Projective spaces as orthogonal modular varieties

2021

Preprint

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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 Q( √ d) for d = −1, −2, −3. As a byproduct, our modular groups provide many explicit examples of finitecovolume reflection groups acting on complex hyperbolic space. Contents 1. Introduction 1 2. Modular forms on ball quotients 3 3. Free algebras of modular forms and the Jacobian 11 4. Twins of free algebras of modular forms 15 5. More free algebras of unitary modular forms 19 6. A free algebra of modular forms on a ball quotient over Z[ √ −2] 34 7. Some interesting non-free algebras of unitary modular forms 36 8. Open questions and conjectures 38 9. Appendix: Tables 38 References 41

Section: Twins Of Free Algebras Of Modular Formsmentioning

confidence: 99%

mentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

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Free algebras of modular forms on ball quotients

Wang¹,

2021

Preprint

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“…In this section we discuss the algebras of modular forms for the lattices L = 2U (3) ⊕ A 1 and L = U ⊕ U (2) ⊕ A 1 (2). It was proved in [33] that for the first lattice M * ( O + (L)) is freely generated by four forms of weight 1. As mentioned in the introduction, M * ( O r (L)) is not free for the second lattice.…”

Section: The U ⊕ S 8 Latticementioning

confidence: 99%

Simple lattices and free algebras of modular forms

Wang¹,

2020

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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 ≤ n ≤ 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. Contents 1. Introduction and statement of results 1 2. Preliminaries 5 3. The U ⊕ U (2) ⊕ R tower 11 4. The 2U (2) ⊕ R tower 19 5. The U ⊕ U (3) ⊕ R tower 25 6. The U ⊕ U (4) ⊕ A 1 lattice 29 7. The U ⊕ S 8 lattice 30 8. The 2U (3) ⊕ A 1 and U ⊕ U (2) ⊕ A 1 (2) lattices 31 9. The U ⊕ U (2) ⊕ 2A 1 lattice 34 References 36

“…The necessary condition yields an explicit classification of free algebras of orthogonal modular forms [61]. Using the sufficient part of the criterion, we constructed a number of free algebras of orthogonal modular forms [60,65,66]. In [67] we extended this approach to modular forms on complex balls attached to unitary groups of signature (l, 1).…”

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confidence: 99%

Modular forms with poles on hyperplane arrangements

Wang¹,

2021

Preprint

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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 ⊕ D11 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 ⊕ D11, 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.