# On the motivic Segal conjecture

## TL;DR

This paper proves the motivic Segal conjecture for algebraic groups of roots of unity by developing new motivic constructions and spectral sequences, extending classical theorems into the motivic setting.

## Contribution

It introduces motivic Singer constructions and a delayed limit Adams spectral sequence to establish the conjecture for all primes.

## Key findings

- Motivic versions of Lin and Gunawardena's theorems confirmed.
- Motivic Singer constructions developed for symmetric groups and roots of unity.
- A new delayed limit Adams spectral sequence introduced.

## Abstract

We establish motivic versions of the theorems of Lin and Gunawardena, thereby confirming the motivic Segal conjecture for the algebraic group $\mu_\ell$ of $\ell$-th roots of unity, where $\ell$ is any prime. To achieve this we develop motivic Singer constructions associated to the symmetric group $S_\ell$ and to $\mu_\ell$, and introduce a delayed limit Adams spectral sequence.