Gross–Zagier type CM value formulas on X0⁎(p)

Revisiting the Gross–Zagier discriminant formula

“…Such a formula is now known as the Gross–Zagier CM value formula, and it has been extensively studied and extended in different directions. See, e.g., [3, 9 12, 15–17]. In view of the resultant interpretation, Gross and Zagier also considered the discriminant of a Hilbert class polynomial associated to the j ‐invariant and an imaginary quadratic field with p prime and congruent to 3 modulo 4, which can be regarded as the complementary case for the “rational norm” of two singular moduli associated to imaginary quadratic points of the same fundamental discriminant, and they showed that [7, Thm.…”