Transformation formulae for classical theta functions - MathOverflow most recent 30 from http://mathoverflow.net2013-05-22T19:42:17Zhttp://mathoverflow.net/feeds/question/19400http://www.creativecommons.org/licenses/by-nc/2.5/rdfhttp://mathoverflow.net/questions/19400/transformation-formulae-for-classical-theta-functionsTransformation formulae for classical theta functionsRobin Chapman2010-03-26T09:39:30Z2010-04-23T13:22:17Z
<p>I am looking for a reference for the transformation formulae
for the classical theta-functions
$$\theta_4(\tau)=\sum_{n=-\infty}^\infty (-1)^n q^{n^2}$$
and
$$\theta_2(\tau)=\sum_{n=-\infty}^\infty q^{(2n+1)^2/4}$$
under the congruence group $\Gamma_0(4)$.
Here $\tau$ lies in the upper-half plane and $q^x$ denotes
$\exp(2\pi i x\tau)$. More precisely I want the exact automorphy
factors for each $A\in\Gamma_0(4)$ (some eighth root of
unity times $\sqrt{c\tau+d}$). I know these can easily
be deduced from those for the basic theta-function
$$\theta_3(\tau)=\sum_{n=-\infty}^\infty q^{n^2}$$
for which a nice reference for the automorphy factors is Koblitz's <em>Introduction
to Elliptic Curves and Modular Forms</em>. However</p>
<ol>
<li><p>a citation would be useful to me,</p></li>
<li><p>I want to check my calculation and</p></li>
<li><p>a reference may give the formulae in a more convenient form than I have.</p></li>
</ol>
<p>Thanks in advance.</p>
<p><strong>EDIT</strong> I have now found a convenient reference: Rademacher's
<em>Topics in Analytic Number Theory</em>.</p>
<p><strong>FURTHER EDIT</strong> Rademacher atcually gives full transformation formula
for the two-variable classical Jacobi theta functions under arbitrary
matrices in $\mathrm{SL}_2(\mathbb{Z})$. From these we can deduce
for $A\in\Gamma_1(4)$ that
$$\frac{\theta_2(A\tau)}{\theta_3(A\tau)}
=i^b\frac{\theta_2(\tau)}{\theta_3(\tau)}$$
and
$$\frac{\theta_4(A\tau)}{\theta_3(A\tau)}
=i^{-c/4}\frac{\theta_4(\tau)}{\theta_3(\tau)}$$
in the usual notation. Once noticed, these relations are easy to prove
from scratch.</p>
<p>Thanks to all who replied to this question.</p>
http://mathoverflow.net/questions/19400/transformation-formulae-for-classical-theta-functions/19512#19512Answer by Wadim Zudilin for Transformation formulae for classical theta functionsWadim Zudilin2010-03-27T13:42:11Z2010-03-27T13:42:11Z<p>A classical sourse could be E.T. Whittaker and G.N. Watson, A course of modern analysis, 4th edn. (Cambridge, Cambridge University Press, 1927).</p>
http://mathoverflow.net/questions/19400/transformation-formulae-for-classical-theta-functions/19602#19602Answer by defgh for Transformation formulae for classical theta functionsdefgh2010-03-28T08:59:13Z2010-03-28T08:59:13Z<p>K. Chandrasekharan "elliptic functions" chapter 5 discuss also 2 variales transformation but theta-{2,4} becomes {1,2} in his notation</p>