Reflective modular forms on lattices of prime level

TL;DR

This paper classifies symmetric reflective modular forms on lattices of prime level, completing the classification of such forms and lattices, and discusses some applications.

Contribution

It provides a full classification of lattices of prime level with reflective modular forms, extending previous work on strongly reflective forms.

Findings

Classified symmetric reflective modular forms on prime level lattices

Identified all lattices of prime level with reflective modular forms

Presented applications of the classification

Abstract

One of the main open problems in the theory of automorphic products is to classify reflective modular forms. In [Sch06] Scheithauer classified strongly reflective modular forms of singular weight on lattices of prime level. In this paper we classify symmetric reflective modular forms on lattices of prime level. This yields a full classification of lattices of prime level which have reflective modular forms. We also present some applications.