% macros.tex \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsthm} \author{William A. Stein\footnote{University of California, Berkeley, U.S.A., {\tt was@math.berkeley.edu}}} \font\cyr=wncyr10 scaled \magstep 1 \newcommand{\myhead}[3]{ \par\noindent {Preprint (\today), Version #2} \vspace{10ex} \par\noindent {\bf \LARGE #1}\\ \vspace{3ex} \par\noindent {\large W.A. Stein}\\ {\small Department of Mathematics, University of California, Berkeley, CA 94720, USA}\vspace{1ex}\\ #3 \vspace{2ex}\par } \newcommand{\myheadauth}[3]{ \par\noindent {Preprint (\today), Version #2} \vspace{10ex} \par\noindent {\bf \LARGE #1}\\ \vspace{3ex} \par\noindent #3 \vspace{5ex}\par } \newcommand{\defn}[1]{{\bf #1}} \newcommand{\solution}[1]{\vspace{1em}% \par\noindent{\bf Solution #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}}} \DeclareMathOperator{\ind}{ind} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\abs}{abs} \DeclareMathOperator{\lcm}{lcm} \DeclareMathOperator{\cores}{cores} \DeclareMathOperator{\coker}{coker} \DeclareMathOperator{\image}{image} \DeclareMathOperator{\prt}{part} \DeclareMathOperator{\Br}{Br} \DeclareMathOperator{\Ann}{Ann} \DeclareMathOperator{\End}{End} \DeclareMathOperator{\Eis}{Eis} \DeclareMathOperator{\Pic}{Pic} \DeclareMathOperator{\Vol}{Vol} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\Sel}{Sel} \DeclareMathOperator{\Mat}{Mat} \DeclareMathOperator{\BSD}{BSD} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\dz}{dz} %\DeclareMathOperator{\Re}{Re} %\DeclareMathOperator{\Im}{Im} \DeclareMathOperator{\Selmer}{Selmer} \newcommand{\pfSel}{\widehat{\Sel}} \newcommand{\qe}{\stackrel{\mbox{\tiny ?}}{=}} \newcommand{\isog}{\simeq} \newcommand{\e}{\mathbf{e}} \newcommand{\bN}{\mathbf{N}} \newcommand{\Sha}{\mbox{\cyr X}} \newcommand{\TS}{Tate-Shafarevich group} \newcommand{\Gam}{\Gamma} \renewcommand{\Im}{\text{Im}} \newcommand{\X}{\mathcal{X}} \newcommand{\cH}{\mathcal{H}} \newcommand{\cA}{\mathcal{A}} \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{\Spec}{Spec} \DeclareMathOperator{\Div}{Div} \DeclareMathOperator{\Jac}{Jac} \DeclareMathOperator{\res}{res} \DeclareMathOperator{\Ker}{Ker} \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{\cross}{\times} \DeclareMathOperator{\md}{mod} \DeclareMathOperator{\toric}{toric} \DeclareMathOperator{\Frac}{Frac} \newcommand{\rb}{\overline{\rho}} \newcommand{\ra}{\rightarrow} \newcommand{\la}{\leftarrow} \newcommand{\lra}{\longrightarrow} \newcommand{\riso}{\xrightarrow{\sim}} \newcommand{\da}{\downarrow} \newcommand{\ua}{\uparrow} \newcommand{\con}{\equiv} \newcommand{\Gm}{\mathbf{G}_m} \newcommand{\pni}{\par\noindent} \newcommand{\set}[1]{\{#1\}} \newcommand{\iv}{^{-1}} \newcommand{\alp}{\alpha} \newcommand{\bq}{\mathbf{q}} \newcommand{\hecke}{{\tt HECKE}} \newcommand{\lidia}{{\tt LiDIA}} \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{array}{rrr}#1\\#4\\#7 % \end{array}\right)} \newcommand{\mthree}[9]{\left( \begin{matrix}#1\\#4\\#7 \end{matrix}\right)} %\newcommand{\mtwo}[4]{\left( % \begin{array}{rr}#1\\#3 % \end{array}\right)} \newcommand{\mtwo}[4]{\left( \begin{matrix}#1\\#3 \end{matrix}\right)} \newcommand{\smallmtwo}[4]{\left( \begin{smallmatrix}#1\\#3 \end{smallmatrix}\right)} \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{\comment}[1]{} \newcommand{\Q}{\mathbf{Q}} \newcommand{\GQ}{G_{\Q}} \newcommand{\R}{\mathbf{R}} \newcommand{\D}{{\mathbb D}} \newcommand{\cC}{\mathcal{C}} \newcommand{\cD}{\mathcal{D}} \newcommand{\cS}{\mathcal{S}} \newcommand{\Sbar}{\overline{S}} \newcommand{\K}{{\mathbb K}} \newcommand{\C}{\mathbf{C}} \newcommand{\Cp}{{\mathbb C}_p} \newcommand{\Sets}{\mbox{\rm\bf Sets}} \newcommand{\bcC}{\boldsymbol{\mathcal{C}}} \renewcommand{\P}{\mathbf{P}} \newcommand{\Qbar}{\overline{\Q}} \newcommand{\kbar}{\overline{k}} \newcommand{\dual}{\bot} \newcommand{\T}{\mathbf{T}} \newcommand{\calT}{\mathcal{T}} \newcommand{\cT}{\mathcal{T}} \newcommand{\cbT}{\mathbf{\mathcal{T}}} \newcommand{\cU}{\mathcal{U}} \newcommand{\Z}{\mathbf{Z}} \newcommand{\F}{\mathbf{F}} \newcommand{\Fl}{\F_{\ell}} \newcommand{\Flbar}{\overline{\F}_{\ell}} \newcommand{\Flnu}{\F_{\ell^{\nu}}} \newcommand{\Fbar}{\overline{{\mathbb F}}} \newcommand{\Fpbar}{\overline{\F}_p} \newcommand{\fbar}{\overline{f}} \newcommand{\Qp}{\Q_p} \newcommand{\Ql}{\Q_{\ell}} \newcommand{\Qlbar}{\overline{\Q}_{\ell}} \newcommand{\Qlnr}{\Q_{\ell}^{\text{nr}}} \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{\Aut}{Aut} \DeclareMathOperator{\Frob}{Frob} \DeclareMathOperator{\Ver}{Ver} \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} \newcommand{\galq}{\Gal(\Qbar/\Q)} \newcommand{\rhobar}{\overline{\rho}} \newcommand{\cM}{\mathcal{M}} \newcommand{\cB}{\mathcal{B}} \newcommand{\cE}{\mathcal{E}} \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{\ur}{ur} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Ext}{Ext} \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}{PARI} %%%% Theoremstyles \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{conjecture}[theorem]{Conjecture} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{question}[theorem]{Question} \theoremstyle{remark} \newtheorem{goal}[theorem]{Goal} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise}