Two-complete stable motivic stems over finite fields

Wilson, Glen Matthew; Østvær, Paul

doi:10.2140/agt.2017.17.1059

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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

DOI: 10.2140/agt.2017.17.1059

Abstract

Let $ℓ$ be a prime and $q = p^{ν}$ , where $p$ is a prime different from $ℓ$ . We show that the $ℓ$ –completion of the $n^{t h}$ stable homotopy group of spheres is a summand of the $ℓ$ –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 $π_{n, 0} {(F_{q})}_{2}^{\land}$ for $0 \leq n \leq 18$ for all finite fields and $π_{19, 0} {(F_{q})}_{2}^{\land}$ and $π_{20, 0} {(F_{q})}_{2}^{\land}$ when $q \equiv 1 mod 4$ assuming Morel’s connectivity theorem for $F_{q}$ 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

References

Publication

Received: 2 February 2016

Revised: 3 October 2016

Accepted: 12 October 2016

Published: 14 March 2017

Authors

Glen Matthew Wilson
Department of Mathematics University of Oslo PO Box 1053 0316 Oslo Norway

Paul Arne Østvær
Department of Mathematics University of Oslo PO Box 1053 0316 Oslo Norway

Volume 17, issue 2 (2017)