On the motivic Segal conjecture

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