Stable A^1-homotopy and R-equivalence

We prove that existence of a k-rational point can be detected by the stable A^1-homotopy category of S^1-spectra, or even a “rationalized” variant of this category.

💡 Research Summary

The paper “Stable A¹‑homotopy and R‑equivalence” establishes a striking bridge between two seemingly distant areas of arithmetic geometry: the existence of rational points on a smooth algebraic variety over a field k, and the stable A¹‑homotopy category SH(k) of S¹‑spectra (and its rationalized version). The authors prove that the mere presence of a k‑rational point can be detected entirely by homotopical data: the morphism group from the sphere spectrum (the unit object 𝟙) to the suspension spectrum Σ^∞_S¹ X₊ of the variety X, after rationalization, is non‑zero if and only if X(k) is non‑empty. Conversely, if X(k) is empty, this morphism group vanishes after tensoring with ℚ.

The argument proceeds in several stages. First, the paper reviews the foundations of Morel‑Voevodsky’s A¹‑homotopy theory, emphasizing the construction of the stable category SH(k) and the role of the S¹‑suspension functor. The authors then focus on the zeroth stable A¹‑homotopy sheaf π₀^{sA¹} and its rationalization π₀^{sA¹}⊗ℚ, showing that this sheaf encodes precisely the set of R‑equivalence classes X(k)/R. Here, R‑equivalence is the classical relation introduced by Manin: two k‑points are R‑equivalent if they can be connected by a rational curve (i.e., a morphism from ℙ¹_k whose image contains both points).

A central technical achievement is the explicit calculation of π₀^{sA¹}⊗ℚ using Milnor‑Witt K‑theory, the motivic Steenrod algebra, and the slice filtration. By identifying the unit map 𝟙 → Σ^∞_S¹ X₊ with an element of the motivic stable homotopy group, the authors construct a natural bijection