Two-complete stable motivic stems over finite fields

TL;DR

This paper establishes a connection between classical stable homotopy groups and motivic stable homotopy groups over finite fields, enabling explicit computations of motivic stems up to degree 20.

Contribution

It demonstrates that the $ ext{ell}$-completion of classical stable homotopy groups is a summand of motivic groups over finite fields and provides explicit calculations for these motivic stems.

Findings

Computed $ ext{2}$-complete stable motivic stems $oldsymbol{_{n,0}(F_q)}$ for $0 \u2264 n \u2264 18$

Extended computations to $oldsymbol{_{19,0}(F_q)}$ and $oldsymbol{_{20,0}(F_q)}$ under certain conditions

Established a structural link between classical and motivic homotopy groups over finite fields

Abstract

Let $\ell$ be a prime and $q = p^{\nu}$ where $p$ is a prime different from $\ell$. We show that the $\ell$-completion of the $n$th stable homotopy group of spheres is a summand of the $\ell$-completion of the $(n, 0)$ motivic stable homotopy group of spheres over the finite field with $q$ elements $F_q$. With this, and assisted by computer calculations, we are able to explicitly compute the two-complete stable motivic stems $\pi_{n, 0}(F_q)^{\wedge}_2$ for $0\leq n\leq 18$. Additionally, we compute $\pi_{19, 0}(F_q)^{\wedge}_2$ and $\pi_{20, 0}(F_q)^{\wedge}_2$ when $q \equiv 1 \bmod 4$ assuming Morel's connectivity theorem for $F_q$ holds.