zbMATH — the first resource for mathematics

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)

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
Full Text: DOI EuDML
[1] Cohen, H.: Sums involving the values at negative integers ofL-functions of quadratic characters. Math. Ann.217, 271-285 (1975) · Zbl 0311.10030 · doi:10.1007/BF01436180
[2] Eichler, M., Zagier, D.: The Theory of Jacobi Forms. Progr. Math. Vol. 55, Boston Basel Stuttgart: Birkhäuser 1985 · Zbl 0554.10018
[3] Kohnen, W., Zagier, D.: Modular forms with rational periods. In: Rankin, R.A. (ed.) Modular Forms, pp. 197-249. Chichester: Ellis Horwood 1984 · Zbl 0618.10019
[4] Lang, S.: Introduction to Modular Forms. Berlin Heidelberg New York: Springer 1976 · Zbl 0344.10011
[5] Mumford, D.: Tata Lectures on Theta I. Prog. Math. Vol. 28, Boston Basel Stuttgart: Birkhäuser 1983 · Zbl 0509.14049
[6] Zagier, D.: Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields. In (eds) Serre, J-P., Zagier, D. Modular Forms of One Variable VI. (Lecture Notes Math. Vol. 627, pp. 105-169) Berlin Heidelberg New York: Springer 1977 · Zbl 0372.10017
[7] Zagier, D.: The Rankin-Selberg method for automorphic functions which are not of rapid decay. J. Fac. Sci. Tokyo28, 415-438 (1982) · Zbl 0505.10011
[8] Zagier, D.: Hecke operators and periods of modular forms. israel Mathematical Conference Proceedings3, 321-336 (1990) · Zbl 0712.11033
[9] Rankin, R.: The scalar product of modular forms. Proc. London Math. Soc.2, 198-217 (1982) · Zbl 0049.33904 · doi:10.1112/plms/s3-2.1.198
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.