\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\A}{\mathbb{A}}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\Qbar}{\overline{\mathbb{Q}}}
\newcommand{\Fbar}{\overline{\mathbb{F}}}
\newcommand{\Kbar}{{\overline{K}}}
\newcommand{\Ql}{\overline{\mathbb{Q}}_\ell}
\newcommand{\OO}{\mathcal{O}}
\newcommand{\p}{{\mathfrak{p}}}
\newcommand{\q}{{\mathfrak{q}}}
\newcommand{\lp}{{\mathfrak{l}}}
\newcommand{\lpb}{{\bar{\mathfrak{l}}}}
\newcommand{\VV}{\mathbb{V}}
\newcommand{\LL}{\mathcal{L}}
\newcommand{\MM}{\mathcal{M}}
\newcommand{\SSS}{\mathcal{S}}
\newcommand{\rhob}{\bar\rho}
\newcommand{\epsb}{\bar\varepsilon_p}
\newcommand{\ninf}{n_\infty}
\newcommand{\Sp}{\mathrm{S}}
\DeclareMathOperator{\Gal}{Gal}
\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\SL}{SL}
\DeclareMathOperator{\SU}{SU}
\DeclareMathOperator{\Art}{Art}
\DeclareMathOperator{\Frob}{Frob}
\DeclareMathOperator{\Res}{Res}
\DeclareMathOperator{\Nm}{N}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\Cl}{Cl}
\DeclareMathOperator{\Aut}{Aut}
\DeclareMathOperator{\Stab}{Stab}
\DeclareMathOperator{\sgn}{sgn}
\DeclareMathOperator{\rk}{rank}
\DeclareMathOperator{\WD}{WD}
\DeclareMathOperator{\rec}{rec}
\DeclareMathOperator{\Ind}{Ind}
\DeclareMathOperator{\ind}{ind}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\Ext}{Ext}
\newcommand{\RR}{\mathrm{R}}
\newcommand{\isom}{\cong}
\newcommand{\cc}{\mathsf{c}}
\newcommand{\eqdef}{\coloneqq}
\DeclareMathOperator{\rd}{rd}
\DeclareMathOperator{\PGL}{PGL}
\DeclareMathOperator{\SO}{SO}
\DeclareMathOperator{\GO}{GO}
\DeclareMathOperator{\mult}{mult}
\DeclareMathOperator{\Fil}{Fil}
\DeclareMathOperator{\gr}{gr}
\DeclareMathOperator{\ad}{ad}
\newcommand{\dK}{\delta_{K/\Q}}
\newcommand{\Rsys}{\mathcal{R}}
\newcommand{\Rb}{\overline{R}}
\newcommand{\FE}{\mathbb{F}_E}
\newcommand{\mE}{\mathfrak{m}_E}
\newcommand{\Qtf}{\Q_{25}}
\newcommand{\epsf}{\bar\varepsilon_5}
\newcommand{\TFL}{\mathbf{T}}
\newcommand{\DFL}{\mathcal{D}}
\newcommand{\Gthree}{\mathcal{G}_3}
\newcommand{\Cthree}{\mathcal{C}_3}
\newcommand{\Cp}{\mathcal{C}_p}
\newcommand{\Fss}{F\textup{-ss}}
\makeatletter\renewcommand{\pod}[1]{\nobreak\if@display\mkern18mu\else\mkern8mu\fi(#1)}\makeatother