Automorphic Borcherds products of singular weight

We describe some new structure results for automorphic products of singular weight. First we give a simple characterisation of the Borcherds function $\Phi_{12}$. Second we show that holomorphic automorphic products of singular weight on lattices of prime level exist only in small signatures and we derive an explicit bound. Finally we give a complete classification of reflective automorphic products of singular weight on lattices of prime level.