Explicit images for the Shimura Correspondence
Matthew Boylan, Swati
TL;DR
The paper delivers a constructive framework for explicit Shimura lifts of half-integer weight modular forms with eta- and theta-multipliers, extending Yang's results to obtain concrete formulas for the r-th lift $\mathcal{S}_r(F)$ when $1\le r\le 23$ is odd and $N=1$, as well as lifts of theta-quotient–weighted eigenforms and Rankin–Cohen brackets. By decomposing eta-quotient twists of Hecke eigenforms into bases of newforms and computing their Shimura images, the authors produce explicit linear combinations in level-6 and level-2 newform spaces with precise Atkin–Lehner eigenvalues. The work includes explicit examples and a detailed background on the theta- and eta-multiplier settings, culminating in proofs that cover both $(r,6)=1$ and $(r,6)=3$ scenarios. These results provide ready-to-use, explicit formulas for Shimura correspondences in the eta- and theta-multiplier contexts, with potential applications to L-value relations and partition congruences. The combination of constructive lifts, operator analysis, and Rankin–Cohen bracket treatment significantly advances computational access to Shimura images in half-integer weight theory.
Abstract
In 2014, Yang showed that for $F \in \mathcal{A}_{r, s, 1, 1_N}$, we have $\textup{Sh}_{r}(F \mid V_{24}) = G \otimes χ_{12}$ where $G\in S^{new}_{r+2s - 1}(Γ_{0}(6), - \left( \frac{8}{r} \right), - \left( \frac{12}{r} \right))$, where $\textup{Sh}_{r}$ 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 $1\leq r\leq 23$ 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.