Reflective modular forms in algebraic geometry

TL;DR

This paper explores the implications of strongly reflective modular forms on the Kodaira dimension of modular varieties, constructing explicit examples and analyzing their geometric properties.

Contribution

It introduces a Jacobi lifting method to construct new strongly reflective modular forms and applies this to specific modular varieties, advancing understanding in algebraic geometry.

Findings

Existence of strongly reflective modular forms implies negative or zero Kodaira dimension.

Constructed three towers of modular forms with minimal divisors.

Produced modular varieties of dimensions 4, 6, and 7 with Kodaira dimension zero.

Abstract

We prove that the existence of a strongly reflective modular form of a large weight implies that the Kodaira dimension of the corresponding modular variety is negative or, in some special case, it is equal to zero. Using the Jacobi lifting we construct three towers of strongly reflective modular forms with the simplest possible divisor. In particular we obtain a Jacobi lifting construction of the Borcherds-Enriques modular form Phi_4 and Jacobi liftings of automorphic discriminants of the K\"ahler moduli of Del Pezzo surfaces constructed recently by Yoshikawa. We obtain also three modular varieties of dimension 4, 6 and 7 of Kodaira dimension 0.