Modular forms with rational periods. (English) Zbl 0618.10019

Modular forms, Symp. Durham/Engl. 1983, 197-249 (1984).
[For the entire collection see Zbl 0546.00010.]
The space of elliptic modular forms carries the usual rational structure defined by the rationality of Fourier coefficients. In the paper under review the authors investigate a different type of rational structure, which is connected with the Eichler-Shimura isomorphism, as for instance described by S. Lang [Introduction to modular forms (1976; Zbl 0344.10011)], and arises from the rationality of periods.
To be more precise let \(S_{2k}\) denote the space of elliptic cusp forms for \(SL_ 2({\mathbb{Z}})\) of weight 2k. Given \(f\in S_{2k}\) one defines the n-th period of f by \(r_ n(f)=\int^{\infty}_{0}f(it)t^ n dt,\) \(0\leq n\leq 2k-2\). Then the Eichler-Shimura isomorphism leads to the two rational structures on \(S_{2k}:\) \[ S^{\pm}_{2k}=\{f\in S_{2k};\quad r_ n(f)\in {\mathbb{Q}}\quad for\quad 0\leq n\leq 2k-2;\quad n\quad even / n\quad odd\}. \] The authors give examples of functions in \(S^{\pm}_{2k}\), whose periods are interesting arithmetical expressions. If \(W^{\pm}_{2k-2}\) denotes the attached space of period polynomials over \({\mathbb{Q}}\), one obtains an isomorphism \(r^-: S^- _{2k}\to W^-_{2k-2}\) and an exact sequence \[ 0\quad \to \quad S^+_{2k}\quad \to^{r^+}\quad W^+_{2k-2}\quad \to^{\lambda}\quad {\mathbb{Q}}\quad \to \quad 0 \] with explicit descriptions of \(r^{\pm}\) as well as of \(\lambda\) involving Bernoulli numbers. It is shown that the spaces \(S^+_{2k}\) and \(S^-_{2k}\) are dual to each other with respect to the Petersson scalar product \((.,.)\). The first interesting example of a modular form with rational periods is the cusp form \(R_ n\) characterized by \(r_ n(f)=(f,R_ n)\) for all \(f\in S_{2k}\), where \(n=0,1,...,2k-2\). Bernoulli polynomials occur in the description of the period polynomials of \(R_ n.\)
In the second section the authors deal with the cusp form \[ f_{k,D,{\mathcal A}}=(1/2\pi)\binom{2k-2}{k-1}^{-1}\quad D^{k- 1/2}\sum_{[a,b,c]\in {\mathcal A}}(az^ 2+bz+c)^{-k}\quad, \] where \({\mathcal A}\) is an equivalence class of quadratic forms of discriminant \(D>0\). The periods of the related functions \(f^{\pm}_{k,D,{\mathcal A}}\) are rational and involve the special value \(\zeta_{{\mathcal A}}(1-k)\), where \(\zeta_{{\mathcal A}}(s)\) equals the usual zeta-function of the ideal class corresponding to \({\mathcal A}\), if D is a fundamental discriminant. This result is applied in order to study the diagonal restriction of Hecke-Eisenstein series associated to \({\mathcal A}.\)
In the third section the periods are considered as integrals along certain geodesics in the upper half plane. If \(Q=[a,b,c]\) is an arbitrary quadratic form in \({\mathcal A}\), let \(C_ Q\) be the geodesic joining the roots of \(az^ 2+bz+c=0\) and \(\Gamma_{{\mathbb{Q}}}\) the subgroup of \(PSL_ 2({\mathbb{Z}})\) corresponding to the group of totally positive units of \({\mathbb{Q}}(\sqrt{D})\), preserving Q. Then the number \[ r_{{\mathcal A}}(f)=\int_{\Gamma_{{\mathbb{Q}}}\setminus C_ Q}f(z)(az^ 2+bz+c)^{k-1} dz \] makes sense for any \(f\in S_{2k}\) and satisfies \(r_{{\mathcal A}}(f)=2^{2k-2}(f,f_{k,D,{\mathcal A}}).\) The authors describe \(r_{{\mathcal A}}(f)\) as an explicit linear combination of \(r_ n(f)\), \(n=0,1,...,2k-2\), and apply their results to zeta-functions of real quadratic fields.
Reviewer: A.Krieg


11F11 Holomorphic modular forms of integral weight
11R42 Zeta functions and \(L\)-functions of number fields