Gross-Zagier Type CM Value Formulas on $X_{0}^{*}(p)$

TL;DR

This paper derives Gross-Zagier type formulas for CM values of Hauptmoduls on Fricke groups and demonstrates their application in computing Hilbert class polynomials.

Contribution

It introduces new CM value formulas for Hauptmoduls on Fricke groups and shows how to use them to compute Hilbert class polynomials.

Findings

Derived explicit Gross-Zagier type CM value formulas

Applied formulas to compute Hilbert class polynomials

Enhanced understanding of CM values on Fricke groups

Abstract

In this work, we derive Gross-Zagier type CM value formulas for Hauptmoduls $j_{p}^{*}(\tau)$ on Fricke groups $\Gamma_{0}^{*}(p)$. We also illustrate how to employ these formulas to obtain certain Hilbert class polynomials.