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\begin{document}

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\section*{function field analogy}

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

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

[[!include arithmetic geometry - contents]]

\hypertarget{analytic_geometry}{}\paragraph*{{Analytic geometry}}\label{analytic_geometry}

[[!include analytic geometry -- contents]]

\hypertarget{complex_geometry}{}\paragraph*{{Complex geometry}}\label{complex_geometry}

[[!include complex geometry - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Formalizations}{Formalizations}\dotfill \pageref*{Formalizations} \linebreak
\noindent\hyperlink{Overview}{Overview}\dotfill \pageref*{Overview} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

There is a noticeable [[analogy]] between phenomena ([[theorems]]) in the theory of [[number fields]] and those in the theory of [[function fields]] over [[finite fields]] (\hyperlink{Weil39}{Weil 39}, \hyperlink{Iwasaw69}{Iwasawa 69}, \hyperlink{MazurWiles83}{Mazur-Wiles 83}), hence between the theories of the two kinds of \emph{[[global fields]]}. When regarding [[number theory]] dually as [[arithmetic geometry]], then one may see that this analogy extends further to include [[complex analytic geometry]], the theory of [[complex curves]] (e.g. \hyperlink{Frenkel05}{Frenkel 05}).

At a very basic level the analogy may be plausible from the fact that both the [[integers]] $\mathbb{Z}$ as well as the [[polynomial rings]] $\mathbb{F}_q[x]$ (over [[finite fields]] $\mathbb{F}_q$) are [[principal ideal domains]] with [[finite group|finite]] [[group of units]], all [[quotients]] being finite rings and with infinitely many [[prime ideals]], which already implies that a lot of [[arithmetic]] over these rings is similar. Since [[number fields]] are the [[finite number|finite]] [[dimension|dimensional]] [[field extensions]] of the [[field of fractions]] of $\mathbb{Z}$, namely the [[rational numbers]] $\mathbb{Q}$, and since [[function fields]] are just the finite-dimensional field extensions of the fields of fractions $\mathbb{F}_q(x)$ of $\mathbb{F}_q[x]$, this similarity plausibly extends to these extensions. (Also the [[entire function|entire]] [[holomorphic functions]] on the [[complex plane]] are, while not quite an principal ideal domain still a [[Bézout domain]]. )

But the analogy ranges much deeper than this similarity alone might suggest. For instance (\hyperlink{Weil39}{Weil 39}) defined an invariant of a [[number field]] -- the \emph{[[genus of a number field]]}-- which is analogous to the [[genus of a curve|genus]] of the [[algebraic curve]] on which a given [[function field]] is the [[rational functions]]. This is such as to make the statement of the [[Riemann-Roch theorem]] for [[algebraic curves]] extend to [[arithmetic geometry]] (\hyperlink{Neukirch92}{Neukirch 92, chapter II, prop.(3.6)}).

Another notable part of the analogy is the fact that there are natural analogs of the [[Riemann zeta function]] in all three columns of the analogy. This aspect has found attention notably through the lens of regarding [[number fields]] as [[rational functions]] on ``[[arithmetic curves]] over the would-be [[field with one element]] $\mathbb{F}_1$''.

The analogy between [[p-adic numbers]] and [[Laurent series]] over $\mathbb{F}_p$ is strengthened by (\hyperlink{FontaineWinterberger79}{Fontaine-Winterberger 79}), which shows that the absolute [[Galois groups]] of the [[perfect field|perfection]] of $\mathbb{F}_p((t))$ and of $\mathbb{Q}_p[p^{\frac{1}{p^\infty}}]$ are [[isomorphism|isomorphic]]. For more review of this see also (\hyperlink{Hartl06}{Hartl 06}). (The generalization of this to higher dimensions is the topic of [[perfectoid spaces]].)

It is also the function field analogy which induces the conjecture of the [[geometric Langlands correspondence]] by analogy from the number-theoretic [[Langlands correspondence]]. Here one finds that the [[moduli stack of bundles]] over a [[complex curve]] is analogous in absolute [[arithmetic geometry]] to the [[coset space]] of the [[general linear group]] with coefficients in the [[ring of adeles]] of a number field, on which [[unramified]] [[automorphic representations]] are functions. Under this analogy the [[Weil conjecture on Tamagawa numbers]] may be regarded as giving the [[groupoid cardinality]] of the [[moduli stack of bundles]] in [[arithmetic geometry]].

In summary then the analogy says that the theory of [[number fields]] and of [[function fields]] both looks much like a [[global analytic geometry]]-version of the theory [[complex curves]].

\hypertarget{Formalizations}{}\subsection*{{Formalizations}}\label{Formalizations}

To date the function field analogy remains just that, an [[analogy]], though various research programs may be thought of as trying to provide a context in which the analogy would become a consequence of a systematic theory (see e.g. the introduction of \hyperlink{vdGeer05}{v.d. Geer et al 05}). This includes

\begin{itemize}%
\item [[Arakelov geometry]];


\item [[global analytic geometry]].


\item geometry ``over [[F1]]''.



\end{itemize}
Regarding the last point, in particular [[Borger's absolute geometry]] (\hyperlink{Borger09}{Borger 09}) makes precise the analogy between [[Spec(Z)]] and the [[polynomial ring]] $k[z]$/[[entire holomorphic function]]-ring $\mathcal{O}_{\mathbb{C}}$ by interpreting the analog of the canonical [[derivation]] $\frac{\partial}{\partial z}$ on the latter two as the [[Fermat quotient]] operation, and more generally by interpreting the lift of this to arithmetic spaces over ${Spec}(\mathbb{Z})$ as lifts of [[Frobenius homomorphisms]] as given by [[Lambda-ring]] structures. See at \emph{\href{Borger%27s+absolute+geometry#Motivation}{Borger's absolute geometry -- Motivation}} for more on this.

In this context the analogy between geometry over [[number fields]] and over [[function fields]] is made precise by showing (\hyperlink{Borger09}{Borger 09, section 7}) that for any smooth connected curve $S/\mathbb{F}_q$ over a [[finite field]] $\mathbb{F}_q$ the standard [[geometric morphism]] of (``big'') [[toposes]]

\begin{displaymath}
Spec(S/\mathbb{F}_q)\longrightarrow Spec(\mathbb{F}_q)
\end{displaymath}
factors through an alternative base topos $\widetilde Spec(\mathbb{F}_q)$

\begin{displaymath}
Spec(S/\mathbb{F}_q)\longrightarrow \widetilde Spec(\mathbb{F}_q) \longrightarrow Spec(\mathbb{F}_q)
\end{displaymath}
which, while different from $Spec(\mathbb{F}_q)$ is ``close'' to $Spec(\mathbb{F}_q)$ in some precise sense, but which has the advantage that its construction does exist for $q = 1$ in that there is directly analogous

\begin{displaymath}
Spec(\mathbb{Z}) \longrightarrow \widetilde Spec(\mathbb{F}_1)
    \,,
\end{displaymath}
where the notation $\widetilde Spec(\mathbb{F}_1)$ here stands for Borger's the topos over [[Lambda-rings]], see at \emph{[[Borger's absolute geometry]]} for the actual details.

\hypertarget{Overview}{}\subsection*{{Overview}}\label{Overview}

[[!include function field analogy -- table]]

$\,$

[[!include Langlands analogies -- table]]

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[MKR analogy]] in [[arithmetic topology]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Original articles includes

\begin{itemize}%
\item [[André Weil]], \emph{Sur l'analogie entre les corps de nombres alg\'e{}brique et les corps de fonctions alg\'e{}brique}, Revue Scient. 77, 104-106, 1939


\item [[Kenkichi Iwasawa]], \emph{Analogies between number fields and function fields}, in \emph{Some Recent Advances in the Basic Sciences}, Vol. 2 (Proc. Annual Sci. Conf., Belfer Grad. School Sci., Yeshiva Univ., New York, 1965-1966), Belfer Graduate School of Science, Yeshiva Univ., New York, pp. 203--208, \href{http://www.ams.org/mathscinet-getitem?mr=0255510}{MR 0255510}

for more on this see: Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Main_conjecture_of_Iwasawa_theory}{Main conjecture of Iwasawa theory}}


\item [[Jean-Marc Fontaine]], [[Jean-Pierre Wintenberger]], \emph{Extensions alg\'e{}brique et corps des normes des extensions APF des corps locaux}, C. R. Acad. Sci. Paris S\'e{}r. A--B 288(8) (1979), A441--A444


\item [[Barry Mazur]], [[Andrew Wiles]], \emph{Analogies between function fields and number fields}, American Journal of Mathematics Vol. 105, No. 2 (Apr., 1983), pp. 507-521 (\href{http://www.jstor.org/stable/2374266}{JStor})



\end{itemize}
Textbook accounts include

\begin{itemize}%
\item [[Jürgen Neukirch]], \emph{Algebraische Zahlentheorie} (1992), English translation \emph{Algebraic Number Theory}, Grundlehren der Mathematischen Wissenschaften 322, 1999 (\href{http://www.plouffe.fr/simon/math/Algebraic%20Number%20Theory.pdf}{pdf})


\item Michael Rosen, \emph{Number theory in function fields}, Graduate texts in mathematics, 2002



\end{itemize}
Tables showing the parallels between number fields and function fields are in

\begin{itemize}%
\item [[David Goss]] \emph{Dictionary}, in David Goss, David R. Hayes, Michael Rosen (eds.) \emph{The Arithmetic of Function Fields}, Ohio State Univ. Math. Res. Inst. Publ., 2, de Gruyter, Berlin, 1992, pp. 475-482,


\item [[Bjorn Poonen]], section 2.6 of \emph{Lectures on rational points on curves}, 2006 (\href{http://math.mit.edu/~poonen/papers/curves.pdf}{pdf})


\item [[Urs Hartl]], \emph{A Dictionary between Fontaine-Theory and its Analogue in Equal Characteristic} (\href{http://arxiv.org/abs/math/0607182}{arXiv:math/0607182})



\end{itemize}
See also

\begin{itemize}%
\item M. Blickle,[[Hélène Esnault]], K. R\"u{}lling, \emph{Characteristic $0$ and $p$ analogies, and somemotivic cohomology} ([[EsnaultAnalogy.pdf:file]])

\end{itemize}
A collection of more recent developments is in

\begin{itemize}%
\item van der Geer et al (eds.) \emph{Number Fields and Function Fields -- Two Parallel Worlds}, Birkh\"a{}user 2005 (\href{http://www.springer.com/birkhauser/mathematics/book/978-0-8176-4397-3}{publisher page})

\end{itemize}
Discussion including also the complex-analytic side includes

\begin{itemize}%
\item [[Edward Frenkel]], section 2 of \emph{Lectures on the Langlands Program and Conformal Field Theory} (\href{http://arxiv.org/abs/hep-th/0512172}{arXiv:hep-th/0512172}).

\end{itemize}
and a comparison of the number theory to that of [[foliations]] is in

\begin{itemize}%
\item [[Christopher Deninger]], \emph{Analogies between analysis on foliated spaces and arithmetic geometry} (\href{http://arxiv.org/abs/0709.2801}{arXiv:0709.2801})

\end{itemize}
An actual formalization of the analogy between geometry over number fields and function fields is in

\begin{itemize}%
\item [[James Borger]], section 7 of \emph{Lambda-rings and the field with one element} (\href{http://arxiv.org/abs/0906.3146}{arXiv/0906.3146})

\end{itemize}


\end{document}