On a Conjecture of Yui and Zagier II
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| Format: | Preprint |
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2025
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| _version_ | 1866929752424579072 |
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| author | Li, Yingkun Yang, Tonghai Ye, Dongxi |
| author_facet | Li, Yingkun Yang, Tonghai Ye, Dongxi |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_18892 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a Conjecture of Yui and Zagier II Li, Yingkun Yang, Tonghai Ye, Dongxi Number Theory 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. |
| title | On a Conjecture of Yui and Zagier II |
| topic | Number Theory |
| url | https://arxiv.org/abs/2502.18892 |