On traces of modular forms and Eisenstein series for congruence classes of the rational modular group (Q1073828)

The work under review is a further development of a classical work of \textit{E. Hecke} [Abh. Math. Semin. Univ. Hamb. 5, 199--224 (1927; JFM 53.0345.02)] on Eisenstein series for subgroups of the rational modular group \(\mathrm{SL}(2,\mathbb{Z})\). Suppose that \(\Gamma \subset \mathrm{SL}(2,\mathbb{Z})\) is a subgroup of finite index, \(-I\in \Gamma,\) and let \(v\) be a unitary multiplier system on \(\Gamma\) of weight \(r\in\mathbb{R}\). Let \(A^{-1} \infty\) \(A\in \mathrm{SL}(2,\mathbb{Z}))\) be a cusp of \(\Gamma\) such that the associated normalized parabolic generator \(P\) satisfies \(v(P)=1\). For \(\text{Re}\, s>2-r\) and \(\text{Im}\, \tau >0\) let \[ G(s,\tau;{\mathcal K},A)=\sum_{M\in {\mathfrak S}(A\Gamma)}v^{-1}(M)\quad (m_ 1\tau +m_2)^{-r}\quad | m_1\tau +m_2|^{-s} \] the Eisenstein series (with factors ensuring convergence) for the space \(\mathcal K = \{\Gamma,-r,v\}\) of modular forms on \(\Gamma\) of weight \(r\) with multiplier system \(v\). The summation extends over a maximal system \({\mathfrak S}(A\Gamma)\) of matrices \(M=\left( \begin{matrix} \cdot & \cdot \\ m_1 & m_2\end{matrix} \right)\in A\Gamma\) with different second rows. For principal congruence subgroups these Eisenstein series were introduced by E. Hecke (loc.cit.). If \(\Gamma\) is a congruence subgroup, the series \(G(1,\tau; \mathcal K,A)\) may be obtained from Hecke's series by means of a trace operator. For \(r>2\) one may simply choose \(s=0\), and the author investigates the spanning and the metric properties of the corresponding series. The cases \(r=2\) and \(r=1\) are dealt with by means of an analytic continuation with respect to \(s\) and forming suitable linear combinations which are holomorphic functions of \(\tau\). The last section contains some classes of examples for the group \(\Gamma_ 0[\ell]\). This paper was edited from the posthumous works of the late Professor Hans Petersson (1902--1984) who started his impressive publishing activity in 1924 with a paper in the Hamburger Abhandlungen. For more than six decades his works added important and nowadays classical contributions to the theory of discontinuous groups and automorphic functions. Sixty-one manuscripts from the literary bequest of H. Petersson are preserved in the Universitätsbibliothek Münster.