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% Copyright 2004 by Till Tantau <tantau@users.sourceforge.net>.
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% In principle, this file can be redistributed and/or modified under
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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}
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{Weizmann Institute of Science}
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%{August 14, 2008}

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\title % (optional, use only with long paper titles)
[Representation Count etc.]
{Representation count of arithmetic groups, moduli spaces of local systems, and pushforward of smooth 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.


%\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{Question}
How many irreducible representations of dimension $n$ does $SL_d(\Z)$ have?
\myP
\begin{definition}
$$\zeta_{G}(s)=\sum_{\pi\in \irr G} \dim \pi ^{-s}$$
\end{definition}
\myP

When does $\zeta_{G}(s)$ converge?
\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 semi-simple group defined over $\Z$ whose $\Q$-split rank is $>1$. \myP
Then, $\zeta_{G(\Z)}(1000)$ converges.
\end{theorem}



\end{frame}
\begin{frame}
\thispagestyle{empty}
%\myFT{Local to Global}
\begin{theorem}[Bass-Milnor-Serre 1967]
Let $d>2$
Any irreducible representation $\pi$ of $SL_d(\Z)$ can be written as $$\pi=\pi_{fin}\otimes \pi_{alg},$$ where $\pi_{fin}$ factors through $SL_d(\Z|_n)$ and $\pi_{alg}$ extends to an algebraic representation of $SL_d(\C)$
\end{theorem}
\myP
\begin{corollary}
$\zeta_{SL_d(\Z)}=\zeta_{SL_d(\C)}\prod_p \zeta_{SL_d(\Z_p)} $
\end{corollary}
\myP
\begin{corollary}
To show that $\zeta_{G(\Z)}(s)$ converges, \myP
enough to show that
$\zeta_{G(\Z_p)}(s)$ converges, \myP $\zeta_{G(\C)}(s)$ and  \myP $\zeta_{G(\Z_p)}(s) \underset{p\to \infty}{\longrightarrow} 1$ fast enough.
\end{corollary}
\myP
%\begin{corollary}
For this it is enough to show that
$\zeta_{G(\Z/p^n\Z)}(s) \underset{p\to \infty}{\longrightarrow} 1$ fast enough (uniformly on $n$).
\myP

Equivalently that
$\zeta_{G(\Z/k\Z)}(s)$  is bounded.
%\end{corollary}

\end{frame}


\begin{frame}
\myFT{Frobenius Formula}
%\thispagestyle{empty}
\myP
\begin{theorem}[Frobenius 1896]
Let $H$ be a finite group. \myP Then %$\forall x\in H:$
%$$\sum_{\pi \in irr G}\frac{\chi_\pi(g)}{\dim \pi^{?}}=\frac{\#\{(g_1,h_1,\dots %g_n,h_n) \in G^{2n}|[g_1,h_1]\cdots [g_1,h_1] =g\}}{\#G^n}$$
\myP
\begin{itemize}
\myIt  $\zeta_H(2)= \sum\limits_{\pi \in \irr H}{(\dim \pi)^{2}}=\# H$
\myIt $\zeta_H(0)=\sum\limits_{\pi \in \irr H}{(\dim \pi)^{0}}=\# (H//H)$ \myP $ =\frac{\#
\{(g,h) \in H^{2}|[g,h]=1\}}{\#H}$
\myIt {$\dots$}
\myIt $\zeta_H(2n-2)=\frac{\#
\{(g_1,h_1,\dots g_n,h_n) \in H^{2n}|[g_1,h_1]\cdots [g_n,h_n] =1\}}{\#H^{2n-1}}$
%\myIt {$\sum \limits_{\pi \in \irr H}\frac{\chi_\pi(1)}{\dim \pi^{2n-1}}=\frac{\#
%\{(g_1,h_1,\dots g_n,h_n) \in H^{2n}|[g_1,h_1]\cdots [g_n,h_n] =1\}}{\#H^{2n-1}}$}
%\myIt
%Let $x \in H$. Then
%$\sum_{\pi \in \irr H}\frac{\chi_\pi(x)}{\dim \pi^{2n-1}}=\frac{\#
%\{(g_1,h_1,\dots g_n,h_n) \in H^{2n}|[g_1,h_1]\cdots [g_n,h_n] =x\}}{\#H^{2n-1}}$
\end{itemize}

\end{theorem}
\end{frame}
\begin{frame}

\myFT{Product of  commutators of random elements}
\myP
The convergence of $\zeta_{G(\Z)}(1000)$ is equivalent to:
\myP
\begin{theorem}[A.-Avni 2014]
Let $n>500$. Then there exists a constant $c$ s.t. for any integer $k$ and $A \subset G$:
  $$ \mathrm{Prob}([g_1,h_1]\cdots [g_n,h_n] \in A)<c \cdot \mathrm{Prob}(g \in A),$$
for random elements $g, g_1 \dots g_n \in G(\Z|_k)$
\end{theorem}

\end{frame}
\begin{frame}

\myFT{Pushforward of smooth measures}
\myP
Let $\Phi_{G,n}:G^{2n} \to G$ be defined by:
$$\Phi_{G,n}(g_1,h_1,\dots,g_n,h_n):=[g_1,h_1]\cdots [g_n,h_n].$$
\myP
Let $\mu$ be the Haar measure on $G(\Z_p)$.
\myP
The convergence of $\zeta_{G(\Z_p)}(2n-2)$ is equivalent to the the fact that $\Phi(\mu)=f\cdot \mu$ for a continuous function $f$.
\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 filed $F$, 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}


\end{frame}

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


\begin{frame}
\myFT{Rationality of the singularities of moduli spaces}
\begin{theorem}[A.-Avni 2013]
%Let $G=SL_d$.
Let $n > 500$.
Then $\Phi_{G, n}$ is \FRS.
In particular, the singularities of the deformation variety $\Deff_{G,\Sigma_n}:=\Phi_{G, n}^{-1}$
%$$\Deff_{G}=\{(g_1,h_1,\dots g_n,h_n) \in G^{2n}|[g_1,h_1]\cdots [g_1,h_1] %=1\}=Hom(\pi_1(\Sigma_n),G).$$
are rational (and complete intersection).
\end{theorem}
\myP
\begin{corollary}[A.-Avni 2013]
The moduli spaces of $G$ local systems on a genus $n$ surface have rational singularities.
\end{corollary}

\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 connected variety. TFAE:
\begin{itemize}
 \myIt $X$ is  irreducible and 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{Uniform bounds}
\begin{theorem}[A.-Avni 2014]
Let $X$ be a local complete intersection scheme of finite type. \myP
Let $n_X(p,k)=\frac{\#X(\Z/p^k\Z)}{p^{k\dim X}}$ and $m_X(p,k)=\frac{\#X(\F_{p}[t]/t^{k})}{p^{k\dim X}}$.
TFAE:
\begin{itemize}
 \myIt $X$ is irreducible and has rational singularities.
 \myIt For any $k$:  $\lim_{p\to \infty} m_X(p,k)=1$.
 \myIt For any $k$: $\lim_{p\to \infty} n_X(p,k)=1$.
\myIt  $n_X(p,k) - 1 =O(\frac{1}{\sqrt{p}})$,  For  almost all $p$.
\myIt  $m_X(p,k) - 1 =O(\frac{1}{\sqrt{p}})$. For  almost all $p$.
\end{itemize}
%
%$ $
%
%$ $
%
%$ $
%
%$ $
\end{theorem}

\end{frame}

%\begin{frame}[noframenumbering]
%%\thispagestyle{empty}
%
%\myFT{Uniform bounds}
%\begin{conjecture}
%Let $X$ be a local complete intersection scheme of finit type.
%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}
% \item $X$ is irreducible and has rational singularities.
% \item For any $k$:  $\lim_{p\to \infty} m_X(p,k)=1$.
%\item For any $k$: $\lim_{p\to \infty} n_X(p,k)=1$.
%\item $n_X(p,k) - 1 =O(\sqrt{p})$,  For  almost all $p$.
%\item  $m_X(p,k) - 1 =O(\sqrt{p})$. For  almost all $p$.
%\item $X_{\bar\Q}$ is connected and, for any $p$:  $n_X(p,k)$ bounded.
%\item  $X_{\bar\Q}$ is connected and, for almost any $p$:  $n_X(p,k)$ bounded.
%\item  $X_{\bar\Q}$ is connected and, for almost any $p$: $m_X(p,k)$ bounded.
%\end{itemize}
%\end{conjecture}
%
%\end{frame}



\begin{frame}
\myFT{Conclusion}
We conclude from the above results:
 \myP
\begin{theorem}[A.-Avni 2014]
We have the following implications:
\begin{itemize}
\myIt  $\zeta_{G(\Z)}(2n-2)<\infty$.
\item[$\Downarrow$]
\myIt  $\zeta_{G(\Z_p)}(2n-2)<\infty$ for any $p$.
\item[$\Updownarrow$]
\myIt  $Def_{G,n}:=\Phi_{G,n}^{-1}(1)$ has rational singularities.
\item[$\Downarrow$]
\myIt  $\zeta_{G(\Z)}(2n-2+\eps)<\infty$.
\end{itemize}
\end{theorem}
\myP
All of the above happens for $n>500$.
\end{frame}
%
%\begin{frame}
%\myFT{Flowchart}
%{\small $$  \xymatrixcolsep{0.1pc}\xymatrix{\Phi_{G,n} \text{ is FRS at } 1 & &\quad \quad \quad  \quad  \quad \quad \quad  \quad  \quad  \\
%&  &  \\
% & \quad \quad \quad  \quad  \quad \quad \quad  \quad  \quad \ \   &
%  }$$}
%\end{frame}
%
%\begin{frame}[noframenumbering]
%\myFT{Flowchart}
%{\small $$  \xymatrixcolsep{0.1pc}\xymatrix{\Phi_{G,n} \text{ is FRS at } 1\ar@{=>}[rr]^{\color{red}{\text{continuity criterion}}} & & \frac{(\Phi_{G,n})_*(\mu_{K_l^{2n}})}{\mu_{K_l}} \text{ is cont.} &  \\
%&  &  \\
% & \quad \quad \quad  \quad  \quad \quad \quad  \quad  \quad \ \   &
%  }$$}
%\end{frame}
%
%
%
%
%\begin{frame}[noframenumbering]
%\myFT{Flowchart}
%{\small $$  \xymatrixcolsep{0.1pc}\xymatrix{\Phi_{G,n} \text{ is FRS at } 1\ar@{=>}[rr]^{\color{red}{\text{continuity criterion}}}_{\underset{\text{motivic int., Hironaka thm., Elkik thm., Grothendieck duality,  G-R thm. }}{\text{Uses:}}} & & \frac{(\Phi_{G,n})_*(\mu_{K_l^{2n}})}{\mu_{K_l}} \text{ is cont.} &  \\
%&  &  \\
% & \quad \quad \quad  \quad  \quad \quad \quad  \quad  \quad \ \   &
%  }$$}
%\end{frame}
%
%
%\begin{frame}[noframenumbering]
%\myFT{Flowchart}
%{\small $$  \xymatrixcolsep{0.1pc}\xymatrix{\Phi_{G,n} \text{ is FRS at } 1\ar@{=>}[rr]^{\color{red}{\text{continuity criterion}}}_{\underset{\text{motivic int., Hironaka thm., Elkik thm., Grothendieck duality,  G-R thm. }}{\text{Uses:}}} & & \frac{(\Phi_{G,n})_*(\mu_{K_l^{2n}})}{\mu_{K_l}} \text{ is cont.} \ar@{=>}[d]^{\text{Frobenius formula}}&  \\
%&  & \zeta_{K_l}(2n-2)<\infty \\
% & \quad \quad \quad  \quad  \quad \quad \quad  \quad  \quad \ \   &
%  }$$}
%\end{frame}
%
%\begin{frame}[noframenumbering]
%\myFT{Flowchart}
%{\small $$  \xymatrixcolsep{0.1pc}\xymatrix{\Phi_{G,n} \text{ is FRS at } 1\ar@{=>}[rr]^{\color{red}{\text{continuity criterion}}}_{\underset{\text{motivic int., Hironaka thm., Elkik thm., Grothendieck duality,  G-R thm. }}{\text{Uses:}}} & & \frac{(\Phi_{G,n})_*(\mu_{K_l^{2n}})}{\mu_{K_l}} \text{ is cont.} \ar@{=>}[d]^{\text{Frobenius formula}}&  \\
%& \zeta_{G(\Z_p)}(2n-2)<\infty & \zeta_{K_l}(2n-2)<\infty\ar@{=>}[l] \\
% & \quad \quad \quad  \quad  \quad \quad \quad  \quad  \quad \ \   &
%  }$$}
%\end{frame}
%
%
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%\begin{frame}[noframenumbering]
%\myFT{Flowchart}
%{\small $$  \xymatrixcolsep{0.1pc}\xymatrix{\Phi_{G,n} \text{ is FRS at } 1\ar@{=>}[rr]^{\color{red}{\text{continuity criterion}}}_{\underset{\text{motivic int., Hironaka thm., Elkik thm., Grothendieck duality,  G-R thm. }}{\text{Uses:}}} & & \frac{(\Phi_{G,n})_*(\mu_{K_l^{2n}})}{\mu_{K_l}} \text{ is cont.} \ar@{=>}[d]^{\text{Frobenius formula}}&  \\
%    \sup\limits_k\zeta_{G(\Z/p^k\Z)}(2n-2)<\infty & \zeta_{G(\Z_p)}(2n-2)<\infty\ar@{=>}[l] & \zeta_{K_l}(2n-2)<\infty\ar@{=>}[l] \\
% & \quad \quad \quad  \quad  \quad \quad \quad  \quad  \quad \ \   &
%  }$$}
%\end{frame}
%\begin{frame}[noframenumbering]
%\myFT{Flowchart}
%{\small $$  \xymatrixcolsep{0.1pc}\xymatrix{\Phi_{G,n} \text{ is FRS at } 1\ar@{=>}[rr]^{\color{red}{\text{continuity criterion}}}_{\underset{\text{motivic int., Hironaka thm., Elkik thm., Grothendieck duality,  G-R thm. }}{\text{Uses:}}} & & \frac{(\Phi_{G,n})_*(\mu_{K_l^{2n}})}{\mu_{K_l}} \text{ is cont.} \ar@{=>}[d]^{\text{Frobenius formula}}&  \\
%    \sup\limits_k\zeta_{G(\Z/p^k\Z)}(2n-2)<\infty\ar@{=>}[d] & \zeta_{G(\Z_p)}(2n-2)<\infty\ar@{=>}[l] & \zeta_{K_l}(2n-2)<\infty\ar@{=>}[l] \\
%  \sup\limits_k \frac{\#\Phi_{G(\Z/p^k\Z),n}^{-1}(1)}{\#G(\Z/p^k\Z)^{2n-2}} < \infty & \quad \quad \quad  \quad  \quad \quad \quad  \quad  \quad \ \   &
%  }$$}
%\end{frame}
%
%\begin{frame}[noframenumbering]
%\myFT{Flowchart}
%{\small $$  \xymatrixcolsep{0.1pc}\xymatrix{\Phi_{G,n} \text{ is FRS at } 1\ar@{=>}[rr]^{\color{red}{\text{continuity criterion}}}_{\underset{\text{motivic int., Hironaka thm., Elkik thm., Grothendieck duality,  G-R thm. }}{\text{Uses:}}} & & \frac{(\Phi_{G,n})_*(\mu_{K_l^{2n}})}{\mu_{K_l}} \text{ is cont.} \ar@{=>}[d]^{\text{Frobenius formula}}&  \\
%    \sup\limits_k\zeta_{G(\Z/p^k\Z)}(2n-2)<\infty\ar@{=>}[d] & \zeta_{G(\Z_p)}(2n-2)<\infty\ar@{=>}[l] & \zeta_{K_l}(2n-2)<\infty\ar@{=>}[l] \\
%  \sup\limits_k \frac{\#\Phi_{G(\Z/p^k\Z),n}^{-1}(1)}{\#G(\Z/p^k\Z)^{2n-2}} < \infty\ar@{=>}[r] & \frac{(\Phi_{G,n})_*(\mu_{G(\Z_p)^{2n}})}{\mu_{G(\Z_p)}}  \text{ is cont.}&
%  }$$}
%\end{frame}
%
%\begin{frame}[noframenumbering]
%\myFT{Flowchart}
%{\small $$  \xymatrixcolsep{0.1pc}\xymatrix{\Phi_{G,n} \text{ is FRS at } 1\ar@{=>}[rr]^{\color{red}{\text{continuity criterion}}}_{\underset{\text{motivic int., Hironaka thm., Elkik thm., Grothendieck duality,  G-R thm. }}{\text{Uses:}}} & & \frac{(\Phi_{G,n})_*(\mu_{K_l^{2n}})}{\mu_{K_l}} \text{ is cont.} \ar@{=>}[d]^{\text{Frobenius formula}}&  \\
%    \sup\limits_k\zeta_{G(\Z/p^k\Z)}(2n-2)<\infty\ar@{=>}[d] & \zeta_{G(\Z_p)}(2n-2)<\infty\ar@{=>}[l] & \zeta_{K_l}(2n-2)<\infty\ar@{=>}[l] \\
%  \sup\limits_k \frac{\#\Phi_{G(\Z/p^k\Z),n}^{-1}(1)}{\#G(\Z/p^k\Z)^{2n-2}} < \infty\ar@{=>}[r] & \frac{(\Phi_{G,n})_*(\mu_{G(\Z_p)^{2n}})}{\mu_{G(\Z_p)}}  \text{ is cont.}\ar@{=>}[r]
% & \Phi_{G,n}\text{ is FRS}
%  }$$}
%\end{frame}
%
%\begin{frame}
%\myFT{Flowchart}
%
%  \xymatrix{Def_{G,n}:= \Phi_{G,n}^{-1}(1) \text{ has rational singularities}\\
%\quad\\
% \quad \\   \quad}
%\end{frame}
%
%\begin{frame}[noframenumbering]
%
%\myFT{Flowchart}
%  \xymatrix{Def_{G,n}:= \Phi_{G,n}^{-1}(1) \text{ has rational singularities}\ar@{=>}[d]^{\text{Mustata Theorem}}\\
%  jet_k(Def_{G,n}) \text{ is irreducible}\\
% \quad \\   \quad}
%\end{frame}
%
%\begin{frame}[noframenumbering]
%\myFT{Flowchart}
%
%  \xymatrix{Def_{G,n}:= \Phi_{G,n}^{-1}(1) \text{ has rational singularities}\ar@{=>}[d]^{\text{Mustata Theorem}}\\
%  jet_k(Def_{G,n}) \text{ is irreducible}\ar@{=>}[d]^{\text{Lang-Weil bounds}}\\
% \frac{\#Def_{G,n}(\F_p[t]/t^k)}{\#G(\F_p[t]/t^k)^{2n-2}}\underset{\underset{k \text{ fixed }}{p\to \infty}}{\longrightarrow} 1 \\   \quad}
%\end{frame}
%
%\begin{frame}[noframenumbering]
%
%\myFT{Flowchart}
%  \xymatrix{Def_{G,n}:= \Phi_{G,n}^{-1}(1) \text{ has rational singularities}\ar@{=>}[d]^{\text{Mustata Theorem}}\\
%  jet_k(Def_{G,n}) \text{ is irreducible}\ar@{=>}[d]^{\text{Lang-Weil bounds}}\\
% \frac{\#Def_{G,n}(\F_p[t]/t^k)}{\#G(\F_p[t]/t^k)^{2n-2}}\underset{\underset{k \text{ fixed }}{p\to \infty}}{\longrightarrow} 1\ar@{=>}[d]^{\text{model theory}}  \\   n_{2n-2}(p,k):=\frac{\#Def_{G,n}(\Z/p^k\Z)}{\#G(\Z/p^k\Z)^{2n-2}}\underset{\underset{k \text{ fixed }}{p\to \infty}}{\longrightarrow} 1}
%\end{frame}
%
%
%\begin{frame}
%\myFT{Flowchart}
%  {\small \xymatrixcolsep{0.1pc}\xymatrix{n_{2m}(p,k) \underset{\underset{k \text{ fixed }}{p\to \infty}}{\longrightarrow} 1 & \quad \quad \quad \quad \quad  \quad  \quad  \quad \  &  \quad \\
%    \sup\limits_k n_{2m}(p,k) < \infty & \quad \quad \quad \quad \quad \quad \quad  \quad  \quad \quad \  & \quad
%    \\ &  &\quad
%    \\ &  \quad\\
%    & & \quad }}
%\end{frame}
%
%
%\begin{frame}[noframenumbering]
%\myFT{Flowchart}
%  {\small \xymatrixcolsep{0.1pc}\xymatrix{n_{2m}(p,k) \underset{\underset{k \text{ fixed }}{p\to \infty}}{\longrightarrow} 1 \ar@{=>}[rr]^{\text{motivic int. } \quad}_{\text{L-W bounds } \quad}& \quad \quad \quad \quad \quad  \quad  \quad  \quad \  &  \quad \\
%    \sup\limits_k n_{2m}(p,k) < \infty \ar@{=>}[rru]^{ \text{Chebotaryov thm. }\quad \quad }& \quad \quad \quad \quad \quad \quad \quad  \quad  \quad \quad \  & \quad
%    \\ &  &\quad
%    \\ &  \quad\\
%    & & \quad }}
%\end{frame}
%
%
%\begin{frame}[noframenumbering]
%\myFT{Flowchart}
%  {\small \xymatrixcolsep{0.1pc}\xymatrix{n_{2m}(p,k) \underset{\underset{k \text{ fixed }}{p\to \infty}}{\longrightarrow} 1 \ar@{=>}[rr]^{\text{motivic int. } \quad}_{\text{L-W bounds } \quad}& \quad \quad \quad \quad \quad \quad \quad  \quad  \quad  \quad \  &  n_{2m}(p,k)=1+O(\frac{1}{\sqrt{p}}) \\
%    \sup\limits_k n_{2m}(p,k) < \infty \ar@{=>}[rru]^{ \text{Chebotaryov thm. }\quad \quad }& \quad \quad \quad \quad \quad  \quad  \quad \quad \  & \quad
%    \\ &  &\quad
%    \\ &  \quad\\
%    & & \quad }}
%\end{frame}
%
%\begin{frame}[noframenumbering]
%\myFT{Flowchart}
%  {\small \xymatrixcolsep{0.1pc}\xymatrix{n_{2m}(p,k) \underset{\underset{k \text{ fixed }}{p\to \infty}}{\longrightarrow} 1 \ar@{=>}[rr]^{\text{motivic int. } \quad}_{\text{L-W bounds } \quad}& \quad \quad \quad \quad \quad  \quad  \quad  \quad \  &  n_{2m}(p,k)=1+O(\frac{1}{\sqrt{p}})\ar@{=>}[d]^{\text{Frobenius formula}} \\
%    \sup\limits_k n_{2m}(p,k) < \infty \ar@{=>}[rru]^{ \text{Chebotaryov thm. }\quad \quad }& \quad \quad \quad \quad \quad  \quad  \quad \quad \  & \zeta_{G(\Z_p)}(2m)= 1+O(\frac{1}{\sqrt{p}})
%    \\ &  &\quad
%    \\ &  \quad\\
%    & & \quad }}
%\end{frame}
%
%
%\begin{frame}[noframenumbering]
%\myFT{Flowchart}
%  {\small \xymatrixcolsep{0.1pc}\xymatrix{n_{2m}(p,k) \underset{\underset{k \text{ fixed }}{p\to \infty}}{\longrightarrow} 1 \ar@{=>}[rr]^{\text{motivic int. } \quad}_{\text{L-W bounds } \quad}& \quad \quad \quad \quad \quad  \quad  \quad  \quad \  &  n_{2m}(p,k)=1+O(\frac{1}{\sqrt{p}})\ar@{=>}[d]^{\text{Frobenius formula}} \\
%    \sup\limits_k n_{2m}(p,k) < \infty \ar@{=>}[rru]^{ \text{Chebotaryov thm. }\quad \quad }& \quad \quad \quad \quad \quad  \quad  \quad \quad \  & \zeta_{G(\Z_p)}(2m)= 1+O(\frac{1}{\sqrt{p}})\ar@{=>}[d]^{\text{motivic integration, ...}}_{\text{Avni-Klopsch-Onn-Voll}}
%    \\ &  &\zeta_{G(\Z_p)}(2m)= 1+O(\frac{1}{p})
%    \\ &  \quad\\
%    & & \quad }}
%\end{frame}
%
%
%
%\begin{frame}[noframenumbering]
%\myFT{Flowchart}
%  {\small \xymatrixcolsep{0.1pc}\xymatrix{n_{2m}(p,k) \underset{\underset{k \text{ fixed }}{p\to \infty}}{\longrightarrow} 1 \ar@{=>}[rr]^{\text{motivic int. } \quad}_{\text{L-W bounds } \quad}& \quad \quad \quad \quad \quad  \quad  \quad  \quad \  &  n_{2m}(p,k)=1+O(\frac{1}{\sqrt{p}})\ar@{=>}[d]^{\text{Frobenius formula}} \\
%    \sup\limits_k n_{2m}(p,k) < \infty \ar@{=>}[rru]^{ \text{Chebotaryov thm. }\quad \quad }& \quad \quad \quad \quad \quad  \quad  \quad \quad \  & \zeta_{G(\Z_p)}(2m)= 1+O(\frac{1}{\sqrt{p}})\ar@{=>}[d]^{\text{motivic integration, ...}}_{\text{Avni-Klopsch-Onn-Voll}}
%    \\ &  &\zeta_{G(\Z_p)}(2m)= 1+O(\frac{1}{p})\ar@{=>}[d]
%    \\ &  &\prod\limits_p \zeta_{G(\Z_p)}(2m+\eps)< \infty
%    \\ \quad  & &  \quad \quad \quad
%     }}
%\end{frame}
%
%\begin{frame}[noframenumbering]
%\myFT{Flowchart}
%  {\small \xymatrixcolsep{0.1pc}\xymatrix{n_{2m}(p,k) \underset{\underset{k \text{ fixed }}{p\to \infty}}{\longrightarrow} 1 \ar@{=>}[rr]^{\text{motivic int. } \quad}_{\text{L-W bounds } \quad}& \quad \quad \quad \quad \quad  \quad  \quad  \quad \  &  n_{2m}(p,k)=1+O(\frac{1}{\sqrt{p}})\ar@{=>}[d]^{\text{Frobenius formula}} \\
%    \sup\limits_k n_{2m}(p,k) < \infty \ar@{=>}[rru]^{ \text{Chebotaryov thm. }\quad \quad }& \quad \quad \quad \quad \quad  \quad  \quad \quad \  & \zeta_{G(\Z_p)}(2m)= 1+O(\frac{1}{\sqrt{p}})\ar@{=>}[d]^{\text{motivic integration, ...}}_{\text{Avni-Klopsch-Onn-Voll}}
%    \\ &  &\zeta_{G(\Z_p)}(2m)= 1+O(\frac{1}{p})\ar@{=>}[d]
%    \\ &  &\prod\limits_p \zeta_{G(\Z_p)}(2m+\eps)< \infty \ar@{=>}[d]^{\text{Weyl character formula}}
%    \\ \quad  & &  \left(\zeta_{G(\C)}\prod\limits_p \zeta_{G(\Z_p)}\right)(2m+\eps)< \infty
%     }}
%\end{frame}
%
%\begin{frame}[noframenumbering]
%\myFT{Flowchart}
%  {\small \xymatrixcolsep{0.1pc}\xymatrix{n_{2m}(p,k) \underset{\underset{k \text{ fixed }}{p\to \infty}}{\longrightarrow} 1 \ar@{=>}[rr]^{\text{motivic int. } \quad}_{\text{L-W bounds } \quad}& \quad \quad \quad \quad \quad  \quad  \quad  \quad \  &  n_{2m}(p,k)=1+O(\frac{1}{\sqrt{p}})\ar@{=>}[d]^{\text{Frobenius formula}} \\
%    \sup\limits_k n_{2m}(p,k) < \infty \ar@{=>}[rru]^{ \text{Chebotaryov thm. }\quad \quad }& \quad \quad \quad \quad \quad  \quad  \quad \quad \  & \zeta_{G(\Z_p)}(2m)= 1+O(\frac{1}{\sqrt{p}})\ar@{=>}[d]^{\text{motivic integration, ...}}_{\text{Avni-Klopsch-Onn-Voll}}
%    \\ &  &\zeta_{G(\Z_p)}(2m)= 1+O(\frac{1}{p})\ar@{=>}[d]
%    \\ &  &\prod\limits_p \zeta_{G(\Z_p)}(2m+\eps)< \infty \ar@{=>}[d]^{\text{Weyl character formula}}
%    \\ \zeta_{G(\Z)}(2m+\eps)< \infty  & &  \left(\zeta_{G(\C)}\prod\limits_p \zeta_{G(\Z_p)}\right)(2m+\eps)< \infty\ar@{=>}[ll]^{\text{Margulis, Raghunathan} \quad  \quad  \quad}
%     }}
%\end{frame}
%

\begin{frame}
\myFT{Summary}
{\small \xymatrixcolsep{0.1pc}\xymatrix{& Def_{G,n+1} \text{ has rat. sing.}&\\ \quad\quad\quad\quad\quad\quad \quad \quad \quad&  & \quad  \quad
   \\
 &
 %\underset{\underset{}{p\to \infty}}
 % {\longrightarrow} 1
&
    \\ &   &\\
     &  &}}
\end{frame}

\begin{frame}[noframenumbering]
\myFT{Summary}
{\small \xymatrixcolsep{0.1pc}\xymatrix{& Def_{G,n+1} \text{ has rat. sing.}  \ar@{=>}[ld]_{\color{red}{\text{continuity criterion}} \quad \quad}^{\quad \quad \quad \quad \quad \quad\text{Frobenius formula}}&\\
\sup\limits_k \zeta_{G(\Z/p^k\Z)}(2n) < \infty &  & \quad  \quad
   \\
 &
 %\underset{\underset{}{p\to \infty}}
 % {\longrightarrow} 1
&
    \\ &   &\\
     &  &}}
\end{frame}

\begin{frame}[noframenumbering]
\myFT{Summary}
{\small \xymatrixcolsep{0.1pc}\xymatrix{& Def_{G,n+1} \text{ has rat. sing.} \ar@{=>}[rd]^{\quad \quad  \quad \text{Mustata Theorem}} \ar@{=>}[ld]_{\color{red}{\text{continuity criterion}} \quad \quad}^{\quad \quad \quad \quad \quad \quad\text{Frobenius formula}}&\\
\sup\limits_k \zeta_{G(\Z/p^k\Z)}(2n) < \infty &  & \quad  \quad \zeta_{G(\Z/p^k\Z)}(2n) \underset{\underset{k \text{ fixed }}{p\to \infty}}{\longrightarrow} 1
   \\
 &
 %\underset{\underset{}{p\to \infty}}
 % {\longrightarrow} 1
&
    \\ &   &\\
     &  &}}
\end{frame}


\begin{frame}[noframenumbering]
\myFT{Summary}
{\small \xymatrixcolsep{0.1pc}\xymatrix{& Def_{G,n+1} \text{ has rat. sing.} \ar@{=>}[rd]^{\quad \quad  \quad \text{Mustata Theorem}} \ar@{=>}[ld]_{\color{red}{\text{continuity criterion}} \quad \quad}^{\quad \quad \quad \quad \quad \quad\text{Frobenius formula}}&\\
\sup\limits_k \zeta_{G(\Z/p^k\Z)}(2n) < \infty  \ar@{=>}@/_1pc/[rd]^{\quad \quad \quad \quad \quad \text{motivic int., L-W bounds, Chebotaryov thm.}}&  & \quad  \quad \zeta_{G(\Z/p^k\Z)}(2n) \underset{\underset{k \text{ fixed }}{p\to \infty}}{\longrightarrow} 1 \ar@{=>}@/^2pc/[ld]
   \\
 & \quad  \quad \quad  \quad \quad  \quad \quad  \quad \quad  \quad
 %\underset{\underset{}{p\to \infty}}
 % {\longrightarrow} 1
&
    \\ &   &\\
     &  &}}
\end{frame}

\begin{frame}[noframenumbering]
\myFT{Summary}
{\small \xymatrixcolsep{0.1pc}\xymatrix{& Def_{G,n+1} \text{ has rat. sing.} \ar@{=>}[rd]^{\quad \quad  \quad \text{Mustata Theorem}} \ar@{=>}[ld]_{\color{red}{\text{continuity criterion}} \quad \quad}^{\quad \quad \quad \quad \quad \quad\text{Frobenius formula}}&\\
\sup\limits_k \zeta_{G(\Z/p^k\Z)}(2n) < \infty  \ar@{=>}@/_1pc/[rd]^{\quad \quad \quad \quad \quad \text{motivic int., L-W bounds, Chebotaryov thm.}}&  & \quad  \quad \zeta_{G(\Z/p^k\Z)}(2n) \underset{\underset{k \text{ fixed }}{p\to \infty}}{\longrightarrow} 1 \ar@{=>}@/^2pc/[ld]
   \\
 & \zeta_{G(\Z_{p})}(2n) = 1+O(\frac{1}{\sqrt{p}})
 %\underset{\underset{}{p\to \infty}}
 % {\longrightarrow} 1
&
    \\ &   &\\
     &  &}}
\end{frame}
\begin{frame}[noframenumbering]
\myFT{Summary}
{\small \xymatrixcolsep{0.1pc}\xymatrix{& Def_{G,n+1} \text{ has rat. sing.} \ar@{=>}[rd]^{\quad \quad  \quad \text{Mustata Theorem}} \ar@{=>}[ld]_{\color{red}{\text{continuity criterion}} \quad \quad}^{\quad \quad \quad \quad \quad \quad\text{Frobenius formula}}&\\
\sup\limits_k \zeta_{G(\Z/p^k\Z)}(2n) < \infty  \ar@{=>}@/_1pc/[rd]^{\quad \quad \quad \quad \quad \text{motivic int., L-W bounds, Chebotaryov thm.}}&  & \quad  \quad \zeta_{G(\Z/p^k\Z)}(2n) \underset{\underset{k \text{ fixed }}{p\to \infty}}{\longrightarrow} 1 \ar@{=>}@/^2pc/[ld]
   \\
 & \zeta_{G(\Z_{p})}(2n) = 1+O(\frac{1}{\sqrt{p}})
 %\underset{\underset{}{p\to \infty}}
 % {\longrightarrow} 1
 \ar@{=>}[d]_{\text{motivic integration, etc}}^{\text{Avni-Klopsch-Onn-Voll}}&
    \\ &  \zeta_{G(\Z_p)}(2n)= 1+O(\frac{1}{p}) &\\
     &  &}}
\end{frame}

\begin{frame}[noframenumbering]
\myFT{Summary}
{\small \xymatrixcolsep{0.1pc}\xymatrix{& Def_{G,n+1} \text{ has rat. sing.} \ar@{=>}[rd]^{\quad \quad  \quad \text{Mustata Theorem}} \ar@{=>}[ld]_{\color{red}{\text{continuity criterion}} \quad \quad}^{\quad \quad \quad \quad \quad \quad\text{Frobenius formula}}&\\
\sup\limits_k \zeta_{G(\Z/p^k\Z)}(2n) < \infty  \ar@{=>}@/_1pc/[rd]^{\quad \quad \quad \quad \quad \text{motivic int., L-W bounds, Chebotaryov thm.}}&  & \quad  \quad \zeta_{G(\Z/p^k\Z)}(2n) \underset{\underset{k \text{ fixed }}{p\to \infty}}{\longrightarrow} 1 \ar@{=>}@/^2pc/[ld]
   \\
 & \zeta_{G(\Z_{p})}(2n) = 1+O(\frac{1}{\sqrt{p}})
 %\underset{\underset{}{p\to \infty}}
 % {\longrightarrow} 1
 \ar@{=>}[d]_{\text{motivic integration, etc}}^{\text{Avni-Klopsch-Onn-Voll}}&
    \\ &  \zeta_{G(\Z_p)}(2n)= 1+O(\frac{1}{p})\ar@{=>}[d]^{\text{Margulis, Raghunathan, Weyl character formula}} &\\
     & \zeta_{G(\Z)}(2n+\eps)< \infty &}}
\end{frame}

\end{document}


\begin{frame}
   \frametitle{flowchart}
Elkik Theorem, $\Phi_{G,n}$ is FRS at 1,    $(\Phi_{G,n})*(\mu_{K_l})$ has continuous density, $\zeta_{K_l}(2m)<\infty$, $\zeta_{G(\Z_p)}(2n-2)<\infty$, $(\Phi_{G,n})*(\mu_{G(\Z_p)})$,
 $\sup_k \frac{\#\Phi_{G(\Z/p^k\Z),n}^{-1}(1)}{\#G(\Z/p^k\Z)^{2n-2}} \infty$ ,
 $\Phi_{G,n}$ is FRS,
$jet_k(\Phi_{G,n}^{-1})$ is irreducible,  for fixed $k$: $\frac{\#\Phi_{G(\F_p(t)/t^k),n}^{-1}(1)}{\#G(\F_p(t)/t^k)^{2n-2}}\longrightarrow_{p\to \infty} 1$, for fixed $k$: $\frac{\#\Phi_{G(\Z/p^k\Z),n}^{-1}(1)}{\#G(\Z/p^k\Z)^{2n-2}}\longrightarrow_{p\to \infty} 1$,  $\frac{\#\Phi_{G(\Z/p^k\Z),n}^{-1}(1)}{\#G(\Z/p^k\Z)^{2n-2}}= 1+O(\frac{1}{\sqrt{p}})$,
$\zeta_{G(\Z/p^k\Z)}(2n-2)= 1+O(\frac{1}{\sqrt{p}})$, $\zeta_{G(\Z_p)}(2n-2)= 1+O(\frac{1}{\sqrt{p}})$,
 $\zeta_{G(\Z_p)}(2n-2)= 1+O(\frac{1}{p})< \infty$ , $\zeta_{G(\C)}(2n-2+\eps)\prod_p \zeta_{G(\Z_p)}(2n-2+\eps)< \infty$, $\zeta_{G(\Z)}(2n-2+\eps)< \infty$
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 needed)



