The classification of reflective modular forms

A holomorphic modular form on an orthogonal group of signature (2,n) is called reflective if it has special zero divisors defined by reflections in the orthogonal group. The Borcherds modular form of weight 12 and the Igusa Siegel cusp forms of weights 10 and 35 are all reflective.  Reflective modular forms have applications in generalized Kac-Moody algebra, algebraic geometry, and number theory.  In 1998, Gritsenko and Nikulin conjectured that the number of lattices having reflective modular forms is finite, and they raised the problem of classifying reflective modular forms. Some progress has been made in the past two decades: Scheithauer and Dittmann in the case of singular weight, and finiteness theorems by Ma. 

 

In this talk, I will introduce some new classification results. I prove the nonexistence of reflective modular forms on lattices of large rank. I give a full classification of 2-reflective modular forms which are the most fundamental class of reflective modular forms. The approach is based on the theory of Jacobi forms of lattice index.