Revisiting the Gross–Zagier discriminant formula

Abstract: We derive a formula for the discriminant of the ring class polynomial of conductor \(N\) associated to the \(j\)-invariant and an imaginary quadratic field of odd discriminant \(-d\) using the theory of Borcherds lifts. When d is a prime and \(N=1\), the formula recovers the Gross–Zagier discriminant formula.