LATTICES WITH MANY BORCHERDS PRODUCTS

We prove that there are only finitely many isometry classes of even lattices L of signature (2, n) for which the space of cusp forms of weight 1 + n/2 for the Weil representation of the discriminant group of L is trivial. We compute the list of these lattices. They have the property that every Heegner divisor for the orthogonal group of L can be realized as the divisor of a Borcherds product. We obtain similar classification results in greater generality for finite quadratic modules.