aa r X i v : . [ m a t h . N T ] F e b ON WEAK JACOBI FORMS OF RANK TWO
HAOWU WANG AND BRANDON WILLIAMS
Abstract.
We study a ring of weak Jacobi forms indexed by integral lattices of rank two. We findan explicit finite set of generators of this ring and give a dimension formula for weak Jacobi formsof rank two lattice index. Introduction
In this paper we will give structure results for weak Jacobi forms indexed by integral lattices ofrank two. These are two-variable analogues of the weak Jacobi forms introduced by Eichler–Zagier[5] that occur in the Fourier-Jacobi expansions of modular forms on O(2 ,
4) [6], on U(2 ,
2) [11] andon Sp(6). Our object of study is the graded ring J := ∞ M a,b,c =0 M k ∈ Z J wk, (cid:16) a + b bb c + b (cid:17) , where J wk,M is the space of weak Jacobi forms of weight k and whose index is the lattice with Grammatrix M . (We do not require M to be positive-definite or have even diagonal.) The main resultsare: Theorem 1.1.
The graded ring J above is finitely generated. Every weak Jacobi form ϕ ( τ, z, w ) in the even subring J = J w ∗ , ( ∗ ∗ ∗ ∗ ) can be written as a polynomial in the rank one forms f ( τ, z ) , f ( τ, w ) , f ( τ, z + w ) , f ∈ { φ − , , φ , , φ − , } , where φ − , , φ , , φ − , are the generators of the ring of scalar-index weak Jacobi forms of Eichler–Zagier [5]. The full ring J is generated by rank-one weak Jacobi forms and by rank-two weakJacobi forms whose indices are lattices of discriminant , , . As an application, we will show that all graded rings of the form J L = L ∞ n =0 J w ∗ ,L ( n ) are finitelygenerated (see Corollary 6.7). Here L is an integral lattice of rank two and L ( n ) is that lattice withits bilinear form multiplied by n . Theorem 1.2.
Let L be an integral lattice of rank two and let ϑ be the theta function ϑ ( τ, z ) = q / ζ / ∞ X n = −∞ q n ( n +1) / ( − ζ ) n , q = e πiτ , ζ = e πiz . Date : February 23, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
Jacobi forms, lattices of rank two.
Every weak Jacobi form of index L can be expressed as a rational function in the forms ϑ ( k ) ( τ, λz ) , k ≤ , where λ ∈ L and where ϑ ( k ) is the k -th derivative with respect to z . Theorem 1.3.
Let L be an integral lattice of rank two and choose a Gram matrix of the form (cid:0) a + b bb c + b (cid:1) with a, b, c ∈ N . Let P a , Q a denote the Laurent polynomials P a ( t ) = t − a + 1 − t − a − t − = t − a + t − a + ... + 1 , a ≥ and Q a ( t ) = 1 − t − a − t − = t − a + t − a + ... + 1 , a ≥ , and define P ( t ) = 1 , P ( t ) = t − and Q − ( t ) = Q ( t ) = 0 . Then the C [ E , E ] -module J w ∗ ,L of weakJacobi forms of index L is free on generators ϕ , ..., ϕ n of weights k , ..., k n that are determined by t k + ... + t k n = P a ( t ) P b ( t ) P c ( t ) + Q a ( t ) Q b ( t ) Q c ( t ) + (2 t − − Q a − ( t ) Q b − ( t ) Q c − ( t ) − t − (cid:16) Q a ( t ) Q b − ( t ) Q c − ( t ) + Q a − ( t ) Q b ( t ) Q c − ( t ) + Q a − ( t ) Q b − ( t ) Q c ( t ) (cid:17) + ( t − a − b − c : abc = 0;0 : abc = 0 . This project was motivated partly by physical applications. Weak Jacobi forms of rank twohave applications to 6 D superconformal field theories (cf. [2, 3, 4, 9, 10]) where they occur aselliptic genera and partition functions of strings. In [10] Babak, Lockhart and Vafa conjectured afusion rule from n pairs of E-strings to n heterotic strings, which yields a new approach to explicitexpressions for the elliptic genus of n E-strings, and is equivalent to certain non-trivial identitiesamong weak Jacobi forms associated to lattices of type E ( n ) ⊕ M n where M n is an integral latticeof rank 2. When n = 1 ,
2, the lattice M n is diagonal so Jacobi forms of index M n reduce to thescalar-index Jacobi forms of Eichler–Zagier [5], and in these cases the conjecture was proved in [10].In general, the lattice M n is not diagonal (for instance, M is Z with Gram matrix ( )), andthere is no general structure theorem for weak Jacobi forms of rank two in the literature. The case n = 3 was proved in [2] by working in a diagonal sublattice. This argument also appears in [9].Passing to a diagonal sublattice creates many redundant generators which causes the computationto be quite difficult; we hope the structure results proved here will simplify such arguments.The layout of this paper is as follows. In § L ⊕ M of two integral latticesis generated by tensor products of the generators of weak Jacobi forms of indices L and M (seeTheorem 2.4). In § § § §
6. Theorem 1.2 follows from Theorem 6.6 and our construction ofthe generators. In § N WEAK JACOBI FORMS OF RANK TWO 3 Preliminaries
Let L be an integral lattice with positive semidefinite bilinear form h− , −i : L ⊗ L → Z andquadratic form Q : L → Z , Q ( x ) = h x, x i / . For every N ∈ N we let L ( N ) denote the lattice L with quadratic form N · Q .A weakly holomorphic Jacobi form of weight k ∈ Z and index L is a holomorphic function f : H × ( L ⊗ C ) −→ C that satisfies f (cid:16) aτ + bcτ + d , z cτ + d (cid:17) = ( cτ + d ) k exp (cid:16) πiccτ + d Q ( z ) (cid:17) f ( τ, z )and f ( τ, z + λτ + µ ) = ( − Q ( λ )+ Q ( µ ) exp (cid:16) − πiτ Q ( λ ) − πi h λ, z i (cid:17) f ( τ, z )for all (cid:0) a bc d (cid:1) ∈ SL ( Z ) and λ, µ ∈ L , and whose Fourier expansion as a function of τ , f ( τ, z ) = X n ≫−∞ c ( n, z ) q n , q = e πiτ involves only finitely many negative powers of q . We call f a weak Jacobi form if its Fourier seriesis supported only on nonnegative exponents: f ( τ, z ) = ∞ X n =0 c ( n, z ) q n , and a holomorphic Jacobi form if every function f ( τ, λτ + µ ) with λ, µ ∈ N L , N ∈ N is a holomorphicmodular form (of level Γ( N )).The spaces of weakly holomorphic Jacobi forms, weak Jacobi forms and holomorphic Jacobiforms of weight k and index L are labelled J ! k,L , J wk,L , J k,L respectively.This definition extends the standard notion of Jacobi forms of lattice index by allowing oddlattices (i.e. vectors may have half-integral norm) and by allowing degenerate lattices. Neither ofthese enlarges the class of Jacobi forms significantly: if f ( τ, z ) has index L , then f ( τ, z ) has evenindex L (4); and if L is a degenerate lattice with kernelker( L ) = { y ∈ L : h x, y i = 0 for all x ∈ L } then we have an identification π ∗ : J w ∗ ,L/ ker( L ) ∼ −→ J w ∗ ,L , f π ∗ f ( τ, z ) := f ( τ, π ( z )) , where π : L ⊗ C → ( L/ ker( L )) ⊗ C is the quotient map.When rank( L ) ≤
1, (weak) Jacobi forms of matrix index M = ( m ) are the same as (weak) Jacobiforms of index m/ Jacobi forms of index f ( τ, z ) that are constant in z and modular formsof level SL ( Z ) in τ . The fundamental example of a weak Jacobi form of half-integral index is theform (see [7]) φ − , / ( τ, z ) = ϑ ( τ, z ) η ( τ ) HAOWU WANG AND BRANDON WILLIAMS of weight − /
2, where ϑ is the Jacobi theta function as defined in Theorem 1.2 and η ( τ ) = q / Q ∞ n =1 (1 − q n ) is the Dedekind eta function, such that η ( τ ) = ϑ ′ ( τ,
0) by Jacobi’sidentity. More generally:
Proposition 2.1.
The graded ring of weak Jacobi forms of rank one and half-integer index isgenerated by the Eisenstein series E , E and by the weak Jacobi forms φ − , / ( τ, z ) = ϑ ( τ, z ) η ( τ ) = ( − ζ − / + ζ / ) + ( ζ − / − ζ − / + 3 ζ / − ζ / ) q + O ( q ) ,φ , ( τ, z ) = ϑ ( τ, z ) η ( τ ) · π ℘ ( τ, z ) = ( ζ − + 10 + ζ ) + (10 ζ − − ζ − + 108 − ζ + 10 ζ ) q + O ( q ) ,φ , / ( τ, z ) = ϑ ( τ, z ) ϑ ( τ, z ) = ( ζ − / + ζ / ) + ( − ζ − / + ζ − / + ζ / − ζ / ) q + O ( q ) . There is a decomposition J w ∗ , ∗ / = C [ φ − , / , φ , ] ⊕ φ , / C [ φ − , / , φ , ] . Here ℘ ( τ, z ) = 1 z + X ω ∈ ( Z + τ Z ) \{ } (cid:16) z + ω ) − ω (cid:17) is the Weierstrass elliptic function. The subring of integer-index weak Jacobi forms is then generatedby the forms φ − , := φ − , / , φ , , and φ − , := φ − , / · φ , / , cf. [5]. Conversely, Proposition 2.1 can be easily derived from the structure theorem of [5].It will be important at many points that the theta function ϑ ( τ, z ) (and therefore also φ − , / ) forfixed τ has simple zeros in the lattice points z ∈ Z + τ Z and nowhere else. This follows immediatelyfrom the Jacobi triple product ϑ ( τ, z ) = q / ζ / ∞ Y n =1 (1 − q n )(1 − q n ζ )(1 − q n − ζ − )or it can be proved more directly.For any fixed index L , the Jacobi forms J ∗ ,L and weak Jacobi forms J w ∗ ,L of index L and all weightscan be viewed as modules over the ring C [ E , E ] of modular forms. Abstractly the structure ofthese modules is well-known, but we include a proof for convenience: Proposition 2.2. J ∗ ,L and J w ∗ ,L are free C [ E , E ] -modules. If L is positive-definite then bothmodules have rank det( L ) . More generally, for any congruence subgroup Γ ≤ SL ( Z ), the M ∗ (Γ)-modules J ∗ ,L and J w ∗ ,L arefree on the same det( L ) generators. Proof.
To see that J ∗ ,L and J w ∗ ,L are free, one can adapt the proof of Theorem 8.4 of [5] (which,as remarked there, applies to a wide class of modules over M ∗ (Γ); the only essential ingredientsare that J k,L and J wk,L are always finite-dimensional, and are zero for sufficiently low k ). Since J ∗ ,M ⊆ J w ∗ ,M and ∆ r J w ∗ ,M ⊆ J ∗ ,M for all large enough r ∈ N , or more precisely whenever r ≥ max γ ∈ L ′ /L min x ∈ γ + L Q ( x ) , N WEAK JACOBI FORMS OF RANK TWO 5 the modules J ∗ ,M and J w ∗ ,M have the same rank.When L is an even index we can identify J ∗ ,L with the module of vector-valued modular formsfor the Weil representation attached to L . Then the Riemann-Roch theorem (through the formulaof section 4 of [1]) implies that, for weights k > L ) / J k +12 ,L = dim J k,L + ( dim span( e γ + e − γ : γ ∈ L ′ /L ) : k ≡ e γ − e − γ : γ ∈ L ′ /L ) : k ≡ e γ ∈ C [ L ′ /L ] is the group ring vector attached to γ ∈ L ′ /L . This implies that the even-weightand odd-weight submodules are free of rankrank J ∗ ,L = rank J w ∗ ,L = dim span( e γ + e − γ )and rank J ∗ +1 ,L = rank J w ∗ +1 ,L = dim span( e γ − e − γ ) , and we immediately obtain the full rankrank J ∗ ,L = rank J ∗ ,L + rank J ∗ +1 ,L = det( L ) . This rank also has the following interpretation. The q -term c (0 , z ) of a weak Jacobi form ofweight k and even index L is a Laurent polynomial c (0 , z ) = X r ∈ L ′ c (0 , r ) ζ r , ζ r = e πi h r, z i where c (0 , r ) may be nonzero only if r has minimal norm among all vectors in its coset r + L ,and where c (0 , r ) = ( − k c (0 , − r ). That J w ∗ ,L has rank det( L ) is equivalent to the fact that everyLaurent polynomial satisfying these two conditions actually occurs as the q -term of a weak Jacobiform (of some weight); in other words, if V is the space spanned by the Fourier expansions of weakJacobi forms (of all weights), then the map V / ∆ V ∼ −→ span (cid:16) ζ r : r ∈ L ′ of minimal norm in its coset (cid:17) sending P n,r c ( n, r ) q n ζ r to its q -term is an isomorphism of vector spaces.Now if L is an odd lattice, we obtainrank J ∗ ,L = rank J w ∗ ,L = det( L )by identifying the Fourier expansions of forms { f ( τ, z ) : f ∈ J w ∗ ,L } with the subspace of V / (∆ V )of q -terms of Jacobi forms of index L (4) satisfying c (0 , r ) = 0 unless r/ ∈ L (4) ′ ;this is seen to be a det( L )-dimensional space, and the claim follows. (cid:3) Corollary 2.3.
For every integral lattice L , there is a polynomial P L and a Laurent polynomial P wL such that ∞ X k =0 dim J k,L t k = P L ( t )(1 − t )(1 − t ) , ∞ X k = −∞ dim J wk,L t k = P wL ( t )(1 − t )(1 − t ) . They have the form P L ( t ) = t k + ... + t k n , P wL ( t ) = t ℓ + ... + t ℓ n , where k , ..., k n are the weights of generators of the C [ E , E ] -module J ∗ ,L and where ℓ , ..., ℓ n arethe weights of generators of the C [ E , E ] -module J w ∗ ,L . HAOWU WANG AND BRANDON WILLIAMS
The following description of weakly holomorphic and weak Jacobi forms associated to a directsum of lattices has no analogue for holomorphic Jacobi forms.
Theorem 2.4.
For any integral lattices
L, M there are isomorphisms of C [ E , E ] -modules J ! ∗ ,L ⊗ J ! ∗ ,M ∼ −→ J ! ∗ ,L ⊕ M ,J w ∗ ,L ⊗ J w ∗ ,M ∼ −→ J w ∗ ,L ⊕ M , given by the direct product( f, g ) ( f ⊗ g )( τ, z L , z M ) := f ( τ, z L ) g ( τ, z M ) , where z L ∈ L ⊗ C , z M ∈ M ⊗ C , and ( z L , z M ) ∈ ( L ⊕ M ) ⊗ C . In terms of the Laurent polynomials P w this implies P wL ⊕ M = P wL · P wM . Proof.
We can assume without loss of generality that L and M are positive-definite, because J ! ∗ ,L , J w ∗ ,L can otherwise be identified with J ! ∗ ,L/ ker( L ) , J w ∗ ,L/ ker( L ) and because( L ⊕ M ) / ker( L ⊕ M ) = ( L ⊕ M ) / (ker( L ) ⊕ ker( M )) ∼ = ( L/ ker( L )) ⊕ ( M/ ker( M )) . Similarly, we can assume without loss of generality that L and M are even lattices; otherwise,substitute z z .Suppose first that M = (2 m ) has rank 1. To any weakly holomorphic Jacobi form h ∈ J ! k,L ⊕ M weassociate as in [5] a sequence of weakly holomorphic Jacobi forms h , h , ... of index L and weights k, k + 1 , ... as the Taylor coefficients of the function (see [7, Proposition 1.5])˜ h ( τ, z L , z ) := e mz G ( τ ) h ( τ, z L , z ) = ∞ X n =0 h n ( τ, z L ) z n about 0. Here G ( τ ) = π (cid:16) − ∞ X n =1 σ ( n ) q n (cid:17) is the non-normalized Eisenstein series of weight two.We split the space J ! k,L ⊕ M into its even and odd subspaces (with respect to M ): J ! , even k,L ⊕ M = { h ∈ J ! k,L ⊕ M : h ( τ, z L , − z ) = h ( τ, z L , z ) } ,J ! , odd k,L ⊕ M = { h ∈ J ! k,L ⊕ M : h ( τ, z L , − z ) = − h ( τ, z L , z ) } . Then the map J ! , even k,L ⊕ M ∼ −→ m M n =0 J ! k +2 n,L , h ( h , h , ..., h m )is injective, because for fixed τ and z L the function h ( τ, z L , z ) has exactly 2 m zeros in any fun-damental parallelogram for C / ( Z + Z τ ), and it is surjective as any sequence ( h , ..., h m ) arises(essentially) as the sequence of coefficients of the form h ( τ, z L , z ) = h ( τ, z L ) φ m , ( τ, z ) + h ( τ, z L ) φ m − , ( τ, z ) φ − , ( τ, z ) + ... + h m ( τ, z L ) φ m − , ( τ, z ) . Under this map the subspace of weak Jacobi forms J w, even k,L ⊕ M is identified with L mn =0 J wk +2 n,L . N WEAK JACOBI FORMS OF RANK TWO 7
It follows that J ! , odd k,L ⊕ M ∼ −→ m − M n =1 J ! k +2 n − ,L and J w, odd k,L ⊕ M ∼ −→ m − M n =1 J wk +2 n − ,L ,h ( h , h , ..., h m − )are isomorphisms: we can apply the result for even forms to the quotient h ( τ, z L , z ) /φ − , ( τ, z ).This remains a weakly-holomorphic Jacobi form, and it is a weak Jacobi form if h was, because any h ∈ J w, odd k,L ⊕ M has forced zeros at the 2-torsion points z = 0 , / , τ / , ( τ + 1) / φ − , ( τ, z ) hasits simple zeros. In particular, every weak Jacobi form of index L ⊕ M can be written as a linearcombination of products of weak Jacobi forms h i ( τ, z L ) of index L by weak Jacobi forms f ( τ, z ) ofrank one.In general let ˜ M be a diagonal sublattice of M . Repeatedly applying the rank one argumentabove yields the isomorphism ⊗ : J ! ∗ ,L ⊗ J ! ∗ , ˜ M ∼ −→ J ! ∗ ,L ⊕ ˜ M . Let R denote the averaging map R : J ! ∗ , ˜ M −→ J ! ∗ ,M , h X ζ ∈H ( M ) / H ( ˜ M ) h (cid:12)(cid:12)(cid:12) ζ, which extends naturally to the map J ! ∗ ,L ⊕ ˜ M → J ! ∗ ,L ⊕ M by acting only on the variable z M , where H ( M ) and H ( ˜ M ) are respectively the integral Heisenberg groups of M and ˜ M (see [6]). Then weobtain J ! ∗ ,L ⊕ M = R (cid:16) J ! ∗ ,L ⊕ ˜ M (cid:17) = R (cid:16) J ! ∗ ,L ⊗ J ! ∗ , ˜ M (cid:17) = J ! ∗ ,L ⊗ J ! ∗ ,M . This averaging argument requires some care for weak Jacobi forms, since the image of a weakJacobi form under R is not necessarily a weak Jacobi form. That J w ∗ ,L ⊕ M is indeed identified with J w ∗ ,L ⊗ J w ∗ ,M under this map follows from the following lemma. (cid:3) Lemma 2.5.
Suppose f , ..., f r ∈ J w ∗ ,L are a C [ E , E ] -basis of weak Jacobi forms of index L , andsuppose g , ..., g r ∈ J ! ∗ ,M are weakly holomorphic Jacobi forms for which h := f ⊗ g + ... + f r ⊗ g r is a weak Jacobi form. Then each of g , ..., g r is a weak Jacobi form.Proof. Let d be the minimal exponent for which some g i has a nonzero coefficient of q d , and suppose d <
0. Write f i = ∞ X n =0 a ( f i , n ) q n , g i = ∞ X n = d b ( g i , n ) q n , where a ( f i , n ) and b ( g i , n ) are Laurent polynomials in independent variables z L and z M , such that h ( τ, z ) = (cid:16) r X i =1 a ( f i , z L ) b ( g i , d )( z M ) (cid:17)| {z } =0 q d + O ( q d +1 ) . Since f , ..., f r are a basis of J w ∗ ,L , their q -terms are C -linearly independent. In particular, b ( g i , d )are Laurent polynomials which are zero at every value of z M and therefore identically zero, contra-dicting the definition of d . (cid:3) HAOWU WANG AND BRANDON WILLIAMS Weak Jacobi forms of matrix index
It will be more convenient to fix the underlying group L = Z and consider Jacobi forms indexedby a varying Gram matrix on L . Throughout the rest of this paper, an index matrix is a matrix ofthe form M = (cid:0) a + b bb c + b (cid:1) ∈ Z × , with a, b, c ∈ N . We do not require M to have full rank. Weak Jacobi forms of index M are then written as functions f ( τ, z, w ) , τ ∈ H , z, w ∈ C . We will study the space of all weak Jacobi forms of all weights and all index matrices: J := M k ∈ Z ∞ M a,b,c =0 J wk, (cid:16) a + b bb c + b (cid:17) . It is not hard to see that every rank two lattice (
L, Q ) is isometric to Z with quadratic form z T M z/ M . The anharmonic group G = h (cid:0) − (cid:1) , ( ) i ⊆ PGL ( Z )acts on the set of index matrices by conjugation. In particular, the ring J is preserved under thefollowing changes of coordinates in the elliptic variables:( z, w ) ( z + w, − w ) , ( w, − z − w ) , ( w, z ) . More precisely, if f ( τ, z, w ) has index (cid:0) a + b bb c + b (cid:1) , then f ( τ, z + w, − w ) has index (cid:0) b + a aa c + a (cid:1) ; and f ( τ, w, − z − w ) has index (cid:0) b + c cc a + c (cid:1) ; and f ( τ, w, z ) has index (cid:0) c + b bb a + b (cid:1) . In other words, if M iswritten in the form (cid:0) a + b bb c + b (cid:1) then G acts as the permutations of the indices ( a, b, c ). If the indicesare such that b ≤ a ≤ c and we do not allow multiples of the identity then the quadratic form( a + b ) x + 2 bxy + ( c + b ) y is the reduced form in its SL ( Z )-equivalence class; however it is notconvenient to make these restrictions.If f ( τ, z ) is any (weak) Jacobi form of index m ∈ N , then we obtain (weak) Jacobi forms ofpositive semidefinite matrix index by substituting values in the elliptic variable in f . In particular,the form F ( τ, z, w ) := f ( τ, z ) has index ( m
00 0 ); F ( τ, z, w ) := f ( τ, w ) has index ( m ); and F ( τ, z, w ) := f ( τ, z + w ) has index ( m m m m ). Taking products of these forms yields Jacobi formsof (true) rank two. When f = φ − , / the forms constructed this way are called theta blocks (see[8]). Note that the forms F , F , F are permuted under the anharmonic group.4. Construction of generators
There are various differential operators that can be used to produce Jacobi forms. We mentiontwo here:(i) Suppose f ( τ, z ) is a (weak) Jacobi form of weight k and index m ∈ N . Then(2 πi ) − f ′ ( τ, z ) f ( τ, z ) + 2 m Im( z )Im( τ )transforms like a Jacobi form of weight 1 and index 0 (i.e. an elliptic function). Here, f ′ denotesdifferentiation with respect to the z variable. In particular, an expression of the form n X i =1 ϑ ′ ( τ, h λ i , z i ) ϑ ( τ, h λ i , z i ) , λ i ∈ L N WEAK JACOBI FORMS OF RANK TWO 9 is an abelian function (or meromorphic Jacobi form of index zero) if and only if P i λ i = 0 . (ii) Suppose f ( τ, z ) is a (weak) Jacobi form of weight k and positive-definite index M . Then the Serre derivative S f ( τ, z ) := H f ( τ, z ) − π (cid:16) k − rank( L )24 (cid:17) G ( τ ) f ( τ, z )is a (weak) Jacobi form of weight k + 2 and the same index M . Here H is the heat operator(2 πi ) − ∂ τ − (2 πi ) − ∆ L /
2, i.e. on Fourier series it acts as H (cid:16) X r ∈ Z n ∞ X n =0 c ( n, r ) q n ζ r (cid:17) = X r ∈ Z n ∞ X n =0 c ( n, r )( n − r T M − r/ q n ζ r . When M has even diagonal, S is obtained by applying the Serre derivative componentwise on thecorresponding vector-valued modular form (cf. [5]). Note that S ϑ = 0. Using this relation onecan express the Serre derivative of a polynomial in the forms ϑ ( τ, λz ) , ..., ϑ ( n ) ( τ, λz ), λ ∈ L as apolynomial in the forms ϑ ( τ, λz ) , ..., ϑ ( n +2) ( τ, λz ).We can now construct a number of examples of Jacobi forms of rank two which are not simplyproducts of Jacobi forms of rank one. The forms we construct here will be useful in our inductionarguments later due to their special values at w = 0 and w = − z . For now we leave theseconstructions somewhat unmotivated. Let q = e πiτ , ζ = e πiz and ω = e πiw . Lemma 4.1. (a corollary of the A -case of Wirthm¨uller’s theorem [12]) The graded C [ E , E ] -module of weak Jacobi forms of matrix index ( ) is free on the basis Φ − , ( )( τ, z, w ) = ϑ ( τ, z ) ϑ ( τ, w ) ϑ ( τ, z + w ) η ( τ )= − ζ − ω − + ζ − + ω − − ζ − ω + ζω + O ( q ) , Φ − , ( )( τ, z, w ) = (cid:16) ϑ ′ ( τ, z ) ϑ ( τ, z ) + ϑ ′ ( τ, w ) ϑ ( τ, w ) − ϑ ′ ( τ, z + w ) ϑ ( τ, z + w ) (cid:17) Φ − , ( )= ζ − ω − + ζ − + ω − − ζ + ω + ζω + O ( q ) , Φ , ( )( τ, z, w ) = − · S (cid:16) Φ − , ( ) (cid:17) = ζ − ω − + ζ − + ω − + 18 + ζ + ω + ζω + O ( q ) . Lemma 4.2.
The weak Jacobi form Φ , ( ) = 52 S (cid:16) ϑ ( τ, z ) ϑ ( τ, w ) φ , ( τ, z + w ) η ( τ ) (cid:17) = − ζ − / ω − / + 12 ζ − / ω − / + 12 ζ − / ω − / + 112 ζ − / ω − / + 112 ζ / ω / + 12 ζ / ω / + 12 ζ / ω / − ζ / ω / + O ( q ) of weight and index ( ) satisfies Φ , ( )( τ, z, − z ) = φ , ( τ, z ) and Φ , ( )( τ, z,
0) = Φ , ( )( τ, , z ) = 6 · φ , / ( τ, z ) . Lemma 4.3.
The weak Jacobi form Φ , ( )( τ, z, w ) = 292 E ( τ ) φ − , / ( τ, z ) φ − , / ( τ, w )Φ − , ( )( τ, z, w )+ 48 · S (cid:16) φ − , / ( τ, z ) φ − , / ( τ, w )Φ , ( )( τ, z, w ) (cid:17) = − ζ − / ω − / + 12 ζ − / ω / + 1032 ζ − / ω − / + 20 ζ − / ω / + 12 ζ − / ω / + 12 ζ / ω − / + 20 ζ / ω − / + 1032 ζ / ω / + 12 ζ / ω − / − ζ / ω / + O ( q ) of weight and index ( ) satisfies Φ , ( )( τ, z, − z ) = φ , ( τ, z ) and Φ , ( )( τ, z,
0) = Φ , ( )( τ, , z ) = 72 φ , / ( τ, z ) . Weak Jacobi forms of even weight
Lemma 5.1.
Let M = (cid:0) a bb c (cid:1) be an index matrix. Suppose either (i) b is even, or (ii) b is odd and b < min( a, c ) .Let k ∈ Z be a weight with k ≡ a + c mod . Then the pullback map P : J wk,M −→ J wk, ( a + c ) / − b , F ( τ, z, w ) F ( τ, z, − z ) is surjective. The condition b < min( a, c ) when b is odd is necessary to rule out the cases ( a, b, c ) = (1 , , , , , , Proof.
First note that the map P satisfies P ( φ , ( τ, z + w ) · F ) = 12 P ( F ) , since φ , ( τ, z + w ) is mapped to φ , ( τ,
0) = 12. In particular, if we have proved that P is surjectivefor some index (cid:0) a bb c (cid:1) then it is also surjective for the index (cid:0) a +2 b +2 b +2 c +2 (cid:1) . Therefore we may assumethat b = 0 or b = 1 .Suppose b = 0. Let ψ ∈ J wk, ( a + c ) / be any Jacobi form of index ( a + c ) /
2. By our assumption onthe weight, we can write ψ as a polynomial ψ = Q ( E , E , φ − , / , φ , ) . In each monomial in Q , we replace any c copies of φ − , / ( τ, z ) and c copies of φ , ( τ, z ) by − φ − , / ( τ, w ) and φ , ( τ, w ) for some indices c , c with 2 c + c = c , to obtain a Jacobi form ˆ Q of index ( a c ) with P ( ˆ Q ) = ψ .Finally suppose b = 1, and by assumption a, c ≥
2. Let ψ ∈ J wk, ( a + c ) / − be any Jacobi form.Since ψ has index at least 1, we can write it in the form ψ = φ − , P + φ , Q, P, Q ∈ C [ E , E , φ − , / , φ , ] . N WEAK JACOBI FORMS OF RANK TWO 11
Let ˆ
P , ˆ Q be any Jacobi forms of index (cid:0) a − c − (cid:1) with P ( ˆ P ) = P and P ( ˆ Q ) = Q as in the previousparagraph. Then the form F = Φ − , ( ) ˆ P + Φ , ( ) ˆ Q satisfies P ( F ) = ψ . (cid:3) This lemma quickly leads to a simple set of generators when both the weight and the off-diagonalindex b is even: Theorem 5.2.
The graded ring J := J w ∗ , ( ∗ ∗ ∗ ∗ ) = M a,b,c ∈ N b ≤ min( a,c ) M k ∈ Z J wk, (cid:16) a b b c (cid:17) is generated by the Eisenstein series E , E , the rank one Jacobi forms φ − , ( τ, z ) , φ − , ( τ, w ) , φ − , ( τ, z + w ) ,φ , ( τ, z ) , φ , ( τ, w ) , φ , ( τ, z + w ) , and the products φ − , ( τ, z ) φ − , ( τ, w ) , φ − , ( τ, z ) φ − , ( τ, z + w ) , φ − , ( τ, w ) φ − , ( τ, z + w ) . Proof.
Let ϕ ( τ, z, w ) be a weak Jacobi form of even weight k and index (cid:0) a b b c (cid:1) . We will show byinduction on b that ϕ can be written as a polynomial in the claimed generators. When b = 0 thisfollows from the classification of diagonal-index forms in Theorem 2.4. If b = min( a, c ) then one of ϕ ( τ, z + w, − w ) and ϕ ( τ, w, − z − w ) is a Jacobi form of index with off-diagonal b = 0. Since thischange-of-variables leaves the ring generated by the Jacobi forms in this claim invariant, we mayalso assume that b < min( a, c ).In general, using Lemma 5.1 we can find a weak Jacobi form F of index (cid:16) a − b ) 00 2( c − b ) (cid:17) with P ( F ) = P ( ϕ ). Then the form ϕ ( τ, z, w ) − ( φ , ( τ, z + w ) / b F ( τ, z, w ) ∈ J wk, (cid:16) a b b c (cid:17) vanishes at every point ( z, w ) with z + w ∈ Z + τ Z and is therefore divisible by ϑ ( τ, z + w ), so ϕ ( τ, z, w ) := (cid:16) ϕ ( τ, z, w ) − ( φ , ( τ, z + w ) / b F ( τ, z, w ) (cid:17) /φ − , / ( τ, z + w ) ∈ J wk +1 , (cid:16) a − b − b − c − (cid:17) is again a weak Jacobi form.The image P ( ϕ ) = ϕ ( τ, z, − z ) ∈ J wk +1 ,a + c − b is a weak Jacobi form of odd weight and integral index and is therefore a multiple of φ − , . Since b < min( a, c ), we may use Lemma 5.1 to find a weak Jacobi form G of index (cid:0) a − b b c − (cid:1) with P ( ϕ ) /φ − , = P ( G ) . Then P (cid:16) ϕ ( τ, z, w ) − φ − , / ( τ, z − w ) G ( τ, z, w ) (cid:17) = 0 , so we can again divide by ϑ ( τ, z + w ) and obtain a weak Jacobi form ϕ := (cid:16) ϕ ( τ, z, w ) − φ − , / ( τ, z − w ) G ( τ, z, w ) (cid:17) /φ − , / ( τ, z + w ) ∈ J wk +2 , (cid:16) a − b − b − c − (cid:17) . By induction, ϕ is a polynomial as in the theorem. The claim follows by writing ϕ = ( φ , ( τ, z + w ) / b F ( τ, z, w )+ φ − , / ( τ, z + w ) φ − , / ( τ, z − w ) G ( τ, z, w )+ φ − , ( τ, z + w ) ϕ ( τ, z, w )and using the identity φ − , / ( τ, z + w ) φ − , / ( τ, z − w ) = 112 (cid:16) φ − , ( τ, z ) φ , ( τ, w ) − φ , ( τ, z ) φ − , ( τ, w ) (cid:17) , which is easily proved using Fourier series since both sides of this equation lie in the two-dimensionalspace of weak Jacobi forms of weight − ) which is spanned by φ − , ( τ, z ) φ , ( τ, w )and φ , ( τ, z ) φ − , ( τ, w ). (cid:3) The larger ring of weak Jacobi forms for even-weight, even index matrices where the off-diagonalindex may be odd is already obtained by including the weak Jacobi forms of A -index: Theorem 5.3.
As a J -module, J w ∗ , (cid:16) ∗ ∗ +12 ∗ +1 2 ∗ (cid:17) = M a,b,c ∈ N b< min( a,c ) M k ∈ Z J wk, (cid:16) a b +12 b +1 2 c (cid:17) is spanned by the forms Φ − , ( ) and Φ , ( ) .Proof. Let ϕ ∈ J w k, (cid:16) a b +12 b +1 2 c (cid:17) be a weak Jacobi form. Write P ϕ ( τ, z ) = ϕ ( τ, z, − z ) ∈ J w k,a + c − b − in the form P ϕ = φ − , u + φ , v, where u, v ∈ C [ φ − , , φ , ] . (Note that the index a + c − b − F and G of index (cid:0) a − b b c − (cid:1) with P ( F ) = u and P ( G ) = v . Moreover, P (cid:16) C · Φ − , ( ) (cid:17) = φ − , and P (cid:16) C · Φ , ( ) (cid:17) = φ , for some nonzero constants C , C . Then ϕ := (cid:16) ϕ − C Φ − , ( ) F − C Φ , ( ) G (cid:17) /φ − , / ( τ, z + w )is a weak Jacobi form of odd weight 2 k + 1 and index (cid:0) a − b b c − (cid:1) . We will now show by inductionon b that ϕ must have a representation of the form f Φ − , ( ) + g Φ , ( ) with f, g ∈ J .(i) Suppose b = 0. Then the form ϕ constructed above is a weak Jacobi form of diagonal index.By Theorem 2.4, J w k − , (cid:16) a +1 00 2 c +1 (cid:17) = φ − , / ( τ, z ) φ , / ( τ, w ) J w k, (cid:16) a
00 2 c − (cid:17) + φ , / ( τ, z ) φ − , / ( τ, w ) J w k, (cid:16) a − c (cid:17) ⊆ J · φ − , / ( τ, z ) φ , / ( τ, w ) + J · φ , / ( τ, z ) φ − , / ( τ, w ) . Using the linear relations φ − , / ( τ, z ) φ , / ( τ, w ) φ − , / ( τ, z + w ) ∈ Span (cid:16) Φ − , ( )( τ, z, w ) φ , ( τ, w ) , Φ , ( ) φ − , ( τ, w ) (cid:17) and φ , / ( τ, z ) φ − , / ( τ, w ) φ − , / ( τ, z + w ) ∈ Span (cid:16) Φ − , ( )( τ, z, w ) φ , ( τ, z ) , Φ , ( ) φ − , ( τ, z ) (cid:17) N WEAK JACOBI FORMS OF RANK TWO 13 (which can be proved by showing that the space of weight − ) and( ) are two-dimensional and comparing Fourier coefficients) we obtain the desired representationfor ϕ .(ii) Suppose 0 < b < min( a, c ) −
1. In this case, the pullback P ( ϕ ) of the above ϕ is again aweak Jacobi form of odd weight and integral index and therefore a multiple of φ − , . By Lemma 5.1and its proof, we can find a weak Jacobi form G ∈ Φ − , ( ) J + Φ , ( ) J of index (cid:0) a − b +12 b +1 2 c − (cid:1) such that P ( G ) = P ( ϕ ) /φ − , . (Note that the condition of Lemma 5.1 is satisfied because b < min( a, c ) − ϕ := (cid:16) ϕ ( τ, z, w ) − φ − , / ( τ, z − w ) G ( τ, z, w ) (cid:17) /φ − , / ( τ, z + w ) ∈ J w k +2 , (cid:16) a − b − b − c − (cid:17) , and the claim follows by induction using the identity for φ − , / ( τ, z + w ) φ − , / ( τ, z − w ) in theproof of Theorem 5.2.(iii) Suppose b = min( a, c ) −
1; by swapping z and w if necessary, we may assume b = a − ϕ has the form (cid:0) a a − a − c (cid:1) . Then ϕ ( τ, z + w, − w ) has index (cid:16) a
11 2( c − a +1) (cid:17) andwe obtain an expression for it using (i). The claim follows because the weak Jacobi forms of index( ) are invariant under ( z, w ) ( z + w, − w ). (cid:3) The full ring of weak Jacobi forms
Lemma 6.1.
Let M = (cid:0) a bb c (cid:1) be an even index matrix. Suppose either that k is even, or k is oddand b < c . Then the map Q : J wk,M −→ J wk,a , ϕ ( τ, z, w ) ϕ ( τ, z, is surjective. The condition on b in odd weights cannot be removed in general; for example there is no weight − ) that could map to φ − , . Proof. (i) Suppose b = 0. Then any weak Jacobi form f ∈ J wk,a arises as the image of the Jacobiform f ( τ, z )( φ , ( τ, w ) / c of index ( a
00 2 c ) . (ii) Suppose b = min(2 a, c ). If b = 2 a then we can pass from the Jacobi form ϕ to the Jacobiform ϕ ( τ, z + w, − w ) of index (cid:16) a
00 2( c − a ) (cid:17) , preserving the value at w = 0, and apply (i). If b = 2 c and by assumption k is even, then J wk,a is spanned by monomials in φ , and φ − , , and these ariseas images of monomials in φ , ( τ, z + w ) , φ − , ( τ, z + w ) , φ , ( τ, z ) , φ − , ( τ, z ) . (iii) Suppose 0 < b < min(2 a, c ). If k is even then J wk,a = φ , · J wk,a − + φ − , · J wk +2 ,a − , and by an induction on b it is enough to find Jacobi forms of index ( ) whose images are (anonzero multiple of) φ , and φ − , . We have seen that Φ , ( ) and Φ − , ( ) have this property.If k is odd then J wk,a = φ − , · J wk +1 ,a −
24 HAOWU WANG AND BRANDON WILLIAMS and we need to find a Jacobi form of index ( ) whose image is a nonzero multiple of φ − , . ButΦ , ( )( τ, z + w, − w ) φ − , / ( τ, z ) has this property. (cid:3) We will combine the following technical lemmas to obtain a system of generators for the algebraof weak Jacobi forms of arbitrary rank and index.
Lemma 6.2.
Let M = (cid:0) a bb c (cid:1) be an even index matrix with a ≥ c > b > and M = ( ) . (i) If b ≤ a − , then for any k ∈ Z , J w k − , (cid:16) a bb c (cid:17) = Φ , ( )( τ, z + w, − w ) φ − , / ( τ, z ) · J w k, (cid:16) a − b − b − c − (cid:17) + φ − , / ( τ, w ) · J w k, (cid:16) a bb c − (cid:17) . (ii) If b = 2 a − (and necessarily a = c ) then J w k − , (cid:16) a a − a − a (cid:17) = φ − , / ( τ, z + w ) φ , / ( τ, w ) · J w k, (cid:16) a − a − a − a − (cid:17) + φ − , / ( τ, z + w ) · J w k, (cid:16) a − a − a − a − (cid:17) . (iii) If b = 2 a − then J w k − , (cid:16) a a − a − a (cid:17) = φ − , / ( τ, z + w ) · J w k, (cid:16) a − a − a − a − (cid:17) . Proof.
Let ϕ be a weak Jacobi form of index (cid:0) a bb c (cid:1) and odd weight 2 k + 1.(i) Suppose first that 2 a ≥ b + 3. Then ϕ ( τ, z,
0) is a weak Jacobi form of index a and odd weightand therefore a multiple of φ − , ( τ, z ). By assumption, 2 a ≥
4, so using Lemma 6.1 we can find aweak Jacobi form F of weight 2 k + 2 and index (cid:0) a − b − b − c − (cid:1) with F ( τ, z,
0) = ϕ ( τ, z, /φ − , ( τ, z ) . Then ˜ ϕ ( τ, z, w ) := ϕ ( τ, z, w ) − C · Φ , ( )( τ, z + w, − w ) φ − , / ( τ, z ) F ( τ, z, w )(for an appropriate constant C ) satisfies ˜ ϕ ( τ, z,
0) = 0 , so G := ˜ ϕ ( τ, z, w ) /φ − , / ( τ, w ) is a weak Jacobi form and ϕ = Φ , ( )( τ, z + w, − w ) φ − , / ( τ, z ) · ( C · F ) + φ − , / ( τ, w ) · G. (ii) Suppose b = 2 a −
2. Then P ( ϕ ) = ϕ ( τ, z, − z ) is a weak Jacobi form of odd weight and index2, so it has the form f ( τ ) φ − , ( τ, z ) with f ∈ C [ E , E ]. Since b >
0, we have a ≥
2. Thus the form F ( τ, z, w ) := f ( τ )( φ , ( τ, z + w ) / a − φ − , / ( τ, z + w ) φ , / ( τ, w )and satisfies P ( F ) = P ( ϕ ), so F − ϕ is a multiple of φ − , / ( τ, z + w ).(iii) Suppose b = 2 a −
1. In this case P ( ϕ ) is a weak Jacobi form of odd weight and index 1, soit is identically zero. Therefore ϕ is divisible by φ − , / ( τ, z + w ). (cid:3) Lemma 6.3.
Let (cid:0) a +1 bb c (cid:1) be an index matrix with < b < min(2 c, a + 1) . (i) For any k ∈ Z , J w k − , (cid:16) a +1 bb c (cid:17) = φ − , / ( τ, z ) · J w k, (cid:16) a bb c (cid:17) + φ − , / ( τ, w ) · J w k, (cid:16) a +1 bb c − (cid:17) . (ii) Suppose b ≤ a − . For any k ∈ Z , J w k, (cid:16) a +1 bb c (cid:17) = φ − , / ( τ, w ) · J w k +1 , (cid:16) a +1 bb c − (cid:17) + φ , / ( τ, z ) · J w k, (cid:16) a − bb c (cid:17) . N WEAK JACOBI FORMS OF RANK TWO 15 (iii)
Suppose b = 2 a − . For any k ∈ Z , J w k, (cid:16) a +1 2 a − a − c (cid:17) = φ − , / ( τ, w ) φ − , / ( τ, z + w ) · J w k +2 , (cid:16) a a − a − c − (cid:17) + φ − , / ( τ, z ) · J w k +1 , (cid:16) a a − a − c (cid:17) + Φ , ( )( τ, z + w, − w ) · J w k, (cid:16) a − a − a − c − (cid:17) . (iv) Suppose b = 2 a . For any k ∈ Z , J w k, (cid:16) a +1 2 a a c (cid:17) = φ − , / ( τ, w ) φ − , / ( τ, z + w ) · J w k +2 , (cid:16) a a − a − c − (cid:17) + φ − , / ( τ, z ) · J w k +1 , ( a a a c )+ Φ , ( )( τ, z + w, − w ) · J w k, (cid:16) a − a − a − c − (cid:17) . Proof.
Let ϕ ( τ, z, w ) be a weak Jacobi form of index (cid:0) a +1 bb c (cid:1) .(i) Suppose ϕ has odd weight. Then ϕ ( τ, z,
0) is a weak Jacobi form of odd weight and half-integral index, and therefore a multiple of φ − , / . By Lemma 6.1 we can find a weak Jacobi form F of index (cid:0) a bb c (cid:1) with F ( τ, z,
0) = ϕ ( τ, z, /φ − , / ( τ, z ). Then G := ( ϕ − F · φ − , / ( τ, z )) /φ − , / ( τ, w )is a weak Jacobi form of index (cid:0) a +1 bb c − (cid:1) , i.e. ϕ = φ − , / ( τ, z ) · F + φ − , / ( τ, w ) · G. (ii) Since ϕ ( τ, z,
0) has even weight and half-integral index, it is a multiple of φ , / . By Lemma5.1 we can find a weak Jacobi form F of index (cid:0) a − bb c (cid:1) with F ( τ, z,
0) = ϕ ( τ, z, /φ , / ( τ, z ) . Then G := ( ϕ − F · φ , / ( τ, z )) /φ − , / ( τ, w )is a weak Jacobi form, i.e. ϕ = φ , / ( τ, z ) · F + φ − , / ( τ, w ) · G. (iii) The form ϕ ( τ, , w ) has even weight and integer index c , so we can write ϕ ( τ, , w ) = φ , ( τ, w ) f ( τ, w ) + φ − , ( τ, w ) g ( τ, w )where f, g ∈ C [ E , E , φ − , , φ , ]. Let F and G be weak Jacobi forms of index (cid:0) a − a − a − c − (cid:1) and (cid:0) a a − a − c − (cid:1) , respectively, such that F ( τ, , w ) = f ( τ, w ) and G ( τ, , w ) = g ( τ, w ) . (The existence follows from Lemma 6.1 after swapping the roles of z and w .) Then the form˜ ϕ ( τ, z, w ) := ϕ ( τ, z, w ) − Φ , ( )( τ, z + w, − w ) F ( τ, z, w ) − φ − , / ( τ, w ) φ − , / ( τ, z + w ) G ( τ, z, w )vanishes at z = 0, so ˜ ϕ ( τ, z, w ) /φ − , / ( τ, z ) is a weak Jacobi form of index (cid:0) a a − a − c (cid:1) , and theclaim follows.(iv) As in (iii) we consider the form ϕ ( τ, , w ) of index c . Here c > a ≥
1, so we can write ϕ ( τ, , w ) = φ , ( τ, w ) f ( τ, w ) + φ − , ( τ, w ) g ( τ, w ) with f, g ∈ C [ E , E , φ − , , φ , ]. Using Lemma 6.1 we can find weak Jacobi forms F and G ofindex (cid:0) a − a − a − c − (cid:1) and (cid:0) a a − a − c − (cid:1) , respectively, such that F ( τ, , w ) = f ( τ, w ) and G ( τ, , w ) = g ( τ, w ) . Then˜ ϕ ( τ, z, w ) := ϕ ( τ, z, w ) − Φ , ( )( τ, z + w, − w ) F ( τ, z, w ) − φ − , / ( τ, w ) φ − , / ( τ, z + w ) G ( τ, z, w )vanishes at z = 0, so ˜ ϕ ( τ, z, w ) /φ − , / ( τ, z ) is a weak Jacobi form of index ( a a a c ). (cid:3) Lemma 6.4.
Let M = (cid:0) a +1 bb c +1 (cid:1) be an index matrix with < b < c + 1 ≤ a + 1 and let k ∈ Z . (i) If b ≤ a − , then J w k − ,M = φ , / ( τ, z ) φ − , / ( τ, w ) · J w k, (cid:16) a − bb c (cid:17) + φ − , / ( τ, z + w ) · J w k, (cid:16) a b − b − c (cid:17) . (ii) If b = 2 a − (so c = a ), then J w k − ,M = Φ , ( )( τ, z + w, − w ) φ − , / ( τ, w ) · J w k, (cid:16) a − a − a − a − (cid:17) + φ − , / ( τ, z + w ) · J w k, (cid:16) a a − a − a (cid:17) . (iii) If b = 2 a (so c = a ), then J w k − ,M = φ − , / ( τ, z + w ) · J w k, (cid:16) a a − a − a (cid:17) . Proof.
Let ϕ ∈ J w k − , (cid:16) a +1 bb c +1 (cid:17) . The pullback P ϕ ( τ, z ) = ϕ ( τ, z, − z )is a weak Jacobi form of odd weight 2 k − a − b + c + 1 and therefore a multipleof φ − , .(i) In this case, we can apply Lemma 5.1 to find a weak Jacobi form F of weight 2 k and index (cid:0) a − bb c (cid:1) with F ( τ, z, − z ) = ϕ ( τ, z, − z ) /φ − , ( τ, z ) . Then ˜ ϕ ( τ, z, w ) := ϕ ( τ, z, w ) − φ , / ( τ, z ) φ − , / ( τ, w ) F ( τ, z, w )vanishes whenever z + w ∈ Z + τ Z , so it is a multiple of φ − , / ( τ, z + w ).(ii) We argue as in (i) but instead take F to have index (cid:0) a − a − a − a − (cid:1) with F ( τ, z, − z ) = ϕ ( τ, z, − z ) /φ − , ( τ, z ) . (This lies in C [ E , E ] as ϕ ( τ, z, − z ) has index 2, so we can take F ∈ C [ E , E ] · φ , ( τ, z + w ) a − .)Then we consider the form˜ ϕ = ϕ − Φ , ( )( τ, z + w, − w ) φ − , / ( τ, w ) · F which vanishes whenever z + w ∈ Z + τ Z so it is a multiple of φ − , / ( τ, z + w ).(iii) In this case P ϕ is a weak Jacobi form of odd weight and index 1 and therefore identicallyzero; so ϕ is already a multiple of φ − , / ( τ, z + w ). (cid:3) Lemma 6.5.
Let < b < min(2 a + 1 , c + 1) . (i) If b ≥ , then J w k, (cid:16) a +1 bb c +1 (cid:17) = φ − , / ( τ, w ) · J w k +1 , (cid:16) a +1 bb c (cid:17) + Φ , ( ) · J w k, (cid:16) a − b − b − c − (cid:17) . N WEAK JACOBI FORMS OF RANK TWO 17 (ii)
When b = 1 , J w k, (cid:16) a +1 11 2 c +1 (cid:17) = φ − , / ( τ, z ) φ − , / ( τ, w ) · J w k +2 , ( a
11 2 c )+ φ − , / ( τ, z + w ) · J w k +1 , ( a
00 2 c )+ Φ , ( ) · J w k, (cid:16) a − c − (cid:17) . Proof. (i) Let ϕ be a weak Jacobi form of even weight 2 k and index (cid:0) a +1 bb c +1 (cid:1) . Then ϕ ( τ, z, a + 1 / φ , / . By Lemma 6.1 we can find a weak Jacobi form F of index (cid:0) a − b − b − c − (cid:1) with F ( τ, z,
0) = ϕ ( τ, z, /φ , / ( τ, z ) . Let C be the nonzero constant with C · Φ , ( )( τ, z,
0) = φ , / ( τ, z ); then ϕ := (cid:16) ϕ − C · Φ , ( ) · F (cid:17) /φ − , / ( τ, w )is a weak Jacobi form of weight 2 k − (cid:0) a +1 bb c (cid:1) , and the claim follows.(ii) The pullback P ϕ ( τ, z ) = ϕ ( τ, z, − z ) ∈ J w k,a + c is an even-weight Jacobi form of index at least two, so we can write P ϕ = φ , f + φ − , g with f, g ∈ C [ E , E , φ − , , φ , ] . By Lemma 5.1 there exist Jacobi forms
F, G of index (cid:0) a − c − (cid:1) and ( a
11 2 c ), respectively, such that P F = f and P G = g. Then the form˜ ϕ ( τ, z, w ) := ϕ ( τ, z, w ) + φ − , / ( τ, z ) φ − , / ( τ, w ) G ( τ, z, w ) − Φ , ( )( τ, z, w ) F ( τ, z, w )vanishes at all points z + w ∈ Z + τ Z , so˜ ϕ ( τ, z, w ) /φ − , / ( τ, z + w )is a weak Jacobi form, and the claim follows. (cid:3) Theorem 6.6.
The graded algebra J := M a,b,c ∈ N b ≤ min( a,c ) M k ∈ Z J wk, (cid:16) a bb c (cid:17) is generated by the Eisenstein series E , E , the rank-one weak Jacobi forms f ( τ, z ) , f ( τ, w ) , f ( τ, z + w ) , f ∈ { φ − , / , φ , , φ , / } , the A -index forms Φ − , ( ) , Φ , ( ) , and by the odd lattice-index forms Φ , ( )( τ, z, w ) , Φ , ( )( τ, z + w, − w ) , Φ , ( )( τ, z + w, − z ) , Φ , ( )( τ, z, w ) , Φ , ( )( τ, z + w, − w ) , Φ , ( )( τ, z + w, − z ) . Proof.
Let ϕ ( τ, z, w ) be a weak Jacobi form of index (cid:0) a bb c (cid:1) . We will prove that ϕ can be expressedas a polynomial in these forms by induction on b . Without loss of generality, we may assume that a ≥ c . (Otherwise, we swap the elliptic variables z and w . This preserves the system of generatorsin the claim.(i) If b = 0, then ϕ has diagonal index and the claim follows from Theorem 2.4.(ii) Suppose 0 < b < min( a, c ). One of the lemmas above (depending on the parity of the weight k and the parity of a and c ) applies to this index and yields a decomposition of ϕ into weak Jacobiforms with lower off-diagonal index. Note that in Lemma 6.2, φ − , / ( τ, z + w ) φ , / ( τ, w ) is a C -linear combination of the three forms φ − , / ( τ, z )Φ , ( )( τ, z + w, − w ) , φ − , / ( τ, z + w )Φ , ( )( τ, z, w ) ,E ( τ )Φ − , ( )( τ, z, w )Φ − , ( )( τ, z, w ) . Therefore we do not need φ − , / ( τ, z + w ) φ , / ( τ, w ) as a generator of J . The claim follows byinduction.(iii) Suppose b = min( a, c ). By applying one of the changes of variables ( z, w ) ( z + w, − w ) or( z, w ) ( w, − z − w ) we obtain a weak Jacobi form of off-diagonal index b = 0 and we can applycase (i). (cid:3) Corollary 6.7.
Let L ⊆ R be a positive-definite integral lattice. Then the ring J L := ∞ M n =0 J w ∗ ,L ( n ) is finitely-generated over C .Proof. Choose a Gram matrix for L of the form (cid:0) a + b bb c + b (cid:1) with a, b, c ∈ N . If 0 ∈ { a, b, c } thenthis reduces to a statement about weak Jacobi forms of rank one by Theorem 2.4, so assume all of a, b, c are positive.The ring J L is generated by all monomials in the generators of Theorem 6.6 that have index L ( n ) for some n ∈ N . Since f ∈ J ⊆ C [ E , E ][ h ( τ, z ) , h ( τ, w ) , h ( τ, z + w ) : h ∈ { φ − , / , φ , , φ , / } ]for every f ∈ J , we only need to consider monomials in which the rank-two generators Φ ∗ , ∗ ofTheorem 6.6 appear with exponent at most one. Suppose f is such a monomial of index L ( n ) with n ≥
24 and factor f in the form f = g · h where g is a product of distinct rank-two generators and h ∈ C [ h ( τ, z ) , h ( τ, w ) , h ( τ, z + w ) : h ∈ { φ − , / , φ , , φ , / } ]. Adding the possible indices showsthat g has index (cid:16) α + β ββ γ + β (cid:17) with α, β, γ ≤
11. Then h has index (cid:0) x + y yy z + y (cid:1) with x ≥ a − y ≥ b − z ≥ c −
11. In particular, h contains either φ , / ( τ, z ) to exponent at least 2 a , or φ , ( τ, z ) to exponent at least 3 a , or φ − , / ( τ, z ) to exponent at least 6 a ; otherwise its index wouldhave x ≤ (2 a − · a − · a − · a − < a − . N WEAK JACOBI FORMS OF RANK TWO 19
In particular, h splits off a factor of index ( a
00 0 ). By the same argument, h splits off factors ofindex (cid:0) b b b b (cid:1) and ( c ), and multiplying these together yields a factor of index L (6).In particular, we have shown that J L is generated by the finitely many monomials that haveindex L ( n ), n <
24. (Of course this bound is not sharp.) (cid:3)
A folklore conjecture states that the graded ring J w ∗ ,L, ∗ := L k ∈ Z ,n ∈ N J wk,L ( n ) is finitely generatedover C for any positive-definite integral lattice L . This is known when L has rank one by [5], andwhen it has rank two by the above Corollary. Theorem 2.4 implies that it is true for L ⊕ M ifit is true for L and M . For Weyl-invariant Jacobi forms for irreducible root lattices other than E , Wirthm¨uller’s theorem [12] provides a stronger result (the rings are polynomial algebras). Forgeneral lattices L little seems to be known.7. The weights of generators
The lemmas of the previous section also make it possible to compute the Hilbert seriesHilb J ( q, r, s, t ) := ∞ X a,b,c =0 X k ∈ Z dim J wk, (cid:16) a + b bb c + b (cid:17) q a r b s c t k in closed form, and therefore the C [ E , E ]-module structure of weak Jacobi forms of every ranktwo index. The ring J fits into an exact sequence of the form0 −→ K −→ J ϕ ϕ ( τ,z, −→ R −→ . Assume that the above ϕ has matrix index (cid:0) a bb c (cid:1) . The kernel consists exactly of multiples φ − , / ( τ, w ) f ( τ, z, w ), where f has index (cid:0) a bb c − (cid:1) . Here one has the minor issue that the inequality c − ≤ b is not satisfied if b = c , i.e. f does not belong to J . In any case, K is quite close to φ − , / ( τ, w ) · J , and due to the additivity of the Hilbert series in short exact sequences our imme-diate goal will be to determine the series Hilb R . Here R must be understood as the multigradedring whose piece of degree ( a, b, c, k ) is the subspace of J wk, ( a + c ) / − b spanned by ϕ ( τ, z,
0) for weakJacobi forms ϕ of weight k and index (cid:0) a bb c (cid:1) .We will compute the dimensions dim R a,b,c,k by induction using the following technical lemmas. Lemma 7.1.
Let k ∈ Z . The map Q : J w k, (cid:16) a bb c (cid:17) −→ J w k,a/ , ϕ ( τ, z, w ) ϕ ( τ, z, is surjective except for the cases ( b, c ) = (0 , and ( a, b, c ) = ( a, a, a + 1) , a ≥ and ( a, b, c ) = (3 , , where the image is trivial, and the cases ( b, c ) = (1 , , a ≥ , where the image is exactly φ − , · J w k +2 ,a/ − .Proof. If a = b then weak Jacobi forms are spanned by monomials f ( τ, z + w ) g ( τ, w ) with f ∈ J w ∗ ,a/ and g ∈ J w ∗ , ( c − a ) / , and setting w = 0 shows that this is surjective unless ( c − a ) = 1 (in whichcase the image is zero). Similarly, if b = c then weak Jacobi forms are spanned by monomials f ( τ, z ) g ( τ, z + w ) with f ∈ J w ∗ , ( a − c ) / , g ∈ J w ∗ ,c/ , so we can read off the image immediately in thiscase also. Suppose that 0 < b < min( a, c ). If a, c are even then this claim is contained in Lemma 6.1. Suppose a is odd and c is even. If0 < b ≤ a − Q ( J w k, (cid:16) a bb c (cid:17) ) ⊆ J w k,a/ = φ , / · J w k, ( a − / = Q (cid:16) φ , / ( τ, z ) · J w k, (cid:16) a − bb c (cid:17) (cid:17) ⊆ Q (cid:16) J w k, (cid:16) a bb c (cid:17) (cid:17) showing surjectivity. When b = a − c is even, setting w = 0 in Lemma 6.3 (iii) and applyingLemma 6.1 yields Q (cid:16) J w k, (cid:16) a a − a − c (cid:17) (cid:17) = φ − , / ( τ, z ) · Q (cid:16) J w k +1 , (cid:16) a − a − a − c (cid:17) (cid:17) + Φ , ( )( τ, z, · Q (cid:16) J w k, (cid:16) a − a − a − c − (cid:17) (cid:17) = φ − , / · J w k +1 , ( a − / + φ , / · J w k, ( a − / = J w k,a/ . When b = a − c ≥ Q (cid:16) J w k, (cid:16) a a − a − c (cid:17) (cid:17) = φ − , / ( τ, z ) · Q (cid:16) J w k +1 , (cid:16) a − a − a − c (cid:17) (cid:17) + Φ , ( )( τ, z, · Q (cid:16) J w k, (cid:16) a − a − a − c − (cid:17) (cid:17) = φ − , / · J w k +1 , ( a − / + φ , / · J w k, ( a − / = J w k,a/ . In the case c = 2, b = 2 we change variables to pass to the index matrix (cid:0) a − (cid:1) to obtainsurjectivity if and only if a ≥
5; and when c = 2 and b = 0 , a is even and c is odd (swapping the role of z and w is Lemma 6.3), and yields surjectivity also. Finally, when both a and c are odd, we useLemma 6.5 to see that Q is surjective except when c = 1, in which case the image is always trivialif b = 0 and consists of multiples of φ − , when b = 1. (cid:3) Lemma 7.2.
Let k ∈ Z . The map Q : J w k +1 , (cid:16) a bb c (cid:17) −→ J w k +1 ,a/ is surjective except when ( b, c ) = (0 , or ( a, b, c ) = (4 , , , in which cases the image is zero.Proof. When b = 0, using Theorem 2.4 we see that Q is surjective unless c = 1, in which caseall weak Jacobi forms are multiples of φ − , / ( τ, w ) and the image of Q is trivial. Similarly, when b = c we can conjugate ( a cc c ) to (cid:0) a − c c (cid:1) to see that Q is surjective when c ≥ c = 1; or when c = 2 and a = 4 (when a = 4 the image is trivial). When b = a we see that Q is surjective forall indices c by a similar argument. Therefore suppose 0 < b < min( a, c ). We will show that Q isalways surjective:(i) When a and c are both even, this is contained in Lemma 6.1.(ii) Suppose a is odd and c is even. By Lemma 5.1, J w k − ,a/ = φ − , / · J w k, ( a − / = Q (cid:16) φ − , / ( τ, z ) · J w k, (cid:16) a − bb c (cid:17) (cid:17) ⊆ Q (cid:16) J w k − , (cid:16) a bb c (cid:17) (cid:17) . (iii) Suppose a, c are both odd. Then J w k − ,a/ = φ − , / · J w k, ( a − / = Q (cid:16) φ − , / ( τ, z + w ) · J w k, (cid:16) a − b − b − c − (cid:17) (cid:17) ⊆ Q (cid:16) J w k − , (cid:16) a bb c (cid:17) (cid:17) by Lemma 6.1.(iv) Suppose a is even and c is odd. Then Lemma 6.3 (after swapping the variables z and w ) yields Q (cid:16) J w k − , (cid:16) a bb c (cid:17) (cid:17) = φ − , / · Q (cid:16) J w k, (cid:16) a − bb c (cid:17) (cid:17) , N WEAK JACOBI FORMS OF RANK TWO 21 which equals φ − , / · J w k, ( a − / = φ − , / φ , / · J w k,a/ − = J w k − ,a/ by Lemma 7.1. (cid:3) Theorem 7.3.
The Hilbert series of J has closed form Hilb J ( q, r, s, t ) = X a,b,c ∈ N k ∈ Z dim J wk, (cid:16) a + b bb c + b (cid:17) q a r b s c t k = F ( q, r, s, t )(1 − t )(1 − t ) , where F ( q, r, s, t ) = qrst − (1 − qt − )(1 − rt − )(1 − st − )+ (1 − q + q )(1 − r + r )(1 − s + s ) − qrst − ( qr + qs + rs − qrs ) + qrs (1 − qrs )(1 − q )(1 − qt − )(1 − r )(1 − rt − )(1 − s )(1 − st − ) Remark 7.4.
It is amusing to check that certain properties of the ring J are reflected in thefunction F . For example, the series F is symmetric in the variables q, r, s , corresponding to theaction of the anharmonic group on J . Setting t = 1 yields(1 − q + q )(1 − r + r )(1 − s + s ) + qrs (2 − q − r − s )(1 − q ) (1 − r ) (1 − s ) , which is the generating function of the determinant det( (cid:0) a + b bb c + b (cid:1) ) = ab + ac + bc (up to somecoefficients in exponents of determinant zero); this corresponds to the rank of J w ∗ , (cid:16) a + b bb c + b (cid:17) as a C [ E , E ]-module. Also, setting r to zero yields the factorization F ( q, , s, t ) = 1 + q (1 − qt − )(1 − q ) · s (1 − st − )(1 − s ) , corresponding to the fact that J w ∗ , ( ∗ ∗ ) is the tensor square of J w ∗ , ∗ . Proof.
The structure theorem for weak Jacobi forms of rank one shows that X k ∈ Z dim J wk,a/ t k = t − a + t − a + ... + 1(1 − t )(1 − t ) = t − a − t − a + t − a − t (1 − t )(1 − t )(1 − t )for every a ∈ N . (When a = 0 this formula is incorrect; it yields the numerator (1 − t ) rather than1 − t .) After correcting for this we find X k ∈ Z ∞ X a,b,c =0 dim J wk, ( a + b ) / q a r b s c t k = 1(1 − t )(1 − t )(1 − t ) (cid:16) ∞ X b =0 r b ∞ X c =0 s c ∞ X a =0 (( qt − ) a (1 − t + t ) − q a t ) + t (1 − t ) ∞ X c =0 s c (cid:17) = 1(1 − s )(1 − t )(1 − t )(1 − t ) h − t + t (1 − qt − )(1 − rt − ) − t (1 − q )(1 − r ) + t (1 − t ) i . Taking the exceptional cases in the above lemmas where the degree ( a, b, c, k ) piece of R doesnot equal J wk,a/ into account, i.e. the indices( a, b, c ) = (0 , a, , (1 , , , (2 , , as well as ( b, c ) = (0 , , (1 , R = 1(1 − s )(1 − t )(1 − t )(1 − t ) h − t + t (1 − qt − )(1 − rt − ) − t (1 − q )(1 − r ) + t (1 − t ) i − − t )(1 − t ) (cid:16) qr + q r t − (cid:17) − ∞ X a =0 X k ∈ Z dim J wk,a/ t k q a s − ∞ X a =1 X k ∈ Z dim J wk,a/ t k r a s − − t )(1 − t ) ∞ X a =1 q a r, where the series of weak Jacobi form dimensions is ∞ X a =0 X k ∈ Z dim J wk,a/ t k q a = (1 + q )(1 − t )(1 − t )(1 − qt − )(1 − q ) . The Hilbert series of the kernel K isHilb K = st − · Hilb J + X k ∈ Z ∞ X c =0 ∞ X a =0 dim ker (cid:16) Q : J wk, ( a + c cc c ) → J wk, ( a + c ) / (cid:17) q a r c t k . This kernel is zero when c = 0 and otherwise has dimensiondim J wk, ( a + c cc c ) − dim J wk, ( a + c ) / except for the special cases c = 1 and c = 2, a ∈ { , } . Considering the weights of the missing C [ E , E ]-module generators gives us(1 − t )(1 − t ) · X k ∈ Z ∞ X c =0 ∞ X a =0 dim ker (cid:16) Q : J wk, ( a + c cc c ) → J wk, ( a + c ) / (cid:17) q a r c t k = ∞ X c =1 ∞ X a =0 (cid:16) P wa/ ( t ) P wc/ ( t ) − P w ( a + c ) / ( t ) (cid:17) q a r c + ∞ X a =1 q a r + qr + q r t − , where P wa/ ( t ) is the weak Hilbert polynomial P wa/ ( t ) = t − a − t − a + t − a − t − t . This simplifies to1 − q + q − r − q r + q r − rt − (1 − q − r )(1 − q )(1 − r )(1 − qt − )(1 − rt − ) + qr − q + qr + q r t − . The Hilbert series of J is now determined byHilb J = Hilb K + Hilb R. Using some elementary algebraic manipulations we obtain the closed form in the claim. (cid:3)
Theorem 1.3 follows quickly from Theorem 7.3. Namely, if L ⊆ R is an integral lattice with Grammatrix of the form (cid:0) a + b bb c + b (cid:1) with a, b, c ≥
0, then J w ∗ ,L is a free C [ E , E ]-module on generators ϕ , ..., ϕ n of weights k , ..., k n , where t k + ... + t k n is the coefficient of q a r b s c in the power seriesexpansion of F ( q, r, s, t ) about q, r, s = 0. N WEAK JACOBI FORMS OF RANK TWO 23
As an additional corollary, we obtain a description of weak Jacobi forms of small weight.
Corollary 7.5.
The minimal weight of a weak Jacobi form of index M = (cid:0) a + b bb c + b (cid:1) is k min = − ǫ = − ( a + b + c ) . The space J wk min ,M is always one-dimensional, spanned by the theta block ϑ ( τ, z ) a ϑ ( τ, z + w ) b ϑ ( τ, w ) c η ( τ ) ǫ . The space J wk min +1 ,M is nonzero if and only if abc = 0 , in which case it is also one-dimensional,spanned by the form (cid:16) ϑ ′ ( τ, z ) ϑ ( τ, z ) + ϑ ′ ( τ, w ) ϑ ( τ, w ) − ϑ ′ ( τ, z + w ) ϑ ( τ, z + w ) (cid:17) · ϑ ( τ, z ) a ϑ ( τ, z + w ) b ϑ ( τ, w ) c η ( τ ) ǫ . The space J wk min +2 ,M has dimension dim J wk min +2 ,M = δ a ≥ + δ b ≥ + δ c ≥ ≤ . It is spanned by the subset of φ , ( τ, z ) ϑ ( τ, z ) a − ϑ ( τ, z + w ) b ϑ ( τ, w ) c η ( τ ) ǫ − , φ , ( τ, z + w ) ϑ ( τ, z ) a ϑ ( τ, z + w ) b − ϑ ( τ, w ) c η ( τ ) ǫ − ,φ , ( τ, w ) ϑ ( τ, z ) a ϑ ( τ, z + w ) b ϑ ( τ, w ) c − η ( τ ) ǫ − which are holomorphic in z and w .Proof. From the expression of Theorem 1.3 we see that the Laurent polynomial P ( t ) with ∞ X k = −∞ dim J wk,M t k = P ( t )(1 − t )(1 − t )has the form P ( t ) = t − ( a + b + c ) + t − ( a + b + c ) · δ abc =0 + t − ( a + b + c ) · ( δ a ≥ + δ b ≥ + δ c ≥ ) + O ( t − ( a + b + c ) ) . (cid:3) Acknowledgements
H. Wang would like to thank Kaiwen Sun for valuable discussions, and he isgrateful to Max Planck Institute for Mathematics in Bonn for its hospitality and financial support.
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Max-Planck-Institut f¨ur Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
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