CoCalc Public Fileswww / talks / MSRI-touching / tex / maninstrick.texOpen with one click!
Author: William A. Stein
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\documentclass{slides}
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\usepackage{graphicx}
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\usepackage{psfrag}
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\usepackage{amsmath}
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\newcommand{\C}{\mathbf{C}}
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\newcommand{\F}{\mathbf{F}}
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\newcommand{\Q}{\mathbf{Q}}
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\newcommand{\Qbar}{\bar \Q}
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\newcommand{\Z}{\mathbf{Z}}
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\newcommand{\tors}{\mbox{\rm\small tors}}
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\font\cyr=wncyr10 scaled \magstep 4
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\newcommand{\Sha}{\mbox{\cyr X}}
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\newcommand{\Gal}{\mbox{\rm Gal}}
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\newcommand{\mtwo}[4]{\left(
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\begin{matrix}#1&#2\\#3&#4
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\end{matrix}\right)}
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\newcommand{\SL}{\mbox{\rm SL}}
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\newcommand{\union}{\cup}
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\usepackage{color}
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\begin{document}
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\begin{slide}
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{\bf Manin's trick}: Writes {\em any} symbol
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$\{\alpha, \beta\}$ as a linear combination of generating
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symbols of the form $r_i \{0,\infty\}$.
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\noindent
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The trick implies that the symbols $r_i \{0,\infty\}$ generate
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\noindent{\bf The trick:}\\
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Suffices to consider $\{0,b/a\}$.
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Expand $b/a$ as a continued fraction and
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consider the successive convergents in lowest terms:
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{\small
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$$
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\frac{b}{a}
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= \frac{b_n}{a_n},\quad
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\frac{b_{n-1}}{a_{n-1}},\,\,
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\ldots\,
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,\quad\frac{b_0}{a_0} = \frac{b_0}{1},\quad
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\frac{b_{-1}}{a_{-1}} = \frac{1}{0},\quad
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\frac{b_{-2}}{a_{-2}} = \frac{0}{1}$$}\vspace{-2ex}
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(the last two are added formally).
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\end{slide}
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\begin{slide}
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Then
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$$b_k a_{k-1} - b_{k-1} a_k = (-1)^{k-1},$$
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so that
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$$g_k = \mtwo{b_k}{\hfill (-1)^{k-1}b_{k-1}}{a_k}{(-1)^{k-1}a_{k-1}}\in \SL_2(\Z).$$
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Hence
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$$\left\{\frac{b_{k-1}}{a_{k-1}}, \frac{b_k}{a_k}\right\}
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= g_k \{ 0, \infty\} = r_i \{0,\infty\},$$
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for some~$i$, is of the required special form.
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\end{slide}
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\begin{slide}
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{\bf Example:}
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Let $N=11$, and consider $\{0,4/7\}$.
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We have
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$$\frac{4}{7}
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= 0 + \frac{1}{1+\frac{1}{1+\frac{1}{3}}},$$
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so partial convergents are
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{\small
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$$\frac{b_{-2}}{a_{-2}} = \frac{0}{1},\quad
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\frac{b_{-1}}{a_{-1}} = \frac{1}{0},\quad
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\frac{b_0}{a_0} = \frac{0}{1},\quad
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\frac{b_1}{a_1} = \frac{1}{1},\quad
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\frac{b_2}{a_2} = \frac{1}{2},\quad
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\frac{b_3}{a_3} = \frac{4}{7}.
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$$}
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\par\noindent Thus
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{\small
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\begin{eqnarray*}
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\{0,4/7\}
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&=& \{0,\infty\} +\{\infty,0\} + \{0,1\}+ \{1,1/2\} + \{1/2,4/7\}\\
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&=&
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\mtwo{1}{-1}{2}{-1}\{0,\infty\}
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+ \mtwo{4}{1}{7}{2}\{0,\infty\}\\
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&=&
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2 \cdot \left[\mtwo{1}{4}{1}{5}\{0,\infty\}\right]
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\end{eqnarray*}
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}
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\end{slide}
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\end{document}
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