\newcommand{\Z}{{\bf{Z}}}
\newcommand{\Q}{{\bf{Q}}}
\newcommand{\C}{{\bf{C}}}
\newcommand{\pr}{{\mathbf{P}}}
\newcommand{\A}{{\mathbb{A}}}
\newcommand{\R}{{\mathbf R}}
\newcommand{\OO}{{\mathcal O}}
\newcommand{\G}{\mathbb{G}_{\text{m}}}
\newcommand{\from}{{\colon}}
\newcommand{\injects}{\hookrightarrow}
\newcommand{\surjects}{\twoheadrightarrow}
\newcommand{\isom}{\cong}
\newcommand{\eps}{\varepsilon}
\DeclareMathOperator{\ord}{ord} \DeclareMathOperator{\Div}{Div}
\DeclareMathOperator{\divi}{div} \DeclareMathOperator{\Pic}{Pic}
\DeclareMathOperator{\End}{End} \DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\Gal}{Gal}\DeclareMathOperator{\dR}{dR}
\DeclareMathOperator{\GL}{GL}\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\an}{an}\DeclareMathOperator{\Aut}{Aut}
\DeclareMathOperator{\rk}{rk}\DeclareMathOperator{\Sym}{Sym}
\DeclareMathOperator{\RR}{R}\DeclareMathOperator{\tr}{Tr}
\DeclareMathOperator{\Tr}{Tr}
\DeclareMathOperator{\Ind}{Ind}\DeclareMathOperator{\PGL}{PGL}
\DeclareMathOperator{\SL}{SL}
\DeclareMathOperator{\coker}{coker}


\newtheorem{Theorem}{Theorem}[section]
\newtheorem{lemma}[Theorem]{Lemma}
\newtheorem{Cor}[Theorem]{Corollary}
\newtheorem{prop}[Theorem]{Proposition}
\newtheorem{conj}[Theorem]{Conjecture}

\makeatletter

\def\eqnalign{%
   \stepcounter{equation}%
   \def\@currentlabel{\p@equation\theequation}%
   \global\@eqnswtrue
   \m@th
   \global\@eqcnt\z@
   \tabskip\@centering
   \let\\\@eqncr
   $$\everycr{}\halign to\displaywidth\bgroup
       \global\@eqcnt\@ne
         \hskip\@centering$\displaystyle\tabskip\z@skip{##}$
      &\global\@eqcnt\tw@
         $\displaystyle{##}$\hfil\tabskip\@centering
      &\global\@eqcnt\thr@@ \hb@xt@\z@\bgroup\hss##\egroup
         \tabskip\z@skip
      \cr
}
\def\endeqnalign{%
      \@@eqncr
      \egroup
      \global\advance\c@equation\m@ne
   $$\@ignoretrue
}
\@namedef{eqnalign*}{\def\@eqncr{\nonumber\@seqncr}\eqnalign}
\@namedef{endeqnalign*}{\nonumber\endeqnalign}

\def\eqnlines{%
   \stepcounter{equation}%
   \def\@currentlabel{\p@equation\theequation}%
   \global\@eqnswtrue
   \m@th
   \global\@eqcnt\z@
   \tabskip\@centering
   \let\\\@eqncr
   $$\everycr{}\halign to\displaywidth\bgroup
       \global\@eqcnt\tw@
         \hfil$\displaystyle{##}$\hfil\tabskip\@centering
      &\global\@eqcnt\thr@@ \hb@xt@\z@\bgroup\hss##\egroup
         \tabskip\z@skip
      \cr
}
\def\endeqnlines{%
      \@@eqncr
      \egroup
      \global\advance\c@equation\m@ne
   $$\@ignoretrue
}
\@namedef{eqnlines*}{\def\@eqncr{\nonumber\@seqncr}\eqnlines}
\@namedef{endeqnlines*}{\nonumber\endeqnlines}

\makeatother



\theoremstyle{definition}
\newtheorem{defn}[Theorem]{Definition}
\newtheorem{question}[Theorem]{Question}
\newtheorem{exmp}[Theorem]{Example}
\newtheorem{examples}[Theorem]{Examples}
\newtheorem{rmk}[Theorem]{Remark}


%\theoremstyle{remark}

%\newtheorem{remarks}[Theorem]{Remarks}



%\theoremstyle{definition} \newtheorem{definition}{Definition}

%\theoremstyle{plain}% default
%\newtheorem{Theorem}{Theorem}[section]
%\newtheorem{lemma}[Theorem]{Lemma}
%\newtheorem{prop}[Theorem]{Proposition}
%\newtheorem{Cor}[Theorem]{Corollary}
%\theoremstyle{definition}
%\newtheorem{defn}{Definition}[section]
%\newtheorem{conj}{Conjecture}[section]
%\newtheorem{exmp}{Example}[section]
%\newtheorem{rmk}{Remark}[section]


%with arguments
\newcommand{\abs}[1]{\left\lvert #1 \right\rvert}
\newcommand{\class}[1]{\left< #1 \right>}
\newcommand{\set}[1]{\left\{ #1 \right\}}
\newcommand{\norm}[1]{\left\lVert #1 \right\rVert}
\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor}
\newcommand{\ceil}[1]{\left\lceil #1\right\rceil}
\newcommand{\un}[1]{\underline{#1}}
\newcommand{\tbt}[4]{\left(\begin{matrix}#1 & #2\\#3 & #4\end{matrix}\right)}
\newcommand{\tbo}[2]{\left(\begin{matrix}#1\\ #2\end{matrix}\right)}
\newcommand{\ilim}{\mathop{\varinjlim}\limits}

%paper-specific

\newcommand{\X}{{\mathfrak{X}}}
\newcommand{\nr}{{\text{nr}}}
\newcommand{\Et}{{\text{\'et}}}
\newcommand{\E}{{\mathcal{E}}}
\newcommand{\T}{{\mathbf{T}}}
\newcommand{\F}{{\mathbf{F}}}
\newcommand{\FF}{{\overline{\mathbf{F}}}}
\newcommand{\Fc}{{\mathcal{F}}}
\newcommand{\gp}{{\mathfrak{p}}}
\newcommand{\gP}{{\mathfrak{P}}}
\newcommand{\FS}{{\mathfrak{S}}}
\newcommand{\QQ}{\overline{\mathbf{Q}}}
\DeclareMathOperator{\Sh}{Sh}
\DeclareMathOperator{\Frob}{Frob}
\DeclareMathOperator{\Ad}{Ad}
\DeclareMathOperator{\Spm}{Spm}
\DeclareMathOperator{\rec}{rec}
\DeclareMathOperator{\ab}{ab}
\DeclareMathOperator{\N}{N}
\DeclareMathOperator{\univ}{univ}
\DeclareMathOperator{\fin}{fin}
\DeclareMathOperator{\Art}{Art}
\DeclareMathOperator{\Stab}{Stab}
\DeclareMathOperator{\Id}{Id}
\DeclareMathOperator{\Sp}{Sp}
\DeclareMathOperator{\Res}{Res}
\DeclareMathOperator{\St}{St}
\DeclareMathOperator{\PSL}{PSL}
\DeclareMathOperator{\Jac}{Jac}
\DeclareMathOperator{\sgn}{sgn}
\DeclareMathOperator{\Ig}{Ig}
\DeclareMathOperator{\Ver}{Ver}
\DeclareMathOperator{\JL}{JL}
\DeclareMathOperator{\cInd}{c-Ind}
\DeclareMathOperator{\red}{red}
\DeclareMathOperator{\cond}{cond}
%%%%