# Transformation formulae for classical theta functions

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 Introduction to Elliptic Curves and Modular Forms. However

1. a citation would be useful to me,

2. I want to check my calculation and

3. a reference may give the formulae in a more convenient form than I have.

EDIT I have now found a convenient reference: Rademacher's Topics in Analytic Number Theory.

FURTHER EDIT Rademacher actually 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.

Thanks to all who replied to this question.

• Aah curses my first thought was "the Serre-Stark paper in Antwerp VI" (Springer LNM 627) but they only carefully state the factors for theta_3 and you have a reference for that already. They mention Shimura Ann Math 97 (1973) "On modular forms of half integral weight", and Shimura is often very careful about that sort of thing, but I don't know if he'll have what you need. Mar 26, 2010 at 11:46
• Thanks for that. Alas, Shimura also has the formula for $\theta_3$ but not for $\theta_2$ or $\theta_4$. Maybe one can deduce the formulae from his more general considerations but he doesn't give the sort of explicit formulae I want. :-( Mar 26, 2010 at 12:20

• Alas, although Whittaker and Watson prove the formula for the substitution $\tau\mapsto-1/\tau$ they do not give explicit formulae for the general transformation from $\Gamma_0(4)$. Mar 27, 2010 at 18:13
• But this is really exotic to have the general formula for each of the thetanulls in one book. I have to check whether Yoshida's "Hypergeometric functions--my love" includes this material; the book involves a special "treatment" of $\Gamma_0(4)$ and its modular forms, although it mostly discusses the lambda invariant. Mar 28, 2010 at 3:52
• Like W and W Chandraskeharan gives only the transformation law for $\tau\mapsto−1/\tau$ and not for general elements of $\Gamma_0(4)$. But they do give a reference to the old treatise of Tannery and Molk. Mar 28, 2010 at 10:40