\usepackage{color} \definecolor{dredcolor}{rgb}{.5,0.0,0.0} \definecolor{dgreencolor}{rgb}{0,0.4,0} \definecolor{dbluecolor}{rgb}{.02,.02,.908} \newcommand{\dred}{\color{dredcolor}\bf} \newcommand{\dblue}{\color{dbluecolor}\bf} \newcommand{\dgreen}{\color{dgreencolor}\bf} % macros.tex \usepackage[active]{srcltx}\usepackage[active]{srcltx}\usepackage[active]{srcltx} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsthm} % You change everything, by adding \usepackage{times} to the document % Preamble. Now all the roman letters will be set in times and all the % sans serif stuff will be set in Helvetica. If you don't like times, % you can try the packages: palatcm, charter, helvet, palatino, avant, % newcent and bookman % If you want to change explicitly to a certain font, use the command % \fontfamily{XYZ}\selectfont whereby XYZ can be set to: pag for Adobe % AvantGarde, pbk for Adobe Bookman, pcr for Adobe Courier, phv for % Adobe Helvetica, pnc for Adobe NewCenturySchoolbook, ppl for Adobe % Palatino, ptm for Adobe Times Roman, pzc for Adobe ZapfChancery \newcommand{\courier}{\fontfamily{pcr}\selectfont} \newcommand{\edit}[1]{\footnote{[[#1]]}\marginpar{\hfill {\sf[[\thefootnote]]}}} %\newcommand{\edit}[1]{{\sl\small [[Todo: #1]]}} %\author{William~A. Stein} \newcommand{\Hbar}{\overline{H}} \newcommand{\myhead}[3]{ \par\noindent {Version #2} \vspace{10ex} \par\noindent {\bf \LARGE #1}\\ \vspace{3ex} \par\noindent {\large W.\thinspace{}A. Stein}\\ {\small Department of Mathematics, Harvard University}\vspace{1ex}\\ #3 \vspace{2ex}\par } \newcommand{\myheadauth}[3]{ \par\noindent {Version #2} \vspace{10ex} \par\noindent {\bf \LARGE #1}\\ \vspace{3ex} \par\noindent #3 \vspace{5ex}\par } \newcommand{\zbar}{\overline{z}} \newcommand{\dzbar}{d\overline{z}} \usepackage{xspace} % to allow for text macros that don't eat space \newcommand{\SAGE}{{\sf SAGE}\xspace} \newcommand{\sage}{\SAGE} \newcommand{\gzero}{\Gamma_0(N)} \newcommand{\esM}{\overline{\sM}} \newcommand{\sM}{\boldsymbol{\mathcal{M}}} \newcommand{\sS}{\boldsymbol{\mathcal{S}}} \newcommand{\sB}{\boldsymbol{\mathcal{B}}} \newcommand{\eE}{\mathbb{E}} \newcommand{\bA}{\mathbb{A}} \newcommand{\Adual}{A^{\vee}} \newcommand{\Bdual}{B^{\vee}} \newcommand{\defn}[1]{{\em\dred #1}} \newcommand{\todo}[1]{\noindent$\bullet$ {\small \textsf{#1}} $\bullet$\\} \newcommand{\done}[1]{\noindent {\small \textsf{Done: #1}}\\} \newcommand{\danger}[1]{\marginpar{\small \textsl{#1}}} \renewcommand{\div}{\mbox{\rm div}} \DeclareMathOperator{\GCD}{GCD} \DeclareMathOperator{\can}{can} \DeclareMathOperator{\Cl}{Cl} \DeclareMathOperator{\Mod}{Mod} \DeclareMathOperator{\chr}{char} \DeclareMathOperator{\charpoly}{charpoly} \DeclareMathOperator{\cris}{cris} \DeclareMathOperator{\dR}{dR} \DeclareMathOperator{\Fil}{Fil} \DeclareMathOperator{\ind}{ind} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\oo}{\infty} \DeclareMathOperator{\abs}{abs} \DeclareMathOperator{\lcm}{lcm} \DeclareMathOperator{\cores}{cores} \DeclareMathOperator{\coker}{coker} \DeclareMathOperator{\image}{image} \DeclareMathOperator{\prt}{part} \DeclareMathOperator{\proj}{proj} \DeclareMathOperator{\Br}{Br} \DeclareMathOperator{\Ann}{Ann} \DeclareMathOperator{\End}{End} \DeclareMathOperator{\Tan}{Tan} \DeclareMathOperator{\Eis}{Eis} \newcommand{\unity}{\mathbb{1}} \DeclareMathOperator{\Pic}{Pic} \DeclareMathOperator{\Tate}{Tate} \DeclareMathOperator{\Vol}{Vol} \DeclareMathOperator{\Vis}{Vis} \DeclareMathOperator{\Reg}{Reg} %\DeclareMathOperator{\myRes}{Res} %\newcommand{\Res}{\myRes} \DeclareMathOperator{\Res}{Res} \newcommand{\an}{{\rm an}} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\Sel}{Sel} \DeclareMathOperator{\Mat}{Mat} \DeclareMathOperator{\BSD}{BSD} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\dz}{dz} %\DeclareMathOperator{\Re}{Re} \renewcommand{\Re}{\mbox{\rm Re}} \DeclareMathOperator{\Imm}{Im} \renewcommand{\Im}{\Imm} \DeclareMathOperator{\Selmer}{Selmer} \newcommand{\pfSel}{\widehat{\Sel}} \newcommand{\qe}{\stackrel{\mbox{\tiny ?}}{=}} \newcommand{\isog}{\simeq} \newcommand{\e}{\mathbf{e}} \newcommand{\bN}{\mathbf{N}} % ---- SHA ---- \DeclareFontEncoding{OT2}{}{} % to enable usage of cyrillic fonts \newcommand{\textcyr}[1]{% {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}% \selectfont #1}} \newcommand{\Sha}{{\mbox{\textcyr{Sh}}}} %\font\cyr=wncyr10 scaled \magstep 1 %\font\cyr=wncyr10 %\newcommand{\Sha}{{\cyr X}} \newcommand{\Shaan}{\Sha_{\mbox{\tiny \rm an}}} \newcommand{\TS}{Shafarevich-Tate group} \newcommand{\Gam}{\Gamma} \newcommand{\X}{\mathcal{X}} \newcommand{\cH}{\mathcal{H}} \newcommand{\cA}{\mathcal{A}} \newcommand{\cF}{\mathcal{F}} \newcommand{\cG}{\mathcal{G}} \newcommand{\cJ}{\mathcal{J}} \newcommand{\cL}{\mathcal{L}} \newcommand{\cV}{\mathcal{V}} \newcommand{\cO}{\mathcal{O}} \newcommand{\cQ}{\mathcal{Q}} \newcommand{\cX}{\mathcal{X}} \newcommand{\ds}{\displaystyle} \newcommand{\M}{\mathcal{M}} \newcommand{\E}{\mathcal{E}} \renewcommand{\L}{\mathcal{L}} \newcommand{\J}{\mathcal{J}} \DeclareMathOperator{\new}{new} \DeclareMathOperator{\Morph}{Morph} \DeclareMathOperator{\old}{old} \DeclareMathOperator{\Sym}{Sym} \DeclareMathOperator{\Symb}{Symb} %\newcommand{\Sym}{\mathcal{S}{\rm ym}} \newcommand{\dw}{\delta(w)} \newcommand{\dwh}{\widehat{\delta(w)}} \newcommand{\dlwh}{\widehat{\delta_\l(w)}} \newcommand{\dash}{-\!\!\!\!-\!\!\!\!-\!\!\!\!-} \DeclareMathOperator{\tor}{tor} \newcommand{\Frobl}{\Frob_{\ell}} \newcommand{\tE}{\tilde{E}} \renewcommand{\l}{\ell} \renewcommand{\t}{\tau} \DeclareMathOperator{\cond}{cond} \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\Div}{Div} \DeclareMathOperator{\Jac}{Jac} \DeclareMathOperator{\res}{res} \DeclareMathOperator{\Ker}{Ker} \DeclareMathOperator{\Coker}{Coker} \DeclareMathOperator{\sep}{sep} \DeclareMathOperator{\sign}{sign} \DeclareMathOperator{\unr}{unr} \newcommand{\N}{\mathcal{N}} \newcommand{\U}{\mathcal{U}} \newcommand{\Kbar}{\overline{K}} \newcommand{\Lbar}{\overline{L}} \newcommand{\gammabar}{\overline{\gamma}} \newcommand{\q}{\mathbf{q}} %\renewcommand{\star}{\times} \newcommand{\gM}{\mathfrak{M}} \newcommand{\gA}{\mathfrak{A}} \newcommand{\gP}{\mathfrak{P}} \newcommand{\bmu}{\boldsymbol{\mu}} \newcommand{\union}{\cup} \newcommand{\Tl}{T_{\ell}} \newcommand{\into}{\rightarrow} \newcommand{\onto}{\rightarrow\!\!\!\!\!\!\rightarrow} \newcommand{\intersect}{\cap} \newcommand{\meet}{\cap} \newcommand{\cross}{\times} \DeclareMathOperator{\md}{mod} \DeclareMathOperator{\toric}{toric} \DeclareMathOperator{\tors}{tors} \DeclareMathOperator{\Frac}{Frac} \DeclareMathOperator{\corank}{corank} \newcommand{\kr}[2]{\left(\frac{#1}{#2}\right)} \newcommand{\rb}{\overline{\rho}} \newcommand{\ra}{\rightarrow} \newcommand{\xra}[1]{\xrightarrow{#1}} \newcommand{\hra}{\hookrightarrow} \newcommand{\la}{\leftarrow} \newcommand{\lra}{\longrightarrow} \newcommand{\riso}{\xrightarrow{\sim}} \newcommand{\da}{\downarrow} \newcommand{\ua}{\uparrow} \newcommand{\con}{\equiv} \newcommand{\Gm}{\mathbb{G}_m} \newcommand{\pni}{\par\noindent} \newcommand{\set}[1]{\{#1\}} \newcommand{\iv}{^{-1}} \newcommand{\alp}{\alpha} \newcommand{\bq}{\mathbf{q}} \newcommand{\cpp}{{\tt C++}} \newcommand{\tensor}{\otimes} \newcommand{\bg}{{\tt BruceGenus}} \newcommand{\abcd}[4]{\left( \begin{smallmatrix}#1\\#3\end{smallmatrix}\right)} \newcommand{\mthree}[9]{\left( \begin{matrix}#1\\#4\\#7 \end{matrix}\right)} \newcommand{\mtwo}[4]{\left( \begin{matrix}#1\\#3 \end{matrix}\right)} \newcommand{\vtwo}[2]{\left( \begin{matrix}#1\\#2 \end{matrix}\right)} \newcommand{\smallmtwo}[4]{\left( \begin{smallmatrix}#1\\#3 \end{smallmatrix}\right)} \newcommand{\twopii}{2\pi{}i{}} \newcommand{\eps}{\varepsilon} \newcommand{\vphi}{\varphi} \newcommand{\gp}{\mathfrak{p}} \newcommand{\W}{\mathcal{W}} \newcommand{\oz}{\overline{z}} \newcommand{\Zpstar}{\Zp^{\star}} \newcommand{\Zhat}{\widehat{\Z}} \newcommand{\Zbar}{\overline{\Z}} \newcommand{\Zl}{\Z_{\ell}} \newcommand{\comment}[1]{} \newcommand{\Q}{\mathbb{Q}} \newcommand{\GQ}{G_{\Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\D}{{\mathbf D}} \newcommand{\cC}{\mathcal{C}} \newcommand{\cD}{\mathcal{D}} \newcommand{\cP}{\mathcal{P}} \newcommand{\cS}{\mathcal{S}} \newcommand{\Sbar}{\overline{S}} \newcommand{\K}{{\mathbb K}} \newcommand{\C}{\mathbb{C}} \newcommand{\Cp}{{\mathbb C}_p} \newcommand{\Sets}{\mbox{\rm\bf Sets}} \newcommand{\bcC}{\boldsymbol{\mathcal{C}}} \renewcommand{\P}{\mathbb{P}} \newcommand{\Qbar}{\overline{\Q}} \newcommand{\kbar}{\overline{k}} \newcommand{\dual}{\bot} \newcommand{\T}{\mathbb{T}} \newcommand{\calT}{\mathcal{T}} \newcommand{\cT}{\mathcal{T}} \newcommand{\cbT}{\mathbb{\mathcal{T}}} \newcommand{\cU}{\mathcal{U}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\F}{\mathbb{F}} \newcommand{\Fl}{\F_{\ell}} \newcommand{\Fell}{\Fl} \newcommand{\Flbar}{\overline{\F}_{\ell}} \newcommand{\Flnu}{\F_{\ell^{\nu}}} \newcommand{\Fbar}{\overline{\F}} \newcommand{\Fpbar}{\overline{\F}_p} \newcommand{\fbar}{\overline{f}} \newcommand{\Qp}{\Q_p} \newcommand{\Ql}{\Q_{\ell}} \newcommand{\Qell}{\Q_{\ell}} \newcommand{\Qlbar}{\overline{\Q}_{\ell}} \newcommand{\Qlnr}{\Q_{\ell}^{\text{nr}}} \newcommand{\Qlur}{\Q_{\ell}^{\text{ur}}} \newcommand{\Qltm}{\Q_{\ell}^{\text{tame}}} \newcommand{\Qv}{\Q_v} \newcommand{\Qpbar}{\Qbar_p} \newcommand{\Zp}{\Z_p} \newcommand{\Fp}{\F_p} \newcommand{\Fq}{\F_q} \newcommand{\Fqbar}{\overline{\F}_q} \newcommand{\Ad}{Ad} \newcommand{\adz}{\Ad^0} \renewcommand{\O}{\mathcal{O}} \newcommand{\A}{\mathcal{A}} \newcommand{\Og}{O_{\gamma}} \newcommand{\isom}{\cong} \newcommand{\ncisom}{\approx} % noncanonical isomorphism \DeclareMathOperator{\ab}{ab} \DeclareMathOperator{\Aut}{Aut} \DeclareMathOperator{\Frob}{Frob} \DeclareMathOperator{\Fr}{Fr} \DeclareMathOperator{\Ver}{Ver} \DeclareMathOperator{\Norm}{Norm} \DeclareMathOperator{\Ind}{Ind} \DeclareMathOperator{\norm}{norm} \DeclareMathOperator{\disc}{disc} \DeclareMathOperator{\ord}{ord} \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\PSL}{PSL} \DeclareMathOperator{\PGL}{PGL} \DeclareMathOperator{\Gal}{Gal} \DeclareMathOperator{\SL}{SL} \DeclareMathOperator{\SO}{SO} \DeclareMathOperator{\WC}{WC} \newcommand{\galq}{\Gal(\Qbar/\Q)} \newcommand{\rhobar}{\overline{\rho}} \newcommand{\cM}{\mathcal{M}} \newcommand{\cB}{\mathcal{B}} \newcommand{\cE}{\mathcal{E}} \newcommand{\cR}{\mathcal{R}} \newcommand{\et}{\text{\rm\'et}} \newcommand{\sltwoz}{\SL_2(\Z)} \newcommand{\sltwo}{\SL_2} \newcommand{\gltwoz}{\GL_2(\Z)} \newcommand{\mtwoz}{M_2(\Z)} \newcommand{\gltwoq}{\GL_2(\Q)} \newcommand{\gltwo}{\GL_2} \newcommand{\gln}{\GL_n} \newcommand{\psltwoz}{\PSL_2(\Z)} \newcommand{\psltwo}{\PSL_2} \newcommand{\h}{\mathfrak{h}} \renewcommand{\a}{\mathfrak{a}} \newcommand{\p}{\mathfrak{p}} \newcommand{\m}{\mathfrak{m}} \newcommand{\trho}{\tilde{\rho}} \newcommand{\rhol}{\rho_{\ell}} \newcommand{\rhoss}{\rho^{\text{ss}}} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\order}{order} \DeclareMathOperator{\ur}{ur} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Mor}{Mor} \DeclareMathOperator{\HH}{H} \renewcommand{\H}{\HH} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\Tor}{Tor} \newcommand{\smallzero}{\left(\begin{smallmatrix}0&0\\0&0 \end{smallmatrix}\right)} \newcommand{\smallone}{\left(\begin{smallmatrix}1&0\\0&1 \end{smallmatrix}\right)} \newcommand{\pari}{{\sc Pari}} \newcommand{\magma}{{\sc Magma}} \newcommand{\hecke}{{\sc Hecke}} \newcommand{\lidia}{{\sc LiDIA}} %%%% Theoremstyles \theoremstyle{plain} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{claim}[theorem]{Claim} \newtheorem{hypothesis}[theorem]{Hypothesis} \newtheorem{conjecture}[theorem]{Conjecture} \theoremstyle{definition} \newtheorem{question}[theorem]{Question} \newtheorem{alg}[theorem]{Algorithm} \newtheorem{openproblem}[theorem]{Open Problem} %\theoremstyle{remark} \newtheorem{goal}[theorem]{Goal} \newtheorem{remark}[theorem]{Remark} \newtheorem{remarks}[theorem]{Remarks} \newtheorem{exercise}[theorem]{Exercise} \numberwithin{equation}{section} \numberwithin{figure}{section} \numberwithin{table}{section} % bulleted list environment \newenvironment{bulletlist} { \begin{list} {$\bullet$} { \setlength{\itemsep}{.5ex} \setlength{\parsep}{0ex} \setlength{\parskip}{0ex} \setlength{\topsep}{.5ex} } } { \end{list} } %end newenvironment % bulleted list environment \newenvironment{dashlist} { \begin{list} {---} { \setlength{\itemsep}{.5ex} \setlength{\parsep}{0ex} \setlength{\parskip}{0ex} \setlength{\topsep}{.5ex} } } { \end{list} } %end newenvironment % numbered list environment \newcounter{listnum} \newenvironment{numlist} { \begin{list} {{\em \thelistnum.}}{ \usecounter{listnum} \setlength{\itemsep}{.5ex} \setlength{\parsep}{0ex} \setlength{\parskip}{0ex} \setlength{\topsep}{.5ex} } } { \end{list} } %end newenvironment \newcommand{\hd}[1]{\vspace{1ex}\noindent{\bf #1} } \newcommand{\nf}[1]{\underline{#1}} \newcommand{\cbar}{\overline{c}} \DeclareMathOperator{\rad}{rad} \newenvironment{algorithm}[1]{% \begin{alg}[#1]\index{algorithm!#1}\sf }% {\end{alg}} \newenvironment{steps}% {\begin{enumerate}\setlength{\itemsep}{0.1ex}}{\end{enumerate}} %\usepackage[hypertex]{hyperref} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End: