Theory of newforms for Eisenstein series of half-integral weight (Q2800155)

For natural numbers \(k > 1\) and \(N \geq 1\), let \(M_{k+1/2}(4N)\) (respectively \(S_{k+1/2}(4N)\)) be the space of modular forms (respectively cusp forms) of weight \(k+1/2\) for \(\Gamma_0(N)\). Also let \(M_{k+1/2}^{\text{Eis}}(4N)\) be the orthogonal complement of the space of cusp forms with respect to the Petersson scalar product. When \(N\) is square-free, the theory of new forms have been studied for \(S_{k+1/2}(4N)\)). If \(k >1\), this theory is compatible with the theory of new forms for the space of cusp forms \(S_{2k}(2N)\) of weight \(2k\) for \(\Gamma_0(2N)\) via Shimura correspondence.NEWLINENEWLINEIn this article, the authors study the theory of new forms for \(M_{k+1/2}^{\text{Eis}}(4N)\) when \(k > 1\) and \(2 | N\) and show that it is compatible with the theory of newforms for the space of Eisenstein series of weight \(2k\) for \(\Gamma_0(2N)\). This work extends their earlier work on the theory of newforms on \(M_{k+1/2}^{\text{Eis}}(4N)\) when \(k > 1\) and \(N\) is necessarily an odd integer.