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

Let be a prime and q = pν, where p is a prime different from . We show that the –completion of the nth stable homotopy group of spheres is a summand of the –completion of the (n,0) motivic stable homotopy group of spheres over the finite field with q elements, Fq. With this, and assisted by computer calculations, we are able to explicitly compute the two-complete stable motivic stems πn,0(Fq)2 for 0 n 18 for all finite fields and π19,0(Fq)2 and π20,0(Fq)2 when q 1 mod 4 assuming Morel’s connectivity theorem for Fq holds.

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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
Received: 2 February 2016
Revised: 3 October 2016
Accepted: 12 October 2016
Published: 14 March 2017
Glen Matthew Wilson
Department of Mathematics
University of Oslo
PO Box 1053
0316 Oslo
Paul Arne Østvær
Department of Mathematics
University of Oslo
PO Box 1053
0316 Oslo