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\section*{Weil conjecture on Tamagawa numbers}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{arithmetic_geometry}{}\paragraph*{{Arithmetic geometry}}\label{arithmetic_geometry}

[[!include arithmetic geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{the_statement}{The statement}\dotfill \pageref*{the_statement} \linebreak
\noindent\hyperlink{NumberFieldCase}{Number field case}\dotfill \pageref*{NumberFieldCase} \linebreak
\noindent\hyperlink{FunctionFieldCase}{Function field case}\dotfill \pageref*{FunctionFieldCase} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{work_of_gaitsgorylurie}{Work of Gaitsgory-Lurie}\dotfill \pageref*{work_of_gaitsgorylurie} \linebreak
\noindent\hyperlink{previous_work}{Previous work}\dotfill \pageref*{previous_work} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Let $q$ be a positive-definite [[quadratic form]] over the ring of [[integers]] $\mathbf{Z}$. The \textbf{mass} of $q$ is a weighted count of the number of quadratic forms in the [[genus]] of $q$, up to isomorphism (weighted by multiplicity). The [[Smith-Minkowski-Siegel mass formula]] gives a (complicated but computable) formula for the mass of $q$.

Over [[number fields]], ideas of Tamagawa and Weil allow a reformulation of this formula as the statement that the [[Tamagawa number]] of a certain [[algebraic group]] associated to $q$ is equal to 1. \emph{Weil's conjecture} is then the statement, now a theorem of [[Robert Langlands]], K. F. Lai and Robert Kottwitz, that the [[Tamagawa number]] of any semisimple simply-connected [[algebraic group]] is equal to 1.

There is [[function field analogy|analogue]] of the conjecture for [[function fields]], and it has been proved by [[Dennis Gaitsgory]] and [[Jacob Lurie]].

\hypertarget{the_statement}{}\subsection*{{The statement}}\label{the_statement}

\hypertarget{NumberFieldCase}{}\subsubsection*{{Number field case}}\label{NumberFieldCase}

Let $q$ be a positive-definite [[quadratic form]] over the ring of [[integers]] $\mathbf{Z}$.

\begin{defn}
\label{}\hypertarget{}{}
The \textbf{mass} of $q$ is the sum

\begin{displaymath}
\sum_{q'} \frac{1}{|O_{q'}(\mathbf{Z})|}
\end{displaymath}
taken over the positive-definite [[quadratic forms]] $q'$ in the genus of $g$.

\end{defn}
Let $\mathbf{A}$ be the ring of [[adeles]], a [[locally compact]] [[commutative ring]] containing $\mathbf{Q}$ as a [[discrete]] subring.

\begin{defn}
\label{}\hypertarget{}{}
Let $O_q(\mathbf{A})$ denote the [[automorphism group]] of $q_\mathbf{A}$, the [[base change]] of $q$ to $\mathbf{A}$. Let $SO_q(\mathbf{A}) \subset O_q(\mathbf{A})$ denote the subgroup of [[automorphisms]] with [[determinant]] 1.

\end{defn}
$SO_q(\mathbf{A})$ is a [[locally compact]] [[topological group]] containing $SO_q(\mathbf{Q})$ as a [[discrete]] subgroup and $SO_q(\hat{\mathbf{Z}} \times \mathbf{R})$ as a [[compact]] [[open]] subgroup.

\begin{theorem}
\label{}\hypertarget{}{}
\textbf{(Tamagawa-Weil reformulation of Siegel mass formula)}. Let $\mu_{\mathrm{Tam}}$ denote the [[Tamagawa measure]]. Then

\begin{displaymath}
\mu_{\mathrm{Tam}}(SO_q(\mathbf{Q}) \backslash SO_q(\mathbf{A})) = 2.
\end{displaymath}
Equivalently,

\begin{displaymath}
\mu_{\mathrm{Tam}}(Spin_q(\mathbf{Q})\backslash Spin_q(\mathbf{A})) = 1,
\end{displaymath}
where $\Spin_q$ is the 2-fold universal cover of $SO_q$.

\end{theorem}
\begin{theorem}
\label{}\hypertarget{}{}
\textbf{(Langlands-Lai-Kottwitz, ``[[Weil conjecture]]'')}. Let $G$ be a semisimple simply-connected [[algebraic group]] over $\mathbf{Q}$. Then

\begin{displaymath}
\mu_{\mathrm{Tam}}(G(\mathbf{Q})\backslash G(\mathbf{A})) = 1.
\end{displaymath}
\end{theorem}
\hypertarget{FunctionFieldCase}{}\subsubsection*{{Function field case}}\label{FunctionFieldCase}

Let $X$ be a [[smooth]] [[projective variety|projective]] [[curve]] over the [[finite field]] $F_q$, for some prime $q$. Let $K_X$ denote the [[function field]] of $X$. For $x \in X$, write $O_x$ for the [[completion]] of the [[local ring]] at $x$ and $K_x$ for its [[fraction field]].

\begin{defn}
\label{}\hypertarget{}{}
The \textbf{[[ring of adeles]] of $K_X$} is defined as

\begin{displaymath}
\mathbf{A}_X = \product^{res}_x K_x \subset \product_x K_x,
\end{displaymath}
i.e. the subgroup consisting of elements $\{g_x\}_{x \in X}$ such that $g \in G(O_x)$ for all but finitely many $x$.

\end{defn}
$\mathbf{A}_X$ is a [[locally compact]] [[commutative ring]] with [[discrete]] subring $K_X \subset \mathbf{A}_X$.

Let $G_0$ be a semisimple simply-connected linear [[algebraic group]] over $K_X$. Then $G_0(K_X) \subset G_0(\mathbf{A})$ is a [[discrete]] subgroup of the [[locally compact]] group $G_0(\mathbf{A})$. One defines a [[Tamagawa measure]] on $G(\mathbf{A})$ in a similar way as usual, i.e. by choosing a [[differential form]] and multiplying the forms on $G_0(K_x)$ ($x \in X$). Then the [[function field]] version of [[Weil's conjecture]] is

\begin{theorem}
\label{}\hypertarget{}{}
\textbf{(Gaitsgory-Lurie, ``Weil conjecture for function fields'')}. Let $X$ be a [[smooth]] [[projective variety|projective]] [[curve]] over the [[finite field]] $F_q$, for some prime $q$. Then

\begin{displaymath}
\mu_{\mathrm{Tam}}(G_0(K_X)\backslash G_0(\mathbf{A}_X)) = 1.
\end{displaymath}
\end{theorem}
This was proved by [[Dennis Gaitsgory]] and [[Jacob Lurie]]. They reformulated the conjecture as a statement about the [[cohomology]] of the [[moduli stack of G-principal bundles]] $Bun_G(X)$ on $X$, in view of the [[function field analogy]]:

\begin{remark}
\label{RelationToModuliStack}\hypertarget{RelationToModuliStack}{}
Under the [[function field analogy]], a [[global field]] $K_X$ such as a [[function field]] or a [[number field]] is interpreted as the field of [[global sections]] of the [[rational functions]] on an [[arithmetic curve]] $X$ over a [[finite field]] $\mathbb{F}_q$ or ``over $\mathbb{F}_1$'' (the would-be [[field with one element]]), respectively. Moreover, under this [[analogy]]

\begin{itemize}%
\item the [[ring of adeles]] $\mathbb{A}_X$ is the ring of functions on all punctured [[formal disks]] in $X$ subject to the condition that all but at most finitely many of them extend to the un-punctured disk;


\item accordingly $G(\mathbb{A}_X)$ is the group of $G$-valued such functions;


\item the [[quotient]] $K_{X}\backslash \mathbb{A}_{X}$ is hence the quotient of such functions on punctured formal disks around finitely many points by the functions on $\Sigma$ with these finitely many points removed; and similarly $G(K_X)\backslash G(\mathbb{A}_X)$ is the quotient of group-valued such function;


\item the ring $\mathcal{O}$ is the ring of functions on all formal disks in $\Sigma$;


\item hence the further double [[quotient stack]]

\begin{displaymath}
Bun_G(X) = G(K_X)\backslash G(\mathbb{A}_X)//G(\mathcal{O})
\end{displaymath}
is the [[groupoid]] of [[Cech cohomology|Cech cocycles]] with Cech coboundaries between them for $G$-[[principal bundles]] relative to [[covers]] of $\Sigma$ with patches being the complement of finitely many points and the formal disks around these points.



\end{itemize}
For more on this see at \emph{\href{http://ncatlab.org/nlab/show/moduli+space+of+bundles#OverCurvesAndTheLanglandsCorrespondence}{moduli space of bundles -- over curves}}.

\end{remark}
First they proved a [[Grothendieck-Lefschetz trace formula]] for $Bun_G(X)$, generalizing work of [[Kai Behrend]]:

\begin{theorem}
\label{}\hypertarget{}{}
\textbf{(Gaitsgory-Lurie, ``Grothendieck-Lefschetz trace formula for $Bun_G(X)$'')}. Let $X$ be a [[smooth]] [[projective variety|projective]] [[curve]] over the [[finite field]] $F_q$, for some prime $q$. Then

\begin{displaymath}
\frac{|Bun_G(X)(\mathbf{F}_q)|}{q^{\dim(Bun_G(X))}} = \sum_{i \ge 0} (-1)^i \mathrm{Tr}(Frob^{-1} \mid H^i(\overline{Bun}_G(X) ; \mathbf{Q}_\ell)
\end{displaymath}
where $\overline{Bun}_G(X)$ denotes the [[base change]] of $Bun_G(X)$ to the [[algebraic closure]] of $\mathbf{F}_q$, where ${\vert -\vert}$ denotes [[groupoid cardinality]], and where $Frob : \overline{Bun}_G(X) \to \overline{Bun}_G(X)$ denotes the [[Frobenius map]].

\end{theorem}
Then they proved the following result, via [[nonabelian Poincaré duality]] which provides a [[local-global principle]].

\begin{theorem}
\label{}\hypertarget{}{}
\textbf{(Gaitsgory-Lurie)}. Let $X$ be a [[smooth]] [[projective variety|projective]] [[curve]] over the [[finite field]] $F_q$, for some prime $q$. Then

\begin{displaymath}
\sum_{i \ge 0} (-1)^i \mathrm{Tr}(Frob^{-1} \mid H^i(\overline{Bun}_G(X) ; \mathbf{Q}_\ell) = \prod_{x \in X} \frac{q^{d \cdot \deg(x)}}{|G(\kappa(x))|}
\end{displaymath}
\end{theorem}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Weil_conjecture_on_Tamagawa_numbers}{Weil conjecture on Tamagawa numbers}}

\end{itemize}
\hypertarget{work_of_gaitsgorylurie}{}\subsubsection*{{Work of Gaitsgory-Lurie}}\label{work_of_gaitsgorylurie}

A proof of the [[function field]] case is discussed in

\begin{itemize}%
\item [[Dennis Gaitsgory]], [[Jacob Lurie]], \emph{Weil's conjecture for function fields}, \href{http://www.math.harvard.edu/~lurie/papers/tamagawa.pdf}{pdf}.

\end{itemize}
The proof was announced in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{Tamagawa Numbers via Nonabelian Poincar\'e{} Duality}, talk at \href{http://people.mpim-bonn.mpg.de/teichner/Older/FRG-3.html}{FRG Chern-Simons workshop}, Jan. 15-17, 2011

\end{itemize}
and is outlined in the lecture notes

\begin{itemize}%
\item [[Jacob Lurie]], \emph{Tamagawa Numbers via Nonabelian Poincare Duality (282y)}, lecture notes, 2014 (\href{http://www.math.harvard.edu/~lurie/282y.html}{web})

\end{itemize}
See also the shorter lecture notes

\begin{itemize}%
\item [[Jacob Lurie]], \emph{Tamagawa numbers via nonabelian Poincare duality}, 5 lectures at \href{http://www.math.ku.dk/english/research/conferences/2014/ytm2014}{Young Topologists Meeting 2014}, notes taken by [[Aaron Mazel-Gee]], \href{http://math.berkeley.edu/~aaron/livetex/lurie-tamagawa-poincare.pdf}{pdf}.

\end{itemize}
\hypertarget{previous_work}{}\subsubsection*{{Previous work}}\label{previous_work}

The idea of the relationship between [[Tamagawa numbers]] and [[moduli spaces]] of [[vector bundles]] goes back to [[Günter Harder]], who primarily considered the case $G = SL_n$.

\begin{itemize}%
\item [[Günter Harder]], \emph{Eine Bemerkung zu einer Arbeit von P. E. Newstead.}, Journal f\"u{}r die reine und angewandte Mathematik, 242 (1970): 16-25, \href{http://eudml.org/doc/151010}{eudml}.


\item [[Günter Harder]], M. S. Narasimhan, \emph{On the cohomology groups of moduli spaces of vector bundles on curves}, Mathematische Annalen, 212 (1975), Issue 3, pp 215-248.



\end{itemize}
The [[moduli stack]] of [[principal bundles]] was studied in more generality in

\begin{itemize}%
\item [[Kai Behrend]], Ajneet Dhillon, \emph{On the Motive of the Stack of Bundles}, 2005, \href{http://arxiv.org/abs/math/0512640}{arXiv}.


\item [[Kai Behrend]], Ajneet Dhillon, \emph{Connected components of moduli stacks of torsors via Tamagawa numbers.}, Canad. J. Math, 2009, \href{http://arxiv.org/abs/math/0503383}{arXiv}.



\end{itemize}
See also

\begin{itemize}%
\item Aravind Asok, Brent Doran, Frances Kirwan. \emph{Yang--Mills theory and Tamagawa numbers: the fascination of unexpected links in mathematics.} Bulletin of the London Mathematical Society (2008), \href{http://arxiv.org/abs/0801.4733}{arXiv}.

\end{itemize}
[[!redirects Weil conjecture on Tamagawa number]]



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