On Shimura Correspondence for Non-cusp forms of Half-Integral Weight

In this paper we construct a generalization of Cohen- Eisenstein series which generates the space of Eisenstein series in Kohnen plus space M⁺ _k+1/2 (Γ₀(4N)) of weight k + 1/2 on Γ0(4N) (k ≥ 2 an integer, N ≥ 1 an odd integer). Using these new Eisenstein series we extend the Shimura-Kohnen correspondence given by the maps 𝓁D, where (−1)^kD >0 and D is a fundamental discriminant.