A generalization of Cohen-Eisenstein series and Shimura liftings and some congruences between cusp forms and Eisenstein series (Q1429139)

The authors of the paper under review construct some explicit modular forms in the space of the Eisenstein series \(E^+_{k+{1\over 2}}(4N,\chi_\ell)\), \(k\geq 1\) which can be viewed as a generalization of the Cohen-Eisenstein series (the case \(N= p\), a prime number) and consist of a basis of \(E^+_{k+{1\over 2}}(4N,\chi_\ell)\). After that they prove that the Shimura lifting is an isomorphism from \(E^+_{k+{1\over 2}}(4N,\chi_\ell)\) to \(E_{2k}(N,\text{id})\). At the end they give ``class-number congruences'' like the classical case of Cohen-Eisenstein series. The calculations are very extensive, the results interesting but too complicated to be given here explicitly.