On a Conjecture of Yui and Zagier II

2025/02/26 by Li, Yingkun, Yang, Tonghai, Ye, Dongxi
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2502.18892

Abstract

Yui and Zagier made some fascinating conjectures on the factorization on the norm of the difference of Weber class invariants f(\mathfrak a₁) - f(\mathfrak a₂) based on their calculation in \citeYZ. Here \mathfrak a_i belong two diferent ideal classes of discrimants D_i in imagainary quadratic fields ℚ(√(D_i)). In \citeLY, we proved these conjectures and their generalizations when (D₁, D₂) =1 using the so-called big CM value formula of Borcherds lifting. In this sequel, we prove the conjectures when ℚ(√(D₁)) =ℚ(√(D₂)) 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.

Related

• On a Conjecture of Yui and Zagier
• Difference of modular functions and their CM value factorization
• Generalised Weber Functions
• The lambda invariants at CM points
• A Gross-Zagier formula for quaternion algebras over totally real fields