preamble.tex

\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{amsmath}
\usepackage{amscd}
\usepackage[mathscr]{eucal}
\usepackage{verbatim}

\textheight8.75in \topmargin+0.5in \textwidth6in
\oddsidemargin.125in \evensidemargin.125in

\numberwithin{equation}{section}

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section] 
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}

\theoremstyle{remark}
\newtheorem*{notation}{Notation}
\newtheorem{problem}[theorem]{Problem}







\newcommand{\R}{\mathbb{R}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\h}{\mathbb{H}}
\renewcommand{\H}{\mathbb{H}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\A}{\mathbb{A}}

\newcommand{\G}{\Gamma}
\newcommand{\g}{\gamma}
\newcommand{\s}{\sigma}
\newcommand{\la}{\lambda}
\newcommand{\La}{\Lambda}
\newcommand{\x}{\mathbf{x}}
\newcommand{\y}{\mathbf{y}}
\newcommand{\tb}{\mathbf{t}}

\newcommand{\back}{\backslash}

\newcommand{\V}{V(\Q)}

\newcommand{\wwedge}[1]{\sideset{}{^{#1}}\bigwedge}




% Jan's Macros

\newcommand{\zxz}[4]{\begin{pmatrix} #1 & #2 \\ #3 & #4 \end{pmatrix}}
\newcommand{\abcd}{\zxz{a}{b}{c}{d}}

\newcommand{\kzxz}[4]{\left(\begin{smallmatrix} #1 & #2 \\ #3 & #4\end{smallmatrix}\right) }
\newcommand{\kabcd}{\kzxz{a}{b}{c}{d}}
\newcommand{\veps}{\varepsilon}
\newcommand{\lo}{\longrightarrow}
\newcommand{\imag}{{\rm i}}
\newcommand{\set}[1]{\left\{ #1 \right\} }
\newcommand{\im}{\operatorname{Im}}
\newcommand{\re}{\operatorname{Re}}
\newcommand{\Res}{\operatorname{Res}}

\newcommand{\vl}{v_{\ell}}
\newcommand{\calA}{\mathcal{A}}
\newcommand{\calB}{\mathcal{B}}
\newcommand{\calC}{\mathcal{C}}
\newcommand{\calD}{\mathcal{D}}
\newcommand{\calE}{\mathcal{E}}
\newcommand{\calF}{\mathcal{F}}
\newcommand{\calG}{\mathcal{G}}
\newcommand{\calH}{\mathcal{H}}
\newcommand{\calJ}{\mathcal{J}}
\newcommand{\calK}{\mathcal{K}}
\newcommand{\calL}{\mathcal{L}}
\newcommand{\calM}{\mathcal{M}}
\newcommand{\calN}{\mathcal{N}}
\newcommand{\calO}{\mathcal{O}}
\newcommand{\calP}{\mathcal{P}}
\newcommand{\calQ}{\mathcal{Q}}
\newcommand{\calR}{\mathcal{R}}
\newcommand{\calS}{\mathcal{S}}
\newcommand{\calV}{\mathcal{V}}
\newcommand{\calW}{\mathcal{W}}
\newcommand{\calZ}{\mathcal{Z}}

\newcommand{\fraka}{\mathfrak a}
\newcommand{\frakc}{\mathfrak c}
\newcommand{\frakd}{\mathfrak d}
\newcommand{\frake}{\mathfrak e}
\newcommand{\frakS}{\mathfrak S}
\newcommand{\frakt}{\mathfrak t}
\newcommand{\frakx}{\mathfrak x}
\newcommand{\fraky}{\mathfrak y}
\newcommand{\frakE}{\mathfrak E}
\newcommand{\fraknull}{\mathfrak 0}

\newcommand{\eps}{\varepsilon}
\newcommand{\bs}{\backslash}
\newcommand{\norm}{\operatorname{N}}
\newcommand{\vol}{\operatorname{vol}}
\newcommand{\tr}{\operatorname{tr}}
\newcommand{\Log}{\operatorname{Log}}
\newcommand{\sgn}{\operatorname{sgn}}
\newcommand{\erf}{\operatorname{erf}}
\newcommand{\Kern}{\operatorname{Kern}}
\newcommand{\Span}{\operatorname{span}}
\newcommand{\Stab}{\operatorname{Stab}}
\newcommand{\Cl}{\operatorname{Cl}}
\newcommand{\Sl}{\operatorname{SL}}
\newcommand{\Gl}{\operatorname{GL}}
\newcommand{\SL}{\operatorname{SL}}
\newcommand{\GL}{\operatorname{GL}}
\newcommand{\Spin}{\operatorname{Spin}}
\newcommand{\PSL}{\operatorname{PSL}}
\newcommand{\Sp}{\operatorname{Sp}}
\newcommand{\Symp}{\operatorname{Sp}}
\newcommand{\Mp}{\operatorname{Mp}}
\newcommand{\Orth}{\operatorname{O}}
\newcommand{\Uni}{\operatorname{U}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\Aut}{\operatorname{Aut}}
\newcommand{\Gr}{\operatorname{Gr}}
\newcommand{\SO}{\operatorname{SO}}
\newcommand{\Gal}{\operatorname{Gal}}
\newcommand{\Clh}{\Cl_{\text{\rm \scriptsize H}}}
\newcommand{\CL}{\operatorname{\tilde{Cl}}}
\newcommand{\Ureg}{U_{\text{\rm \scriptsize reg}}}
\newcommand{\supp}{\operatorname{supp}}
\newcommand{\J}{\calJ^{\text{\rm \scriptsize cusp}}}
\newcommand{\Pic}{\operatorname{Pic}}
\newcommand{\Div}{\operatorname{Div}}
\newcommand{\Bil}{\operatorname{Bil}}
\newcommand{\End}{\operatorname{End}}
\newcommand{\tg}{\operatorname{tg}}
\newcommand{\rank}{\operatorname{rank}}
\newcommand{\id}{\operatorname{id}}
\newcommand{\TF}{\tilde{F}}
\newcommand{\dv}{\operatorname{div}}
\newcommand{\Ei}{\operatorname{Ei}}
\newcommand{\erfc}{\operatorname{erfc}}
\newcommand{\Iso}{\operatorname{Iso}}
\newcommand{\ch}{\operatorname{CH}}
\newcommand{\z}{\operatorname{Z}}
\newcommand{\za}{\operatorname{\widehat{Z}}}
\newcommand{\rata}{\operatorname{\widehat{Rat}}}
\newcommand{\dega}{\operatorname{\widehat{deg}}}
\newcommand{\pica}{\operatorname{\widehat{Pic}}}
\newcommand{\diva}{\operatorname{\widehat{div}}}
\newcommand{\cha}{\operatorname{\widehat{CH}}}
\newcommand{\cc}{\operatorname{c}}
\newcommand{\cca}{\widehat{\operatorname{c}}}
\newcommand{\fh}{\operatorname{ht}}
\newcommand{\curv}{\operatorname{curv}}
\newcommand{\Lk}{\operatorname{Lk}}
\newcommand{\PD}{\operatorname{PD}}

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