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The D8-Tower of Weak Jacobi Forms and Applications

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The D8-Tower of Weak Jacobi Forms and Applications The D8-tower of weak Jacobi forms and applications Dmitry Adler and Valery Gritsenko ∗ October 14, 2019 Abstract We construct a tower of arithmetic generators of the bigraded poly- w,O nomial ring J∗,∗ (Dn) of weak Jacobi modular forms invariant with respect to the full orthogonal group O(Dn) of the root lattice Dn for 2 ≤ n ≤ 8. This tower corresponds to the tower of strongly re- flective modular forms on the orthogonal groups of signature (2,n) which determine the Lorentzian Kac-Moody algebras related to the BCOV (Bershadsky-Cecotti-Ooguri-Vafa)-analytic torsions. We prove that the main three generators of index one of the graded ring satisfy a special system of modular differential equations. We found also a general modular differential equation of the generator of weight 0 and index 1 which generates the automorphic discriminant of the moduli space of Enriques surfaces. 1 Introduction In this paper we construct the tower of arithmetic generators of the bigraded w,O polynomial ring J∗,∗ (Dn) of O(Dn)-invariant weak Jacobi modular forms with respect to the root lattice Dn for 2 ≤ n ≤ 8, corresponding to the D8- tower of strongly reflective modular forms on the orthogonal groups O+(2U⊕ Dn(−1)) (see [G3]). The strongly reflective modular forms of this tower + of the orthogonal groups O (2U ⊕ Dn(−1)) (3 ≤ n ≤ 8) determine the arXiv:1910.05226v1 [math.AG] 11 Oct 2019 Lorentzian Kac–Moody algebras corresponding to the BCOV (Bershadsky- e Cecotti-Ooguri-Vafa)-analytic torsions (see [Y] and [G3]). W The fact that the bigraded ring J∗,∗(Dn) of Jacobi forms invariants with respect to the Weyl group of Dn is polynomial was proved in [W]. The Wirtm¨uller theorem is the analog of the Chevalley theorem for invariant polynomials of a Coxeter group (see [Lo1]–[Lo2], [SYS] and [S1]). However, the Wirtm¨uller’s proof does not give a construction of generators. Moreover, ∗The authors are supported by the Laboratory of Mirror Symmetry NRU HSE (RF government grant, ag. no. 14.641.31.0001). 1 this proof in the case of Dn is less clear than for the case of An. An explicit W construction of generators of the polynomial ring J∗,∗(R) (where the root system R 6= E8) is important for determining the flat coordinates in the theory of Frobenius varieties (see [S2], [DZ], [Sat] and [Be1]–[Be2]). For example, in [Sat] one can find (without detailed proof) a construction of the generators in the E6 case. In [Be1]–[Be2] the cases of An, Bn and G2 were considered independently from [W]. We note that H. Wang proved recently W that J∗,∗(E8) is not polynomial (see [Wa]). w,W (Dn) As we mentioned above, our interest to generators of J∗,∗ is ex- plained by existence of the D8-tower of the reflective automorphic discrimi- nants starting with the Borcherds–Enriques modular form + + Φ4 ∈ M4(O (U ⊕ U(2) ⊕ E8(−2)),χ2)= M4(O (U ⊕ U ⊕ D8(−1)),χ2) which is the discriminant of the moduli space of Enriques surfaces (see [Bo], [K] and [G3, §5] or Arxiv.1005.3753). Due to this relation we are interested in the generators of the O(Dn)-invariant weak Jacobi modular forms with respect to the root lattice Dn where O(Dn) is the full orthogonal group of the lattice Dn. We note that our construction is quite arithmetic, and it gives us a rather interesting system of non-linear differential equations related to the main arithmetic generators of the bigraded ring of Jacobi forms (see §5). All reflective automorphic discriminants of the D8-tower are determined O(Dn) by the last generators ϕ0,1 ∈ J0,1 (2 ≤ n ≤ 8) of the bigraded ring of O(Dn) J∗,∗ , which contains two types of generators of index one and two. See Dn Dn Dn §3. All generators of index 2, namely ϕ−2n+2,2, ϕ−2n+4,2,...,ϕ−6,2, come, in fact, from the sublattice nA1 = A1 ⊕ . ⊕ A1 of Dn. We can easily construct them using only Jacobi modular forms ϕ−2,1 and ϕ0,1 introduced by Eichler and Zagier (see [EZ]). Dn 2 Dn Dn Dn There are four generators of index 1, namely (ω−n,1) , ϕ−4,1, ϕ−2,1, ϕ0,1 . Dn The Jacobi form ω−n,1 has the simplest possible divisor. It corresponds to the orbit of the miniscule weights of the root system Dn. So it is proportional Dn to the products of odd Jacobi theta-series. This form ω−n,1 defines the O(Dn) W (Dn) difference between two graded rings J∗,∗ and J∗,∗ (see more details in §4). Dn Dn Dn We prove in §5 that the last three generators ϕ−4,1, ϕ−2,1, ϕ0,1 satisfy a special system of modular differential equations. Moreover we found a Dn general modular differential equation for ϕ0,1 (see (6) in §5). We restrict ourselves in this paper only to the Dn case with n ≤ 8 because of existence of the D8-tower of reflective automorphic forms. Our arithmetic construction of the generators works with small modification for any n. We plan to study different variations of generators and the corresponding non- linear differential equations for arbitrary n in a separate publication. 2 2 Jacobi modular forms and root systems: defini- tions and examples In this paper we use the following definition of weak Jacobi forms in many variables. See [G1], [CG] and [G3] for more details. By a lattice we mean a free Z-module equipped with a non-degenerate symmetric bilinear form (·, ·) with values in Z. A lattice is even if (l, l) is even for all its elements. Definition 2.1. Let L be the a positive definite even lattice with the inner product (· , ·), a variable τ be from the upper half-plane H and let z = (z1,...,zn) ∈ L ⊗ C. Then a weak Jacobi form of weight k ∈ Z and index m ∈ Z for the lattice L is a holomorphic function ϕ : H× (L ⊗ C) → C which satisfies the functional equations c(z,z) aτ + b z k πim ϕ , = (cτ + d) e cτ+d ϕ(τ, z) for any a b ∈ SL (Z), cτ + d cτ + d c d 2 ϕ(τ, z + λτ + µ)= e−2πim(λ, z)−πim(λ, λ)τ ϕ(τ, z) for all λ, µ ∈ L and has a Fourier expansion ϕ(τ, z)= a(n, l)qnζl, n>0X, l∈L∨ where q denotes e2πiτ and ζl denotes e2πi(z, l) for any l ∈ L∨, where L∨ is the dual lattice of L. Remark 2.1. 1) If the Fourier expansion of ϕ(τ, z) satisfies the stronger condition a(n, l) 6= 0 ⇒ 2mn > (l, l), then such form is called holomorphic Jacobi form. And if it satisfies even more strong condition a(n, l) 6= 0 ⇒ 2mn > (l, l), then this form is called cusp Jacobi form. We denote the respective finite dimensional spaces by c w Jk,m(L) ⊂ Jk,m(L) ⊂ Jk,m(L). 2) In fact, these spaces depend only on the discriminant group of the renormalised lattice L(m). The space of holomorphic Jacobi forms is iso- morphic to the holomorphic vector-valued modular forms defined by Weil representation of L(m) (see [G1, Lemma 2.3]). 3) It is easy to show that there are no non-zero Jacobi forms of negative index m. For m = 0 a weak Jacobi form of weight k is a SL2(Z)-modular form f(τ) of weight k. Definition 2.2. Let G < O(L) be a subgroup of the integral orthogonal group of a positive definite lattice L. A weak Jacobi form ϕ(τ, z) for the lattice L is called G-invariant, if for any g ∈ G ϕ(τ, g(z)) = ϕ(τ, z). 3 The finite-dimensional space of all G-invariant weak Jacobi forms of weight w,G k and index m is denoted by Jk,m (L); an analogous notation is used for the holomorphic and cusp forms.) In the special case of the full integral orthogonal group O(L) of lattice L and its subgroup W (L) we have w,O w,W Jk,m (L) ⊂ Jk,m (L). The set of all weak G-invariant Jacobi forms has the structure of a bi- graded ring G w,G J∗,∗(L)= Jk,m (L) k∈Z,M m∈Z≥0 This ring can be considered as an algebra over the graded ring of all holo- morphic modular forms with respect to the full modular group M∗ = M2k(SL2(Z)) = C[E4(τ), E6(τ)] Mk≥0 which is the polynomial ring generated by two Eisenstein series n n E4(τ) = 1 + 240 σ3(n)q , E6(τ) = 1 − 504 σ5(n)q nX≥1 nX≥1 k where σk(n)= d|n d . In this paperP we consider the case L = Dn, the root lattice of Dn type. Following [Bou], let us recall the main prorepties of root lattices Dn. n Let ε1,...,εn be the standard orthonormal basis in Z . We can choose α1 = ε1 − ε2, ..., αn−1 = εn−1 − εn and αn = εn−1 + εn as the basis of root system Dn. These vectors generate even positive definite lattice n Dn = {(x1,...,xn) ∈ Z | xi ≡ 0 mod2}, X n which is a sublattice of index 2 of Z . The root systems of Dn-type are well-defined for all n > 2. We note that the lattice D2 is isomorphic to the direct sum A1 ⊕ A1 and D3 is isomorphic to root lattice A3.
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