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\begin{document}

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\section*{Jacobi form}

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

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

[[!include complex geometry - contents]]

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

[[!include elliptic cohomology -- 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{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{JacobiThetaFunctions}{Jacobi theta-functions}\dotfill \pageref*{JacobiThetaFunctions} \linebreak
\noindent\hyperlink{weierstrass_function}{Weierstrass function}\dotfill \pageref*{weierstrass_function} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Jacobi forms are [[power series]] of two variables which in one variable behave like a [[modular form]] and in the other have an ``elliptic'' nature. They arise naturally as the characteristic series of the [[elliptic genus]]/[[Witten genus]] (\hyperlink{Zagier86}{Zagier 86, pages 8-9}).

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

For $k, n \in \mathbb{Z}$, a \emph{Jacobi form} of \emph{weight} $k$ and \emph{index} $n$ is a [[function]] of the form

\begin{displaymath}
\phi \;\colon\; H \times \mathbb{C} \longrightarrow \mathbb{C}
\end{displaymath}
hence from the [[product]] of the [[upper half plane]] with the full [[complex plane]] which transforms under

\begin{displaymath}
\left(
    \itexarray{
       a & b
       \\
       c & d
    }
  \right)
  \in 
  SL_2(\mathbb{Z})
\end{displaymath}
as

\begin{displaymath}
\phi
  \left(
     \frac{a \tau + b}{c \tau + d},
     \frac{z}{c \tau + d}
  \right)
   =
  (c \tau+ d)^k
  \exp(2 \pi i n c z^2 / (c \tau + d))
  \phi(\tau, z)
  \,.
\end{displaymath}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{JacobiThetaFunctions}{}\subsubsection*{{Jacobi theta-functions}}\label{JacobiThetaFunctions}

The most important examples are the [[Jacobi theta-functions]]. The four Jacobi $\theta$-functions are (with $q = e^{2\pi i \tau}$)

\begin{displaymath}
\theta(z,\tau) 
    \coloneqq
  2 q^{1/8} sin(\pi z)
  \prod_{j = 1}^{\infty}
  \left(   
    \left(
      1 - q^{j}
    \right)
    \left(
      1 - e^{2\pi i z} q^{j}
    \right)
    \left(
      1 - e^{-2 \pi i z} q^{j}
    \right)
  \right)
\end{displaymath}
\begin{displaymath}
\theta_1(z,\tau) 
    \coloneqq
  2 q^{1/8} cos(\pi z)
  \prod_{j = 1}^{\infty}
  \left(   
    \left(
      1 - q^{j}
    \right)
    \left(
      1 + e^{2\pi i z} q^{j}
    \right)
    \left(
      1 + e^{-2 \pi i z} q^{j}
    \right)
  \right)
\end{displaymath}
\begin{displaymath}
\theta_2(z,\tau) 
    \coloneqq
   \;\;\;\;\;\;\;\;\;
  \prod_{j = 1}^{\infty}
  \left(   
    \left(
      1 - q^{j}
    \right)
    \left(
      1 - e^{2\pi i z} q^{j - 1/2}
    \right)
    \left(
      1 - e^{-2 \pi i z} q^{j - 1/2}
    \right)
  \right)
\end{displaymath}
\begin{displaymath}
\theta_3(z,\tau) 
    \coloneqq
   \;\;\;\;\;\;\;\;\;
  \prod_{j = 1}^{\infty}
  \left(   
    \left(
      1 - q^{j}
    \right)
    \left(
      1 + e^{2\pi i z} q^{j - 1/2}
    \right)
    \left(
      1 + e^{-2 \pi i z} q^{j - 1/2}
    \right)
  \right)
\end{displaymath}
See for instance (\hyperlink{KL95}{KL 95, section 2.4}, \hyperlink{ChenHanZhang10}{Chen-Han-Zhang 10, section 2}) for a review in the context of [[elliptic genera]].

As part of this, the [[Kac-Weyl character]] of an integral highest-weight [[loop group representation]] is a Jacobi form (\hyperlink{KL95}{KL 95, section 2.2}).

The \textbf{Jacobi identity} (see at \emph{[[Jacobi triple product]]}) asserts that these are related by

\begin{displaymath}
\theta'(0,\tau) \coloneqq \frac{\partial}{\partial z}\theta(0,\tau)
  =
  \pi \theta_1(0,\tau) \theta_2(0,\tau) \theta_3(0,\tau)
  \,.
\end{displaymath}
\hypertarget{weierstrass_function}{}\subsubsection*{{Weierstrass function}}\label{weierstrass_function}

(\ldots{})

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

The original canonical account is

\begin{itemize}%
\item Martin Eichler, [[Don Zagier]], \emph{The theory of Jacobi forms}, Progress in Mathematics 55, Boston, MA: Birkh\"a{}user Boston (1985), ISBN 978-0-8176-3180-2, MR 781735

\end{itemize}
Discussion of Jacobi forms as coefficients of the [[elliptic genus]]/[[Witten genus]] includes

\begin{itemize}%
\item [[Don Zagier]], pages 8,9 of \emph{Note on the Landweber-Stong elliptic genus} 1986 (\href{http://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/BFb0078047/fulltext.pdf}{pdf})


\item [[Kefeng Liu]], \emph{On modular invariance and rigidity theorems}, J. Differential Geom. Volume 41, Number 2 (1995), 247-514 (\href{http://projecteuclid.org/euclid.jdg/1214456221}{EUCLID}, \href{http://www.math.ucla.edu/~liu/Research/loja.pdf}{pdf})


\item [[Matthew Ando]], Christopher French, [[Nora Ganter]], \emph{The Jacobi orientation and the two-variable elliptic genus}, Algebraic and Geometric Topology 8 (2008) p. 493-539 (\href{http://www.msp.warwick.ac.uk/agt/2008/08-01/agt-2008-08-016s.pdf}{pdf})


\item Qingtao Chen, [[Fei Han]], [[Weiping Zhang]], \emph{Generalized Witten Genus and Vanishing Theorems}, Journal of Differential Geometry 88.1 (2011): 1-39. (\href{http://arxiv.org/abs/1003.2325}{arXiv:1003.2325})



\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Jacobi_form}{Jacobi form}}

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

[[!redirects Jacobi theta-function]]



\end{document}