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\section*{automorphic form}

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

\hypertarget{theta_functions}{}\paragraph*{{Theta functions}}\label{theta_functions}

[[!include theta functions - contents]]

\hypertarget{representation_theory}{}\paragraph*{{Representation theory}}\label{representation_theory}

[[!include representation theory - contents]]

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

\noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak
\noindent\hyperlink{ModularForms}{Modular forms as classical automorphic forms on $PSL(2,\mathbb{R})$}\dotfill \pageref*{ModularForms} \linebreak
\noindent\hyperlink{ModularFormsAdAdelicAutomorphicForms}{Modular forms as adelic automorphic forms on $GL(2,\mathbb{A})$}\dotfill \pageref*{ModularFormsAdAdelicAutomorphicForms} \linebreak
\noindent\hyperlink{InNumberTheory}{General adelic automorphic forms}\dotfill \pageref*{InNumberTheory} \linebreak
\noindent\hyperlink{dirichlet_characters}{Dirichlet characters}\dotfill \pageref*{dirichlet_characters} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{function_field_analogy}{Function field analogy}\dotfill \pageref*{function_field_analogy} \linebreak
\noindent\hyperlink{application_in_string_theory}{Application in string theory}\dotfill \pageref*{application_in_string_theory} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{ReferencesInStringTheory}{In string theory}\dotfill \pageref*{ReferencesInStringTheory} \linebreak


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

Quite generally, \emph{automorphic forms} are suitably well-behaved [[functions]] on a [[quotient]] space $K\backslash X$ where $K$ is typically a [[discrete group]], hence suitable functions on $X$ which are [[invariant]] under the [[action]] of a discrete group. The precise definition has evolved a good bit through time.

[[Henri Poincaré]] considered [[analytic functions]] invariant under a discrete infinite group of [[fractional linear transformations]] and called them \emph{[[Fuchsian functions]]} (after his advisor [[Lazarus Fuchs]]).

More generally, automorphic forms in the modern sense are suitable functions on a [[coset space]] $K \backslash G$, hence functions on [[groups]] $G$ which are [[invariant]] with respect to the [[action]] of the [[subgroup]] $K \hookrightarrow G$. The archetypical example here are [[modular forms]] regarded as functions on $K\backslash PSL(2,\mathbb{R})$ where $K$ is a [[congruence subgroup]], and for some time the terms ``modular form'' and ``automorphic form'' were used essentially synonymously, see \hyperlink{ModularForms}{below}. Based on the fact that a [[modular form]] is a [[section]] of some [[line bundle]] on the [[moduli stack of elliptic curves]], [[Pierre Deligne]] defined an automorphic form to be a section of a line bundle on a [[Shimura variety]].

By pullback of functions the linear space of such functions hence constitutes a [[representation]] of $G$ and such representations are then called \emph{automorphic representations} (e.g. \hyperlink{Martin13}{Martin 13, p. 9}) , specifically so if $G = GL_n(\mathbb{A}_K)$ is the [[general linear group]] with [[coefficients]] in a [[ring of adeles]] of some [[global field]] and $K = GL_n(K)$. This is the subject of the \emph{[[Langlands program]]}. There one also considers [[unramified]] such representations, which are constituted by functions that in addition are invariant under the action of $GL_n$ with coefficients in the [[integral adeles]], see \hyperlink{InNumberTheory}{below}.

\hypertarget{ModularForms}{}\subsubsection*{{Modular forms as classical automorphic forms on $PSL(2,\mathbb{R})$}}\label{ModularForms}

By a standard definition, a \emph{[[modular form]]} is a [[holomorphic function]] on the [[upper half plane]] $\mathfrak{H}$ satisfying a specified transformation property under the [[action]] of a given [[congruence subgroup]] $\Gamma$ of the [[modular group]] $G = PSL(2,\mathbb{Z})$ (e.g. \hyperlink{Martin13}{Martin 13, definition 1}, \hyperlink{Litt}{Litt, def. 1}).

But the [[upper half plane]] is itself the [[coset]] of the [[projective linear group]] $G = PSL(\mathbb{R})$ by the subgroup $K = Stab_G(\{i\}) \simeq SO(2)/\{\pm I\}$

\begin{displaymath}
f\colon \mathfrak{H} \simeq PSL(2,\mathbb{R})/K
  \,.
\end{displaymath}
In view of this, one finds that every modular function $f \colon \mathfrak{H} \to \mathbb{C}$ lifts to a function

\begin{displaymath}
\tilde f \colon \Gamma\backslash PSL(2,\mathbb{R})
  \longrightarrow \mathbb{C}
  \,,
\end{displaymath}
hence to a function on $G$ which is actually \emph{invariant} with respect to the $\Gamma$-action (``automorphy''), but which instead now satisfies some transformation property with respect to the action of $K$, as well as some well-behavedness property

This $\tilde f$ is the incarnation as an \emph{automorphic function} of the modular function $f$ (e.g. \hyperlink{Martin13}{Martin 13, around def. 3}, \hyperlink{Litt}{Litt, section 2}). For emphasis these automorphic forms on $PSL(2,\mathbb{R})$ equivalent to modular forms are called \emph{classical modular forms}.

This is where the concept of automorphic forms originates (for more on the history see e.g. \href{http://mathoverflow.net/a/124785/381}{this MO comment} for the history of terminology) \href{http://mathoverflow.net/a/21556/381}{and this one}.

\hypertarget{ModularFormsAdAdelicAutomorphicForms}{}\subsubsection*{{Modular forms as adelic automorphic forms on $GL(2,\mathbb{A})$}}\label{ModularFormsAdAdelicAutomorphicForms}

Where by the \hyperlink{ModularForms}{above} an ordinary [[modular form]] is equivalently a suitably periodic function on $SL(2,\mathbb{R})$, one may observe that the [[real numbers]] $\mathbb{R}$ appearing as [[coefficients]] in the latter are but one of many [[p-adic number]] completions of the [[rational numbers]]. Hence it is natural to consider suitably periodic functions on $SL(2,\mathbb{Q}_p)$ of all these completions at once. This means to consider functions on $SL(2,\mathbb{A})$, for $\mathbb{A}$ the [[ring of adeles]]. These are the \emph{adelic automorphic forms}. They may be thought of as subsuming ordinary modular forms for all [[level structures]]. (e.g. \hyperlink{Martin13}{Martin 13, p. 8}, also \hyperlink{GoldfeldHundley11}{Goldfeld-Hundley 11, lemma 5.5.10}, \hyperlink{Bump}{Bump, section 3.6}, \hyperlink{Gelbhart84}{Gelbhart 84, p. 22}): we have

\begin{displaymath}
\Gamma \backslash PSL(2,\mathbb{R})
  \simeq
  Z(\mathbb{A}) GL_2(\mathbb{Q})\backslash GL_2(\mathbb{A})/ GL_2(\mathbb{A}_{\mathbb{Z}})
  \,,
\end{displaymath}
where $\mathbb{A}_{\mathbb{Z}}$ are the [[integral adeles]]. (The [[double coset]] on the right is analogous to that which appears in the [[Weil uniformization theorem]], see the discussion there and at \emph{[[geometric Langlands correspondence]]} for more on this analogy.)

This leads to the more general concept of \emph{adelic automorphic forms} \hyperlink{InNumberTheory}{below}.

\hypertarget{InNumberTheory}{}\subsubsection*{{General adelic automorphic forms}}\label{InNumberTheory}

More generally, for the [[general linear group]] $G = GL_n(\mathbb{A}_F)$, for any $n$ and with [[coefficients]] in a [[ring of adeles]] $\mathbb{A}_F$ of some [[number field]] $F$, and for the subgroup $GL_n(F)$, then sufficiently well-behaved functions on $GL_n(F)\backslash GL_n(\mathbb{A}_F)$ form [[representations]] of $GL_n(\mathbb{A}_{F})$ which are called \emph{[[automorphic representations]]}. Here ``well-behaved'' typically means

\begin{enumerate}%
\item \textbf{finiteness} -- the functions [[invariant]] under the [[action]] of the [[maximal compact subgroup]] [[span]] a [[finite number|finite]] [[dimension|dimensional]] [[vector space]];


\item \textbf{central character} -- the action by the [[center of a group|center]] is is controled by (\ldots{}something\ldots{});


\item \textbf{growth} -- the functions are [[bounded functions]];


\item \textbf{cuspidality} -- (\ldots{})



\end{enumerate}
(e.g. \hyperlink{Frenkel05}{Frenkel 05, section 1.6}, \hyperlink{Loeffler11}{Loeffler 11, page 4}, \hyperlink{Martin13}{Martin 13, definition 4}, \hyperlink{Litt}{Litt, def.4}).

(These conditions are not entirely set in stone, they are being varied according to application (see e.g. \href{http://mathoverflow.net/a/66598/381}{this MO comment})).

In particular one considers subspaces of ``[[unramified]]'' such functions, namely those which are in addition trivial on the subgroup of $GL_n$ of the [[integral adeles]] $\mathcal{O}_F$ (\hyperlink{GoldfeldHundley11}{Goldfeld-Hundley 11, def. 2.1.12}). This means that that unramified automorphic representations are spaces of functions on a [[double

coset]] of the form

\begin{displaymath}
GL_n(F)\backslash GL_n(\mathbb{A}_F) / GL_n(\mathcal{O}_F)
  \,.
\end{displaymath}
See at \emph{[[Langlands correspondence]]} for more on this. Such double cosets are [[analogy|analogous]] to those appearing in the [[Weil uniformization theorem]] in [[complex analytic geometry]], an analogy which leads to the conjecture of the [[geometric Langlands correspondence]].

\hypertarget{dirichlet_characters}{}\subsubsection*{{Dirichlet characters}}\label{dirichlet_characters}

For the special case of $n = 1$ in the discussion of adelic automorphic forms \hyperlink{InNumberTheory}{above}, the group

\begin{displaymath}
GL_1(\mathbb{A}_F) = (\mathbb{A}_F)^\times = \mathbb{I}_F
\end{displaymath}
is the [[group of ideles]] and the quotient

\begin{displaymath}
GL_1(F) \backslash GL_1(\mathbb{A}_F)
  = 
  F^\times \backslash (\mathbb{A}_F)^\times
\end{displaymath}
is the [[idele class group]]. Automorphic forms in this case are effectively [[Dirichlet characters]] in disguise\ldots{} (\hyperlink{GoldfeldHundley11}{Goldfeld-Hundley 11, theorem 2.1.9}).

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{function_field_analogy}{}\subsubsection*{{Function field analogy}}\label{function_field_analogy}

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

\hypertarget{application_in_string_theory}{}\subsubsection*{{Application in string theory}}\label{application_in_string_theory}

In [[string theory]] [[partition functions]] tend to be automorphic forms for [[U-duality]] groups. See the \hyperlink{ReferencesInStringTheory}{references below}

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[modular form]], [[topological modular form]], [[topological automorphic form]]


\item [[automorphic L-function]]


\item [[Langlands duality]]



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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

Introductions and surveys include

\begin{itemize}%
\item [[Pierre Deligne]], \emph{Fromed Modulaires et representations de $GL(2)$} (\href{http://publications.ias.edu/sites/default/files/Number21.pdf}{})


\item [[Stephen Gelbart]], starting on p. 20 (196) of \emph{An elementary introduction to the Langlands program}, Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177--219 (\href{http://www.ams.org/journals/bull/1984-10-02/S0273-0979-1984-15237-6/}{web})


\item Nolan Wallach, \emph{Introductory lectures on automorphic forms} (\href{http://math.ucsd.edu/~nwallach/luminy-port2.pdf}{pdf}) [[!redirects automorphic forms]]


\item E. Kowalski, section 3 of \emph{Automorphic forms, L-functions and number theory (March 12--16) Three Introductory lectures} (\href{http://www.math.ethz.ch/~kowalski/lectures.pdf}{pdf})


\item [[Dorian Goldfeld]], [[Joseph Hundley]], chapter 2 of \emph{Automorphic representations and L-functions for the general linear group}, Cambridge Studies in Advanced Mathematics 129, 2011 (\href{https://www.maths.nottingham.ac.uk/personal/ibf/text/gl2.pdf}{pdf})


\item Daniel Bump, \emph{Automorphic forms and representations}


\item David Loeffler, \emph{Computing with algebraic automorphic forms}, 2011 ([[LoefflerAutomorphic.pdf:file]])


\item [[Kimball Martin]], \emph{A brief overview of modular and automorphic forms},2013 \href{http://www2.math.ou.edu/~kmartin/papers/mfs.pdf}{pdf}


\item [[Daniel Litt]], \emph{Automorphic forms notes, part I} (\href{http://math.stanford.edu/~dlitt/Talks/automorphicformspt1.pdf}{pdf})


\item \href{http://math.stanford.edu/~conrad/modseminar/pdf/L10.pdf}{pdf}


\item \href{http://www.math.uni-bonn.de/people/mueller/skripte/specauto.pdf}{pdf}


\item Toshitsune Miyake's \emph{Modular Forms} 1976 (English version 1989) (\href{projecteuclid.org/euclid.bams/1183556263}{review pdf})



\end{itemize}
Review in the context of the [[geometric Langlands correspondence]] is in

\begin{itemize}%
\item [[Edward Frenkel]], \emph{Lectures on the Langlands Program and Conformal Field Theory}, in \emph{Frontiers in number theory, physics, and geometry II}, Springer Berlin Heidelberg, 2007. 387-533. (\href{http://arxiv.org/abs/hep-th/0512172}{arXiv:hep-th/0512172})

\end{itemize}
The generalization of theta functions to [[automorphic forms]] is due to

\begin{itemize}%
\item [[André Weil]], \emph{Sur certaines groups d'operateur unitaires}, Acta. Math. 111 (1964), 143-211

\end{itemize}
see \href{Langlands+program#Gelbhart84}{Gelbhart 84, page 35 (211)} for review.

Further developments here include

\begin{itemize}%
\item [[Stephen Kudla]], \emph{Relations between automorphic forms produced by theta-functions}, in \emph{Modular Functions of One Variable VI}, Lecture Notes in Math. 627, Springer, 1977, 277--285.


\item [[Stephen Kudla]], \emph{Theta functions and Hilbert modular forms},Nagoya Math. J. 69 (1978) 97-106


\item [[Jeffrey Stopple]], \emph{Theta and $L$-function splittings}, Acta Arithmetica LXXII.2 (1995) (\href{http://matwbn.icm.edu.pl/ksiazki/aa/aa72/aa7221.pdf}{pdf})



\end{itemize}
\hypertarget{ReferencesInStringTheory}{}\subsubsection*{{In string theory}}\label{ReferencesInStringTheory}

The relation between [[string theory]] on [[Riemann surfaces]] and automorphic forms was first highlighted in

\begin{itemize}%
\item [[Edward Witten]], \emph{Quantum field theory, Grassmannians, and algebraic curves}, Comm. Math. Phys. Volume 113, Number 4 (1988), 529-700 (\href{http://projecteuclid.org/euclid.cmp/1104160350}{Euclid})

\end{itemize}
See also

\begin{itemize}%
\item [[Michael Green]], Jorge G. Russo, Pierre Vanhove, \emph{Automorphic properties of low energy string amplitudes in various dimensions} (\href{http://arxiv.org/abs/1001.2535}{arXiv:1001.2535})

\end{itemize}
[[!redirects automorphic forms]]

[[!redirects automorphic representation]] [[!redirects automorphic representations]]

[[!redirects Fuchsian function]] [[!redirects Fuchsian functions]]



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