##
**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.

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

### MSC:

11F11 | Holomorphic modular forms of integral weight |

11R42 | Zeta functions and \(L\)-functions of number fields |