Lattices with many Borcherds products
LLATTICES WITH MANY BORCHERDS PRODUCTS
JAN HENDRIK BRUINIER, STEPHAN EHLEN, AND EBERHARD FREITAG
Abstract.
We prove that there are only finitely many isometry classes of even lattices L of signature (2 , n ) for which the space of cusp forms of weight 1 + n/ L is trivial. We compute the list of theselattices. They have the property that every Heegner divisor for the orthogonal group of L can be realized as the divisor of a Borcherds product. We obtain similar classificationresults in greater generality for finite quadratic modules. Introduction
Let L be an even lattice of signature (2 , n ) and write O( L ) for its orthogonal group.In his celebrated paper [Bo1] R. Borcherds constructed a map from vector valued weaklyholomorphic elliptic modular forms of weight 1 − n/ L ) whose zeros and poles are supported on Heegner divisors. Since modular forms arisingin this way have particular infinite product expansions, they are often called Borcherdsproducts . They play important roles in different areas such as Algebraic and ArithmeticGeometry, Number Theory, Lie Theory, Combinatorics, and Mathematical Physics.By Serre duality, the obstructions for the existence of weakly holomorphic modular formswith prescribed principal part at the cusp at ∞ are given by vector valued cusp forms of dualweight 1 + n/ L [Bo2]. In particular, if there are no non-trivial cusp forms of this type, thenthere 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 isomorphismclasses of such simple lattices. Under the assumptions that n ≥ L (i.e. the dimension of a maximal totally isotropic subspace of L ⊗ Z Q ) is 2, it wasproved by M. Bundschuh that there is an upper bound on the determinant of a simplelattice [Bu]. Unfortunately, this bound is very large and therefore not feasible to obtainany classification results. The argument of [Bu] is based on volume estimates for Heegnerdivisors and the singular weight bound for holomorphic modular forms for O( L ).The purpose of the present paper is twofold. First, we show that for any n ≥ , n ), see Theorem 4.5 and Corollary 4.7. Second,we develop an efficient algorithm to determine all of these lattices. It turns out that thereare exactly 362 isomorphism classes. Table 1 shows how many of those occur in the different Date : January 24, 2018.2010
Mathematics Subject Classification. a r X i v : . [ m a t h . N T ] F e b JAN HENDRIK BRUINIER, STEPHAN EHLEN, AND EBERHARD FREITAG signatures. The corresponding genus symbols (see Section 2) of these lattices are listed inTables 7, 8 . Table 1.
Number of simple lattices of signature (2 , n ). n ≤ n ≤
17 18 19 ≤ n ≤
25 26 n > n = 1, we have 26anisotropic lattices. The corresponding modular varieties are Shimura curves while theremaining 230 modular varieties for n = 1 are modular curves. For n = 2, there are 24 ofWitt rank 1 and 43 of Witt rank 2 but no anisotropic lattices. Finally, if n ≥
3, all simplelattices have Witt rank 2.Along the way, we obtain several results on modular forms associated with finite quadraticmodules which are of independent interest and which we now briefly describe. A finitequadratic module is a pair consisting of a finite abelian group A together with a Q / Z -valued non-degenerate quadratic form Q on A , see [Ni], [Sk2]. Important examples of finitequadratic modules are obtained from lattices. If L is an even lattice with dual lattice L (cid:48) ,then the quadratic form on L induces a Q / Z -valued quadratic form on the discriminantgroup L (cid:48) /L .Recall that there is a Weil representation ρ A of the the metaplectic extension Mp ( Z ) ofSL ( Z ) on the group ring C [ A ] of a finite quadratic module A , see Section 3. If k ∈ Z ,we write S k,A for the space of cusp forms of weight k and representation ρ A for the groupMp ( Z ). We say that a finite quadratic module A is k -simple if S k,A = { } . With thisterminology, an even lattice L is simple if and only if L (cid:48) /L is (1 + n/ S k,A can be computed by means of the Riemann-Roch theo-rem. Therefore a straightforward approach to showing that there are nontrivial cusp formsconsists in finding lower bounds for the dimension of S k,A . Unfortunately, the dimensionformula (3.3) involves rather complicated invariants of ρ A at elliptic and parabolic elements,and it is a non-trivial task to obtain sufficiently strong bounds. In the present paper weresolve this problem. For instance, we obtain the following result (see Theorem 4.5 andCorollary 4.6). Theorem.
For every ε > , there is a C ε > , such that (cid:12)(cid:12)(cid:12)(cid:12) dim( S k,A ) − dim( M − k,A ( − ) − | A/ {± }| · k − (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ε · | A/ {± }| · N ε − A for every finite quadratic module A and every weight k ≥ with k ≡ − sig( A ) (mod 4) .Here N A is the level of A , and A ( − denotes the abelian group A equipped with the qua-dratic form − Q . Tables with global realizations can be obtained from [Ehl].
ATTICES WITH MANY BORCHERDS PRODUCTS 3
In Corollary 4.7 we conclude that there exist only finitely many isomorphism classes offinite quadratic modules A with bounded number of generators such that S k,A = { } forsome weight satisfying the condition of the theorem. In particular, there are only finitelymany isomorphism classes of simple lattices. Note that there do exist infinitely manyisomorphism 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 adifficult task to compute the list of all k -simple finite quadratic modules for a boundednumber of generators. We develop an efficient algorithm to address this problem. The ideais to first compute all anisotropic finite quadratic modules that are k -simple for some k . Tothis end we derive an explicit formula for dim( S k,A ) in terms of class numbers of imaginaryquadratic 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 A has a uniqueanisotropic quotient A , and that there are intertwining operators for the correspondingWeil representations. For the difference dim S k,A − dim S k,A very efficient bounds can beobtained. This can be used to classify all k -simple finite quadratic modules with a boundednumber of generators, see Algorithm 5.5 and the tables in Section 5.To resolve the problem of finding all simple lattices of signature (2 , n ), it remains to testwhich of these simple discriminant forms arise as discriminant groups L (cid:48) /L of even lattices L of signature (2 , n ). This is done in Section 6 by applying a criterion of [CS]. Finally, inSection 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. Allcockand A. Mark for their help with an earlier version of this paper. Some of the computationswere performed on computers at the University of Washington, USA, supported by W. Steinand the National Science Foundation Grant No. DMS-0821725. We would also like to thankthe anonymous referee for many helpful comments.2. Finite quadratic modules
Let (
A, Q ) be a finite quadratic module (also called a finite quadratic form or discriminantform in the literature), that is, a pair consisting of a finite abelian group A togetherwith a Q / Z -valued non-degenerate quadratic form Q on A . We denote the bilinear formcorresponding to Q by ( x, y ) = Q ( x + y ) − Q ( x ) − Q ( y ). Recall that Q is called degenerateif there exists an x ∈ A \ { } , such that ( x, y ) = 0 for all y ∈ A . Otherwise, Q is callednon-degenerate.The morphisms in the category of finite quadratic modules are group homomorphismsthat preserve the quadratic forms. In particular, two finite quadratic modules ( A, Q A ) and( B, Q B ) are isomorphic if and only if there is an isomorphism of groups ϕ : A → B , suchthat Q B ◦ ϕ = Q A .In this section we collect some important facts about finite quadratic modules, whichare well known among experts but not easily found in the literature. We mainly followSkoruppa [Sk3]. Other good references include [Ni], [No], [Sk2], [Str].If L is an even lattice, the quadratic form Q on L induces a Q / Z -valued quadratic formon the discriminant group L (cid:48) /L of L . The pair ( L (cid:48) /L, Q ) defines a finite quadratic module, JAN HENDRIK BRUINIER, STEPHAN EHLEN, AND EBERHARD FREITAG which we call the discriminant module of L . According to [Ni], any finite quadratic modulecan be obtained as the discriminant module of an even lattice L . If ( b + , b − ) then denotesthe real signature of L , the difference b + − b − is determined by its discriminant module A modulo 8 by Milgram’s formula1 (cid:112) | A | (cid:88) a ∈ A e ( Q ( a )) = e (( b + − b − ) / . Here and throughout we abbreviate e ( z ) = e πiz for z ∈ C . We call sig( A ) := b + − b − ∈ Z / Z the signature of A .We let N be the level of A defined by N = min { n ∈ Z > | nQ ( x ) ∈ Z for all x ∈ A } . It is easily seen that N is a divisor of 2 | A | .If ( A, Q A ) and ( B, Q B ) are two finite quadratic modules then the orthogonal direct sum ( A ⊕ B, Q A + Q B ) also defines a finite quadratic module. Here ( Q A + Q B )( a + b ) = Q A ( a ) + Q A ( b ) for a ∈ A and b ∈ B . We call a finite quadratic module indecomposable if it is notisomorphic to such a direct sum with non-zero A and B .The finite quadratic module A is isomorphic to the orthogonal sum of its p -components A p = A ⊗ Z Z p with p running through the primes.Next we describe a list of indecomposable finite quadratic modules. Definition 2.1.
Let p be a prime and t be an integer not divisible by p . We define thefollowing elementary finite quadratic modules. A tp k = (cid:18) Z /p k Z , tx p k (cid:19) for p > , A t k = (cid:18) Z / k Z , tx k +1 (cid:19) , B k = (cid:18) Z / k Z ⊕ Z / k Z , x + 2 xy + y k (cid:19) , C k = (cid:16) Z / k Z ⊕ Z / k Z , xy k (cid:17) , 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 quadraticmodule as in Definition 2.1.(3) Moreover, every finite quadratic module is isomorphic to a direct sum of indecom-posable finite quadratic modules.Proof.
Statement (1) is clear. The other two statements follow from the classification of p -adic lattices. See [Ni] for details, in particular Proposition 1.8.1. (cid:3) Consider a decomposition of the p -components of a finite quadratic module A as a directsum A p = A p, ⊕ . . . ⊕ A p,l p , ATTICES WITH MANY BORCHERDS PRODUCTS 5 where each A p,i is a direct sum of elementary finite quadratic modules A t i q i , B q i , C q i , with q i = p r i , r i ≥ q < . . . < q r .Such a decomposition is called a Jordan decomposition of A and the direct summands A p,i are called Jordan components of A . We will also call any finite quadratic module thatis isomorphic to a direct sum of elementary finite quadratic modules A tq , B q , C q for a fixed q a Jordan component.We will now describe a handy notation for such a Jordan decomposition. The symbolswe use are essentially those introduced by Conway and Sloane [CS] for the genus of anintegral 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 L and M that have the same real signatures haveisomorphic finite quadratic modules if and only if L and M are in the same genus. The following two lemmas are straightforward to prove (see also [Ni], Proposition 1.8.2).
Lemma 2.4.
Let p > be a prime and let q = p r for a positive integer r .(1) We have A tq ∼ = A sq if and only if (cid:16) sp (cid:17) = (cid:16) tp (cid:17) .(2) Suppose that (cid:16) sp (cid:17) = 1 and (cid:16) tp (cid:17) = − . Then A sq ⊕ A sq ∼ = A tq ⊕ A tq .(3) In particular, if A is a Jordan component of the form A = n (cid:77) i =1 A t i q , then A is isomorphic to A tq ⊕ n − (cid:77) i =1 A sq , with n (cid:89) i =1 (cid:18) t i p (cid:19) = (cid:18) tp (cid:19) for any s with (cid:16) sp (cid:17) = 1 . Lemma 2.5.
Let q = 2 r for a positive integer r . Moreover, let s, t be odd integers.(1) We have A s ≡ A t if and only if s ≡ t (mod 4) .(2) If r > , then A sq ≡ A tq if and only if s ≡ t (mod 8) .(3) Let s , . . . , s n , t , . . . , t n ∈ Z such that n (cid:88) i =1 s i ≡ n (cid:88) i =1 t i (mod 8) and n (cid:89) i =1 (cid:16) s i (cid:17) = n (cid:89) i =1 t i . Then n (cid:77) i =1 A s i q ∼ = n (cid:77) i =1 A t i q . (4) We have B q ⊕ B q ∼ = C q ⊕ C q . JAN HENDRIK BRUINIER, STEPHAN EHLEN, AND EBERHARD FREITAG (5) Moreover, we have A tq ⊕ B q ∼ = A t q ⊕ A t q ⊕ A t q with t + t + t ≡ t (mod 8) and (cid:89) i =1 (cid:18) t i (cid:19) = − (cid:18) t (cid:19) . (6) Finally, A tq ⊕ C q ∼ = A t q ⊕ A t q ⊕ A t q with t + t + t ≡ t (mod 8) and (cid:89) i =1 (cid:18) t i (cid:19) = (cid:18) t (cid:19) . Definition 2.6.
Using the preceding lemmas, we define a symbol for a Jordan decompo-sition of a finite quadratic module as follows. First of all, by convention, we write 1 +1 or1 − for the trivial module A = { } with the 0-map as quadratic form.Now let A be a Jordan component, p be a prime and q = p r .(1) If p is odd, the two isomorphism classes of Jordan components in Lemma 2.4, (3)are denoted q ± n , where (cid:16) tp (cid:17) = ± p = 2.(a) We write q ± nt if A is isomorphic to A t q ⊕ . . . ⊕ A t n q with t + . . . + t n ≡ t (mod 8)and (cid:0) t (cid:1) · · · (cid:0) t n (cid:1) = ±
1. We normalize t to be contained in the set { , , , } and if q = 2, we take t ∈ { , } .(b) We write q +2 n if A is isomorphic to n copies of C q .(c) And we write q − n if A is isomorphic to n − C q and one copy of B q .For a general finite quadratic module, we concatenate the symbols of the Jordan compo-nents as defined above. Example 2.7.
The Jordan decomposition A ⊕ ( A ⊕ A ) has the symbol 2 +11 +2 and2 − +33 − is the symbol for the Jordan decomposition B ⊕ ( A ⊕ A ⊕ A ) ⊕ A . Proposition 2.8.
Let A and B be finite quadratic modules and let p > be a prime. If A p ∼ = B p , then the corresponding p -components of the genus symbols of A and B coincide.Proof. This follows from the uniqueness of the Jordan decomposition in the case of an oddprime p (see, for instance, Theorem 5.3.2 in [Ki]). (cid:3) Remark . In contrast to Proposition 2.8, note that the symbol (and the Jordan decom-position) for p = 2 is not uniquely determined by the isomorphism class. For instance,the Jordan decompositions 2 +11 +11 and 2 +17 − correspond to isomorphic finite quadraticmodules. See also Theorem 2.14 in the next section.For an integer n we define the n -torsion subgroup of a finite quadratic module A by A [ n ] = { γ ∈ A | nγ = 0 } . Moreover, we let A n be the image of the multiplication by n map. Then we have the exactsequence 0 −→ A [ n ] −→ A −→ A n −→ , ATTICES WITH MANY BORCHERDS PRODUCTS 7 and A is the orthogonal sum of A [ n ] and A n . It follows from the theorem of elementarydivisors and the Jordan decompositon (Theorem 2.2) that(2.1) | A [ n ] | ≤ n | A | N .
The quantity d = d A := | A/ {± }| can be expressed in term of the 2-torsion as d = | A | | A [2] | . (2.2)2.1. Gauss sums and divisor sums.
We now collect some facts about Gauss sums anddivisor sums associated to finite quadratic modules which we will need later. For an integer n ∈ Z the Gauss sum G ( n, A ) is defined by(2.3) G ( n, A ) = (cid:88) γ ∈ A e ( nQ ( γ )) . We have the elementary properties G ( − n, A ) = G ( n, A ) , (2.4) G ( n + N, A ) = G ( n, A ) , (2.5) G ( n, A ⊕ B ) = G ( n, A ) G ( n, B ) . (2.6)The following lemma is a consequence of [Bo3, Lemma 3.1]. Lemma 2.10. If ( n, N ) = 1 then | G ( n, A ) | = (cid:112) | A | . For general n we have the estimate | G ( n, A ) | ≤ (cid:112) | A | (cid:112) | 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 G ( n, A ) to the standard Gauss sums (see, for instance[Str]). Proposition 2.11.
Let p > be a prime. We have G ( n, p ± ) = p if p | n and G ( n, p ± ) = ±√ p (cid:16) p (cid:17) (cid:18) np (cid:19) e (cid:18) − p (cid:19) if ( n, p ) = 1 . Proposition 2.12.
Let q = 2 r . For n ∈ Z we put n (cid:48) = n/ ( n, q ) and q (cid:48) = q/ ( n, q ) . Wehave G ( n, q ± t ) = √ q (cid:112) ( n, q ) (cid:18) tn (cid:48) q (cid:48) (cid:19) · e (cid:0) tn (cid:48) (cid:1) , if q (cid:45) n, , if q || n, , if q | n. Proposition 2.13.
Let q = 2 r with r ≥ . We have G ( n, q − ) = q ( q, n ) (cid:18) q (cid:48) (cid:19) and G ( n, q +2 ) = q ( q, n ) . JAN HENDRIK BRUINIER, STEPHAN EHLEN, AND EBERHARD FREITAG
The following theorem is used later on to decide when two given genus symbols correspondto isomorphic finite quadratic modules.
Theorem 2.14 ([Sk3]) . Let A and B be finite quadratic modules. Then A and B areisomorphic if and only if their underlying abelian groups are isomorphic and G ( n, A ) = G ( n, B ) for all divisors n of the level of A and B .Proof. It is clear that the condition G ( n, A ) = G ( n, B ) is necessary. Using this, it is easy toprove the theorem for p -components with p >
2. If A p and B p are finite quadratic modulesof prime power level p r with G ( p i , A p ) = G ( p i , B p ) for all i ∈ , . . . , r , then A p ∼ = B p followsfrom Lemma 2.4 and the explicit formula for the Gauss sum in Proposition 2.11. Thegeneral case is treated in [Sk3] in detail. (cid:3) For s ∈ C we define a divisor sum σ ( s, A ) associated to A by σ ( s, A ) = (cid:88) a | N a s (cid:112) | A [ a ] | . (2.7)Here the sum runs over all positive divisors a of the level N of A . If B is another finitequadratic module of level N (cid:48) coprime to N , then σ ( s, A ⊕ B ) = σ ( s, A ) σ ( s, B ) . Consequently, σ ( s, A ) is the product of the σ ( s, A p ) for p running through the primesdividing N . Lemma 2.15.
For s ∈ R we have the estimate σ ( s, A ) ≤ (cid:114) | A | N · σ s +1 / ( N ) , where σ s ( N ) = (cid:80) a | N a s denotes the usual divisor sum.Proof. This is a direct consequence of the estimate (2.1). (cid:3) Vector valued modular forms
We write Mp ( Z ) for the metaplectic extension of SL ( Z ), realized as the group of pairs( M, φ ( τ )), where M = ( a bc d ) ∈ SL ( Z ) and φ is a holomorphic function on the uppercomplex half plane H with φ ( τ ) = cτ + d (see e. g. [Bo1], [Br1]). It is well known thatMp ( Z ) is generated by T = (cid:32)(cid:32) (cid:33) , (cid:33) and S = (cid:32)(cid:32) −
11 0 (cid:33) , √ τ (cid:33) . One has the relations S = ( ST ) = Z , where Z = (cid:0)(cid:0) − − (cid:1) , i (cid:1) is the standard generatorof the center of Mp ( Z ).The Weil representation associated with A is a unitary representation ρ A of Mp ( Z ) onthe group algebra C [ A ]. If we denote the standard basis of C [ A ] by ( e γ ) γ ∈ A then ρ A can ATTICES WITH MANY BORCHERDS PRODUCTS 9 be defined by the action of the generators
S, T ∈ Mp ( Z ) as follows (see also [Sk2], [Bo1],[Br1], where the dual of ρ A is used): ρ A ( T ) e γ = e ( − Q ( γ )) e γ , (3.1) ρ A ( S ) e γ = e (sig( A ) / (cid:112) | A | (cid:88) δ ∈ A e (( γ, δ )) e δ . (3.2)Let k ∈ Z . We denote by M k,A the vector space of C [ A ]-valued modular forms of weight k with representation ρ A for the group Mp ( Z ). The subspace of cusp forms is denoted by S k,A . It is easily seen that M k,A = 0, if 2 k (cid:54)≡ sig( A ) (mod 2).The dimension of the vector space M k,A can be computed using the Riemann-Roch the-orem or the Selberg trace formula. This is carried out in [Fr] and [Fi] in a more generalsituation. In our special case the following formula holds (see [Bo2] p. 228 and [Fr] Chap-ter 8.5, Theorem 5.1). For simplicity we assume that 2 k ≡ − sig( A ) (mod 4), since ourapplication to simple lattices will only concern this case. Then the d -dimensional subspace W = span { e γ + e − γ ; γ ∈ A } of C [ A ] is preserved by ρ A , and ρ A ( Z ) acts by multiplicationwith e ( − k/
2) on W . We denote by ρ the restriction of ρ A to W . If M is a unitary matrixof size d with eigenvalues e ( ν j ) and 0 ≤ ν j < j = 1 , . . . , d ), we define α ( M ) = d (cid:88) j =1 ν j . If k ≥ , the dimension of M k,A is given bydim( M k,A ) = d + dk/ − α (cid:0) e πik/ ρ ( S ) (cid:1) − α (cid:16)(cid:0) e πik/ ρ ( ST ) (cid:1) − (cid:17) − α ( ρ ( T ))(3.3) + dim( S − k,A ( − ) . Furthermore, the dimension of S k,A is given bydim( S k,A ) = first line of (3.3) − |{ γ ∈ A/ {± } ; Q ( γ ) ∈ Z }| + dim( M − k,A ( − ) . (3.4)Here A ( −
1) denotes the finite quadratic module given by the abelian group A equippedwith the quadratic form − Q . If k >
2, then M − k,A ( − vanishes. If k = 2, then M ,A ( − isequal to the space of Mp ( Z )-invariants in C [ A ] for the dual representation of ρ A . Finally,when k = , according to the Serre-Stark theorem, the space M ,A ( − is generated byunary theta series. It was explicitly computed by Skoruppa in [Sk1] and [Sk2] as follows.For every non-zero l ∈ Z we write V ( l ) for the finite quadratic module of level 4 | l | given by Z / l Z equipped with the quadratic form Q ( x ) = l x . Let (cid:15) be the automorphism of V ( l )given by multiplication by −
1, and write C [ V ( l )] (cid:15) for the corresponding space of invariants.According to [Sk2, Theorem 8] we have M ,A ( − ∼ = (cid:77) l> , l | NN/ l squarefree (cid:0) C [ V ( − l )] (cid:15) ⊗ C [ A ( − (cid:1) Mp ( Z ) . (3.5)Here the action of Mp ( Z ) on the tensor products on the right hand side is given by theWeil representation. Dimension estimates
In this section we derive lower bounds for the dimension of S k,A . In view of (3.3) and(3.4) we have to estimate the quantities α := α (cid:0) e πik/ ρ ( S ) (cid:1) ,α := α (cid:16)(cid:0) e πik/ ρ ( ST ) (cid:1) − (cid:17) ,α := α ( ρ ( T )) ,α := (cid:12)(cid:12) { γ ∈ A/ {± } ; Q ( γ ) ∈ Z } (cid:12)(cid:12) . We begin by recalling some trivial bounds from [Bu, Bemerkungen 2.2.1 and 2.2.5]. Wehave α ≤ d, α ≤ d, α + α ≤ d. If we insert these bounds into (3.4) we obtain the following corollary.
Corollary 4.1. If k > and k ≡ − sig( A ) (mod 4) , then S k,A (cid:54) = { } . Note that this bound on k is sharp, since there are no nontrivial scalar valued cusp formsof weight 14 for SL ( Z ).To prove the existence of non-trivial cusp forms for smaller values of k by means of thedimension formula, we need much better estimates for the α i . The quantities α and α can be expressed in terms of Gauss sums associated with A . By means of the estimate inLemma 2.10, we obtain the following result (see Lemma 2 and Corollary 3 in [Br2]). Lemma 4.2.
The quantities α and α satisfy the estimates | α − d/ | ≤ (cid:112) | A [2] | , (4.1) | α − d/ | ≤ √ (cid:16) (cid:112) | A [3] | (cid:17) . (4.2) Lemma 4.3.
We have | α | ≤ | A [2] | (cid:112) | A | σ ( − , A ) , where σ ( − , A ) is the divisor sum defined in (2.7) .Proof. We write α as α = 12 (cid:88) γ ∈ A [2] Q ( γ ) ∈ Z (cid:88) γ ∈ AQ ( γ ) ∈ Z . The second term on the right hand side is equal to12 N (cid:88) γ ∈ A (cid:88) ν ( N ) e ( Q ( γ ) ν ) = 12 N (cid:88) ν ( N ) G ( ν, A ) . ATTICES WITH MANY BORCHERDS PRODUCTS 11
Using Lemma 2.10, we obtain | α | ≤ | A [2] | N (cid:88) ν ( N ) (cid:112) | A | (cid:112) | A [ ν ] |≤ | A [2] | (cid:112) | A | N (cid:88) a | N (cid:88) µ ( N/a )( µ,N/a )=1 (cid:112) | A [ aµ ] |≤ | A [2] | (cid:112) | A | N (cid:88) a | N Na (cid:112) | A [ a ] |≤ | A [2] | (cid:112) | A | σ ( − , A ) . This concludes the proof of the lemma. (cid:3)
Before we consider α , we introduce some additional notation. If x ∈ R , we write[ x ] = max { n ∈ Z ; n ≤ x } for the greatest-integer function. Moreover, we let(4.3) B ( x ) = x − ([ x ] − [ − x ]) . be the 1-periodic function on R with B ( x ) = 0 for x = 0 , B ( x ) = x − / < x < α = (cid:88) γ ∈ A/ {± } ( − Q ( γ ) − [ − Q ( γ )]) . Using B ( x ) and α we may rewrite this in the form α = d − α − (cid:88) γ ∈ A/ {± } B ( Q ( γ ))= d − α − (cid:88) γ ∈ A [2] B ( Q ( γ )) − β , where β = (cid:88) γ ∈ A B ( Q ( γ )) . (4.4)For γ ∈ A [2] we have Q ( γ ) ∈ Z , and therefore | B ( Q ( γ )) | ≤ /
4. Hence | α − d/ α / | ≤ | A [2] | / | β | / . (4.5) Lemma 4.4.
The quantity β satisfies | β | ≤ (cid:112) | A | π (cid:18)
32 + ln( N ) (cid:19) (cid:32) σ ( − , A ) − (cid:112) | A | N (cid:33) . Proof.
Exactly as in the proof of [Br2, Lemma 5], we derive | β | ≤ (cid:112) | A | π N − (cid:88) ν =1 ν (cid:112) | A [ ν ] | + (cid:112) | A | πN N − (cid:88) ν =1 (cid:112) | A [ ν ] | . Rewriting the sum over ν , we obtain | β | ≤ (cid:112) | A | π (cid:88) a | Na (cid:54) = N N/a (cid:88) µ =1( µ,N/a )=1 aµ (cid:112) | A [ a ] | + (cid:112) | A | πN (cid:88) a | Na (cid:54) = N N/a (cid:88) µ =1( µ,N/a )=1 (cid:112) | A [ a ] |≤ (cid:112) | A | π (cid:88) a | Na (cid:54) = N (1 + ln( N/a )) 1 a (cid:112) | A [ a ] | + (cid:112) | A | πN (cid:88) a | Na (cid:54) = N Na (cid:112) | A [ a ] |≤ (cid:112) | A | π (cid:18)
32 + ln( N ) (cid:19) (cid:32) σ ( − , A ) − (cid:112) | A | N (cid:33) . Here we have also used the estimate (cid:80) nν =1 1 ν ≤ n ). (cid:3) Putting the above lemmas together, we obtain the following estimate for the dimensionof the space S k,A . Theorem 4.5. If k ≥ and k ≡ − sig( A ) (mod 4) , then (cid:12)(cid:12)(cid:12)(cid:12) dim( S k,A ) − dim( M − k,A ( − ) − d ( k − (cid:12)(cid:12)(cid:12)(cid:12) ≤ R ( A ) , where R ( A ) = (cid:112) | A [2] | (cid:112) | A [3] | √ | A [2] | + (cid:112) | A | σ ( − , A ) + (cid:112) | A | π (cid:18)
32 + ln( N ) (cid:19) (cid:32) σ ( − , A ) − (cid:112) | A | N (cid:33) is independent of k .Proof. The dimension formula (3.4) states thatdim( S k,A ) − dim( M − k,A ( − ) = d ( k + 12)12 − α − α − α − α = d ( k − − ( α − d − ( α − d − ( α − d α − α . Employing (4.5), Lemma 4.2, Lemma 4.3, and Lemma 4.4, we obtain the assertion. (cid:3)
Corollary 4.6.
For every ε > there exists a constant C (independent of k and A ) suchthat (cid:12)(cid:12)(cid:12)(cid:12) dim( S k,A ) − dim( M − k,A ( − ) − d ( k − (cid:12)(cid:12)(cid:12)(cid:12) ≤ CdN ε − . for every finite quadratic module A and every weight k ≥ with k ≡ − sig( A ) (mod 4) . ATTICES WITH MANY BORCHERDS PRODUCTS 13
Proof.
Using Theorem 4.5, the bound (2.1) for | A [ a ] | , and Lemma 2.15, we find that thereare constants C , C > k and A ) such that R ( A ) ≤ C | A | N + C | A |√ N σ − / ( N )(1 + ln( N )) . By means of the estimate σ − / ( N ) (cid:28) ε N ε we see that there exists a C > ε ) such that R ( A ) ≤ C · dN ε − . This proves the assertion. (cid:3)
Corollary 4.7.
Let r ∈ Z ≥ . There exist only finitely many isomorphism classes of finitequadratic modules A with minimal number of generators ≤ r such that S k,A = { } forsome weight k ≥ with k ≡ − sig( A ) (mod 4) .Proof. Since for any N ∈ Z > there are only finitely many isomorphism classes of finitequadratic modules A with bounded minimal number of generators and level N ≤ N , weobtain the assertion from Corollary 4.6. (cid:3) Remark . i) In Corollary 4.7, the bound r on the minimal number of generators isessential. For instance, if A = 3 (cid:15)n with n ∈ Z > odd and (cid:15) = ( − n − , then sig( A ) ≡ S ,A = { } . This follows for instance from the dimension formula in [Ha],Chapter 5.2.1, p. 93.ii) Note that if k = , it follows from [Sk2, Theorem 7] that there exist infinitely manyisomorphism classes of finite quadratic modules A such that S ,A = { } . It would beinteresting to understand what happens in weight 1.Under the assumptions of Corollary 4.7 it is possible to make the constants appearing inthe proof explicit and to derive an explicit lower bound N such that S k,A is nontrivial forall finite quadratic modules A with level larger than N . However, it turns out that such abound is very large, and therefore not useful for a computer computation of the finite list ofsimple finite quadratic modules A . As an example, for ε = 1 / C = 45 .
38 andthis would give the bound N ≥ . · for k = 2. Therefore, a search for finite quadraticmodules with order up to 3 . · would be required in the case of signature (2 , anisotropic simple finite quadratic modules, and then construct allremaining ones by means of isotropic quotients.4.1. Anisotropic finite quadratic modules.
A finite quadratic module (
A, Q ) is called isotropic , if there exists an x ∈ A \ { } such that Q ( x ) = 0 ∈ Q / Z . Otherwise it is called anisotropic . In this subsection we now consider anisotropic finite quadratic modules. Weshow that there are only finitely many isomorphism classes of anisotropic finite quadraticmodules A for which S k,A is trivial. The following Lemma is a direct consequence ofTheorem 2.2 and the theory of quadratic forms over finite fields. Lemma 4.9.
Let ( A, Q ) be an anisotropic finite quadratic module of level N . Then N =2 t N (cid:48) , where N (cid:48) is an odd square-free number and t ∈ { , , , } . If p is a prime dividing N , then the p -component A p of A belongs to the finite quadratic modules given in Table 2. Table 2.
The non-trivial isomorphism classes of anisotropic finite quadraticmodules of prime-power order. The isomorphism classes of the finite qua-dratic modules in the last line depend only on the sum s + t . Here, d ( A ) isthe discriminant of A , equal to | A | ∈ Q × / ( Q × ) . p genus symbol of A sig( A ) d ( A ) p ≡ p ± ± (cid:0) p (cid:1) ) pp − p ≡ p ± ± (cid:0) p (cid:1) pp +2 p = 2 2 − ± nnt , t = 1 , , n = 1 , , nt n ± t , t = 1 , , , t ± s ± t , s = 1 , , t = 1 , , , s + t β defined in (4.4) in terms of class numbers. Before wecan state the precise result we need to introduce some more notation.Let A be an anisotropic finite quadratic module of level N and write A = (cid:76) p | N A p for itsdecomposition into p -components. For each prime divisor p of N , we denote the minimalnumber of generators of A p by r p . If A p = q ε · n with q = p r we write ε A ( p ) = ε . We definea divisor M of N by M = (cid:89) p | N odd r p =2 p · (cid:40) , if A = 2 − , , otherwise.For d | N we define the following auxiliary quantities: S ( d ) = { p prime; p | ( d, M ) } ,ε ( d ) = ( − | S ( d ) | ,a ( d ) = , if r = 0 , r − , if r > d is odd , r ] − , if r > d is even, ATTICES WITH MANY BORCHERDS PRODUCTS 15 S ( d ) = { p prime ; p | d ( d, M ) , p ≡ } ,ε ( d ) = (cid:40) ( − | S d ) |− , if | S ( d ) | is odd,( − | S d ) | , if | S ( d ) | is even.For the p -components corresponding to odd primes, we define a sign ε odd ( d ) = (cid:89) p | d ( d,M ) odd ε A ( p ) (cid:16) p (cid:17) (cid:18) N/dp (cid:19) . We let N be the even part of N and put N ,d = N / ( N , N/d ), N d = N/ ( d · ( N/d, N ))and d (cid:48) = d/ ( d, M · N ). Note that d (cid:48) is odd. If d is odd, we let ε ( d ) = 1. For even d , wedefine ε ( d ) in Table 3. For simplicity, we also define ε ( d ) = ε odd ( d ) ε ( d ) ε ( d ) ε ( d ). Table 3. ε ( d ) for even d . A d (cid:48) ≡ d (cid:48) ≡ − + r r t , ± t (cid:16) − N ,d tN d (cid:17) δ ( r ) (cid:16) N ,d tN d (cid:17) +11 +11 , 2 +11 − (cid:16) − N/d (cid:17) +11 − , 2 +11 +17 (cid:16) N/d (cid:17)
Theorem 4.10.
Let ( A, Q ) be an anisotropic finite quadratic module of level N . We have β = − (cid:88) d | Nd ( d,M ) ≡ , ε ( d ) a ( d ) · ( N/d, M ) H ( − d ( d, M )) . Here, H ( − n ) is equal to the class number of primitive positive definite integral binaryquadratic forms of discriminant − n for n > and H ( −
3) = 1 / and H ( −
4) = 1 / .Proof of Theorem 4.10. Using the pointwise convergent Fourier expansion B ( x ) = − πi (cid:88) n ∈ Z −{ } e ( nx ) n we find β = − π ∞ (cid:88) n =1 n (cid:61) ( G ( n, A )) = − π (cid:88) d | N (cid:88) n ≥ n,N/d )=1 dn (cid:61) ( G ( dn, A )) . First, we assume that N is odd. For a discriminant D , we write χ D for the quadraticDirichlet character modulo | D | given by n (cid:55)→ (cid:0) Dn (cid:1) . Inserting the formula for G ( n, A ) from Proposition 2.11 and substituting
N/d for d , we obtain β = − √ Nπ (cid:88) d | N (cid:88) n ≥ n,d )=1 ( M, N/d ) (cid:112) ( M, d ) n (cid:112) N/d (cid:61) (cid:18) (cid:89) p | dr p =1 ε A ( p ) (cid:16) p (cid:17) (cid:18) nN/dp (cid:19) e (cid:18) − p (cid:19) (cid:89) p | dr p =2 ( − (cid:19) = − √ Nπ (cid:88) d | Nd ( d,M ) ≡ ε ( d ) · ( M, N/d ) (cid:112) ( M, d ) (cid:112) N/d (cid:88) n ≥ n,d )=1 χ − d · ( M,d ) ( n ) n = − (cid:88) d | Nd · ( d,M ) ≡ ε ( d ) · ( M, N/d ) (cid:112) d · ( M, d ) π L ( χ − d · ( M,d ) , . Here, we used that e (cid:0) − p (cid:1) = (cid:0) p (cid:1) for p ≡ e (cid:0) − p (cid:1) = (cid:0) p (cid:1) i for p ≡ L ( χ D , /π = H ( D ) / (cid:112) | D | (cf. [Za], Teil II, §
8, Satz 5) for a negative discriminant D ,we obtain the statement of the theorem in this case.If N is even, we have to consider the different 2-adic components separately. The case A = 2 − is easy to obtain. We give a proof for A = 2 + r rt . The remaining cases are doneanalogously. Using the same argument as before together with the results in Proposition2.12, we obtain β = − √ Nπ (cid:88) d | NN/d odd ε odd ( d ) ε ( d ) · ( M, N/d ) (cid:112) ( M, d ) (cid:112) N/d × (cid:88) n ≥ n,d )=1 n √ r − (cid:61) (cid:18)(cid:18) tnN/d (cid:19) r e (cid:18) r tnN/d (cid:19) (cid:89) p | d odd r p =1 (cid:18) np (cid:19) γ p (cid:33) − π (cid:88) d | NN/d ≡ ε ( d ) · ( M, N/d ) (cid:112) N · ( M, d ) (cid:112) N/d (cid:88) n ≥ n,d )=1 n (cid:18) − d · ( M, d ) n (cid:19) . Here, γ p = 1 for p ≡ γ p = i for p ≡ √ (cid:0) m (cid:1) e (cid:0) m (cid:1) =1 + (cid:0) − m (cid:1) i , we obtain √ r − (cid:61) (cid:18)(cid:18) tnN/d (cid:19) r e (cid:18) r tnN/d (cid:19) (cid:89) p | d odd r p =1 (cid:18) np (cid:19) γ p (cid:19) = a ( N/d ) (cid:89) p | d odd r p =1 (cid:18) np (cid:19) · (cid:16) − tnN/d (cid:17) , if d/ (4 · ( M, d )) ≡ δ ( r ) (cid:16) tnN/d (cid:17) , if d/ (4 · ( M, d )) ≡ , which yields the statement of the theorem for A = 2 + r nt . (cid:3) ATTICES WITH MANY BORCHERDS PRODUCTS 17
The following upper bound for the class number is well known.
Lemma 4.11.
Let − D be a negative discriminant. We have H ( − D ) ≤ √ D ln Dπ .
Proof.
We use again L ( χ D , /π = H ( D ) / (cid:112) | D | and argue as in the proof of Lemma 5.6on page 172 in [Ge] to obtain L ( χ − D , ≤ ln D for D >
4. Note that the bound for H ( − D ) is also valid for D = 3 and D = 4 with our normalization that H ( −
3) = 1 / H ( −
4) = 1 / (cid:3) Lemma 4.12.
Let A be an anisotropic finite quadratic module. We have | β | ≤ . · | A | ln(2 | A | ) . Proof.
We have by Theorem 4.10 and Lemma 4.11 that | β | ≤ π (cid:88) d | Nd ( d,M ) ≡ , a ( d )( N/d, M ) (cid:112) d ( d, M ) ln( d ( d, M )) ≤ π M ln( N M ) c ( A ) (cid:88) d | N (cid:115) d ( d, M ) , where c ( A ) = 2 if r = 3 and c ( A ) = 1, otherwise. We obtain | β | ≤ π M ln( N M ) c ( A ) σ ( M ) σ / ( N/M ) . If the order of A is odd, we have c ( A ) = 1 and M σ ( M ) σ / ( N/M ) = 1 . · | A | . Moreover, if the order of A is a power of 2, then c ( A ) M σ ( M ) σ / ( N/M ) ≤ . | A | . Using the multiplicativity of the divisor sum function, we see that c ( A ) M σ ( M ) σ / ( N/M ) ≤ . · | A | . Finally, using that
N M ≤ | A | implies the statement of the lemma. (cid:3) Corollary 4.13.
Let ( A, Q ) be an anisotropic finite quadratic module. If k ≥ , then dim( S k,A ) ≥ ( | A | + 1)( k − − . − . · | A | ln(2 | A | ) . Proof.
Since A is anisotropic, we have α = 1. Hence the estimate of Theorem 4.5 can berefined to give (cid:12)(cid:12)(cid:12)(cid:12) dim( S k,A ) − dim( M − k,A ( − ) − d ( k − (cid:12)(cid:12)(cid:12)(cid:12) ≤ R (cid:48) ( A ) , where R (cid:48) ( A ) = (cid:112) | A [2] | (cid:112) | A [3] | √ | A [2] | + 12 + | β | . Since A is anisotropic, Lemma 4.9 implies that | A [2] | ≤ , | A [3] | ≤ . Using in addition (2.2) and the estimates N (cid:48) ≤ | A | and N ≤ | A | , we obtaindim( S k,A ) ≥ ( | A | + 1)( k − − √ − √ − − | β | . Together with Lemma 4.12 this proves the corollary. (cid:3)
Corollary 4.14.
Let ( A, Q ) be an anisotropic finite quadratic module such that sig( A ) ≡− k (mod 4) . If | A | ≥ . · , then S ,A (cid:54) = { } and if k ≥ , then S k,A (cid:54) = { } for | A | ≥ . · . We implemented the dimension formula and some of the estimates used here in pythonusing sage [S + k = 2 and k = , we need to calculate thedimension of the invariants of the Weil representation. N. Skoruppa and S. Ehlen wrote aprogram that determines the invariants explicitly and we included this implementation inour repository [Ehl]. Our complete software package, together with all required libraries,examples and documentation is available online [Ehl].We used our program to obtain a list of all anisotropic finite quadratic modules suchthat S k,A = { } for k ≥ . Corollary 4.15.
Let ( A, Q ) be an anisotropic finite quadratic module such that sig( A ) ≡− k (mod 4) . Then S k,A = { } for k ≥ exactly if A belongs to the lists given in Tables4 and 5.Remark . The bound in Corollary 4.14 can be improved substantially for higher weights.However, all of the bounds obtained this way are far away from the correct bounds (themaximal order is 238 for k = and 60 for k = 2) found in Tables 4 and 5.4.2. Differences of dimensions.
Here we give lower bounds for the difference of thedimensions of S k,A ⊕ B and S k,A , where A is an arbitrary finite quadratic module and B isan isotropic finite quadratic module of order p for large primes p . We have to estimatethe differences of the quantities occurring in the dimension formula (3.3). To indicate thedependency of the finite quadratic module, we write α i ( A ) for the quantities α i associatedto A defined at the beginning of Section 4. We will make use of the following principle. Definition 4.17. If A is a finite quadratic module and U ⊂ A is a subgroup, we let U ⊥ = { a ∈ A | ( a, u ) = 0 for all u ∈ U } be the orthogonal complement of U . ATTICES WITH MANY BORCHERDS PRODUCTS 19
Table 4.
The 75 anisotropic finite quadratic modules A with S ,A = { } .Out of these, 59 have signature 1.sig( A ) -simple finite quadratic modules1 (cid:16) Z / N Z , x N (cid:17) for 1 ≤ N <
37 square-free and N ∈ { , , , , , , , , , , , , , , , , , , , , } , +33 +1 , +35 +2 , 2 +35 +1 , 4 +17 − , 4 +11 − , 4 − − , − − , 2 +17 +1 − , 2 +17 − − ,2 +11 +1 +1 , +17 +2 − , 4 +11 − , 4 +17 +1 +1 , +17 − +1 , 2 +17 +1 +1 − , 2 +17 +1 , 2 +35 , 2 +11 +1 , 4 +17 +1 , 4 − − , 2 +17 − , 4 − − , 4 +11 +1 , 2 +33 − ,2 +11 +1 , 4 +11 +2 , 4 +17 +1 , 2 +11 − , 2 +33 +1 , 2 +33 − − If U is a totally isotropic subgroup , that is, we have Q ( u ) = 0 for all u ∈ U , then thepair ( U ⊥ /U, Q ) also defines a finite quadratic module. We have | A | = | U ⊥ /U || U | andsig( A ) = sig( U ⊥ /U ). Proposition 4.18.
Let A be a finite quadratic module and let B = U ⊥ /U for some totallyisotropic subgroup U ⊂ A . We have an injection S k,B (cid:44) → S k,A given by f (cid:55)→ F with F α = 0 for α (cid:54)∈ U ⊥ and F α = f α + U for α ∈ U ⊥ .Proof. See Theorem 4.1 in [Sch2]. (cid:3)
Proposition 4.19. If U ⊂ A is a maximal totally isotropic subgroup, then A = U ⊥ /U isanisotropic. The isomorphism class of A is independent of the choice of U and we call A the anisotropic reduction of A .Proof. It is easy to see that U ⊥ /U is anisotropic for a maximal totally isotropic subgroup:Suppose that x ∈ U ⊥ /U is isotropic, x (cid:54) = 0. Then x = a + U for some a ∈ U ⊥ with Q ( a ) = 0. However, since a ∈ U ⊥ and a (cid:54)∈ U , this implies that the subgroup U (cid:48) of A generated by U and a is isotropic and strictly larger than U .The uniqueness follows from the classification of the anisotropic finite quadratic modules(see Table 2) and the fact that d ( U ⊥ /U ) = d ( A ) and sig( U ⊥ /U ) = sig( A ). (cid:3) Lemma 4.20.
Let A be an arbitrary finite quadratic module, and let B be an isotropicfinite quadratic module of order p , where p is a prime not dividing | A | . Then d A ⊕ B − d A = | A | p − , (4.6) Table 5.
Anisotropic finite quadratic modules A with S k,A = { } for k ≥ k sig( A ) genus symbols2 0 1 +1 , 5 − , 2 +11 +17 , 3 +1 − , 2 − +1 , 2 +22 +1 , 2 +26 − , 13 − , 2 +26 +1 , 17 +1 , 3 − − ,2 +11 +11 +1 , 2 +26 +1 +1 − , 3 +2 , 5 +1 , 5 − , 2 +11 − , 3 − − , 2 − − , 2 +22 − , 2 +26 +1 , 13 +1 , 2 +22 +1 ,17 − , 3 +1 − , 2 +11 +11 − , 2 +22 − − +33 , 4 − , 4 − − , 2 +11 − , 2 +17 +2 , 2 +11 +1 , 2 +11 − , 4 +11 +1 , 2 +17 +1 , 4 +11 − ,4 − +1 , 2 +11 − − +17 , 4 +17 , 2 +11 +1 k sig( A ) genus symbols sig( A ) genus symbols3 2 3 − , 2 +22 , 2 − +1 , 7 +1 , 2 +11 +11 , 11 − , 3 +1 +1 ,3 − − , 2 +22 − , 23 +1 +172 +11 , 4 +11 , 2 +17 − , 2 +11 − , 4 − +1 − +1 , 5 − +1 , 2 − − , 2 +11 − +17 − , 2 +22 , 7 +1 +1112 +11 , 4 +11 +1 − +11
8, 10, 14 0 1 +1 and | α ( A ⊕ B ) − α ( A ) − | A | p − | ≤ (cid:112) | A [2] | , | α ( A ⊕ B ) − α ( A ) − | A | p − | ≤ √ (cid:16) (cid:112) | A [3] | (cid:17) , | α ( A ⊕ B ) − α ( A ) − | A | p − | ≤ p − | A | ,α ( A ⊕ B ) − α ( A ) ≤ ( p − | A | . ATTICES WITH MANY BORCHERDS PRODUCTS 21
Proof.
To prove (4.6), we use (2.2). Since p (cid:54) = 2, we have B [2] = { } , and therefore d A ⊕ B = | A ⊕ B | | A [2] ⊕ B [2] | p | A | | A [2] | . This implies the stated formula.The bounds for the differences of α and α directly follow from (4.6) and Lemma 4.2combined with the fact that B [3] = { } .Let κ ( A ) denote the number of isotropic vectors in A . For α we use that α ( A ) = 12 ( κ ( A ) + κ ( A [2])) . Since the 2-torsion of B is trivial and since p does not divide | A | , we find α ( A ⊕ B ) − α ( A ) = 12 ( κ ( A ⊕ B ) − κ ( A ))= 12 κ ( A ) ( κ ( B ) − . The quantity κ ( B ) is bounded by 2 p − κ ( A ) is trivially bounded by | A | . This gives the claimed bound.We now turn to the estimate for α . For x ∈ R we write { x } = x − [ x ] ∈ [0 ,
1) for thefractional part of of x . By definition we have α ( A ) = (cid:88) γ ∈ A/ {± } {− Q ( γ ) } = 12 (cid:88) γ ∈ A {− Q ( γ ) } + 12 (cid:88) γ ∈ A [2] {− Q ( γ ) } . Consequently, α ( A ⊕ B ) − α ( A ) = 12 (cid:88) γ ∈ A ⊕ B {− Q ( γ ) } − (cid:88) γ ∈ A {− Q ( γ ) } . (4.7)For an arbitrary x ∈ R , we now estimate the sum S ( x, B ) = (cid:88) γ ∈ B { x − Q ( γ ) } . (4.8) If B is the level p isotropic finite quadratic module (which has genus symbol p ε · with ε = ( − p − ), we have S ( x, B ) = (cid:88) a,b ∈ Z /p Z { x − abp } = p { x } + ( p − (cid:88) b ∈ Z /p Z { x + bp } = p { x } + ( p − p − (cid:88) b =0 (cid:18) p { px } + bp (cid:19) = p { x } + ( p − { px } + ( p − α ( A ⊕ B ) − α ( A ) = 12 ( p − (cid:88) γ ∈ A (cid:18) {− Q ( γ ) } + {− pQ ( γ ) } + ( p − (cid:19) . and therefore | α ( A ⊕ B ) − α ( A ) − | A | p − | ≤ p − | A | . (4.9)If B is a finite quadratic module of level q = p , we slightly modify the above argument asfollows. Let ε ∈ Z with (cid:16) εp (cid:17) = ±
1, such that B has the genus symbol q ± . In this case wehave S ( x, B ) = (cid:88) a ∈ Z /p Z { x − ε a p } = (cid:88) a ∈ Z /p Z (cid:88) b ∈ Z /p Z { x − ε ( a + pb ) p } = p { x } + ( p − (cid:88) a ∈ ( Z /p Z ) × { p ( x − ε a p ) } . By means of this identity, we obtain the same bound (4.9) as in the earlier case. (cid:3)
Theorem 4.21.
Let A be an arbitrary finite quadratic module, and let B be an isotropicfinite quadratic module of order p , where p is a prime not dividing | A | . Then for k ≥ ,we have dim( S k,A ⊕ B ) − dim( S k,A ) ≥ | A | ( p − (cid:18) k − − pp − (cid:19) . Proof.
Let U be any totally isotropic subgroup of B ⊂ A ⊕ B . Then we have | B | = p , U ⊥ = A ⊕ U and A ∼ = U ⊥ /U . Therefore, Proposition 4.18 implies that the left hand side ATTICES WITH MANY BORCHERDS PRODUCTS 23 is non-negative. We use the dimension formuladim( S k,A ) = d A ( k + 12)12 − α ( A ) − α ( A ) − α ( A ) − α ( A ) + dim( M − k,A ( − ) . Because of Proposition 4.18, we have dim( M − k, ( A ⊕ B )( − ) ≥ dim( M − k,A ( − ). EmployingLemma 4.20, we obtaindim( S k,A ⊕ B ) − dim( S k,A ) ≥ | A | ( p − k − − (cid:112) | A [2] | − (cid:112) | A [3] | √ −
32 ( p − | A | . The claim now follows by the trivial estimates | A [2] | ≤ | A | and √ | A [3] | √ ≤ | A | . (cid:3) Corollary 4.22.
With the same assumptions as in Theorem 4.21, we have S k,A ⊕ B (cid:54) = { } for p ≥ p k given in Table 6. Table 6.
Bounds on p in Corollary 4.22. k ≤ k ≤ ≤ k ≤ k ≥ p k
73 37 29 19 17 13 11 7 55.
Simple finite quadratic modules If A is a finite quadratic module and k is an integer, we say that A is k -simple if S k,A = { } . We will now develop an algorithm that allows us to easily iterate over all finitequadratic modules starting from anisotropic ones.For a finite quadratic module A and an integer n , consider the finite set of finite quadraticmodules B ( A, n ) = { A (cid:48) | A = U ⊥ /U for a totally isotropic subgroup U ⊂ A (cid:48) with | U | = n } . For simplicity, we define a subset C ( A, n ) ⊂ B ( A, n ) using the following formal rules. Let p be an odd prime and r ≥ +1 (cid:55)→ p (cid:15) p · , where (cid:15) p = 1 if p ≡ (cid:15) p = −
1, otherwise,(O) ( p r ) ± (cid:55)→ ( p r +2 ) ± .A rule can be applied to A if the module on the left hand side of the rule is a directsummand of A . It is important to note that we can always apply the rule starting with thetrivial module 1 +1 . We should also remark that if r = 0 above, the left hand side of therule ( O ) is the trivial module. Therefore, for r = 0, we have the rules 1 +1 (cid:55)→ ( p ) +1 and1 +1 (cid:55)→ ( p ) − . The application of any of these rules to the genus symbol of A yields thegenus symbol of a finite quadratic module in B ( A, p ). Example 5.1.
Consider the finite quadratic module A given by the genus symbol 3 +1 − − .Applying rule (To) to A for p = 3, we obtain 3 − − − . Note that we can also apply rule(O) for both signs for p = 7 by writing 3 +1 − − = 3 +1 − +2 − (cid:55)→ +1 − +2 − and similarly 3 +1 − − = 3 +1 − − +1 (cid:55)→ +1 − − +1 . By applying both rules once inall possible cases, we obtain C ( A,
3) = { − − − , +1 − − , +1 +2 − , − +1 − , +1 − − } C ( A,
7) = { +1 − +5 , +1 − − +1 , +1 − − − , +1 − − +1 , +1 − +2 − } ,C ( A, p ) = { +1 − − p (cid:15) p · , +1 − − ( p ) +1 , +1 − − ( p ) − } for p (cid:54)∈ { , } . For p = 2, the rules we require are more complicated. Let q = 2 r with r ≥ +1 (cid:55)→ +2 ,(Te2) 1 +1 (cid:55)→ +20 ,(E1) q +2 (cid:55)→ (2 q ) +2 ,(E2) q − (cid:55)→ (2 q ) − ,(E3) q ± t (cid:55)→ (4 q ) ± t ,(E4) 2 +2 (cid:55)→ +20 ,(E5) 2 − (cid:55)→ − ,(E6) 2 +22 t (cid:55)→ +22 t for t ∈ { , } ,(E7) 2 +22 t (cid:55)→ − t for t ∈ { , } . Remark . Note that 2 +20 ∼ = 2 − and therefore rule (E2) applies to 2 +20 , as well. Definition 5.3.
We define C ( A, p ) to be the set of finite quadratic modules obtained from A after application of a single rule as listed above, only involving operations for p . For aprime power n = p r , we define C ( A, p r ) to be the set that is obtained from r consecutiveapplications of rules only involving p . Finally, we define C ( A, n ) for any positive integer n by induction on the number of different primes dividing n by putting C ( A, p r m ) = (cid:91) B ∈ C ( A,m ) C ( B, p r )for ( m, p ) = 1.We use these formal rules because it is very easy to implement them on a computer. Theorem 5.4.
Let A be a finite quadratic module and let A be its anisotropic reduction.Then A can be obtained from A in finitely many steps using the rules given above. Moreprecisely, we have A ∈ C ( A , n ) for n = | A | / | A | .Proof. It is enough to prove the claim for a p -module, that is a finite quadratic module ofprime-power order p n . Let us first assume that p is odd and that A has a genus symbol ofthe form q ± n with q = p r . Then it is easy to see that if r is even, we can obtain the symbol q ± starting from the trivial finite quadratic module1 +1 (cid:55)→ ( p ) ± (cid:55)→ . . . (cid:55)→ ( p r ) ± . Applying the same rule n times, we obtain the symbol q ± n . If r is odd instead, we startwith the anisotropic symbol p ± . We obtain p ± (cid:55)→ ( p ) ± (cid:55)→ . . . (cid:55)→ ( p r ) ± . ATTICES WITH MANY BORCHERDS PRODUCTS 25
We have now seen that we can obtain any finite quadratic p -module from a symbol of theform p ± n . Applying rule (To) several times reduces this symbol either to the trivial moduleor to the anisotropic finite quadratic module p − ε p · .For p = 2 we have to distinguish a few more cases. Suppose we are given a symbol of theform q ± rt . We can obtain q ± rt from a symbol that is a direct sum of symbols of the form 2 ± r (cid:48) s or 4 ± r (cid:48) s by applying rule (E3). Using rules (E4-E7), any even number of odd summands oflevel 8 can be reduced to level 4, leaving possibly a rank one odd component of level 8.Now let 2 +1 t . . . +1 t r be any odd discriminant form of level 4 with t , . . . , t r ∈ { , } . If { , } ⊂ { t , . . . , t r } , then 2 +20 = 2 +11 +17 is a summand. Now suppose that the rank is atleast equal to four and the symbol does not contain both, 2 +11 and − . Then 2 +44 is a directsummand of A . However, 2 +44 ∼ = 2 − − ∼ = 2 − +20 . Then, we can apply (Te2) to reduce therank of the level 4 part to at most 3.Finally, if we are given an even 2-adic symbol (2 n ) ± r which is not anisotropic (i.e. is not2 − ), it always contains (2 n ) +2 or (2 n ) − as a direct summand. Therefore, using rules (E1),(E2) and (E5), we can reduce to the case of level 2 or 1. Combining this with the strategyfor the odd symbols gives the result. (cid:3) We now describe the algorithm used to compute all simple lattices.
Algorithm 5.5.
Given integers r, s and a half-integer k , the following algorithm determinesthe isomorphism classes of all k -simple finite quadratic modules of signature s with aminimal number of generators less than or equal to r .(A) Compute all anisotropic k -simple finite quadratic modules satisfying the conditions(see Table 5).(B) For each previously computed k -simple finite quadratic module A compute the set C ( A, p ) for all primes p ≤ p k with p k given in Table 6.(C) Repeat step (B) until no further k -simple finite quadratic modules have been found.The correctness of the algorithm follows from Theorem 5.4 and Proposition 4.18 togetherwith the results from the last sections. Moreover, that the algorithm terminates followsfrom Corollary 4.7. Remark . In each iteration, the bound on the primes can be reduced to the maximalprime such that there is a newly discovered finite quadratic module A (cid:48) in C ( A, p ) for some k -simple finite quadratic module obtained one iteration earlier.The algorithm can be nicely illustrated in a graph. Figure 1 shows the output of thealgorithm with the parameters corresponding to signature (2 , A to some B above A indicates that B is contained in C ( A, p ) for a prime p .The color of the edge is the same as that of A . If an edge is green (and dotted), then thetwo modules connected by the edge are both non-simple (and therefore colored green inthe graph). Note that the graph does not contain all green edges for simplicity. It containsat most one green “incoming” edge per vertex.In Tables 7-8, we list all − n -simple finite quadratic modules of signature 2 − n withminimal number of generators r ≤ n for n ≥ Figure 1.
The graph illustrates Algorithm 5.5 for k = 3 , r = 6 and s = 6. ATTICES WITH MANY BORCHERDS PRODUCTS 27
Table 7.
The table shows the 70 finite quadratic modules A of signature 0with minimal number of generators r ≤ S ,A = { } .level genus symbols level genus symbols1 1 +1
18 2 +2 +1 , 2 +2 − , 2 +4 +1 +2 , 2 +4
20 2 +20 − − , 3 +4
21 3 − − +20 , 4 − , 4 +2 , 2 +40 , 2 +2 +2 , [2 +2 − ], 2 +20 +2 , [4 − ],4 +4
24 2 +11 +11 +1 , 4 − +1 − , 5 +2 , 5 − , 5 +4
25 25 +1 , 25 − +2 − , 2 +4 − , 2 +2 +4
27 3 +1 − −
28 2 +26 +1 +11 +17 , 4 +20 , 2 +31 +17 , 2 +2 +20 , 8 +2 , 2 +2 +2
32 4 +17 +11 +1 , 9 − , 3 − − , 9 −
33 3 +1 −
10 2 − +1 , 2 +2 − , [2 +4 − ], 2 − +1 , 2 +2 +2
36 2 +20 −
12 2 +22 +1 , 2 +26 − , 2 +20 − , 2 +46 − , 2 +26 +3 , 4 +2 −
45 9 +1 −
13 13 −
48 2 − − −
16 2 +17 +11 , 2 − − , 4 +17 +11 , 2 +37 +11 , 8 +20 , 2 +17 +2 +11
49 49 +1
17 17 +1
60 2 +26 +1 +1
64 2 +17 +11 For n = 1, a large family of -simple finite quadratic modules is given by the cyclic finitequadratic modules A ( N ) = ( Z / N Z , x / N ) for N ∈ Z > . The corresponding orthogonalmodular varieties are the modular curves Γ ( N ) \ H ∗ , where H ∗ = H ∪ P ( Q ). Moreover,each of these finite quadratic modules has a global realization given by L N = Z withquadratic form N x + x x . We have that L N is simple if and only if 1 ≤ N ≤
36 or if N is in the following list:38 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , . The remaining finite quadratic modules for n = 1 are included in [Ehl].It is interesting to observe that for N squarefree, L N is simple if and only if X ∗ ( N ) =Γ ∗ ( N ) \ H ∗ has genus zero. Here, Γ ∗ ( N ) is the extension of Γ ( N ) by all Atkin-Lehnerinvolutions. The situation for non-squarefree N is more complicated. It would be interestingto find a similar geometric interpretation in the general case. Table 8.
The table shows all finite quadratic modules of signature 2 − n modulo 8 with minimal number of generators r ≤ n such that S n = { } n level : genus symbols3 : 2 +17 , 2 +37 , 2 +17 +2 , 2 +57 , 2 +37 +2 , 2 +17 +4 : 4 +17 , 2 +2 +17 , 2 +4 +17 , : 2 +11 +1 , 2 +17 − , 2 +17 +4 , : 8 − , 8 +17 , 2 +2 − , [2 +4 − ]4 : 3 +1 , 3 − , 3 +5 , : 2 +4 +1 , [2 +6 +1 ], 2 +2 +1 : 4 − , 2 +2 − , 2 +4 − , [2 +6 − ]6 : 2 − , − , − , [2 − ], : 5 +1 : 4 − , : 2 +11 − : 3 − , : 2 +22 , : 7 +1 : 2 +11 , : 4 +11 , : 8 +11 : 1 +1 , : 2 +2 , : 1 +1 Simple lattices
Let L be an even lattice of signature (2 , n ). We will say that L is simple if S k,L (cid:48) /L = { } for k = n . In this section we determine all isomorphism classes of simple lattices ofsignature (2 , n ). If L is simple, then the finite quadratic module L (cid:48) /L is k -simple for k = n . Thus, we are interested in all k -simple finite quadratic modules with minimalnumber of generators r ≤ n that actually correspond to a lattice of signature (2 , n ). Proposition 6.1.
Let A be a finite quadratic module and write ε q for the sign of the Jordancomponent of A of order q . Let r p be the minimal number of generators of A p . There is aneven lattice L of signature ( r, s ) with L (cid:48) /L = A if and only if all of the following conditionshold.(1) We have sig( A ) ≡ r − s (mod 8) .(2) For all primes p , we have r + s ≥ r p .(3) For all odd primes p with r + s = r p , write ( − s | A | = p α a p with ( a p , p ) = 1 . Thenwe have (6.1) (cid:89) q ε q = (cid:18) ap (cid:19) , where the product runs over all powers q of p .(4) If r + s = r and A does not contain an odd direct summand of the form ± mt with m ≥ , then (6.1) holds for p = 2 and ( − s | A | = 2 α a with ( a,
2) = 1 , as well.Proof.
See [Ni], Theorem 1.10.1. (cid:3)
ATTICES WITH MANY BORCHERDS PRODUCTS 29
Using Proposition 6.1, we determined all genus symbols that do not correspond to latticesof signature (2 , n ). We enclosed them in parentheses [ · ] in Tables 7–8. For instance, themodule M = 2 +2 − in Table 7 does not correspond to any lattice of signature (2 , − a = 1 and M does not contain an oddcomponent of level 4. It does, however, correspond to a (non-simple) lattice of signature(2 , m ) for all m ≥ , n )for n ≥ Proposition 6.2. If L is a simple lattice of signature (2 , n ) and L is not contained inthe genus +11 − − , then its genus contains a unique isomorphism class. The genus +11 − − contains two isomorphism classes.Proof. For n ≥
2, Corollary 22 in Chapter 15 of [CS] states that if there is more than oneclass in the genus of an indefinite lattice L , then | det( L ) | ≥ . There is no finite quadraticmodule of this size in our list in Tables 7-8.The lattices in signature (2 ,
1) are slightly more complicated to treat. Using Theorem 21in Chapter 15 of [CS], we find that only the lattices of discriminant d with 4 d divisible by5 or 8 might contain more than one class in their genus. Theorem 19 ibid. finally leavesus with the following list of genera that might contain more than one class:2 +17 +11 +11 , +11 +17 +11 , +17 +11 +11 , +11 − − . For the first 3 genera, we can use [EH], Theorem 2.2, to see that these also only containone class. The last one is treated in [Wa] in Chapter 7, Section 5. This genus contains twoisomorphism classes represented by the integral ternary quadratic forms φ ( x , x , x ) = x + x x − x + 25 x ,φ ( x , x , x ) = 5( x + x x − x ) + x . (cid:3) Applications to Borcherds products.
Let (
L, Q ) be an even lattice of signature(2 , n ) and let A = L (cid:48) /L be its the discriminant module. We write O( L ) for the orthogonalgroup of L and O( L ) + for the subgroup of index 2 consisting of those elements whosedeterminant has the same sign as the spinor norm. We consider the kernel Γ L of thenatural homomorphism O( L ) + → Aut( A ), sometimes referred to as the stable orthogonalgroup of L .Let D be the hermitian symmetric space associated to the group O( L ⊗ Z R ). It can berealized as a tube domain in C n . The group Γ L acts on D and the quotient X L = Γ L \ D has the structure of a quasi-projective algebraic variety. For suitable choices of L , importantfamilies of classical modular varieties can be obtained in this way, including Shimura curves,Hilbert modular surfaces and Siegel modular threefolds. For every µ ∈ A and every negative m ∈ N Z , there is a Heegner divisor Z ( m, µ ) on X L (sometimes also referred to as special divisor or rational quadratic divisor), see e.g. [Bo2],[Br1]. We denote by Pic Heeg ( X L ) the subgroup of the Picard group Pic( X L ) of X L generatedby all such Heegner divisors.If L is simple, then for every pair ( m, µ ) as above there is a weakly holomorphic modularform f m,µ ∈ M !1 − n/ ,A ( − of weight 1 − n/ f m,µ ) (in the sense ofTheorem 13.3 in [Bo1]) is a meromorphic modular form for Γ L whose divisor is Z ( m, µ ).In particular, the vector space Pic Heeg ( X L ) ⊗ Z Q is one-dimensional and generated by theHodge bundle, hence it is as small as it can be.The weight of the Borcherds product Ψ( f m,µ ) is given by half of the constant term ofthe component of f m,µ corresponding to the characteristic function φ of the zero elementof A . Equivalently, it can be expressed in terms of the coefficient of index ( − m, µ ) of theunique normalized Eisenstein series E n/ ,A ∈ M n/ ,A whose constant term is φ , see e.g.Theorem 12 of [BK]. The coefficients of such Eisenstein series can be explicitly computed,see Theorem 7 of [BK] or [KY]. It would be interesting to use the list of simple latticesto search systematically for holomorphic Borcherds products of singular weight n/ − L . Such Borcherds products are often denominator identities of generalized Kac-Moodyalgebras, see [Sch1]. References [AS]
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E-mail address , J. H. Bruinier: [email protected]
E-mail address , S. Ehlen: [email protected]
Fachbereich Mathematik, Technische Universit¨at Darmstadt, Schlossgartenstraße 7,D–64289 Darmstadt, Germany
E-mail address , E. Freitag: [email protected]@mathi.uni-heidelberg.de