Reflective Modular Forms: A Jacobi Forms Approach

Wang, Haowu

doi:10.1093/imrn/rnz070

Cited by 14 publications

(25 citation statements)

References 35 publications

(98 reference statements)

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“…This gives a generalization of one result in [Loo03]. In §9 we answer some questions proposed in [Wan18] and formulate many new open questions related to this paper.…”

Section: Introductionmentioning

confidence: 63%

“…Known results. We first review some results proved in [Wan18]. Let M = 2U ⊕ L(−1) be an even lattice of signature (2, rank(L) + 2).…”

Section: Classification Of 2-reflective Modular Formsmentioning

confidence: 99%

“…In our previous work [Wan18], we proved the nonexistence of 2-reflective and reflective modular forms on lattices of large rank by constructing certain holomorphic Jacobi forms of small weights using differential operators. In particular, we showed that the only 2-reflective lattices of signature (2, n) satisfying n ≥ 15 and n = 19 are II 2,18 and II 2,26 .…”

Section: Introductionmentioning

confidence: 99%

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The classification of 2-reflective modular forms

Wang¹

2019

Preprint

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The classification of reflective modular forms is an important problem in the theory of automorphic forms on orthogonal groups. In this paper, we develop an approach based on the theory of Jacobi forms to give a full classification of 2-reflective modular forms. We prove that there are only 3 lattices of signature (2, n) having 2-reflective modular forms when n ≥ 14. We show that there are exactly 51 lattices of type 2U ⊕ L(−1) which admit 2-reflective modular forms and satisfy that L has 2-roots. We further determine all 2-reflective modular forms giving arithmetic hyperbolic 2-reflection groups. This is the first attempt to classify reflective modular forms on lattices of arbitrary level.

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“…This gives a generalization of one result in [Loo03]. In §9 we answer some questions proposed in [Wan18] and formulate many new open questions related to this paper.…”

Section: Introductionmentioning

confidence: 63%

“…Known results. We first review some results proved in [Wan18]. Let M = 2U ⊕ L(−1) be an even lattice of signature (2, rank(L) + 2).…”

Section: Classification Of 2-reflective Modular Formsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The classification of 2-reflective modular forms

Wang¹

2019

Preprint

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“…In particular, is called 2-reflective if its support of zero divisor is contained in , and is called a modular form with complete 2-divisor if . Reflective modular forms are very rare (see [Ma17, Ma18, Wan21]) and have many applications on the theory of generalized Kac–Moody algebras, reflection groups and in algebraic geometry (see e.g. [Bor00, Sch06, GN18, Gri18]).…”

Section: Automorphic Forms On Symmetric Domains Of Type IVmentioning

confidence: 99%

“…By Theorem 3.5, the decomposition of the Jacobian determinant will give a modular form with complete 2-divisor. We know from [Wan21, Theorem 3.4] that if there exists a modular form with complete 2-divisor for then either or is a unimodular lattice of rank 16 or 24. By Lemma 4.1, we only need to consider the two cases and .…”

Section: Classification Of Free Algebras Of Orthogonal Modular Formsmentioning

confidence: 99%

The classification of free algebras of orthogonal modular forms

Wang¹

2021

Compositio Math.

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We prove a necessary and sufficient condition for the graded algebra of automorphic forms on a symmetric domain of type IV being 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 $\Gamma$ is a subgroup of the integral orthogonal group of $M$ containing the discriminant kernel. It is proved that there are exactly 26 groups $\Gamma$ such that the space of modular forms for $\Gamma$ is a free algebra. Using the sufficient condition, we recover some well-known results.

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Consistency of eight-dimensional supergravities: anomalies, lattices and counterterms

Lao,

Minasian

2024

J. High Energ. Phys.

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We reexamine the question of quantum consistency of supergravities in eight dimensions. Theories with 16 supercharges suffer from the anomalies under the action of its discrete modular groups. In minimally supersymmetric theory coupled to Yang-Mills multiples of rank l with the moduli space given by SO(2, l)/(U(1) × SO(l)), the existence of a counterterm together with the requirement that its poles and zeros correspond to the gauge symmetry enhancement imposes nontrivial constraints on the lattice. The counterterms needed for anomaly cancellation for all cases, that are believed to lead to consistent theories of quantum gravity (l = 2, 10, 18), are discussed.

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Reflective Modular Forms: A Jacobi Forms Approach

Cited by 14 publications

References 35 publications

The classification of 2-reflective modular forms

The classification of 2-reflective modular forms

The classification of free algebras of orthogonal modular forms

Consistency of eight-dimensional supergravities: anomalies, lattices and counterterms

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