1% macros.tex
2\usepackage[active]{srcltx}
3\usepackage{amsmath}
4\usepackage{amsfonts}
5\usepackage{amssymb}
6\usepackage{amsthm}
7
8
9
10
11%\newcommand{\edit}[1]{\footnote{#1}\marginpar{\hfill {\sf\thefootnote}}}
12\newcommand{\edit}[1]{{\bf [[Todo: #1]]}}
13
14
15%\author{William~A. Stein}
16
17
19\par\noindent
20{Version #2}
21\vspace{10ex}
22\par\noindent
23{\bf \LARGE #1}\\
24\vspace{3ex}
25\par\noindent
26{\large W.\thinspace{}A. Stein}\\
27{\small Department of Mathematics, Harvard University}\vspace{1ex}\\
28#3
29\vspace{2ex}\par
30}
31
33\par\noindent
34{Version #2}
35\vspace{10ex}
36\par\noindent
37{\bf \LARGE #1}\\
38\vspace{3ex}
39\par\noindent
40#3
41\vspace{5ex}\par
42}
43
44
45\newcommand{\gzero}{\Gamma_0(N)}
46\newcommand{\esM}{\overline{\sM}}
47\newcommand{\sM}{\boldsymbol{\mathcal{M}}}
48\newcommand{\sS}{\boldsymbol{\mathcal{S}}}
49\newcommand{\sB}{\boldsymbol{\mathcal{B}}}
50\newcommand{\eE}{\mathbf{E}}
52\newcommand{\Bdual}{B^{\vee}}
53
54\newcommand{\defn}[1]{{\em #1}}
55\newcommand{\solution}[1]{\vspace{1em}%
56  \par\noindent{\bf Solution #1.} }
57\newcommand{\todo}[1]{\noindent$\bullet$ {\small \textsf{#1}} $\bullet$\\}
58\newcommand{\done}[1]{\noindent {\small \textsf{Done: #1}}\\}
59\newcommand{\danger}[1]{\marginpar{\small \textsl{#1}}}
60\renewcommand{\div}{\mbox{\rm div}}
61\DeclareMathOperator{\chr}{char}
62\DeclareMathOperator{\ind}{ind}
63\DeclareMathOperator{\im}{im}
64\DeclareMathOperator{\oo}{\infty}
65\DeclareMathOperator{\abs}{abs}
66\DeclareMathOperator{\lcm}{lcm}
67\DeclareMathOperator{\cores}{cores}
68\DeclareMathOperator{\coker}{coker}
69\DeclareMathOperator{\image}{image}
70\DeclareMathOperator{\prt}{part}
71\DeclareMathOperator{\proj}{proj}
72\DeclareMathOperator{\Br}{Br}
73\DeclareMathOperator{\Ann}{Ann}
74\DeclareMathOperator{\End}{End}
75\DeclareMathOperator{\Tan}{Tan}
76\DeclareMathOperator{\Eis}{Eis}
77\newcommand{\unity}{\mathbf{1}}
78\DeclareMathOperator{\Pic}{Pic}
79\DeclareMathOperator{\Vol}{Vol}
80\DeclareMathOperator{\Vis}{Vis}
81\DeclareMathOperator{\Reg}{Reg}
82%\DeclareMathOperator{\myRes}{Res}
83%\newcommand{\Res}{\myRes}
84\DeclareMathOperator{\Res}{Res}
85\newcommand{\an}{{\rm an}}
86\DeclareMathOperator{\rank}{rank}
87\DeclareMathOperator{\Sel}{Sel}
88\DeclareMathOperator{\Mat}{Mat}
89\DeclareMathOperator{\BSD}{BSD}
90\DeclareMathOperator{\id}{id}
91\DeclareMathOperator{\dz}{dz}
92%\DeclareMathOperator{\Re}{Re}
93\renewcommand{\Re}{\mbox{\rm Re}}
94%\DeclareMathOperator{\Im}{Im}
95\DeclareMathOperator{\Selmer}{Selmer}
96\newcommand{\pfSel}{\widehat{\Sel}}
97\newcommand{\qe}{\stackrel{\mbox{\tiny ?}}{=}}
98\newcommand{\isog}{\simeq}
99\newcommand{\e}{\mathbf{e}}
100\newcommand{\bN}{\mathbf{N}}
101
102% ---- SHA ----
103\DeclareFontEncoding{OT2}{}{} % to enable usage of cyrillic fonts
104  \newcommand{\textcyr}[1]{%
105    {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}%
106     \selectfont #1}}
107\newcommand{\Sha}{{\mbox{\textcyr{Sh}}}}
108
109%\font\cyr=wncyr10 scaled \magstep 1
110%\font\cyr=wncyr10
111
112%\newcommand{\Sha}{{\cyr X}}
113\newcommand{\Shaan}{\Sha_{\mbox{\tiny \rm an}}}
114\newcommand{\TS}{Shafarevich-Tate group}
115
116\newcommand{\Gam}{\Gamma}
117\renewcommand{\Im}{\text{Im}}
118\newcommand{\X}{\mathcal{X}}
119\newcommand{\cH}{\mathcal{H}}
120\newcommand{\cA}{\mathcal{A}}
121\newcommand{\cF}{\mathcal{F}}
122\newcommand{\cG}{\mathcal{G}}
123\newcommand{\cJ}{\mathcal{J}}
124\newcommand{\cL}{\mathcal{L}}
125\newcommand{\cV}{\mathcal{V}}
126\newcommand{\cO}{\mathcal{O}}
127\newcommand{\cQ}{\mathcal{Q}}
128\newcommand{\cX}{\mathcal{X}}
129\newcommand{\ds}{\displaystyle}
130\newcommand{\M}{\mathcal{M}}
131\newcommand{\E}{\mathcal{E}}
132\renewcommand{\L}{\mathcal{L}}
133\newcommand{\J}{\mathcal{J}}
134\DeclareMathOperator{\new}{new}
135\DeclareMathOperator{\Morph}{Morph}
136\DeclareMathOperator{\old}{old}
137\DeclareMathOperator{\Sym}{Sym}
138\DeclareMathOperator{\Symb}{Symb}
139%\newcommand{\Sym}{\mathcal{S}{\rm ym}}
140\newcommand{\dw}{\delta(w)}
141\newcommand{\dwh}{\widehat{\delta(w)}}
142\newcommand{\dlwh}{\widehat{\delta_\l(w)}}
143\newcommand{\dash}{-\!\!\!\!-\!\!\!\!-\!\!\!\!-}
144\DeclareMathOperator{\tor}{tor}
145\newcommand{\Frobl}{\Frob_{\ell}}
146\newcommand{\tE}{\tilde{E}}
147\renewcommand{\l}{\ell}
148\renewcommand{\t}{\tau}
149\DeclareMathOperator{\cond}{cond}
150\DeclareMathOperator{\Spec}{Spec}
151\DeclareMathOperator{\Div}{Div}
152\DeclareMathOperator{\Jac}{Jac}
153\DeclareMathOperator{\res}{res}
154\DeclareMathOperator{\Ker}{Ker}
155\DeclareMathOperator{\Coker}{Coker}
156\DeclareMathOperator{\sep}{sep}
157\DeclareMathOperator{\sign}{sign}
158\DeclareMathOperator{\unr}{unr}
159\newcommand{\N}{\mathcal{N}}
160\newcommand{\U}{\mathcal{U}}
161\newcommand{\Kbar}{\overline{K}}
162\newcommand{\Lbar}{\overline{L}}
163\newcommand{\gammabar}{\overline{\gamma}}
164\newcommand{\q}{\mathbf{q}}
165\renewcommand{\star}{\times}
166\newcommand{\gM}{\mathfrak{M}}
167\newcommand{\gA}{\mathfrak{A}}
168\newcommand{\gP}{\mathfrak{P}}
169\newcommand{\bmu}{\boldsymbol{\mu}}
170\newcommand{\union}{\cup}
171\newcommand{\Tl}{T_{\ell}}
172\newcommand{\into}{\rightarrow}
173\newcommand{\onto}{\rightarrow\!\!\!\!\rightarrow}
174\newcommand{\intersect}{\cap}
175\newcommand{\meet}{\cap}
176\newcommand{\cross}{\times}
177\DeclareMathOperator{\md}{mod}
178\DeclareMathOperator{\toric}{toric}
179\DeclareMathOperator{\tors}{tors}
180\DeclareMathOperator{\Frac}{Frac}
181\DeclareMathOperator{\corank}{corank}
182\newcommand{\rb}{\overline{\rho}}
183\newcommand{\ra}{\rightarrow}
184\newcommand{\xra}[1]{\xrightarrow{#1}}
185\newcommand{\hra}{\hookrightarrow}
186\newcommand{\la}{\leftarrow}
187\newcommand{\lra}{\longrightarrow}
188\newcommand{\riso}{\xrightarrow{\sim}}
189\newcommand{\da}{\downarrow}
190\newcommand{\ua}{\uparrow}
191\newcommand{\con}{\equiv}
192\newcommand{\Gm}{\mathbf{G}_m}
193\newcommand{\pni}{\par\noindent}
194\newcommand{\set}[1]{\{#1\}}
195\newcommand{\iv}{^{-1}}
196\newcommand{\alp}{\alpha}
197\newcommand{\bq}{\mathbf{q}}
198\newcommand{\cpp}{{\tt C++}}
199\newcommand{\tensor}{\otimes}
200\newcommand{\bg}{{\tt BruceGenus}}
201\newcommand{\abcd}[4]{\left(
202        \begin{smallmatrix}#1&#2\\#3&#4\end{smallmatrix}\right)}
203\newcommand{\mthree}[9]{\left(
204        \begin{matrix}#1&#2&#3\\#4&#5&#6\\#7&#8&#9
205        \end{matrix}\right)}
206\newcommand{\mtwo}[4]{\left(
207        \begin{matrix}#1&#2\\#3&#4
208        \end{matrix}\right)}
209\newcommand{\vtwo}[2]{\left(
210        \begin{matrix}#1\\#2
211        \end{matrix}\right)}
212\newcommand{\smallmtwo}[4]{\left(
213        \begin{smallmatrix}#1&#2\\#3&#4
214        \end{smallmatrix}\right)}
215\newcommand{\twopii}{2\pi{}i{}}
216\newcommand{\eps}{\varepsilon}
217\newcommand{\vphi}{\varphi}
218\newcommand{\gp}{\mathfrak{p}}
219\newcommand{\W}{\mathcal{W}}
220\newcommand{\oz}{\overline{z}}
221\newcommand{\Zpstar}{\Zp^{\star}}
222\newcommand{\Zhat}{\widehat{\Z}}
223\newcommand{\Zbar}{\overline{\Z}}
224\newcommand{\Zl}{\Z_{\ell}}
225\newcommand{\comment}[1]{}
226\newcommand{\Q}{\mathbf{Q}}
227\newcommand{\GQ}{G_{\Q}}
228\newcommand{\R}{\mathbf{R}}
229\newcommand{\D}{{\mathbf D}}
230\newcommand{\cC}{\mathcal{C}}
231\newcommand{\cD}{\mathcal{D}}
232\newcommand{\cP}{\mathcal{P}}
233\newcommand{\cS}{\mathcal{S}}
234\newcommand{\Sbar}{\overline{S}}
235\newcommand{\K}{{\mathbf K}}
236\newcommand{\C}{\mathbf{C}}
237\newcommand{\Cp}{{\mathbf C}_p}
238\newcommand{\Sets}{\mbox{\rm\bf Sets}}
239\newcommand{\bcC}{\boldsymbol{\mathcal{C}}}
240\renewcommand{\P}{\mathbf{P}}
241\newcommand{\Qbar}{\overline{\Q}}
242\newcommand{\kbar}{\overline{k}}
243\newcommand{\dual}{\bot}
244\newcommand{\T}{\mathbf{T}}
245\newcommand{\calT}{\mathcal{T}}
246\newcommand{\cT}{\mathcal{T}}
247\newcommand{\cbT}{\mathbf{\mathcal{T}}}
248\newcommand{\cU}{\mathcal{U}}
249\newcommand{\Z}{\mathbf{Z}}
250\newcommand{\F}{\mathbf{F}}
251\newcommand{\Fl}{\F_{\ell}}
252\newcommand{\Fell}{\Fl}
253\newcommand{\Flbar}{\overline{\F}_{\ell}}
254\newcommand{\Flnu}{\F_{\ell^{\nu}}}
255\newcommand{\Fbar}{\overline{\F}}
256\newcommand{\Fpbar}{\overline{\F}_p}
257\newcommand{\fbar}{\overline{f}}
258\newcommand{\Qp}{\Q_p}
259\newcommand{\Ql}{\Q_{\ell}}
260\newcommand{\Qell}{\Q_{\ell}}
261\newcommand{\Qlbar}{\overline{\Q}_{\ell}}
262\newcommand{\Qlnr}{\Q_{\ell}^{\text{nr}}}
263\newcommand{\Qlur}{\Q_{\ell}^{\text{ur}}}
264\newcommand{\Qltm}{\Q_{\ell}^{\text{tame}}}
265\newcommand{\Qv}{\Q_v}
266\newcommand{\Qpbar}{\Qbar_p}
267\newcommand{\Zp}{\Z_p}
268\newcommand{\Fp}{\F_p}
269\newcommand{\Fq}{\F_q}
270\newcommand{\Fqbar}{\overline{\F}_q}
273\renewcommand{\O}{\mathcal{O}}
274\newcommand{\A}{\mathcal{A}}
275\newcommand{\Og}{O_{\gamma}}
276\newcommand{\isom}{\cong}
277\newcommand{\ncisom}{\approx}   % noncanonical isomorphism
278\DeclareMathOperator{\ab}{ab}
279\DeclareMathOperator{\Aut}{Aut}
280\DeclareMathOperator{\Frob}{Frob}
281\DeclareMathOperator{\Fr}{Fr}
282\DeclareMathOperator{\Ver}{Ver}
283\DeclareMathOperator{\Norm}{Norm}
284\DeclareMathOperator{\norm}{norm}
285\DeclareMathOperator{\disc}{disc}
286\DeclareMathOperator{\ord}{ord}
287\DeclareMathOperator{\GL}{GL}
288\DeclareMathOperator{\PSL}{PSL}
289\DeclareMathOperator{\PGL}{PGL}
290\DeclareMathOperator{\Gal}{Gal}
291\DeclareMathOperator{\SL}{SL}
292\DeclareMathOperator{\SO}{SO}
293\newcommand{\galq}{\Gal(\Qbar/\Q)}
294\newcommand{\rhobar}{\overline{\rho}}
295\newcommand{\cM}{\mathcal{M}}
296\newcommand{\cB}{\mathcal{B}}
297\newcommand{\cE}{\mathcal{E}}
298\newcommand{\cR}{\mathcal{R}}
299\newcommand{\et}{\text{\rm\'et}}
300
301\newcommand{\sltwoz}{\SL_2(\Z)}
302\newcommand{\sltwo}{\SL_2}
303\newcommand{\gltwoz}{\GL_2(\Z)}
304\newcommand{\mtwoz}{M_2(\Z)}
305\newcommand{\gltwoq}{\GL_2(\Q)}
306\newcommand{\gltwo}{\GL_2}
307\newcommand{\gln}{\GL_n}
308\newcommand{\psltwoz}{\PSL_2(\Z)}
309\newcommand{\psltwo}{\PSL_2}
310\newcommand{\h}{\mathfrak{h}}
311\renewcommand{\a}{\mathfrak{a}}
312\newcommand{\p}{\mathfrak{p}}
313\newcommand{\m}{\mathfrak{m}}
314\newcommand{\trho}{\tilde{\rho}}
315\newcommand{\rhol}{\rho_{\ell}}
316\newcommand{\rhoss}{\rho^{\text{ss}}}
317\DeclareMathOperator{\tr}{tr}
318\DeclareMathOperator{\order}{order}
319\DeclareMathOperator{\ur}{ur}
320\DeclareMathOperator{\Tr}{Tr}
321\DeclareMathOperator{\Hom}{Hom}
322\DeclareMathOperator{\Mor}{Mor}
323\DeclareMathOperator{\HH}{H}
324\renewcommand{\H}{\HH}
325\DeclareMathOperator{\Ext}{Ext}
326\DeclareMathOperator{\Tor}{Tor}
327\newcommand{\smallzero}{\left(\begin{smallmatrix}0&0\\0&0
328                        \end{smallmatrix}\right)}
329\newcommand{\smallone}{\left(\begin{smallmatrix}1&0\\0&1
330                        \end{smallmatrix}\right)}
331
332\newcommand{\pari}{{\sc Pari}}
333\newcommand{\magma}{{\sc Magma}}
334\newcommand{\hecke}{{\sc Hecke}}
335\newcommand{\lidia}{{\sc LiDIA}}
336
337%%%% Theoremstyles
338\theoremstyle{plain}
339\newtheorem{theorem}{Theorem}[section]
340\newtheorem{proposition}[theorem]{Proposition}
341\newtheorem{corollary}[theorem]{Corollary}
342\newtheorem{claim}[theorem]{Claim}
343\newtheorem{lemma}[theorem]{Lemma}
344\newtheorem{conjecture}[theorem]{Conjecture}
345
346\theoremstyle{definition}
347\newtheorem{definition}[theorem]{Definition}
348\newtheorem{algorithm}[theorem]{Algorithm}
349\newtheorem{question}[theorem]{Question}
350\newtheorem{problem}[theorem]{Problem}
351
352%\theoremstyle{remark}
353\newtheorem{goal}[theorem]{Goal}
354\newtheorem{remark}[theorem]{Remark}
355\newtheorem{remarks}[theorem]{Remarks}
356\newtheorem{example}[theorem]{Example}
357\newtheorem{exercise}[theorem]{Exercise}
358
359\numberwithin{equation}{section}
360\numberwithin{figure}{section}
361\numberwithin{table}{section}
362
363
364% bulleted list environment
365\newenvironment{bulletlist}
366   {
367      \begin{list}
368         {$\bullet$}
369         {
370            \setlength{\itemsep}{.5ex}
371            \setlength{\parsep}{0ex}
372            \setlength{\parskip}{0ex}
373            \setlength{\topsep}{.5ex}
374         }
375   }
376   {
377      \end{list}
378   }
379%end newenvironment
380
381% bulleted list environment
382\newenvironment{dashlist}
383   {
384      \begin{list}
385         {---}
386         {
387            \setlength{\itemsep}{.5ex}
388            \setlength{\parsep}{0ex}
389            \setlength{\parskip}{0ex}
390            \setlength{\topsep}{.5ex}
391         }
392   }
393   {
394      \end{list}
395   }
396%end newenvironment
397
398% numbered list environment
399\newcounter{listnum}
400\newenvironment{numlist}
401   {
402      \begin{list}
403            {{\em \thelistnum.}}{
404            \usecounter{listnum}
405            \setlength{\itemsep}{.5ex}
406            \setlength{\parsep}{0ex}
407            \setlength{\parskip}{0ex}
408            \setlength{\topsep}{.5ex}
409         }
410   }
411   {
412      \end{list}
413   }
414%end newenvironment
415
416\newcommand{\hd}[1]{\vspace{1ex}\noindent{\bf #1} }
417\newcommand{\nf}[1]{\underline{#1}}
418\newcommand{\cbar}{\overline{c}}
419