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1 | \documentclass[9pt,blackandwhite,roman,handout]{beamer} |
2 | \usepackage{etex} | |
3 | ||
4 | \usefonttheme{professionalfonts} | |
5 | ||
6 | ||
7 | % Setup appearance: | |
8 | %\usetheme{Warsaw} | |
9 | \usetheme{Darmstadt} | |
10 | ||
11 | %\usecolortheme{seahorse} | |
12 | %\usecolortheme{dove} | |
13 | \usecolortheme{seagull} | |
14 | %\usefonttheme[onlylarge]{structurebold} | |
15 | %\setbeamerfont*{frametitle}{size=\normalsize,series=\bfseries} | |
16 | %\setbeamertemplate{navigation symbols}{} | |
17 | ||
18 | ||
19 | ||
20 | % Standard packages | |
21 | %\usepackage[frenchb]{babel} | |
22 | %\usepackage[english]{babel} | |
23 | \usepackage[utf8]{inputenc} | |
24 | %\usepackage[T1]{fontenc} | |
25 | \usepackage{cmbright} | |
26 | \usepackage[sans]{dsfont} | |
27 | \usepackage{pifont} | |
28 | %calscaled=.94, | |
29 | %frakscaled=.97, | |
30 | \usepackage[cal=euler,scr=rsfso]{mathalfa}%bb=cm, | |
31 | %frak=mma, | |
32 | %scr=esstix | |
33 | \usepackage{mathrsfs} % pour faire des majuscules calligraphiées \mathcal{blabla} | |
34 | %\usepackage[french,lined]{algorithm2e} % rajouter linesnumbered dans les arguments pour numéroter les lignes des algos, boxed pour l'encadrement | |
35 | %\usepackage{extsizes} | |
36 | %\usepackage{MnSymbol} | |
37 | \usepackage{graphicx} | |
38 | \usepackage{mathabx} | |
39 | \usepackage[all]{xy} | |
40 | \usepackage{ulem} | |
41 | ||
42 | \usepackage{DotArrow} | |
43 | %\usepackage[varg]{txfonts} | |
44 | %\usepackage{matptmx} | |
45 | \usepackage{extpfeil} | |
46 | %\usepackage{MyMnSymbol} | |
47 | \usepackage{comment} | |
48 | %\usepackage{etex} | |
49 | %usepackage{mathtools} | |
50 | %\usepackage{fourier} | |
51 | ||
52 | \usepackage{ragged2e} | |
53 | \justifying | |
54 | ||
55 | % Setup TikZ | |
56 | \usepackage{tikz} | |
57 | \usetikzlibrary{matrix,arrows} | |
58 | \tikzstyle{block}=[draw opacity=0.7,line width=1.4cm] | |
59 | ||
60 | \newcommand{\gk}{$g_k = \left \{ \begin{array}{ll} | |
61 | q^k+q^{k-1}-q^\frac{k+1}{2} - 2q^\frac{k-1}{2}+1 & \mbox{si } k \equiv 1 \pmod 2,\\ | |
62 | q^k+q^{k-1}-\frac{1}{2}q^{\frac{k}{2}+1} - \frac{3}{2}q^{\frac{k}{2}}-q^{\frac{k}{2}-1} +1& \mbox{si } k \equiv 0 \pmod 2. | |
63 | \end{array} \right .$} | |
64 | ||
65 | \newcommand{\ext}{\xymatrix{ | |
66 | \FF= \Fq(x)(y) \ar@{-}[d]^{<\infty} \\ \Fq(x) \ar@{-}[d] \\ \Fq} } | |
67 | ||
68 | \newcommand{\twoheaddownarrow}{% | |
69 | \mathrel{\reflectbox{\rotatebox[origin=c]{-90}{$\xtwoheadrightarrow[ \; ]{ \reflectbox{\rotatebox[origin=c]{-90}{$\pi_Q \ $}} \ \ }$}}}} | |
70 | ||
71 | \newcommand{\longdownmapsto}{% | |
72 | \mathrel{\reflectbox{\rotatebox[origin=c]{-90}{$\xmapsto{ \ \ \ \reflectbox{\rotatebox[origin=c]{-90}{$\pi_Q \ $}} \ \ \ }$}}}} | |
73 | ||
74 | \newcommand{\bigubrace}[1]{\underbrace{\mbox{}f(P_i), f'(P_i), \ldots, f^{(\ell_i-1)}(P_i)\vphantom{\sum_0^0}\hspace{#1}\mbox{}}_?} | |
75 | ||
76 | \newcommand{\myubrace}[2]{\rotatebox[origin=c]{90}{$ | |
77 | \rotatebox[origin=c]{-90}{#1} \left \{ | |
78 | \begin{array}{l} | |
79 | \vspace{#2} \\ | |
80 | \end{array} | |
81 | \right . \hspace{-1em} | |
82 | $}} | |
83 | ||
84 | \newcommand{\bluebrace}{{\color{blue}\left\{}} | |
85 | ||
86 | ||
87 | \definecolor{mycvblue}{rgb}{0.302,0.537,0.737} | |
88 | ||
89 | ||
90 | \setbeamercolor{structure}{fg=mycvblue} | |
91 | \setbeamercolor{palette primary}{fg=mycvblue} | |
92 | \setbeamercolor{subsection in head/foot}{parent=palette primary} | |
93 | ||
94 | \setbeamertemplate{navigation symbols}{ | |
95 | % \insertslidenavigationsymbol | |
96 | % \insertframenavigationsymbol | |
97 | % \insertsubsectionnavigationsymbol | |
98 | % \insertsectionnavigationsymbol | |
99 | % \insertdocnavigationsymbol | |
100 | % \insertbackfindforwardnavigationsymbol | |
101 | } | |
102 | ||
103 | %\renewcommand{\item}{\item[$\bullet$]} | |
104 | ||
105 | ||
106 | ||
107 | \title[]{\LARGE{\textsc{Familles denses de courbes modulaires, nombres premiers\\ et \\rang de tenseur symétrique uniforme de la multiplication dans les corps finis}}} | |
108 | ||
109 | \author[Alexey {\textsc Zykin}]{\textbf{Alexey {\textsc Zykin}$^{\dag}$} \\ (\textbf{1984 - 2017}) \\Laboratoire GAATI \\Université de la Polynésie Française\\ | |
110 | {\small National Research University Higher School of Economics} \\ Institute for Information Transmission Problems of the Russian Academy of Sciences\\\vspace{1em}\textbf{\textcolor{mycvblue}{en collaboration avec}}\\ \vspace{1em} \textbf{Stéphane {\textsc Ballet}}\\ Equipe Arithmétique et Théorie de l'Information\\ Institut de Mathématiques de Marseille \\ Aix-Marseille Université} | |
111 | ||
112 | ||
113 | \date[]{\\ \vspace{2em} {\bf Séminaire GAATI}\\ {\bf UPF} \\{\small Avril 2017}} | |
114 | ||
115 | \newtheorem{defin}{Définition} | |
116 | \newtheorem{theoreme}{Théorème} | |
117 | \newtheorem{lemme}{Lemme} | |
118 | \newtheorem{corollaire}{Corollaire} | |
119 | \newtheorem{proposition}{Proposition} | |
120 | \newtheorem{propriete}{Propriété} | |
121 | %\newtheorem{exemple}[definition]{Exemple} | |
122 | ||
123 | \newcommand{\NN}{\ensuremath{\mathbb{N}}} | |
124 | \newcommand{\CC}{\ensuremath{\mathbb{C}}} | |
125 | \newcommand{\ZZ}{\ensuremath{\mathbb{Z}}} | |
126 | \newcommand{\PF}{\mathbf{P}_F} | |
127 | \newcommand{\DF}{\mathsf{Div}(F)} | |
128 | \newcommand{\Jac}{\ensuremath{\mathcal{J}\mathsf{ac}(F/\Fq)}} | |
129 | \newcommand{\Fqr}[2][q]{\mathds{F}_{\!{#1}^{#2}}} | |
130 | \newcommand{\Fq}{\Fqr{}} | |
131 | \newcommand{\F}{\mathds{F}} | |
132 | \newcommand{\FF}{\mathsf{F}} | |
133 | \newcommand{\Fqn}{\Fqr{n}} | |
134 | \newcommand{\D}[1][D]{\ensuremath{\mathcal{#1}}} | |
135 | \newcommand{\Ld}[1][\D]{\ensuremath{\mathscr{L}\!\!\left(#1\right)}} % utilisation : \Ld ou \Ld[A] pour un diviseur A au lieu de D | |
136 | \newcommand{\Ak}[1][k]{\ensuremath{\mathbb{A}_{#1}}} | |
137 | \newcommand{\A}{\ensuremath{\mathsf{A}}} | |
138 | \newcommand{\Cl}{\ensuremath{\mathsf{Cl}}} | |
139 | \newcommand{\mus}{\ensuremath{\mu^\mathsf{sym}}} | |
140 | \newcommand{\Ms}{\ensuremath{M^\mathsf{sym}}} | |
141 | \newcommand{\ms}{\ensuremath{m^\mathsf{sym}}} | |
142 | \newcommand{\chch}{{C}hudnovsky-{C}hudnovsky} | |
143 | \newcommand{\ch}{{C}hudnovsky} | |
144 | ||
145 | ||
146 | ||
147 | \addtobeamertemplate{footline}{\texttt{\hfill\insertframenumber/{\inserttotalframenumber}}} | |
148 | % | |
149 | %\AtBeginSubsection[] { | |
150 | %\begin{frame}<beamer> | |
151 | %\frametitle{Plan} | |
152 | %\tableofcontents[currentsection,currentsubsection] | |
153 | %\end{frame} | |
154 | %} | |
155 | %\AtBeginSection[] { | |
156 | %\begin{frame}<beamer> | |
157 | %\frametitle{Plan} | |
158 | %\tableofcontents[currentsection]%,currentsubsection] | |
159 | %\end{frame} | |
160 | %} | |
161 | ||
162 | \setbeamertemplate{sections/subsections in toc}[sections numbered] | |
163 | %\setbeamertemplate{sections in toc}[sections numbered] | |
164 | ||
165 | \begin{document} | |
166 | ||
167 | % \begin{frame}[plain] | |
168 | % | |
169 | % \begin{center} | |
170 | % | |
171 | % {\bf Institut de Mathématiques de Marseille}\\ | |
172 | % | |
173 | % {\bf Equipe Analyse, Géométrie et Topologie (AGT) }\\ | |
174 | % | |
175 | % \vspace{2em} | |
176 | % | |
177 | % {\bf Séminaire de Géométrie Complexe}\\ | |
178 | % Mardi 12 Juin 2018 | |
179 | % \end{center} | |
180 | % | |
181 | % \begin{center} | |
182 | % | |
183 | % \end{center} | |
184 | % | |
185 | % \end{frame} | |
186 | ||
187 | \begin{frame}[plain] | |
188 | ||
189 | \titlepage | |
190 | ||
191 | \end{frame} | |
192 | ||
193 | \begin{frame}[plain]{Plan} | |
194 | ||
195 | \tableofcontents | |
196 | ||
197 | \end{frame} | |
198 | ||
199 | \section{Introduction} | |
200 | ||
201 | \subsection{Définitions} | |
202 | ||
203 | %%%%% SLIDE 1 | |
204 | \begin{frame}{Définition formelle I} | |
205 | ||
206 | \end{frame} | |
207 | ||
208 | %%%%% SLIDE 2 | |
209 | \begin{frame}{Définition formelle II} | |
210 | ||
211 | \end{frame} | |
212 | ||
213 | %%%%% SLIDE 3 | |
214 | \begin{frame}{Définition formelle III} | |
215 | ||
216 | \end{frame} | |
217 | ||
218 | \subsection{Quantités asymptotiques} | |
219 | ||
220 | %%%%% SLIDE 6 | |
221 | \begin{frame} | |
222 | ||
223 | \end{frame} | |
224 | ||
225 | \section{Algorithme de D.V. et G.V. Chudnovsky (1987)} | |
226 | ||
227 | \subsection{Avec des places rationnelles} | |
228 | ||
229 | %%%%% SLIDE 9 | |
230 | \begin{frame}{Algorithme original de Chudnovsky et Chudnovsky} | |
231 | ||
232 | \end{frame} | |
233 | ||
234 | \subsection{Principe} | |
235 | ||
236 | %%%%% SLIDE 8 | |
237 | \begin{frame}{Principe pour multiplier avec l'algorithme de Chudnovsky} | |
238 | ||
239 | \end{frame} | |
240 | ||
241 | \subsection{Avec des places de degré un et deux} | |
242 | ||
243 | %%%%% SLIDE 10 | |
244 | \begin{frame}{Evaluations sur des places de degré 1 et 2} | |
245 | ||
246 | \end{frame} | |
247 | ||
248 | \section{Conditions permettant l'utilisation de l'algorithme} | |
249 | ||
250 | \subsection{Conditions principales} | |
251 | ||
252 | %%%%% SLIDE 11 | |
253 | \begin{frame}{Conditions suffisantes pour appliquer l'algorithme} | |
254 | ||
255 | \end{frame} | |
256 | ||
257 | \subsection{Applications} | |
258 | ||
259 | %%%%% SLIDE 11 | |
260 | \begin{frame}{Le cas des extensions de petits degré} | |
261 | ||
262 | \end{frame} | |
263 | ||
264 | \begin{frame}{Algorithme de Chudnovsky sur un corps de fonctions hyperelliptique de genre 2} | |
265 | ||
266 | \end{frame} | |
267 | ||
268 | ||
269 | \begin{frame}{Exemple pour les \textit{petites} extensions $\F_{16^n}$ de $\F_{16}$} | |
270 | ||
271 | \end{frame} | |
272 | ||
273 | \setbeamercovered{transparent} | |
274 | ||
275 | \begin{frame} | |
276 | ||
277 | \end{frame} | |
278 | ||
279 | %%%%%%%%%%%%% T %%%%%%%%%%%%%%% | |
280 | ||
281 | ||
282 | %%%%%%%%%%%%% T %%%%%%%%%%%%%%% | |
283 | ||
284 | ||
285 | %%%%%%%%%%%%% T %%%%%%%%%%%%%%% | |
286 | ||
287 | \section{Nouveau résultats} | |
288 | ||
289 | \subsection{Bornes uniformes connues} | |
290 | ||
291 | \begin{frame} | |
292 | ||
293 | \end{frame} | |
294 | ||
295 | \subsection{Nouvelles bornes uniformes} | |
296 | ||
297 | \begin{frame} | |
298 | ||
299 | \end{frame} | |
300 | ||
301 | %%%%% SLIDE 14 | |
302 | \begin{frame}{Corps de fonctions sur $\F_{p^2}$} | |
303 | ||
304 | \end{frame} | |
305 | ||
306 | \begin{frame} | |
307 | ||
308 | \end{frame} | |
309 | ||
310 | %%%%%%%%%%%%% T %%%%%%%%%%%%%%% | |
311 | ||
312 | \begin{frame} | |
313 | ||
314 | \end{frame} | |
315 | ||
316 | \begin{frame} | |
317 | ||
318 | \end{frame} | |
319 | ||
320 | \section{Conclusions et perspectives} | |
321 | ||
322 | \subsection{Problèmes et/ou travail en cours} | |
323 | ||
324 | %%%%% SLIDE 15 | |
325 | ||
326 | \begin{frame}{Conclusion} | |
327 | ||
328 | \end{frame} | |
329 | ||
330 | \begin{frame} | |
331 | ||
332 | \end{frame} | |
333 | ||
334 | %%%%% SLIDE DE FIN | |
335 | ||
336 | \begin{frame}[plain] | |
337 | \begin{beamerboxesrounded}% | |
338 | [lower=block title, % | |
339 | upper=block title,% | |
340 | shadow=true]{} | |
341 | \begin{center} | |
342 | {\Large \textbf{{\color{mycvblue}Thank you for your attention.}}}\\ | |
343 | \vspace{3em} | |
344 | {\Large \textbf{{\color{mycvblue}Questions?}}}\\ | |
345 | \end{center} | |
346 | \end{beamerboxesrounded} | |
347 | ||
348 | \end{frame} | |
349 | ||
350 | \end{document} |