Topological models for stable motivic invariants of regular number rings

Abstract

For an infinity of number rings we express stable motivic invariants in terms of topological data determined by the complex numbers, the real numbers and finite fields. We use this to extend Morel’s identification of the endomorphism ring of the motivic sphere with the Grothendieck–Witt ring of quadratic forms to deeper base schemes.

Logarithmic motivic homotopy theory

“…In particular, the endomorphism ring of the motivic sphere 1 -the unit for the tensor product -is isomorphic to the endomorphism ring of ω * (1) in logSH. Over fields, this is the Grothendieck-Witt ring of quadratic forms via Morel's theorem on the motivic π 0 of the sphere spectrum [60] ( [5] shows the same result for rings of integers in certain number fields).…”