Periods of modular forms and Jacobi theta functions. (English) Zbl 0742.11029

Zagier, Don

Invent. Math. 104, No. 3, 449-465 (1991).

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

Keywords:

periods; Eichler-Shimura theory; Jacobi form; elliptic modular forms; Jacobi theta function; Eisenstein series

Citations:

Zbl 0618.10019

References:

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