
TL;DR
This paper introduces theta type Jacobi forms associated with positive definite even lattices, constructs three towers of such forms with a simple pullback structure, and explains the existence of a specific cusp form related to the $D_4$ lattice.
Contribution
It defines theta type Jacobi forms for lattices, constructs new towers of these forms, and links them to a special cusp form with symmetries of the $D_4$ lattice.
Findings
Constructed three towers of theta type Jacobi forms
Established a pullback-structure for these forms
Explained the existence of a weight 24 cusp form for $D_4$
Abstract
Let be a positive definite even lattice. We introduce theta type Jacobi forms and construct three towers of Jacobi forms with a particular easy pullback-structure. We use theta type Jacobi forms to explain the existence of a cusp form of weight 24 with respect to the irreducible root lattice which is based on additional symmetries of the Coxeter diagram.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Algebra and Geometry · Algebraic structures and combinatorial models · Advanced Combinatorial Mathematics
Jacobi forms of theta type
Martin Woitalla
Abstract
Let be a positive definite even lattice. We introduce theta type Jacobi forms and construct three towers of Jacobi forms with a particular easy pullback-structure. We use theta type Jacobi forms to explain the existence of a cusp form of weight 24 with respect to the irreducible root lattice which is based on additional symmetries of the Coxeter diagram.
1 Introduction
We give a short presentation of the contents. Let be a real quadratic space of signature where . We will always assume that . The bilinear form of is denoted by . The group of all isometries of is called the orthogonal group of and is given by
[TABLE]
By we denote the subgroup of index two which equals the kernel of the determinant character. We obtain another subgroup of index two as the kernel of the real spinor norm. The intersection of the groups and is denoted by . This is the connected component of the identity and is well-known to be a semisimple and noncompact Lie group, compare for ex. [14]. Its maximal compact subgroup is given by . One can construct a projective model for the Hermitian symmetric space namely
[TABLE]
Here the superscript means that we have chosen one of the two connected components. The associated affine cone is defined as
[TABLE]
Definition 1.1**.**
A subset is called a lattice in if there exist linearly independent vectors in such that . The quantity is called the rank of and is denoted by . We call even if for all the condition is satisfied. A lattice is called a full lattice in if . A lattice is called positive definite if for all we have . For any we define the rescaled lattice to be the set equipped with the bilinear form .
Let be a full lattice in such that is even and splits two hyperbolic planes. In view of the Gramian of this means that there exists a basis such that
[TABLE]
where denotes the Gramian of a positive definite even lattice . We consider the arithmetic subgroup
[TABLE]
For any subgroup of finite index we consider the modular variety . This is a noncompact space. We can add cusps to such that
[TABLE]
is compact, compare e.g. [3]. Here runs through the finitely many -orbits of isotropic lines and runs through the finitely many -orbits of isotropic planes in . Each corresponds to a modular curve and is called a one-dimensional cusp of and each corresponds to a point and is called a zero-dimensional cusp. The one-dimensional cusps are represented by rational two-dimensional totally isotropic subspaces of and the zero-dimensional cusps are represented by primitive isotropic vectors .
Definition 1.2**.**
Let be a subgroup of . A modular form of weight and character with respect to is a holomorphic function such that
[TABLE]
A modular form is called a cusp form if it vanishes at every cusp. The space of modular forms of weight and character for the group will be denoted by . For the subspace of cusp forms we will write .
Orthogonal type modular forms are very important tools to investigate the modular variety . In this text we will focus our interest on the Jacobi group which is a distinguished parabolic type subgroup of , see section 2 for the definition. Modular forms with respect to are called Jacobi forms and can be considered as a natural generalization of the classical Jacobi forms appearing in [7].
Definition 1.3**.**
Let be an even lattice in . The dual lattice is the -module
[TABLE]
Since in this case we can define the discriminant group as the finite abelian group
[TABLE]
The lattice is called maximal if for any even lattice satisfying both and we already have .
The group acts on the discriminant group . The kernel of this action is denoted by . The following construction is due to Gritsenko, see [11] for the details.
Theorem 1.4** (Gritsenko’93).**
Let be a Jacobi form with trivial character and weight where . There exists a linear operator defined on the space of Jacobi forms of weight into the space of modular forms of weight such that
[TABLE]
*For maximal even lattices this lift maps cusp forms to cusp forms. *
In section 2 we will give a review of Jacobi forms in many variables. In section 3 we use Jacobi’s theta-series and the weak Jacobi form in two variables of weight [math] and index to present pullback constructions for several series of lattices. The forms obtained in this way are called theta type Jacobi forms. In the last section we give a new explanation for the existence of the cusp form of weight 24 for the lattice which is based on its representation as an automorphic product defined by theta type Jacobi forms.
2 Jacobi forms
The presentation of the theory is influenced by the classical book on Jacobi forms in one variable of Eichler and Zagier [7]. For the description of Jacobi forms in several variables we follow [11] and [6]. Let be a real quadratic space of signature where . Let be a positive definite even lattice of rank with bilinear form such that . By construction contains two perpendicular hyperbolic planes and . Let be the totally isotropic plane spanned by and . We define the parabolic subgroup of fixing as
[TABLE]
The matrix realization of with respect to our standard basis of is well-known. The integral Heisenberg group of the lattice is generated by the elements
[TABLE]
where and such that . The group law of is
[TABLE]
We define an embedding of into as
[TABLE]
The group acts on the Heisenberg group by conjugation
[TABLE]
Note that since . The integral Jacobi group is defined as the subgroup of which acts trivially on the sublattice . This group is isomorphic to the semi-direct product
[TABLE]
Let be a finite character of the Jacobi group . The structure of the Jacobi group implies
[TABLE]
where is a finite character of and is a finite character for the group . The character satisfies the equation
[TABLE]
We recall the definition of Dedekind‘s eta function where denotes the upper half plane in . This function is defined by the product expansion
[TABLE]
The transformation law for the generators of is
[TABLE]
where we have chosen the branch of the square root which is positive if . There exists a multiplier system of order 24 whose square is a finite character satisfying
[TABLE]
For more details on the function we refer to [1], section 3. We define the quantities as the generators of the ideal in generated by or , respectively, where . These quantities are called scale and norm of the lattice . Obviously For a proof of the next Proposition see [6] and [17].
Proposition 2.1**.**
- (a)
The group of finite characters of is a cyclic group of order 12 and is generated by . 2. (b)
Let be a character of finite order such that
[TABLE]
for all and Then is of the shape
[TABLE]
where such that .
We extend to by -linearity. Let and . We define an action of the Jacobi group on the space of holomorphic functions on by means of the generators of :
[TABLE]
where , and . The two assignments jointly define an action of the Jacobi group on the space of holomorphic functions. Let be a holomorphic function satisfying
[TABLE]
where is a finite character for the Jacobi group. Then is periodic in and since the identities
[TABLE]
hold true for any and for some . Thus the function has a Fourier expansion of the shape
[TABLE]
We now introduce the notion of a Jacobi form.
Definition 2.2**.**
Let and . A holomorphic function
[TABLE]
is called a weak Jacobi form of weight and index with character if it satisfies
[TABLE]
and has a Fourier expansion as in (2) where additionally
[TABLE]
We call a holomorphic Jacobi form if the Fourier expansion ranges over all with and is called a Jacobi cusp form if it ranges over all satisfying . We denote by the vector space of weak Jacobi forms and the corresponding spaces of holomorphic forms and cusp forms are denoted by and , respectively. If the character is trivial we also write for short.
Remark 2.3**.**
- (a)
The action can be extended for and being a multiplier system for . Here we have to replace by the metaplectic cover , see e.g. [4]. In this situation we also use the notation for the spaces of Jacobi forms. 2. (b)
A Jacobi form is called singular if it has weight . 3. (c)
The notion of a Jacobi form is compatible with Definition 1.2. To see this we note that we have an affine model for the homogeneous domain given by
[TABLE]
where we have used the abbreviations
[TABLE]
Let where we assume . We define a holomorphic function on by
[TABLE]
Since and are biholomorphically equivalent we can interpret as an element of . 4. (d)
Suppose is a non-vanishing Jacobi form of index and weight with character . Consider the extended Jacobi form . By part (b) we know that can be interpreted as a modular form. If we transform this function by an investigation of the modular behaviour on the affine domain yields and Proposition 2.1 implies
[TABLE] 5. (e)
The Jacobi forms for a fixed lattice and with trivial character form a bigraded ring. The transformation behaviour of elements belonging to is that of a Jacobi form of weight and index . If both factors are holomorphic then the product is also a holomorphic Jacobi form by virtue of the identity
[TABLE]
which holds for all , and .
We finish this section with several examples of Jacobi forms for the root lattice which correspond to Jacobi forms in the sense of [7]. In the sequel we denote by the standard scalar product in . The standard basis of is denoted by
[TABLE]
For we set .
Example 2.4**.**
- (a)
Dedekind’s eta function is a Jacobi form of weight and index [math] for every positive definite even lattice , thus where is not a character but a multiplier system for . For one defines
[TABLE]
We want to mention the following infinite sum-expansion for
[TABLE]
which follows from the Jacobi triple identity. 2. (b)
In [7] the authors constructed the Jacobi-Eisenstein series . One defines a Jacobi cusp form as
[TABLE]
where
[TABLE]
denotes the classical Eisenstein series of weight for the group , confer e.g. [15], p. 161. 3. (c)
We define
[TABLE]
In accordance with the classical theory of elliptic modular forms we write
[TABLE]
for the first cusp form for . One has since
[TABLE]
regarding the convention
[TABLE]
Moreover the function defined by is a holomorphic Jacobi form of weight and index 1 for the lattice with Fourier expansion
[TABLE] 4. (d)
The Jacobi theta-series of characteristic is given by
[TABLE]
where and . This function was originally discovered by Carl Gustav Jacob Jacobi. In [13] the authors reinterpreted this function as a modular form of half-integral weight and index. The same proof as in [15], section I.6 can be used to show that defines a holomorphic function on . The transformation behaviour under the action of is
[TABLE]
Moreover we have
[TABLE]
and
[TABLE]
by the theta transformation formula. This shows that transforms like a Jacobi form of weight and index . If we consider the extended Jacobi form we obtain
[TABLE]
From the infinite sum expansion we obtain
[TABLE]
by (4). This shows that transforms like a Jacobi form with mutiplier system . Moreover Jacobi’s triple identity yields
[TABLE]
Since each summand in the infinite sum defining is of the shape we see that is a holomorphic Jacobi form. The function has the properties
[TABLE]
for all . Hence the divisor of contains Finally a classical argument from the theory of complex functions yields
[TABLE]
compare for example [5], section 5.6.
3 Theta type Jacobi forms for several root lattices
In this section we give several pullback constructions for Jacobi forms. The constructions are motivated by [10] where the author constructed towers of reflective modular forms by means of Jacobi forms. All lattices appearing in this section are even. The next Lemma can be found in [6], Proposition 3.1.
Lemma 3.1**.**
Let be a sublattice of such that and be a Jacobi form of weight and index for the character . Consider the decomposition . We define the pullback of to as the function on defined by
[TABLE]
*Then and the pullback maps cusp forms to cusp forms. *
Definition 3.2**.**
Let be a sublattice of such that and be a Jacobi form of weight and index . Let . We say that is a pullback of if there exists some such that In this case we use the notation . We set if .
Let be a positive definite even lattice. We define
[TABLE]
We now introduce Jacobi forms of theta type.
Definition 3.3**.**
Let be a positive definite even lattice and . We say that is of theta type if there exists a sublattice , and integers such that
[TABLE]
Proposition 3.4**.**
Let be a positive definite even lattice. We assume that its bilinear form is .
- (a)
The function transforms like a Jacobi form of index and weight for the lattice with a character having trivial Heisenberg part. The Fourier expansion of this function is of the shape
[TABLE]
for some . 2. (b)
Let and suppose such that . We assume that there exists a such that
[TABLE]
We define a function by
[TABLE]
Then transforms like a Jacobi form of index 2 for the lattice with a character having trivial Heisenberg part and satisfies .
Proof**.**
- (a)
For any the identity
[TABLE]
is true. Hence and for any we obtain
[TABLE]
From Example 2.4 (d) we deduce that has the transformation behaviour and character claimed above. The same example yields that each summand of the Fourier expansion is of the shape
[TABLE]
where and . Note that . Now define
[TABLE]
to write the above summand as
[TABLE]
This completes the proof of part (a). 2. (b)
The statement follows from the identity
[TABLE]
and an analogue investigation as in part (a). The statement on the pullback follows from the facts and if .
We now restrict ourselves to some special types of (root) lattices. We first note that for any the space of classical Jacobi forms of weight and index as defined in [7] coincides with the space . For the next Lemma we consider the lattice which can be realized as equipped with the bilinear form . Note that every can be written in the form
[TABLE]
Lemma 3.5**.**
Let and . We define a function
[TABLE]
where
[TABLE]
is an element of . Moreover we have
[TABLE]
Proof**.**
Let and . We consider the decomposition for
[TABLE]
and decompose in the same manner. For every and one has
[TABLE]
which shows that has the correct transformation behaviour. Since we conclude that has a Fourier expansion of the shape
[TABLE]
where .
Following [10] and [6] we can construct Jacobi forms for the special case and .
Proposition 3.6**.**
Let and consider the coordinates (5).
- (a)
We have where
[TABLE] 2. (b)
We define the functions
[TABLE]
Then belongs to and belongs to .
Proof**.**
Since by our construction the ambient vector space of is we can derive the holomorphicity of by the Fourier expansion given in Proposition 3.4 part (a). The Heisenberg part of these functions is a binary character since the bilinear form of is instead of . The last fact is also responsible for the appearance of a half-integral index in this case. The other statements are direct consquences of Proposition 3.4 Example 2.4 part (a) and (d).
The previous considerations show that there exists an operator
[TABLE]
which extends to Jacobi forms of index zero being isomorphic to the space of elliptic modular forms of weight by the same construction. This enables us to build a tower of theta type Jacobi forms for .
Proposition 3.7**.**
Let . We have the following diagram of Jacobi forms
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Delta_{12}}$${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\varphi_{12,A_{1}}}$${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\varphi_{12,2A_{1}}}$${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\varphi_{12,3A_{1}}}$$\varphi_{12,4A_{1}}$${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\psi_{10,A_{1}}}$${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\varphi_{10,2A_{1}}}$${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\varphi_{10,3A_{1}}}$$\varphi_{10,4A_{1}}$${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\psi_{8,2A_{1}}}$${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\varphi_{8,3A_{1}}}$$\varphi_{8,4A_{1}}$${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\psi_{6,3A_{1}}}$$\varphi_{6,4A_{1}}$$\psi_{4,4A_{1}}$$\left.\frac{\partial^{2}}{\partial z_{4}^{2}}\right|_{z_{4}=0}$$\left.\frac{\partial^{2}}{\partial z_{3}^{2}}\right|_{z_{3}=0}$$\left.\frac{\partial^{2}}{\partial z_{2}^{2}}\right|_{z_{2}=0}$$\left.\frac{\partial^{2}}{\partial z_{1}^{2}}\right|_{z_{1}=0}
*where . Except for the last line all forms are cusp forms. The functions at the beginning of each arrow are defined by application of to the function on the arrowhead. Here the quasi-pullbacks on the diagonal are taken with respect to the coordinates (5). *
Proof**.**
The pullback-structure of the diagram follows from Proposition 3.6 and Proposition 3.4. If we rewrite the Fourier expansion of given in Example 2.4, part (c) according to Definition 2.2, we obtain that the hyperbolic norm of all indices belonging to non-vanishing Fourier coefficients is bounded from below by . Note that multiplication with adds to this bound. If we take into account that consists of perpendicular copies of we find that this proves the assertion on the holomorphicity and on cusp forms. Part (d) of the same Example yields the quasi-pullback structure of the diagonal.
We consider another series of root lattices. For the root system is described by
[TABLE]
and can be considered as or , respectively. The first of the following two constructions also appeared in [10] but in different coordinates.
Proposition 3.8**.**
- (a)
We have and for there are Jacobi forms given by
[TABLE]
For these functions define Jacobi cusp forms. These forms are obtained by iterative quasi-pullbacks of , namely
[TABLE] 2. (b)
We have the following diagram of holomorphic Jacobi forms where all forms except for the first column are cups forms of the weight indicated by the index
\psi_{4,D_{8}}$$\phi_{10,D_{8}(2)}$$\phi_{12,D_{8}(2)}$$\phi_{14,D_{8}(2)}$$\phi_{16,D_{8}(2)}$$\phi_{18,D_{8}(2)}$$\phi_{20,D_{8}(2)}$$\phi_{22,D_{8}(2)}$$\psi^{2}_{5,D_{7}}$$\phi_{12,D_{7}(2)}$$\phi_{14,D_{7}(2)}$$\phi_{16,D_{7}(2)}$$\phi_{18,D_{7}(2)}$$\phi_{20,D_{7}(2)}$$\phi_{22,D_{7}(2)}$$\psi^{2}_{6,D_{6}}$$\phi_{14,D_{6}(2)}$$\phi_{16,D_{6}(2)}$$\phi_{18,D_{6}(2)}$$\phi_{20,D_{6}(2)}$$\phi_{22,D_{6}(2)}$$\psi^{2}_{7,D_{5}}$$\phi_{16,D_{5}(2)}$$\phi_{18,D_{5}(2)}$$\phi_{20,D_{5}(2)}$$\phi_{22,D_{5}(2)}$$\psi^{2}_{8,D_{4}}$$\phi_{18,D_{4}(2)}$$\phi_{20,D_{4}(2)}$$\phi_{22,D_{4}(2)}$$\psi^{2}_{9,D_{3}}$$\phi_{20,D_{3}(2)}$$\phi_{22,D_{3}(2)}$$\psi^{2}_{10,D_{2}}$$\phi_{22,D_{2}(2)}$$\psi^{2}_{11,D_{1}}
where the functions at the beginning of each arrow are defined by application of to the function on the arrowhead. More precisely we have
[TABLE]
and is a cusp form if and only if . All the functions except for the first line have the property
[TABLE]
regarding the convention
The arithmetic Lifting is the cusp form of weight 11 for the Siegel paramodular group of level 2.
Proof**.**
- (a)
We first note that our model for is compatible with the requirements of Proposition 3.4. Hence the statements on the modular behaviour and the identification of the character follow directly as well as the property under quasi-pullbacks. It remains to investigate the holomorphicity. Since is a full lattice we conclude from part (a) of Proposition 3.4 that is indeed holomorphic at infinity. Moreover for the forms are cuspidal because they equal the product of a holomorphic Jacobi form of singular weight and a cusp form of index 0. 2. (b)
From part (a) and the iterative construction we can easily derive that is a holomorphic Jacobi form whose Fourier coefficients are parametrized by all pairs such that . Now Proposition 3.4 completes the proof.
We finish this section considering root lattices of type . Here we use the model
[TABLE]
for where . Note that in this case . Hence is not a full lattice according to our realization.
Proposition 3.9**.**
- (a)
Let . We define
[TABLE]
Then and there are Jacobi forms given by
[TABLE]
For these functions define Jacobi cusp forms. These forms are obtained by iterative quasi-pullbacks of , namely
[TABLE] 2. (b)
We have the following diagram of holomorphic Jacobi forms where all forms except for the first column are cups forms of the weight indicated by the index
\psi_{4,A_{7}}$$\phi_{10,A_{7}(2)}$$\phi_{12,A_{7}(2)}$$\phi_{14,A_{7}(2)}$$\phi_{16,A_{7}(2)}$$\phi_{18,A_{7}(2)}$$\psi^{2}_{5,A_{6}}$$\phi_{12,A_{6}(2)}$$\phi_{14,A_{6}(2)}$$\phi_{16,A_{6}(2)}$$\phi_{18,A_{6}(2)}$$\psi^{2}_{6,A_{5}}$$\phi_{14,A_{5}(2)}$$\phi_{16,A_{5}(2)}$$\phi_{18,A_{5}(2)}$$\psi^{2}_{7,A_{4}}$$\phi_{16,A_{4}(2)}$$\phi_{18,A_{4}(2)}$$\psi^{2}_{8,A_{3}}$$\phi_{18,A_{3}(2)}$$\psi^{2}_{9,A_{2}}
where the functions at the beginning of each arrow are defined by application of to the function on the arrowhead. More precisely we have
[TABLE]
and is a cusp form if and only if . All the functions except for the first line have the property
[TABLE]
regarding the convention
Proof**.**
- (a)
We start with the treatment of . As before the transformation behaviour under modular substitutions and the shape of the character follow directly from Proposition 3.4. The only thing which remains to show is the holomorphicity. In this case the proof of this fact is less obvious than in the previous cases since the dimension of the ambient vector space of is strictly greater than . We start with the Fourier expansion of the function . Each summand is of the shape
[TABLE]
where . Since the last term can be rewritten as
[TABLE]
where
[TABLE]
The hyperbolic norm of the last expression equals
[TABLE]
since the parity of is odd. This shows that is indeed a holomorphic Jacobi form since
[TABLE]
Now let . We decompose as
[TABLE]
With respect to this basis decomposes as
[TABLE]
where is realized as a lattice in . Now an induction and the holomorphicity of show that is a holomorphic Jacobi form. The statements on are obvious. 2. (b)
The proof of this assertion is completely analogue to the proof of Proposition 3.8 part (b).
The length of the towers can be arbitrarily increased if we multiply them by powers of to preserve the holomorphicity.
4 Additional symmetries for the lattice
In this section we use theta type Jacobi forms to explain the existence of a particular modular form for the lattice . For any we define the reflection at the hyperplane as
[TABLE]
Let such that . The rational quadratic divisor with respect to is given as
[TABLE]
In particular
[TABLE]
Note that We now state a variant of Borcherds multiplicative lifting which can be found in [9].
Theorem 4.1** (Borcherds, Gritsenko).**
Let be a weakly holomorphic Jacobi form with Fourier coefficients . Assume that if . Then there exists a modular form of weight with respect to satisfying
[TABLE]
For the details of the construction of we refer to [9] and [2]. The root lattices of type have been defined in the last section. Using the same coordinates as before let . In general the only symmetries of the Coxeter diagram of are given by the permutation of and :
If the diagram has additional symmetries:
In this case we can define two additional singular Jacobi forms
[TABLE]
If we use our standard coordinates we obtain:
[TABLE]
For any we define to be the generator of the ideal and we put . A vector is called primitive if
[TABLE]
The following Proposition is very useful if one likes to determine the orbits of the divisor of a multiplicative lifting under the orthogonal group. A proof can be found in [12].
Proposition 4.2** (Eichler criterion).**
*If are primitive, and then there exists a such that . *
The next Theorem can be found in [10].
Theorem 4.3** (Gritsenko ’10).**
Let . There exists a modular form where
[TABLE]
In the last case is a subgroup of index three and the character is defined as
[TABLE]
The divisor of is given by
[TABLE]
if and for it is given by the orbit
[TABLE]
*If then is a cusp form. *
The proof of this Theorem uses Theorem 4.1. The case plays a particular role in this tower because the maximal modular group and the divisor of are smaller. In the next Theorem we use theta type Jacobi forms to construct a modular form of weight 24 which can be considered as the correct replacement for .
Theorem 4.4**.**
There is a cusp form with a binary character
[TABLE]
Its divisor is equal to
[TABLE]
This function appeared in [8] first and later in [16] where the author determined the graded ring of modular forms for the Hurwitz order.
Proof**.**
For the proof we will first verify that
[TABLE]
are multiplicative liftings as in Theorem 4.1. The discriminant group of has the four classes
[TABLE]
where was defined in the diagram above. The group is generated by and the natural embeddings of and into . We start by investigating . From the Fourier-Jacobi criterion for automorphic products, compare [9], Corollary 3.3 we deduce that
[TABLE]
where is the Hecke operator defined in [11], if . A direct computation yields
[TABLE]
where
[TABLE]
Hence is a weak Jacobi form which satisfies the preliminaries of Theorem 4.1. If the hyperbolic norm in the Fourier exansion of a weak Jacobi form is bounded from below by the quantity
[TABLE]
Now assume there is a pair such that . Then one observes
[TABLE]
Since the lattice represents all such values under we know that all the information about the divisor is completely contained in the part of since the Fourier coefficients of a Jacobi form depend only on the hyperbolic norm and the class of . Now Theorem 4.1 and Proposition 4.2 yield that the divisor of is exactly
[TABLE]
By constrcution this divisor is contained in the divisor of . By Koecher’s principle we can divide the last function by and conclude that they coincide up to a multiple in and a second application of the Fourier-Jacobi criterion for automorphic products yields that this constant is actually one in accordance with Theorem 4.3. So far we have just repeated the proof of Theorem 4.3 for the case . Now the same construction can be applied to which yields again modular forms of weight where the divisor is determined by the orbit of or , respectively. We consider the product
[TABLE]
Then has the representation
[TABLE]
From this representation and the above representation in coordinates we deduce
[TABLE]
Since these two elements generate the group of the graph automorphisms of the Coxeter diagram of which is isomorphic to the symmetric group in three letters we see that is -modular with respect to the -character. If we define as the product
[TABLE]
the definition of yields a cusp form whose divisor is the composition of the divisors of the factors which is -modular with respect to a binary character induced by the sign-character.
Acknowledgements
I would like to thank Valery Gritsenko for many valuable discussions and for his help.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] T.M. Apostol. Modular Functions and Dirichlet Series in Number Theory , volume 21. Springer-Verlag New York, Berlin, Heidelberg, 1976.
- 2[2] R.E. Borcherds. Automorphic forms with singularities on Grassmannians. Invent. Math. , 123:491–562, 1998.
- 3[3] A. Borel and L. Ji. Compactifications of symmetric and locally symmetric spaces . Mathematics: Theory & Applications. Birkhäuser, Boston, 2006.
- 4[4] J. H. Bruinier. Borcherds Products on O ( 2 , l ) O 2 𝑙 \operatorname{O}(2,l) and Chern classes of Heegner Divisors , volume 1. Springer-Verlag, Berlin, Heidelberg, 2002.
- 5[5] R. Busam and E. Freitag. Funktionentheorie , volume 1. Springer-Verlag, Berlin, Heidelberg, 1993.
- 6[6] F. Clery and V. Gritsenko. Modular forms of orthogonal type and Jacobi Theta-series. ar Xiv:1106.4733 [math.AG] .
- 7[7] M. Eichler and D. Zagier. The Theory of Jacobi Forms . Birkhäuser, Boston, Basel, Stuttgart, 1985.
- 8[8] E. Freitag and C.F. Hermann. Some modular varieties of low dimension. Advances in Mathematics , 152:203–287, 2000.
