Transformation formulae for classical theta functions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T19:42:17Z http://mathoverflow.net/feeds/question/19400 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/19400/transformation-formulae-for-classical-theta-functions Transformation formulae for classical theta functions Robin Chapman 2010-03-26T09:39:30Z 2010-04-23T13:22:17Z <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#19512 Answer by Wadim Zudilin for Transformation formulae for classical theta functions Wadim Zudilin 2010-03-27T13:42:11Z 2010-03-27T13:42:11Z <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#19602 Answer by defgh for Transformation formulae for classical theta functions defgh 2010-03-28T08:59:13Z 2010-03-28T08:59:13Z <p>K. Chandrasekharan "elliptic functions" chapter 5 discuss also 2 variales transformation but theta-{2,4} becomes {1,2} in his notation</p>