Non-existence of reflective modular forms

A non-constant holomorphic modular form for an even lattice M of signature (2,n) is called reflective (resp. 2-reflective) if the support of its divisor is contained in the union of rational quadratic divisors determined by reflective vectors (resp. 2-reflective vectors) of M. Classification of reflective modular forms is an old open problem and has been investigated by several mathematicians (Borcherds, Gritsenko, Nikulin, Looijenga, Scheithauer, Ma, ...). In this talk, I will prove two non-existence results based on the theory of Jacobi forms. The first one is that if M admits a 2-reflective modular form then n<15, or n=19, or M is isomorphic to the unimodular lattice of signature (2,18) or (2,26). The second is that if M admits a reflective modular form then n<23 or M is isomorphic to the unimodular lattice of signature (2,26).