest weight (1, . . . , 1). If ρ : GL(g, C) → GL(W ) is a finite-dimensional complex representation, the induced vector bundle on A g (C) is defined by the factor of automorphy j(γ, τ ) = ρ(cτ + d) .
A scalar-valued Siegel modular form of degree g > 1 and weight k is a holomorphic function f : for all γ ∈ Γ g , while for g = 1 we also need a growth condition at infinity. If W is a finite-dimensional complex vector space and ρ : GL(g, C) → GL(W ) a representation, then a vector-valued Siegel modular form of degree g > 1 and weight ρ is a holomorphic map f : Siegel modular forms of weight ρ can be interpreted as sections of the vector bundle E ρ , and conversely. Sections of the kth power L k of the determinant line bundle L = det(E) correspond to scalar-valued Siegel modular forms of degree g and weight k. The vector bundle E and the bundles E ρ extend in a canonical way to Faltings-Chai type toroidal compactifications of A g and the Koecher principle says that their sections do so too.
A Siegel modular form F of weight ρ admits a Fourier expansion F = n a(n) q n with q n = e 2πi Tr(nτ ) , where n runs over the half-integral symmetric positive semi-definite g × g matrices and a(n) ∈ W . (Half-integral means that 2n is integral with even entries on the diagonal.) We are interested in the case g = 3. For an irreducible representation ρ of highest weight (ρ 1 , ρ 2 , ρ 3 ) of GL(3, C) we denote the weight of the corresponding Siegel modular forms by The vector space of Siegel modular forms of weight (i, j, k) on Γ 3 is denoted by M i,j,k . The space of cusp forms is denoted by S i,j,k . For scalar-valued Siegel modular forms we abbreviate the weight (0, 0, k) often by k.
Scalar-valued Siegel modular forms of degree 3 form a graded ring: Vector-valued modular forms of degree 3 form a graded module M = ⊕ i,j,k M i,j,k over the ring R. The ring R was described by Tsuyumine [41]. He gave 34 generators and the generating function of R. His work used results by Igusa [29] and Shioda [39]. Igusa showed that there is an exact sequence where χ 18 is a cusp form of weight 18 (see Section 3) and I (2,8) is the ring of invariants of binary octics. The map r is induced by the restriction map to the zero locus in A 3 of χ 18 . This locus is the closure of the image of the hyperelliptic locus H 3 under the Torelli map. Shioda determined the ring of invariants of binary octics [39]. In the recent paper [28], Lercier and Ritzenthaler reduce the number of generators of R to 19.
For Siegel modular forms, both the notion of degree and that of genus are used; we tend to use degree for Siegel modular forms and genus for Teichmüller modular forms, but are not strict in this respect.

The Scalar-valued Siegel Modular Form χ 18
The scalar-valued Siegel modular form χ 18 of degree 3 is up to a normalization defined as the product of the 36 even theta constants. It is a cusp form of weight 18. Its Fourier expansion starts as where we use q j = e 2πiτ jj , u = e 2πiτ 12 , v = e 2πiτ 13 and w = e 2πiτ 23 , and σ i is the ith elementary symmetric function in u, v, w. Thus χ 18 vanishes with multiplicity 2 at infinity. It is well-known that the divisor of χ 18 in the toroidal com-pactificationÃ 3 is H + 2D with H the hyperelliptic locus and D the divisor ofÃ 3 at infinity: div(χ 18 ) = H + 2 D .
(3) We now show that χ 18 is up to a scalar the unique cusp form of weight 18 vanishing twice at infinity. Proof. The dimension of S 0,0,18 is 4. In order to construct a basis we consider the Eisenstein series E 4 and E 6 whose Fourier expansions start as and cusp forms F 12 and F 14 of weight 12 and 14; note that dim S 0,0,12 = 1 = dim S 0,0,14 (cf. [41]). We normalize these forms such that the following table gives their first Fourier coefficients: Here we write the Fourier series of a modular form as before as n≥0 a(n) q n (using q n = e 2πiTr(nτ ) ) with n running over the half-integral positive semi-definite symmetric matrices; A i refers to the Gram matrix of the standard root lattice. Recall that for a scalar-valued modular form we have a(n) = a(u t nu) for all u ∈ GL(3, Z). We look at the induced action of GL(3, Z). The set N of half-integral positive definite symmetric matrices with 1's on the diagonal contains elements from three distinct orbits: the orbit of 1 3 (one element), the orbit of A 1 (1/2) ⊕ A 2 (1/2) (sixteen elements), and the orbit of A 3 (1/2) (six elements). This implies that we can read off the Fourier coefficient of q 1 q 2 q 3 of F 12 and F 14 from the table above. We claim that a basis of S 0,0,18 is given by with T 2 the Hecke operator of the prime 2. We can calculate the Fourier coefficient of χ 18 |T 2 using [12]. We get the following Fourier coefficients: This shows that the four cusp forms of weight 18 are linearly independent and that χ 18 is up to a nonzero scalar the unique cusp form of weight 18 that vanishes twice at infinity.
Remark 3.2. In §11, we will obtain a considerably more general result, with a different method of proof (Thm. 11.6 and Cor. 11.8). The modular form generating S 4,0,8 is the unique nonzero cusp form (up to scalar) for which i + 2j + 3k is minimal (equal to 28) (cf. [40]). This form χ 4,0,8 will play a central role in this paper. Its Fourier expansion can be obtained as in [12] by taking the Schottky form of weight 8 on Γ 4 and developing it in the normal directions to H 1 × H 3 in H 4 . The lowest order term gives a nonzero multiple of ∆ ⊗ χ 4,0,8 in S 12 (Γ 1 ) ⊗ S 4,0,8 (Γ 3 ).
We normalize χ 4,0,8 so that its Fourier expansion starts as follows where we use the same variables as in (2). This modular form is a lift of ∆ = τ (n)q n ∈ S 12 (Γ 1 ) and its Hecke eigenvalues are by the results of [4] predicted to be Another way to see this is by looking at the expansion of χ 4,0,8 given in [12, p. 39]. We thus see that the coordinates of χ 4,0,8 lie in I 2 , but not all in I 3 .
We can trivialize the pullback of E to H 3 as H 3 × C 3 . We choose coordinates z 1 , z 2 , z 3 on C 3 . Since the pullback of Sym 2 (E) can be identified with the cotangent bundle of H 3 the coordinates τ ij correspond to z i z j . In particular, a basis of Sym 4 (E) corresponds to the monomials of degree 4 in z 1 , z 2 and z 3 . Therefore, we can write χ 4,0,8 as where for a multi-index I = (i 1 , i 2 , i 3 ) we inserted a normalization factor n I = 4!/i 1 !i 2 !i 3 !. We take the sum over the lexicographically ordered monomials z I = z i 1 1 z i 2 2 z i 3 3 of degree 4 and α I is a holomorphic function on H 3 which we can present as a Fourier series living in The z I are just dummy variables to indicate the coordinates n I α I of the vector-valued modular form χ 4,0,8 .
The symmetric group S 3 acts on C 3 by permuting z 1 , z 2 , z 3 and hence induces an This action of S 3 derives from an action of a subgroup of Γ 3 by taking a 3×3 permutation matrix a = (a −1 ) t and using ( a 0 0 a ). To give the action on the Fourier expansion, note that we have an induced action of S 3 on q 1 , q 2 , q 3 and on u, v, w.
The action on the fifteen coordinates has one orbit of length 6 and three orbits of length 3. The reader may check that the action of (23) ∈ S 3 on Sym 4 (E) is given by sending the transpose of v I = (v 1 , v 2 , . . . , v 15 ) to the transpose of and that of (13) ∈ S 3 is given by For χ 4,0,8 it thus suffices to give the coordinates v 1 , v 2 , v 4 and v 5 only; the other coordinates can then be calculated by using the action of Lemma 4.2.

The Fourier-Jacobi Expansion of the Schottky Form
In the preceding section the form χ 4,0,8 was defined by developing the Schottky form of degree 4 and weight 8 along H 1 × H 3 and observing that the first nonzero term is ∆ ⊗ χ 4,0,8 in S 12 (Γ 1 ) ⊗ S 4,0,8 (Γ 3 ). For our application we need to be able to calculate the Fourier expansion of χ 4,0,8 quite far and for that we need the first Fourier-Jacobi coefficient of the scalar-valued form J 8 .
Recall that the Schottky form J 8 can be expressed as follows: where we use the theta series with characteristics [ a b ] with a and b row vectors of size g ∈ Z 1 with coordinates in Z: l∈Z g e πi(l+a/2)τ (l+a/2) t e 2πi(l+a/2)·(z+b/2) t , τ ∈ H g , z = (z 1 , . . . , z g ) ∈ C g .
When z = 0, we denote ϑ [ a b ] (τ, 0) simply by ϑ [ a b ] (τ ). We thus need to calculate the first Fourier-Jacobi coefficient of these theta functions. For any τ = (τ ij ) ∈ H 4 we write and we abbreviate τ ii as τ i . We write the Fourier-Jacobi expansion of ϑ µ 0 0 0 So we get where q 4 = e 2πiτ 4 . These first terms correspond to l = 0 and l = ±1 in the formula (5) and the next term is given by l = ±2 which gives rise to O(q 2 4 ). Since we are dealing with even characteristics, the latter formula simplifies to We introduce for τ ∈ H 3 and z ∈ C 3 the notation By using formulas (5) and (5 ′ ) and this notation we get the following result.
Recall now that we write χ 4,0,8 as n I α I z I where the sum is over the fifteen lexicographically ordered multi-indices of the monomials of degree 4 in the three variables z 1 , z 2 , z 3 and each α I is a holomorphic function on H 3 .
As a check on our calculations we use the fact that the Hecke eigenvalues of χ 4,0,8 for a prime p are by the results of [4] predicted to be . By applying the explicit formulas for the Hecke operators for Γ 3 given in [12] we can calculate the Fourier coefficients a(n) and hence the eigenvalues for small p; for example with the notation used in [12, p In this section we derive another expression for χ 4,0,8 that connects it to the geometry of curves of genus three. For a principally polarized abelian variety (X, Θ) of dimension g we consider the space where m 0 means the order of vanishing at the origin, see [20]. For a principally polarized abelian variety (X, Θ) that is not decomposable (that is, not a product of nontrivial lower-dimensional principally polarized abelian subvarieties) it is known that For an indecomposable principally polarized abelian threefold the space Γ 00 has dimension 1. In [19] Frobenius constructed a non-zero element of Γ 00 by defining a function ϕ(τ, z) on H 3 × C 3 . A somewhat different construction, leading to the same result, is as follows. Let Γ 3 [2,4] be the usual congruence subgroup of Γ 3 of level (2, 4). The theta functions of the second order Θ σ (τ, 0) on H 3 define a morphism which is an injective map and an immersion along the locus A 0 3 [2,4] of indecomposable abelian varieties, see [37,Thm. 3.2]. The closure of the image is a hypersurface given by an explicit equation F (. . . , x σ , . . .) = 0 of degree 16 in the coordinates x σ of P 7 , see [20, p. 632]. Then the expression equals up to a non-zero multiplicative constant the function constructed by Frobenius, see [20, p. 624]. In fact, the functions ϕ(τ, z) and φ(τ, z) differ multiplicatively by a function that depends only on τ and is invariant under Γ 3 [2,4] and descends to a holomorphic function on A 0 3 [2,4] and thus is constant since the locus of decomposable abelian threefolds has codimension 2. For a τ that corresponds to the periods of a smooth projective nonhyperelliptic curve Y the first term of φ in the Taylor expansion as a function in z 1 , z 2 , z 3 gives the quartic polynomial that defines the canonical image of Y as Frobenius showed [19, p. 37]. In fact, the zero locus of φ on the abelian variety Jac( the diagonal is contracted and the tangent cone to the singularity is the cone over the canonical image of Y , see [20]. It follows from the properties of the theta functions Θ σ and the fact that φ vanishes to order 4 along the zero section in the universal abelian threefold X 3 over A 3 that the degree 4 term in the normal expansion of φ along the zero section defines a section of Sym 4 (E) ⊗ det 8 (E). Proof. Over the locus of indecomposable abelian threefolds the section φ vanishes along the zero section of X 3 → A 3 to order 4. Since its zero locus for a smooth curve Y is the surface {(x − y) : x, y ∈ Y } it vanishes with exact order 4. Therefore the first term in the development is a non-zero section of Sym 4 (E) ⊗ det 8 (E) over this locus. Since the complement has codimension two, the section extends over all of A 3 to give a holomorphic modular form. It vanishes on the locus of decomposable abelian varieties, hence it defines a cusp form.

Teichmüller Modular Forms
The Torelli map is a morphism t = t g : M g → A g of Deligne-Mumford stacks obtained by associating to a smooth projective curve of genus g its Jacobian. It can be extended to a morphism from the Deligne-Mumford compactification M g to the toroidal compactification defined by the second Voronoi fan [33]. In the case of genus 3 this compactificationÃ 3 is the standard compactification of A 3 (cf. [14]).
The moduli space M 3 carries a vector bundle E ′ of rank 3, the Hodge bundle. It is the pullback under the Torelli morphism of the Hodge bundle E onÃ 3 . Moreover, for each irreducible representation ρ of GL(3) of highest weight (ρ 1 , ρ 2 , ρ 3 ) we have the vector bundle E ′ ρ on M 3 , again pulled back fromÃ 3 . A Teichmüller modular form of weight w(ρ) = (ρ 1 − ρ 2 , ρ 2 − ρ 3 , ρ 3 ) and degree 3 is a section of E ′ ρ on M 3 . In Prop. 7.3 below, we show that a Teichmüller modular form extends to M 3 . We denote the space of Teichmüller modular forms of weight w(ρ) by T w(ρ) . If w(ρ) = (0, 0, k) we are dealing with scalar-valued Teichmüller modular forms (of weight k). Scalar-valued Teichmüller modular forms form a ring T = ⊕ k T 0,0,k . Ichikawa showed that there exists a Teichmüller modular form χ 9 of weight (0, 0, 9) on M 3 . It is a square root of the pullback of χ 18 . In particular, χ 9 vanishes on the hyperelliptic locus, with class h ∼ 9λ − δ 0 − 3δ 1 . The divisor of χ 9 is therefore known: Ichikawa also showed that the ring T is generated over the ring R of scalar-valued Siegel modular forms of degree 3 by χ 9 . See [24,25,26,27]. The map M 3 → A 3 of stacks is a double cover of its image, the locus of indecomposable abelian threefolds. It is ramified along the hyperelliptic locus. The covering involution ι can be understood by using a fine moduli space M 3 [N] of curves with level-N structure (N ≥ 3): it sends the isomorphism class of a pair (C, α) of a curve C and a level structure α to the class of (C, −α). Note that (A, α) ∼ = (A, −α) for a polarized abelian variety A.
Further, ι preserves the Hodge bundle E ′ and acts as −1 on the fiber. Hence the bundle E ′ ρ is preserved as well and the action on the fiber is as ( Call ρ odd (resp. even) when ρ 1 + ρ 2 + ρ 3 is odd (resp. even). Let us also call a Teichmüller (or Siegel) modular form of weight w(ρ) odd (resp. even) when ρ is odd (resp. even). It is clear that an odd Siegel modular form is identically zero and that an odd Teichmüller modular form vanishes on the hyperelliptic locus. Further, a Siegel modular form of weight w(ρ) pulls back to a Teichmüller modular form of the same weight. Moreover, we have the following result. Proof. Note that the complement of the image of M 3 in A 3 is A 2,1 , the locus of products of principally polarized abelian varieties, of codimension two in A 3 . An even Teichmüller modular form of weight w(ρ) is invariant under ι and descends to a section of E ρ over t(M 3 ). Since A 3 is a smooth stack and the codimension of A 2,1 is two, this section extends to A 3 and yields the Siegel modular form that pulls back to the Teichmüller modular form.
We also have the following extension of Ichikawa's result.
Lemma 7.2. An odd Teichmüller modular form is divisible by χ 9 , i.e., it is the product of χ 9 and an even Teichmüller modular form.
Proof. Let f be an odd Teichmüller modular form. Consider g = f /χ 9 , a priori a meromorphic Teichmüller modular form with possibly a simple pole along H 3 . Since f vanishes on H 3 , it follows that g is a regular Teichmüller modular form, which clearly is even. (So g is the pullback of a Siegel modular form.) A Teichmüller modular form extends holomorphically to M 3 : 3. Let f be a Teichmüller modular form of weight w(ρ). Then f extends to a holomorphic section of E ′ ρ on M 3 . Proof. If f is even, then it is the pullback of a Siegel modular form F . By the Koecher principle, F extends toÃ 3 . The pullback of the extension gives the extension of f to M 3 . If f is odd, it is the product of χ 9 and an even Teichmüller modular form. Both extend, so f extends as well.
Corollary 7.4. An odd Teichmüller modular form is a cusp form, i.e., it vanishes along δ 0 . It also vanishes with multiplicity at least three along δ 1 . If it is nonzero and of weight w(ρ), then ρ 3 ≥ 9.
Consider the modular form χ 4,0,8 . We define a Siegel modular cusp form of weight (4i, 0, 8i) by taking the ith symmetric power of χ 4,0,8 and projecting:  Proof. The vanishing orders of χ 18 along D and H were given in Section 3, the order along A 2,1 is given in [12,Prop. 5.2]. If the form χ (2) would vanish on the hyperelliptic locus, we could divide it by χ 18 , obtaining a holomorphic Siegel modular form of weight (8, 0, −2), which has to be zero. So χ (2) does not vanish along H. It follows that ord H χ 4,0,8 = 0. For the remaining entries, see §4, in particular Lemma 4.1.
Note that we order the monomials lexicographically and normalize x a y b z c by the factor 4!/a! b! c!.
An element A = (a ij ) of the group GL(3, C) acts on Sym 4 (V ) via f (x, y, z) → f (a 11 x + a 21 y + a 31 z, a 12 x + a 22 y + a 32 z, a 13 x + a 23 y + a 33 z) .
We also consider the induced actions of PGL(3, C) and SL(3, C) on Q = P(Sym 4 (V )). We take (a 0 , a 1 , . . . , a 14 ) as coordinates on Sym 4 (V ) and Q. The natural ample line bundle L = O Q (1) on Q admits an action of SL(3, C) compatible with the action on the projectivized space Q of ternary quartics, cf. [32].
Before we come to the notion of concomitant, we will fix coordinates on ∧ 2 V . We takê then a basis of Sym i (V )⊗Sym j (∧ 2 V ) is given by the lexicographically ordered monomials of degree i in x, y, z and degree j inx,ŷ,ẑ. We recall the notion of concomitants for ternary quartics. Write W = V ∨ . Consider an equivariant inclusion of SL(3, C)-representations ϕ : A → Sym d (Sym 4 (W )), or equivalently C ֒→ Sym d (Sym 4 (W )) ⊗ A ∨ , where C denotes the trivial representation; write Φ for the image of 1 under this map. If A = W [ρ] is the irreducible representation of highest weight ρ we call Φ a concomitant of type (d, ρ). It can be viewed as a form of degree d in the coordinates a i of the ternary quartic and of degree ρ 1 − ρ 2 (resp. ρ 2 − ρ 3 ) in x, y, z (resp.x,ŷ,ẑ). In essence, we follow here Chipalkatti's set-up in [9], but the notation is the dual one and our basis of Sym 4 (V ) includes the multinomial coefficients. Also, although we work with SL(3, C) here, we have used the notation introduced earlier for the irreducible representations of GL(3, C), since we will soon need to work with the latter group.
The simplest nontrivial example is the case where d = 1 and A = Sym 4 (W ) and the inclusion is the identity. Then the corresponding concomitant Φ is the universal ternary quartic f given in the beginning of this section.
For d = 2 we have the decomposition The concomitants corresponding to the first and last isotypical component are f 2 and a classical concomitant denoted σ by Salmon (see [36, p. 264]). It is a quartic polynomial inx,ŷ,ẑ with coefficients of degree 2 in the a i , see Section 12.
The concomitants form a module C over the ring I of invariants.

Quartic Curves and Moduli
Let To describe M nh 3 we consider a 3-dimensional vector space V over a field k, say generated by x, y, z. We may view Sym 4 (V ) as the space of ternary quartics. There is a natural left action of GL(V ) on Sym 4 (V ): view f ∈ Sym 4 (V ) as a homogeneous polynomial map from the dual V ∨ to k of degree 4; the action of A ∈ GL(V ) on f is defined as be the partial compactification of M nh 3 consisting of nonhyperelliptic stable curves of genus 3 with at most one node, which is nondisconnecting. Essentially the same proof as above gives: Proposition 9.2. There exists an isomorphism of algebraic stacks For f ∈ U ′ , the elements x, y, z give a basis of the space of global sections of the dualizing sheaf on the quartic in P 2 defined by f . This globalizes and we obtain the following result. Hence the pullback under q of Sym 4 (E ′ ) ⊗ det −1 (E ′ ) is the GL(V )-equivariant bundle U ′ × Q, which admits the diagonal section, the universal (at most 1-nodal) ternary quartic.
Corollary 9.4. The diagonal section of U ′ × Q descends to a canonical section of We revisit the notion of concomitants for ternary quartics. Working with GL(3, C) instead of SL(3, C), it is best to work with Q or its dual R, as explained above. We consider an equivariant inclusion of GL(3, C)-representations where C denotes the trivial representation; write Φ for the image of 1 under this map. If A = W [ρ] is the irreducible representation of highest weight ρ we call Φ a concomitant of type (d, ρ). It can be viewed as a form of degree d in the coordinates a i of the twisted ternary quartic and of degree ρ 1 − ρ 2 (resp. ρ 2 − ρ 3 , resp. ρ 3 ) in x, y, z (resp. x,ŷ,ẑ, resp. x ∧ y ∧ z). Note that d = ρ 1 + ρ 2 + ρ 3 , so d is determined by ρ in the GL(3, C)-setting. Sometimes, we will speak of concomitants of type ρ.

Invariants, Concomitants and Modular Forms
We consider a scalar-valued Teichmüller modular form F of weight k, that is, a section of det k (E ′ ) on M 3 , cf. Prop. 7.3. By Corollary 9.3, the pullback of det k (E ′ ) to U ′ under q is det k (V ). The pullback of F is a section of det k (V ) over U ′ that extends to a section of det k (V ) over Q (since the complement of U ′ has codimension two). The corresponding irreducible representation of GL(V ) has highest weight (k, k, k); it occurs in Sym 3k (Q). Thus F defines an invariant ι F ∈ I of degree 3k (for SL (3, C)).
Conversely, if we have a homogeneous invariant, necessarily of degree 3k with k ∈ N, it gives a section of det k (V ). It descends to a holomorphic section of det k (E ′ ) over the open subset M nh 3 that extends to a meromorphic section over M 3 . Since χ 9 vanishes exactly once on the hyperelliptic locus, we conclude that we get inclusions with T the graded ring of scalar-valued Teichmüller modular forms and T χ 9 the ring obtained by inverting χ 9 . Under the first map above, the modular form χ 9 maps to (a nonzero multiple of) the discriminant, an invariant of degree 27. (In the recent paper [28], Lercier and Ritzenthaler study the relation between scalar-valued Siegel modular forms and invariants of ternary quartics in detail. Their approach differs from ours.) We now generalize this to concomitants for GL(3, C). A concomitant Φ of type (d, ρ) gives an equivariant section of V ρ . It descends to a holomorphic section of E ′ ρ on M nh 3 that extends to a meromorphic section γ(Φ) on M 3 . After multiplication with the rth power of χ 9 for r large enough, it becomes a holomorphic section of E ′ ρ ⊗ det 9r (E ′ ) (as we shall see r = d suffices). Conversely, if F is a Teichmüller modular form of weight ρ, that is, a section of E ′ ρ , then by pulling it back from M nh, * 3 to U ′ we get an equivariant section of V ρ that extends to Q: a concomitant β(F ) of type ρ.
We thus find a commutative diagram where Σ = ⊕ ρ T ρ is the T -module of vector-valued Teichmüller modular forms (with ρ running over the irreducible representations of GL(3, C)) and C is the I-module of concomitants. A modular form F of weight (i, j, k) is sent to a concomitant of type and a concomitant of type (d, ρ) to a (meromorphic) form of weight Note that the parity of the form equals the parity of the degree d of the concomitant. A basic observation is the following.
Then each α I is a holomorphic function on H 3 or also on Teichmüller space T 3 . The map γ is given by substituting in a concomitant the meromorphic functions α I /χ 9 on T 3 for the a I . Sometimes we prefer to work with holomorphic modular forms. Then we don't use the map γ, but a slightly adapted map γ ′ that maps the tautological f to γ(f )χ 9 and that is defined by substituting in a concomitant c of degree d the α I for the a I . The result is a holomorphic vector-valued modular form γ ′ (c) = γ(c)χ d 9 = c • χ 4,0,8 . For a concomitant of type (d, ρ) it is a form of weight (ρ 1 − ρ 2 , ρ 2 − ρ 3 , ρ 3 + 9d), i.e., a section of E ρ ⊗ det 9d E. We can calculate the Fourier expansion of γ ′ (c) from the Fourier expansion of χ 4,0,8 .
Depending on the parity of ρ, the form γ(c) is a (meromorphic) Siegel or Teichmüller modular form, but γ ′ (c) is (the pullback of) a holomorphic Siegel modular form. If it vanishes along the hyperelliptic locus, we can divide it by χ 18 . Note that the order of divisibility by χ 18 is bounded above by 1/2 of the order along δ 0 and also by 1/6 of the order along δ 1 .

The Order along the Locus of Double Conics
Inside the space P 14 of ternary quartics there is the locus DC of double conics. The order of a concomitant c of ternary quartics along the locus DC is determined as the order in the parameter t ∈ C of the evaluation of c on the quartic tf + g 2 , where f is a general ternary quartic and g a general ternary conic. Proof. The proof is completely analogous to that of Thm. 1 in [11], so we only mention the necessary changes. The meromorphic form γ(c) is obtained by substituting χ 4,0,−1 in c and χ 4,0,−1 has a simple pole along H 3 , so the result holds for the universal f (with d = 1 and v = 0). After multiplying with f or f 2 if necessary, we may assume that d is divisible by three, equal to 3e. We now let A be the invariant of degree 3: A = a 0 a 10 a 14 − 4a 0 a 11 a 13 + 3a 0 a 2 12 − 4a 1 a 6 a 14 + 12a 1 a 7 a 13 − 12a 1 a 8 a 12 + . . . . One easily checks that A doesn't vanish along DC, so γ(A) has a pole of order 3 along   Proof. This follows from the proposition, since the composition γ • β of the maps in diagram (10) above is the identity map onto Σ ⊂ Σ χ 9 . That the parities agree was already known: it is obvious in the even case and follows in the odd case from the fact that χ 9 has multiplicity 1 along H 3 .
Notation 11.5. Let m ≥ 0 be an integer. We denote by C d,ρ (−m DC) the vector space of concomitants of type (d, ρ) that have order ≥ m along the locus DC of double conics.
We denote by S m i,j,k the vector space of Siegel modular forms of type (i, j, k) that have order ≥ m along the boundary divisor D. So S 0 i,j,k = M i,j,k and S 1 i,j,k = S i,j,k .
These notations also make sense when m < 0, but in most cases this doesn't give anything new, since we consider regular or holomorphic concomitants and modular forms here. However, see Corollary 11.8 below.
Theorem 11.6. Notation as above. There exists an isomorphism Proof. Let c ∈ C d,ρ (−mDC). By Proposition 11.1, the order along H 3 of γ(c) is at least 2m − d. So ϕ(c) is regular along H 3 . Since d and n have the same parity, ϕ(c) is (the pullback of) a Siegel modular form. It follows immediately that ϕ(c) ∈ S n ρ 1 −ρ 2 ,ρ 2 −ρ 3 ,ρ 3 +9n . Now ϕ is certainly injective, but also surjective: take F ∈ S n ρ 1 −ρ 2 ,ρ 2 −ρ 3 ,ρ 3 +9n , then β(F )/Ξ n is a meromorphic concomitant that has order at least m along DC and order at least 0 along the locus Ξ = 0 of singular quartics. So it is in fact a regular concomitant and an element of C d,ρ (−mDC).
(2) Let d = 2 and take ρ = (4, 0, −2) and m = 0, so n = 2. We get an identification between the space of concomitants of degree 2 and type (4, 0, −2) and the space S There are seven other nonzero concomitants of degree 3; they don't vanish along DC. Taking m = 0, we find nine one-dimensional spaces of cusp forms with order at least three along D (two from the cusp forms just mentioned). Finally, seventeen spaces S i,j,k+6 and ten spaces S 3 i,j,k+24 with i + 2j + 3k = 12 vanish. (4) Take ρ = (5, 4, −4), so d = 5, and m = 2, so n = 1. There is an isomorphism C 5,(10,9,1) (−2DC) ∼ = S 1,8,5 . The space S 1,8,5 is one-dimensional and a generator of it has been constructed in the paper of Ibukiyama and Takemori [23]. There is a unique concomitant of type ρ and indeed it vanishes to order 2 along DC. We give more details in Section 12.3. (5) For d = 27 and ρ = (9,9,9), take m = 14, so n = −1. On the right, the space S −1 0,0,0 is just C. Hence the space of invariants of degree 27 with order at least 14 along DC is one-dimensional; it is generated by the discriminant Ξ, which is mapped to 1, since γ(Ξ) = χ 9 .

Constructing Modular Forms from Concomitants
12.1. Degree 1 and 2. In this section, we will use the SL(3, C)-notation for concomitants as in §8, mainly because it is easier to read. As long as the degree d is known, this should not lead to confusion. As mentioned in §9, the GL(3, C)-notation for a concomitant of degree d is obtained by lowering the three entries by d.
There is one concomitant of degree d = 1, the universal ternary quartic. As we have seen, the image under the map γ ′ of the basic concomitant f is χ 4,0,8 .
Under the map γ ′ the three concomitants corresponding to the three terms in the decomposition of Sym 2 (Sym 4 (W )) give rise to Siegel modular forms of weights χ 8,0,16 , χ 4,2,16 and χ 0,4,16 respectively. The form χ 8,0,16 is the (symmetric) square of χ 4,0,8 . Since χ 4,0,8 vanishes with multiplicity 1 along D, these three modular forms vanish with multiplicity ≥ 2 along D. In order to give the beginning of the Fourier expansion of χ 0,4,16 , we notice that it suffices to give the coordinates v 1 , v 2 , v 4 and v 5 , since under the action of S 3 given in (3) the 15 coordinates satisfy the same relations as those of χ 4,0,8 . We have for the entries of the coefficient of q 2 1 q 2 2 q 2 3 : where as before σ i is the elementary symmetric function of degree i in u, v, w.
As an example, we take a closer look at the case of the concomitant c provided by W [7,4,1] in Sym 3 (Sym 4 (W )). The modular form γ ′ (c) is divisible by χ 18 and yields a cusp form χ 3,3,7 = a(N)q N of weight (3, 3, 7) on Γ 3 . We can calculate some of its Fourier coefficients: we find the irreducible representation of highest weight (13,10,7) inside Sym 3 (W )⊗Sym 3 (∧ 2 (W ))⊗det 7 (W ) with W the standard representation of GL(3). gives the eigenvalue λ 2 = 1080. The eigenvalues can be checked against the data provided by [5]. The invariant ι of type (3, (4, 4, 4)) is given by a 0 a 10 a 14 − 4 a 0 a 11 a 13 + 3 a 0 a 2 12 + 4 a 1 a 11 a 9 − 12 a 1 a 12 a 8 + 12 a 1 a 13 a 7 − 4 a 1 a 14 a 6 −4 a 10 a 2 a 9 + 3 a 10 a 2 5 + 12 a 11 a 2 a 8 − 12 a 11 a 4 a 5 − 12 a 12 a 2 a 7 + 6 a 12 a 3 a 5 + 12 a 12 a 2 4 +4 a 13 a 2 a 6 − 12 a 13 a 3 a 4 + 3 a 14 a 2 3 − 12 a 3 a 7 a 9 + 12 a 3 a 2 8 + 12 a 4 a 6 a 9 − 12 a 4 a 7 a 8 −12 a 5 a 6 a 8 + 12 a 5 a 2 7 in the coefficients a i of the ternary quartic f . It defines a cusp form χ 28 of weight 28 vanishing with multiplicity 3 at D. Using the pairing induced by the pairing of W and ∧ 2 W , we get f, σ = ι or in terms of modular forms where the pairing of χ 4,0,8 = I n I α I x I and χ 0,4,16 = I β Ix I is I n I α I β I . The Fourier expansion of χ 28 starts as . .))q 3 1 q 3 2 q 3 3 + . . . . This Siegel modular form vanishes with order 3 at infinity and along A 2,1 with order 8. Indeed, expanding F as a Taylor series along H 2 × H 1 as done in [12,Prop. 2.1] we get as first term a tensor product F ′ ⊗ F ′′ ∈ S 8,28 (Γ 2 ) ⊗ S 36 (Γ 1 ) because F ′′ will be a cusp form on Γ 1 vanishing with multiplicity 3 at the cusp and the first such form is ∆ 3 . By looking around H 3 1 we see that F ′ vanishes along H 2 1 with multiplicity 4; dividing F ′ by χ 2 10 we get a cusp form of weight (8,8) on Γ 2 . Now S 8,8 (Γ 2 ) has dimension 1 and is generated by the form χ 8,8 (see [10, p. 11]) and we find F ′ = (1/104230) 2 χ 2 10 χ 8,8 . We thus see that the modular form F , viewed as Teichmüller form, vanishes with order 3 along δ 0 and order 8 along δ 1 . It can be seen as a section of O(28λ − 3 δ 0 − 8 δ 1 ); see the last remark of [35, p. 1766  This gives the Hecke eigenvalue at p = 2: λ 2 = −2880 = −24(216 + 2 2 (−24)), in agreement with the fact that this cusp form is predicted to be a lift from Γ 1 with Hecke eigenvalues of the shape λ p = τ (p)(b(p) + p 2 τ (p)), where τ (p) is the Fourier coefficient of ∆ at p, while b(p) is that of the unique normalized cusp form of weight 16 on Γ 1 . This modular form is also given in [23, §5.6], where it is constructed using theta functions.
12.4. Degree 6. The so-called catalecticant is an invariant of degree d = 6 and is associated to W [8,8,8] occurring in Sym 6 (Sym 4 (W )). It is given as a 0 a 3 a 5 a 4 a 2 a 1 a 3 a 10 a 12 a 11 a 7 a 6 a 5 a 12 a 14 a 13 a 9 a 8 a 4 a 11 a 13 a 12 a 8 a 7 a 2 a 7 a 9 a 8 a 5 a 4 a 1 a 6 a 8 a 7 a 4 a 3 and gives rise to a Siegel modular form of weight 56 vanishing with order 6 at D and with order at least 16 along A 2,1 since the first cusp form vanishing with order 6 at ∞ on Γ 1 is ∆ 6 of weight 72 and 16 = 72 − 56. This modular form can be interpreted as a section of O(56 λ − 6 δ 0 − 16 δ 1 ) on M 3 , in agreement with [35, p. 1766].
Remark 12.1. Some of the modular forms constructed above have a geometric meaning. Chipalkatti proves in [9] that a ternary quartic f is the sum of the fourth powers of s linear forms f = ℓ 4 1 + ℓ 4 2 + · · · + ℓ 4 s for 1 ≤ s ≤ 5 if and only if the concomitants in a certain set U s vanish, where the set U s is given by

Teichmüller modular forms and the cohomology of local systems
It is well-known that Siegel modular forms of degree g occur in the cohomology of local systems on the moduli space A g of principally polarized abelian varieties of dimension g. Denoting by π : X g → A g the universal abelian variety, we let V = R 1 π * Q ℓ be the standard local system of rank 2g on A g . This comes with a symplectic pairing V × V → Q ℓ (−1). For every irreducible representation of the symplectic group GSp(2g, Q) with highest weight µ, we have a local system V µ obtained from V by applying a Schur functor. We consider the 'motivic' Euler characteristic of compactly supported cohomology. The cohomology group H i c (A g ⊗ C, V µ ⊗ C) (resp. H i (A g ⊗ C, V µ ⊗ C)) is provided with a mixed Hodge structure of weights ≤ |µ| + i (resp. ≥ |µ| + i) and the sums of the elements of the 2 g subsets of {µ g + 1, µ g−1 + 2, . . . , µ 1 + g} yield the degrees at which nontrivial steps in the Hodge filtration may occur, see [17] or [4] and references there. So the last step is F |µ|+g(g+1)/2 and it is here that we find Siegel modular forms: there is an isomorphism where n(µ) = (µ 1 − µ 2 , µ 2 − µ 3 , . . . , µ g−1 − µ g , µ g + g + 1) .
We denote by H i The above results are due to Faltings [16] and Faltings-Chai [17]; a key role is played by the (dual) BGG-complex.
We are interested in a similar interpretation of Teichmüller modular forms for g ≥ 2. So far, we have only considered these for g = 3. With a Teichmüller modular form of type ρ (or weight w(ρ)), we mean here a section over M g of E ′ ρ , the bundle obtained by applying the Schur functor associated to an irreducible representation ρ of GL(g) to the Hodge bundle E ′ .
By pulling back under the Torelli map t, we obtain local systems V ′ µ = t * V µ on M g and M c g = M g − ∆ 0 , the moduli space of curves of compact type. We consider the motivic Euler characteristic and similarly we can consider e c (M c g ; V ′ µ ). We obtain the following partial analogue of the above results.
Theorem 13.1. For the natural mixed Hodge structures on the middle cohomology groups, we have the following isomorphisms: (16) where m = 3g − 3 + |µ| is the maximum possible Hodge degree.
Proof. It is not clear whether the BGG-complex can be adapted to this setting and we resort to the logarithmic de Rham-complex. The local system V ′ µ ⊗ C on M g (or M c g ) corresponds to a holomorphic vector bundle V µ with flat connection ∇ (the Gauss-Manin connection). The boundary D = M g − M g is a divisor with normal crossings. The bundle V µ admits a canonical extension V µ to M g and ∇ extends to a map Mg (log D) with nilpotent polar part. Then the hypercohomology of the logarithmic de Rhamcomplex computes the cohomology of V ′ µ ⊗ C on M g : Cf. [38]. Twisting with O(−D), we obtain a complex that computes the compactly supported cohomology: ). Analogous statements hold for M c g , after replacing D by ∆ 0 . These complexes admit natural Hodge filtrations, which induce the Hodge filtrations of the mixed Hodge structures on H p , considered as a complex supported in degree 3g − 3. As is well-known [22], the canonical bundle of the stack M g equals O(13λ − 2δ). As an immediate consequence, we obtain the stated isomorphisms.
Thus the final steps in the Hodge filtrations on the middle cohomology groups are isomorphic to spaces of Teichmüller modular forms of highest weight µ + (13, 13, . . . , 13) with prescribed vanishing behaviour along components of the boundary. This is entirely analogous to the relation between Siegel modular forms and the cohomology of A g . In particular, we have in genus 2 the isomorphism M c 2 ∼ = A 2 and the relation 10λ ∼ δ 0 +2δ 1 ; the isomorphisms for M c 2 above agree with those for A 2 discussed earlier. However, note that the Teichmüller modular forms found here are cusp forms in a strong sense: they need to vanish (at least) once or twice along each component of the boundary.
The isomorphisms continue to hold if we change the cohomological degrees on both sides by the same amount; in particular, F m H p (c) = (0) for p < 3g − 3. As a trivial example, take µ = 0, so m = 3g − 3; the top compactly supported cohomology of M g and M c g is spanned by L 3g−3 and coincides with F 3g−3 ; on the other hand, H 3g−3 (M g , O(13λ − 2δ)) ∼ = C, as follows from Serre duality.
We now return to g = 3. Assume first that µ = (a, b, c) is even, i.e., a + b + c is even. Then µ + (13, 13, 13) is odd and so are the Teichmüller modular forms in the isomorphisms above. As we saw in §7, odd Teichmüller modular forms of genus 3 are divisible by χ 9 , and the even quotients are pullbacks of Siegel modular forms. So for g = 3 and µ even, the isomorphisms above may be rewritten as follows: with m = a + b + c + 6. Therefore, we also have isomorphisms with F m H 6 (A 3 , V µ ) resp. F m H 6 c (A 3 , V µ ). An interesting particular case of equation (17) is obtained by taking µ = 0. We find that F 6 H 6 (M 3 , C) and F 6 H 6 (M c 3 , C) are onedimensional and naturally isomorphic to M 0,0,4 , which is spanned by the Eisenstein series α 4 (see [41]). This is Looijenga's class of type (6, 6) (cf. [30]). (The dual class in H 6 c is of type (0, 0) and F 6 H 6 c = (0), since α 4 is not a cusp form.) Assume next that µ is odd, so ν = µ + (13, 13, 13) is even. Equations (14)-(16) say the following. The classes in F a+b+c+6 of the middle cohomology of V ′ µ ⊗ C on M 3 correspond to Teichmüller forms of type ν that vanish along the entire boundary. Only those corresponding to forms that vanish at least twice along the boundary divisors parametrizing reducible curves are restrictions from M c 3 . Only the forms vanishing at least twice along the entire boundary correspond to compactly supported classes.
For concrete examples, note that Bergström [2] has determined e c (M 3 , V ′ µ ⊗ C), hence e(M 3 , V ′ µ ⊗ C) for all µ with |µ| = a + b + c ≤ 7. E.g., for µ = (1, 1, 1), so m = 9, one has e c = −L 7 − L 2 + L + 1, so e = L 9 + L 8 − L 7 − L 2 . Then F m e = L 9 , but it is not yet clear which cohomology groups contribute to this. Since the virtual cohomological dimension of M 3 equals 7 (cf. [21]), only classes in H 6 and H 7 can contribute, and there must be at least one class in F 9 H 6 . Now O(14λ − δ)). According to [41], the space S 0,0,14 is spanned by the cusp form β 14 . We conclude that β 14 vanishes along A 2,1 and that F 9 H 7 = (0). In fact, by loc. cit., β 14 vanishes twice along A 2,1 , so the corresponding class is a restriction from M c 3 . In [4], Bergström and two of the present authors have counted curves of genus 2 and 3 over finite fields and their numbers of points and in this way determined the F q -traces of e c (M 3 , V ′ µ ) and e c (A 3 , V µ ) for q ≤ 25. As discussed in loc. cit., this has led to a complete conjecture for these motivic Euler characteristics in the case of A 3 . In the case of M 3 , we have obtained precise conjectures for nearly all µ with |µ| ≤ 20 (for three µ with |µ| = 19, some information is missing). The work of Chenevier-Renard [8], Chenevier-Lannes [7], Taïbi [40], and Mégarbané [31] has played a particularly important role here. Below, we use some of the results to provide further examples (in effect, we obtain further evidence for the conjectures). Note that the 'motivic' conjectures tell us in particular where terms of the maximum possible Hodge degree are to be expected.
This space is a subspace of S 2 a−b,b−c,c+13 , which by Thm. 11.6 is isomorphic to More precisely, the subspace consists of those forms vanishing to order at least 2 along A 2,1 (or along δ 1 when considered as Teichmüller forms). Now the isomorphism is given by c → γ(c)χ 18 and we know that χ 18 has order 6 along A 2,1 . So whenever γ(c) has order at least −4 along δ 1 , we obtain an element of the subspace of interest. Since χ 4,0,−1 has a simple pole along δ 1 (by Lemma 4.1 and Prop. 10.1), this is automatically the case for d ≤ 4.
The conjectural formulas for the Euler characteristics e c (M 3 , V ′ µ ) with µ odd are rather complicated and contain many more terms than just the motives mentioned above. The techniques and computations that were used to obtain the formulas, and the formulas themselves, will be discussed elsewhere.