Modular forms with poles on hyperplane arrangements

TL;DR

This paper investigates algebras of meromorphic modular forms with poles on hyperplane arrangements, constructing free algebras and analyzing their properties across various lattices and symmetric domains.

Contribution

It provides a uniform construction of 147 hyperplane arrangements with free modular form algebras and explores their geometric and algebraic properties, including the theta block conjecture.

Findings

Identified 147 arrangements with free modular form algebras

Constructed 8 free algebras on complex balls with hyperplane poles

Proved modularity of formal Fourier--Jacobi series for certain lattices

Abstract

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.