On a Conjecture of Yui and Zagier II

TL;DR

This paper proves conjectures by Yui and Zagier on the factorization of differences of Weber class invariants in imaginary quadratic fields, extending previous results to cases where the fields are the same, and provides explicit factorization formulas.

Contribution

It extends the proof of Yui-Zagier conjectures to cases with identical quadratic fields using small CM value formulas and offers explicit factorization formulas for Weber class invariants.

Findings

Proved Yui-Zagier conjectures for identical quadratic fields.

Derived explicit factorization formulas for Weber class invariants.

Extended previous results using small CM value formulas.

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.