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arXiv · 2502.18892

On a Conjecture of Yui and Zagier II

Yingkun Li·Tonghai Yang·Dongxi Ye
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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.

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Yingkun Li, Tonghai Yang, Dongxi Ye. 2025-02-26. On a Conjecture of Yui and Zagier II. https://arxiv.org/abs/2502.18892

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