Splitting of abelian varieties in motivic stable homotopy category

TL;DR

This paper investigates how abelian varieties decompose in the motivic stable homotopy category, providing explicit splitting results over real fields and exploring their relation to Chow-Witt correspondences.

Contribution

It establishes a general splitting result for abelian varieties in motivic stable homotopy categories and offers explicit descriptions over real fields, connecting to Chow-Witt correspondences.

Findings

Abelian varieties always split off a top-dimensional cell in $ ext{SH}(k)$.

Explicit splitting determined by motives and real points over $ ext{SH}( eal)$.

Connection between motivic splitting and Chow-Witt correspondences.

Abstract

In this paper, we discuss the motivic stable homotopy type of abelian varieties. For an abelian variety over a perfect field $k$ with a rational point, it always splits off a top-dimensional cell in motivic stable homotopy category $\text{SH}(k)$. Let $k=\mathbb{R}$, there is a concrete splitting which is determined by the motive of X and the real points $X(\mathbb{R})$ in $\text{SH}(\mathbb{R})_\Lambda$ for some $\mathbb{Z}\subset\Lambda\subset\mathbb{Q}$. We will also discuss this splitting from a viewpoint of the Chow-Witt correspondences.