Explicit images for the Shimura Correspondence

Abstract

In 2014, Yang showed that for $F\in {\mathcal{A}}_{r,s,1,1_N}$, we have ${\text{Sh}}_r(F\mid V_{24})=G\otimes \ddot{I}{‡}_{12}$ where $G\in S_{r+2s-1}^{new}(\hat{I}{“}_0(6),-\left(\frac{8}{r}\right),-\left(\frac{12}{r}\right))$, where ${\text{Sh}}_r$ is the $r$-th Shimura lift associated to the theta-multiplier. He proved a similar result for $(r,6)=3$.\:His proofs rely on trace computations in integral and half-integral weights. In this paper, we provide a constructive proof of Yang's result. We obtain explicit formulas for ${\mathcal{S}}_r(F)$, the $r$-th Shimura lift associated to the eta-multiplier defined by Ahlgren, Andersen, and Dicks, when $1\le r\le 23$ is odd and $N=1$. We also obtain formulas for lifts of Hecke eigenforms multiplied by theta-function eta-quotients and lifts of Rankin-Cohen brackets of Hecke eigenforms with theta-function eta-quotients.

35 pages, Comments are welcome