On a conjecture of Chen and Yui: Resultants and discriminants

Abstract

AbstractIn [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$\Gamma _{0}(p)$forpprime, 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.