In [5], Chen and Yui conjectured that Gross–Zagier type formulas may also exist for Thompson series. In this work, we verify Chen and Yui’s conjecture for the cases for Thompson series $j_{p}(\tau )$ for $\Gamma _{0}(p)$ for p prime, and equivalently establish formulas for the prime decomposition of the resultants of two ring class polynomials associated to $j_{p}(\tau )$ and imaginary quadratic fields and the prime decomposition of the discriminant of a ring class polynomial associated to $j_{p}(\tau )$ and an imaginary quadratic field. Our method for tackling Chen and Yui’s conjecture on resultants can be used to give a different proof to a recent result of Yang and Yin. In addition, as an implication, we verify a conjecture recently raised by Yang, Yin, and Yu.

在[5]中，Chen 和Yui 推测Thompson 级数也可能存在Gross-Zagier 类型的公式。在这项工作中，我们验证了 Chen 和 Yui 对于 Thompson 级数 $j_{p}(\tau )$ 对于 $\Gamma_{0}(p)$ 对于p素数的情况的猜想，并等效地建立了素数分解的公式与 $j_{p}(\tau )$ 和虚二次域 相关联的两个环类多项式的结果以及与 $j_{p}(\tau ) $ 相关联的环类多项式的判别式的素数分解和一个假想的二次场。我们处理 Chen 和 Yui 关于结果的猜想的方法可以用来对最近的阳和阴结果给出不同的证明。此外，作为暗示，我们验证了杨、尹和于最近提出的一个猜想。