CoCalc Public Fileswww / tables / hecke-tutorial / macros.tex
Author: William A. Stein
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{\marginalfootnote}[1]{%
12   \footnote{#1}
13   \marginpar{\hfill {\sf\thefootnote}}%
14}
15\newcommand{\edit}[1]{\marginalfootnote{#1}}
16
17
18\author{William~A. Stein}
19
20
22\par\noindent
23{Version #2}
24\vspace{10ex}
25\par\noindent
26{\bf \LARGE #1}\\
27\vspace{3ex}
28\par\noindent
29{\large W.\thinspace{}A. Stein}\\
30{\small Department of Mathematics, Harvard University}\vspace{1ex}\\
31#3
32\vspace{2ex}\par
33}
34
36\par\noindent
37{Version #2}
38\vspace{10ex}
39\par\noindent
40{\bf \LARGE #1}\\
41\vspace{3ex}
42\par\noindent
43#3
44\vspace{5ex}\par
45}
46
47
48\newcommand{\gzero}{\Gamma_0(N)}
49\newcommand{\esM}{\overline{\sM}}
50\newcommand{\sM}{\boldsymbol{\mathcal{M}}}
51\newcommand{\sS}{\boldsymbol{\mathcal{S}}}
52\newcommand{\sB}{\boldsymbol{\mathcal{B}}}
53\newcommand{\eE}{\mathbf{E}}
55\newcommand{\Bdual}{B^{\vee}}
56
57\newcommand{\defn}[1]{{\em #1}}
58\newcommand{\solution}[1]{\vspace{1em}%
59  \par\noindent{\bf Solution #1.} }
60\newcommand{\todo}[1]{\noindent$\bullet$ {\small \textsf{#1}} $\bullet$\\}
61\newcommand{\done}[1]{\noindent {\small \textsf{Done: #1}}\\}
62\newcommand{\danger}[1]{\marginpar{\small \textsl{#1}}}
63\renewcommand{\div}{\mbox{\rm div}}
64\DeclareMathOperator{\chr}{char}
65\DeclareMathOperator{\ind}{ind}
66\DeclareMathOperator{\im}{im}
67\DeclareMathOperator{\oo}{\infty}
68\DeclareMathOperator{\abs}{abs}
69\DeclareMathOperator{\lcm}{lcm}
70\DeclareMathOperator{\cores}{cores}
71\DeclareMathOperator{\coker}{coker}
72\DeclareMathOperator{\image}{image}
73\DeclareMathOperator{\prt}{part}
74\DeclareMathOperator{\proj}{proj}
75\DeclareMathOperator{\Br}{Br}
76\DeclareMathOperator{\Ann}{Ann}
77\DeclareMathOperator{\End}{End}
78\DeclareMathOperator{\Eis}{Eis}
79\newcommand{\unity}{\mathbf{1}}
80\DeclareMathOperator{\Pic}{Pic}
81\DeclareMathOperator{\Vol}{Vol}
82\DeclareMathOperator{\Vis}{Vis}
83\DeclareMathOperator{\Reg}{Reg}
84%\DeclareMathOperator{\myRes}{Res}
85%\newcommand{\Res}{\myRes}
86\DeclareMathOperator{\Res}{Res}
87\newcommand{\an}{{\rm an}}
88\DeclareMathOperator{\rank}{rank}
89\DeclareMathOperator{\Sel}{Sel}
90\DeclareMathOperator{\Mat}{Mat}
91\DeclareMathOperator{\BSD}{BSD}
92\DeclareMathOperator{\id}{id}
93\DeclareMathOperator{\dz}{dz}
94%\DeclareMathOperator{\Re}{Re}
95\renewcommand{\Re}{\mbox{\rm Re}}
96%\DeclareMathOperator{\Im}{Im}
97\DeclareMathOperator{\Selmer}{Selmer}
98\newcommand{\pfSel}{\widehat{\Sel}}
99\newcommand{\qe}{\stackrel{\mbox{\tiny ?}}{=}}
100\newcommand{\isog}{\simeq}
101\newcommand{\e}{\mathbf{e}}
102\newcommand{\bN}{\mathbf{N}}
103
104% ---- SHA ----
105\DeclareFontEncoding{OT2}{}{} % to enable usage of cyrillic fonts
106  \newcommand{\textcyr}[1]{%
107    {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}%
108     \selectfont #1}}
109\newcommand{\Sha}{{\mbox{\textcyr{Sh}}}}
110
111%\font\cyr=wncyr10 scaled \magstep 1
112%\font\cyr=wncyr10
113
114%\newcommand{\Sha}{{\cyr X}}
115\newcommand{\Shaan}{\Sha_{\mbox{\tiny \rm an}}}
116\newcommand{\TS}{Shafarevich-Tate group}
117
118\newcommand{\Gam}{\Gamma}
119\renewcommand{\Im}{\text{Im}}
120\newcommand{\X}{\mathcal{X}}
121\newcommand{\cH}{\mathcal{H}}
122\newcommand{\cA}{\mathcal{A}}
123\newcommand{\cF}{\mathcal{F}}
124\newcommand{\cG}{\mathcal{G}}
125\newcommand{\cJ}{\mathcal{J}}
126\newcommand{\cL}{\mathcal{L}}
127\newcommand{\cV}{\mathcal{V}}
128\newcommand{\cO}{\mathcal{O}}
129\newcommand{\cQ}{\mathcal{Q}}
130\newcommand{\cX}{\mathcal{X}}
131\newcommand{\ds}{\displaystyle}
132\newcommand{\M}{\mathcal{M}}
133\newcommand{\E}{\mathcal{E}}
134\renewcommand{\L}{\mathcal{L}}
135\newcommand{\J}{\mathcal{J}}
136\DeclareMathOperator{\new}{new}
137\DeclareMathOperator{\Morph}{Morph}
138\DeclareMathOperator{\old}{old}
139\DeclareMathOperator{\Sym}{Sym}
140\DeclareMathOperator{\Symb}{Symb}
141%\newcommand{\Sym}{\mathcal{S}{\rm ym}}
142\newcommand{\dw}{\delta(w)}
143\newcommand{\dwh}{\widehat{\delta(w)}}
144\newcommand{\dlwh}{\widehat{\delta_\l(w)}}
145\newcommand{\dash}{-\!\!\!\!-\!\!\!\!-\!\!\!\!-}
146\DeclareMathOperator{\tor}{tor}
147\newcommand{\Frobl}{\Frob_{\ell}}
148\newcommand{\tE}{\tilde{E}}
149\renewcommand{\l}{\ell}
150\renewcommand{\t}{\tau}
151\DeclareMathOperator{\cond}{cond}
152\DeclareMathOperator{\Spec}{Spec}
153\DeclareMathOperator{\Div}{Div}
154\DeclareMathOperator{\Jac}{Jac}
155\DeclareMathOperator{\res}{res}
156\DeclareMathOperator{\Ker}{Ker}
157\DeclareMathOperator{\Coker}{Coker}
158\DeclareMathOperator{\sep}{sep}
159\DeclareMathOperator{\sign}{sign}
160\DeclareMathOperator{\unr}{unr}
161\newcommand{\N}{\mathcal{N}}
162\newcommand{\U}{\mathcal{U}}
163\newcommand{\Kbar}{\overline{K}}
164\newcommand{\Lbar}{\overline{L}}
165\newcommand{\gammabar}{\overline{\gamma}}
166\newcommand{\q}{\mathbf{q}}
167\renewcommand{\star}{\times}
168\newcommand{\gM}{\mathfrak{M}}
169\newcommand{\gA}{\mathfrak{A}}
170\newcommand{\gP}{\mathfrak{P}}
171\newcommand{\bmu}{\boldsymbol{\mu}}
172\newcommand{\union}{\cup}
173\newcommand{\Tl}{T_{\ell}}
174\newcommand{\into}{\rightarrow}
175\newcommand{\onto}{\rightarrow\!\!\!\!\rightarrow}
176\newcommand{\intersect}{\cap}
177\newcommand{\meet}{\cap}
178\newcommand{\cross}{\times}
179\DeclareMathOperator{\md}{mod}
180\DeclareMathOperator{\toric}{toric}
181\DeclareMathOperator{\tors}{tors}
182\DeclareMathOperator{\Frac}{Frac}
183\DeclareMathOperator{\corank}{corank}
184\newcommand{\rb}{\overline{\rho}}
185\newcommand{\ra}{\rightarrow}
186\newcommand{\xra}[1]{\xrightarrow{#1}}
187\newcommand{\hra}{\hookrightarrow}
188\newcommand{\la}{\leftarrow}
189\newcommand{\lra}{\longrightarrow}
190\newcommand{\riso}{\xrightarrow{\sim}}
191\newcommand{\da}{\downarrow}
192\newcommand{\ua}{\uparrow}
193\newcommand{\con}{\equiv}
194\newcommand{\Gm}{\mathbf{G}_m}
195\newcommand{\pni}{\par\noindent}
196\newcommand{\set}[1]{\{#1\}}
197\newcommand{\iv}{^{-1}}
198\newcommand{\alp}{\alpha}
199\newcommand{\bq}{\mathbf{q}}
200\newcommand{\cpp}{{\tt C++}}
201\newcommand{\tensor}{\otimes}
202\newcommand{\bg}{{\tt BruceGenus}}
203\newcommand{\abcd}[4]{\left(
204        \begin{smallmatrix}#1&#2\\#3&#4\end{smallmatrix}\right)}
205\newcommand{\mthree}[9]{\left(
206        \begin{matrix}#1&#2&#3\\#4&#5&#6\\#7&#8&#9
207        \end{matrix}\right)}
208\newcommand{\mtwo}[4]{\left(
209        \begin{matrix}#1&#2\\#3&#4
210        \end{matrix}\right)}
211\newcommand{\vtwo}[2]{\left(
212        \begin{matrix}#1\\#2
213        \end{matrix}\right)}
214\newcommand{\smallmtwo}[4]{\left(
215        \begin{smallmatrix}#1&#2\\#3&#4
216        \end{smallmatrix}\right)}
217\newcommand{\twopii}{2\pi{}i{}}
218\newcommand{\eps}{\varepsilon}
219\newcommand{\vphi}{\varphi}
220\newcommand{\gp}{\mathfrak{p}}
221\newcommand{\W}{\mathcal{W}}
222\newcommand{\oz}{\overline{z}}
223\newcommand{\Zpstar}{\Zp^{\star}}
224\newcommand{\Zhat}{\widehat{\Z}}
225\newcommand{\Zl}{\Z_{\ell}}
226\newcommand{\comment}[1]{}
227\newcommand{\Q}{\mathbf{Q}}
228\newcommand{\GQ}{G_{\Q}}
229\newcommand{\R}{\mathbf{R}}
230\newcommand{\D}{{\mathbf D}}
231\newcommand{\cC}{\mathcal{C}}
232\newcommand{\cD}{\mathcal{D}}
233\newcommand{\cP}{\mathcal{P}}
234\newcommand{\cS}{\mathcal{S}}
235\newcommand{\Sbar}{\overline{S}}
236\newcommand{\K}{{\mathbf K}}
237\newcommand{\C}{\mathbf{C}}
238\newcommand{\Cp}{{\mathbf C}_p}
239\newcommand{\Sets}{\mbox{\rm\bf Sets}}
240\newcommand{\bcC}{\boldsymbol{\mathcal{C}}}
241\renewcommand{\P}{\mathbf{P}}
242\newcommand{\Qbar}{\overline{\Q}}
243\newcommand{\kbar}{\overline{k}}
244\newcommand{\dual}{\bot}
245\newcommand{\T}{\mathbf{T}}
246\newcommand{\calT}{\mathcal{T}}
247\newcommand{\cT}{\mathcal{T}}
248\newcommand{\cbT}{\mathbf{\mathcal{T}}}
249\newcommand{\cU}{\mathcal{U}}
250\newcommand{\Z}{\mathbf{Z}}
251\newcommand{\F}{\mathbf{F}}
252\newcommand{\Fl}{\F_{\ell}}
253\newcommand{\Fell}{\Fl}
254\newcommand{\Flbar}{\overline{\F}_{\ell}}
255\newcommand{\Flnu}{\F_{\ell^{\nu}}}
256\newcommand{\Fbar}{\overline{\F}}
257\newcommand{\Fpbar}{\overline{\F}_p}
258\newcommand{\fbar}{\overline{f}}
259\newcommand{\Qp}{\Q_p}
260\newcommand{\Ql}{\Q_{\ell}}
261\newcommand{\Qell}{\Q_{\ell}}
262\newcommand{\Qlbar}{\overline{\Q}_{\ell}}
263\newcommand{\Qlnr}{\Q_{\ell}^{\text{nr}}}
264\newcommand{\Qlur}{\Q_{\ell}^{\text{ur}}}
265\newcommand{\Qltm}{\Q_{\ell}^{\text{tame}}}
266\newcommand{\Qv}{\Q_v}
267\newcommand{\Qpbar}{\Qbar_p}
268\newcommand{\Zp}{\Z_p}
269\newcommand{\Fp}{\F_p}
270\newcommand{\Fq}{\F_q}
271\newcommand{\Fqbar}{\overline{\F}_q}
274\renewcommand{\O}{\mathcal{O}}
275\newcommand{\A}{\mathcal{A}}
276\newcommand{\Og}{O_{\gamma}}
277\newcommand{\isom}{\cong}
278\newcommand{\ncisom}{\approx}   % noncanonical isomorphism
279\DeclareMathOperator{\ab}{ab}
280\DeclareMathOperator{\Aut}{Aut}
281\DeclareMathOperator{\Frob}{Frob}
282\DeclareMathOperator{\Fr}{Fr}
283\DeclareMathOperator{\Ver}{Ver}
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