Complete systems of invariants of $m$-tuples for fundamental groups of a two-dimensional bilinear-metric space over the field of rational numbers

Abstract

Let $Q$ be the field of rational numbers and $Q^{2}$ be the $2$-dimensional linear space over $Q$. A classification of all non-degenerate symmetric bilinear-metric forms over $Q^{2}$ have obtained. Let $\varphi $ be a non-degenerate symmetric bilinear form on $Q^{2}$. Denote by $O(2,\varphi, Q)$ the group of all $\varphi$-orthogonal (that is the form $\varphi$ preserving) transformations of $Q^{2}$. Put $MO( 2,\varphi, Q)=\left\{F:Q^{2}\rightarrow Q^{2}\mid Fx=gx+b, g\in O(2,\varphi, Q), b\in Q^{2}\right\}$, $SO(2,\varphi, Q)=\left\{ g\in O(2, \varphi, Q)|detg=1\right\}$ and $MSO(2, \varphi, Q)= \left\{F\in M(2, \varphi, Q)|detg=1\right\}$.The present paper is devoted to solutions of problems of $G$-equivalence of $m$-tuples in $Q^{2}$ for groups $G=O(2,\varphi, Q), SO(2,\varphi, Q)$, $MO(2,\varphi, Q)$, $MSO(2,\varphi, Q)$. Complete systems of $G$-invariants of $m$-tuples in $Q^{2}$ for these groups are obtained.