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\section*{modular form}

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

\hypertarget{elliptic_cohomology}{}\paragraph*{{Elliptic cohomology}}\label{elliptic_cohomology}

[[!include elliptic cohomology -- 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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_components}{In components}\dotfill \pageref*{in_components} \linebreak
\noindent\hyperlink{AsSections}{As sections of a line bundle over the moduli stack}\dotfill \pageref*{AsSections} \linebreak
\noindent\hyperlink{ForCongruenceSubgroups}{For congruence subgroups}\dotfill \pageref*{ForCongruenceSubgroups} \linebreak
\noindent\hyperlink{AsAutomorphicForms}{As automorphic forms}\dotfill \pageref*{AsAutomorphicForms} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_of__to_the_elliptic_genus}{Relation of $MF_0(2)$ to the elliptic genus}\dotfill \pageref*{relation_of__to_the_elliptic_genus} \linebreak
\noindent\hyperlink{relation_to_elliptic_cohomology}{Relation to elliptic cohomology}\dotfill \pageref*{relation_to_elliptic_cohomology} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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}

A \emph{modular form} is a [[holomorphic function]] on the [[upper half-plane]] that satisfies certain transformation property under the [[action]] of the [[modular group]]. Abstractly this transformation property makes the function a [[section]] of a certain [[line bundle]] on the [[quotient]] of the [[upper half plane]] that makes it the [[moduli stack of elliptic curves]] (over the [[complex numbers]]) or more generally the [[Riemann moduli space]] for [[Riemann surfaces]] of given [[genus]].

Modular forms are also often called \emph{classical [[automorphic forms]]}, see \href{}{below}

Modular forms appear as the [[coefficient]] ring of the [[Witten genus]] on manifolds with rational [[string structure]]. For manifolds with actual [[string structure]] this refines to [[topological modular forms]], which are the [[homotopy groups]] of the [[spectrum]] [[tmf]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{in_components}{}\subsubsection*{{In components}}\label{in_components}

\begin{defn}
\label{}\hypertarget{}{}
An \textbf{(integral) [[modular form]]} of weight $w$ is a [[holomorphic function]] on the [[upper half-plane]]

\begin{displaymath}
f : (\mathbb{R}^2)_+ \hookrightarrow \mathbb{C}
\end{displaymath}
(complex numbers with strictly positive imaginary part)

such that

\begin{enumerate}%
\item if $A = \left( \itexarray{a & b \\ c& d}\right) \in SL_2(\mathbb{Z})$ acting by $A : \tau \mapsto = \frac{a \tau + b }{c \tau + d}$ we have

\begin{displaymath}
f(A(\tau)) = (c \tau + d)^w f(\tau)
\end{displaymath}
(notice that for $A = \left( \itexarray{1 & 1 \\ 0& 1}\right)$ then $f(\tau + 1) = f(\tau)$)


\item $f$ has at worst a pole at $\{0\}$ (for \emph{weak} modular forms this condition is relaxed)

it follows that $f = f(q)$ with $q = e^{2 \pi i \tau}$ is a meromorphic funtion on the open disk.


\item \textbf{integrality} $\tilde f(q) = \sum_{k = -N}^\infty a_k \cdot q^k$ then $a_k \in \mathbb{Z}$



\end{enumerate}
\end{defn}
More generally there is such a definition for $SL(2,\mathbb{Z})$ replaced by any other arithmetic subgroup $\Gamma \subset SL(2,\mathbb{R})$ (e.g. \hyperlink{Litt}{Litt, def.1}), giving modular forms on a [[Riemann moduli spaces]].

\hypertarget{AsSections}{}\subsubsection*{{As sections of a line bundle over the moduli stack}}\label{AsSections}

More abstractly, for $\mathcal{M}_{ell}$ the [[moduli stack of elliptic curves]] (or rather its [[Deligne-Mumford compactification]]) and $A \to \mathcal{M}_{ell}$ the corresponding universal bundle, write $\Omega^1_{A/S}$ for the [[line bundle]] of fiberwise [[Kähler differential forms]]. Write $e$ for the 0-[[section]] of this line bundle. Then

\begin{displaymath}
\omega \coloneqq e^\ast \Omega^1_{A/S}
\end{displaymath}
is a [[line bundle]] over the [[moduli stack of elliptic curves]]. A modular form of weight $k$ is a [[section]] of $\omega^{\otimes k}$

\hypertarget{ForCongruenceSubgroups}{}\subsubsection*{{For congruence subgroups}}\label{ForCongruenceSubgroups}

Similarly one considers modular forms for [[congruence subgroups]] of the full [[modular group]], hence on the space of [[elliptic curves with level structure]].

\hypertarget{AsAutomorphicForms}{}\subsubsection*{{As automorphic forms}}\label{AsAutomorphicForms}

Instead of regarding, as \hyperlink{AsSections}{above}, modular forms as [[sections]] of a [[line bundle]] on a [[quotient]] of the [[upper half plane]], one may regard them alternatively as plain functions, but on the ([[Möbius group|projective]]) [[special linear group]] $SL(2,\mathbb{R})$. (e.g. \href{Martin13}{Martin 13, section 2}, \hyperlink{Litt}{Litt, section 2}).

As such these functions are then invariant under the [[action]] of the [[modular group|modular]] [[subgroup]] $SL(2,\mathbb{Z})\hookrightarrow SL(2,\mathbb{R})$ and hence are really functions on the [[coset space]] $SL(2,\mathbb{R})/SL(2,\mathbb{Z})$ (for forms on moduli of elliptic curves) or more generally $\Gamma \backslash SL(2,\mathbb{Z})$ (for forms on moduli of more general [[Riemann surfaces]]).

This generalizes to the case of other [[congruence subgroups]] (as \href{ForCongruenceSubgroups}{above}). Generally such functions on [[coset spaces]] like this are called \emph{[[automorphic forms]]}. See there for more.

For the history of the terminology ``modular form''/``automorphic form'' see also \href{http://mathoverflow.net/a/124785/381}{this MO comment}.

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

\hypertarget{relation_of__to_the_elliptic_genus}{}\subsubsection*{{Relation of $MF_0(2)$ to the elliptic genus}}\label{relation_of__to_the_elliptic_genus}

Write $\Gamma_0(2) \hookrightarrow SL_2(\mathbb{Z})$ for the [[subgroup]] of the [[modular group]] on those elements $\left(\itexarray{a & b \\ c & d}\right)$ for which $c = 0\, mod\, 2$.

A modular function for $\Gamma_0(2)$ is a [[meromorphic function]] on the [[upper half plane]] which transforms as a modular form under the action of $\Gamma_0(2) \hookrightarrow SL_2(\mathbb{Z})$. Write $MF_\bullet(\Gamma_0(2))$ for the ring of these.

There is a natural isomorphism

\begin{displaymath}
MF_\bullet(\Gamma_0(2)) \simeq \mathbb{C}[\epsilon, \delta]
\end{displaymath}
(see at [[elliptic genus]]) for the notation.

(\hyperlink{LandweberRavenelStong93}{Landweber-Ravenel-Stong 93, theorem 1.5 and sections 5.3, 5.8})

\hypertarget{relation_to_elliptic_cohomology}{}\subsubsection*{{Relation to elliptic cohomology}}\label{relation_to_elliptic_cohomology}

For $E$ the [[elliptic cohomology theory]] associated to the [[elliptic curve]] $C$, then

\begin{displaymath}
E_{2n} \simeq \omega(C)^{\otimes n}
\end{displaymath}
(where $\omega$ is the line bundle from \hyperlink{AsSections}{above})

and

\begin{displaymath}
E_{2n+1}\simeq 0
  \,.
\end{displaymath}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item [[Eisenstein series]]


\item [[j-invariant]]


\item [[Weierstrass sigma-function]]


\item [[Dedekind eta function]]



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

\begin{itemize}%
\item [[Jacobi form]]


\item [[elliptic genus]], [[Witten genus]]


\item [[topological modular form]], [[tmf]]


\item generalization to functions on moduli of higher dimensional [[abelian varieties]]: [[Hilbert modular form]], [[Siegel modular forms]]


\item [[modularity theorem]]



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

A basic and handy reference is

\begin{itemize}%
\item [[Pierre Deligne]], \emph{Courbes elliptiques: formulaire d'apres J. Tate}, In \emph{Modular functions of one variable}, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pages 53\{73. Lecture Notes in Math., Vol. 476. Springer, Berlin, 1975 (\href{http://modular.math.washington.edu/Tables/antwerp/deligne/}{web})

\end{itemize}
Textbook accounts include

\begin{itemize}%
\item N. Koblitz, \emph{Introduction to Elliptic Curves and Modular Forms}, Graduate Texts in Math, vol. 97, Springer-Verlag, 1984, 1993 (2nd ed.).

\end{itemize}
Lecture notes and reviews include

\begin{itemize}%
\item Richard Hain, section 4 of \emph{Lectures on Moduli Spaces of Elliptic Curves} (\href{http://arxiv.org/abs/0812.1803}{arXiv:0812.1803})


\item [[Charles Rezk]], section 10 of \href{http://www.math.uiuc.edu/~rezk/512-spr2001-notes.pdf}{pdf}


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


\item Jan Hendrik Bruinier, Gerard van der Geer, [[Günter Harder]], Don Zagier, \emph{The 1-2-3 of modular forms}, Lectures at a Summer School 2004 in Nordfjordeid, Norway; Universitext, Springer 2008.


\item [[Nora Ganter]], \emph{\href{http://www.ms.unimelb.edu.au/~nganter/talbot/index.html}{Topological modular forms literature list}}


\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 Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Modular_form}{Modular form}}



\end{itemize}
Original discussion in the context of [[elliptic genera]] and [[elliptic cohomology]] includes

\begin{itemize}%
\item [[Peter Landweber]], [[Douglas Ravenel]], [[Robert Stong]], \emph{Periodic cohomology theories defined by elliptic curves}, in [[Haynes Miller]] et. al. (eds.), \emph{The Cech centennial: A conference on homotopy theory}, June 1993, AMS (1995) (\href{http://www.math.sciences.univ-nantes.fr/~hossein/GdT-Elliptique/Landweber-Ravenel-Stong.pdf}{pdf})


\item [[Peter Landweber]], \emph{Elliptic Cohomology and Modular Forms}, in \emph{Elliptic Curves and Modular Forms in Algebraic Topology}, Lecture Notes in Mathematics Volume 1326, 1988, pp 55-68 ([[LandweberEllipticModular.pdf:file]])



\end{itemize}
Reviews of this include

\begin{itemize}%
\item [[Kefeng Liu]], \emph{Modular forms and topology}, 1996 (\href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.130.9779}{citeseer})

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

[[!redirects integral modular form]] [[!redirects weak modular form]] [[!redirects modular function]] [[!redirects integral modular function]] [[!redirects weak integral modular form]]

[[!redirects integral modular forms]] [[!redirects weak modular forms]] [[!redirects modular functions]] [[!redirects integral modular functions]] [[!redirects weak integral modular forms]]



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