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% $Header: /cvsroot/latex-beamer/latex-beamer/solutions/generic-talks/generic-ornate-15min-45min.en.tex,v 1.4 2004/10/07 20:53:08 tantau Exp $
 %\texhash

% This file is a solution template for:

% - Giving a talk on some subject.
% - The talk is between 15min and 45min long.
% - Style is ornate.



% Copyright 2004 by Till Tantau <tantau@users.sourceforge.net>.
%
% In principle, this file can be redistributed and/or modified under
% the terms of the GNU Public License, version 2.
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% However, this file is supposed to be a template to be modified
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% notice.


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%\title % (optional, use only with long paper titles)
%{Fourier transform of algebraic measures}
%
%%\subtitle
%%{$F$ a $p$-adic field} % (optional)
%
\author % (optional, use only with lots of authors)
{A. Aizenbud}
% - Use the \inst{?} command only if the authors have different
%   affiliation.


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{Weizmann Institute of Science}
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\date[] % (optional)
%{August 14, 2008}

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\title % (optional, use only with long paper titles)
[Points of algebraic varieties over finite rings.]
{Points of algebraic varieties over finite rings.}
%
%%\subtitle
%%{$F$ a $p$-adic field} % (optional)
%
\author % (optional, use only with lots of authors)
{A. Aizenbud}
% - Use the \inst{?} command only if the authors have different
%   affiliation.


%\institute[Massachusetts Institute of Technology] % (optional, but mostly needed)
%{Massachusetts Institute of Technology}

%\myD
\begin{document}


\begin{frame}
\myTP
\center{Joint with Nir Avni}

$ $

$ $

\center{\url{http://www.wisdom.weizmann.ac.il/~aizenr/}}

\end{frame}
\begin{frame}

\myFT{Questions}
%How many irreducible representations of dimension $n$ does $SL_d(\Z)$ have?
Let $X$ be a scheme of finite type over $\Z$.
\myP
\begin{question}
How many points does $X(\Z)$ have?
\end{question}
\myP
\begin{question}
How many points does $X(\Z/n\Z)$ have?
\end{question}
\myP
\begin{question}
How many points does $X(\Z/p^k\Z)$ have?
\end{question}
\myP
\begin{notation}
Let $R$ be a finite ring. \myP Denote $$h_X(R):=\frac{\#X(R)}{\#R^{\dim X_\Q}}.$$
\end{notation}
\end{frame}

\begin{frame}
\myFT{Finite Rings}
\begin{examples}
\begin{itemize}
\myIt $\Z/p^k\Z$
\myIt $\F_p[t]/t^k$
\myIt $\F_q[t]/t^k$
\myIt Let $\Q_{p^n}$ be the unique unramified extension of $\Q_p$ of order $n$. Let $\Z_{p^n}$ be its ring of integers and $\pi$ be its uniformizer. \myP Consider $R:=\Z_q/\pi^k$.
\end{itemize}
\end{examples}
\end{frame}

\begin{frame}
\myFT{Zero vs. positive characteristic}
\myP
\begin{notation}
For a finite set of primes $S$ denote by $\mathcal P_S$ the set of powers of primes outside $S$. \myP Denote also $\cP:=\cP_{\emptyset}$.
\myP Let $q\in \cP$ and $k \in \N.$ Denote:
\begin{itemize}
\myIt $h_X(q,k)=h_X(\Z_q/\pi^k)$
\myIt $h'_X(q,k)=h_X(\F_p[t]/t^k)$
\end{itemize}
\end{notation}
\myP
\begin{theorem}[Cluckers-Loeser $\sim$ 2005]
There exists a finite set $S$ of primes s.t. for any $q\in \cP_S$ $$h_X(q,k)=h'_X(q,k)$$
\end{theorem}
\end{frame}



%\begin{frame}
%\myFT{Subsets of $\cP$}
%\begin{theorem}
%There exist a finite set $S$ of primes s.t. for any $q\in \cP_S$ $$h_X(q,k)=h'_X(q,k)$$
%\end{theorem}
%\myP
%\begin{definition}
%We say that a subset $\mathcal L \subset  \cP$ is  large if any polynomial splits over $\F_q$ for some $q\in \mathcal L$
%\end{definition}
%\begin{examples}
%\begin{itemize}
%\myIt powers of a prime.
%\myIt all primes
%\myIt all primes but finitely many
%\myIt all primes in $1+n \Z$.
%\end{itemize}
%\end{examples}
%\end{frame}


\begin{frame}
%\thispagestyle{empty}

\myFT{Main Results}
\begin{theorem}[A.-Avni 2014]
Assume that $X_{\bar \Q}$ is irreducible local complete intersection. \myP
TFAE:
\begin{enumerate}
 \myIt $X$  has rational singularities.
 \myIt For any $k$:
% there exist large $\mathcal L \in \cP$ s.t.
% $\lim_{\mathcal L \ni q\to \infty} h_X(q,k)=1$.
$\lim_{p\to \infty} h_X(p,k)=1$.
\myIt There exists a finite set of primes $S$ s.t. $h_X(q,k) - 1 =O(\frac{1}{\sqrt{q}})$ for $q\in \cP_{S}$ (uniformly on $k$).
\end{enumerate}

$ $

$ $

$ $

$$ $$


\end{theorem}

\end{frame}

\begin{frame}[noframenumbering]
%\thispagestyle{empty}

\myFT{Main Results}
\begin{conjecture}
Assume that $X_\Q$ is irreducible local complete intersection.
TFAE:
\begin{enumerate}
\item $X$ has rational singularities.
 \item For any $k$:
% there exist large $\mathcal L \in \cP$ s.t.
% $\lim_{\mathcal L \ni q\to \infty} h_X(q,k)=1$.
$\lim_{p\to \infty} h_X(p,k)=1$.
\item  There exists a finite set of primes $S$ s.t. $h_X(q,k) - 1 =O(\frac{1}{\sqrt{q}})$ for $p\in \cP_{S}$ (uniformly on $k$).
{\color{red}
%\myIt  For a large set of $q$:  $n_X(p,k)$ bounded.
\item For almost any prime $p$: $h_X(p,k)$ is bounded.
 \myIt For any $q\in \cP$: $h_X(q,k)$ is bounded.
\myIt There exists a finite set of primes $S$ s.t. $h_X(q,k) - h_X(q,1) =O(\frac{1}{q})$ for $q\in \cP_{S}$.
}
\end{enumerate}
\end{conjecture}

\end{frame}


\begin{frame}
%\thispagestyle{empty}

\myFT{Jet schemes and rational singularities}
\myP
\begin{definition}
For a scheme $X$ defined over $k$, the jet scheme $jet_n(X) $ is the natural scheme defined over $k$ s.t. $X(k[t]/t^n) \cong jet_n(X)(k)$.
\end{definition}
\myP
\begin{theorem}[Mustata 2001]
Assume that $X$ is a local complete intersection irreducible variety. TFAE:
\begin{itemize}
 \myIt $X$  has rational singularities.
 \myIt The jet schemes of $X$ are irreducible.
\end{itemize}
\end{theorem}
%\begin{theorem}[A.-Avni 2014]
%Let $X$ be as above. \myP
%Let $n_X(p,k)=\frac{\#X(\Z/p^k\Z)}{p^{k\dim X}}$ and $m_X(p,k)=\frac{\#X(\F_{p}[t]/t^{k})}{p^{k\dim X}}$.
%TAE:
%\begin{itemize}
% \myIt $X$ has rational singularities.
% \myIt For any $k$: $\lim_{p\to \infty} n_X(p,k)=1$.
% \myIt For any $k$: $\lim_{p\to \infty} m_X(p,k)=1$.
%\myIt For any $p$: $n_X(p,k)$ bounded.
%\myIt For any $p$: $m_X(p,k)$ bounded.
%\myIt  $n_X(p,k) - 1 =O(\sqrt{p})$.  %uniformly on $k$
%\myIt  $m_X(p,k) - 1 =O(\sqrt{p})$.  %uniformly on $k$
%\end{itemize}
%\end{theorem}

\end{frame}

\begin{frame}
%\thispagestyle{empty}
\myFT{Lang Weil bounds and application of the Chebotarev theorem}
\begin{thm}[Lang-Weil]
Let $X$ be an absolutely irreducible scheme.
$$\#X(\F_p)=p^{\dim X}+O(p^{\dim X-\frac{1}{2}}).$$
\end{thm}
\myP
\begin{thm}[Chebotarev]
Let $X$ be a scheme.
Then, for almost any $p$, the set of absolute irreducibility components of $X_{\Q}$ is bijective to the set of absolute irreducibility components of $X_{\F_p}$,  and, \myP  for positive percentage of $p$, they are all defined over $\F_p$.
\end{thm}

\end{frame}


\begin{frame}
%\thispagestyle{empty}
\myFT{1 $\Longleftrightarrow$ 2}
\begin{enumerate}
\item $X$ has rational singularities.
%\item   $\lim_{\mathcal L \ni q\to \infty} h'_X(q,k)=1$.
\item $\lim_{p\to \infty} h'_X(p,k)=1$.
\end{enumerate}
\myP
For a scheme $Y$, let $c_p(Y)$ be the number of absolute irreducibility components defined over $\F_{p}$.
\myP
$$h'_X(p,k)=h'_{Jet_k(X)}(p,1)=c_p(Jet_k(X))+O(\frac{1}{\sqrt{p}}).$$
\end{frame}
\begin{frame}
\myFT{Rational singularities}% and conversion of integrals}
\myTPSE
%\thispagestyle{empty}
 \myP
\begin{definition}
%Let $X$ be an algebraic variety over $k$.  \myP  Let $\pi: \tilde X \to X$ be it resolution of singularities.
Let $\tilde X \overset{\pi}{\to} X$ be a resolution of singularities of an $F$-algebraic variety.
\myP  $X$ is said to be of rational singularities if the following equivalent conditions hold: \myP
\begin{enumerate}
\myIt $R \pi_*(O_{\tilde X})=O_{X}$  \com{original definition}
\myIt $R\pi_*(\Omega_{\tilde X})=\Omega_{X}$ \myIt $X$ is Cohen-Macaulay and  \myP  $\pi_*(\Omega_{\tilde X})=\Omega_{X}$ \myIt $X$ is normal Cohen-Macaulay and \myP $\pi_*(\Omega_{\tilde X})=i_*(\Omega_{X^{reg}})$
\end{enumerate}
\end{definition}
$$\xymatrix{
 & \tilde X \ar@{->}^\pi[d]\\
X^{reg} \ar@{->}^i[r] \ar@{->}[ru] &X
}
$$

\end{frame}


\begin{frame}
\myFT{Pushforward of smooth measures}
\myP
\begin{theorem}[A.-Avni, 2013]
%\begin{center}
%\center
%\endcenter
%\left-align
%\end{center}
%{\center $\overset{m}{X}\overset{\phi}{\to} Y$}
%????????????

Let:

{\centering $\overset{m}{X}\overset{\phi}{\to} Y$

}
s.t.
\begin{itemize}
\myIt $\phi$ is a flat morphism of smooth algebraic varieties over a local field $F$, s.t. all its fibers are of rational singularities \myP (in what follows: \FRS morphism).
\myIt $m$ is a Schwartz (i.e. compactly supported locally Haar) measure on $X(F)$.
\end{itemize}
\myP
Then $\phi_*(m)$ has continuous density.
\end{theorem}

{
\setbeamercolor{background canvas}{bg=}
\includepdf[pages=-]{AV_3.pdf}
}


\end{frame}
\begin{frame}
%\thispagestyle{empty}
\myFT{1 $\Longrightarrow$ 4}
\begin{enumerate}
\item $X$ has rational singularities.
\setcounter{enumi}{3}
\item  $\sup_k h_X(p,k)< \infty$.
\end{enumerate}
\myP
WLOG we may assume that $X= \phi^{-1}(0)$ for a flat $\phi:\A^a \to \A^b$.

\myP Let $\mu, \nu$ be the Haar measures on $\Z_q^a$ and $\Z_q^b$.
$$h_X(q,k)=\frac{\mu(\phi^{-1}(\pi^k\Z_q^b))}{\nu(\pi^k\Z_q^b)}\myP=\frac{\phi_*(\mu)(\pi^k\Z_q^b)}{\nu(\pi^k\Z_q^b)}.$$
\end{frame}

\begin{frame}
%\thispagestyle{empty}
\myFT{Motivic integration}
\begin{thm}[Hrushovski-Kazhdan, et al.]
Let $X$ be a definable set in an algebraic variety over a valued field, and $\omega$ be a definable top differential form on $X$ depending on a parameter $n$ in the valuation group. Let $$f(q,n):=\int_{X(\Q_q)} |\omega|.$$
Then
%$f(p,n)$ can be written as combination of functions the most
%complicated of which is
$$f(q,n)=\sum f_i(q,n) \#X_i(\F_q),$$
where $X_i$ are schemes and $f_i(q,n)$ are "simple-minded" non-negative functions.
\end{thm}

\end{frame}

\begin{frame}
%\thispagestyle{empty}
\myFT{2\&4 $\Longrightarrow$ 3}
\begin{enumerate}
\setcounter{enumi}{1}
\item   $\lim_{ p\to \infty} h_X(p,k)=1$.
\setcounter{enumi}{3}
\item  $\sup_k h_X(p,k)< \infty$.
\setcounter{enumi}{2}
\item $h_X(q,k) - 1 =O(\frac{1}{\sqrt{q}})$, (uniformly on $k$).
\end{enumerate}
\myP

$$h_X(q,k)=\sum f_i(q,k) \#X_i(\F_q)\myP=1 +O(\frac{1}{\sqrt{q}}).$$

\end{frame}


\begin{frame}
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        \end{tikzpicture}
\end{frame}


%\begin{frame}
%\myFT{Summary}
%{%\small
%\xymatrixcolsep{1.4pc}
%\xymatrix{X \text{ has rat. sing.}\ar@{=>}@/^1pc/[rr]^{\quad \text{Mustata Thm., L-W Bounds}}
%\ar@{=>}[dd]_{\text{Continuity Criterion}}& &\lim \limits_{q\to \infty}h'_X(p,k)  \ar@{=>}@/^1pc/[ll]^{\text{Mustata, L-W, Chebotarev}}\ar@{-^{>}}@<1ex>@/_2.5pc/[rddd] &
%\\
% & &  &
%\\ \sup\limits_k h_X(p,k)\ar@{-}@/_3pc/[rruu]\ar@{-_{>}}@<-0.7ex>@/^1pc/[rrrd]   &  &  \quad \quad \parbox{50pt}{\tiny{Motivic int.,\\ L-W Bounds,\\ Chebotarev thm.}}&
%\\ & & & \lim \limits_{p\to \infty}\sup\limits_k h_X(p,k) \ar@{=>}@/_1pc/[uuul]_{Obvious} \ar@{=>}@/^1pc/[lllu]^{Obvious}
%     }
%     }
%
%\end{frame}

%\includegraphics{Samery_slide2.png}
%\includepdf{Samery_slide3.pdf}
% \begin{frame}[noframenumbering]
 %\thispagestyle{empty}
 %\begin{figure}
 %\includegraphics[width=12.5cm,center]{Samery_slide2.png}
 %\end{figure}
% \end{frame}

\begin{frame}
\myFT{Applications}
\begin{theorem}[A.-Avni 2014]
Let $n>500$ and $G$ be a semisimple group defined over $\Z$. Then there exists a constant $c$ s.t. for any integer $k$ and $A \subset G(\Z/k\Z)$:
  $$ \mathrm{Prob}([g_1,h_1]\cdots [g_n,h_n] \in A)<c \cdot \mathrm{Prob}(g \in A),$$
for random elements $g, g_1 \dots g_n, h_1 \dots h_n \in G(\Z/k\Z)$
\end{theorem}
\myP
\begin{theorem}[A.-Avni 2014]
%Let $G=SL_d(O_F)$, where $F$ is a non-Archimedean local field of characteristic
% $0$
Let $G$ be a semisimple group defined over $\Z$ whose $\Q$-split rank is $>1$. \myP
Then $$\# \{\pi \in \irr(G(\Z))|\dim \pi <n\}<Cn^{1000}.$$
\end{theorem}
\end{frame}


\end{document}






