Convers Theorems of Borcherds Products

Abstract

In his paper, Borcherds introduced a theta lift which allowed him to lift classical modular forms with poles at cusps to automorphic forms on the orthogonal group O(2, l). The resulting automorphic forms, called Borcherds products, possess an infinite product expansion and have their singularities located along certain arithmetic divisors, the so-called Heegner divisors. Mainly based on the work of Bruinier, we study the question whether every automorphic form having its divisor along the Heegner divisors can be realized as a Borcherds product.