kbuzzard/macros.tex Secret

## macros.tex
\documentclass[english]{amsart}


\usepackage{ulem}
\normalem
\usepackage{graphicx}

\usepackage{amsmath}

\usepackage{amssymb}


    \DeclareFontFamily{U}{wncy}{}
    \DeclareFontShape{U}{wncy}{m}{n}{<->wncyr10}{}
    \DeclareSymbolFont{mcy}{U}{wncy}{m}{n}
    \DeclareMathSymbol{\Sha}{\mathord}{mcy}{"58}


\newcommand{\sepa}{\operatorname{sep}}
\newcommand{\ca}{\mathcal{a}}
\newcommand{\height}{\operatorname{ht}}
\usepackage{amscd}
\usepackage[all,cmtip,line]{xy}

\newcommand{\Zhat}{\widehat{\Z}}
\newcommand{\SO}{\operatorname{SO}}

\newcommand{\ave}{\mathrm{Average}}
\newcommand{\aveup}{\overline{\mathrm{Average}}}
\newcommand{\avedown}{\underline{\mathrm{Average}}}
\newcommand{\Sel}{\mathrm{Sel}}

\newcommand{\Abar}{\overline{A}}
\newcommand{\Bbar}{\overline{B}}
\newcommand{\cl}{\operatorname{cl}}
\newcommand{\codim}{\operatorname{codim}}
\newcommand{\Pic}{\operatorname{Pic}}
\newcommand{\NS}{\operatorname{NS}}
\newcommand{\Loc}{\operatorname{Loc}}
\newcommand{\lambdaH}{{}^\lambda H}
\newcommand{\LH}{{}^L H}
\newcommand{\lambdaG}{{}^\lambda G}
\newcommand{\uK}{\underline{K}}
\newcommand{\uf}{\underline{f}}
\newcommand{\upi}{\underline{\pi}}

\newcommand{\supp}{\operatorname{supp}}
\newcommand{\Hecke}{\mathcal{H}ecke}
\newcommand{\That}{\widehat{T}}
\newcommand{\Ghat}{\widehat{G}}
\newcommand{\contains}{\ni}
\newcommand{\Xbar}{\overline{X}}
\newcommand{\ur}{{\operatorname{ur}}}
\newcommand{\Ob}{\operatorname{Ob}}
\newcommand{\Bun}{\operatorname{Bun}}
\newcommand{\Sets}{\operatorname{Sets}}
\newcommand{\perv}{\operatorname{Perv}}
\newcommand{\Groups}{\operatorname{Groups}}
\newcommand{\Sh}{\operatorname{Sh}}
\newcommand{\im}{\operatorname{im}}
\newcommand{\Sch}{\operatorname{Sch}}
\newcommand{\id}{\operatorname{id}}
\newcommand{\Mor}{\operatorname{Mor}}
\newcommand{\Zar}{{\operatorname{Zar}}}
\newcommand{\an}{{\operatorname{an}}}
\newcommand{\PreSh}{\operatorname{PreSh}}
\newcommand{\Map}{\operatorname{Map}}
\newcommand{\open}{{\operatorname{open}}}


\newcommand{\lonto}{\longrightarrow \hspace{-1.44em} \longrightarrow}
\newcommand{\onto}{\twoheadrightarrow}
\newcommand{\ra}{\rightarrow}
\newcommand{\lra}{\longrightarrow}
\newcommand{\la}{\leftarrow}
\newcommand{\lla}{\longleftarrow}
\newcommand{\into}{\hookrightarrow}
\newcommand{\linto}{\hookleftarrow}
\newcommand{\mt}{\mapsto}
\newcommand{\lmt}{\longmapsto}
\newcommand{\ua}{\uparrow}
\newcommand{\da}{\downarrow}
\newcommand{\iso}{\stackrel{\sim}{\ra}}
\newcommand{\liso}{\stackrel{\sim}{\lra}}
\newcommand{\To}{\longrightarrow}
\newcommand{\isoto}{\stackrel{\sim}{\To}}


\newcommand{\BrMod}{\operatorname{BrMod}}
\newcommand{\dR}{\mathrm{dR}}
\newcommand{\textB}{\mathrm{B}}
\newcommand{\Bcris}{\textB_{\cris}}
 \newcommand{\BdR}{\textB_{\dR}}
\newcommand{\BHT}{\textB_{\HT}}
\newcommand{\Bst}{\textB_{\st}}
\newcommand{\Dcris}{\textD_{\cris}}
\newcommand{\DdR}{\textD_{\dR}}
\newcommand{\DHT}{\textD_{\HT}}
\newcommand{\Dst}{\textD_{\st}}


\newcommand{\pfbegin}{{{\em Proof:}\;}}
\newcommand{\pfend}{$\Box$ \medskip}
\newcommand{\mat}[4]{\left( \begin{array}{cc} {#1} & {#2} \\ {#3} & {#4}
\end{array} \right)}
\newcommand{\galois}[1]{\Gal ( \overline{ {#1} }/ {#1})}
\newcommand{\smat}[4]{{\mbox{\scriptsize $\mat{{#1}}{{#2}}{{#3}}{{#4}}$}}}
\newlength{\ownl}
\newcommand{\stacktwo}[2]{\settowidth{\ownl}{\scriptsize{${#1}$}}
\parbox[t]{1\ownl}{{\scriptsize{${#1}$}} \\[-.9ex] {\scriptsize{${#2}$}}}}
\newcommand{\stackthree}[3]{\settowidth{\ownl}{\scriptsize{${#1}$}}
\parbox[t]{1\ownl}{{\scriptsize{${#1}$}} \\[-.9ex] {\scriptsize{${#2}$}}
\\[-.9ex] {\scriptsize{${#3}$}}}}
\newcommand{\<}{\langle}
\newcommand{\norm}{{\mbox{\bf N}}}
\newcommand{\ndiv}{{\mbox{$\not| $}}}
\newcommand{\hotimes}{\widehat{\otimes}}

\newcommand{\obs}{{\operatorname{obs}}}
\newcommand{\ad}{{\operatorname{ad}\,}}
\newcommand{\Aff}{{\operatorname{Aff}}}
\newcommand{\Ann}{{\operatorname{Ann}\,}}
\newcommand{\Art}{{\operatorname{Art}\,}}
\newcommand{\Aut}{{\operatorname{Aut}\,}}
\newcommand{\uAut}{\underline{\operatorname{Aut}}}
\newcommand{\BC}{{\operatorname{BC}\,}}
\newcommand{\Br}{{\operatorname{Br}\,}}
\newcommand{\chara}{{\operatorname{char}\,}}
\newcommand{\cind}{{\operatorname{c-Ind}\,}}
\newcommand{\Cl}{{\operatorname{Cl}}}
\newcommand{\coker}{{\operatorname{coker}\,}}
\newcommand{\conju}{{\operatorname{conj}}}
\newcommand{\Cusp}{{\operatorname{Cusp}\,}}
\newcommand{\DC}{{\operatorname{DC}}}
\newcommand{\Def}{{\operatorname{Def}}}
\newcommand{\diag}{{\operatorname{diag}}}
\newcommand{\End}{{\operatorname{End}\,}}
\newcommand{\ep}{{\operatorname{ep}\,}}
\newcommand{\es}{\emptyset}
\newcommand{\Ext}{{\operatorname{Ext}\,}}
\newcommand{\Fil}{{\operatorname{Fil}\,}}
\newcommand{\Fitt}{{\operatorname{Fitt}\,}}
\newcommand{\Fix}{{\operatorname{Fix}}}
\newcommand{\Frob}{{\operatorname{Frob}}}
\newcommand{\Fr}{{\operatorname{Fr}}}
\newcommand{\Gal}{{\operatorname{Gal}\,}}
\newcommand{\uGal}{\underline{\operatorname{Gal}}}
\newcommand{\gen}{{\operatorname{gen}}}
\newcommand{\gr}{{\operatorname{gr}\,}}
\newcommand{\Groth}{{\operatorname{Groth}\,}}
\newcommand{\hdg}{{\operatorname{hdg}}}
\newcommand{\Hom}{{\operatorname{Hom}\,}}
\newcommand{\RHom}{{\operatorname{RHom}\,}}
\newcommand{\CHom}{{\CH om\,}}
\newcommand{\ICP}{{\operatorname{IPC}}}
\newcommand{\Id}{{\operatorname{Id}}}
\renewcommand{\Im}{{\operatorname{Im}\,}}
\newcommand{\Ind}{{\operatorname{Ind}\,}}
\newcommand{\ind}{{\operatorname{ind}}}
\newcommand{\inv}{{\operatorname{inv}\,}}
\newcommand{\Irr}{{\operatorname{Irr}}}
\newcommand{\Isom}{{\operatorname{Isom}\,}}
\newcommand{\Iw}{{\operatorname{Iw}}}
\newcommand{\JL}{{\operatorname{JL}\,}}
\newcommand{\Lie}{{\operatorname{Lie}\,}}
\newcommand{\nind}{{\operatorname{n-Ind}\,}}
\newcommand{\nred}{{\operatorname{n-red}}}
\newcommand{\Out}{{\operatorname{Out}\,}}
\newcommand{\PC}{{\operatorname{PC}}}
\newcommand{\pic}{{\operatorname{PIC}}}
\newcommand{\pr}{{\operatorname{pr}\,}}
\renewcommand{\Re}{{\operatorname{Re}\,}}
\newcommand{\rec}{{\operatorname{rec}}}
\newcommand{\redu}{{\operatorname{red}}}
\newcommand{\Red}{{\operatorname{Red}}}
\newcommand{\Rep}{{\operatorname{Rep}}}
\newcommand{\REP}{{\operatorname{GG}}}
\newcommand{\GrG}{{\operatorname{Rep}}}
\newcommand{\res}{{\operatorname{res}}}
\newcommand{\Res}{{\operatorname{Res}}}
\newcommand{\rk}{{\operatorname{rk}\,}}
\newcommand{\RS}{{\operatorname{RS}}}
\newcommand{\DR}{{\operatorname{DR}}}
\newcommand{\WD}{{\operatorname{WD}}}
\newcommand{\Sd}{{\operatorname{Sd}}}
\newcommand{\sgn}{{\operatorname{sgn}\,}}
\newcommand{\Spec}{{\operatorname{Spec}\,}}
\newcommand{\Spf}{{\operatorname{Spf}\,}}
\newcommand{\Spp}{{\operatorname{Sp}\,}}
\newcommand{\Symm}{{\operatorname{Symm}\,}}
\newcommand{\sym}{{\operatorname{Sym}\,}}
\newcommand{\tr}{{\operatorname{tr}\,}}
\newcommand{\Twist}{{\operatorname{Twist}\,}}
\newcommand{\val}{{\operatorname{val}\,}}
\newcommand{\vol}{{\operatorname{vol}\,}}
\newcommand{\WDRep}{{\operatorname{WDRep}}}
\newcommand{\wt}{{\operatorname{wt}}}


\newcommand{\GL}{\operatorname{GL}}
\newcommand{\GSp}{\operatorname{GSp}}
\newcommand{\Gdual}{{\hat{G}}}
\newcommand{\Gm}{{\mathbb{G}_m}}
\newcommand{\LG}{{{}^{L}G}}
\newcommand{\PGL}{\operatorname{PGL}}
\newcommand{\PSL}{\operatorname{PSL}}
\newcommand{\SL}{\operatorname{SL}}
\newcommand{\Sp}{\operatorname{Sp}}
\newcommand{\Tdual}{{\hat{T}}}

\newcommand{\APHT}{{\operatorname{APHT}}}
\newcommand{\AV}{{\operatorname{AV}}}
\newcommand{\FP}{{\operatorname{FP}}}
\newcommand{\Fsemis}{{\operatorname{F-ss}}}
\newcommand{\GB}{{\operatorname{GB}}}
\newcommand{\LA}{{\operatorname{LA}}}
\newcommand{\PHT}{{\operatorname{PHT}}}
\newcommand{\SC}{{\operatorname{SC}}}
\newcommand{\Tam}{{\operatorname{Tam}}}
\newcommand{\ab}{{\operatorname{ab}}}
\newcommand{\ac}{{\operatorname{ac}}}
\newcommand{\alg}{{\operatorname{alg}}}
\newcommand{\cris}{{\operatorname{cris}}}
\newcommand{\crord}{{\operatorname{cr-ord}}}
\newcommand{\cts}{{\operatorname{cts}}}
\newcommand{\der}{{\operatorname{der}}}
\newcommand{\disc}{{\operatorname{disc}}}
\newcommand{\dr}{{\operatorname{DR}}}
\newcommand{\et}{{\operatorname{et}}}
\newcommand{\jl}{{\operatorname{JL}}}
\newcommand{\loc}{{\operatorname{loc}}}
\newcommand{\new}{{\operatorname{new}}}
\newcommand{\noeth}{{\operatorname{noeth}}}
\newcommand{\nr}{{\operatorname{nr}}}
\newcommand{\old}{{\operatorname{old}}}
\newcommand{\opp}{{\operatorname{op}}}
\newcommand{\op}{{\operatorname{op}}}
\newcommand{\pnr}{{\operatorname{p-nr}}}
\newcommand{\red}{{\operatorname{red}}}
\newcommand{\reg}{{\operatorname{reg}}}
\newcommand{\rig}{{\operatorname{rig}}}
\newcommand{\semis}{{\operatorname{ss}}}
\newcommand{\ssord}{{\operatorname{ss-ord}}}
\newcommand{\st}{{\operatorname{st}}}
\newcommand{\tf}{{\operatorname{tf}}}
\newcommand{\topo}{{\operatorname{top}}}
\newcommand{\tor}{{\operatorname{tor}}}
\newcommand{\triv}{{\operatorname{triv}}}
\newcommand{\univ}{{\operatorname{univ}}}

\newcommand{\A}{{\mathbb{A}}}
\newcommand{\B}{{\mathbb{B}}}
\newcommand{\C}{{\mathbb{C}}}
\newcommand{\D}{{\mathbb{D}}}
\newcommand{\E}{{\mathbb{E}}}
\newcommand{\F}{{\mathbb{F}}}
\newcommand{\G}{{\mathbb{G}}}
\newcommand{\HH}{{\mathbb{H}}}
\newcommand{\I}{{\mathbb{I}}}
\newcommand{\J}{{\mathbb{J}}}
\newcommand{\K}{{\mathbb{K}}}
\newcommand{\LL}{{\mathbb{L}}}
\newcommand{\M}{{\mathbb{M}}}
\newcommand{\N}{{\mathbb{N}}}
\newcommand{\OO}{{\mathbb{O}}}
\newcommand{\PP}{{\mathbb{P}}}
\newcommand{\Q}{{\mathbb{Q}}}
\newcommand{\R}{{\mathbb{R}}}
\newcommand{\mathbbS}{{\mathbb{S}}}
\newcommand{\T}{{\mathbb{T}}}
\newcommand{\U}{{\mathbb{U}}}
\newcommand{\V}{{\mathbb{V}}}
\newcommand{\W}{{\mathbb{W}}}
\newcommand{\X}{{\mathbb{X}}}
\newcommand{\Y}{{\mathbb{Y}}}
\newcommand{\Z}{{\mathbb{Z}}}

\newcommand{\CA}{{\mathcal{A}}}
\newcommand{\CB}{{\mathcal{B}}}
\newcommand{\CC}{{\mathcal{C}}}
\newcommand{\calD}{{\mathcal{D}}}
\newcommand{\CE}{{\mathcal{E}}}
\newcommand{\CF}{{\mathcal{F}}}
\newcommand{\CG}{{\mathcal{G}}}
\newcommand{\CH}{{\mathcal{H}}}
\newcommand{\CI}{{\mathcal{I}}}
\newcommand{\CJ}{{\mathcal{J}}}
\newcommand{\CK}{{\mathcal{K}}}
\newcommand{\CL}{{\mathcal{L}}}
\newcommand{\CM}{{\mathcal{M}}}
\newcommand{\CN}{{\mathcal{N}}}
\newcommand{\CO}{{\mathcal{O}}}
\newcommand{\CP}{{\mathcal{P}}}
\newcommand{\CQ}{{\mathcal{Q}}}
\newcommand{\CR}{{\mathcal{R}}}
\newcommand{\CS}{{\mathcal{S}}}
\newcommand{\CT}{{\mathcal{T}}}
\newcommand{\CU}{{\mathcal{U}}}
\newcommand{\CV}{{\mathcal{V}}}
\newcommand{\CW}{{\mathcal{W}}}
\newcommand{\CX}{{\mathcal{X}}}
\newcommand{\CY}{{\mathcal{Y}}}
\newcommand{\CZ}{{\mathcal{Z}}}

\newcommand{\cA}{\mathcal{A}}
\newcommand{\cB}{\mathcal{B}}
\newcommand{\cC}{\mathcal{C}}
\newcommand{\cD}{\mathcal{D}}
\newcommand{\cE}{\mathcal{E}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\cG}{\mathcal{G}}
\newcommand{\cH}{\mathcal{H}}
\newcommand{\cI}{\mathcal{I}}
\newcommand{\cJ}{\mathcal{J}}
\newcommand{\cK}{\mathcal{K}}
\newcommand{\cM}{\mathcal{M}}
\newcommand{\cN}{\mathcal{N}}
\newcommand{\cO}{\mathcal{O}}
\newcommand{\cP}{\mathcal{P}}
\newcommand{\cQ}{\mathcal{Q}}
\newcommand{\cR}{\mathcal{R}}
\newcommand{\cS}{\mathcal{S}}
\newcommand{\cT}{\mathcal{T}}
\newcommand{\cU}{\mathcal{U}}
\newcommand{\cV}{\mathcal{V}}
\newcommand{\cW}{\mathcal{W}}
\newcommand{\cX}{\mathcal{X}}
\newcommand{\cY}{\mathcal{Y}}
\newcommand{\cZ}{\mathcal{Z}}

\newcommand{\gA}{{\mathfrak{A}}}
\newcommand{\gB}{{\mathfrak{B}}}
\newcommand{\gC}{{\mathfrak{C}}}
\newcommand{\gD}{{\mathfrak{D}}}
\newcommand{\gE}{{\mathfrak{E}}}
\newcommand{\gF}{{\mathfrak{F}}}
\newcommand{\gG}{{\mathfrak{G}}}
\newcommand{\gH}{{\mathfrak{H}}}
\newcommand{\gI}{{\mathfrak{I}}}
\newcommand{\gJ}{{\mathfrak{J}}}
\newcommand{\gK}{{\mathfrak{K}}}
\newcommand{\gL}{{\mathfrak{L}}}
\newcommand{\gM}{{\mathfrak{M}}}
\newcommand{\gN}{{\mathfrak{N}}}
\newcommand{\gO}{{\mathfrak{O}}}
\newcommand{\gP}{{\mathfrak{P}}}
\newcommand{\gQ}{{\mathfrak{Q}}}
\newcommand{\gR}{{\mathfrak{R}}}
\newcommand{\gS}{{\mathfrak{S}}}
\newcommand{\gT}{{\mathfrak{T}}}
\newcommand{\gU}{{\mathfrak{U}}}
\newcommand{\gV}{{\mathfrak{V}}}
\newcommand{\gW}{{\mathfrak{W}}}
\newcommand{\gX}{{\mathfrak{X}}}
\newcommand{\gY}{{\mathfrak{Y}}}
\newcommand{\gZ}{{\mathfrak{Z}}}

\newcommand{\ga}{{\mathfrak{a}}}
\newcommand{\gb}{{\mathfrak{b}}}
\newcommand{\gc}{{\mathfrak{c}}}
\newcommand{\gd}{{\mathfrak{d}}}
\newcommand{\goe}{{\mathfrak{e}}}
\newcommand{\gf}{{\mathfrak{f}}}
\newcommand{\gog}{{\mathfrak{g}}}
\newcommand{\gh}{{\mathfrak{h}}}
\newcommand{\gi}{{\mathfrak{i}}}
\newcommand{\gj}{{\mathfrak{j}}}
\newcommand{\gk}{{\mathfrak{k}}}
\newcommand{\gl}{{\mathfrak{l}}}
\newcommand{\gm}{{\mathfrak{m}}}
\newcommand{\gn}{{\mathfrak{n}}}
\newcommand{\go}{{\mathfrak{o}}}
\newcommand{\gp}{{\mathfrak{p}}}
\newcommand{\gq}{{\mathfrak{q}}}
\newcommand{\gor}{{\mathfrak{r}}}
\newcommand{\gs}{{\mathfrak{s}}}
\newcommand{\gt}{{\mathfrak{t}}}
\newcommand{\gu}{{\mathfrak{u}}}
\newcommand{\gv}{{\mathfrak{v}}}
\newcommand{\gw}{{\mathfrak{w}}}
\newcommand{\gx}{{\mathfrak{x}}}
\newcommand{\gy}{{\mathfrak{y}}}
\newcommand{\gz}{{\mathfrak{z}}}

\newcommand{\barA}{\overline{{A}}}
\newcommand{\barB}{\overline{{B}}}
\newcommand{\barC}{\overline{{C}}}
\newcommand{\barD}{\overline{{D}}}
\newcommand{\barE}{\overline{{E}}}
\newcommand{\barF}{\overline{{F}}}
\newcommand{\barG}{\overline{{G}}}
\newcommand{\barH}{\overline{{H}}}
\newcommand{\barI}{\overline{{I}}}
\newcommand{\barJ}{\overline{{J}}}
\newcommand{\barK}{\overline{{K}}}
\newcommand{\barL}{\overline{{L}}}
\newcommand{\barM}{\overline{{M}}}
\newcommand{\barN}{\overline{{N}}}
\newcommand{\barO}{\overline{{O}}}
\newcommand{\barP}{\overline{{P}}}
\newcommand{\barQ}{\overline{{Q}}}
\newcommand{\barR}{\overline{{R}}}
\newcommand{\barS}{\overline{{S}}}
\newcommand{\barT}{\overline{{T}}}
\newcommand{\barU}{\overline{{U}}}
\newcommand{\barV}{\overline{{V}}}
\newcommand{\barW}{\overline{{W}}}
\newcommand{\barX}{\overline{{X}}}
\newcommand{\barY}{\overline{{Y}}}
\newcommand{\barZ}{\overline{{Z}}}

\newcommand{\barAA}{\overline{{\A}}}
\newcommand{\barBB}{\overline{{\B}}}
\newcommand{\barCC}{\overline{{\C}}}
\newcommand{\barDD}{\overline{{\D}}}
\newcommand{\barEE}{\overline{{\E}}}
\newcommand{\barFF}{\overline{{\F}}}
\newcommand{\barGG}{\overline{{\G}}}
\newcommand{\barHH}{\overline{{\HH}}}
\newcommand{\barII}{\overline{{\I}}}
\newcommand{\barJJ}{\overline{{\J}}}
\newcommand{\barKK}{\overline{{\K}}}
\newcommand{\barLL}{\overline{{\LL}}}
\newcommand{\barMM}{\overline{{\M}}}
\newcommand{\barNN}{\overline{{\N}}}
\newcommand{\barOO}{\overline{{\OO}}}
\newcommand{\barPP}{\overline{{\PP}}}
\newcommand{\barQQ}{\overline{{\Q}}}
\newcommand{\barRR}{\overline{{\R}}}
\newcommand{\barSS}{\overline{{\SS}}}
\newcommand{\barTT}{\overline{{\T}}}
\newcommand{\barUU}{\overline{{\U}}}
\newcommand{\barVV}{\overline{{\V}}}
\newcommand{\barWW}{\overline{{\W}}}
\newcommand{\barXX}{\overline{{\X}}}
\newcommand{\barYY}{\overline{{\Y}}}
\newcommand{\barZZ}{\overline{{\Z}}}

\newcommand{\bara}{\overline{{a}}}
\newcommand{\barb}{\overline{{b}}}
\newcommand{\barc}{\overline{{c}}}
\newcommand{\bard}{\overline{{d}}}
\newcommand{\bare}{\overline{{e}}}
\newcommand{\barf}{\overline{{f}}}
\newcommand{\barg}{\overline{{g}}}
\newcommand{\barh}{\overline{{h}}}
\newcommand{\bari}{\overline{{i}}}
\newcommand{\barj}{\overline{{j}}}
\newcommand{\bark}{\overline{{k}}}
\newcommand{\barl}{\overline{{l}}}
\newcommand{\barm}{\overline{{m}}}
\newcommand{\barn}{\overline{{n}}}
\newcommand{\baro}{\overline{{o}}}
\newcommand{\barp}{\overline{{p}}}
\newcommand{\barq}{\overline{{q}}}
\newcommand{\barr}{\overline{{r}}}
\newcommand{\bars}{\overline{{s}}}
\newcommand{\bart}{\overline{{t}}}
\newcommand{\baru}{\overline{{u}}}
\newcommand{\barv}{\overline{{v}}}
\newcommand{\barw}{\overline{{w}}}
\newcommand{\barx}{\overline{{x}}}
\newcommand{\bary}{\overline{{y}}}
\newcommand{\barz}{\overline{{z}}}

\newcommand{\tA}{\widetilde{{A}}}
\newcommand{\tB}{\widetilde{{B}}}
\newcommand{\tC}{\widetilde{{C}}}
\newcommand{\tD}{\widetilde{{D}}}
\newcommand{\tE}{\widetilde{{E}}}
\newcommand{\tF}{\widetilde{{F}}}
\newcommand{\tG}{\widetilde{{G}}}
\newcommand{\tH}{\widetilde{{H}}}
\newcommand{\tI}{\widetilde{{I}}}
\newcommand{\tJ}{\widetilde{{J}}}
\newcommand{\tK}{\widetilde{{K}}}
\newcommand{\tL}{\widetilde{{L}}}
\newcommand{\tM}{\widetilde{{M}}}
\newcommand{\tN}{\widetilde{{N}}}
\newcommand{\tO}{\widetilde{{O}}}
\newcommand{\tP}{\widetilde{{P}}}
\newcommand{\tQ}{\widetilde{{Q}}}
\newcommand{\tR}{\widetilde{{R}}}
\newcommand{\tS}{\widetilde{{S}}}
\newcommand{\tT}{\widetilde{{T}}}
\newcommand{\tU}{\widetilde{{U}}}
\newcommand{\tV}{\widetilde{{V}}}
\newcommand{\tW}{\widetilde{{W}}}
\newcommand{\tX}{\widetilde{{X}}}
\newcommand{\tY}{\widetilde{{Y}}}
\newcommand{\tZ}{\widetilde{{Z}}}


\newcommand{\ta}{\widetilde{{a}}}
\newcommand{\tb}{\widetilde{{b}}}
\newcommand{\tc}{\widetilde{{c}}}
\newcommand{\td}{\widetilde{{d}}}
\newcommand{\te}{\widetilde{{e}}}
\newcommand{\tif}{\widetilde{{f}}}
\newcommand{\tg}{\widetilde{{g}}}
\newcommand{\tih}{\widetilde{{h}}}
\newcommand{\ti}{\widetilde{{i}}}
\newcommand{\tj}{\widetilde{{j}}}
\newcommand{\tk}{\widetilde{{k}}}
\newcommand{\tl}{\widetilde{{l}}}
\newcommand{\tm}{\widetilde{{m}}}
\newcommand{\tn}{\widetilde{{n}}}
\newcommand{\tio}{\widetilde{{o}}}
\newcommand{\tp}{\widetilde{{p}}}
\newcommand{\tq}{\widetilde{{q}}}
\newcommand{\tir}{\widetilde{{r}}}
\newcommand{\ts}{\widetilde{{s}}}
\newcommand{\tit}{\widetilde{{t}}}
\newcommand{\tu}{\widetilde{{u}}}
\newcommand{\tv}{{\widetilde{{v}}}}
\newcommand{\tw}{\widetilde{{w}}}
\newcommand{\tx}{\widetilde{{x}}}
\newcommand{\ty}{\widetilde{{y}}}
\newcommand{\tz}{\widetilde{{z}}}

\newcommand{\baralpha   }{\overline{\alpha  }}
\newcommand{\barbeta         }{\overline{\beta}}
\newcommand{\bargamma   }{\overline{\gamma}}
\newcommand{\bardelta        }{\overline{\delta}}
\newcommand{\barepsilon    }{\overline{\epsilon}}
\newcommand{\barvarepsilon  }{\overline{\varepsilon}}
\newcommand{\barzeta       }{\overline{\zeta}}
\newcommand{\bareta         }{\overline{\eta}}
 \newcommand{\bartheta    }{\overline{\theta}}
\newcommand{\barvartheta   }{\overline{\vartheta}}
\newcommand{\bariota     }{\overline{\iota}}
\newcommand{\barkappa     }{\overline{\kappa}}
 \newcommand{\barlambda   }{\overline{\lambda}}
 \newcommand{\barmu    }{\overline{\mu}}
 \newcommand{\barnu    }{\overline{\nu}}
 \newcommand{\barxi    }{\overline{\xi}}
 \newcommand{\barpi    }{\overline{\pi}}
 \newcommand{\barvarpi   }{\overline{\varpi}}
 \newcommand{\barrho   }{{\overline{\rho}}}
 \newcommand{\barvarrho   }{\overline{\varrho}}
 \newcommand{\barsigma   }{\overline{\sigma}}
 \newcommand{\barvarsigma   }{\overline{\varsigma}}
 \newcommand{\bartau     }{\overline{\tau}}
 \newcommand{\barupsilon   }{\overline{\upsilon}}
 \newcommand{\barphi    }{\overline{\phi}}
 \newcommand{\barvarphi   }{\overline{\varphi}}
 \newcommand{\barchi   }{\overline{\chi}}
 \newcommand{\barpsi   }{\overline{\psi}}
 \newcommand{\baromega   }{\overline{\omega}}
 \newcommand{\barGamma     }{\overline{\Gamma}}
\newcommand{\barDelta         }{\overline{\Delta}}
\newcommand{\barTheta       }{\overline{\Theta}}
\newcommand{\barLambda   }{\overline{\Lambda}}
\newcommand{\barXi   }{\overline{\Xi}}
\newcommand{\barPi   }{\overline{\Pi}}
\newcommand{\barSigma   }{\overline{\Sigma}}
\newcommand{\barUpsilon   }{\overline{\Upsilon}}
\newcommand{\barPhi    }{\overline{\Phi}}
\newcommand{\barPsi   }{\overline{\Psi}}
\newcommand{\barOmega   }{\overline{\Omega}}


\newcommand{\alphabar   }{\overline{\alpha  }}
\newcommand{\betabar         }{\overline{\beta}}
\newcommand{\gammabar   }{\overline{\gamma}}
\newcommand{\deltabar        }{\overline{\delta}}
\newcommand{\epsilonbar    }{\overline{\epsilon}}
\newcommand{\varepsilonbar  }{\overline{\varepsilon}}
\newcommand{\zetabar       }{\overline{\zeta}}
\newcommand{\etabar         }{\overline{\eta}}
 \newcommand{\thetabar    }{\overline{\theta}}
\newcommand{\varthetabar   }{\overline{\vartheta}}
\newcommand{\iotabar     }{\overline{\iota}}
\newcommand{\kappabar     }{\overline{\kappa}}
 \newcommand{\lambdabar   }{\overline{\lambda}}
 \newcommand{\mubar    }{\overline{\mu}}
 \newcommand{\nubar    }{\overline{\nu}}
 \newcommand{\xibar    }{\overline{\xi}}
 \newcommand{\pibar    }{\overline{\pi}}
 \newcommand{\varpibar   }{\overline{\varpi}}
 \newcommand{\rhobar   }{{\overline{\rho}}}
 \newcommand{\varrhobar   }{\overline{\varrho}}
 \newcommand{\sigmabar   }{\overline{\sigma}}
 \newcommand{\varsigmabar   }{\overline{\varsigma}}
 \newcommand{\taubar     }{\overline{\tau}}
 \newcommand{\upsilonbar   }{\overline{\upsilon}}
 \newcommand{\phibar    }{\overline{\phi}}
 \newcommand{\varphibar   }{\overline{\varphi}}
 \newcommand{\chibar   }{\overline{\chi}}
 \newcommand{\psibar   }{\overline{\psi}}
 \newcommand{\omegabar   }{\overline{\omega}}
 \newcommand{\Gammabar     }{\overline{\Gamma}}
\newcommand{\Deltabar         }{\overline{\Delta}}
\newcommand{\Thetabar       }{\overline{\Theta}}
\newcommand{\Lambdabar   }{\overline{\Lambda}}
\newcommand{\Xibar   }{\overline{\Xi}}
\newcommand{\Pibar   }{\overline{\Pi}}
\newcommand{\Sigmabar   }{\overline{\Sigma}}
\newcommand{\Upsilonbar   }{\overline{\Upsilon}}
\newcommand{\Phibar    }{\overline{\Phi}}
\newcommand{\Psibar   }{\overline{\Psi}}
\newcommand{\Omegabar   }{\overline{\Omega}}

\newcommand{\talpha   }{\widetilde{\alpha  }}
\newcommand{\tbeta         }{\widetilde{\beta}}
\newcommand{\tgamma   }{\widetilde{\gamma}}
\newcommand{\tdelta        }{\widetilde{\delta}}
\newcommand{\tepsilon    }{\widetilde{\epsilon}}
\newcommand{\tvarepsilon  }{\widetilde{\varepsilon}}
\newcommand{\tzeta       }{\widetilde{\zeta}}
\newcommand{\teta         }{\widetilde{\eta}}
 \newcommand{\ttheta    }{\widetilde{\theta}}
\newcommand{\tvartheta   }{\widetilde{\vartheta}}
\newcommand{\tiota     }{\widetilde{\iota}}
\newcommand{\tkappa     }{\widetilde{\kappa}}
 \newcommand{\tlambda   }{\widetilde{\lambda}}
 \newcommand{\tmu    }{\widetilde{\mu}}
 \newcommand{\tnu    }{\widetilde{\nu}}
 \newcommand{\txi    }{\widetilde{\xi}}
 \newcommand{\tpi    }{\widetilde{\pi}}
 \newcommand{\tvarpi   }{\widetilde{\varpi}}
 \newcommand{\trho   }{\widetilde{\rho}}
 \newcommand{\tvarrho   }{\widetilde{\varrho}}
 \newcommand{\tsigma   }{\widetilde{\sigma}}
 \newcommand{\tvarsigma   }{\widetilde{\varsigma}}
 \newcommand{\ttau     }{\widetilde{\tau}}
 \newcommand{\tupsilon   }{\widetilde{\upsilon}}
 \newcommand{\tphi    }{\widetilde{\phi}}
 \newcommand{\tvarphi   }{\widetilde{\varphi}}
 \newcommand{\tchi   }{\widetilde{\chi}}
 \newcommand{\tpsi   }{\widetilde{\psi}}
 \newcommand{\tomega   }{\widetilde{\omega}}
 \newcommand{\tGamma     }{\widetilde{\Gamma}}
\newcommand{\tDelta         }{\widetilde{\Delta}}
\newcommand{\tTheta       }{\widetilde{\Theta}}
\newcommand{\tLambda   }{\widetilde{\Lambda}}
\newcommand{\tXi   }{\widetilde{\Xi}}
\newcommand{\tPi   }{\widetilde{\Pi}}
\newcommand{\tSigma   }{\widetilde{\Sigma}}
\newcommand{\tUpsilon   }{\widetilde{\Upsilon}}
\newcommand{\tPhi    }{\widetilde{\Phi}}
\newcommand{\tPsi   }{\widetilde{\Psi}}
\newcommand{\tOmega   }{\widetilde{\Omega}}

\newcommand{\alphat   }{\widetilde{\alpha  }}
\newcommand{\betat         }{\widetilde{\beta}}
\newcommand{\gammat   }{\widetilde{\gamma}}
\newcommand{\deltat        }{\widetilde{\delta}}
\newcommand{\epsilont    }{\widetilde{\epsilon}}
\newcommand{\varepsilont  }{\widetilde{\varepsilon}}
\newcommand{\zetat       }{\widetilde{\zeta}}
\newcommand{\etat         }{\widetilde{\eta}}
 \newcommand{\thetat    }{\widetilde{\theta}}
\newcommand{\varthetat   }{\widetilde{\vartheta}}
\newcommand{\iotat     }{\widetilde{\iota}}
\newcommand{\kappat     }{\widetilde{\kappa}}
 \newcommand{\lambdat   }{\widetilde{\lambda}}
 \newcommand{\mut    }{\widetilde{\mu}}
 \newcommand{\nut    }{\widetilde{\nu}}
 \newcommand{\xit    }{\widetilde{\xi}}
 \newcommand{\pit    }{\widetilde{\pi}}
 \newcommand{\varpit   }{\widetilde{\varpi}}
 \newcommand{\rhot   }{\widetilde{\rho}}
 \newcommand{\varrhot   }{\widetilde{\varrho}}
 \newcommand{\sigmat   }{\widetilde{\sigma}}
 \newcommand{\varsigmat   }{\widetilde{\varsigma}}
 \newcommand{\taut     }{\widetilde{\tau}}
 \newcommand{\upsilont   }{\widetilde{\upsilon}}
 \newcommand{\phit    }{\widetilde{\phi}}
 \newcommand{\varphit   }{\widetilde{\varphi}}
 \newcommand{\chit   }{\widetilde{\chi}}
 \newcommand{\psit   }{\widetilde{\psi}}
 \newcommand{\omegat   }{\widetilde{\omega}}
 \newcommand{\Gammat     }{\widetilde{\Gamma}}
\newcommand{\Deltat         }{\widetilde{\Delta}}
\newcommand{\Thetat       }{\widetilde{\Theta}}
\newcommand{\Lambdat   }{\widetilde{\Lambda}}
\newcommand{\Xit   }{\widetilde{\Xi}}
\newcommand{\Pit   }{\widetilde{\Pi}}
\newcommand{\Sigmat   }{\widetilde{\Sigma}}
\newcommand{\Upsilont   }{\widetilde{\Upsilon}}
\newcommand{\Phit    }{\widetilde{\Phi}}
\newcommand{\Psit   }{\widetilde{\Psi}}
\newcommand{\Omegat   }{\widetilde{\Omega}}


\newcommand{\hatotimes}{{\widehat{\otimes}}}
\newcommand{\barCG}{\overline{{\CG}}}
\newcommand{\proj}{{\operatorname{proj}}}
\newcommand{\MK}{{\underline{M}_{\CO_K}}}
\newcommand{\MF}{{\CM\CF}}
\newcommand{\BT}{{Barsotti-Tate $\CO_K$-module}}
\newcommand{\Qbar}{{\overline{\Q}}}
\newcommand{\Fbar}{{\overline{\F}}}
\newcommand{\tbarr}{{\widetilde{\barr}}}

\newcommand{\kbar}{{\overline{k}}}
\def\RCS$#1: #2 ${\expandafter\def\csname RCS#1\endcsname{#2}}
\RCS$Revision: 327 $
\RCS$Date: 2010-11-15 13:46:16 -0600 (Mon, 15 Nov 2010) $
\newcommand{\versioninfo}{Version \RCSRevision; Last commit \RCSDate}


\DeclareMathOperator{\RE}{Re} \DeclareMathOperator{\IM}{Im}
\DeclareMathOperator{\ess}{ess}

\newcommand{\notdiv}{\nmid}


\newcommand{\m}{\mathfrak{m}}


 \newcommand{\h}{\mathcal{H}}
\newcommand{\s}{\mathcal{S}} \newcommand{\St}{\operatorname{St}}
\newcommand{\cL}{\mathcal{L}}
\newcommand{\bigO}{\mathcal{O}} \newcommand{\calO}{\mathcal{O}}
\newcommand{\BOP}{\mathbb{B}}
\newcommand{\BH}{\mathbb{B}(\mathcal{H})}
\newcommand{\KH}{\mathcal{K}(\mathcal{H})} \newcommand{\p}{\mathfrak{p}} \newcommand{\q}{\mathfrak{q}}
\newcommand{\frakl}{\mathfrak{l}}
\newcommand{\fp}{\mathfrak{p}}
\newcommand{\fq}{\mathfrak{q}} \newcommand{\fQ}{\mathfrak{Q}}
\newcommand{\Field}{\mathbb{F}}
\newcommand{\Gn}{\mathcal{G}_n}
\newcommand{\sq}{\square}

\newcommand{\tensor}{\otimes}
\newcommand{\thbarpr}{\bar{\theta'}}

\newcommand{\Favoid}{F^{(\mathrm{avoid})}}

\newcommand{\mcal}{\mathcal}
\newcommand{\bb}{\mathbb}
\newcommand{\mc}{\mathcal}
\newcommand{\mf}{\mathfrak}

\newcommand{\Uop}{U_{\lambda,\varpi_{\wt{v}}}^{(j)}}
\newcommand{\TUniv}{\widehat{\bb{T}}^{T,\ord}(U(\mf{l}^{\infty}),\mc{O})}
\newcommand{\TT}{\bb{T}^{T}_{\lambda}(U,\mc{O})}

\DeclareMathOperator{\ord}{ord}

\DeclareMathOperator{\lcm}{lcm}
\DeclareMathOperator{\charpol}{char}
\DeclareMathOperator{\sesi}{ss}
\DeclareMathOperator{\ssg}{ss}
\DeclareMathOperator{\sd}{sd}

\DeclareMathOperator{\GO}{GO}
\DeclareMathOperator{\FL}{FL}

\newcommand{\RPlus}{\Real^{+}} \newcommand{\Polar}{\mathcal{P}_{\s}}
\newcommand{\Poly}{\mathcal{P}(E)} \newcommand{\EssD}{\mathcal{D}}


\newcommand{\rbar}{\bar{r}}
\newcommand{\Rbar}{\bar{R}}
\newcommand{\rbarwtv}{\rbar|_{G_{F_{\wt{v}}}}}
\newcommand{\rbarwtvL}{\rbar|_{G_{L_{\wt{v}}}}}
\newcommand{\rmbarwtv}{\rbar_{\mf{m}}|_{G_{F_{\wt{v}}}}}

\newcommand{\Lom}{\mathcal{L}}
\newcommand{\States}{\mathcal{T}}
\newcommand{\abs}[1]{\left\vert#1\right\vert}
\newcommand{\set}[1]{\left\{#1\right\}}
\newcommand{\seq}[1]{\left<#1\right>}


\newcommand{\HT}{\operatorname{HT}}
\newcommand{\Pj}{\mathbb{P}}


\newcommand{\FI}{\operatorname{FI}}
\newcommand{\Gr}{\operatorname{Gr}}
\newcommand{\Zbar}{\overline{\Z}}
\newcommand{\GQ}{\Gal(\Qbar/\Q)}
\newcommand{\Qp}{\Q_p}
\newcommand{\Zp}{\Z_p}
\newcommand{\Zl}{\Z_l}
\newcommand{\Ql}{\Q_l} \newcommand{\Cp}{\C_p}
\newcommand{\Qpbar}{\overline{\Q}_p}
\newcommand{\Qlbar}{\overline{\Q}_{l}}
\newcommand{\Fpbar}{\overline{\F}_p}
\newcommand{\Fqbar}{\overline{\F}_q}
\newcommand{\Fl}{{\F_l}}
\newcommand{\Fp}{{\F_p}}
\newcommand{\Fq}{{\F_q}}
\newcommand{\Flbar}{\overline{\F}_l}
\newcommand{\Zpbar}{\overline{\Z}_p}

\newcommand{\gfv}{{G_{F_{\tilde{v}}}}}

\newcommand{\Jac}{\operatorname{Jac}}

\newcommand{\Inn}{\operatorname{Inn}}

\newcommand{\rank}{\operatorname{rank}}

\newcommand{\Ver}{\operatorname{Ver}}
\newcommand{\Sym}{\operatorname{Sym}}

\newcommand{\Mod}{\operatorname{Mod}}
\newcommand{\rmod}{R-\Mod_{cris,dd}^{k-1}}

\newcommand{\bigs}{\mathfrak{S}}

\newcommand{\labelbp}[1]
{\refstepcounter{equation}\hfill(\theequation)\label{#1}}

\newcommand{\mar}[1]{\marginpar{\tiny #1}}

\def\smallmat#1#2#3#4{\bigl(\begin{smallmatrix}{#1}&{#2}\\{#3}&{#4}\end{smallmatrix}\bigr)}


\usepackage{amsthm}

\newtheorem*{thmn}{Theorem}
\newtheorem{ithm}{Theorem}
\newtheorem{thm}{Theorem}[subsection]
\newtheorem{corollary}[thm]{Corollary}
\newtheorem{cor}[thm]{Corollary}
 \newtheorem{lemma}[thm]{Lemma}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{conj}[thm]{Conjecture} \theoremstyle{definition}
\newtheorem{conjecture}[thm]{Conjecture} \theoremstyle{definition}
\newtheorem{defn}[thm]{Definition} \theoremstyle{remark}
\newtheorem{rem}[thm]{Remark}
\newtheorem{remark}[thm]{Remark}
\newtheorem{example}[thm]{Example}
\newtheorem{fact}[thm]{Fact}
\newtheorem{observation}[thm]{Observation}
\newtheorem{question}[thm]{Question}
\numberwithin{equation}{subsection}
\renewcommand{\theithm}{\Alph{ithm}}
\theoremstyle{definition}\newtheorem*{notation}{Notation}
\newtheorem{situation}[thm]{Situation}
\newtheorem*{sublem}{Sublemma}


\setcounter{tocdepth}{1}

\usepackage[bookmarksopen,bookmarksdepth=2,breaklinks=true]{hyperref}
%\usepackage[anythingbreaks]{breakurl}

\begin{document}