Sharedwww / tables / Notes / macros.texOpen in CoCalc
Author: William A. Stein
1
% macros.tex
2
\usepackage{amsmath}
3
\usepackage{amsfonts}
4
\usepackage{amssymb}
5
\usepackage{amsthm}
6
7
\author{W.\thinspace{}A. Stein\footnote{University of California, Berkeley, U.S.A.,
8
{\tt was@math.berkeley.edu}}}
9
10
%\font\cyr=wncyr10 scaled \magstep 1
11
\font\cyr=wncyr10
12
13
\newcommand{\myhead}[3]{
14
\par\noindent
15
{Preprint (\today), Version #2}
16
\vspace{10ex}
17
\par\noindent
18
{\bf \LARGE #1}\\
19
\vspace{3ex}
20
\par\noindent
21
{\large W.\thinspace{}A. Stein}\\
22
{\small Department of Mathematics, University of California, Berkeley,
23
CA 94720, USA}\vspace{1ex}\\
24
#3
25
\vspace{2ex}\par
26
}
27
28
\newcommand{\myheadauth}[3]{
29
\par\noindent
30
{Preprint (\today), Version #2}
31
\vspace{10ex}
32
\par\noindent
33
{\bf \LARGE #1}\\
34
\vspace{3ex}
35
\par\noindent
36
#3
37
\vspace{5ex}\par
38
}
39
40
41
%\newcommand{\sM}{\underline{\mathcal{M}}}
42
\newcommand{\sM}{\boldsymbol{\mathcal{M}}}
43
\newcommand{\sS}{\boldsymbol{\mathcal{S}}}
44
\newcommand{\sB}{\boldsymbol{\mathcal{B}}}
45
\newcommand{\eE}{\mathbf{E}}
46
\newcommand{\Adual}{\hat{A}}
47
\newcommand{\Bdual}{\hat{B}}
48
49
\newcommand{\defn}[1]{{\em #1}}
50
\newcommand{\solution}[1]{\vspace{1em}%
51
\par\noindent{\bf Solution #1.} }
52
\newcommand{\todo}[1]{\noindent$\bullet$ {\small \textsf{#1}} $\bullet$\\}
53
\newcommand{\done}[1]{\noindent {\small \textsf{Done: #1}}\\}
54
\newcommand{\danger}[1]{\marginpar{\small \textsl{#1}}}
55
\DeclareMathOperator{\ind}{ind}
56
\DeclareMathOperator{\im}{im}
57
\DeclareMathOperator{\oo}{\infty}
58
\DeclareMathOperator{\abs}{abs}
59
\DeclareMathOperator{\lcm}{lcm}
60
\DeclareMathOperator{\cores}{cores}
61
\DeclareMathOperator{\coker}{coker}
62
\DeclareMathOperator{\image}{image}
63
\DeclareMathOperator{\prt}{part}
64
\DeclareMathOperator{\proj}{proj}
65
\DeclareMathOperator{\Br}{Br}
66
\DeclareMathOperator{\Ann}{Ann}
67
\DeclareMathOperator{\End}{End}
68
\DeclareMathOperator{\Eis}{Eis}
69
\newcommand{\unity}{\mathbf{1}}
70
\DeclareMathOperator{\Pic}{Pic}
71
\DeclareMathOperator{\Vol}{Vol}
72
\DeclareMathOperator{\Reg}{Reg}
73
\newcommand{\an}{{\rm an}}
74
\DeclareMathOperator{\rank}{rank}
75
\DeclareMathOperator{\Sel}{Sel}
76
\DeclareMathOperator{\Mat}{Mat}
77
\DeclareMathOperator{\BSD}{BSD}
78
\DeclareMathOperator{\id}{id}
79
\DeclareMathOperator{\dz}{dz}
80
%\DeclareMathOperator{\Re}{Re}
81
%\DeclareMathOperator{\Im}{Im}
82
\DeclareMathOperator{\Selmer}{Selmer}
83
\newcommand{\pfSel}{\widehat{\Sel}}
84
\newcommand{\qe}{\stackrel{\mbox{\tiny ?}}{=}}
85
\newcommand{\isog}{\simeq}
86
\newcommand{\e}{\mathbf{e}}
87
\newcommand{\bN}{\mathbf{N}}
88
\newcommand{\Sha}{\mbox{\cyr X}}
89
\newcommand{\Shaan}{\Sha_{\mbox{\tiny \rm an}}}
90
\newcommand{\TS}{Tate-Shafarevich group}
91
\newcommand{\Gam}{\Gamma}
92
\renewcommand{\Im}{\text{Im}}
93
\newcommand{\X}{\mathcal{X}}
94
\newcommand{\cH}{\mathcal{H}}
95
\newcommand{\cA}{\mathcal{A}}
96
\newcommand{\cJ}{\mathcal{J}}
97
\newcommand{\ds}{\displaystyle}
98
\newcommand{\M}{\mathcal{M}}
99
\newcommand{\E}{\mathcal{E}}
100
\renewcommand{\L}{\mathcal{L}}
101
\newcommand{\J}{\mathcal{J}}
102
\DeclareMathOperator{\new}{new}
103
\DeclareMathOperator{\Morph}{Morph}
104
\DeclareMathOperator{\old}{old}
105
\DeclareMathOperator{\Sym}{Sym}
106
\DeclareMathOperator{\Symb}{Symb}
107
%\newcommand{\Sym}{\mathcal{S}{\rm ym}}
108
\newcommand{\dw}{\delta(w)}
109
\newcommand{\dwh}{\widehat{\delta(w)}}
110
\newcommand{\dlwh}{\widehat{\delta_\l(w)}}
111
\newcommand{\dash}{-\!\!\!\!-\!\!\!\!-\!\!\!\!-}
112
\DeclareMathOperator{\tor}{tor}
113
\newcommand{\Frobl}{\Frob_{\ell}}
114
\newcommand{\tE}{\tilde{E}}
115
\renewcommand{\l}{\ell}
116
\renewcommand{\t}{\tau}
117
\DeclareMathOperator{\Spec}{Spec}
118
\DeclareMathOperator{\Div}{Div}
119
\DeclareMathOperator{\Jac}{Jac}
120
\DeclareMathOperator{\res}{res}
121
\DeclareMathOperator{\Ker}{Ker}
122
\DeclareMathOperator{\sign}{sign}
123
\DeclareMathOperator{\unr}{unr}
124
\newcommand{\N}{\mathcal{N}}
125
\newcommand{\U}{\mathcal{U}}
126
\newcommand{\Kbar}{\overline{K}}
127
\newcommand{\Lbar}{\overline{L}}
128
\newcommand{\gammabar}{\overline{\gamma}}
129
\newcommand{\q}{\mathbf{q}}
130
\renewcommand{\star}{\times}
131
\newcommand{\gM}{\mathfrak{M}}
132
\newcommand{\gA}{\mathfrak{A}}
133
\newcommand{\gP}{\mathfrak{P}}
134
\newcommand{\bmu}{\boldsymbol{\mu}}
135
\newcommand{\union}{\cup}
136
\newcommand{\Tl}{T_{\ell}}
137
\newcommand{\into}{\rightarrow}
138
\newcommand{\onto}{\rightarrow\!\!\!\!\rightarrow}
139
\newcommand{\intersect}{\cap}
140
\newcommand{\cross}{\times}
141
\DeclareMathOperator{\md}{mod}
142
\DeclareMathOperator{\toric}{toric}
143
\DeclareMathOperator{\Frac}{Frac}
144
\newcommand{\rb}{\overline{\rho}}
145
\newcommand{\ra}{\rightarrow}
146
\newcommand{\la}{\leftarrow}
147
\newcommand{\lra}{\longrightarrow}
148
\newcommand{\riso}{\xrightarrow{\sim}}
149
\newcommand{\da}{\downarrow}
150
\newcommand{\ua}{\uparrow}
151
\newcommand{\con}{\equiv}
152
\newcommand{\Gm}{\mathbf{G}_m}
153
\newcommand{\pni}{\par\noindent}
154
\newcommand{\set}[1]{\{#1\}}
155
\newcommand{\iv}{^{-1}}
156
\newcommand{\alp}{\alpha}
157
\newcommand{\bq}{\mathbf{q}}
158
\newcommand{\cpp}{{\tt C++}}
159
\newcommand{\tensor}{\otimes}
160
\newcommand{\bg}{{\tt BruceGenus}}
161
\newcommand{\abcd}[4]{\left(
162
\begin{smallmatrix}#1&#2\\#3&#4\end{smallmatrix}\right)}
163
\newcommand{\mthree}[9]{\left(
164
\begin{matrix}#1&#2&#3\\#4&#5&#6\\#7&#8&#9
165
\end{matrix}\right)}
166
\newcommand{\mtwo}[4]{\left(
167
\begin{matrix}#1&#2\\#3&#4
168
\end{matrix}\right)}
169
\newcommand{\vtwo}[2]{\left(
170
\begin{matrix}#1\\#2
171
\end{matrix}\right)}
172
\newcommand{\smallmtwo}[4]{\left(
173
\begin{smallmatrix}#1&#2\\#3&#4
174
\end{smallmatrix}\right)}
175
\newcommand{\eps}{\varepsilon}
176
\newcommand{\vphi}{\varphi}
177
\newcommand{\gp}{\mathfrak{p}}
178
\newcommand{\W}{\mathcal{W}}
179
\newcommand{\oz}{\overline{z}}
180
\newcommand{\Zpstar}{\Zp^{\star}}
181
\newcommand{\Zhat}{\widehat{\Z}}
182
\newcommand{\Zl}{\Z_{\ell}}
183
\newcommand{\comment}[1]{}
184
\newcommand{\Q}{\mathbf{Q}}
185
\newcommand{\GQ}{G_{\Q}}
186
\newcommand{\R}{\mathbf{R}}
187
\newcommand{\D}{{\mathbb D}}
188
\newcommand{\cC}{\mathcal{C}}
189
\newcommand{\cD}{\mathcal{D}}
190
\newcommand{\cS}{\mathcal{S}}
191
\newcommand{\Sbar}{\overline{S}}
192
\newcommand{\K}{{\mathbb K}}
193
\newcommand{\C}{\mathbf{C}}
194
\newcommand{\Cp}{{\mathbb C}_p}
195
\newcommand{\Sets}{\mbox{\rm\bf Sets}}
196
\newcommand{\bcC}{\boldsymbol{\mathcal{C}}}
197
\renewcommand{\P}{\mathbf{P}}
198
\newcommand{\Qbar}{\overline{\Q}}
199
\newcommand{\kbar}{\overline{k}}
200
\newcommand{\dual}{\bot}
201
\newcommand{\T}{\mathbf{T}}
202
\newcommand{\calT}{\mathcal{T}}
203
\newcommand{\cT}{\mathcal{T}}
204
\newcommand{\cX}{\mathcal{X}}
205
\newcommand{\cbT}{\mathbf{\mathcal{T}}}
206
\newcommand{\cU}{\mathcal{U}}
207
\newcommand{\Z}{\mathbf{Z}}
208
\newcommand{\F}{\mathbf{F}}
209
\newcommand{\Fl}{\F_{\ell}}
210
\newcommand{\Fell}{\Fl}
211
\newcommand{\Flbar}{\overline{\F}_{\ell}}
212
\newcommand{\Flnu}{\F_{\ell^{\nu}}}
213
\newcommand{\Fbar}{\overline{\F}}
214
\newcommand{\Fpbar}{\overline{\F}_p}
215
\newcommand{\fbar}{\overline{f}}
216
\newcommand{\Qp}{\Q_p}
217
\newcommand{\Ql}{\Q_{\ell}}
218
\newcommand{\Qlbar}{\overline{\Q}_{\ell}}
219
\newcommand{\Qlnr}{\Q_{\ell}^{\text{nr}}}
220
\newcommand{\Qltm}{\Q_{\ell}^{\text{tame}}}
221
\newcommand{\Qv}{\Q_v}
222
\newcommand{\Qpbar}{\Qbar_p}
223
\newcommand{\Zp}{\Z_p}
224
\newcommand{\Fp}{\F_p}
225
\newcommand{\Fq}{\F_q}
226
\newcommand{\Fqbar}{\overline{\F}_q}
227
\newcommand{\Ad}{Ad}
228
\newcommand{\adz}{\Ad^0}
229
\renewcommand{\O}{\mathcal{O}}
230
\newcommand{\A}{\mathcal{A}}
231
\newcommand{\Og}{O_{\gamma}}
232
\newcommand{\isom}{\cong}
233
\newcommand{\ncisom}{\approx} % noncanonical isomorphism
234
\DeclareMathOperator{\Aut}{Aut}
235
\DeclareMathOperator{\Frob}{Frob}
236
\DeclareMathOperator{\Ver}{Ver}
237
\DeclareMathOperator{\Norm}{Norm}
238
\DeclareMathOperator{\disc}{disc}
239
\DeclareMathOperator{\ord}{ord}
240
\DeclareMathOperator{\GL}{GL}
241
\DeclareMathOperator{\PSL}{PSL}
242
\DeclareMathOperator{\PGL}{PGL}
243
\DeclareMathOperator{\Gal}{Gal}
244
\DeclareMathOperator{\SL}{SL}
245
\DeclareMathOperator{\SO}{SO}
246
\newcommand{\galq}{\Gal(\Qbar/\Q)}
247
\newcommand{\rhobar}{\overline{\rho}}
248
\newcommand{\cM}{\mathcal{M}}
249
\newcommand{\cB}{\mathcal{B}}
250
\newcommand{\cE}{\mathcal{E}}
251
252
\newcommand{\sltwoz}{\SL_2(\Z)}
253
\newcommand{\sltwo}{\SL_2}
254
\newcommand{\gltwoz}{\GL_2(\Z)}
255
\newcommand{\mtwoz}{M_2(\Z)}
256
\newcommand{\gltwoq}{\GL_2(\Q)}
257
\newcommand{\gltwo}{\GL_2}
258
\newcommand{\gln}{\GL_n}
259
\newcommand{\psltwoz}{\PSL_2(\Z)}
260
\newcommand{\psltwo}{\PSL_2}
261
\newcommand{\h}{\mathfrak{h}}
262
\renewcommand{\a}{\mathfrak{a}}
263
\newcommand{\p}{\mathfrak{p}}
264
\newcommand{\m}{\mathfrak{m}}
265
\newcommand{\trho}{\tilde{\rho}}
266
\newcommand{\rhol}{\rho_{\ell}}
267
\newcommand{\rhoss}{\rho^{\text{ss}}}
268
\DeclareMathOperator{\tr}{tr}
269
\DeclareMathOperator{\ur}{ur}
270
\DeclareMathOperator{\Tr}{Tr}
271
\DeclareMathOperator{\Hom}{Hom}
272
\DeclareMathOperator{\Ext}{Ext}
273
\DeclareMathOperator{\Tor}{Tor}
274
\newcommand{\smallzero}{\left(\begin{smallmatrix}0&0\\0&0
275
\end{smallmatrix}\right)}
276
\newcommand{\smallone}{\left(\begin{smallmatrix}1&0\\0&1
277
\end{smallmatrix}\right)}
278
279
\newcommand{\pari}{{\sc Pari}}
280
\newcommand{\magma}{{\sc Magma}}
281
\newcommand{\hecke}{{\sc Hecke}}
282
\newcommand{\lidia}{{\sc LiDIA}}
283
284
%%%% Theoremstyles
285
\theoremstyle{plain}
286
\newtheorem{theorem}{Theorem}[section]
287
\newtheorem{proposition}[theorem]{Proposition}
288
\newtheorem{corollary}[theorem]{Corollary}
289
\newtheorem{lemma}[theorem]{Lemma}
290
\newtheorem{conjecture}[theorem]{Conjecture}
291
292
\theoremstyle{definition}
293
\newtheorem{definition}[theorem]{Definition}
294
\newtheorem{algorithm}[theorem]{Algorithm}
295
\newtheorem{question}[theorem]{Question}
296
297
\theoremstyle{remark}
298
\newtheorem{goal}[theorem]{Goal}
299
\newtheorem{remark}[theorem]{Remark}
300
\newtheorem{example}[theorem]{Example}
301
\newtheorem{exercise}[theorem]{Exercise}
302
303
304
305