Modular forms of half-integral weights on SL(2,Z)

Abstract

In this paper, we prove that, for an integer $r$ with $(r,6)=1$ and $0<r<24$ and a nonnegative even integer $s$, the set {\eta(24\tau)^rf(24\tau):f(\tau)\in M_s(1)} is isomorphic to S_{r+2s-1}^{\text{new}}(6,-(\frac8r),-(\frac{12}r))\otimes(\frac{12}\cdot) as Hecke modules under the Shimura correspondence. Here $M_s(1)$ denotes the space of modular forms of weight $s$ on $\Gamma_0(1)=\mathrm{SL}(2,\Z)$, $S_{2k}^{\text{new}}(6,\epsilon_2,\epsilon_3)$ is the space of newforms of weight $2k$ on $\Gamma_0(6)$ that are eigenfunctions with eigenvalues $\epsilon_2$ and $\epsilon_3$ for Atkin-Lehner involutions $W_2$ and $W_3$, respectively, and the notation $\otimes(\frac{12}\cdot)$ means the twist by the quadratic character $\frac{12}\cdot)$. There is also an analogous result for the cases $(r,6)=3$.