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Speaker unknown : On the converse theorem in the theory of Borcherds products

R. Borcherds constructed a lifting from elliptic modular forms of weight $1-n/2$ to meromorphic modular forms on the orthogonal group $O(2,n)$. The lifted modular forms can be written as infinite products analogous to the Dedekind $\eta$-function (``Borcherds products''). Moreover, their divisors are always linear combinations of Heegner divisors; these are algebraic divisors that come from embedded quotients of $O(2,n-1)$. We address the natural question, whether every meromorphic modular form on $O(2,n)$, whose divisor is a linear combination of Heegner divisors, is a Borcherds product? We discuss some recent results that answer this question in the affirmative in a large class of cases.

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