arXiv subjects

Haowu Wang

Publications and source records attributed to Haowu Wang.

At least 37 records · Page 2Linked to original sources

arXiv2022

Mathieu moonshine and Borcherds products

The twisted elliptic genera of a $K3$ surface associated with the conjugacy classes of the Mathieu group $M_{24}$ are known to be weak Jacobi forms of weight $0$. In 2010, Cheng constructed formal infinite products from the twisted elliptic genera and conjectured that they define Siegel modular forms of degree two. In this paper we prove that for each conjugacy class of level $N_g$ the associated product is a meromorphic Borcherds product on the lattice $U(N_g)\oplus U \oplus A_1$ in a strict sense. We also compute the divisors of these products and determine for which conjugacy classes the product can be realized as an additive (generalized Saito--Kurokawa) lift.

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arXiv2021

Theta blocks related to root systems

Gritsenko, Skoruppa and Zagier associated to a root system $R$ a theta block $\vartheta_R$, which is a Jacobi form of lattice index. We classify the theta blocks $\vartheta_R$ of $q$-order $1$ and show that their Gritsenko lift is a strongly-reflective Borcherds product of singular weight, which is related to Conway's group $\operatorname{Co}_0$. As a corollary we obtain a proof of the theta block conjecture by Gritsenko, Poor and Yuen for the pure theta blocks obtained as specializations of the functions $\vartheta_R$.

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arXiv2021

Reflective modular forms on lattices of prime level

One of the main open problems in the theory of automorphic products is to classify reflective modular forms. In [Sch06] Scheithauer classified strongly reflective modular forms of singular weight on lattices of prime level. In this paper we classify symmetric reflective modular forms on lattices of prime level. This yields a full classification of lattices of prime level which have reflective modular forms. We also present some applications.

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arXiv2021

Modular forms with poles on hyperplane arrangements

We study algebras of meromorphic modular forms whose poles lie on Heegner divisors for orthogonal and unitary groups associated to root lattices. We give a uniform construction of $147$ hyperplane arrangements on type IV symmetric domains for which the algebras of modular forms with constrained poles are free and therefore the Looijenga compactifications of the arrangement complements are weighted projective spaces. We also construct $8$ free algebras of modular forms on complex balls with poles on hyperplane arrangements. The most striking example is the discriminant kernel of the $2U\oplus D_{11}$ lattice, which admits a free algebra on $14$ meromorphic generators. Along the way, we determine minimal systems of generators for non-free algebras of orthogonal modular forms for $26$ reducible root lattices and prove the modularity of formal Fourier--Jacobi series associated to them. By exploiting an identity between weight one singular additive and multiplicative lifts on $2U\oplus D_{11}$, we prove that the additive lift of any (possibly weak) theta block of positive weight and $q$-order one is a Borcherds product; the special case of holomorphic theta blocks of one elliptic variable is the theta block conjecture of Gritsenko, Poor and Yuen.

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arXiv2021

Siegel modular forms of degree two and level five

We construct a ring of meromorphic Siegel modular forms of degree 2 and level 5, with singularities supported on an arrangement of Humbert surfaces, which is generated by four singular theta lifts of weights 1, 1, 2, 2 and their Jacobian. We use this to prove that the ring of holomorphic Siegel modular forms of degree 2 and level $Γ_0(5)$ is minimally generated by eighteen modular forms of weights 2, 4, 4, 4, 4, 4, 6, 6, 6, 6, 10, 11, 11, 11, 13, 13, 13, 15.

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arXiv2021

Graded rings of Hermitian modular forms with singularities

We study graded rings of meromorphic Hermitian modular forms of degree two whose poles are supported on an arrangement of Heegner divisors. For the group $\mathrm{SU}_{2,2}(\mathcal{O}_K)$ where $K$ is the imaginary-quadratic number field of discriminant $-d$, $d \in \{4, 7, 8, 11, 15, 19, 20, 24\}$ we obtain a polynomial algebra without relations. In particular the Looijenga compactifications of the arrangement complements are weighted projective spaces.

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arXiv2021

On some free algebras of orthogonal modular forms II

In this paper we construct 16 free algebras of modular forms on symmetric domains of type IV for some reflection groups related to the eight lattices $A_1(2)$, $A_1(3)$, $A_1(4)$, $2A_1(2)$, $A_2(2)$, $A_2(3)$, $A_3(2)$, $D_4(2)$. As a corollary, we prove the modularity of formal Fourier--Jacobi expansions for these reflection groups.

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arXiv2021

Weyl invariant Jacobi forms: a new approach

The weak Jacobi forms of integral weight and integral index associated to an even positive definite lattice form a bigraded algebra. In this paper we prove a criterion for this type of algebra being free. As an application, we give an automorphic proof of K. Wirthmüller's theorem which asserts that the bigraded algebra of weak Jacobi forms invariant under the Weyl group is a polynomial algebra for any irreducible root system not of type $E_8$. This approach is also applicable to $E_8$. Even if the algebra of $E_8$ Jacobi forms is known to be non-free, we still derive a new structure result.

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arXiv2021

The classification of free algebras of orthogonal modular forms

We prove a necessary and sufficient condition for the graded algebra of automorphic forms on a symmetric domain of type IV to be free. From the necessary condition, we derive a classification result. Let $M$ be an even lattice of signature $(2,n)$ splitting two hyperbolic planes. Suppose $Γ$ is a subgroup of the integral orthogonal group of $M$ containing the discriminant kernel. It is proved that there are exactly 26 groups $Γ$ such that the space of modular forms for $Γ$ is a free algebra. Using the sufficient condition, we recover some well-known results.

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arXiv2021

Free algebras of modular forms on ball quotients

In this paper we study algebras of modular forms on unitary groups of signature $(n,1)$. We give a necessary and sufficient condition for an algebra of unitary modular forms to be free in terms of the modular Jacobian. As a corollary we obtain a criterion that guarantees in many cases that, if $L$ is an even lattice with complex multiplication and the ring of modular forms for its orthogonal group is a polynomial algebra, then the ring of modular forms for its unitary group is also a polynomial algebra. We prove that a number of rings of unitary modular forms are freely generated by applying these criteria to Hermitian lattices over the rings of integers of $\mathbb{Q}(\sqrt{d})$ for $d=-1,-2,-3$. As a byproduct, our modular groups provide many explicit examples of finite-covolume reflection groups acting on complex hyperbolic space.

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arXiv2021

Weyl invariant $E_8$ Jacobi forms

We investigate the Jacobi forms for the root system $E_8$ invariant under the Weyl group. This type of Jacobi forms has significance in Frobenius manifolds, Gromov--Witten theory and string theory. In 1992, Wirthmüller proved that the space of Jacobi forms for any irreducible root system not of type $E_8$ is a polynomial algebra. But very little has been known about the case of $E_8$. In this paper we show that the bigraded ring of Weyl invariant $E_8$ Jacobi forms is not a polynomial algebra and prove that every such Jacobi form can be expressed uniquely as a polynomial in nine algebraically independent Jacobi forms introduced by Sakai with coefficients which are meromorphic SL(2,Z) modular forms. The latter result implies that the space of Weyl invariant $E_8$ Jacobi forms of fixed index is a free module over the ring of SL(2,Z) modular forms and that the number of generators can be calculated by a generating series. We determine and construct all generators of small index. These results give a proper extension of the Chevalley type theorem to the case of $E_8$.

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arXiv2021

On weak Jacobi forms of rank two

We study a ring of weak Jacobi forms indexed by integral lattices of rank two. We find an explicit finite set of generators of this ring and give a dimension formula for weak Jacobi forms of rank two lattice index.

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arXiv2020

Simple lattices and free algebras of modular forms

We study the algebras of modular forms on type IV symmetric domains for simple lattices; that is, lattices for which every Heegner divisor occurs as the divisor of a Borcherds product. For every simple lattice $L$ of signature $(n,2)$ with $3\leq n \leq 10$, we prove that the graded algebra of modular forms for the maximal reflection subgroup of the orthogonal group of $L$ is freely generated. We also show that, with five exceptions, the graded algebra of modular forms for the maximal reflection subgroup of the discriminant kernel of $L$ is also freely generated.

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arXiv2020

Projective spaces as orthogonal modular varieties

We construct $16$ reflection groups $Γ$ acting on symmetric domains $\mathcal{D}$ of Cartan type IV, for which the graded algebras of modular forms are freely generated by forms of the same weight, and in particular the Satake--Baily--Borel compactification of $\mathcal{D} / Γ$ is isomorphic to a projective space. Four of these are previously known results of Freitag--Salvati Manni, Matsumoto, Perna and Runge. In addition we find several new modular groups of orthogonal type whose algebras of modular forms are freely generated.

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arXiv2020

Borcherds products of half-integral weight

We give a necessary and sufficient criterion for the existence of Borcherds products of half-integral weight associated to even lattices that split two hyperbolic planes. In particular we prove that half-integral weight Borcherds products exist for lattices of arbitrary rank.

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arXiv2020

Graded rings of integral Jacobi forms

We determine the structure of the bigraded ring of weak Jacobi forms with integral Fourier coefficients. This ring is the target ring of a map generalising the Witten and elliptic genera and a partition function of $(0,2)$-model in string theory. We also determine the structure of the graded ring of all weakly holomorphic Jacobi forms of weight zero and integral index with integral Fourier coefficients. These forms are the data for Borcherds products for the Siegel paramodular groups.

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arXiv2020

On some free algebras of orthogonal modular forms

For 25 orthogonal groups of signature $(2,n)$ related to the root lattices $A_1$, $2A_1$, $3A_1$, $4A_1$, $A_2$, $A_3$, $A_4$, $A_5$, $A_6$, $A_7$, $D_4$, $D_5$, $D_6$, $D_7$, $D_8$, $E_6$, $E_7$, we prove that the algebras of modular forms on symmetric domains of type IV are freely generated by the additive lifts of some special Jacobi forms. The proof is universal and elementary.

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arXiv2019

Theta block conjecture for Siegel paramodular forms

The theta-block conjecture proposed by Gritsenko--Poor--Yuen in 2013 characterizes Siegel paramodular forms which are simultaneously Borcherds products and additive Jacobi lifts. In this paper, we prove this conjecture for two new infinite series of theta blocks of weights 2 and 3. The proof is based on Scheithauer's classification of reflective modular forms of singular weight.

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