Computing $n^{\mathrm{th}}$ roots in $S{L}_2$ and Fibonacci polynomials

Abstract

Let $k$ be a field of characteristic $\ne 2$. In this paper we study squares, cubes and their products in split and anisotropic groups of type $A_1$. In split case, we show that computing $n^{\mathrm{th}}$ roots is equivalent to finding solutions of certain polynomial equations in at most two variables over the base field $k$. The description of these polynomials involves generalised Fibonacci polynomials. Using this we obtain asymptotic proportions of $n^{\mathrm{th}}$ powers, and conjugacy classes which are $n^{\mathrm{th}}$ powers, in ${\mathrm{SL}}_2({\mathbb{F}}_q)$ when $n$ is a prime or $n=4$. We also extend already known Waring type result for ${\mathrm{SL}}_2({\mathbb{F}}_q)$, that every element of ${\mathrm{SL}}_2({\mathbb{F}}_q)$ is a product of two squares, to ${\mathrm{SL}}_2(k)$ for an arbitrary $k$. For anisotropic groups of type $A_1$, namely ${\mathrm{SL}}_1(Q)$ where $Q$ is a quaternion division algebra, we prove that when $2$ is a square in $k$, every element of ${\mathrm{SL}}_1(Q)$ is a product of two squares if and only if $-1$ is a square in ${\mathrm{SL}}_1(Q)$.

References in corpus (1)

• Powers of a matrix and combinatorial identities