Equivalent Binary Quadratic Form and the Extended Modular Group

TL;DR

This paper explores the relationship between binary quadratic forms and the extended modular group, proving equivalence criteria and reduction procedures using group actions, with extensions to specific imaginary quadratic fields.

Contribution

It establishes that two positive definite forms are equivalent if their base points are related by the extended modular group, and shows how forms can be reduced via this group's action, extending to certain quadratic fields.

Findings

Two positive definite forms are equivalent iff their base points are related by the extended modular group.

Any positive definite integral form can be reduced to a canonical form using the extended modular group.

Results are extended to quadratic fields () with specific subset considerations.

Abstract

Extended modular group $\bar{\Pi}=<R,T,U:R^2=T^2=U^3=(RT)^2=(RU)^2=1>$, where $ R:z\rightarrow -\bar{z}, \sim T:z\rightarrow\frac{-1}{z},\simU:z\rightarrow\frac{-1}{z +1} $, has been used to study some properties of the binary quadratic forms whose base points lie in the point set fundamental region $F_{\bar{\Pi}}$ (See \cite{Tekcan1, Flath}). In this paper we look at how base points have been used in the study of equivalent binary quadratic forms, and we prove that two positive definite forms are equivalent if and only if the base point of one form is mapped onto the base point of the other form under the action of the extended modular group and any positive definite integral form can be transformed into the reduced form of the same discriminant under the action of the extended modular group and extend these results for the subset $\QQ^*(\sqrt{-n})$ of the imaginary quadratic field $\QQ(\sqrt{-m})$.