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Periods of modular forms and Jacobi theta functions. (English) Zbl 0742.11029
In an earlier paper W. Kohnen and D. Zagier [Modular forms, Symp. Durham 1983, 197-249 (1983; Zbl 0618.10019)] introduced the period polynomial $$r_ f(X)=\int_ 0^{i\infty}f(\tau)(\tau-X)^{k-2}d\tau$$ for a cusp form $$f$$ of weight $$k$$ in the context of the Eichler-Shimura isomorphism. There they also derived a formula for the (rational) coefficients of a related polynomial in two variables.
In the paper under review the author gives a more attractive formula by introducing a generating function. First of all the definition of $$r_ f(X)$$ is extended to $$f\in M_ k$$, the space of elliptic modular forms of weight $$k$$. Then the generating function is \begin{aligned} C(X,Y;\tau,T) & = {(XY-1)(X+Y)\over X^ 2Y^ 2}T^{-2} \\ & +\sum^ \infty_{k=2}\sum_{{f\in M_ k\atop\text{eigenform}}}{r_ f(X)r_ f(Y)-r_ f(-X)r_ f(-Y)\over 2(2i)^{k-3}(f,f)(k-2)!} f(\tau)T^{k- 2},\end{aligned} where $$(f,f)$$ is the Petersson scalar product. If $$\Theta(u)=\Theta_ \tau(u)$$ denotes the Jacobi theta function, one obtains the surprising identity $C(X,Y;\tau,T)=\Theta'(0)^ 2{\Theta((XY-1)T) \Theta((X+Y)T)\over \Theta(XYT) \Theta(XT) \Theta(YT) \Theta(T)}.$ The right hand side can also be rewritten, where the Eisenstein series $$G_ k$$, $$k\geq 2$$, are involved in place of the theta function.
Reviewer: A.Krieg (Münster)

##### MSC:
 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11F27 Theta series; Weil representation; theta correspondences 11F11 Holomorphic modular forms of integral weight
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##### References:
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