Finiteness of 2-reflective lattices of signature (2,n)

Abstract

A modular form for an even lattice L of signature (2,n) is said to be 2-reflective if its zero divisor is set-theoretically contained in the Heegner divisor defined by the (-2)-vectors in L. We prove that there are only finitely many even lattices with n>6 which admit 2-reflective modular forms. In particular, there is no such lattice in n>25 except the even unimodular lattice of signature (2,26). This proves a conjecture of Gritsenko and Nikulin in the range n>6.