MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

Thanks in advance.

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

FURTHER EDIT 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.

Thanks to all who replied to this question.

share|cite|improve 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. – Kevin Buzzard Mar 26 '10 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. :-( – Robin Chapman Mar 26 '10 at 12:20

A classical sourse could be E.T. Whittaker and G.N. Watson, A course of modern analysis, 4th edn. (Cambridge, Cambridge University Press, 1927).

share|cite|improve this answer
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 $\Gamma_0(4)$. – Robin Chapman Mar 27 '10 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. – Wadim Zudilin Mar 28 '10 at 3:52
Thanks in advance, Wadim. I'm not at all familiar with Yoshida's book. – Robin Chapman Mar 28 '10 at 10:41
It depends on why are you looking for the transformation law... I never pay attention to such kind of things, whenever the form of transformation is clear. The major part of my library is packed, so I can hardly be explicit enough. Ken Ono's book discusses this in an explicit form, it could be in Iwaniec's book as well. I am surprised to know that T & M beat W & W at this. – Wadim Zudilin Mar 29 '10 at 7:36

K. Chandrasekharan "elliptic functions" chapter 5 discuss also 2 variales transformation but theta-{2,4} becomes {1,2} in his notation

share|cite|improve this answer
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. – Robin Chapman Mar 28 '10 at 10:40

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.