\(\newcommand{\AA}{\mathcal{A}} \newcommand{\CH}{\mathcal{H}} \newcommand{\CL}{\mathcal{L}} \newcommand{\CO}{\mathcal{O}} \newcommand{\FF}{\mathbb{F}} \newcommand{\NN}{\mathbb{N}} \newcommand{\CC}{\mathbb{C}} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\RR}{\mathbb{R}} \newcommand{\ZZ}{\mathbb{Z}}\) \(\DeclareMathOperator{\an}{an} \DeclareMathOperator{\char}{char} \DeclareMathOperator{\colim}{colim} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Id}{Id} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\Isom}{Isom} \DeclareMathOperator{\Gal}{Gal} \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\Ob}{Ob} \DeclareMathOperator{\Res}{Res} \DeclareMathOperator{\sh}{sh} \DeclareMathOperator{\SL}{SL} \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\Sym}{Sym}\)
Theorem 3.7.   Let \(K_{n, \ell}\) be the largest unramified extension of \(\overline{\QQ}\) away from \(n\) and \(\ell\), \(\varphi_p\) a relative Frobenious to \(p\) in \(\Gal(K_{n, \ell}/\QQ)\), \(F\) the endomorphism \(\varphi_p^{-1}\) of \(^k_n W_{\ell}\), and \(V\) the transpose of \(F\). Then

\[T_p = F + I_p^* V, \qquad FV = p^{k+1},\] \[1 - T_p + p R_p X^2 = (1-FX)(1+I_p^*VX).\]