Modulformen auf \(\Gamma_ 0(N)\) mit rationalen Perioden. (Modular forms for \(\Gamma_ 0(N)\) with rational periods) (Q1191442)

Let \(S_{2k}(\Gamma_ 0(N))\) be the vector space of cusp forms of even weight \(2k\) on the congruence group \(\Gamma_ 0(N)\). For \(f\) in \(S_{2k}(\Gamma_ 0(N))\) and \(A\) in \(SL_ 2(\mathbb{Z})\), consider the polynomials \[ \rho_ A(f)(X)=\int_ 0^{i\infty}(f|_{2k}A)(z)\cdot(X-z)^ w dz=\sum_{n=0}^ w (- 1)^ n {w\choose n}r_{n,A}(f)X^{w-n}, \] with \(w=2k-2\) and the ``periods'' \(r_{n,A}(f)=\int_ 0^{i\infty}(f|_{2k}A)(z)z^ n dz\). Similarly, consider \[ \rho_ A^ \pm(f)(X)={1\over2}(\rho_ A(f)(X)\pm\rho_{\varepsilon A\varepsilon}(f)(-X))=\sum_{n=0}^ w(- 1)^ n{w\choose n}r_{n,a}^ \pm(f)X^{w-n},\;\text{with} \varepsilon=\left({-1\atop 0}{0\atop 1}\right). \] The author shows that, for \(N\) squarefree, the Eichler-Shimura isomorphism leads to two rational structures on \(S_{2k}(\Gamma_ 0(N))\), namely, the \(\mathbb{Q}\)-vector spaces \(S_{2k}^ \pm(\Gamma_ 0(N))\) of all those \(f\) in \(S_{2k}(\Gamma_ 0(N))\) for which \(\rho_ A^ \pm(f)\) has rational coefficients for all \(A\) in \(SL_ 2(\mathbb{Z})\). Theorem A states that each of the spaces \(S_{2k}^ \pm(\Gamma_ 0(N))\) contains a basis of \(S_{2k}(\Gamma_ 0(N))\) and is invariant with respect to Hecke operators and Atkin-Lehner involutions. Let \(R_{n,A}^ \pm\) be the kernel function for the \(n\)th coefficient of \(\rho_ A^ \pm\) with respect to the Petersson inner product \(\langle.,.\rangle\); this means that \(\langle f,R_{n,a}^ \pm\rangle\) is the \(n\)th coefficient of \(\rho_ A^ \pm(f)(X)\) for all \(f\) in \(S_{2k}(\Gamma_ 0(N))\). In Theorem \(B\), an explicit formula is given for \(r_{m,B}^ +(R_{n,A}^ -)\), for arbitrary \(A\), \(B\) in \(SL_ 2(\mathbb{Z})\) and \(m\), \(n\) in \(\{0,\dots,w\}\), involving values of Bernoulli polynomials at rational points and generalized Dedekind sums. The case \(N=1\) has been treated by \textit{W. Kohnen} and \textit{D. Zagier} [Modular forms, Symp. Durham/Engl. 1983, 197-249 (1983; Zbl 0618.10019)]. The restriction to \(N\) squarefree is made for simplicity only; most of the results hold for arbitrary \(N\).