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 $$\left\{\eta \right(24\tau {)}^{r}f\left(24\tau \right):f\left(\tau \right)\in M_s\left(1\right)\}$$ is isomorphic to $$S_{r+2s-1}^{\mathrm{new}}(6,-(\frac{8}{r}),-(\frac{12}{r}\left)\right)\otimes \left(\frac{12}{\cdot }\right)$$ as Hecke modules under the Shimura correspondence. Here $M_s\left(1\right)$ denotes the space of modular forms of weight $s$ on ${\Gamma }_0\left(1\right)=SL(2,\mathbb{Z})$, $S_{2k}^{\mathrm{new}}(6,{ϵ}_2,{ϵ}_3)$ is the space of newforms of weight $2k$ on ${\Gamma }_0\left(6\right)$ that are eigenfunctions with eigenvalues ${ϵ}_2$ and ${ϵ}_3$ for Atkin–Lehner involutions $W_2$ and $W_3$, respectively, and the notation $\otimes (12/\cdot )$ means the twist by the quadratic character $(12/\cdot )$. There is also an analogous result for the cases $(r,6)=3$.