Transformation equations and the special values of Shimura's zeta functions
Let a(n) resp. b(n) be the Fourier coefficients of a primitive cusp form f of weight k on \(\Gamma_ 0(N)\) resp. of a modular form g of weight \(\ell <k\) on \(\Gamma_ 0(N)\). For \(D(s,f,g)=\sum^{\infty}_{n=1}a(n) b(n) n^{-s}\) \textit{G. Shimura} [Commun. Pure Appl. Math. 29, 783-804 (1976; Zbl 0348.10015)] proved that the values D(m,f,g) at integers m, \((k+\ell)-1<m<k\), are algebraic numbers times the Petersson self inner product of f and a power of \(\pi\). For a cusp form h on \(\Gamma_ 0(N)\) the trace Tr(h) is the sum of the transforms of f by a set of representatives of \(\Gamma_ 0(N)\) modulo \(\Gamma =SL_ 2({\mathbb{Z}})\) or, equivalently, the sum of all roots of the transformation equation for h. The authors consider the traces \(Tr(gE^*_{\lambda,N})^{\mu}\) where g is a cusp form and \(E^*_{\lambda,N}\) is an Eisenstein series of weight \(\lambda >2\) on \(\Gamma_ 0(N)\). They prove that these traces are linear combinations of the primitive cusp forms f of weight \(k\mu =(\ell +\lambda)\mu\) on \(\Gamma\) with coefficients which are essentially the above mentioned algebraic numbers at \(m=k\mu -1\).
scientific article; zbMATH DE number 3900861