Siegel modular forms of degree two and three and invariant theory
aa r X i v : . [ m a t h . AG ] F e b SIEGEL MODULAR FORMS OF DEGREE TWO AND THREEAND INVARIANT THEORY
GERARD VAN DER GEER
Abstract.
This is a survey based on the construction of Siegel modular forms ofdegree 2 and 3 using invariant theory in joint work with Fabien Cl´ery and Carel Faber. Introduction
Modular forms are sections of naturally defined vector bundles on arithmetic quotientsof bounded symmetric domains. Often such quotients can be interpreted as moduli spacesand sometimes this moduli interpretation allows a description as a stack quotient underthe action of an algebraic group like GL n . In such cases classical invariant theory can beused for describing modular forms.In the 1960s Igusa used the close connection between the moduli of principally polar-ized complex abelian surfaces and the moduli of algebraic curves of genus two to describethe ring of scalar-valued Siegel modular forms of degree two (and level 1) in terms ofinvariants of the action of GL acting on binary sextics, see [23, 24]. Igusa used thetafunctions and a crucial step in Igusa’s approach were Thomae’s formulas from the 19thcentury that link theta constants for hyperelliptic curves to the cross ratios of the branchpoints of the canonical map of the hyperelliptic curve to P .In the 1980s Tsuyumine, a student of Igusa, used the connection between the moduli ofabelian threefolds and curves of genus 3 to describe generators for the ring of scalar-valuedSiegel modular forms of degree 3 (and level 1). He used the moduli of hyperelliptic curvesof genus 3 as an intermediate step and used theta functions and the invariant theory ofbinary octics as developed by Shioda, see [33, 29].The description of the moduli of curves of genus 2 (resp. 3) in terms of a stack quotientof GL acting on binary sextics (resp. of GL acting on ternary quartics) makes it possibleto construct the modular forms directly from the stack quotient without the recourse totheta functions or cross ratios. This applies not only to scalar-valued modular forms,but to vector-valued modular forms as well. Covariants (or concomitants) yield explicitmodular forms in an efficient way. This in contrast to earlier and more laborious methodsof constructing vector-valued Siegel modular forms of degree 2 and 3 that use thetafunctions.In joint work with Fabien Cl´ery and Carel Faber [4, 5, 6] we exploited this for theconstruction of Siegel modular forms of degree 2 and 3. In degree 2 the universal binarysextic, the most basic covariant, defines a meromorphic Siegel modular form χ , − of Mathematics Subject Classification. weight (6 , − χ , − in a covariant produces a mero-morphic modular form that becomes holomorphic after multiplication by an appropriatepower of χ , a cusp form of weight 10 associated to the discriminant. For degree 3 wecan play a similar game, now involving the universal ternary quartic and a meromorphicTeichm¨uller modular form χ , , − of weight (4 , , −
1) that becomes a holomorphic Siegelmodular form χ , , of weight (4 , ,
8) after multiplication with χ , a Teichm¨uller formof weight 9 related to the discriminant.With this approach it is easy to retrieve Igusa’s result on the ring of scalar modularforms of degree 2. Another advantage of this direct approach is that one can treatmodular forms in positive characteristic as well. Thus it enabled the determination ofthe rings of scalar valued modular forms of degree 2 in characteristic 2 and 3, two casesthat were unaccounted for so far, see [17, 9].In this survey we sketch the approach and indicate how one constructs Siegel modularforms of degree 2 and 3. We show how to derive the results on the rings of scalar-valuedmodular forms of degree 2. 2. Siegel Modular Forms
Classically, Siegel modular forms are described as functions on the Siegel upper halfspace. We recall the definition.For g ∈ Z ≥ we set L = Z g with generators e , . . . , e g , f , . . . , f g and define a sym-plectic form h , i on L via h e i , f j i = δ ij . The Siegel modular group Γ g = Aut( L , h , i ) ofdegree g is the automorphism group of this symplectic lattice. Here we write an element γ ∈ Γ g as a matrix (cid:18) a bc d (cid:19) of four g × g blocks using the basis e i and f i ; we oftenabbreviate this as γ = ( a, b ; c, d ). The group Γ g acts on the Siegel upper half space H g = { τ ∈ Mat( g × g, C ) : τ t = τ, Im( τ ) > } via τ γ ( τ ) = ( aτ + b )( cτ + d ) − .A scalar-valued Siegel modular form of weight k and degree g > f : H g → C satisfying f ( γ ( τ )) = det( cτ + d ) k f ( τ ) for all γ = ( a, b ; c, d ) ∈ Γ g .If ρ : GL( g ) → GL( V ) is a complex representation of GL g then a vector-valued Siegelmodular form of weight ρ and degree g > f : H g → V satisfying f ( γ ( τ )) = ρ ( cτ + d ) f ( τ ) for all γ = ( a, b ; c, d ) ∈ Γ g (1)We may restrict to irreducible representations ρ . For g = 1 we have to require anadditional growth condition for y = Im( τ ) → ∞ .However, for an algebraic geometer modular forms are sections of vector bundles. Let A g be the moduli space of principally polarized abelian varieties of dimension g . This isa Deligne-Mumford stack of relative dimension g ( g + 1) / Z . It carries a universalprincipally polarized abelian variety π : X g → A g . This provides A g with a naturalvector bundle E = E ( g ) , the Hodge bundle, defined as E = π ∗ (Ω X g / A g ) . IEGEL MODULAR FORMS AND INVARIANT THEORY 3
Starting from E we can create new vector bundles. Each irreducible representation ρ of GL g defines a vector bundle E ρ by applying a Schur functor (or just by applying ρ to the transition functions of E ). In particular, we have the determinant line bundle L = det( E ). Scalar-valued modular forms of weight k are sections of L ⊗ k and these forma graded ring. In fact, for g ≥ F we have the ring R g ( F ) = ⊕ k H ( A g ⊗ F, L ⊗ k ) . The moduli space A g can be compactified. There is the Satake compactification,in some sense a minimal compactification, based on the fact that L is an ample linebundle on A g . This compactification A ∗ g is defined as Proj( R g ) and satisfies the inductiveproperty A ∗ g = A g ⊔ A ∗ g − . Restriction to the ‘boundary’ A ∗ g − induces a map called the Siegel operatorΦ : R g ( F ) → R g − ( F ) . We will also use (smooth) Faltings-Chai type compactifications ˜ A g and over these theHodge bundle extends ([14]). We will denote the extension also by E .For g > E ρ over A g extend to regular sectionsof the extension of E ρ over ˜ A g , see [14, Prop 1.5, p. 140]. For g = 1 this does not holdsince the boundary in A ∗ is a divisor, and we define modular forms of weight k as sectionsof L ⊗ k over ˜ A . If D denote the divisor added to ˜ A g to compactify A g , then elements of H ( ˜ A g , E ρ ⊗ O ( − D )) are called cusp forms.We will write M ρ (Γ g )( F ) for H ( ˜ A g ⊗ F, E ρ ) or simply M ρ (Γ g ) when F is clear. Thespace of cusp forms is denoted by S ρ (Γ g ). By the Koecher principle the spaces M ρ (Γ g )( F )and S ρ (Γ g )( F ) do not depend on the choice of a Faltings-Chai compactification.Over the complex numbers if ρ : GL( g ) → GL( V ) is an irreducible representation,elements of H ( ˜ A g ⊗ C , E ρ ) correspond to holomorphic functions f : H g → V satisfying(1). Such a function allows a Fourier expansion f ( τ ) = X n ≥ a ( n ) q n , where the sum is over symmetric g × g half-integral matrices (meaning 2 n is integral andeven on the diagonal) which are positive semi-definite, a ( n ) ∈ V and q n is shorthand for e πi Tr( nτ ) .The definition R g ( F ) = ⊕ k H ( ˜ A g ⊗ F, L ⊗ k )for a commutative ring F allows one speak of modular forms in positive characteristicby taking F = F p . One cannot define such modular forms by Fourier series.We summarize what is known about the rings R g ( F ). It is a classical result that thering R ( C ) is freely generated by two Eisenstein series E and E of weights 4 and 6.Deligne determined in [10] the ring R ( Z ) and the rings R ( F p ). He showed that R ( Z ) = Z [ c , c , ∆] / ( c − c − , GERARD VAN DER GEER where ∆ is a cusp form of weight 12 and c and c are of weight 4 and 6. Reductionmodulo p gives a surjection of R ( Z ) to R ( F p ) for p ≥
5. Moreover, Deligne showedthat R ( F p ) in characteristic 2 and 3 is given by R ( F ) = F [ a , ∆] , R ( F ) = F [ b , ∆] , where in each case ∆ is a cusp form of weight 12 and a (resp. b ) is a modular form ofweight 1 (resp. 2).In [23] Igusa determined the ring R ( C ). He showed that the subring R ev2 ( C ) of evenweight modular forms is generated freely by modular forms of weight 4 , ,
10 and 12 and R ( C ) is generated over R ev2 ( C ) by a cusp form of weight 35 whose square lies in R ev2 ( C );see also [24]. Later ([25]) he also determined the ring R ( Z ); it has 15 generators ofweights ranging from 4 to 48.In characteristic p ≥ R ( F p ) is similar to that of R ( C ),see [27, 2]; these are generated by forms of weight 4 , , ,
12 and 35. The structure of R ( F p ) for p = 2 and 3 was determined recently in [9, 17]. All these cases can be dealtwith easily using the approach with invariant theory.In degree 2 one can provide the R -module M = ⊕ j,k M j,k (Γ ) with M j,k (Γ ) = H ( ˜ A , Sym j ( E ) ⊗ det( E ) k )with the structure of a ring using the projection of GL -representations Sym m ( V ) ⊗ Sym n ( V ) → Sym m + n ( V ) with V the standard representation by interpreting Sym j ( V )as the space of homogeneous polynomials of degree j in two variables, say x , x andperforming multiplication of polynomials. The ring M is not finitely generated asGrundh showed, see [3, p. 234]. The dimensions of the spaces S j,k (Γ )( C ) are knownby Tsushima[32] for k ≥
4; for k = 3 they were obtained independently by Petersen andTa¨ıbi [28, 31].For fixed j the R ev2 ( C )-modules ⊕ k M j, k (Γ )( C ) and ⊕ k M j, k +1 (Γ )( C )are finitely generated modules and their structure has been determined in a number ofcases by Satoh, Ibukiyama and others, see the references in [5]. Invariant theory makesit easier to obtain such results.For g = 3 the results are less complete. Tsuyumine showed in 1985 ([33]) that thering R ( C ) is generated by 34 generators. Recently Lercier and Ritzenthaler showed in[26] that 19 generators suffice.3. Moduli of Curves of Genus two as a Stack Quotient
We start with g = 2. Let F be a field of characteristic = 2 and V = h x , x i the F -vector space with basis x , x . The algebraic group GL acts on V via ( x , x ) ( ax + bx , cx + dx ) for ( a, b ; c, d ) ∈ GL ( F ). We will write V j,k = Sym j ( V ) ⊗ det( V ) ⊗ k for j ∈ Z ≥ and k ∈ Z . This is an irreducible representation of GL . The underlyingvector space can be identified with the space of homogeneous polynomials of degree j IEGEL MODULAR FORMS AND INVARIANT THEORY 5 in x , x . We will denote by V j,k the open subspace of polynomials with non-vanishingdiscriminant.The moduli space M of smooth projective curves of genus 2 over F allows a presen-tation as an algebraic stack M ∼ −→ [ V , − / GL ]Here the action of ( a, b ; c, d ) ∈ GL ( F ) is by f ( x , x ) ( ad − bc ) − f ( ax + bx , cx + dx ).Indeed, if C is a curve of genus 2 the choice of a basis ω , ω of H ( C, K ) with K = Ω C defines a canonical map C → P . Let ι denote the hyperelliptic involution of C . Choosinga non-zero element η ∈ H ( C, K ) ι = − yields eight elements η , ω , ω ω , . . . , ω in the7-dimensional space H ( C, K ) ι =1 and thus a non-trivial relation.In inhomogeneous terms this gives us an equation y = f with f ∈ F [ x ] of degree 6with non-vanishing discriminant. The space H ( C, K ) has a basis xdx/y, dx/y . If we letGL act on ( x, y ) via ( x, y ) (( ax + b ) / ( cx + d ) , y ( ad − bc ) / ( cx + d ) ) then this actionpreserves the form of the equation y = f if we take f in V , − . Then λ Id V acts via λ on V , − . Thus the stabilizer of a generic element f is of order 2. Moreover − Id V acts by y
7→ − y on y and the action of GL on the differentials is by the standard representation. Conclusion 3.1.
The pull back of the Hodge bundle E on M under the composition V , − → [ V , − / GL ] ∼ −→ M is the equivariant bundle V . The moduli space M can be constructed from the projectivized space P ( V , − ) ofbinary sextics. The discriminant defines a hypersurface D whose singular locus hascodimension 1 in D . The locus of binary sectics with three coinciding roots forms anirreducible component D ′ of the singular locus. To illustrate the relation between P ( V , − )at a general point of D ′ and M at a point of the locus δ in M of stable curves whoseJacobian is a product of two elliptic curves, we reproduce the picture of [12, p. 80]. E E E Here we look at a plane Π intersecting D transversally at a general point of D ′ . Oneblows up three times, starting at Π ∩ D ′ , and then blows down the exceptional divisors E and E ; after that E corresponds to the locus δ in M ; in A this corresponds tothe locus A , of product of elliptic curves.4. Invariant Theory of Binary Sextics
We review the invariant theory of GL acting on binary sextics. Let V = h x , x i bea 2-dimensional vector space over a field F . By definition an invariant for the action GERARD VAN DER GEER of GL acting on the space Sym ( V ) of binary sextics is an element invariant underSL ( F ) ⊂ GL ( F ). If we write f = X i =0 a i x − i x i (2)for an element of Sym ( V ) and thus take ( a , . . . , a ) as coordinates on Sym ( V ) thenan invariant is a polynomial in a , . . . , a invariant under SL ( F ). The discriminant of abinary sextic, a polynomial of degree 10 in the a i , is an example.For F = C the ring of invariants was determined by Clebsch, Bolza and others in the19th century. It is generated by invariants A, B, C, D, E of degrees 2 , , ,
10 and 15 inthe a i . Also for F = F p we have generators generators of these degrees. We refer to[18, 23].A covariant for the action of GL on binary sextics is an element of V ⊕ Sym ( V )invariant under the action of SL . Such an element is a polynomial in a , . . . , a and x , x . One way to make such covariants is to consider equivariant embeddings of anirreducible GL -representation U into Sym d (Sym ( V )). Equivalently, we consider anequivariant embedding ϕ : C ֒ → Sym d (Sym ( V )) ⊗ U ∨ . Then Φ = ϕ (1) is a covariant. If U has highest weight ( λ ≥ λ ) then Φ is homogeneousof degree d in a , . . . , a and degree λ − λ in x , x . We say that Φ has degree d andorder λ − λ .The simplest example is the universal binary sextic f given by (2); it corresponds totaking U = Sym ( V ).Another example is the Hessian of f . Indeed, we decompose in irreducible represen-tations Sym (Sym ( V )) = V [12 , ⊕ V [10 , ⊕ V [8 , ⊕ V [6 , , where V [ a, b ] = Sym a − b ( V ) ⊗ det( V ) b is the irreducible representation of highest weight( a, b ). By taking U = V [12 ,
0] we find the covariant Φ = f and by taking U = V [10 , U = V [6 ,
6] gives the invariant A .The covariants form a ring C and the invariants form a subring I = I (2 , C was studied intensively at the end of the 19th century and thebeginning of the 20th century. The ring C is finitely generated and Grace and Youngpresented 26 generators for the ring C , see [19]. These 26 covariants are constructed astransvectants by differentiating in a way similar to the construction of the Hessian. The k th transvectant of two forms g ∈ Sym m ( V ), h ∈ Sym n ( V ) is defined as( g, h ) k = ( m − k )!( n − k )! m ! n ! k X j =0 ( − j (cid:18) kj (cid:19) ∂ k g∂x k − j ∂x j ∂ k h∂x j ∂x k − j and the index k is usually omitted if k = 1. Examples of the generators are C , = f , C , = ( f, f ) , C , = ( f, f ) , C , = ( f, C , ) . We refer to [5] for a list of these 26generators. IEGEL MODULAR FORMS AND INVARIANT THEORY 7 Covariants of Binary Sextics and Modular Forms
The Torelli morphism induces an embedding M ֒ → A . The complement of theimage is the locus A , of products of elliptic curves. As a compactification we can take˜ A = M .We now fix the field F to be C or a finite prime field F p .In the Chow ring CH ∗ Q ( ˜ A ) ⊗ F we have the cycle relation10 λ = 2[ A , ] + [ D ]with λ = c ( E ) the first Chern class of E and D the divisor that compactifies A ⊗ F .This implies that there exists a modular form of weight 10 with divisor 2 A , + D , hence acusp form. It is well-defined up to a non-zero multiplicative constant. We will normalizeit later. We denote it by χ ∈ R ( F ).We let V be the F -vector space with basis x , x . The fact that the pullback of theHodge bundle E under V , − → [ V , − / GL ] → M ⊗ F ֒ → A ⊗ F (3)is the equivariant bundle V implies that a section of L k = det( E ) k pulls back to aninvariant of degree k . We thus get an embedding of the ring of scalar-valued modularforms of degree 2 into the ring of invariants R ( F ) ֒ → I (2 , F ) . Conversely, an invariant of degree d defines a section of L d on M ⊗ F , hence a rational(meromorphic) modular form of weight d that is holomorphic outside A , ⊗ F . Bymultiplying it with an appropriate power of χ it becomes holomorphic on A ⊗ F ,hence on all of ˜ A ⊗ F . We thus get maps R ( F ) ֒ → I (2 , F ) −→ R ( F )[1 /χ ] (4)the composition of which is the identity on R ( F ).From the description of the moduli M given above one sees that the image of a cuspform is an invariant divisible by the discriminant D . The image of χ is a non-zeromultiple of the discriminant D . We may fix χ by requiring that ν ( D ) = χ .This extends to the case of vector-valued modular forms. Let M ( F ) = ⊕ j,k M j,k (Γ )( F )denote the ring of vector-valued modular forms of degree 2. Proposition 5.1.
Pullback via (3) defines homomorphisms M ( F ) ֒ → C (2 , F ) ν −→ M ( F )[1 /χ ] , the composition of which is the identity. A modular form of weight ( j, k ) corresponds to a covariant of degree d = j/ k andorder j . A covariant of degree d and order r gives rise to a meromorphic modular formof weight ( r, d − r/ GERARD VAN DER GEER
The most basic covariant is the universal binary sextic f . By construction ν ( f ) is ameromorphic modular form of weight (6 , − Whichrational modular form is ν ( f ) ? Let A , ⊂ A be the locus of products of elliptic curves. Under the map A × A → A , → A the pullback of the Hodge bundle E = E (2) is p ∗ E (1) ⊕ p ∗ E (1) with p and p the projectionsof A ×A on its factors. The pullback of an element h ∈ M j,k (Γ ) thus can be indentifiedwith an element of j M i =0 M k + j − i (Γ ) ⊗ M k + i (Γ ) . Near a point of A , we can write such an element symbolically as h = j X i =0 η j X j − i X i , where the X i are dummy variables to indicate the vector coordinates, and such that thecoefficient η j defines the element of M k + j − i (Γ ) ⊗ M k + i (Γ ).In particular we have ν ( f ) = X i =0 α i X − i X i , where α i are rational functions near a point of A , . By interchanging x and x (thatcorresponds to the element γ ∈ Γ that interchanges e and e ) we see that α − i = α i for i = 0 , . . . , Proposition 5.2. If char( F ) = 2 and = 3 , then dim S , (Γ )( F ) = 1 and χ ν ( f ) is agenerator of S , (Γ )( F ) .Proof. We shall use that dim S , (Γ )( C ) ≥
1. Indeed, we know an explicit cusp formof weight (6 , S j,k (Γ )( C ) for k ≥
4, see [32]; in particular we know dim S , (Γ )( C ) = 1.) By semi-continuity thisimplies that dim S , (Γ )( F ) ≥ S , (Γ )( F ) to the locus A , ⊗ F lands in M i =0 S − i (Γ )( F ) ⊗ S i (Γ )( F ) , and as we have dim S k (Γ )( F ) = 0 for k <
12 it vanishes on A , ⊗ F .The tangent space to A at a point [ X = X × X ] of A , , with X i elliptic curves,can be identified withSym ( T X ) = Sym ( T X ) ⊕ ( T X ⊗ T X ) ⊕ Sym ( T X )with T X (resp T X i ) the tangent space at the origin of X (resp. X i ), and with the middleterm corresponding to the normal space. Thus we see that the pullback of the conormal IEGEL MODULAR FORMS AND INVARIANT THEORY 9 bundle of A , to A × A is the tensor product of the pullback of the Hodge bundles onthe two factors A .Let h ∈ S , (Γ )( F ) and write h as h = X i =0 η i X − i X i locally at a general point of A , ⊗ F . If we consider the Taylor development in thenormal direction of A , of the form h that vanishes on A , ⊗ F then the first non-zeroTaylor term of η i , say the r th term, is an element of S − i + r (Γ )( F ) ⊗ S i + r (Γ )( F ) . Since S k (Γ )( F ) = (0) for k <
12, a non-zero r th Taylor term of η i can occur only for14 − i + r ≥
12 and 8 + i + r ≥
12. We thus find:ord A , ( η , . . . , η ) ≥ (4 , , , , , , . Lemma 5.3.
We have ord A , ( η ) = 1 .Proof. If ord A , ( η ) ≥ h/χ is a regular form in S , − (Γ ) and we write it as h/χ = P i =0 ξ i X − i X i with ξ i = η i /χ regular. Then the invariant A = 120 a a − a a + 8 a a − a defines a non-zero regular modular form ν ( A ) = 120 ξ ξ − ξ ξ + 8 ξ ξ − ξ in M (Γ )( F ). But restriction to A , gives for even k an exact sequence0 → M k − (Γ )( F ) → M k (Γ )( F ) → Sym ( M k (Γ )( F )) (4)with the second arrow multiplication by χ . This implies that dim M (Γ )( F ) = 0 forchar( F ) = 2 and = 3. This proves the lemma. (cid:3) The image of a non-zero element χ , of S , in C (2 ,
6) is a covariant of degree 11 andorder 6. But since χ , is a cusp form, this covariant is divisible by the discriminantwhich is of degree 10. Therefore, χ , /χ corresponds to a covariant of degree 1, henceis a non-zero multiple of f . This implies that dim S , (Γ )( F ) = 1. (cid:3) Corollary 5.4.
If we write ν ( f ) = P i =0 α i X − i X i then ord A , ( α , . . . , α ) ≥ (2 , , , − , , , and ord A , ( α ) = − . Constructing vector-valued modular forms of degree ν ( f ) by Proposition 5.2 we can describe the map ν : C (2 , → M [1 /χ ] explicitly. Recall that a covariant is a polynomial in a , . . . , a and x , x . Wearrive at the following conclusion. Proposition 6.1.
The map ν : C (2 , → M [1 /χ ] is substitution of α i for a i (and X i for x i ). In order to efficiently apply the proposition we need to know the coordinates of agenerator χ , of S , very explicity. Remark . If F = F the moduli space A [2] of level 2 is a Galois cover of A withgroup Sp(2 , Z / Z ). This group is isomorphic to the symmetric group S . The signcharacter of S defines a character ǫ of Γ . The pullback of χ under π : A [2] → A is a square χ since the pullback of D under ˜ A [2] → ˜ A is divisible by 2 as a divisor.Thus χ is a modular form of weight 5 with character ǫ .Let now F = C . Recall that χ , vanishes on A , . Dividing χ , by χ provides aholomorphic vector-valued modular form χ , ∈ M , (Γ , ǫ )( F ) with character ǫ . Such aform can be constructed as follows.We consider the six odd order two theta functions ϑ i ( τ, z ) with ( τ, z ) ∈ H × C . Thegradient G i = ( ∂ϑ i /∂z , ∂ϑ i /∂z ) is a modular form of weight (1 , /
2) on some congruencesubgroup, but the product of the transposes of these six gradients defines a vector-valuedmodular form of weight (6 ,
3) on Γ with character ǫ . The product χ , = χ χ , is acusp form of weight (6 ,
8) on Γ . A non-zero multiple of its Fourier expansion starts with(with q = e πiτ , q = e πiτ and r = e πiτ ) χ , ( τ ) = r − − r r − r − ) r − − r q q + − r − +8 r − − r + r )8( r − +4 r − − r − r ) − r − − r − − − r +7 r )12( r − − r − +2 r − r ) − r − − r − +6 − r + r ) q q + − r − − r − +6 − r + r )12( r − − r − +2 r − r ) − r − − r − − − r +7 r )8( r − +4 r − − r − r ) − r − +8 r − − r + r )00 q q + r − − r − +16 − r + r ) − r − − r − +3 r − r )+128( r − − r ) − r − +5 r − − r − r )+128( r − − r ) − r − − r − +3 r − r )16( r − − r − +16 − r + r ) q q + . . . Proposition 6.1 provides an extremely effective way of constructing complex vector-valued Siegel modular forms of degree 2. Let us give a few examples. In the decomposi-tion Sym (Sym ( V )) = V [12 , ⊕ V [10 , ⊕ V [8 , ⊕ V [6 , (Sym ( V )) the covariant H defined by V [10 ,
2] is the Hessian and by Corollary5.4 gives rise to a form χ , = ν ( H ) χ ∈ S , (Γ ) and using the Fourier expansion of χ , we obtain the Fourier expansion of χ , . Similarly, the covariant corresponding to V [8 ,
4] gives a form χ , after multiplication with χ . Finally, the covariant defined by V [6 ,
6] is the invariant A and defines the cusp form χ = ν ( A ) χ . We refer to [4] formore details.As illustration of this we refer to the website [1] that gives the Fourier series forgenerators for all cases where dim S j,k (Γ ) = 1.Another illustration of the efficacity of the construction of modular forms appearswhen one considers the modules ⊕ k M j,k (Γ ) and ⊕ k M j,k (Γ , ǫ ). Let R ev2 be the ring of IEGEL MODULAR FORMS AND INVARIANT THEORY 11 scalar-valued modular forms of even weight. The structure of the R ev2 -modules ⊕ k M j, k (Γ ) , ⊕ k M j, k +1 (Γ )has been determined for j = 2 , , , ,
10 by Satoh, Ibukiyama, Kiyuna, van Dorp andTakemori using various methods. Using covariants one can uniformly treat these casesand the cases of modular forms with character for the same values of j ⊕ k M j, k (Γ , ǫ ) , ⊕ k M j, k +1 (Γ , ǫ ) . For example, the R -module ⊕ k M , k +1 (Γ , ǫ ) is free with generators of weight (2 , ,
11) and (2 ,
17) and the module ⊕ k M , k (Γ , ǫ ) is free with 10 generators. We referto [5].Yet another application of the construction of modular forms via covariants deals withsmall weights. It is known by Skoruppa ([30]) that dim S j, (Γ ) = 0. He proved thisusing Fourier-Jacobi forms. We conjecture dim S j, (Γ ) = 0 and proved this for j ≤ k = 3 the smallest j such that dim S j, (Γ ) = 0 is 36. It isnot difficult to construct a generator of S , (Γ ) using covariants, see [8].7. Rings of Scalar-Valued Modular Forms
The approach explained in the preceding section makes it easy to find generators forthe rings R ( F ) = ⊕ k M k (Γ )( F ) of modular forms of degree 2 for F = C or F = F p . Wewrite ν F for the map I (2 , F ) → R ( F )[1 /χ ]. We denote by R ev2 ( F ) the subring ofeven weight modular forms.The degree 2 invariant A of a binary sextic f = P i =0 a i x − i x i can be written as120 a a − a a + 8 a a − a . Corollary 5.4 implies that ν F ( A ) cannot be regular for F = C or F p with p ≥
5, but alsothat ν F ( AD ) is a cusp form χ ∈ S (Γ )( F ) of weight 12.In degree 4 there is the invariant B given by(81 a a + 9 a a ) a − a a a + 15 a a a + a a + a a ) a + · · · + a a and Corollary 5.4 implies that it defines a regular modular form ψ = ν F ( B ) of weight 4.The invariant C of degree 6 is given by18 (9 a a + 4 a a ) a − a a a + 33 a a a + 4 a a + 4 a a ) a + · · · and in a similar way one sees that AB − C starts with1458 a a a −
486 ( a a a + a a a ) a + · · · and defines a regular modular form ψ = ν F ( AB − C ) of weight 6.The discriminant D starts as729 a a a − a a a a − a a + 9 a a a a − a a ) a + · · · and is seen to have order 2 along A , . It defines a cusp form that is a non-zero multipleof χ . Proposition 7.1.
For F = C or F = F p with p ≥ the modular forms ψ , ψ , χ and χ generate R ev2 ( F ) .Proof. The algebraic independence of
A, B, C, D shows that the generators are alge-braically independent. Therefore ψ , ψ , χ , χ generate a graded subring T ( F ) ⊆ R ev2 ( F ) such that for even k we havedim T k ( F ) = k O ( k ) . Now by Riemann-Roch we have for even k dim M k (Γ )( F ) = c ( L ) k + O ( k )since c ( L ) = 1 / · · ·
12 = 2880. (cid:3)
Remark . Restriction to A , shows that ψ , ψ , χ , χ generate M k (Γ )( F ) for k ≤
12. Let d ( k ) = dim F M k (Γ )( F ) and t ( k ) = dim F T k ( F ). Then t ( k ) ≤ d ( k ) and for even k the exact sequence (4) yields d ( k ) ≤ d ( k −
10) + c ( k )( c ( k ) + 1)2with c ( k ) = dim F M k (Γ )( F ). Now one easily sees t ( k ) − t ( k −
10) = c ( k )( c ( k ) + 1) / d ( k −
10) = t ( k −
10) we get t ( k ) ≤ d ( k ) ≤ d ( k −
10) + c ( k )( c ( k ) + 1)2 = t ( k )and this provides via induction another proof that ψ , ψ , χ and χ generate R ev2 ( F )for F = C or F p with p ≥ E (of degree 15) of binary sextics starts with − a a − a a ) a + . . . and one checks that it has order − A , . So χ ν F ( E ) defines a regular cusp form χ ∈ S (Γ )( F ) with order 1 along A , .Let now char( F ) = 2. The locus in A ⊗ F of principally polarized abelian surfaces X with Aut( X ) containing Z / Z × Z / Z consists of two irreducible divisors H = A , and H , the Humbert surface of degree 4 of abelian surfaces isogenous with a productby an isogeny of degree 4. In terms of moduli of curves, H is the locus of curves thatare double covers of elliptic curves. We know that the cycle class of H + H is 35 λ inPic Q ( A ), see [15, p. 218]. Lemma 7.3.
Suppose that char( F ) = 2 . A modular form f ∈ M k (Γ )( F ) with k oddvanishes on H and H .Proof. An abelian surface [ X ] ∈ H or [ X ] ∈ H possesses an involution that acts by − H ( X, Ω X ). (cid:3) Corollary 7.4.
The form χ has as divisor H + H + D . IEGEL MODULAR FORMS AND INVARIANT THEORY 13
We can now easily derive the results of Igusa and Nagaoka (see [23, 27], and also [20]).
Theorem 7.5.
Let F = C or F = F p with p ≥ . Then the ring R ( F p ) is generatedover R ev2 ( F p ) = F [ ψ , ψ , χ , χ ] by the cusp form χ of weight with χ ∈ R ev2 ( F ) .Proof. Any odd weight modular form vanishes on H and H , hence is divisible by χ . (cid:3) Remark . The same argument proves Theorem 7.5 for any commutative ring F inwhich 6 is invertible. It can also be used to obtain Igusa’s result on the ring R ( Z ).Now positive characteristic sometimes allows more modular forms than characteristiczero. We know that the locus in A g ⊗ F p of abelian varieties of p -rank < g has cycleclass ( p − λ , [16, 13]. This implies that there is a non-zero modular form of weight p − p . This modular form is called the Hasse invariant of degree g andweight p −
1. The image of the Hasse invariant of degree g under the Siegel operator isthe Hasse invariant of degree g − a and b in R ( F ) = F [ a , ∆] , R ( F ) = F [ b , ∆] . The degree 2 invariant A of binary sextics reduces to a a − a a modulo 3 and in viewof Conclusion 5.4 defines a form ν F ( A ) ∈ M (Γ )( F ) and it must agree with the Hasseinvariant (up to a non-zero multiplicative scalar) as there is only one invariant of degree2 (up to multiplicative scalars). A careful analysis of the invariants in characteristic 3leads to the description of the ring R ( F ) given in [17]. Theorem 7.7.
The subring R ev2 ( F ) of modular forms of even weight is generated byforms of weights , , , and and has the form R ev2 ( F ) = F [ ψ , χ , ψ , χ , χ ] /J with J the ideal generated by the relation ψ χ − χ ψ − ψ χ χ + χ . Moreover, R ( F ) is generated over R ev2 by a form χ of weight whose square lies in R ev2 ( F ) .The ideal of cusp forms is generated by χ , χ , χ , χ . The case of characteristic 2 was treated in joint work with Cl´ery in [8]. In the case ofcharacteristic 2 a curve of genus 2 is not described by a binary sextic. Instead we findan equation y + a y + b = 0with a (resp. b ) in k [ x ] of degree ≤ ≤
6) and the hyperelliptic involution is y y + a . It comes with a basis xdx/a, dx/a of regular differentials. In this case we lookat pairs ( a, b ) ∈ V , − × V , − with V n,m = Sym n ( V ) ⊗ det( V ) m . Let V ⊂ V , − × V , − be the open subset defining smooth hyperelliptic curves. Now we have an action of GL and an action of Sym ( V ) via ( a, b ) ( a, b + v + va ) Together this defines a stack quotient[ V / GL ⋉ V , − ]Now by an invariant we mean a polynomial in the coefficients a , . . . , a and b , . . . , b that is invariant under SL( V ) ⋉ Sym ( V ). Let K be the ring of invariants. A first exampleis the square root of the discriminant of a : K = a a + a a . As an analogue of (4) we now get homomorphisms R ( F ) ֒ → K ν −→ R ( F )[1 /χ ]the composition of which is the identity.In order to construct characteristic 2 invariants one can still uses binary sextics asIgusa suggested in [23]. Indeed, one lifts the curve given by y + ay + b = 0 to the Wittring, say defined by y + ˜ ay + ˜ b = 0 and takes an invariant of the binary sextic given by˜ a + 4˜ b , then divides these by the appropriate power of 2 and reduces modulo 2.For example, the degree 2 invariant of binary sextics yields in this way an invariant K that equals K . A degree 4 invariant yields an invariant K that turns out to bedivisible by K . We thus find an invariant K of degree 3.The Hasse invariant ψ must map to K . As in characteristic 3 a careful analysis givesthe orders of a i and b i along A , and we can deduce for an invariant K the order of ν ( K ) along A , . The ring R ( F ) was described in [8]. Theorem 7.8.
The ring R ( F ) is generated by modular forms of weights , , , , satisfying one relation of weight : R ( F ) = F [ ψ , χ , ψ , χ , χ ] / ( R ) with R = χ + ψ χ χ + ψ χ + χ ψ . The ideal of cusp forms is generated by χ , χ and χ . Moduli of Curves of Genus Three and Invariant Theory of TernaryQuartics
Now we turn to genus 3 treated in [6] and consider the moduli space M nh3 of non-hyperelliptic curves of genus 3 over a field F . This is an open part of the modulispace M with as complement the divisor H of hyperelliptic curves. Let now V = h x , x , x i be the 3-dimensional F -vector space with basis x , x , x . We let V , , − bethe irreducible representation Sym ( V ) ⊗ det( V ) − . The underlying space is the spaceof ternary quartics. It contains the open subset V , , − of ternary quartics with non-vanishing discriminant; that is, the ternary quartics that define smooth plane quarticcurves.It is known that M nh3 has a description as stack quotient M nh3 ∼ −→ [ V , , − / GL ]Indeed, if C is a non-hyperelliptic curve of genus 3 then a choice of basis of H ( C, K )defines an embedding of C into P and the image satisfies an equation f ( x , x , x ) = 0 IEGEL MODULAR FORMS AND INVARIANT THEORY 15 with f homogeneous of degree 4. In order that the action on the space of differentialswith basis x i ( x dx − x dx ) / ( ∂f /∂x ) , i = 0 , , V we need to twist Sym ( V ) by det( V ) − . Then λ Id ∈ GL ( F ) acts by λ on V , , − and we arrive at the familiar stack quotient [ Q/ PGL ]with Q the space of smooth projective curves of degree 4 in P by first dividing by themultiplicative group of multiples of the diagonal. Conclusion 8.1.
The pull back of the Hodge bundle E on M nh3 under V , , − → [ V , , − ] / GL ] ∼ −→ M nh3 is the equivariant bundle V . Therefore we now look at the invariant theory of GL acting on ternary quarticsSym ( V ) with V = h x, y, z i the standard representation of GL ( V ). We write the uni-versal ternary quartic f as f = a x + a x y + · · · + a z in a lexicographic way. We fix coordinates for ∧ V ˆ x = y ∧ z, ˆ y = z ∧ x, ˆ z = x ∧ y . Recall that an irreducible representation ρ of GL is determined by its highest weight( ρ ≥ ρ ≥ ρ ). This representation appears inSym ρ − ρ ( V ) ⊗ Sym ρ − ρ ( ∧ V ) ⊗ det( V ) An invariant for the action of GL on Sym ( V ) is a polynomial in a , . . . , a invariantunder SL . Instead of the notion of covariant we consider here the notion of a con-comitant. A concomitant is a polynomial in a , . . . , a and in x, y, z and ˆ x, ˆ y, ˆ z thatis invariant under the action of SL . The most basic example is the universal ternaryquartic f .Concomitants can be obtained as follows. One takes an equivariant map of GL -representations U ֒ → Sym d (Sym ( V ))or equivalently the equivariant embedding ϕ : C −→ Sym d (Sym ( V )) ⊗ U ∨ Then Φ = ϕ (1) is a concomitant. If U is an irreducible representation of highest weight ρ ≥ ρ ≥ ρ then Φ is of degree d in a , . . . , a , of degree ρ − ρ in x, y, z and degree ρ − ρ in ˆ x, ˆ y, ˆ z .The invariants form a ring I (3 ,
4) and the concomitants C (3 ,
4) form a module over I (3 , I (3 ,
4) see [11]. Concomitants of ternary quartics and modular forms of degree t : M → A defined by associating to a curve of genus 3 its Jacobian. This is a morphism of Deligne-Mumford stacks of degree 2 ramified along the hyperelliptic locus H . Indeed, everyabelian variety has an automorphism of order 2, but a generic curve of genus 3 does nothave non-trivial automorphisms. Hyperelliptic curves have an automorphism of order 2that induces − Jac on the Jacobian.There is a Siegel modular form χ ∈ S (Γ ) constructed by Igusa [24]. It is defined asthe product of the 36 even theta constants of order 2. The divisor of χ in the standardcompactification ˜ A is H + 2 D with D the divisor at infinity.The pullback under the Torelli morphism of the Hodge bundle E on A is the Hodgebundle of M . The Hodge bundle on M extends to the Hodge bundle over M , denotedagain by E . For each irreducible representation ρ of GL have a bundle E ρ on M constructed by applying a Schur functor. We thus can consider T ρ = H ( M , E ρ )and elements of it are called Teichm¨uller modular forms of weight ρ and genus (ordegree) 3. There is an involution ι acting on the stack M associated to the doublecover M → A . If the characteristic is not 2 we can thus split T ρ into ± ι T ρ = T + ρ ⊕ T − ρ . We can identify the invariants under ι with Siegel modular forms T + ρ = M ρ (Γ ) (5)while the space T − ρ consists of the genuine Teichm¨uller modular forms.The pullback of χ to M is a square χ with χ a Teichm¨uller modular form ofweight 9 constructed by Ichikawa [21, 22].Using the identification (5) We have χ T − ρ ⊂ S ρ ′ (Γ ) with ρ ′ = ρ ⊗ det .We will now use invariant theory of ternary quartics Conclusion 8.1 implies that thepullback of a scalar-valued Teichm¨uller modular form of weight k is an invariant of weight3 k in I (3 , d defines a meromorphic Teichm¨uller modular formof weight d on M that becomes holomorphic after multiplication by an appropriatepower of χ . Indeed, an invariant of degree 3 d is defined by an equivariant embeddingdet( V ) d ֒ → Sym d (Sym ( V )) or taking care of the necessary twisting bydet( V ) d ֒ → Sym d (Sym ( V )) ⊗ det( V ) − d . IEGEL MODULAR FORMS AND INVARIANT THEORY 17
We thus get T −→ I (3 , −→ T [1 /χ ] , where the composition of the arrows is the identity. In particular, the Teichm¨ullermodular form χ maps to an invariant of degree 27 and since it is a cusp form one cancheck that it must be divisible by the discriminant, hence is a multiple of the discriminant.We can extend this to vector-valued Teichm¨uller modular formsΣ −→ C (3 , ν −→ Σ[1 /χ ]with the T -module Σ defined as Σ = ⊕ ρ T ρ with ρ running through the irreducible representations of GL .We can ask what the image ν ( f ) of the universal ternary quartic is. By constructionit is a meromorphic modular form of weight (4 , , − ( V ) ⊗ det( V ) − of GL .We know that there exists a holomorphic modular cusp form χ , , of weight (4 , , Proposition 9.1.
Over C the Siegel modular modular form χ ν ( f ) is a generator of S , , (Γ )( C ) .Proof. The cusp form χ , , maps to a concomitant of degree 28 that is divisible by thediscriminant. Therefore, χ , , − = χ , , /χ corresponds to a concomitant of degree 1.This must be a non-zero multiple of f . (cid:3) If we write the universal ternary quartic lexicographically as f = a x + a x y + · · · + a z and we write the meromorphic Teichm¨uller form χ , , − similarly lexicographically as χ , , − = α X + α X Y + · · · + α Z with dummy variables X, Y, Z to indicate the coordinates of χ , , − , we arrive at theanalogue for degree 3: Proposition 9.2.
The map ν : C (3 , → T [1 /χ ] is given by substituting α i for a i (and X, Y, Z for x, y, z and ˆ X, ˆ Y , ˆ Z for ˆ x, ˆ y, ˆ z ). In the following we restrict to F = C . One way to construct a generator of S , , (Γ )( C )is to take the Schottky form of degree 4 and weight 8 that vanishes on the Torelli locus.We can develop it along A , , the locus in A of products of abelian threefolds and ellipticcurves. It restriction to A , is a form in S (Γ ) ⊗ S (Γ ) and thus vanishes. The firstnon-zero term in the Taylor expansion along A , is χ , , ⊗ ∆ ∈ S , , (Γ ) ⊗ S (Γ )Since the Schottky form can be constructed explicitly with theta functions we can easilyobtain the beginning of the Fourier expansion. We refer to [7] for the details.In [6] we formulated a criterion that tells us which elements of C (3 ,
4) will give holo-morphic modular forms. We can associate to a concomitant its order along the locus of double conics by looking at its order in t when we evaluate it on the ternary quartic t f + q where q is a sufficiently general quadratic form in x, y, z . Then the result is thefollowing, see [6]. Theorem 9.3.
Let c be a concomitant of degree d and v ( c ) its order along the locus ofdouble conics. If d is odd then ν ( c ) χ is a Siegel modular form with order v ( c ) − ( d − / along the hyperelliptic locus. If d is even, then the order of ν ( c ) is v ( c ) − d/ . We formulate a corollary. Let M i,j,k (Γ ) ( m ) be the space of Siegel modular forms ofweight ( i, j, k ) vanishing with multiplicity ≥ m at infinity. (The weight ( i, j, k ) corre-sponds to the irreducible representation of GL of highest weight ( i + j + k, j + k, k ).)Moreover, let C d,ρ ( − m DC ) be the vector space of concomitants of type ( d, ρ ) that haveorder ≥ m along the locus of double conics. (Type ( d, ρ ) means belonging to an irre-ducible representation U of highest weight ρ occurring in Sym d (Sym ( V )).) Corollary 9.4.
The exists an isomorphism C d,ρ ( − m DC ) ∼ −→ M ( d − m ) ρ − ρ ,ρ − ρ ,ρ +9( d − m ) given by c ν ( c ) χ d − m . This allows now the construction of Siegel modular forms and Teichm¨uller modularforms of degree 3. In fact, in principle, all of them. As a simple example we decomposeSym (Sym ( V )) = V [8 , ,
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