Difference of the modular function $\omega_{2}(\tau)$, revisited

Abstract

Adapting the analytic method of Gross and Zagier, Roskam proved a prime-factorization formula for the norm of the difference of two level-two Weber singular moduli. Independently, Yang and Yin obtained an equivalent formula using Borcherds lifts. More precisely, the formula concerns the norm of \[ \omega_{2}\left(\frac{-1+\sqrt{d_{1}}}{2}\right) - \omega_{2}\left(\frac{-1+\sqrt{d_{2}}}{2}\right) \] for coprime negative fundamental quadratic discriminants $d_{1},d_{2}\equiv1\pmod 8$, where \[ \omega_{2}(\tau) = 2^{12}\frac{\eta(2\tau)^{24}}{\eta(\tau)^{24}}, \] and $\eta(\tau)$ denotes the Dedekind eta function. In this work, we revisit this formula from the perspective of arithmetic intersection theory and give a new proof.