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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.

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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

2 Answers 2

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

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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

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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

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