The $\mathbb{C}$-motivic Adams Spectral Sequence for $s\leq5$

TL;DR

This paper analyzes the $ au^n$-torsion in the early stages of the $C$-motivic Adams spectral sequence, revealing torsion patterns and filtration bounds using advanced techniques.

Contribution

It determines the $ au^n$-torsion in the first five lines of the $E_2$ page, providing new insights into torsion behavior and filtration bounds in the $C$-motivic setting.

Findings

Elements are either $ au^1$-torsion or $ au$-free in the specified range.

$ au^n$-torsion appears only at filtration ≥ $2n+2$.

Evidence suggests a possible $3n$ bound for the appearance of $ au^n$-torsion.

Abstract

We determine the $\tau^n$-torsion in the first 5-lines of the $E_2$ page of the $\mathbb{C}$-motivic Adams spectral sequence using the techniques of Burklund-Xu. In particular, every element in this range is either $\tau^1$-torsion or $\tau$-free. We also show that $\tau^n$-torsion elements can appear only in Adams filtration at least $2n+2$ and give further evidence of a possible $3n$ bound.