On a conjecture of Chen and Yui: Resultants and discriminants

Abstract: In [Chen and Yui, 1993], 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.