标题： 等价的二元二次型和扩展模群 显示英文标题

标题： Equivalent Binary Quadratic Form and the Extended Modular Group

摘要： 扩展模群$\bar{\Pi}=<R,T,U:R^2=T^2=U^3=(RT)^2=(RU)^2=1>$，其中$ R:z\rightarrow -\bar{z}, \sim T:z\rightarrow\frac{-1}{z},\simU:z\rightarrow\frac{-1}{z +1} $，已被用于研究一些二元二次型的性质，其基点位于基本区域点集$F_{\bar{\Pi}}$中（参见\cite{Tekcan1, Flath}）。 在本文中，我们研究基点是如何被用于等价二元二次型的研究的，并证明两个正定形式等价当且仅当一个形式的基点在扩展模群的作用下被映射到另一个形式的基点，并且任何正定整系数形式都可以在扩展模群的作用下转化为相同判别式的约简形式，并将这些结果扩展到虚二次域$\QQ(\sqrt{-m})$的子集$\QQ^*(\sqrt{-n})$。 显示英文摘要

摘要： 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 $\QQ(\sqrt{-m})$.