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# Lattices with many Borcherds products

Jan Hendrik Bruinier [ Stephan Ehlen [ Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstraße 7, D–64289 Darmstadt, Germany  and  Eberhard Freitag [ Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 288, D–69120 Heidelberg, Germany
May 10, 2020
###### Abstract.

We prove that there are only finitely many isometry classes of even lattices of signature for which the space of cusp forms of weight for the Weil representation of the discriminant group of is trivial. We compute the list of these lattices. They have the property that every Heegner divisor for the orthogonal group of can be realized as the divisor of a Borcherds product. We obtain similar classification results in greater generality for finite quadratic modules.

###### 2010 Mathematics Subject Classification:
11F12, 11E20, 11–04, 14C22
The first and the second author are partially supported by DFG grant BR-2163/4-1.

J. H. Bruinier] S. Ehlen] E. Freitag]

## 1. Introduction

Let be an even lattice of signature and write for its orthogonal group. In his celebrated paper [Bo1] R. Borcherds constructed a map from vector valued weakly holomorphic elliptic modular forms of weight to meromorphic modular forms for whose zeros and poles are supported on Heegner divisors. Since modular forms arising in this way have particular infinite product expansions, they are often called Borcherds products. They play important roles in different areas such as Algebraic and Arithmetic Geometry, Number Theory, Lie Theory, Combinatorics, and Mathematical Physics.

By Serre duality, the obstructions for the existence of weakly holomorphic modular forms with prescribed principal part at the cusp at are given by vector valued cusp forms of dual weight transforming with the Weil representation associated with the discriminant group of [Bo2]. In particular, if there are no non-trivial cusp forms of this type, then there are no obstructions, and every Heegner divisor is the divisor of a Borcherds product. A lattice with this property is called simple.

It was conjectured by the third author that there exist only finitely many isomorphism classes of such simple lattices. Under the assumptions that and that the Witt rank of (i.e. the dimension of a maximal totally isotropic subspace of ) is , it was proved by M. Bundschuh that there is an upper bound on the determinant of a simple lattice [Bu]. Unfortunately, this bound is very large and therefore not feasible to obtain any classification results. The argument of [Bu] is based on volume estimates for Heegner divisors and the singular weight bound for holomorphic modular forms for .

The purpose of the present paper is twofold. First, we show that for any (without any additional assumption on the Witt rank) there exist only finitely many isomorphism classes of even simple lattices of signature , see Theorem 4.5 and Corollary 4.7. Second, we develop an efficient algorithm to determine all of these lattices. It turns out that there are exactly isomorphism classes. Table 1 shows how many of those occur in the different signatures. The corresponding genus symbols (see Section 2) of these lattices are listed in Tables 7, 8111Tables with global realizations can be obtained from [Ehl]..

It is also interesting to record the Witt ranks of these lattices. For , we have anisotropic lattices. The corresponding modular varieties are Shimura curves while the remaining modular varieties for are modular curves. For , there are of Witt rank and of Witt rank but no anisotropic lattices. Finally, if , all simple lattices have Witt rank .

Along the way, we obtain several results on modular forms associated with finite quadratic modules which are of independent interest and which we now briefly describe. A finite quadratic module is a pair consisting of a finite abelian group together with a -valued non-degenerate quadratic form on , see [Ni], [Sk2]. Important examples of finite quadratic modules are obtained from lattices. If is an even lattice with dual lattice , then the quadratic form on induces a -valued quadratic form on the discriminant group .

Recall that there is a Weil representation of the the metaplectic extension of on the group ring of a finite quadratic module , see Section 3. If , we write for the space of cusp forms of weight and representation for the group . We say that a finite quadratic module is -simple if . With this terminology, an even lattice is simple if and only if is -simple.

The dimension of the space can be computed by means of the Riemann-Roch theorem. Therefore a straightforward approach to showing that there are nontrivial cusp forms consists in finding lower bounds for the dimension of . Unfortunately, the dimension formula (3.3) involves rather complicated invariants of at elliptic and parabolic elements, and it is a non-trivial task to obtain sufficiently strong bounds. In the present paper we resolve this problem. For instance, we obtain the following result (see Theorem 4.5 and Corollary 4.6).

###### Theorem.

For every , there is a , such that

 ∣∣∣dim(Sk,A)−dim(M2−k,A(−1))−|A/{±1}|⋅k−112∣∣∣≤Cε⋅|A/{±1}|⋅Nε−12A

for every finite quadratic module and every weight with . Here is the level of , and denotes the abelian group equipped with the quadratic form .

In Corollary 4.7 we conclude that there exist only finitely many isomorphism classes of finite quadratic modules with bounded number of generators such that for some weight satisfying the condition of the theorem. In particular, there are only finitely many isomorphism classes of simple lattices. Note that there do exist infinitely many isomorphism classes of -simple finite quadratic modules, see Remark 4.8.

Since the constant implied in the Landau symbol in the above theorem is large, it is a difficult task to compute the list of all -simple finite quadratic modules for a bounded number of generators. We develop an efficient algorithm to address this problem. The idea is to first compute all anisotropic finite quadratic modules that are -simple for some . To this end we derive an explicit formula for in terms of class numbers of imaginary quadratic fields and dimension bounds that are strong enough to obtain a classification (Theorem 4.10 and Table 5).

Next we employ the fact that an arbitrary finite quadratic module has a unique anisotropic quotient , and that there are intertwining operators for the corresponding Weil representations. For the difference very efficient bounds can be obtained. This can be used to classify all -simple finite quadratic modules with a bounded number of generators, see Algorithm 5.5 and the tables in Section 5.

To resolve the problem of finding all simple lattices of signature , it remains to test which of these simple discriminant forms arise as discriminant groups of even lattices of signature . This is done in Section 6 by applying a criterion of [CS]. Finally, in Section 6.1 we explain some applications of our results in the context of Borcherds products.

We thank R. Schulze-Pillot and N.-P. Skoruppa for their help. We also thank D. Allcock and A. Mark for their help with an earlier version of this paper. Some of the computations were performed on computers at the University of Washington, USA, supported by W. Stein and the National Science Foundation Grant No. DMS-0821725. We would also like to thank the anonymous referee for many helpful comments.

Let be a finite quadratic module (also called a finite quadratic form or discriminant form in the literature), that is, a pair consisting of a finite abelian group together with a -valued non-degenerate quadratic form on . We denote the bilinear form corresponding to by . Recall that is called degenerate if there exists an , such that for all . Otherwise, is called non-degenerate.

The morphisms in the category of finite quadratic modules are group homomorphisms that preserve the quadratic forms. In particular, two finite quadratic modules and are isomorphic if and only if there is an isomorphism of groups , such that .

In this section we collect some important facts about finite quadratic modules, which are well known among experts but not easily found in the literature. We mainly follow Skoruppa [Sk3]. Other good references include [Ni], [No], [Sk2], [Str].

If is an even lattice, the quadratic form on induces a -valued quadratic form on the discriminant group of . The pair defines a finite quadratic module, which we call the discriminant module of . According to [Ni], any finite quadratic module can be obtained as the discriminant module of an even lattice . If then denotes the real signature of , the difference is determined by its discriminant module modulo by Milgram’s formula

 1√|A|∑a∈Ae(Q(a))=e((b+−b−)/8).

Here and throughout we abbreviate for . We call the signature of .

We let be the level of defined by

 N=min{n∈Z>0∣nQ(x)∈Z for all x∈A}.

It is easily seen that is a divisor of .

If and are two finite quadratic modules then the orthogonal direct sum also defines a finite quadratic module. Here for and . We call a finite quadratic module indecomposable if it is not isomorphic to such a direct sum with non-zero and .

The finite quadratic module is isomorphic to the orthogonal sum of its -components with running through the primes.

Next we describe a list of indecomposable finite quadratic modules.

###### Definition 2.1.

Let be a prime and be an integer not divisible by . We define the following elementary finite quadratic modules.

 Atpk =(Z/pkZ,tx2pk) for p>2, At2k =(Z/2kZ,tx22k+1), B2k =(Z/2kZ⊕Z/2kZ,x2+2xy+y22k), C2k =(Z/2kZ⊕Z/2kZ,xy2k),
###### Theorem 2.2.
1. The finite quadratic modules listed in Definition 2.1 are indecomposable.

2. Every indecomposable finite quadratic module is isomorphic to a finite quadratic module as in Definition 2.1.

3. Moreover, every finite quadratic module is isomorphic to a direct sum of indecomposable finite quadratic modules.

###### Proof.

Statement (1) is clear. The other two statements follow from the classification of -adic lattices. See [Ni] for details, in particular Proposition 1.8.1. ∎

Consider a decomposition of the -components of a finite quadratic module as a direct sum

 Ap=Ap,1⊕…⊕Ap,lp,

where each is a direct sum of elementary finite quadratic modules , with , and .

Such a decomposition is called a Jordan decomposition of and the direct summands are called Jordan components of . We will also call any finite quadratic module that is isomorphic to a direct sum of elementary finite quadratic modules for a fixed a Jordan component.

We will now describe a handy notation for such a Jordan decomposition. The symbols we use are essentially those introduced by Conway and Sloane [CS] for the genus of an integral lattice, that is, its class under rational equivalence. The following statement (see [Ni], Corollary 1.16.2) motivates the use of their symbols for us.

###### Proposition 2.3.

Two even lattices and that have the same real signatures have isomorphic finite quadratic modules if and only if and are in the same genus.

The following two lemmas are straightforward to prove (see also [Ni], Proposition 1.8.2).

###### Lemma 2.4.

Let be a prime and let for a positive integer .

1. We have if and only if .

2. Suppose that and . Then .

3. In particular, if is a Jordan component of the form

 A=n⨁i=1Atiq,

then is isomorphic to

 Atq⊕n−1⨁i=1Asq, with % n∏i=1(2tip)=(2tp)

for any with .

###### Lemma 2.5.

Let for a positive integer . Moreover, let be odd integers.

1. We have if and only if .

2. If , then if and only if .

3. Let such that

 n∑i=1si≡n∑i=1ti(mod8) and n∏i=1(si2)=n∏i=1ti.

Then

 n⨁i=1Asiq≅n⨁i=1Atiq.
4. We have .

5. Moreover, we have with and

 3∏i=1(ti2)=−(t2).
6. Finally, with and

 3∏i=1(ti2)=(t2).
###### Definition 2.6.

Using the preceding lemmas, we define a symbol for a Jordan decomposition of a finite quadratic module as follows. First of all, by convention, we write or for the trivial module with the -map as quadratic form.

Now let be a Jordan component, be a prime and .

1. If is odd, the two isomorphism classes of Jordan components in Lemma 2.4, (3) are denoted , where .

2. Let .

1. We write if is isomorphic to with and . We normalize to be contained in the set and if , we take .

2. We write if is isomorphic to copies of .

3. And we write if is isomorphic to copies of and one copy of .

For a general finite quadratic module, we concatenate the symbols of the Jordan components as defined above.

###### Example 2.7.

The Jordan decomposition has the symbol and is the symbol for the Jordan decomposition .

###### Proposition 2.8.

Let and be finite quadratic modules and let be a prime. If , then the corresponding -components of the genus symbols of and coincide.

###### Proof.

This follows from the uniqueness of the Jordan decomposition in the case of an odd prime (see, for instance, Theorem 5.3.2 in [Ki]). ∎

###### Remark 2.9.

In contrast to Proposition 2.8, note that the symbol (and the Jordan decomposition) for is not uniquely determined by the isomorphism class. For instance, the Jordan decompositions and correspond to isomorphic finite quadratic modules. See also Theorem 2.14 in the next section.

For an integer we define the -torsion subgroup of a finite quadratic module by

 A[n]={γ∈A∣nγ=0}.

Moreover, we let be the image of the multiplication by map. Then we have the exact sequence

 0⟶A[n]⟶A⟶An⟶0,

and is the orthogonal sum of and . It follows from the theorem of elementary divisors and the Jordan decompositon (Theorem 2.2) that

 (2.1) |A[n]|≤2n|A|N.

The quantity can be expressed in term of the -torsion as

 (2.2) d=|A|2+|A[2]|2.

### 2.1. Gauss sums and divisor sums

We now collect some facts about Gauss sums and divisor sums associated to finite quadratic modules which we will need later. For an integer the Gauss sum is defined by

 (2.3) G(n,A)=∑γ∈Ae(nQ(γ)).

We have the elementary properties

 (2.4) G(−n,A) =¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯G(n,A), (2.5) G(n+N,A) =G(n,A), (2.6) G(n,A⊕B) =G(n,A)G(n,B).

The following lemma is a consequence of [Bo3, Lemma 3.1].

###### Lemma 2.10.

If then . For general we have the estimate

 |G(n,A)|≤√|A|√|A[n]|.

We will also need explicit formulas for the Gauss sums in some cases in Section 4.1. These are easily proven by relating to the standard Gauss sums (see, for instance [Str]).

###### Proposition 2.11.

Let be a prime. We have if and

 G(n,p±1)=±√p(p2)(np)e(1−p8)

if .

###### Proposition 2.12.

Let . For we put and . We have

 G(n,q±1t)=√q√(n,q)(tn′q′)⋅⎧⎪ ⎪⎨⎪ ⎪⎩e(tn′8),if q∤n,0,if q∣∣n,1,if 2q∣n.
###### Proposition 2.13.

Let with . We have

 G(n,q−2)=q(q,n)(3q′)

and

 G(n,q+2)=q(q,n).

The following theorem is used later on to decide when two given genus symbols correspond to isomorphic finite quadratic modules.

###### Theorem 2.14 ([Sk3]).

Let and be finite quadratic modules. Then and are isomorphic if and only if their underlying abelian groups are isomorphic and

 G(n,A)=G(n,B)

for all divisors of the level of and .

###### Proof.

It is clear that the condition is necessary. Using this, it is easy to prove the theorem for -components with . If and are finite quadratic modules of prime power level with for all , then follows from Lemma 2.4 and the explicit formula for the Gauss sum in Proposition 2.11. The general case is treated in [Sk3] in detail. ∎

For we define a divisor sum associated to by

 (2.7) σ(s,A)=∑a∣Nas√|A[a]|.

Here the sum runs over all positive divisors of the level of . If is another finite quadratic module of level coprime to , then

 σ(s,A⊕B)=σ(s,A)σ(s,B).

Consequently, is the product of the for running through the primes dividing .

###### Lemma 2.15.

For we have the estimate

 σ(s,A)≤√2|A|N⋅σs+1/2(N),

where denotes the usual divisor sum.

###### Proof.

This is a direct consequence of the estimate (2.1). ∎

## 3. Vector valued modular forms

We write for the metaplectic extension of , realized as the group of pairs , where and is a holomorphic function on the upper complex half plane with (see e. g. [Bo1], [Br1]). It is well known that is generated by

 T=((1101),1)andS=((0−110),√τ).

One has the relations , where is the standard generator of the center of .

The Weil representation associated with is a unitary representation of on the group algebra . If we denote the standard basis of by then can be defined by the action of the generators as follows (see also [Sk2], [Bo1], [Br1], where the dual of is used):

 (3.1) ρA(T)eγ =e(−Q(γ))eγ, (3.2) ρA(S)eγ =e(sig(A)/8)√|A|∑δ∈Ae((γ,δ))eδ.

Let . We denote by the vector space of -valued modular forms of weight with representation for the group . The subspace of cusp forms is denoted by . It is easily seen that , if .

The dimension of the vector space can be computed using the Riemann-Roch theorem or the Selberg trace formula. This is carried out in [Fr] and [Fi] in a more general situation. In our special case the following formula holds (see [Bo2] p. 228 and [Fr] Chapter 8.5, Theorem 5.1). For simplicity we assume that , since our application to simple lattices will only concern this case. Then the -dimensional subspace of is preserved by , and acts by multiplication with on . We denote by the restriction of to . If is a unitary matrix of size with eigenvalues and (for ), we define

 α(M)=d∑j=1νj.

If , the dimension of is given by

 (3.3) dim(Mk,A) =+dim(S2−k,A(−1)).

Furthermore, the dimension of is given by

 (3.4) dim(Sk,A)=first line of (???)−|{γ∈A/{±1};Q(γ)∈Z}|+dim(M2−k,A(−1)).

Here denotes the finite quadratic module given by the abelian group equipped with the quadratic form . If , then vanishes. If , then is equal to the space of -invariants in for the dual representation of . Finally, when , according to the Serre-Stark theorem, the space is generated by unary theta series. It was explicitly computed by Skoruppa in [Sk1] and [Sk2] as follows. For every non-zero we write for the finite quadratic module of level given by equipped with the quadratic form . Let be the automorphism of given by multiplication by , and write for the corresponding space of invariants. According to [Sk2, Theorem 8] we have

 (3.5) M12,A(−1)≅⨁l>0,4l∣NN/4l squarefree(C[V(−l)]ϵ⊗C[A(−1)])Mp2(Z).

Here the action of on the tensor products on the right hand side is given by the Weil representation.

## 4. Dimension estimates

In this section we derive lower bounds for the dimension of . In view of (3.3) and (3.4) we have to estimate the quantities

 α1 :=α(eπik/2ρ(S)), α2 :=α((eπik/3ρ(ST))−1), α3 :=α(ρ(T)), α4 :=∣∣{γ∈A/{±1};Q(γ)∈Z}∣∣.

We begin by recalling some trivial bounds from [Bu, Bemerkungen 2.2.1 and 2.2.5]. We have

 α1≤12d,α2≤23d,α3+α4≤d.

If we insert these bounds into (3.4) we obtain the following corollary.

###### Corollary 4.1.

If and , then .

Note that this bound on is sharp, since there are no nontrivial scalar valued cusp forms of weight for .

To prove the existence of non-trivial cusp forms for smaller values of by means of the dimension formula, we need much better estimates for the . The quantities and can be expressed in terms of Gauss sums associated with . By means of the estimate in Lemma 2.10, we obtain the following result (see Lemma 2 and Corollary 3 in [Br2]).

###### Lemma 4.2.

The quantities and satisfy the estimates

 (4.1) |α1−d/4| ≤14√|A[2]|, (4.2) |α2−d/3| ≤13√3(1+√|A[3]|).
###### Lemma 4.3.

We have

 |α4|≤|A[2]|2+√|A|2σ(−1,A),

where is the divisor sum defined in (2.7).

###### Proof.

We write as

 α4=12∑γ∈A[2]Q(γ)∈Z1+12∑γ∈AQ(γ)∈Z1.

The second term on the right hand side is equal to

 12N∑γ∈A∑ν(N)e(Q(γ)ν)=12N∑ν(N)G(ν,A).

Using Lemma 2.10, we obtain

 |α4| ≤|A[2]|2+12N∑ν(N)√|A|√|A[ν]| ≤|A[2]|2+√|A|2N∑a∣N∑μ(N/a)(μ,N/a)=1√|A[aμ]| ≤|A[2]|2+√|A|2N∑a|NNa√|A[a]| ≤|A[2]|2+√|A|2σ(−1,A).

This concludes the proof of the lemma. ∎

Before we consider , we introduce some additional notation. If , we write for the greatest-integer function. Moreover, we let

 (4.3) B(x)=x−12([x]−[−x]).

be the -periodic function on with for and for . By definition

 α3=∑γ∈A/{±1}(−Q(γ)−[−Q(γ)]).

Using and we may rewrite this in the form

 α3 =d2−α42−∑γ∈A/{±1}B(Q(γ)) =d2−α42−12∑γ∈A[2]B(Q(γ))−β2,

where

 (4.4) β =∑γ∈AB(Q(γ)).

For we have , and therefore . Hence

 (4.5) |α3−d/2+α4/2| ≤|A[2]|/8+|β|/2.
###### Lemma 4.4.

The quantity satisfies

 |β|≤√|A|π(32+ln(N))(σ(−1,A)−√|A|N).
###### Proof.

Exactly as in the proof of [Br2, Lemma 5], we derive

 |β| ≤√|A|πN−1∑ν=11ν√|A[ν]|+√|A|2πNN−1∑ν=1√|A[ν]|.

Rewriting the sum over , we obtain

 |β| ≤√|A|π∑a|Na≠NN/a∑μ=1(μ,N/a)=11aμ√|A[a]|+√|A|2πN∑a|Na≠NN/a∑μ=1(μ,N/a)=1√|A[a]| ≤√|A|π∑a|Na≠N(1+ln(N/a))1a√|A[a]|+√|A|2πN∑a|Na≠NNa√|A[a]| ≤√|A|π(32+ln(N))(σ(−1,A)−√|A|N).

Here we have also used the estimate . ∎

Putting the above lemmas together, we obtain the following estimate for the dimension of the space .

###### Theorem 4.5.

If and , then

 ∣∣∣dim(Sk,A)−dim(M2−k,A(−1))−d(k−1)12∣∣∣≤R(A),

where

 R(A) =√|A[2]|4+1+√|A[3]|3√3+38|A[2]| =+√|A|4σ(−1,A)+√|A|2π(32+ln(N))(σ(−1,A)−√|A|N)

is independent of .

###### Proof.

The dimension formula (3.4) states that

 dim(Sk,A)−dim(M2−k,A(−1)) =d(k+12)12−α1−α2−α3−α4 =d(k−1)12−(α1−d4)−(α2−d3)−(α3−d2+α42)−α42.

Employing (4.5), Lemma 4.2, Lemma 4.3, and Lemma 4.4, we obtain the assertion. ∎

###### Corollary 4.6.

For every there exists a constant (independent of and ) such that

 ∣∣∣dim(Sk,A)−dim(M2−k,A(−1))−d(k−1)12∣∣∣≤CdNε−12.

for every finite quadratic module and every weight with .

###### Proof.

Using Theorem 4.5, the bound (2.1) for , and Lemma 2.15, we find that there are constants (independent of and ) such that

 R(A)≤C1|A|N+C2|A|√Nσ−1/2(N)(1+ln(N)).

By means of the estimate we see that there exists a (depending on ) such that

 R(A)≤C⋅dNε−12.

This proves the assertion. ∎

###### Corollary 4.7.

Let . There exist only finitely many isomorphism classes of finite quadratic modules with minimal number of generators such that for some weight with .

###### Proof.

Since for any there are only finitely many isomorphism classes of finite quadratic modules with bounded minimal number of generators and level , we obtain the assertion from Corollary 4.6. ∎

###### Remark 4.8.

i) In Corollary 4.7, the bound on the minimal number of generators is essential. For instance, if with odd and , then and . This follows for instance from the dimension formula in [Ha], Chapter 5.2.1, p. 93.

ii) Note that if , it follows from [Sk2, Theorem 7] that there exist infinitely many isomorphism classes of finite quadratic modules such that . It would be interesting to understand what happens in weight .

Under the assumptions of Corollary 4.7 it is possible to make the constants appearing in the proof explicit and to derive an explicit lower bound such that is nontrivial for all finite quadratic modules with level larger than . However, it turns out that such a bound is very large, and therefore not useful for a computer computation of the finite list of simple finite quadratic modules . As an example, for we can get and this would give the bound for . Therefore, a search for finite quadratic modules with order up to would be required in the case of signature , which is not feasible. Even though it might be possible to find better parameters, we did not try to optimize this. Instead, we use a more systematic approach to this computational task. We first compute all anisotropic simple finite quadratic modules, and then construct all remaining ones by means of isotropic quotients.

### 4.1. Anisotropic finite quadratic modules

A finite quadratic module is called isotropic, if there exists an such that . Otherwise it is called anisotropic. In this subsection we now consider anisotropic finite quadratic modules. We show that there are only finitely many isomorphism classes of anisotropic finite quadratic modules for which is trivial. The following Lemma is a direct consequence of Theorem 2.2 and the theory of quadratic forms over finite fields.

###### Lemma 4.9.

Let be an anisotropic finite quadratic module of level . Then , where is an odd square-free number and . If is a prime dividing , then the -component of belongs to the finite quadratic modules given in Table 2.
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