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