aa r X i v : . [ m a t h . N T ] M a r REFLECTIVE MODULAR FORMS: A JACOBI FORMSAPPROACH
HAOWU WANG
Abstract.
We give an explicit formula to express the weight of 2-reflectivemodular forms. We prove that there is no 2-reflective lattice of signature (2 , n )when n ≥
15 and n = 19 except the even unimodular lattices of signature(2 ,
18) and (2 , U ⊕ E ( − ⊕ h− n i is not 2-reflective if n >
1. Wealso classify reflective modular forms on lattices of large rank and the modularforms with the simplest reflective divisors. Introduction
Let M be an even lattice of signature (2 , n ). A non-constant holomorphic mod-ular form for M is called reflective if the support of its divisor is contained inthe union of quadratic divisors determined by reflective vectors of M . This typeof modular forms first appeared in the works of Borcherds [2, 3] and Gritsenko-Nikulin [19]. They have many applications in various related topics, such as theclassification of Lorentzian Kac-Moody algebras [1, 20, 21, 23, 24, 30, 31], search ofhyperbolic reflection groups [1, 4] and the theory of moduli spaces [17, 18, 25, 27].Reflective modular forms seem to be exceptional and very rare. The classificationof reflective modular forms has been widely studied by several mathematicians. In1998, Gritsenko and Nikulin first conjectured that the number of lattices havingreflective modular forms is finite [20] and gave a complete classification for n = 3[21, 23]. Later Scheithauer gave a complete classification of reflective modular formsof singular weight on lattices of prime level [30, 31, 32, 34]. Looijenga [25] provedone part of the arithmetic mirror symmetry conjecture formulated in [22], whichmight give a new approach to classify reflective modular forms. Recent work of Ma[27] showed that there are only finitely many lattices of signature (2 , n ) which carrya strongly reflective modular form that vanishes at order one along the reflectivedivisors when n ≥ M is called 2-reflective if the support of its zero divisor is contained in theHeegner divisor defined by the ( − M . A lattice M is called 2-reflectiveif it admits a 2-reflective modular form. In 2017, Ma [26] showed that there areonly finitely many 2-reflective lattices of signature (2 , n ) with n ≥ n ≥
26 except the even unimodular lattice II , ofsignature (2 , Date : March 15, 2019.2010
Mathematics Subject Classification.
Primary 11F50, 11F55; Secondary 17B67, 14J28.
Key words and phrases.
Jacobi forms, reflective modular forms, Borcherds products.
Theorem 1.1 (Theorem 3.8) . Let M be a -reflective lattice of signature (2 , n ) with n ≥ . Then n = 19 or M is isomorphic to the unique even unimodularlattice of signature (2 , or (2 , . Our approach is based on the Gritsenko-Nikulin representation of Borcherdsproducts in terms of Jacobi forms [15, 21]. When M contains two integral hyperbolicplanes (i.e. M = 2 U ⊕ L ( − L such that its Borcherds product gives the above 2-reflectivemodular form. Then the identity (Lemma 2.3) related to q -term of Jacobi formsof weight 0 yields a formula expressing the weight of 2-reflective modular forms(Theorem 3.2). Furthermore, we can construct holomorphic Jacobi forms of certainsmall weights from the above Jacobi form of weight 0 by using the weight raisingdifferential operators (Lemma 2.2). The existence of such Jacobi forms implies thenon-existence of 2-reflective modular forms with respect to lattices of large rank.Our main theorem gives us a necessary condition for a lattice of signature (2 , T n = 2 U ⊕ E ( − ⊕ h− n i are not 2-reflective for n ≥ Theorem 1.2 (Theorem 4.9) . Let M be a reflective lattice of signature (2 , n ) andof prime level p . (1) If p = 2 and n > and n = 22 , then M is isomorphic to II , (2) . (2) If p = 3 and n > and n = 20 , then M is isomorphic to II , (3) or II , (3) . (3) If p > and n >
10 + 24 / ( p + 1) , then M is isomorphic to II , ( p ) or II , ( p ) . In [26], Ma showed that there is no reflective lattice of signature (2 , n ) with n ≥
26 containing 2 U except the even unimodular lattice of signature (2 , Theorem 1.3 (Theorem 4.11) . There is no reflective lattice of signature (2 , n ) with ≤ n ≤ . Our approach can be applied to some other questions. For instance, it providesus a straightforward way to classify the modular forms with the simplest reflectivedivisors, i.e. the dd-modular forms (see Section 5). This gives a generalization ofthe main results in [8, 16] (see Theorem 5.2).The paper is organized as follows. In section 2 we introduce briefly Jacobi formsand differential operators. In section 3 we define 2-reflective modular forms andprove our first main theorem. An application is also presented. In section 4 weclassify reflective modular forms. Section 5 is devoted to the classification of dd-modular forms.
EFLECTIVE MODULAR FORMS: A JACOBI FORMS APPROACH 3 Preliminaries: Jacobi forms
In this section, some standard facts about Jacobi forms are recalled. We refer to[9, 12] for more details. From now on, L always denotes an even positive-definitelattice with bilinear form ( · , · ) and dual lattice L ∨ . The rank of L is denoted asrank( L ). We define the Jacobi forms in the following way. Definition 2.1.
Let ϕ : H × ( L ⊗ C ) → C be a holomorphic function and k ∈ Z .If ϕ satisfies the functional equations ϕ (cid:18) aτ + bcτ + d , z cτ + d (cid:19) = ( cτ + d ) k e iπ c ( z , z ) cτ + d ϕ ( τ, z ) , (2.1) ϕ ( τ, z + xτ + y ) = e − iπ (( x,x ) τ +2( x, z )) ϕ ( τ, z )(2.2)for any (cid:18) a bc d (cid:19) ∈ SL ( Z ) and any x, y ∈ L and ϕ admits a Fourier expansion as(2.3) ϕ ( τ, z ) = X n ≥ n X l ∈ L ∨ f ( n, l ) q n ζ l where n ∈ Z , q = e πiτ and ζ l = e πi ( l, z ) , then ϕ is called a weakly holomorphicJacobi form of weight k and index L . If ϕ further satisfies the condition f ( n, l ) = 0 = ⇒ n ≥ ϕ is called a weak Jacobi form. If ϕ further satisfies the stronger condition f ( n, l ) = 0 = ⇒ n − ( l, l ) ≥ ϕ is called a holomorphic Jacobi form. We denote by J w.h.k,L (resp. J wk,L , J k,L )the vector space of weakly holomorphic Jacobi forms (resp. weak Jacobi forms,holomorphic Jacobi forms) of weight k and index L .Similarly, we can also define Jacobi forms with character. For more details, werefer to [9, 12, 21].The Fourier coefficient f ( n, l ) depends only on the number 2 n − ( l, l ) and the classof l modulo L . The fact implies that for fixed n the sum P l ∈ L ∨ f ( n, l ) q n ζ l in (2.3)is finite. The number 2 n − ( l, l ) is called the hyperbolic norm of Fourier coefficient f ( n, l ). The Fourier coefficients f ( n, l ) with negative hyperbolic norm are calledsingular Fourier coefficients, which play a crucial role in the theory of Borcherdsproducts. By definition, a weakly holomorphic Jacobi form without singular Fouriercoefficient is a holomorphic Jacobi form.We next explain the isomorphism between vector-valued modular forms andJacobi forms. More details can be found in [9, Section 2] and [12, Lemma 2.3]. Let U be a hyperbolic plane, i.e. U = Z e ⊕ Z f with ( e, f ) = 1, ( e, e ) = ( f, f ) = 0.Then the lattice 2 U ⊕ L ( −
1) is an even lattice of signature (2 , rank( L ) + 2), whosediscriminant group is isomorphic to D ( L ) = L ∨ /L . Let { e γ : γ ∈ D ( L ) } be thebasis of the group ring C [ D ( L )]. Let Mp ( Z ) be the metaplectic group which is adouble cover of SL ( Z ). We denote the Weil representation of Mp ( Z ) on C [ D ( L )]by ρ D ( L ) (see [5, Section 1.1]). Let F be a nearly holomorphic vector-valued modular HAOWU WANG form for ρ D ( L ) of weight k with Fourier expansion of the form F ( τ ) = X γ ∈ D ( L ) X n ∈ Z − ( γ,γ )2 c ( γ, n ) q n e γ = X γ ∈ D ( L ) F γ ( τ ) e γ . Here, the sum X γ ∈ D ( L ) X n ∈ Z − ( γ,γ )2 n< c ( γ, n ) q n e γ is called the principal part of F . We define the theta-function for the lattice L as(2.4) Θ Lγ ( τ, z ) = X l ∈ γ + L exp ( πi ( l, l ) τ + 2 πi ( l, z )) , γ ∈ D ( L ) . Then the function(2.5) X γ ∈ D ( L ) F γ ( τ )Θ Lγ ( τ, z )is a weakly holomorphic Jacobi form of weight k + rank( L ) and index L . Theprincipal part of F corresponds to the singular Fourier coefficients of the aboveJacobi form. The map M k,ρ D ( L ) → J k + rank( L ) ,L F ( τ ) X γ ∈ D ( L ) F γ ( τ )Θ Lγ ( τ, z )is an isomorphism sending holomorphic vector-valued modular forms to holomor-phic Jacobi forms. Since the weights of holomorphic vector-valued modular formsare non-negative, the space J k,L of holomorphic Jacobi forms of weight k and index L is trivial if k < rank( L ) /
2. The minimum possible weight k = rank( L ) / ( Z ) to modular forms on orthogonal groups (see [2, 3]). Bymeans of the isomorphism between vector-valued modular forms for the Weil repre-sentation and Jacobi forms, Gritsenko and Nikulin ([15, Theorem 3.1] or [19, The-orem 2.1]) gave a variant of Borcherds products, which lifts weakly holomorphicJacobi forms of weight 0 to modular forms on orthogonal groups. We denote theBorcherds product of a weakly holomorphic Jacobi form φ of weight 0 by Borch( φ ).We now recall the following weight raising differential operator which will beused later. Such technique can also be found in [10] for the general case or in [11]for classical Jacobi forms. Lemma 2.2.
Let ψ ( τ, z ) = P a ( n, l ) q n ζ l be a weakly holomorphic Jacobi form ofweight k and index L . Then H k ( ψ ) is a weakly holomorphic Jacobi form of weight k + 2 and index L , where H k ( ψ ) = H ( ψ ) + (2 k − rank( L )) G ψ, (2.6) H ( ψ )( τ, z ) = 12 X n ∈ Z X l ∈ L ∨ [2 n − ( l, l )] a ( n, l ) q n ζ l , (2.7) EFLECTIVE MODULAR FORMS: A JACOBI FORMS APPROACH 5 and G ( τ ) = − + P n ≥ σ ( n ) q n is the Eisenstein series of weight on SL ( Z ) .Proof. Let { α , ..., α n } be a basis of L and { α ∗ , ..., α ∗ n } be its dual basis. We write z = P ni =1 z i α i ∈ L ⊗ C , z i ∈ C . We define ∂∂ z = P ni =1 α ∗ i ∂∂z i . Then we have that (cid:18) ∂∂ z , ∂∂ z (cid:19) e πi ( l, z ) = − π ( l, l ) e πi ( l, z ) and the operator H ( · ) is equal to the heat operator H = 12 πi ∂∂τ + 18 π (cid:18) ∂∂ z , ∂∂ z (cid:19) . By formulas (3.5) and (3.7) in [10, Lemma 3.3], the transformations of the function H ( ψ ) with respect to the actions of SL ( Z ) and the Heisenberg group of L areknown. From these transformations, we see that H ( ψ ) does not transform likea Jacobi form, so we make an automorphic correction by considering the quasi-modular Eisenstein series G of weight 2. By direct calculations, we can showthat H k ( ψ ) is invariant under SL ( Z ) and the Heisenberg group. Therefore, it is aweakly holomorphic Jacobi form of weight k + 2 and index L . (cid:3) The next lemma plays a key role in our discussions on the weight of 2-reflectivemodular forms and in the classification of dd-modular forms. It is a particular caseof [15, Proposition 2.2].
Lemma 2.3.
Assume that φ is a weakly holomorphic Jacobi form of weight andindex L with the Fourier expansion φ ( τ, z ) = aq − + X l ∈ L ∨ c (0 , l ) ζ l + O ( q ) . Then we have the following identity X l ∈ L ∨ c (0 , l ) − L ) X l ∈ L ∨ c (0 , l )( l, l ) − a = 0 . Proof.
From Lemma 2.2, it follows that H ( φ ) is a weakly holomorphic Jacobi formof weight 2. Therefore H ( φ )( τ,
0) is a nearly holomorphic modular form of weight2 for the full modular group SL ( Z ). By [2, Lemma 9.2], H ( φ )( τ,
0) has zeroconstant term, which establishes the desired formula. (cid:3) In this section we study 2-reflective modular forms. We prove our first maintheorem and give an application of our results.3.1.
Non-existence of 2-reflective modular forms.
Let M be an even integrallattice of signature (2 , n ), n ≥
3, and let(3.1) D ( M ) = { [ ω ] ∈ P ( M ⊗ C ) : ( ω, ω ) = 0 , ( ω, ¯ ω ) > } + be the associated Hermitian symmetric domain of type IV (here + denotes one of itstwo connected components). Let us denote the index 2 subgroup of the orthogonalgroup O( M ) preserving D ( M ) by O + ( M ). We define(3.2) H = [ l ∈ M ( l,l )= − l ⊥ ∩ D ( M ) HAOWU WANG as the Heegner divisor of D ( M ) generated by the ( − M . Definition 3.1.
Let F be a non-constant holomorphic modular form on D ( M )with respect to a finite index subgroup Γ < O + ( M ) and a character (of finiteorder) χ : Γ → C . The function F is called 2-reflective if the support of its zerodivisor is contained in H . A lattice M is called 2-reflective if it admits a 2-reflectivemodular form.Following [26], we consider the decomposition (3.3) of the ( − H . Let A M = M ∨ /M be the discriminant group of M . We denote the mostimportant subgroup of O + ( M ) acting trivially on A M by e O + ( M ). The invariants of e O + ( M )-orbit of a primitive vector l ∈ M are its norm ( l, l ) and its image l/ div( l ) ∈ A M where div( l ) is the positive integer generating the ideal ( l, M ). From this pointof view, we choose the following notations. For λ ∈ A M and m ∈ Q we put H ( λ, m ) = [ l ∈ M + λ ( l,l )=2 m l ⊥ ∩ D ( M )as the Heegner divisor of discriminant ( λ, m ). In particular, H = H (0 , − π M ⊂ A M be the subset of elements of order 2 and norm − /
2. For each µ ∈ π M we abbreviate H ( µ, − /
4) by H µ . We also set H = [ l ∈ M, ( l,l )= − l )=1 l ⊥ ∩ D ( M ) . Then we have the following decomposition(3.3) H = H + X µ ∈ π M H µ . By [26, Lemma 2.2], if M admits a 2-reflective modular form with respect to someΓ < O + ( M ), then M also has a 2-reflective modular form with respect to any otherfinite index subgroup Γ < O + ( M ). Thus, the lattice M is 2-reflective if and onlyif it admits a 2-reflective modular form with respect to e O + ( M ). Throughout thissection, we only consider 2-reflective modular forms with respect to e O + ( M ).Next, we assume that the lattice M contains 2 U and M = 2 U ⊕ L ( − L is a positive-definite even lattice and U is a hyperbolic plane. In this case, each H ∗ is an e O + ( M )-orbit of a single quadratic divisor l ⊥ ∩ D ( M ) and it is irreducible.We can write each element of π M in the form µ = (0 , n µ , µ / , , n µ ∈ Z , µ ∈ L and 2 n µ − ( µ , µ ) = − . If M admits a 2-reflective modular form F ofweight k , then its divisor can be written asdiv( F ) = β H + X µ ∈ π M β µ H µ = β H + X µ ∈ π M ( β µ − β ) H µ , (3.4)where β ∗ are non-negative integers. By [5, Theorem 5.12] or [6, Theorem 1.2], thereexists a nearly holomorphic vector-valued modular form f of weight − rank( L ) / ρ M of Mp ( Z ) on the group ring C [ A M ] EFLECTIVE MODULAR FORMS: A JACOBI FORMS APPROACH 7 with principal part β q − e + X µ ∈ π M ( β µ − β ) q − / e µ , such that F is the Borcherds product of f . In view of the isomorphism betweenvector-valued modular forms and Jacobi forms, there exists a weakly holomorphicJacobi form φ L of weight 0 and index L with singular Fourier coefficients of theform (see [9])(3.5) sing ( φ L ) = β X r ∈ L q ( r,r ) / − ζ r + X µ ∈ π M ( β µ − β ) X s ∈ L + µ / q ( s,s ) / − / ζ s where ζ l = e πi ( l, z ) . Thus, we have(3.6) φ L ( τ, z ) = β q − + β X r ∈ R ( L ) ζ r + 2 k + X u ∈ π M ( β µ − β ) X s ∈ R µ ( L ) ζ s + O ( q )here and subsequently, R ( L ) denotes the set of 2-roots in L and(3.7) R µ ( L ) = { s ∈ L ∨ : 2 s ∈ R ( L ) , s − µ / ∈ L } . With the help of equation (3.6) and Lemma 2.3, we get the following theorem.
Theorem 3.2.
Let L be a positive-definite even lattice and M = 2 U ⊕ L ( − .Suppose that F is a -reflective modular form of weight k with divisor of the form (3.4) . Then the weight k of F is given by the following formula k = β (cid:20)
12 + | R ( L ) | (cid:18) L ) − (cid:19)(cid:21) + (cid:18) L ) − (cid:19) X µ ∈ π M ( β µ − β ) | R µ ( L ) | . (3.8) Remark . From the above theorem, we have(1) If R ( L ) is empty, then the weight of 2-reflective modular form is 12 β .(2) When rank( L ) ≥
6, there is no 2-reflective modular form with β = 0.This fact can also be proved by Riemann–Roch theorem as the proof of [34,Proposition 6.1].(3) When rank( L ) = 6, the weight of 2-reflective modular form is β (12 + | R ( L ) | ) and the modular form is not of singular weight.We next study modular forms with complete 2-divisor (i.e. div( F ) = H ), whichare the simplest 2-reflective modular forms. Theorem 3.4.
If there exists a modular form with complete -divisor for M =2 U ⊕ L ( − , then either rank( L ) ≤ , or L is a unimodular lattice of rank or . Moreover, the weight of the corresponding modular form is (3.9) k = 12 + | R ( L ) | (cid:18) L ) − (cid:19) . Proof.
Firstly, the above formula is a direct result of Theorem 3.2. Let F be amodular form with complete 2-divisor. Then there exists a weakly holomorphicJacobi form of weight 0 and index L such that φ ( τ, z ) = q − + X r ∈ R ( L ) ζ r + 2 k + O ( q ) HAOWU WANG whose singular Fourier coefficients are (see formula (3.5)) sing ( φ ) = X n ≥− X l ∈ L ( l,l )=2 n +2 q n ζ l , with Borch( φ ) = F . By [26], it is known that rank( L ) <
24 or L is a unimodularlattice of rank 24. Let us assume that rank( L ) ≤
23 and let us construct twospecial Jacobi forms by using the differential operators introduced in Lemma 2.2.For simplicity, we set R = | R ( L ) | and n = rank( L ). f ( τ, z ) = 24 n − H ( φ )( τ, z )= q − + X r ∈ R ( L ) ζ r − R + O ( q ) ∈ J w.h. ,L f ( τ, z ) = 24 n − H ( f )( τ, z )= q − + X r ∈ R ( L ) ζ r − ( R + 24)( n − n −
28 + O ( q ) ∈ J w.h. ,L Let E and E denote the Eisenstein series on SL ( Z ) of weight 4 and 6, respectively.Then we can check that g ( τ, z ) = n − E ( τ ) φ ( τ, z ) − f ( τ, z )]= R (cid:18) − n (cid:19) + 6( n −
26) + O ( q ) ∈ J ,L and h ( τ, z ) = E ( τ ) φ ( τ, z ) − E ( τ ) f ( τ, z )= 24 Rn −
720 + O ( q ) ∈ J ,L are holomorphic Jacobi forms of weight 4 and 6, respectively. In fact, the singularFourier coefficients are stable under the actions of the differential operators, so thesingular Fourier coefficients of f and f come from sing ( φ ). In order to check g and h are holomorphic Jacobi forms, we only need to check that g and h have nosingular Fourier coefficient i.e. the singular part sing ( φ ) has been cancelled by theabove combinations of φ , f and f .Since the singular weight of holomorphic Jacobi form of index L is n , we deducethat g = 0 if n > h = 0 if n >
12. By direct calculations, we have • when R = 0, g = 0 if n < • when R > g = 0 if n ≤ • when n ≥ g = 0 and h = 0 if and only if n = 16 and R = 480.When n = 16, from h = 0, it follows that the Fourier coefficients of φ satisfy: c ( n, l ) = 0 if 2 n − ( l, l ) = 0 and l L . Otherwise, there exists a Fourier coefficient c ( n, l ) = 0 with 2 n − ( l, l ) = 0 and l L . Assume that c ( n, l ) is such Fouriercoefficient with the smallest n . Then the coefficient of q n ζ l in E φ is c ( n, l ) andthe coefficient of q n ζ l in E f is − c ( n, l ). Thus, the coefficient of q n ζ l in h isnot zero and then h = 0, which leads to a contradiction. Therefore, the following EFLECTIVE MODULAR FORMS: A JACOBI FORMS APPROACH 9 holomorphic Jacobi form of singular weight 8 E φ − E f = 1728 + X n> ,l ∈ L ∨ n =( l,l ) a ( n, l ) q n ζ l ∈ J ,L satisfies the same condition: a ( n, l ) = 0 if 2 n − ( l, l ) = 0 and l L . We then obtain E φ − E f = 1728 X l ∈ L q ( l,l )2 ζ l and L has to be unimodular. The proof is completed. (cid:3) Remark . It is worth pointing out that there exist lattices L which admit amodular form with complete 2-divisor when 1 ≤ rank( L ) ≤ χ is a modular form with complete 2-divisor.We are going to generalize our method to prove the non-existence of 2-reflectivemodular forms in higher dimensions. Theorem 3.6.
Suppose that M = 2 U ⊕ L ( − is a -reflective lattice satisfying rank( L ) ≥ . Then either rank( L ) = 17 , or L is a unimodular lattice of rank or . Furthermore, when rank( L ) = 17 , the weight of the corresponding -reflectivemodular form is β , where β is the multiplicity of the divisor H .Proof. If M has a 2-reflective modular form F of weight k with divisor of theform (3.4), then there exists a weakly holomorphic Jacobi form φ of weight 0 andindex L with singular Fourier coefficients of the form (3.5). Let us assume thatrank( L ) ≤
23. We next construct a holomorphic Jacobi form of weight 6 from φ .We write φ = S + d + S + · · · , where S and S are the first and second terms in(3.5), respectively, and d = 2 k . It is clear that φ − S − S does not have the termwith negative hyperbolic norm. We can construct ( n = rank( L )) f = 24 n − H ( φ ) = S + d + c S + · · · ∈ J w.h. ,L ,f = 24 n − H ( f ) = S + d + c S + · · · ∈ J w.h. ,L ,f = 24 n − H ( f ) = S + d + c S + · · · ∈ J w.h. ,L , where d = n ( d − β ) n − , c = n − n − ,d = ( n − d − β ) n − , c = c n − n − ,d = ( n − d − β ) n − , c = c n − n − . The function ϕ = ( c − c ) E φ + ( c − E f + (1 − c ) f = u + O ( q ) ∈ J ,L where u = ( d − β )( c − c ) + ( d + 240 β )( c −
1) + d (1 − c )is a holomorphic Jacobi form of weight 6 because the potential singular Fouriercoefficients S and S have been cancelled. In view of the singular weight, ϕ = 0 if rank( L ) ≥
13. From Remark 3.3, weknow that if F exists then d = 2 k ≥ rank( L ) and β > L ≥
6. Bydirect calculations, when n = 13 or 14, u = 0, which is impossible.We next assume that 15 ≤ rank( L ) ≤
23. We construct g = E φ − f = ( d + 240 β ) − d + (1 − c ) S + · · · ∈ J w.h. ,L . Since g only has singular Fourier coefficients of the form S , the minimum possiblehyperbolic norm of its Fourier coefficients is − . Therefore, η g is a holomorphicJacobi form of weight 7 and index L with character, where η is the Dedekind etafunction. In view of the singular weight, we have that η g = 0 and then g = 0.If S = 0, then F is a 2-reflective modular form with complete 2-divisor, whichgives that L is a unimodular lattice of rank 16 by Theorem 3.4. If S = 0, then1 − c = 0, which gives n = 17. By ( d + 240 β ) − d = 0 and n = 17, we get d = 150 β . We hence complete the proof. (cid:3) Remark . Firstly, there exist 2-reflective lattices when 1 ≤ rank( L ) ≤
8. Whenrank( L ) = 11 ,
12, we do not know if there exists any 2-reflective lattice. Whenrank( L ) = 9 , ,
17, there exist 2-reflective lattices. They are constructed as follows:(a) L = E ⊕ A : Borch( E E , ⊗ ϑ E / ∆) is a 2-reflective modular form ofweight 195.(b) L = E ⊕ A : Borch( E , ⊗ E , ⊗ ϑ E / ∆) is a 2-reflective modular formof weight 138.(c) L = 2 E ⊕ A : Borch( E , ⊗ ϑ E ⊗ ϑ E / ∆) is a 2-reflective modular formof weight 75.Here the Borcherds products are constructed from weakly holomorphic Jacobi formsof weight 0 (see [15] or [21]). The function ϑ E denotes the theta series for the rootlattice E which is a holomorphic Jacobi form of weight 4 and index E and thefunction E , denotes the Jacobi-Eisenstein series of weight 4 and index 1 introducedin [11]. The function ∆ denotes the holomorphic cusp form of weight 12 on SL ( Z ).We now consider the general case, this means that M does not contain twohyperbolic planes. Theorem 3.8.
Let M be a -reflective lattice of signature (2 , n ) with n ≥ . Then n = 19 or M is isomorphic to the unique even unimodular lattice of signature (2 , or (2 , .Proof. The proof is similar to the proof of [26, Proposition 3.1]. By [26] and [27,Lemma 1.7], we know If M has a 2-reflective modular form, then any even overlattice M ′ of M has a2-reflective modular form too. One can choose an even overlattice M ′ of M such that M ′ contains 2 U .We thus complete the proof by Theorem 3.6. (cid:3) Application: moduli space of K3 surfaces.
As a first application, weconsider the family of lattices(3.10) T n = U ⊕ U ⊕ E ( − ⊕ E ( − ⊕ h− n i EFLECTIVE MODULAR FORMS: A JACOBI FORMS APPROACH 11 where n ∈ N . The modular variety e O + ( T n ) \D ( T n ) is the moduli space of polarizedK3 surfaces of degree 2 n . The subset(3.11) Discr = [ l ∈ T n ( l,l )= − l ⊥ ∩ D ( T n )is the discriminant of this moduli space. Nikulin [29] asked the question whetherthe discriminant is equal to the set of zeros of certain automorphic form. Thisquestion is equivalent to whether T n is 2-reflective. Nikulin showed that for any N there exists n > N such that T n is not 2-reflective. Gritsenko and Nikulin [23]proved that the lattice T n is not 2-reflective if n > (cid:0) + √
128 + 8 N (cid:1) , where N isthe integer such that any even integer larger than N can be represented as the sumof the squares of eight different positive integers. Finally, Looijenga [25] provedthat T n is not 2-reflective if n ≥
2. As a direct consequence of Theorem 3.2 andTheorem 3.6, we present a pretty simple proof of the result.
Theorem 3.9.
The lattice T n is -reflective if and only if n = 1 .Proof. By case (c) of Remark 3.7, T is 2-reflective. By contradiction, we assumethat T n is 2-reflective with n ≥
2. On the one hand, by Theorem 3.2, the weight ofthe corresponding 2-reflective modular form is k = β (cid:20)
12 + 480 (cid:18) − (cid:19)(cid:21) = β (cid:18)
110 + 1417 (cid:19) > β . On the other hand, Theorem 3.6 tells that k = 75 β , which leads to a contradiction.Hence T n is not 2-reflective when n ≥ (cid:3) Non-existence of reflective modular forms
In this section we introduce reflective modular forms and use similar argumentsto show the non-existence of reflective modular forms of large rank.Let M be an even lattice of signature (2 , n ), n ≥
3. The level of M is the smallestpositive integer N such that N ( x, x ) ∈ Z for all x ∈ M ∨ . We remark that anyeven lattice of squarefree level is of even rank. A primitive vector l ∈ M of negativenorm is called reflective if the reflection(4.1) σ l ( x ) = x − l, x )( l, l ) l, x ∈ M is in O + ( M ). Definition 4.1.
A non-constant holomorphic modular form on D ( M ) is calledreflective if the support of its divisor is set-theoretically contained in the union ofquadratic divisors l ⊥ ∩ D ( M ) determined by reflective vectors l of M . A lattice M is called reflective if it admits a reflective modular form.A primitive vector l ∈ M with ( l, l ) = − d is reflective if and only if div( l ) = 2 d or d . Let us fix λ = [ l/ div( l )] ∈ A M . Then l ⊥ ∩D ( M ) is contained in H ( λ, − / (4 d ))in the first case, and is contained in H ( λ, − /d ) − X ν = λ H ( ν, − / (4 d ))in the second case. Note that all ( − M = 2 U ⊕ L ( −
1) be an even lattice of prime level p and F be a reflectivemodular form of weight k with respect to e O + ( M ). By [34, Section 6], the divisorof F can be represented as(4.2) div( F ) = β H + X γ ∈ π M,p β γ H ( γ, − /p ) , where π M,p ⊂ A M is the subset of elements of norm − /p . By [6], there exists anearly holomorphic modular form with principal part β q − e + X γ ∈ π M,p β γ q − /p e γ . Then there exists a weakly holomorphic Jacobi form of weight 0 with singularFourier coefficients(4.3) sing ( ψ L ) = β X r ∈ L q ( r,r ) / − ζ r + X γ ∈ π M,p β γ X s ∈ L + γ q ( s,s ) / − /p ζ s . Then the q -term of ψ L can be written as ψ L ( τ, z ) = β q − + β X r ∈ R ( L ) ζ r + 2 k + X γ ∈ π M,p β γ X s ∈ C γ ( L ) ζ s + O ( q ) , where(4.4) C γ ( L ) = { s ∈ L ∨ : ( s, s ) = 2 /p, s − γ ∈ L } . Thus, we get a formula related to the weight of the above reflective modular form k = β (cid:20)
12 + | R ( L ) | (cid:18) L ) − (cid:19)(cid:21) + (cid:18) p · rank( L ) − (cid:19) X γ ∈ π M,p β γ | C γ ( L ) | . (4.5)It is possible to find a similar formula for the weight of reflective modular forms forgeneral lattices. Remark . Let M = 2 U ⊕ L ( −
1) be an even lattice of prime level p and F bea reflective modular form of weight k for M . From (4.5), when rank( L ) = 12 and p = 2, then k = β (12 + | R ( L ) | ) so the function F is not of singular weight. Whenrank( L ) = 8 and p = 3, k = β (12 + | R ( L ) | ) and F is not of singular weight.By [26, Proposition 3.2], when a reflective modular form F exists, we have thateither rank( L ) ≤
23 or L is a unimodular lattice of rank 24. We next give a finerclassification of reflective modular forms on lattices of prime level. Theorem 4.3.
Let M = 2 U ⊕ L ( − be an even lattice of prime level p . If M admits a reflective modular form of weight k for e O + ( M ) , then we have (1) when p = 2 , either rank( L ) ≤ or rank( L ) = 20 and k = 24 β . EFLECTIVE MODULAR FORMS: A JACOBI FORMS APPROACH 13 (2) when p = 3 , either rank( L ) ≤ or rank( L ) = 18 and k = 48 β . (3) when p ≥ , rank( L ) ≤ / ( p + 1) .Proof. Similar to the proof of Theorem 3.6, there exists a weakly holomorphicJacobi form φ of weight 0 and index L with singular Fourier coefficients of the form(4.3). Assume that rank( L ) ≤
23. We write φ = S + d + S + · · · , where S and S are the first and second terms in (4.3), respectively, and d = 2 k . We can construct( n = rank( L ) and a = 24 /p ) f = 24 n − H ( φ ) = S + d + c S + · · · ∈ J w.h. ,L ,f = 24 n − H ( f ) = S + d + c S + · · · ∈ J w.h. ,L ,f = 24 n − H ( f ) = S + d + c S + · · · ∈ J w.h. ,L , where d = n ( d − β ) n − , c = n − an − ,d = ( n − d − β ) n − , c = c n − a − n − ,d = ( n − d − β ) n − , c = c n − a − n − . We can check that the function ϕ = ( c − c ) E φ + ( c − E f + (1 − c ) f = u + O ( q ) ∈ J ,L has no term of the form S or S and then it has no singular Fourier coefficient, soit is a holomorphic Jacobi form of weight 6, where u = ( d − β )( c − c ) + ( d + 240 β )( c −
1) + d (1 − c ) . We also construct a weakly holomorphic Jacobi form of weight 4 g = E φ − f = ( d + 240 β ) − d + (1 − c ) S + · · · ∈ J w.h. ,L . By Theorem 3.4, we have S = 0 when n >
8. By direct calculations, we get(4.6) c = 1 ⇐⇒ rank( L ) = 14 + 12 p . Therefore, when p = 2, c = 1 if and only if n = 20, when p = 3, c = 1 if andonly if n = 18, when p > c − = 0. We thus obtain • when p = 2, if g = 0 then n = 20 and d = 48 β ; • when p = 3, if g = 0 then n = 18 and d = 96 β ; • when p ≥ g = 0.Suppose g = 0 and n >
8. Then c = 1, otherwise g will be a holomorphic Jacobiform of weight 4, which contradicts the singular weight. The weakly holomorphicJacobi form g corresponds to a nearly holomorphic vector-valued modular form F = X γ ∈ A M F γ e γ of weight 4 − n /
2. Note that F ( τ ) has no term q n with negative n and for anynonzero γ ∈ A M , the possible term q n with negative n of F γ is q − /p . We knowfrom [33, Proposition 5.3] that F = 0 because the function P σ ∈ O( A M ) σ · F is invariant under the orthogonal group O( A M ) of the discriminant group A M and itis not zero. In addition, F is a nearly holomorphic modular form of weight 4 − n / ( p ) and its expansion at the cusp 0 is a linear combination of F γ .As in the proof of [34, Proposition 6.1], the Riemann-Roch theorem applied to F gives − ≤ pν ( F ) + ν ∞ ( F ) ≤ (cid:16) − n (cid:17) p + 112 . This implies n ≤ p + 1 . It remains to prove that M is not reflective if n = 14 and p = 3. But in this case, u = 0 and then ϕ = 0, which gives a contradiction. The proof is completed. (cid:3) Note that when rank( L ) = 16 and p = 2, we have u ≡
0. Therefore, our argumentcannot determine the weight of the corresponding reflective modular form.
Remark . (1) The rank of an even positive-definite lattice of level 2 is known to be divisibleby 4 (see [35]). Thus, by Theorem 4.3, if 2 U ⊕ L ( −
1) is a reflective lattice oflevel 2, then rank( L ) can only be 4, 8, 12, 16 or 20. We note that there existreflective lattices of such ranks. When L = D , 2 D , E ⊕ D , E ⊕ D or 2 E ⊕ D , the lattice 2 U ⊕ L ( −
1) admits a reflective modular form.(2) If L is an even positive-definite lattice of level 3 and of rank n with deter-minant det( L ) = | L ∨ /L | = 3 r , then either r ∈ { , n } and 8 | n , or 0 < r < n and 2 r ≡ n mod 4. Theorem 4.3 says that 2 U ⊕ L ( −
1) is a reflective latticeof level 3 only if rank( L ) = 2 , , , , ,
12 or 18. Reflective lattices of suchranks exist. When L = A , 2 A , 3 A , 4 A , E ⊕ A , E ⊕ A or 2 E ⊕ A ,the lattice 2 U ⊕ L ( −
1) admits a reflective modular form.By the above theorem and the weight formula (4.5), it is easy to prove thefollowing criterions.
Corollary 4.5.
Suppose that the even lattice M = 2 U ⊕ L ( − of prime level p isreflective. (1) When rank( L ) = 20 and p = 2 , we have | R ( L ) | ≥ . (2) When rank( L ) = 18 and p = 3 , we have | R ( L ) | ≥ . The above result can be used to decide whether a given lattice is reflective or not.For instance, we see at once that 2 U ⊕ L ( −
1) is not reflective when L = E (2) ⊕ D or 2 E ⊕ A .We next extend the above classification results to the general case. The followinglemma introduced in [27, Corollary 3.2] is very useful for our purpose. Lemma 4.6.
Let M be a lattice of signature (2 , n ) with n ≥ . There exists alattice M on M ⊗ Q such that O + ( M ) ⊂ O + ( M ) and that M is a scaling of aneven lattice containing U . We remark that the above lattice M is usually not an overlattice of M and it isconstructed as a sublattice of an overlattice of M . We next use the above lemmato extend Ma’s result [26, Proposition 3.2] to the general case. Theorem 4.7.
EFLECTIVE MODULAR FORMS: A JACOBI FORMS APPROACH 15 (1)
There is no reflective lattice of signature (2 , n ) with n > . (2) Let M be an even lattice of signature (2 , . If it admits a reflective mod-ular form which can be constructed as a Borcherds product, then it is iso-morphic to the unique even unimodular lattice of signature (2 , .Proof. We first prove the statement (1). By contradiction, assume that there is areflective lattice M of signature (2 , n ) with n >
26 and we denote the correspondingreflective modular form by F . By Lemma 4.6, there exists a lattice M on M ⊗ Q such that O + ( M ) ⊂ O + ( M ) and that M is a scaling of an even lattice M containing 2 U . Here, we have a natural isomorphism D ( M ) ∼ = D ( M ) ∼ = D ( M ) , where the first comes from the equality M ⊗ Q = M ⊗ Q and the second fromthe identification M = M as Z -modules. Moreover, the inclusion O + ( M ) ⊂ O + ( M ) ∼ = O + ( M ) is compatible with this isomorphism and the isomorphismpreserves the reflective divisors. Thus, F is a reflective modular form for M andthen also a reflective modular form for M , which contradicts [26, Proposition 3.2].We next prove the statement (2). The proof is similar to [26, Proposition 3.2].Assume that the corresponding reflective modular form F is a Borcherds productof a nearly holomorphic modular form f . Then ∆ f is a holomorphic modular formof weight 0 and hence must be an Mp ( Z )-invariant vector in C [ A M ]. Then weget M = II , because ∆ f does not transform correctly under the matrix S when | A M | 6 = 1. This completes the proof. (cid:3) Remark . We do not know if there is any other reflective lattice of signature(2 ,
26) except the scalings of II , . By the second statement in Theorem 4.7 and[5], such reflective lattice is not of the form U ⊕ U ( m ) ⊕ L ( − Theorem 4.9.
Let M be a reflective lattice of signature (2 , n ) and of prime level p . (1) If p = 2 and n > and n = 22 , then M is isomorphic to II , (2) . (2) If p = 3 and n > and n = 20 , then M is isomorphic to II , (3) or II , (3) . (3) If p > and n >
10 + 24 / ( p + 1) , then M is isomorphic to II , ( p ) or II , ( p ) .Proof. Let F be a reflective modular form for M . In this case, we have n ≥ M is p a , where 1 ≤ a ≤ n + 2 is an integer. Since M is of prime level p , the discriminant group of M is isomorphic to ( Z /p Z ) a andthe minimum number of generators of this group is l ( M ) = a .If n > a + 2, by [28, Corollary 1.10.2], there exists an even lattice L of sinature(0 , n −
2) having discriminant form M ∨ /M because n − > l ( M ). Then the lattice2 U ⊕ L is an even lattice of signatute (2 , n ) with the same discriminant form as M .From [28, Corollary 1.13.3], it follows that M and 2 U ⊕ L are isomorphic. Thus,the function F is also a reflective modular form for 2 U ⊕ L . We then prove thistheorem by Theorem 4.3. If n ≤ a + 2, then a ≥ n ≥
11. Since M is of prime level p , the lattice M ∨ ( p ), which is a scaling of the dual lattice of M , is even and of determinant p n − a . Moreover, if M ∨ ( p ) is not unimodular, then it is of level p . In view of thenatural isomorphism O + ( M ) ∼ = O + ( M ∨ ) ∼ = O + ( M ∨ ( p )) , the function F is a reflective modular form for M ∨ ( p ). Since n > (2 + n − a ) +2, we can prove this case as the previous case. If M ∨ ( p ) is unimodular, then M = ( M ∨ ( p )) ∨ ( p ) is a scaling of an even unimodular lattice. Thus, the proof iscompleted. (cid:3) Remark . If a reflective modular form can be constructed as a Borcherds prod-uct of a vector-valued modular form invariant under O( A M ), it is called symmetric,otherwise it is called non-symmetric. [34, Theorem 6.5] gives bounds on the signa-ture for non-symmetric reflective modular forms. But the bounds do not hold inthe symmetric case. However, the above result gives bounds in the symmetric case. Theorem 4.11.
There is no reflective lattice of signature (2 , n ) with ≤ n ≤ .Proof. Let M be an even lattice of signature (2 , n ) with 23 ≤ n ≤
25. Firstly,we assume that M contains 2 U . By contradiction, assume that M is reflective.Then, there exists a weakly holomorphic Jacobi form φ of weight 0 and we canconstruct a weakly holomorphic Jacobi form f of weight 4 from φ as in the proofof Theorem 4.3. We define g = E φ − f . Then g has no singular Fourier coefficientof hyperbolic norm −
2. From (4.6), we get g = 0. Moreover, the minimum possiblehyperbolic norm of the Fourier coefficients of g is −
1. Then η g is a holomorphicJacobi form of weight 10 with character, which leads to a contradiction due to thesingular weight. The general case can be proved as the proof of Theorem 4.7. (cid:3) Remark . When 1 ≤ rank( L ) ≤
20 and rank( L ) = 15 or 19, there exist reflectivelattices 2 U ⊕ L ( − A n for 1 ≤ n ≤ D , E ⊕ A , E ⊕ A , E ⊕ A ⊕ A , E ⊕ D , E ⊕ D ⊕ A , E ⊕ D ⊕ A , E ⊕ D , 2 E ⊕ A , 2 E ⊕ A , 2 E ⊕ D .But we do not know if there exists reflective lattice 2 U ⊕ L ( −
1) with rank( L ) = 15or 19. Questions . Here, we would like to formulate some interesting questions relatedto our work:(1) Are there 2-reflective lattices of signature (2 ,
13) or (2 , ,
17) or (2 , M = 2 U ⊕ L ( −
1) be a 2-reflective lattice of signature (2 , L equalto 2 E ⊕ A up to isomorphism?(4) Classify the following interesting lattices: • , • reflective lattices of signature (2 ,
22) and of level 2, • reflective lattices of signature (2 ,
20) and of level 3.5.
Another application: dd-modular forms
Our arguments in the previous section are also applicable to some other ques-tions. In this section we use similar arguments to classify the modular forms withthe simplest reflective divisors, i.e. the dd-modular forms defined in [8].
EFLECTIVE MODULAR FORMS: A JACOBI FORMS APPROACH 17
Let nA denote the lattice of n copies of A = h i , n ∈ N . Let { e , ..., e n } denote the standard basis of R n with standard scalar product ( · , · ). We choose thefollowing model for the lattice nA ( m ):(5.1) ( h e , ..., e n i Z , m ( · , · ))and set z n = P ni =1 z i e i ∈ nA ⊗ C , ζ i = e πiz i , for 1 ≤ i ≤ n . We define(5.2) Γ n,m = O + (2 U ⊕ nA ( − m ))and the definition of dd-modular forms is as follows. Definition 5.1.
A holomorphic modular form with respect to Γ n,m is called a dd-modular form if it vanishes exactly along the Γ n,m -orbit of the diagonal { z n = 0 } .The Γ n,m -orbit of the diagonal { z n = 0 } , denoted by Γ n,m { z n = 0 } , is called thediagonal divisor.It is well known that the Igusa form ∆ which is the product of the ten eventheta constants vanishes precisely along the diagonal divisor { z = 0 } . Therefore,the dd-modular form is a natural generalization of ∆ . Gritsenko and Hulek [16]proved that the dd-modular form exists for the lattice A ( m ) if and only if 1 ≤ m ≤
4. Cl´ery and Gritsenko [8] developed the arguments in [16] and gave thefull classification of the dd-modular forms with respect to the Hecke subgroups ofthe Siegel paramodular groups. But their approach is hard to generalize to higherdimensions. Since dd-modular forms are crucial in determining the structure ofthe fixed space of modular forms and have applications in physics, as an importantapplication of our arguments, we prove the following classification results for alldd-modular forms for lattices of the shape nA . Theorem 5.2.
The dd-modular form exists if and only if the pair ( n, m ) takes oneof the eight values (1 , , (1 , , (1 , , (1 , , (2 , , (2 , , (3 , , (4 , . Proof.
Suppose that F c is a modular form of weight k with respect to Γ n,m with thedivisor c · Γ n,m { z n = 0 } , where c is the multiplicity of the diagonal divisor and it isa positive integer. The diagonal divisor Γ n,m { z n = 0 } is the union of the primitiveHeegner divisors P ( ± e i / (2 m ) , − / (4 m )), 1 ≤ i ≤ n , where the primitive Heegnerdivisor of discriminant ( µ, d ) is defined as P ( µ, d ) = [ M + µ ∋ l primitive ( l,l )=2 d l ⊥ ∩ D ( M ) . It is clear that we have P ( µ, y ) = H ( µ, y ) − X d>y x d H ( λ d , d ) , where x d are integers and λ d ∈ M ∨ (we refer to [7, Lemma 4.2] for an explicitformula). For arbitrary Heegner divisor H ( λ, d ) with λ = (0 , n , λ , n , ∈ A M ,the principal part of the corresponding nearly holomorphic modular form of weight − rank( L ) / ρ M of Mp ( Z ) is q d e λ . Hencethe singular Fourier coefficients of the corresponding weakly holomorphic Jacobiform of weight 0 are represented as X r ∈ L + λ q ( r,r ) / d e πi ( r, z ) . Since {± e i / (2 m ) : 1 ≤ i ≤ n } is the set of vectors in nA ( m ) ∨ with the minimumnorm 1 / (2 m ) in nA ( m ) ∨ /nA ( m ), through the previous explanations, there existsa weak Jacobi form f nA ,m of weight 0 and index nA ( m ) satisfying f nA ,m = c · X ≤ i ≤ n ζ ± i + 2 k + O ( q )such that F c is the Borcherds product of f nA ,m . We next apply Lemma 2.3 to ourcase. In the case, a = 0, rank( L ) = n . For each term cζ i or cζ − i , the corresponding l = ± m e i , c (0 , l )( l, l ) = c · m · m = c m . Therefore, we get m (2 nc + 2 k ) = 12 c, then nm ≤
5. It is not hard to show that a weak Jacobi form for nA ( m ) hasintegral Fourier coefficients if its q -term is integral when nm ≤ m ≤
4, 2 k is integral if c = 1. Hence the existence of F c is equivalent tothe existence of F . In view of k ≥ n/
2, then the triplet ( m, n, k ) can only take oneof the eight values(1 , , , (1 , , , (1 , , , (1 , , , (2 , , , (2 , , , (3 , , , (4 , ,
12 ) . When m = 5, we only need to consider the case of c = 5, and we obtain theunique solution (5 , , A is ψ (1)0 , = 5 ζ ± +2+ q ( − ζ + · · · ) (see [13, formula (1.12)]). The correspondingBorcherds product is not holomorphic, that is, F c does not exist in the case. Wehave thus proved the theorem. (cid:3) Remark . Similarly, we can define dd-modular forms with respect to the lattices A n ( m ) or D n ( m ). Using the same methodology, we can easily classify these dd-modular forms. In fact, dd-modular forms with respect to the lattices L ( m ), where L = A n , n ≥ L = D n , n ≥
4, exist if and only if the pair (
L, m ) takes one ofthe following fifteen values( A ,
1) ( A ,
1) ( A ,
1) ( A ,
1) ( A ,
1) ( A ,
1) ( A ,
2) ( A , A ,
2) ( D ,
1) ( D ,
1) ( D ,
1) ( D ,
1) ( D ,
1) ( D , . Note that all dd-modular forms in Theorem 5.2 and in the above list do exist andcan be found in [8, 14, 21, 24].
Acknowledgements
The author would like to thank his supervisor Valery Grit-senko for helpful discussions and constant encouragement. The author thanksShouhei Ma for explaining the proof of Lemma 4.6 to him. The author also thanksthe anonymous referees for their careful reading and useful suggestions. This workwas supported by the Labex CEMPI (ANR-11-LABX-0007-01) in the University ofLille.
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