% %http://jorsansa.webs.upv.es/blog/?q=intro-beamer
\documentclass[color=usenames, spanish]{beamer}
\usepackage[usenames]{color}
%\usepackage{amssymb, amsmath} 
%\usepackage{latexsym} 
\usepackage{beamerthemesplit}
%\usepackage{color} 
\usepackage{graphics} 
%\usepackage{wrapfig}
%\usepackage[pdftex]{graphicx}
\usepackage{graphicx}
%http://oregonstate.edu/~peterseb/tex/samples/color-package.html


%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% 	SPANISH
%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage[spanish]{babel}
\usepackage[latin1]{inputenc}
%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% 	MACROS
%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\R}{\mathbb{R}} 
\newcommand{\C}{\mathbb{C}} 
\newcommand{\N}{\mathbb{N}} 
\newcommand{\Z}{\mathbb{Z}} 
\newcommand{\Q}{\mathbb{Q}} 
\newcommand{\F}{\mathbb{F}} 


%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% 	COLORES
%%%%%%%%%%%%%%%%%%%%%%%%%%
\definecolor{MyDarkBlue}{rgb}{0,0.08,0.45}
\definecolor{MedioRojo}{rgb}{0.5,0,0}
\definecolor{masrojo}{rgb}{0.7,0,0}
%  \colorlet{darkred}{red!80!black}
%  \colorlet{darkblue}{blue!80!black}
\definecolor{darkgreen}{rgb}{0.1,0.5,0.2}
\definecolor{Gris}{rgb}{0.5,0.5,0.5}
\definecolor{pink}{cmyk}{0.0,0.7,0,0}
\definecolor{Cyan}{rgb}{0,1,1}
\definecolor{Gold}{rgb}{1.,0.84,0.}
\definecolor{Pink}{rgb}{1.,0.75,0.8}
\definecolor{Blue3}{rgb}{0.53,0.81,.92}
\definecolor{Blue5}{rgb}{.8,1.,1.}
\definecolor{subblue}{rgb}{0.27, 0.51, 0.71} 

\mode<presentation>{
%\usetheme[numbers]{Warsaw} %Rebordes y n\'{u}meros
\usetheme{Frankfurt} %Rebordes y n\'{u}meros indice con puntos
\usecolortheme{seahorse} % Rebordes malvas
%\usecolortheme{rose} %Reborde azules m\'{a}s claros en bloques
%\usefonttheme[onlylarge]{structuresmallcapsserif} %\sc en t\'{\i}tulos
%\usefonttheme[onlysmall]{structurebold} % t\'{\i}tulo de arriba en negrita ??
%\setbeamerfont{title}{shape=\itshape,family=\rmfamily} %t\'{\i}tulo en it\'{a}lica
%\setbeamercolor{title}{fg=blue!80!black,bg=green!20!white} %t\'{\i}tulo en rojo con algo de negro y fondo rojiblanco
}

\setbeamertemplate{navigation symbols}{} 

\setbeamertemplate{headline}
{%
\begin{beamercolorbox}{section in head/foot}
%\alert{\ \insertframenumber\ de \inserttotalframenumber}
\ \vskip6.5pt\par \insertnavigation{\paperwidth}\vskip2pt
\end{beamercolorbox}%
}


\setbeamercolor{footline}{fg=black!40!white}
\setbeamertemplate{footline}{%
%\leavevmode\hbox{\insertshortauthor\qquad \insertshorttitle\qquad \insertshortdate\qquad \insertshortinstitute\qquad page %\insertframenumber{} of \inserttotalframenumber} 
}



\title{Kloosterman\'{\i}a}
\author{Fernando}
\institute[Aut\'{o}noma]{Universidad Aut\'{o}noma}


\begin{document}



%%%%%%%%%%%%%%%%%%%%
% SLIDE 0  %
%%%%%%%%%%%%%%%%%%%%
%P\'{a}gina del t\'{\i}tulo
\begin{frame}[plain] %Plain quita los estilos de esta p\'{a}gina
\begin{center}
\begin{tabular}{c}\includegraphics[scale=0.36]{./images/title.jpg}\end{tabular}
\\[8pt] Fernando Chamizo  \\ {\tiny http://www.uam.es/fernando.chamizo}
\\[17pt] \textrm{Universidad Aut\'{o}noma de Madrid}
\\[5pt]
\textrm{\begin{small}Madrid, 20 de enero de 2010        \end{small}}
\end{center}
\end{frame}


%%%%%%%%%%%%%%%%%%%%
% SLIDE 0+ %
%%%%%%%%%%%%%%%%%%%%
%Crea el \'{\i}ndice
%\section*{\'{I}ndice}
\begin{frame}
\frametitle{\'{I}ndice}
%\tableofcontents[pausesections]
\tableofcontents
\end{frame}




%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\begin{block}{Resumen}
La teor\'{\i}a espectral de formas automorfas fue desarrollada por A.~Selberg pasada la mitad del siglo~XX siendo uno de sus resultados m\'{a}s conocidos la f\'{o}rmula de traza. A principios de los a\~{n}os~80, N.V.~Kuznetsov prob\'{o} otra f\'{o}rmula espectral con potenciales aplicaciones aritm\'{e}ticas que conten\'{\i}a unas sumas introducidas por Kloosterman m\'{a}s de cincuenta a\~{n}os atr\'{a}s. La expectaci\'{o}n causada por esta f\'{o}rmula y los desarrollos iniciales en torno a ella se llamaron entonces \textit{Kloosterman\'{\i}a}.  

En esta charla intentamos dar una introducci\'{o}n muy b\'{a}sica de la teor\'{\i}a espectral de formas automorfas. No se supone ning\'{u}n conocimiento previo del tema.
\end{block}

\end{frame}


%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\section[S. de Kloosterman]{Sumas de Kloosterman}
\begin{frame}
\frametitle{?\lq Qu\'{e} son las sumas de Kloosterman?}

\begin{block}{Sumas de Kloosterman}
\[
 S(m,n;c)=
\sum_{a\cdot d\equiv 1\pmod{c}}e
\bigg(
\frac{ma+nd}{c}
\bigg)
\qquad\text{con }\ 
{\color{masrojo}e(x):=e^{2\pi ix}}
\]
\end{block}


\

{\color{black}Equivalentemente
\[
S(m,n;c)=
\sideset{}{^*}\sum_{a= 1}^c
e
\bigg(
\frac{ma+n\bar{a}}{c}
\bigg)
\]
donde $\bar{a}$ es el inverso de $a$ m\'{o}dulo $c$ y consecuentemente la suma est\'{a} restringida a $\text{mcd}(a,c)=1$.
}
\end{frame}


%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}

{\setbeamercolor{block title}{bg=green}
\begin{block}{}
Las sumas de Kloosterman sirven para medir la distribuci\'{o}n de los inversos m\'{o}dulo $c$.
\end{block}
}

\begin{block}{}
\[
\mathcal N_{IJ}
=\#\{a\in I\;:\; \bar{a}\in J\}
=\sideset{}{^*}\sum_{a= 1}^c\chi_I(a)\chi_J(\bar{a})
\]
\end{block}

\pause

\begin{block}{}
\[
\chi_J(x)=\raisebox{-10pt}{\includegraphics[scale=0.4]{./images/fcar.png}}
=\sum \alpha_n e(nx/c)
\]
\end{block}

\begin{block}{}
\[
\mathcal N_{IJ}
=
\sideset{}{^*}\sum_{a= 1}^c
\chi_I(a)\chi_J(\bar{a})
=
\sum_n\sum_m
\alpha_n\beta_m
\sideset{}{^*}\sum_{a= 1}^c
e
\bigg(
\frac{ma+n\bar{a}}{c}
\bigg)
\]
\end{block}


\end{frame}




%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}

{\setbeamercolor{block title}{bg=Gold}
\begin{block}{{\color{masrojo}\textbf{La historia no es siempre l\'{o}gica} (Parte I)}}
Lo m\'{a}s natural es que las sumas de Kloosterman hubieran aparecido en alg\'{u}n problema directamente relacionado con la distribuci\'{o}n de los inversos pero no fue as\'{\i}. Kloosterman las introdujo en 1926 al crear una versi\'{o}n del m\'{e}todo del c\'{\i}rculo que permitiera aproximar el n\'{u}mero de representaciones por formas cuadr\'{a}ticas de cuatro variables.

\

Ej: ?\lq Cu\'{a}ntas soluciones tiene $N=x^2+3y^2+5z^2+7t^2$?
\end{block}
}

\end{frame}


%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{El m\'{e}todo del c\'{\i}rculo}

\vspace{-11.5pt}
\begin{block}{}
\vspace{-13pt}
\[
\#\{(a_1,\dots,a_n)\in\mathcal A^n\;:\; N=a_1+\dots+a_n\}
=\int_0^1 S^n(x)e(-Nx)\;dx
\]
\[
\text{{\color{blue}$n=2$, $\mathcal A=\{\text{primos}\le N\}\leftrightarrow \text{Goldbach}$ }}
\qquad\quad\ 
S(x)=\sum_{a\in\mathcal A}e(ax)
\]
\end{block}

\vspace{-7.1pt}
\begin{block}{}
$S(x)$ con $x\approx\frac aq$, $\leftrightarrow$ distribuci\'{o}n de $\mathcal A$ en p. aritm\'{e}ticas

\begin{small}$q$ peque\~{n}o $\rightarrow$ cuenta directa
\end{small}
\qquad
\begin{small}$q$ grande $\rightarrow$ sumas trigonom\'{e}tricas                                                  \end{small}
\end{block}


\vspace{-7.1pt}
{\setbeamercolor{block title}{bg=green}
\begin{block}{}
Para problemas {\color{darkgreen}binarios} como Goldbach en que la integral deber\'{\i}a ser como $|\mathcal A|$ el m\'{e}todo no funciona porque una indeterminaci\'{o}n de~$C |\mathcal A|^{1/2+\epsilon}$ en $S(x)$ es lo mejor que podr\'{\i}amos so\~{n}ar en general.
\end{block}
}

{\setbeamercolor{block title}{bg=red}
\begin{block}{}
Kloosterman logr\'{o} resolver con sus sumas un problema binario relacionado con formas cuadr\'{a}ticas.
\end{block}
}

\end{frame}



%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{El m\'{e}todo del c\'{\i}rculo seg\'{u}n Kloosterman}

\vspace{-13pt}
\begin{tabular}{lll}
	{\color{MedioRojo}\fbox{\begin{tabular}{l}
	Divisi\'{o}n
	\\
	de Farey
	\end{tabular}}}
\hspace{-18pt}
&
	\begin{tabular}{l}
	\includegraphics[scale=0.43]{./images/arcs.png}
	\end{tabular}
&
\hspace{-15pt}
	{\color{black}\begin{tabular}{l}
	$b=a+A$
	\\
	$d=c+C$
	\end{tabular}}
\end{tabular}

\vspace{-3pt}
\begin{block}{}
\[
\begin{matrix}
cA-Ca=1
\\[-2pt]
\Downarrow
\\[-2pt]
r=1/cd
,\ 
a=-\bar{d}
\end{matrix}
\qquad
\smash{\int_0^rf\big(\frac ac+x\big)\;dx=\int_0^{1/cd}f\big(-\frac{\bar{d}}{c}+x\big)}
\]
\end{block}

{\setbeamercolor{block title}{bg=green}
\begin{block}{}
\begin{center}
\begin{tabular}{lll}
Cambio de $d$
&$\longrightarrow$
& cambio de arco
\\
Sumas de Kloosterman
&$\longrightarrow$
&cancelaci\'{o}n entre arcos
\end{tabular}
\end{center}
\end{block}
}

\begin{block}{}
\[
{\color{subblue}
\genfrac{[}{]}{0pt}{}{f\text{ real y par}}{N\in\Z^+}}
\ \ \ \ 
\smash{\int_0^1} f(x)\;dx=2
\mathop{\sum \sideset{}{^*}\sum}_{c\le N<d\le c+N}
\smash{\int_0^{1/cd}f\big(-\frac{\bar{d}}{c}+x\big)} 
\]
\end{block}


\end{frame}



%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Genealog\'{\i}a de la Kloosterman\'{\i}a}

%\begin{center}
{\color{black}
\begin{tabular}{ccc}
	\begin{tabular}{l}
		\begin{tabular}{ll}
		\begin{tabular}{l}
		\includegraphics[scale=0.2]{./images/kloosterman.jpg}
		\end{tabular}
		&
		\hspace{-25pt}
		\begin{footnotesize}\begin{tabular}{l}
		\textbf{H.D. Kloosterman}
		\\[4pt]
		1926 
		\\
		M\'{e}todo del c\'{\i}rculo
		\\[4pt]
		1927 
		\\
		Coef. f. modulares
		\end{tabular}
		\end{footnotesize}
		\end{tabular}
		\\[-3pt]
		\hspace{1.3cm}$|$
		\\[1pt]
		\begin{tabular}{ll}
		\begin{tabular}{l}
		\includegraphics[scale=0.19]{./images/weil.jpg}
		\end{tabular}
		&
		\hspace{-25pt}
		\begin{footnotesize}\begin{tabular}{l}
		\textbf{A. Weil}
		\\[4pt]
		1948
		\\
		Estimaci\'{o}n \'{o}ptima
		\\
		H. Riemann
		en $\mathbb F_q$
		\end{tabular}
		\end{footnotesize}
		\end{tabular}
	\end{tabular}
&
\ \hspace{-50pt}
&
\begin{tabular}{l}
	\begin{tabular}{ll}
	\begin{tabular}{l}
	\includegraphics[scale=0.35]{./images/maass.jpg}
	\end{tabular}
	&
	\hspace{-25pt}
	\begin{footnotesize}\begin{tabular}{l}
	\textbf{H. Maass}
	\\[4pt]
	1949 
	\\
	F. modulares
	\\
	no holomorfas 
	\end{tabular}
	\end{footnotesize}
	\end{tabular}
	\\[-3pt]
	\hspace{1.3cm}$|$
	\\[1pt]
	\begin{tabular}{ll}
	\begin{tabular}{l}
	\includegraphics[scale=0.19]{./images/selberg.jpg}
	\end{tabular}
	&
	\hspace{-25pt}
	\begin{footnotesize}\begin{tabular}{l}
	\textbf{A. Selberg}
	\\[4pt]
	1965
	\\
	F. de traza
	\\[8pt]
	T{\textordfeminine}. espectral
	\end{tabular}
	\end{footnotesize}
	\end{tabular}
\end{tabular}
\\
{$\ddots$}
&
\ \hspace{-50pt}
&
\ \hspace{-150pt}
$\swarrow$
\end{tabular}
%\end{center}

\ \hspace{50pt}
\begin{footnotesize}\textbf{N.V. Kuznetsov}  1980, 1981 F. de Kuznetsov                                                 \end{footnotesize}

}






\end{frame}




%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Dos citas}



\begin{block}{D.R. Heath-Brown}
Kuznetsov's paper sparked a tremendous flurry of work,
%principally by Kuznetsov and Iwaniec, and their co-authors. A
%multitude of different kinds of averages of Kloosterman sums were
%investigated, 
[\dots ] 
and a host of applications were produced.
This activity was nicknamed \lq Kloostermania\rq, which reflected
the excitement generated in the field at large.
\end{block}


\begin{block}{P. Michel}
The word Kloostermania of the title was invented by M. Huxley in the    80's
to highlight a series of striking developments that took place at that time in analytic number
theory and which build further of \textit{(sic)} on the modular nature of Kloosterman sums.
\end{block}


\end{frame}




%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\section[Bases]{Elementos b\'{a}sicos}
\begin{frame}
\frametitle{Filosof\'{\i}a b\'{a}sica (\textit{a posteriori})}


\begin{block}{}
La teor\'{\i}a anal\'{\i}tica de n\'{u}meros ha utilizado el an\'{a}lisis arm\'{o}nico en $\R/\Z$ para estudiar la distribuci\'{o}n m\'{o}dulo~1.
\end{block}

\begin{block}{}
Para estudiar la distribuci\'{o}n de los inversos o el algoritmo de Euclides quiz\'{a} deber\'{\i}amos hacer an\'{a}lisis arm\'{o}nico en el espacio base $\text{SL}_2(\R)$ y con $\text{SL}_2(\Z)$ como subgrupo discreto
\[
\text{SL}_2(K)=
\left\{
\begin{pmatrix}
a&b\\ c&d
\end{pmatrix}
\;:\; ad-bc=1
\right\}.
\]
\end{block}


{\setbeamercolor{block title}{bg=Cyan}
\begin{block}{}
En cierto sentido $\text{SL}_2(\R)$ es muy similar al semiplano de Poincar\'{e} $\mathbb H$ m\'{a}s familiar geom\'{e}tricamente y que ya aparece en las formas modulares cl\'{a}sicas.
\end{block}}


\end{frame}



%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{El semiplano de Poincar\'{e}}


\vspace{-16pt}
{\color{black}
\[
\hspace{-1pt}
\begin{matrix}
\mathbb H\text{ semiplano de Poincar\'{e}}
\\
\raisebox{-10pt}{\includegraphics[scale=0.25]{./images/spoin.png}}
\end{matrix}
\hspace{-3pt}
\begin{matrix}
\text{\begin{small}M\'{e}trica\end{small}}
\\
\scriptstyle ds^2=y^{-2}(dx^2+dy^2)
\\[4pt]
\text{\begin{small}Medida invariante\end{small}}
\\
\scriptstyle d\mu=y^{-2}dxdy
\end{matrix}
\begin{matrix}
\text{ geod\'{e}sicas}
\\
\raisebox{-10pt}{\includegraphics[scale=0.25]{./images/gpoin.png}}
\end{matrix}
\]
}


\begin{block}{}
Dado $\gamma=
\big(
\begin{smallmatrix}
a&b\\ c&d
\end{smallmatrix}
\big)
\in \text{SL}_2(\R)$ y $z\in\mathbb H$ se define
$\gamma z=
\frac{az+b}{cz+d}$.
Con esta acci\'{o}n $\text{SL}_2(\R)$ son las isometr\'{\i}as directas de $\mathbb H$.
\end{block}


\vspace{5pt}
\begin{footnotesize}
{\setbeamercolor{block title}{bg=Cyan}
\begin{block}{}
A partir de $\gamma i$ es posible recuperar $\gamma\in\text{SL}_2(\R)$ excepto por la matriz de un giro (eucl\'{\i}deo), de este modo $\mathbb H$ es como  $\text{SL}_2(\R)$ olvid\'{a}ndose de las rotaciones.
\end{block}
}
\end{footnotesize}

\end{frame}



%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}

{\color{black}
\begin{small}$D=\text{SL}_2(\Z)\backslash \mathbb H=\text{pegar los puntos $z$ y $w$ si $z=\gamma w$ con $\gamma\in \text{SL}_2(\Z)$}$\end{small} 

\vspace{-5pt}
\[
\hspace{-1pt}
\begin{matrix}
\raisebox{-10pt}{\includegraphics[scale=0.25]{./images/dfun.png}}
\end{matrix}
\hspace{-3pt}
\begin{matrix}
\text{SL}_2(\Z)=\langle
\big(
\begin{smallmatrix}
 1&1\\ 0&1
\end{smallmatrix}
\big)
,
\big(
\begin{smallmatrix}
 0&-1\\ 1&0
\end{smallmatrix}
\big)
\rangle
\\[6pt]
\ \ \ \ \ \ \ \ =\langle
{\scriptstyle z\mapsto z+1},
{\scriptstyle z\mapsto -\frac 1z}
\rangle
\end{matrix}
\begin{matrix}
\raisebox{-10pt}{\includegraphics[scale=0.25]{./images/pmod.png}}
\end{matrix}
=D
\]
}





\vspace{-5pt}
\begin{block}{}
Las funciones en $D$ son funciones {\color{MedioRojo}automorfas} en $\mathbb H$, es decir, verifican $f(\gamma z)=f(z)$ para $\gamma\in \text{SL}_2(\Z)$.

Un m\'{e}todo b\'{a}sico para crear funciones automorfas es considerar $F(z)=\sum f(\gamma z)$ con $f$ una funci\'{o}n en $\mathbb H$. 
\end{block}



\begin{footnotesize}
{\setbeamercolor{block title}{bg=Cyan}
\begin{block}{}
Desde el punto de vista complejo, $D$ es una superficie de Riemann (variedad 1-dim sobre $\C$). A diferencia del caso eucl\'{\i}deo, $\R/\Z$, no es compacta.\end{block}
}
\end{footnotesize}

\end{frame}



%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}

{\setbeamercolor{block title}{bg=Gold}
\begin{block}{}
?\lq Es posible encontrar problemas aritm\'{e}ticos que den lugar a sumas sobre $\gamma\in \text{SL}_2(\Z)$?
\end{block}
}

{\setbeamercolor{block title}{bg=Gold}
\begin{block}{}
Aqu\'{\i} mencionaremos dos de ellos:

\begin{itemize}
 \item 
El momento de orden 4 de la funci\'{o}n $\zeta$ de Riemann en la l\'{\i}nea cr\'{\i}tica.
\item
La equidistribuci\'{o}n de ra\'{\i}ces de un polinomio cuadr\'{a}tico m\'{o}dulo $p$ variable.
\end{itemize}


\end{block}
}


\end{frame}




%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Momentos de la funci\'{o}n $\zeta$ de Riemann}

\begin{block}{}
Los momentos $\int_0^T|\zeta(\frac 12+it)|^{2k}\;dt$ son cantidades naturales en relaci\'{o}n con la \textit{hip\'{o}tesis de Lindel\"of} \begin{small}($\zeta$ crece lo menos posible en la l\'{\i}nea cr\'{\i}tica $\Leftarrow$ Hip\'{o}tesis de Riemann)\end{small}. 
\end{block}


{\setbeamercolor{block title}{bg=green}
\begin{block}{}
\[
\int |\zeta(\frac 12+it)|^2\;dt
=\sum\frac{1}{\sqrt{mn}}
\int 
\big( m/n\big)^{it}\;dt,
\ \quad m\approx n \rightarrow\text{principal}
\]
\end{block}}

\vspace{-9pt}
{\color{black}
\[
\zeta^2(s)=
\sum \frac{1}{n^s}
\sum \frac{1}{m^s}
=
\sum \frac{d(n)}{n^s}
\]}

\vspace{-12pt}
{\setbeamercolor{block title}{bg=green}
\begin{block}{}
\[
\int |\zeta(\frac 12+it)|^4\;dt
=\sum\frac{d(m)d(n)}{\sqrt{mn}}
\int 
\big( m/n\big)^{it}\;dt,
\quad m\approx n \rightarrow\text{princ.}
\]
\end{block}}


\end{frame}


%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}


{\setbeamercolor{block title}{bg=green}
\begin{block}{}
\[
\sum d(n+k)d(n)
=
\sum_{ad=n+k}\
\sum_{bc=n}
1
\]
\end{block}}


{\setbeamercolor{block title}{bg=green}
\begin{block}{}
\begin{tabular}{lll}
$k=1$ & $\Rightarrow$ &
$
\begin{pmatrix}
a&b\\ c&d
\end{pmatrix}
\in\text{SL}_2(\Z)
$
\\
$k>1$ & $\Rightarrow$ &
Se relaciona con
$
\text{SL}_2(\Z)
$ 
(operadores de Hecke)
\end{tabular}

\

\vspace{8pt}

\underline{Ej}.
\begin{small}\[
\left\{
\begin{matrix}
\begin{pmatrix}
a&b\\ c&d
\end{pmatrix}\;:\;
\\
ad-bc=2
\end{matrix}
\right\}
=
\text{SL}_2(\Z)
\begin{pmatrix}
1&0\\ 0&2
\end{pmatrix}
\cup
\text{SL}_2(\Z)
\begin{pmatrix}
2&0\\ 0&1
\end{pmatrix}
\cup
\text{SL}_2(\Z)
\begin{pmatrix}
1&1\\ 0&2
\end{pmatrix}
\]\end{small}
\end{block}}



\end{frame}



%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Equidistribuci\'{o}n de ra\'{\i}ces de polinomios cuadr\'{a}ticos}

\begin{block}{}
Sabemos que la ecuaci\'{o}n $x^2\equiv -1\pmod{p}$ tiene soluci\'{o}n si $p=4k+1$ pero ?\lq c\'{o}mo est\'{a}n distribuidas las ra\'{\i}ces $\pm r_p$ cuando el primo $p$ var\'{\i}a? 
\end{block}


{\setbeamercolor{block title}{bg=green}
\begin{block}{}
\[
\text{Distribuci\'{o}n de }
\frac{r_p}{p}
\text{ en $[0,1]$}
\overset{\text{Fourier}}{\longleftrightarrow}
\text{Cancelaci\'{o}n en }
\sum_p e\big(\frac{nr_p}{p}\big)
\]
\end{block}}

{\setbeamercolor{block title}{bg=red}
\begin{block}{}
Un tratamiento anal\'{\i}tico de estas sumas se muestra dif\'{\i}cil porque no parece posible parametrizar cantidades tan aritm\'{e}ticas como $p$ (los primos) y $r_p$. 
T\'{e}cnicas \lq\lq de criba'' permiten cambiar primos por enteros pero el problema con la dependencia en $r_p$ persiste.
\end{block}}


\end{frame}


%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}


{\setbeamercolor{block title}{bg=green}
\begin{block}{}
\begin{small}
\begin{align*}
\gamma=
\begin{pmatrix}
a&b\\ c&d
\end{pmatrix}\in
\text{SL}_2(\Z)
&\Leftrightarrow
\gamma\gamma^t=
\begin{pmatrix}
a^2+b^2& ac+bd\\ ac+bd & c^2+d^2
\end{pmatrix}\in
\text{SL}_2(\Z)
\\
&\Leftrightarrow
(a^2+b^2)(c^2+d^2)=(ac+bd)^2+1
\end{align*}
\end{small}
\vspace{-12pt}
\begin{align*}
x^2\equiv -1\pmod{m}\text{ tiene sol.}\ \Leftrightarrow\ &
m=c^2+d^2\text{ con $c$, $d$ coprimos}
\\
\text{Sol. }x=ac+bd&\text{ con } ad-bc=1
\end{align*}
\end{block}}



{\setbeamercolor{block title}{bg=green}
\begin{block}{}
\[
\text{SL}_2(\Z) \longleftrightarrow 
\text{ soluciones de }x^2\equiv -1\pmod{m}
\]
\end{block}}


{\setbeamercolor{block title}{bg=green}
\begin{block}{}
Para $\gamma=
\big(
\begin{smallmatrix}
a&b\\ c&d
\end{smallmatrix}
\big)
\in \text{SL}_2(\Z)$
se tiene 
$\gamma i=\frac{ac+bd}{c^2+d^2}+\frac{i}{c^2+d^2}$.

Entonces
\[
\sum_p e\big(\frac{nr_p}{p}\big)
\text{\lq\lq}\longleftarrow\text{''}
\sum_\gamma e(n\text{Re}\gamma i).
\]
\end{block}}




\end{frame}





%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}


\begin{alertblock}{}
Las sumas de Kloosterman son inherentes al tratamiento con an\'{a}lisis de Fourier de sumas sobre $\gamma\in\text{SL}_2(\Z)$.
\end{alertblock}

\begin{block}{}
\[
F(x,y)=\smash{\sum_\gamma} f(\gamma z,w)\qquad
\text{con $z=x+x_0i$, $w=y+y_0i$}
\]

\vspace{5pt}
\begin{small}$f(z+t,w+t)=f(z,w)$ para $t\in \R$ $\Rightarrow$ $F$ peri\'{o}dica en $x,\;y$  $\Rightarrow$  el coef. de Fourier $(m,-n)$ es $\smash{\sum_c S(m,n;c)\widetilde{F}(c)}$ con $\smash{\widetilde{F}}$ cierta transformada.\end{small}
\end{block}

\begin{block}{}
Usando $\gamma z=1/c-c^{-2}(z+d/c)^{-1}$, para $e(x,y):=e(-mx+ny)$
\vspace{-4pt}
\begin{small}\[
\int_0^1
\int_0^1
f(\gamma z,w)e(x,y)dxdy
=
\int_0^1
\int_0^1
f\big(\frac ac-\frac{1}{c^2(z+d/c)},w)e(x,y)dxdy
\]  \end{small}
y con el cambio $x\mapsto x-d/c$, $y\mapsto y+a/c$ 
\vspace{-4pt}
\begin{small}\[
e\big(\frac{md+na}c\big)
\int_0^1
\int_0^1
f\big({\color{Gris}\frac ac-}\frac{1}{c^2z},{\color{Gris}\frac ac+}w)e(x,y)dxdy .
\]  \end{small}
\end{block}

\end{frame}





%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\section{Teor\'{\i}a espectral}
\begin{frame}
\frametitle{Teor\'{\i}a espectral}


\vspace{-6pt}
\begin{block}{Descomposici\'{o}n espectral \hfill $f\in L^2(D)$}
\[
f(z)=\sum_j \langle f,u_j\rangle u_j(z)+\dots \text{(contr. espectro continuo)}
\] 


\begin{tabular}{lll}
$u_j(z)$&$=$&\textit{{\color{masrojo}formas de Maass}}
\\
&$=$&autofunciones en $L^2(D)$ del laplaciano hiperb\'{o}lico
\\[3pt]
$\langle f,g\rangle$&$=$&$\int_D \bar{f}{g}d\mu$
\end{tabular}
\vspace{-3pt}
\[
\Delta u_j=-\lambda_j u_j,
\qquad
\Delta=y^2\bigg(
\frac{\partial^2}{\partial x^2}
+\frac{\partial^2}{\partial y^2}
\bigg)
\]
\end{block}

\vspace{-2pt}
\begin{small}{\setbeamercolor{block title}{bg=green}
\begin{block}{}
$(\text{Im}(z))^s=y^s$ cumple $\Delta y^s=-s(1-s)y^s$  y considerando $\sum (\text{Im}(\gamma z))^s$ se puede forzar a que sea una autofunci\'{o}n que vive en $D$ pero no est\'{a} en $L^2(D)$ y despu\'{e}s de una continuaci\'{o}n anal\'{\i}tica da lugar a la contribuci\'{o}n del espectro continuo.
\end{block}
} \end{small}


\end{frame}




%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}


\begin{block}{}
Las formas de Maass son  1-peri\'{o}dicas $u_j(z+1)=u_j(z)$,  por tanto tienen un desarrollo de Fourier
\[
 u_j(z)=\sum_n a_n(y) e(nx)\qquad \text{donde }z=x+iy.
\]
Como no son holomorfas no hay un desarrollo $\sum c_n e(nz)$ pero la propiedad $\Delta u_j=-\lambda_j u_j$ determina $a_n(y)$ salvo una constante
\[
 u_j(z)=\sum_n \rho_j(n)\cdot\text{f. de Bessel}(y) e(nx). 
\]
\end{block}


\begin{block}{}
Estos \lq\lq coeficientes de Fourier'' $\rho_j(n)$ se ajustan, al igual que el caso holomorfo, a la teor\'{\i}a de Hecke, en particular son multiplicativos una vez normalizados. 
\end{block}


\end{frame}



%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}

{\setbeamercolor{block title}{bg=Gold}
\begin{block}{{\color{masrojo}\textbf{La historia no es siempre l\'{o}gica} (Parte II)}}
Ser\'{\i}a natural que las formas de Maass hubieran nacido en el seno del an\'{a}lisis arm\'{o}nico o las ecuaciones en derivadas parciales. Sin embargo Maass las introdujo para resolver un problema relacionado con ideales en cuerpos cuadr\'{a}ticos $K=\Q(\sqrt{d})$.



\end{block}
}


{\setbeamercolor{block title}{bg=Gold}
\begin{block}{}

\vspace{3pt}
\textbf{Caso $\mathbf{d<0}$:} 
normas de ideales 
$\leftrightarrow$ formas cuadr\'{a}ticas 
$\leftrightarrow$ formas modulares 
$\leftrightarrow$ ecuaciones funcionales

\vspace{3pt}
Ej. $K=\Q(i)\ \Rightarrow\ \zeta_K(s)=\sum N(I)^{-s}=\sum (a^2+b^2)^{-s}$ 

\

$f(z)=\sum e\big((a^2+b^2)z\big)\overset{\text{Poisson}}{\leftrightarrow} f(-1/4z)\overset{\text{Mellin}}{\rightarrow}\text{ec. func. } \zeta_K$


\vspace{13pt}
\begin{footnotesize}(En realidad el caso de inter\'{e}s es $L(s)=\sum \chi(I)N(I)^{-s}$)\end{footnotesize}

\end{block}



}


\end{frame}





%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}

{\setbeamercolor{block title}{bg=Gold}
\begin{block}{}

\vspace{5pt}
\textbf{Caso $\mathbf{d>0}$:} ?\lq ?  $N(a+b\sqrt{7})=|a^2-7b^2|$ no es forma cuadr\'{a}tica y no se puede aplicar Poisson como antes.


\vspace{5pt}
Maass en 1949 consigui\'{o} salvar el esquema para el caso $d>0$ utilizando sustitutos no holomorfos para $f(z)$, las formas de Maass, que permit\'{\i}an probar las ecuaciones funcionales esperadas. Las funciones de Bessel de su desarrollo de Fourier se comportaban bien bajo transformadas de Mellin.


\vspace{13pt}
\begin{footnotesize}(En realidad las funciones de Maass eran invariantes por cierto $\Gamma<\text{SL}_2(\Z)$)\end{footnotesize}

\end{block}
}

\begin{block}{} 
Poco despu\'{e}s la \textit{Tesis de Tate} permiti\'{o} aplicar directamente \lq\lq sumaci\'{o}n de Poisson sobre los adeles'' a $\zeta_K(s)$ y otras funciones $L$.  
\end{block}


\end{frame}



%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\section[Traza de Selberg]{La f\'{o}rmula de traza de Selberg}
\begin{frame}
\frametitle{La f\'{o}rmula de traza de Selberg}


\begin{block}{F\'{o}rmula de Traza para $\text{SL}_2(\Z)$}
\[
\sum_j h(t_j)=
\frac{1}{12}
\int_{-\infty}^\infty 
\hspace{-3pt}
th(t)\tanh(\pi t)\; dt
+\frac{1}{4\pi}
\sum_\ell
\frac{\ell^*}{\sinh (\ell/2)}
\widehat{h}(\frac{\ell}{2\pi})+\cdots 
\] 
donde 
\vspace{-12pt}
\begin{align*}
&{\color{MedioRojo}{h}}=\text{funci\'{o}n par}
\quad\ {\color{MedioRojo}\widehat{h}}(\xi)=
\int_{-\infty}^\infty h(t)e(-\xi t)\; dt
\\
&1/4+{\color{MedioRojo}t_j^{{\color{black}2}}}=\text{autovalores de }-\Delta
\\
&{\color{MedioRojo}\ell}=\text{Longitudes de las geod\'{e}sicas}
\\
&{\color{MedioRojo}\ell^*}=\text{Longitudes primitivas }(\ell=k\ell^*)
\\
&{\color{MedioRojo}\dots}=\text{contribuci\'{o}n de algunos puntos especiales}
\end{align*}
\end{block}


\end{frame}



%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}


\begin{block}{}
La f\'{o}rmula de traza es una f\'{o}rmula esencialmente geom\'{e}trica pero tambi\'{e}n admite interpretaciones aritm\'{e}ticas. Se puede probar que las longitudes $\ell^*$ son de la forma $2\log\epsilon_d$ con $\epsilon_d$ la soluci\'{o}n \lq\lq m\'{a}s peque\~{n}a'' de la ecuaci\'{o}n de Pell $x^2-dy^2=4$. Adem\'{a}s cada longitud aparece $h(d)$ veces, el n\'{u}mero de clases. Sarnak dedujo de la f\'{o}rmula de traza
\[
 \smash{\sum_{\epsilon_d\le x}}h(d)=\text{li}(x^2)+O\big(x^{3/2}\log^2x\big).
\]
\vspace{1pt}
Por otro lado, la asint\'{o}tica de $\sum_{d\le x}h(d)$ se desconoce.
\end{block}

\begin{small}
{\setbeamercolor{block title}{bg=Cyan}
\begin{block}{}
Selberg prob\'{o} que $Z(s)=\prod_{\ell^*}\prod_{k=0}^\infty\big(1-e^{-(s+k)\ell^*}\big)$ tiene una continuaci\'{o}n anal\'{\i}tica y una ecuaci\'{o}n funcional. Adem\'{a}s !\lq cumple la hip\'{o}tesis de Riemann porque $\Delta$ es autoadjunto! (sue\~{n}o de P\'{o}lya-Hilbert). Con ello se puede hacer una teor\'{\i}a de \lq\lq geod\'{e}sicas primas'' en superficies de Riemann.
\end{block}
}
\end{small}



\end{frame}



%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Probando la f\'{o}rmula de {\color{red}Poisson} y la de {\color{darkgreen}Selberg}}


\vspace{-15pt}
{\setbeamercolor{block title}{bg=red}
\begin{block}{}
\vspace{-15pt}
\[
\text{\begin{tabular}{l}
Poisson
\\
($f$ par)
\end{tabular}
}
\quad
\sum_n f(n)
=
\sum_n \widehat{f}(n),
\qquad
\widehat{f}(\xi)=\int_{-\infty}^\infty f(x)e(-\xi x)\;dx
\] 
\end{block}
}

{\setbeamercolor{block title}{bg=green}
\begin{block}{}
\vspace{-15pt}
\[
\text{\begin{tabular}{l}
Selberg
\\
($f$ par)
\end{tabular}
}
\quad
\sum_j f(t_j)=
\text{(principal)}
+\frac{1}{4\pi}
\sum_\ell
\frac{\ell^*}{\sinh (\ell/2)}
\widehat{f}(\frac{\ell}{2\pi})+\cdots 
\] 
\end{block}
}



{\setbeamercolor{block title}{bg=red}
\begin{block}{}
\vspace{-15pt}
\[
P(x,y)
\hspace{-3pt}
=
\hspace{-3pt}
\sum f\big(d(x+n,y)\big)
\hspace{-3pt}
=
\hspace{-3pt}
\sum f\big(|n+x-y|\big),
\text{ \begin{small}$1$-peri\'{o}dica en $x$ e $y$\end{small}}
\] 
\end{block}
}

{\setbeamercolor{block title}{bg=green}
\begin{block}{}
%\vspace{-15pt}
\[
S(z,w)
={\sum_\gamma} f\big(d(\gamma z,w)\big),
\text{ \quad \begin{small}  $\text{SL}_2(\Z)$-peri\'{o}dica en $z$ y $w$\end{small}}
\] 
\begin{footnotesize}$S(\delta z,w)=S(z,w)\leftarrow \widetilde{\gamma}=\gamma\delta,\ S(z,\delta w)=S(z,w)\leftarrow d(z,\delta w)=d(\delta^{-1} z,w)$\end{footnotesize}
\end{block}
}


\end{frame}


%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}


{\setbeamercolor{block title}{bg=red}
\begin{block}{}
\vspace{-15pt}
\[
\text{Desarrollo autofunc. }
P(x,y)
=\sum a_n e(nx)e(-ny),\quad a_n=\widehat{f}(n)
\] 
\end{block}
}

{\setbeamercolor{block title}{bg=green}
\begin{block}{}
\vspace{-15pt}
\[
\rightarrow\ \ 
S(z,w)
=\sum a_j u_j(z)\overline{u_j(w)}+\dots \quad 
\text{(\textit{f\'{o}rmula de pretraza})}
\] 
\begin{small}$a_j=\widetilde{f}(t_j)=\text{cierta transformada integral}$\end{small}
\end{block}
}


{\setbeamercolor{block title}{bg=red, fg=black}
\begin{block}{}
{\color{MyDarkBlue}$\text{Traza}=\int_0^1 P(x,x)$}
\vspace{-12pt}
\[
\int\sum f\big(|n+x-x|\big)dx
=\int \sum \widehat{f}(n) e(nx)e(-nx)dx
\rightarrow\text{Poisson}
\] 
\end{block}
}

{\setbeamercolor{block title}{bg=green}
\begin{block}{}
En teor\'{\i}a deber\'{\i}a ser similar con $\int_D S(z,z) d\mu$
\vspace{-8pt}
\[
\smash{\sum_\gamma}
\int_D f\big(d(\gamma z,z)\big) d\mu
=\int_D\sum \widetilde{f}(t_j) |u_j(z)|^2d\mu
+\dots 
\] 
\vspace{-3pt}
pero $\sum=\dots=\infty$ $\rightarrow$ separar y cancelar lo que da $\infty$
\end{block}
}


\end{frame}



%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}

\vspace{-10pt}
\begin{block}{}
Selberg reinterpret\'{o} la suma resultante en t\'{e}rminos geom\'{e}tricos.
\end{block}


{\setbeamercolor{block title}{bg=red}
\begin{block}{}
%\vspace{-12pt}
En $\R^2$ las traslaciones corresponden a geod\'{e}sicas. Las traslaciones enteras corresponden a geod\'{e}sicas cerradas en $\R^2/\Z^2$.
\end{block}
}

{\setbeamercolor{block title}{bg=green}
\begin{block}{}
%\vspace{-12pt}
Los $\gamma\in\text{SL}_2(\R)$ \lq\lq hiperb\'{o}licos'' corresponden a geod\'{e}sicas en $\mathbb H$ y si $\gamma\in\text{SL}_2(\Z)$ corresponden a geod\'{e}sicas cerradas en $D$.
\end{block}
}

\begin{tabular}{lll}
\includegraphics[scale=0.2]{./images/trasl.png}
&\quad \ 
\includegraphics[scale=0.2]{./images/hiper.png}
&
\includegraphics[scale=0.2]{./images/super.png}
\end{tabular}



\end{frame}








%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\section[F\'{o}rmula de Kuznetsov]{La f\'{o}rmula de Kuznetsov}
\begin{frame}
\frametitle{La f\'{o}rmula de Kuznetsov}


\begin{block}{F\'{o}rmula de Kuznetsov para $\text{SL}_2(\Z)$}
\[
\sum_c\frac{S(m,n;c)}{c}
f\bigg(\frac{4\pi\sqrt{|mn|}}{c}\bigg)
=
\sum_j\widetilde{f}(t_j)\bar{\rho}_j(m)\rho_j(n)
+\dots
\] 
donde 
\vspace{-12pt}
\begin{align*}
&{\color{MedioRojo}{f}}=\text{funci\'{o}n buena con $f(0)=f(\infty)=0$}
\\
&{\color{MedioRojo}\widetilde{f}}=\text{transformada de Bessel complicada}
\\
&1/4+{\color{MedioRojo}t_j^{{\color{black}2}}}=\text{autovalores de }-\Delta
\\
&{\color{MedioRojo}\rho_j}(n)=\text{$n$-\'{e}simo coeficiente de Fourier de $u_j$}
\\
&{\color{MedioRojo}\dots}=\text{contribuci\'{o}n del $\infty$ y de formas modulares cl\'{a}sicas}
\end{align*}
\end{block}


\end{frame}


%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}

{\setbeamercolor{block title}{bg=red}
\begin{block}{}{{\color{MyDarkBlue}\lq\lq Prueba'' ultrarr\'{a}pida de Poisson} \hfill {\color{black}$\delta_p=\delta$ de Dirac peri\'{o}dica}}
\[
 \delta_p(x)\overset{\text{Fourier}}{=}\sum e(nx)
\overset{\int f(x)\cdot}{\Longrightarrow}
\sum f(n)=\sum\widehat{f}(n)
\]
\end{block}
}

{\setbeamercolor{block title}{bg=red}
\begin{block}{}{{\color{MyDarkBlue}Formalizaci\'{o}n con funciones de Green}}
\[
 g(x)=\sum\frac{e(nx)}{4\pi^2n^2+r^2}\qquad
\text{converge para $r\ne 0$}
\]
Formalmente $L[g]=\delta_p$ con $L[g]=-g''+r^2g$. Por otro lado $y(x)=e^{-r|x|}/2r$ es la soluci\'{o}n de $-y''+r^2y=\delta$
\[
 \delta_p(x)=\sum\delta(x+n)\ \Rightarrow\ 
g(x)=\sum\frac{e(nx)}{4\pi^2n^2+r^2}=\frac{1}{2r}\sum e^{-r|x+n|}.
\]
Calculando $f*g(x)=\int f(t)g(x-t) dt$ con las dos expresiones para $g$ y aplicando $L$ despu\'{e}s, se recupera la f\'{o}rmula de Poisson.
\end{block}
}
\end{frame}




%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Prueba de la f\'{o}rmula de Kuznetsov (idea)}

{\setbeamercolor{block title}{bg=green}
\vspace{-9pt}
\begin{block}{}
\begin{small}\[
G(z,w)=\smash{\sum_j}\frac{u_j(z)\overline{u_j(w)}}{\lambda_j+r^2}+\dots 
\qquad
(-\Delta+r^2)G(\cdot,w)=\delta_D(\cdot-w)
\]  \end{small}
y tambi\'{e}n podemos resolver $-\Delta+r^2=\delta$ en $\mathbb H$ y transformar $\delta$ en $\delta_D$ sumando sobre $\text{SL}_2(\Z)$
\vspace{-4pt}
\[
 G(z,w)=\smash{\sum_j}\frac{u_j(z)\overline{u_j(w)}}{\lambda_j+r^2}+\dots =\smash{\sum_\gamma} f(\gamma z,w)
\]
\begin{footnotesize}(en realidad hay un problema de convergencia)\end{footnotesize}.

Calculando el coeficiente de Fourier $(m,-n)$
\vspace{-4pt}
\[
 \rho_j(m)\overline{\rho_j(n)}\cdot\text{f. de Bessel}+\dots=\smash{\sum_c} S(m,n;c)\cdot\text{f. de Bessel}.
\]
\begin{small}Con transformadas se pasa de f. de Bessel a cualquier funci\'{o}n \begin{footnotesize}(dif\'{\i}cil)\end{footnotesize}.\end{small} 
\end{block}
}

\end{frame}


%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}



\begin{block}{}
La consecuencia de la f\'{o}rmula de Kuznetsov es que hay una cancelaci\'{o}n entre sumas de Kloosterman para diferentes m\'{o}dulos.
\end{block}

\begin{block}{}
Weil prob\'{o} con m\'{e}todos de  geometr\'{\i}a algebraica esencialmente que $|S(m,n;c)|\le d(c)\sqrt{c}$ con $d(c)$ el n\'{u}mero de divisores. Esto es \lq\lq \'{o}ptimo'' porque se conoce $\sum_{c\le x}|S(m,n;c)|/\sqrt{c}>Cx$. 

\vspace{4pt}
La f\'{o}rmula de Kuznetsov permite deducir
\[
\smash{\sum_{c\le x}}\frac{S(m,n;c)}{\sqrt{c}}<Cx^{2/3}\log x 
\]
y por tanto cuantifica una variaci\'{o}n en los signos de $S(m,n;c)$ que es crucial en algunas aplicaciones.
\end{block}


\begin{block}{}
Tambi\'{e}n se pueden deducir resultados de teor\'{\i}a espectral a partir de resultados de sumas de Kloosterman. 
\end{block}



\end{frame}







%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\section{Kloosterman\'{\i}a hoy}
\begin{frame}
\frametitle{La Kloosterman\'{\i}a en retrospectiva}



\begin{block}{Algunos logros en los primera d\'{e}cada de Kloosterman\'{\i}a}
\begin{itemize}
 \item 
Acotaciones para la funci\'{o}n $\zeta$ de Riemann \begin{small}(Iwaniec 1980, m\'{a}s adelante Jutila consigui\'{o} el resultado sin teor\'{\i}a espectral)\end{small}.
 \item 
Mejora del teorema de Bombieri-Vinogradov \begin{small}(Bombieri, Fouvry, Friedlander, Iwaniec 1984, 1986, 1989)\end{small}.
 \item 
Relaci\'{o}n local-global en representaciones por formas cuadr\'{a}ticas ternarias \begin{small}(Duke, Schulze-Pillot 1988, 1990)\end{small}.
\end{itemize}

\end{block}


\begin{block}{}
Los m\'{e}todos de la Kloosterman\'{\i}a han participado tambi\'{e}n en importantes resultados posteriores en teor\'{\i}a anal\'{\i}tica de n\'{u}meros y adem\'{a}s han establecido algunas relaciones inesperadas con la F\'{\i}sica (\textit{Arithmetic Quantum Chaos}, Sarnak 1992).
\end{block}


\end{frame}


%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}


%{\setbeamercolor{block title}{bg=masrojo}
%\begin{block}{}
%\color{MedioRojo}
%\begin{center}
%\begin{Large}\bf Atenci\'{o}n: {\color{black}Diapositiva muy opinable}\end{Large} 
%\end{center}\end{block}
%}


%\begin{small}
\begin{block}{}
Actualmente Kloosterman\'{\i}a ha perdido su nombre y se
ha incorporado a la atenci\'{o}n 
que ha recibido el mundo automorfo en la teor\'{\i}a de n\'{u}meros contempor\'{a}nea (sobre todo despu\'{e}s de la prueba del \'{u}ltimo teorema de Fermat). 
\end{block}
%\end{small}


{\setbeamercolor{block title}{bg=green}
\begin{block}{}
?\lq No estaremos haciendo m\'{a}s complicada de lo necesario la teor\'{\i}a anal\'{\i}tica de n\'{u}meros introduciendo la teor\'{\i}a espectral?
\end{block}
}

\begin{block}{Y. Motohashi (1995)}
La funci\'{o}n
\[
F(s)=\int^\infty_1|\zeta(\frac12+it)|^4t^{-s}dt
\]
tiene una extensi\'{o}n meromorfa en $\C$ y sus polos en  $\text{Re}(s)\ge 1/2$  est\'{a}n  exactamente en los $s$ tales que $(1-s)s$ es autovalor del laplaciano en $\text{SL}_2(\Z)\backslash\mathbb H$.
\end{block}


\end{frame}




%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
%\section{Algunas lecturas}
%\begin{frame}
%\frametitle{Algunas lecturas}
 
%\cite{HBKl}, 
%\cite{HuxKl},
%\cite{IwP}, 
%\cite{IwB},
%\cite{IwKo}, 
%\cite{MotoS}
%\cite{Moto}, 
%\cite{Mic}, 
%\cite{Sar}
%%\cite{Spr}, 
%\cite{Ter}.

%\end{frame}


%\bibliography{minibib}{}
%\bibliographystyle{amsplain}
%%\bibliographystyle{alpha}





%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\section[Lecturas]{Algunas lecturas}
\begin{frame}
\frametitle{Algunas lecturas}
 
{\setbeamercolor{block title}{bg=subblue}
\begin{block}{}
\begin{itemize}
 \item 
D.~R. Heath-Brown, \emph{Arithmetic applications of {K}loosterman sums}, Nieuw
  Arch. Wiskd. (5) \textbf{1} (2000), no.~4, 380--384. 
\item
M.~{{N}}. Huxley, \emph{Introduction to {K}loostermania}, Elementary and
  analytic theory of numbers ({W}arsaw, 1982), Banach Center Publ., vol.~17,
  PWN, Warsaw, 1985, pp.~217--306. 
\item
H.~Iwaniec, \emph{Promenade along modular forms and analytic number theory},
  Topics in analytic number theory ({A}ustin, {T}ex., 1982), Univ. Texas Press,
  Austin, TX, 1985, pp.~221--303. 
\item
H.~Iwaniec and E.~Kowalski, \emph{Analytic number theory}, American
  Mathematical Society Colloquium Publications, vol.~53, American Mathematical
  Society, Providence, RI, 2004. 
\end{itemize}
\end{block}
}

\end{frame}




%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Algunas lecturas}
 
{\setbeamercolor{block title}{bg=subblue}
\begin{block}{}
\begin{itemize}
 \item 
P.~Michel, \emph{Some recent applications of {K}loostermania}, Physics and
  number theory, IRMA Lect. Math. Theor. Phys., vol.~10, Eur. Math. Soc.,
  Z\"urich, 2006, pp.~225--251. 
\item
Y.~Motohashi, \emph{The {R}iemann zeta-function and the non-{E}uclidean
  {L}aplacian}, Sugaku Expositions \textbf{8} (1995), no.~1, 59--87. 
\item
Y.~Motohashi, \emph{Spectral theory of the {R}iemann zeta-function}, Cambridge
  Tracts in Mathematics, vol. 127, Cambridge University Press, Cambridge, 1997.
\item
P.~Sarnak, \emph{Kloosterman, quadratic forms and modular forms}, Nieuw Arch.
  Wiskd. (5) \textbf{1} (2000), no.~4, 385--389. 
\item
A.~Terras, \emph{Harmonic analysis on symmetric spaces and applications. {I}},
  Springer-Verlag, New York, 1985. 
\end{itemize}
\end{block}
}

\end{frame}


\end{document}


%%%%%%%%%%%%%%%%%%%%
% SLIDE            %
%%%%%%%%%%%%%%%%%%%%
\begin{frame}

\colorbox{green}{Colores}


\begin{center}
% use packages: array
\begin{tabular}{llll}
{\color{MyDarkBlue}\rule[-1mm]{5mm}{5mm} MyDarkBlue} &  
{\color{MedioRojo}\rule[-1mm]{5mm}{5mm} MedioRojo} &  
{\color{masrojo}\rule[-1mm]{5mm}{5mm} masrojo} &  
{\color{darkgreen}\rule[-1mm]{5mm}{5mm} darkgreen} 
\\[4.23pt]
{\color{Gris}\rule[-1mm]{5mm}{5mm} Gris} &  
{\color{pink}\rule[-1mm]{5mm}{5mm} pink} &  
{\color{Cyan}\rule[-1mm]{5mm}{5mm} Cyan} &  
{\color{Gold}\rule[-1mm]{5mm}{5mm} Gold} 
\\[4.23pt] 
{\color{Pink}\rule[-1mm]{5mm}{5mm} Pink} &  
{\color{Blue3}\rule[-1mm]{5mm}{5mm} Blue3} &  
{\color{Blue5}\rule[-1mm]{5mm}{5mm} Blue5} &  
{\color{subblue}\rule[-1mm]{5mm}{5mm} subblue} 
\end{tabular}
\end{center}



\color{black}
\begin{center}
\color{Gold}
\fcolorbox{Blue3}{Blue5}{\begin{large}\bf {\color{subblue}En Matem\'{a}ticas todo est\'{a} conectado}\end{large}}
\end{center}

\end{frame}



\end{document}
