On a Conjecture of Yui and Zagier II

Abstract

Yui and Zagier made some fascinating conjectures on the factorization on the norm of the difference of Weber class invariants $f({\mathfrak{a}}_1)-f({\mathfrak{a}}_2)$ based on their calculation in \cite{YZ}. Here ${\mathfrak{a}}_i$ belong two diferent ideal classes of discrimants $D_i$ in imagainary quadratic fields $\mathbb{Q}(\sqrt{D_i})$. In \cite{LY}, we proved these conjectures and their generalizations when $(D_1,D_2)=1$ using the so-called big CM value formula of Borcherds lifting. In this sequel, we prove the conjectures when $\mathbb{Q}(\sqrt{D_1})=\mathbb{Q}(\sqrt{D_2})$ using the so-called small CM value formula. In addition, we give a precise factorization formula for the resultant of two different Weber class invariant polynomials for distinct orders.

41 pages