\(\newcommand{\AA}{\mathcal{A}} \newcommand{\CH}{\mathcal{H}} \newcommand{\CL}{\mathcal{L}} \newcommand{\CO}{\mathcal{O}} \newcommand{\FF}{\mathbb{F}} \newcommand{\NN}{\mathbb{N}} \newcommand{\CC}{\mathbb{C}} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\RR}{\mathbb{R}} \newcommand{\ZZ}{\mathbb{Z}}\) \(\DeclareMathOperator{\an}{an} \DeclareMathOperator{\char}{char} \DeclareMathOperator{\colim}{colim} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Id}{Id} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\Isom}{Isom} \DeclareMathOperator{\Gal}{Gal} \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\Ob}{Ob} \DeclareMathOperator{\Res}{Res} \DeclareMathOperator{\sh}{sh} \DeclareMathOperator{\SL}{SL} \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\Sym}{Sym}\)
Proposition 3.6.   \(F\) is the inverse of the Frobenious from the action of \(\Gal(\overline{\FF}_p/\FF_p)\) on

\[\widetilde{H}^1(M_n \otimes \overline{\FF}_p, \Sym^k(R^1 f_{n*} \underline{\ZZ}_{\ell})).\]

Over \(\FF_p\), we have \(T_p = F + I_p^* V\) and \(FV = VF = p^{k+1}\).