Computing n-th roots in SL2 and Fibonacci polynomials

Abstract

Let k be a field of characteristic ≠2. In this paper, we study squares, cubes and their products in split and anisotropic groups of type A1. In the split case, we show that computing n-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-th powers, and conjugacy classes which are n-th powers, in SL2(Fq) when n is a prime or n=4. We also extend the already known Waring type result for SL2(Fq), that every element of SL2(Fq) is a product of two squares, to SL2(k) for an arbitrary k. For anisotropic groups of type A1, namely SL1(Q) where Q is a quaternion division algebra, we prove that when 2 is a square in k, every element of SL1(Q) is a product of two squares if and only if −1 is a square in SL1(Q).