In a pair of Invenitones papers in 1995 and 1998, Borcherds gave a construction of meromorphic modular forms on the hermitian symmetric domain associated to a rational quadratic space V, ( , )  of signature (n,2).  These modular forms have remarkable properties including

(1) an explicit divisor given in terms of special divisors and

(2) product formulas, each valid in a neighborhood of a point boundary component.

In this lecture, after reviewing Borcherds' construction, and under the assumption that V, ( , ) admits 2-dimensional rational isotropic subspaces,  I will describe some new product formulas for these modular forms, each valid in the neighborhood of a 1-dimensional boundary component. The products involve Jacobi theta functions and eta functions.

They should prove useful in understanding the integral theory of Borcherds forms.