On some free algebras of orthogonal modular forms II

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

Section: The U ⊕ S 8 Latticementioning

confidence: 92%

Simple lattices and free algebras of modular forms

Wang¹,

2020

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“…In previous work [36,37,38,39] we constructed a number of free algebras of orthogonal modular forms (some new, some previously known). For each of these the Jacobian J O of the generators is a nonzero cusp form that vanishes exactly on the mirrors of reflections in the modular group with multiplicity one.…”

Section: Twins Of Free Algebras Of Modular Formsmentioning

confidence: 99%

“…This asserts that the algebra of modular forms is free if and only if the modular Jacobian of (potential) generators is a cusp form vanishing precisely along mirrors of reflections with multiplicity one. Using this condition, we obtained a complete classification of such free algebras under some mild conditions (see [35]) and constructed free algebras of orthogonal modular forms for a large class of lattices (see [36,37,38,39]), which also cover most of the (many) previously known examples in the literature.…”

Section: Introductionmentioning

confidence: 99%

“…There is a unique Hermitian lattice II E 5,1 of signature (5, 1) over the Eisenstein integers whose underlying Z-lattice is the even unimodular lattice II 10,2 . Hashimoto and Ueda [21] proved that the algebra of modular forms for the orthogonal group of II 10,2 is freely generated by 11 forms of weights 4,10,12,16,18,22,24,28,30,36,42. Since the weight of a nonzero modular form on U(II E 5,1 ) must be a multiple of 6, the orthogonal generators of weight 4, 10, 16, 22, 28 vanish identically on the unitary symmetric domain.…”

Section: Introductionmentioning

confidence: 99%

“…Since the weight of a nonzero modular form on U(II E 5,1 ) must be a multiple of 6, the orthogonal generators of weight 4, 10, 16, 22, 28 vanish identically on the unitary symmetric domain. The above criterion implies that the algebra of modular forms for U(II E 5,1 ) is freely generated by 6 forms of weight 12,18,24,30,36,42. As a corollary, we see that the group U(II E 5,1 ) is generated by reflections, which gives a new proof of a theorem of Allcock [2].…”

Section: Introductionmentioning

confidence: 99%

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Free algebras of modular forms on ball quotients

Wang¹,

2021

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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

Projective spaces as orthogonal modular varieties

Wang¹,

2020

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We construct 16 reflection groups Γ acting on symmetric domains 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 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.