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\begin{document}

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\section*{Jacobi theta function}

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

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

[[!include theta functions - contents]]

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

[[!include complex geometry - contents]]

\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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{FunctionalEquation}{Functional equation and Reciprocity}\dotfill \pageref*{FunctionalEquation} \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}

The fundamental example of [[theta functions]] is the \emph{Jacobi theta function} given by

\begin{displaymath}
\vartheta(z,\tau)
  =
  \underoverset{n = - \infty}{\infty}{\sum}
   \exp(\pi i n^2 \tau + 2\pi i n z)
 \,.
\end{displaymath}
As a variable of two arguments, this is actually a \emph{[[Jacobi form]]}. These are the local [[coordinate]] expressions of the the [[covariant derivative|covariantly constant]] [[sections]] of the [[Hitchin connection]] (for [[circle group|circle]] [[gauge group]]) on the [[moduli space of elliptic curves]] (\hyperlink{Hitchin90}{Hitchin 90, remark 4.12}). See there for more and see at \emph{[[theta function]]} for the general idea.

At $z =0$ the function

\begin{displaymath}
\vartheta(0,\tau)
  =
  \underoverset{n = - \infty}{\infty}{\sum}
   \exp(\pi i n^2 \tau)
 \,.
\end{displaymath}
is what in [[number theory]] is often just called ``the theta function''. This is the one whose [[Mellin transform]] is the [[Riemann zeta function]], see at \emph{\href{http://ncatlab.org/nlab/show/Riemann+zeta+function#RelationToThetaFunctions}{Riemann zeta function -- Relation to Jacobi theta function}}

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

\hypertarget{FunctionalEquation}{}\subsubsection*{{Functional equation and Reciprocity}}\label{FunctionalEquation}

By the [[Poisson summation formula]] the number-theoretic theta function $\theta(0,z)$ satisfies the following [[functional equation]]:

\begin{displaymath}
\theta(0,\tau) = \frac{1}{\sqrt{\tau}} \theta(0,\frac{1}{\tau}) 
  \,.
\end{displaymath}
Under the [[Mellin transform]] this implies the functional equation of the [[Riemann zeta function]], see at \emph{\href{Riemann+zeta+function#functional_equation}{Riemann zeta function -- Functional equation}}.

It also provides an analytic proof of the [[Landsberg-Schaar relation]]

\begin{displaymath}
\frac{1}{\sqrt{p}}\sum_{n=0}^{p-1}\exp\left(\frac{2\pi i n^2 q}{p}\right)=\frac{e^{\pi i/4}}{\sqrt{2q}}\sum_{n=0}^{2q-1}\exp\left(-\frac{\pi i n^2 p}{2q}\right)
\end{displaymath}
where $p$ and $q$ are arbitrary positive integers. To prove it, apply theta reciprocity to $\tau=2iq/p+\epsilon$, $\epsilon \gt 0$, and then let $\epsilon\to 0$.

This reduces to the formula for the quadratic Gauss sum when $q=1$:

\begin{displaymath}
\sum_{n=0}^{p-1} e^{2 \pi i n^2 / p} =
\left\{
\itexarray{
\sqrt{p} & {if } \; p\equiv 1\mod 4 \\
i\sqrt{p} & {if } \; p\equiv 3\mod 4
}
\right.
\end{displaymath}
(where $p$ is an odd prime). From this, it's not hard to deduce Gauss's ``golden theorem''.

[[quadratic reciprocity]]: $\left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=(-1)^{(p-1)(q-1)/4}$ for odd primes $p$ and $q$.

See e.g. (\hyperlink{Karlsson}{Karlsson}).

\begin{quote}%
some of this material from \href{http://mathoverflow.net/q/120067/381}{this MO discussion}


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

\begin{itemize}%
\item [[Poisson summation formula]], [[functional equation]]

\end{itemize}
[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]

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

Due to [[Carl Jacobi]].

Review includes

\begin{itemize}%
\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Theta_function#Jacobi_theta_function}{Jacobi theta-function}}


\item section 9 in \emph{Analytic theory of modular forms} (\href{http://www.math.harvard.edu/~jbland/ma259x_notes.pdf}{pdf})


\item Anders Karlsson, \emph{Applications of heat kernels on abelian groups: $\zeta(2n)$, quadratic reciprocity, Bessel integrals} (\href{http://www.math.kth.se/~akarl/langmemorial.pdf}{psd})


\item [[Nigel Hitchin]], \emph{Flat connections and geometric quantization}, : Comm. Math. Phys. Volume 131, Number 2 (1990), 347-380. (\href{http://projecteuclid.org/euclid.cmp/1104200841}{Euclid})



\end{itemize}
[[!redirects Jacobi theta functions]]

[[!redirects Jacobi theta-function]] [[!redirects Jacobi theta-functions]]



\end{document}