#### Volume 17, issue 2 (2017)

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Two-complete stable motivic stems over finite fields

### Glen Matthew Wilson and Paul Arne Østvær

Algebraic & Geometric Topology 17 (2017) 1059–1104
##### 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 $\left(n,0\right)$ motivic stable homotopy group of spheres over the finite field with $q$ elements, ${\mathbb{F}}_{q}$. With this, and assisted by computer calculations, we are able to explicitly compute the two-complete stable motivic stems ${\pi }_{n,0}{\left({\mathbb{F}}_{q}\right)}_{2}^{\wedge }$ for $0\le n\le 18$ for all finite fields and ${\pi }_{19,0}{\left({\mathbb{F}}_{q}\right)}_{2}^{\wedge }$ and ${\pi }_{20,0}{\left({\mathbb{F}}_{q}\right)}_{2}^{\wedge }$ when $q\equiv 1\phantom{\rule{0.2em}{0ex}}mod\phantom{\rule{0.2em}{0ex}}4$ assuming Morel’s connectivity theorem for ${\mathbb{F}}_{q}$ holds.

##### Keywords
motivic Adams spectral sequence, stable motivic stems over finite fields, computer-assisted motivic Ext group calculations
##### Mathematical Subject Classification 2010
Primary: 16-04, 14F42, 18G15, 55T15