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\newcommand{\R}{\mathbb R}
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\newcommand{\Q}{\mathbb Q}
\newtheorem{defi}{Definici\'{o}n}
\newtheorem{theo}{Teorema}


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\mode<presentation>{
%\usetheme[numbers]{Warsaw} %Rebordes y n\'{u}meros
\usetheme{Frankfurt} %Rebordes y n\'{u}meros indice con puntos
}

\title{\bf Problemas de puntos del retículo}

%\subtitle{\includegraphics[scale=0.213]{compti.png}}
\author{Fernando Chamizo Lorente}
\date{}
\institute{Universidad Aut\'{o}noma de Madrid \\ {\tiny http://www.uam.es/fernando.chamizo}
\\
\
\\
{\small 19 de julio de 2007}}

\begin{document}

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%\alert{\ \insertframenumber\ de \inserttotalframenumber}
{\color{MedioRojo}\ \insertframenumber\ de \inserttotalframenumber}
\vskip2pt\insertnavigation{\paperwidth}\vskip2pt
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%\end{document}


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% TRANSPARENCIA 0  %
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\begin{frame}
\titlepage
\end{frame}


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% TRANSPARENCIA 0+ %
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\section*{\'{I}ndice}
\begin{frame}
\frametitle{\'{I}ndice}
%\tableofcontents[pausesections]
\tableofcontents
\end{frame}





%%%%%%%%%%%%%%%%%%%%
% TRANSPARENCIA 1  %
%%%%%%%%%%%%%%%%%%%%
\section{Introducción}
\begin{frame}
\frametitle{Buscando en el MSC}

\begin{overprint}
    \onslide<2>
        \includegraphics[scale=0.45]{./images/menu1.png}
    \onslide<3>
        \includegraphics[scale=0.45]{./images/menu2.png}
    \onslide<4>
        \includegraphics[scale=0.45]{./images/menu3.png}
\end{overprint}

\begin{footnotesize}$\uncover<2->{{65}}\uncover<3->{\cdot {27}}
\uncover<4->{\cdot 9=15795}$
\end{footnotesize}
\end{frame}


%%%%%%%%%%%%%%%%%%%%
% TRANSPARENCIA 2  %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{¿Qué tipo de problemas tratamos?}

\begin{tabular}{ll}
    \begin{tabular}{l}
    $
    \displaystyle{\sum_{n=1}^N d(n)=N\log N+(2\gamma-1)N+\text{error}}
    $
    \\
Problema del divisor (Dirichlet)
\\[5pt]
    \uncover<2->{$
    \displaystyle{\sum_{n=1}^N r(n)=\pi N+\text{error}}
    $
    \\
Problema del círculo (Gauss)
\\[5pt]}
    \uncover<3->{$
    \displaystyle{\sum_{d=1}^N h(-d)=\frac{\pi}{18\zeta(3)}N^{3/2}-\frac3{2\pi^2}N+\text{error}}
    $
\\ Promedio del número de clases (Gauss)}
    \end{tabular}
&
    \only<1>{
    \begin{tabular}{l}
    \\[-20pt]
    \includegraphics[scale=0.2]{./images/pdivi.png}
    \end{tabular}
    }
    \only<2>{
    \begin{tabular}{l}
    \\[-20pt]
    \includegraphics[scale=0.2]{./images/pcirc.png}
    \end{tabular}
    }
    \only<3>{
    \begin{tabular}{l}
    \\[-20pt]
    \includegraphics[scale=0.2]{./images/pclas.png}
    \end{tabular}
    }
\end{tabular}



\end{frame}



%%%%%%%%%%%%%%%%%%%%
% TRANSPARENCIA 3  %
%%%%%%%%%%%%%%%%%%%%
\section{Análisis armónico}
\begin{frame}
\frametitle{Análisis armónico}

%http://www.perseus.tufts.edu/cgi-bin/ptext?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3D%236943

¿Teoría analítica de números?
\qquad Análisis$\leftrightsquigarrow$Aritmética$\leftrightarrow$Álgebra


\begin{center}
¿Análisis$\leftrightarrow$continuo,\qquad Álgebra$\leftrightarrow$discreto?
\end{center}


\uncover<2->{\begin{block}{}
\quad\begin{Large}$
\mbox{\boldmath$\alpha$}
\mbox{\boldmath$\nu$}
\mbox{\boldmath$\alpha$}
\mbox{\boldmath$\lambda$}
\mbox{\boldmath$\upsilon$}
\mbox{\boldmath$\sigma$}
\mbox{\boldmath$\iota$}
\mbox{\boldmath$\varsigma$}
$.\end{Large}
liberación, disolución, fin, muerte; solución.
%\textbf{1.} loosing, releasing.
%\textbf{2.} dissolving.
%\textbf{3.} resolution of a problem by the analysis of its conditions.
%\textbf{4.} reduction of the imperfect figures into the perfect one.
%\textbf{5.} solution of a problem.
%\textbf{6.} retrogression; retirement, departure; death.
\end{block}}

\


\uncover<3->{El análisis armónico permite descomponer (analizar) funciones complicadas en armónicos sencillos (tonos puros).}


\


\uncover<4->{Los armónicos muchas veces forman un conjunto discreto.}

\end{frame}


%%%%%%%%%%%%%%%%%%%%
% TRANSPARENCIA 4  %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Ejemplos de análisis armónico}

\begin{center}
% use packages: array
\begin{tabular}{|l|l}
\textbf{Espacio}  & \textbf{Armónicos} (caracteres) \\
\hline
\\[-8pt]
1-periódicas, $L^2$ & $e^{2\pi inx}$ \quad $n\in\Z$ \\
\hline
\\[-8pt]
$L^2(\R^n)$ & $e^{2\pi i\xi \cdot x}$ \quad $\xi \in\R^n$ \\
\hline
\\[-8pt]
$f:\Z/m\Z\longrightarrow\C$ & $e^{2\pi i nx/m}$ \\
\hline
\\[-8pt]
$f:(\Z/m\Z)^*\longrightarrow\C$ & $\chi(x)$ \\
\hline
\\[-8pt]
$\text{SL}_2(\Z)$-periódicas, $L^2(\mathbb{H})$ & formas de Maass, $E(z,s)$ \\
\hline
\\[-8pt]
$f:\mathbb{A}\longrightarrow\C$, adèles & $e^{2\pi i\lambda(\mbox{\boldmath$\xi$}\cdot\mbox{\boldmath$x$})}$,\quad  \begin{small}$\lambda(\mbox{\boldmath$t$})=\sum\pm\text{Tr}_{K_v/\Q_p}(\mbox{\boldmath$t$})$\end{small}
\\
\hline
\end{tabular}
\end{center}

\end{frame}



%%%%%%%%%%%%%%%%%%%%
% TRANSPARENCIA 5  %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}
%\frametitle{Ejemplos de análisis armónico}

\begin{center}
\vskip-30pt
{\color{MedioRojo}
El tipo de análisis depende de la estructura del problema.}
\end{center}

\underline{Ejemplo}: $f(x)=1$ si $x\equiv 1$ (mod $m$) y 0 en el resto.

\vskip10pt
Análisis en $\Z/m\Z$ y en $(\Z/m\Z)^*$

\vskip-20pt
\[
f(x)=\frac 1m \sum_{n=1}^me^{-2\pi in/m}e^{2\pi inx/m}, \qquad
f(x)=\frac 1{\phi(m)} \sum_\chi \chi(x)
\]
\uncover<2->{\[
\log\prod_\chi \sum\frac{\chi(n)}{n^{s}}
=\log\prod_\chi\prod_p\big(1-\frac{\chi(p)}{p^{s}}\big)^{-1}
\stackrel{\scriptscriptstyle{\alert{\log(1-x)^{-1}\sim x}}}{\approx}
\kern-5pt
\phi(m)\sum_p \frac{f(p)}{p^{s}}
\]
\[
\log \big(L(s,\chi_0) \prod_{\chi\ne\chi_0}L(s,\chi)\big)
\stackrel{s\approx 1^+}{\to}
 \log\big(\frac{\phi(m)}{m(s-1)}\cdot \text{cte$\ne 0$}\big)
\to\infty
\]}
\uncover<3->{\quad\ \qquad\ \qquad$\Rightarrow$
{\color{MyDarkBlue}Hay infinitos primos en progresiones aritméticas }

\quad\ \qquad\ \qquad\ \quad (similar para $x\equiv a$ (mod $m$) con $a,m$ coprimos).}

\end{frame}


%%%%%%%%%%%%%%%%%%%%
% TRANSPARENCIA 6  %
%%%%%%%%%%%%%%%%%%%%
\section{F. de sumación}
\begin{frame}
\frametitle{Fórmulas de sumación}
\vskip-20pt
$H$ discreto $<$ $G$ abeliano localmente compacto
\[
\text{Sumación de Poisson}
\longrightarrow
|G/H|\sum_{h\in H}f(h)=\sum_{c\in\widehat{G/H}}\widehat{f}(c)
\]

\vskip-20pt

\uncover<2->{\begin{eqnarray*}
\sum_{n\in\Z}f(n) &=& \sum_{m\in\Z}\widehat{f}(m),\quad
\widehat{f}(\xi)=\int f(x)e^{-2\pi i\xi x}dx
\\[-10pt]
&\downarrow&
\\[-1pt]
\sum e^{-\pi n^2x} &=& x^{-1/2}\sum e^{-\pi n^2/x}
\\[-5pt]
&\downarrow&
\\[-1pt]
\Lambda(s) &=& \Lambda(1-s), \quad
\Lambda(s)=\pi^{-s/2}\Gamma(s/2)\zeta(s)
\end{eqnarray*}}


\uncover<3->{fórmulas de sumación $\leftarrow$ formas modulares $\rightarrow$ ec. funcionales}

\end{frame}


%%%%%%%%%%%%%%%%%%%%
% TRANSPARENCIA 7  %
%%%%%%%%%%%%%%%%%%%%
\section{P. de incertidumbre}
\begin{frame}
\frametitle{Principio de incertidumbre}

Con ondas de frecuencia $<\nu$ vemos borroso en longitudes
$\lambda<1/\nu$

\begin{overprint}
    \onslide<1>
        \begin{tabular}{ll}
        \includegraphics[scale=0.6]{./images/caju0.png}
        &
            \begin{tabular}{l}
            \\[-5cm]
            \includegraphics[scale=0.3]{./images/longo.png}
            \\
            \
            \\
            $\nu<\infty$,\quad
            $\lambda>0$
            \\
            \end{tabular}
        \end{tabular}
    \onslide<2>
        \begin{tabular}{ll}
        \includegraphics[scale=0.6]{./images/caju1.png}
        &
            \begin{tabular}{l}
            \\[-5cm]
            \includegraphics[scale=0.3]{./images/longo.png}
            \\
            \
            \\
            $\nu<100$,\quad
            $\lambda>0.01$
            \\
            \end{tabular}
        \end{tabular}
    \onslide<3>
        \begin{tabular}{ll}
        \includegraphics[scale=0.6]{./images/caju2.png}
        &
            \begin{tabular}{l}
            \\[-5cm]
            \includegraphics[scale=0.3]{./images/longo.png}
            \\
            \
            \\
            $\nu<50$,\quad
            $\lambda>0.02$
            \\
            \end{tabular}
        \end{tabular}
    \onslide<4>
        \begin{tabular}{ll}
        \includegraphics[scale=0.6]{./images/caju5.png}
        &
            \begin{tabular}{l}
            \\[-5cm]
            \includegraphics[scale=0.3]{./images/longo.png}
            \\
            \
            \\
            $\nu<20$,\quad
            $\lambda>0.05$
            \\
            \end{tabular}
        \end{tabular}
    \onslide<5>
        \begin{tabular}{ll}
        \includegraphics[scale=0.6]{./images/caju10.png}
        &
            \begin{tabular}{l}
            \\[-5cm]
            \includegraphics[scale=0.3]{./images/longo.png}
            \\
            \
            \\
            $\nu<10$,\quad
            $\lambda>0.1$
            \\
            \end{tabular}
        \end{tabular}
    \onslide<6>
        \begin{tabular}{ll}
        \includegraphics[scale=0.6]{./images/caju20.png}
        &
            \begin{tabular}{l}
            \\[-5cm]
            \includegraphics[scale=0.3]{./images/longo.png}
            \\
            \
            \\
            $\nu<5$,\quad
            $\lambda>0.2$
            \\
            \end{tabular}
        \end{tabular}
    \onslide<7->
        \begin{tabular}{ll}
        \includegraphics[scale=0.6]{./images/caju50.png}
        &
            \begin{tabular}{l}
            \\[-5cm]
            \includegraphics[scale=0.3]{./images/longo.png}
            \\
            \
            \\
            $\nu<2$,\quad
            $\lambda>0.5$
            \\
            \end{tabular}
        \end{tabular}
\end{overprint}

\uncover<8->{Los detalles a escala $\lambda$ sólo afectan a los armónicos de
frecuencia mayor que $1/\lambda$}

\end{frame}


%%%%%%%%%%%%%%%%%%%%
% TRANSPARENCIA 8  %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{El análisis básico}

\vskip-3.1cm
\begin{tabular}{p{7.8cm} @{\kern-1pt} p{3cm}}
\underline{Ej.} $f=$ f. característica del círculo de radio $R$.


\vskip9pt
$\mathcal{N}=\sum f(\vec{n}) = \sum \widehat{f}(\vec{m})
=\sum\int f(\vec{x})e^{-2\pi i\vec{m}\cdot \vec{x}}d\vec{x}$

\vskip9pt
\uncover<2->{$\widehat{f}$ es radial}\uncover<4->{,
\begin{small}$\widehat{f}(\vec{m})=O(R^{1/2}\|\vec{m}\|^{-3/2})$, $\widehat{f}(\vec{0})=\pi R^2$\end{small}

\vskip3pt
pero $R^{1/2}\sum \|\vec{m}\|^{-3/2}=\infty$ si $\vec{m}$ recorre $\Z^2$.}
&
    \begin{overprint}
            \onslide<1>
            \begin{tabular}{l}
            \\[2cm]
            \includegraphics[scale=0.2]{./images/pcirc.png}
            \end{tabular}
            \onslide<2>
            \begin{tabular}{l}
            \\[2cm]
            \includegraphics[scale=0.2]{./images/ocirc.png}
            \end{tabular}
            \onslide<3-4>
            \begin{tabular}{l}
            \\[2cm]
            \includegraphics[scale=0.2]{./images/ocirc2.png}
            \end{tabular}
            \onslide<5-6>
            \begin{tabular}{l}
            \\[2cm]
            \includegraphics[scale=0.2]{./images/perfi.png}
            \end{tabular}
            \onslide<7>
            \begin{tabular}{l}
            \\[2cm]
            \includegraphics[scale=0.2]{./images/conve.png}
            \end{tabular}
    \end{overprint}
\end{tabular}


\uncover<5->{{Principio de incertidumbre} $\Rightarrow$ el análisis armónico no ve la regularización hasta las frecuencias $H^{-1}$.}

\vskip-18pt
\uncover<6->{\[
\mathcal{N}=\pi(R+O(H))^2+O(R^{1/2}\kern-10pt\sum_{\|\vec{m}\|<H^{-1}}\kern-6pt \|\vec{m}\|^{-3/2})
\kern-3pt
\stackrel{\alert{\scriptscriptstyle{H=R^{-1/3}}}}{=}
\kern-5pt
\pi R^2+O(R^{2/3}).
\]}


\vskip-10pt
\uncover<7->{{\color{MyDarkBlue} Cuerpo convexo en $\R^d$ $\Rightarrow$ $\mathcal{N}=\text{Volumen}+O(R^{(d-1)d/(d+1)})$}}

\end{frame}


%%%%%%%%%%%%%%%%%%%%
% TRANSPARENCIA 9  %
%%%%%%%%%%%%%%%%%%%%
\section{Cuatro problemas}
\begin{frame}
\frametitle{¿Cómo superar el principio de incertidumbre?}


\vskip-6pt
$\bullet$ Estudiar las interferencias debidas a las fases de $\widehat{f}(\vec{m})$.
\begin{center}
Ej. $R^{1/2}\sum \|\vec{m}\|^{-3/2}\ \longrightarrow\  R^{1/2}\sum \|\vec{m}\|^{-3/2} e^{2\pi iR\|\vec{m}\|}$
\end{center}
Método de sumas trigonométricas


\begin{overprint}
    \onslide<2>
    {{\color{MedioRojo}
    (ausencia de resonancias $\leftrightarrow$ consideraciones diofánticas

    \qquad Ej. $|\alpha-p/q|<q^{-2}$, $N\ge q$ $\Rightarrow$
    $\sum_{n=1}^Ne^{2\pi i\alpha n^2}\le CN/\sqrt{q}$)}}
    \onslide<3->

    \vskip-20pt
    $\bullet$ Utilizar otras interpretaciones aritméticas

    \

    \uncover<4->{\qquad Ejemplos en dimensión 3 (trabajos con E. Cristóbal y A. Ubis)
    \begin{center}
    % use packages: array
    \qquad \begin{tabular}{lll}
    1. {\color{MyDarkBlue} Número de clases} &  & 2. {\color{MyDarkBlue} Puntos en esferas} \\
    3. {\color{MyDarkBlue} Puntos visibles} &  & 4. {\color{MyDarkBlue} Elipsoides racionales}
    \end{tabular}
    \end{center}}
\end{overprint}

\end{frame}



%%%%%%%%%%%%%%%%%%%%
% TRANSPARENCIA 10 %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{1. Promedio del número de clases}

\vskip-18pt
\begin{block}{Gauss \hfill Art. 302, 304, \textit{Disquisitiones arithmeticae} (1801)}
\begin{small}Hemos  encontrado por una investigación teórica que el promedio del número de clases hasta determinante $-D$ puede ser expresado aproximadamente por $\gamma\sqrt{D}-\delta$ donde $\gamma=0.7467183115=\frac{2\pi}{7e}$ donde $e$ es la suma de la serie $1+\frac 18+\frac1{27}+\frac1{64}+\text{etc}.$ $\delta=0.2026423673=\frac{2}{\pi^2}$. [\dots ] el décimo millar es 72549 por la tabla, 72572 por la fórmula.\end{small}
\end{block}

\uncover<2->{\begin{small}Para discriminantes positivos, multiplicando por $\log\epsilon_D$

\begin{block}{}
el promedio de este producto se expresa aproximadamente por una fórmula como
$m\sqrt{D}-n$.
Sin embargo hasta ahora no hemos sido capaces de determinar el valor de las constantes $m$, $n$ teóricamente.
\end{block}\end{small}}

\vskip-7pt
\uncover<3->{\begin{small}

\

\qquad Primeras pruebas publicadas:

\qquad$-D$ $\rightarrow$ Lipschitz 1865
\qquad
$+D$ $\rightarrow$ Vinogradov 1919 \begin{scriptsize}(Siegel 1944)\end{scriptsize}\end{small}}


\end{frame}


%%%%%%%%%%%%%%%%%%%%
% TRANSPARENCIA 11 %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{1. Promedio del número de clases}


%\vskip-3.1cm

\begin{tabular}{p{4.8cm} @{\kern-1pt} p{5cm}}
            \begin{tabular}{l}
            \\[-0.8cm]
            \includegraphics[scale=0.35]{./images/pernc.png}
            \end{tabular}
&
    \begin{small}\begin{tabular}{|l|}
    \hline
    $\displaystyle{S=\sum_{d=1}^D h(-d)\ \rightarrow\ S_r}$
    \\
    ec. funcional $\sum h(-d)d^{-s}$ (Shintani)
    \\
    $\displaystyle{S_r\ \rightarrow\ \sum_{d\le H^{-1}}h(-d)e^{2\pi i\sqrt{Dd}}}$
    \\
    $\phantom{S_r}\ \rightarrow\ \sum\sum\sum e^{2\pi i\sqrt{D}\sqrt{4ac-b^2}}$
    \\
    \hline
    \end{tabular}\end{small}
\end{tabular}

\

\vskip6pt
\uncover<2->{\begin{tabular}{lll}
$H$ mayor  & $\Rightarrow$ & Suma trigonométrica más corta $\smile$ \\
 & $\Rightarrow$ & mayor diferencia entre $S$ y $S_r$ $\frown$
\end{tabular}}

\


\vskip6pt
\uncover<3->{\begin{tabular}{ll}
$S-S_r=\sum L(1,\chi_n)$ $\rightarrow$ &  sumas cortas de caracteres \\
   & (Burgess, H. de Riemann sobre $\mathbb F_q$)
\end{tabular}}


\

\uncover<4->{\qquad\begin{tabular}{lll}
Nuevos armónicos  & $\rightarrow$ & nueva forma de atacar al principio \\
 &  & de incertidumbre
\end{tabular}}

\end{frame}

%%%%%%%%%%%%%%%%%%%%
% TRANSPARENCIA 12 %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{2. Puntos en esferas y funciones $L$}

\vskip-20pt
\begin{eqnarray*}
\text{Error}&=&T+C\qquad \text{(Poisson, fórmula del número de clases)}
\\
&=&
\sum_{\|\vec{m}\|<H^{-1}}\text{ondas}+
\sum_{R<\sqrt{n}<R+H}b_nL(1,\chi_n)
\end{eqnarray*}


\vskip-2cm
\uncover<2->{
\begin{tabular}{p{7.2cm} @{\kern-1pt} p{2.5cm}}
    \begin{tabular}{c}
    $T\leftrightarrow$Acotación de sumas trigonométricas \\
    $C\leftrightarrow$Acotación de sumas de caracteres \\
    \uncover<3->{$\updownarrow$}
    \\
    \uncover<3->{Distribución de ceros de funciones $L$}
    \end{tabular}

\vskip9pt
\uncover<4->{$\delta=$distancia a $\Re s=1$ del cero más alejado}

\vskip-3pt
\uncover<5->{\begin{center}
$\delta>2/25$ $\Rightarrow$ mejora
\end{center}}
&
    \begin{overprint}
            \onslide<3>
            \begin{tabular}{l}
            \\[2cm]
            \includegraphics[scale=0.27]{./images/funcL.png}
            \end{tabular}
            \onslide<4->
            \begin{tabular}{l}
            \\[2cm]
            \includegraphics[scale=0.27]{./images/funcLp.png}
            \end{tabular}
    \end{overprint}
\end{tabular}
}

\end{frame}

%%%%%%%%%%%%%%%%%%%%
% TRANSPARENCIA 13 %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{3. Puntos visibles}

\begin{tabular}{l}
Puntos visibles \ $=$ \\
\
\end{tabular}
\begin{tabular}{l}
todos
$-$ múltiplos de 2
$-$ múltiplos de 3
\\
$+$ múltiplos de 6
$-$ \dots\ \uncover<2->{$\sim\frac{4\pi}{3\zeta(3)}R^3$}
\end{tabular}

\[
\uncover<2->{\text{Error}\rightarrow \sum_{d,\; n}\frac Rd\mu(d)
\frac{r_3(n)}n e^{2\pi i\frac Rd\sqrt{n}}} \uncover<3->{, \quad
\text{func. sencilla}\rightarrow \sum_m \frac{ e^{-2\pi
iR\sqrt{m}}}{\sqrt{m}}}
\]
\[
\uncover<4->{\text{Error}\cdot \text{func. sencilla}
\stackrel{\alert{\scriptstyle{m=n/d^2}}}{\longrightarrow}
\sum_n\sum_{d^2|n}R\mu(d)\frac{r_3(n)}{n^{3/2}}+\dots}
\]

\vskip-20pt
\[
\uncover<5->{\left.
\begin{array}{l}
\sum_{d^2|n}\mu(d)=\mu^2(n)
\\
\sum\mu^2(n)\dfrac{r_3(n)}{n^{3/2}}\text{ diverge}
\end{array}
\right\}} \uncover<6->{\quad\Rightarrow\quad
\text{Error}\not=o(R(\log R)^{1/2})}
\]
\end{frame}


\end{document}


%%%%%%%%%%%%%%%%%%%%
% TRANSPARENCIA 14 %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{4. Elipsoides racionales}


\vskip-9pt
\begin{columns}
\column{.3\textwidth}
\
\column{.4\textwidth}
\begin{block}{}
\begin{center}
$r_Q(n)=r_Q(n,\text{gen})+a_n$
\end{center}
\end{block}
\column{.3\textwidth}
\
\end{columns}


\vskip-3.1cm
\begin{tabular}{p{7.8cm} @{\kern-1pt} p{3cm}}
\begin{center}
% use packages: array
\uncover<2->{\begin{tabular}{rll}
$r_Q(n,\text{gen})$ & $\rightarrow$ & producto de factores locales \\
 & $\rightsquigarrow$ & suma de caracteres \\
$a_n$ & $\rightarrow$ & coef. de $f\in S_{3/2}(\Gamma)$. Duke:\\
 &  & $a_n\le Cn^{13/28+\epsilon}$ si $\mu(n)\ne 0$ \\
 & $\rightarrow$ & error aceptable si $\mu(n)\ne 0$
\end{tabular}}
\end{center}
&
            \begin{tabular}{l}
            \\[3cm]
            \includegraphics[width=30mm, height=22mm]{./images/capae.png}
            \end{tabular}
\end{tabular}

\uncover<3->{$f=f_1+f_2$ con $Sh(f_1)$ (corresp. de Shimura) parabólica, $f_1\perp f_2$}

\uncover<4->{\[
\left.
\begin{array}{l}
\text{$\rightsquigarrow$ $f_2$ muchos coeficientes cero}
\\
\text{\phantom{$\rightsquigarrow$} $a_n\le Cn^{13/28+\epsilon}$}
\\
\text{\phantom{$\rightsquigarrow$} cota de Deligne para $Sh(f_1)$}
\end{array}\right\}
\Rightarrow\
\begin{array}{l}
\text{Mejor control}
\\
\text{de la capa}
\end{array}
\]}

\end{frame}




\end{document}
