$\mathbb{C}$-Motivic Eilenberg-Mac Lane Spectra as $\mathrm{M}\mathrm{G}\mathrm{L}$-Motivic Thom Spectra and Normed ${\mathbb{G}}_m$-Hochschild Homology

We exhibit the $p$-complete $\mathbb{C}$-motivic Eilenberg-Mac Lane spectra $\mathrm{M}{\mathbb{F}}_p$ and $(\mathrm{M}\mathbb{Z}{)}_p^{\wedge }$ as ${\mathbb{E}}_{\mathbb{3}}$-${\mathrm{M}\mathrm{G}\mathrm{L}}_p^{\wedge }$-motivic Thom spectra over the motivic ${\mathbb{G}}_m$-loop spaces ${\mathrm{\Omega }}_{{\mathbb{G}}_m}\mathrm{B}\mathrm{G}\mathrm{L}$ and ${\mathrm{\Omega }}_{{\mathbb{G}}_m}\mathrm{B}\mathrm{S}\mathrm{L}$, respectively. Furthermore, for $p=2$, we compute the ${\mathrm{M}\mathrm{G}\mathrm{L}}_2^{\wedge }$-relative ${\mathbb{G}}_m$-Hochschild Homology ${\mathbb{G}}_m{\odot }^{{\mathrm{M}\mathrm{G}\mathrm{L}}_2^{\wedge }}-$, recently introduced by Bachmann-Burklund-Dundas-Østvær, of $\mathrm{M}{\mathbb{F}}_2$ and $(\mathrm{M}\mathbb{Z}{)}_2^{\wedge }$ to be $\mathrm{M}{\mathbb{F}}_2[\mathrm{B}\mathrm{G}\mathrm{L}]$ and $(\mathrm{M}\mathbb{Z}{)}_2^{\wedge }[\mathrm{B}\mathrm{S}\mathrm{L}]$ as ${\mathbb{E}}_2$-$\mathrm{M}{\mathbb{F}}_2$-algebras and ${\mathbb{E}}_2$-$(\mathrm{M}\mathbb{Z}{)}_2^{\wedge }$-algebras, respectively.