Free algebras of modular forms on ball quotients

Abstract

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

Modular forms with poles on hyperplane arrangements

“…Proof. This theorem can be proved in a similar way to Theorem 2.6 and [67,Theorem 3.3]. The essential fact is that the boundary components of the Loojenga compactification of the arrangement complement (D U (M ) − H U )/Γ have codimension at least two.…”

“…As in [67,Theorem 4.2], the above theorem holds when we replace O + (M Z ) and U(M ) with O(M Z ) and U(M ) and the proof is similar. We cannot extend this theorem to other discriminants, because when d / ∈ {−1, −3} the restriction of a reflection in O + (M Z ) is not necessarily a reflection in U(M ).…”

“…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). In this paper we further extend this argument to modular forms with singularities on hyperplane arrangements.…”

Big line bundles on unitary modular varieties

“…Remark 1.1. Note that the Hermitian lattices considered in this paper are different from [36], [37], [51], so we have to pay attention to the notion "unimodular". where p runs over any prime number which divides D and detpLq.…”