The eta-inverted sphere over the rationals

TL;DR

This paper computes the motivic stable homotopy groups of the two-complete sphere spectrum over certain fields after inverting eta, revealing detailed algebraic structures in these motivic homotopy groups.

Contribution

It provides the first calculation of the eta-inverted motivic stable homotopy groups over fields of cohomological dimension at most 2, including the rationals, with explicit ring structures.

Findings

Computed motivic stable homotopy groups over specified fields

Determined the ring structure of these groups

Extended understanding of motivic homotopy over fields of low cohomological dimension

Abstract

We calculate the motivic stable homotopy groups of the two-complete sphere spectrum after inverting multiplication by the Hopf map eta over fields of cohomological dimension at most 2 with characteristic different from 2 (this includes the p-adic fields and the finite fields of odd characteristic) and the field of rational numbers; the ring structure is also determined.