The Shimura lift and congruences for modular forms with the eta multiplier (Q6540260)

The Shimura lifts maps modular forms of half-integral weight~\(k\) to modular forms of integral weight~\(2k - 1\). As devised by Shimura it requires the input to be a modular form with integral Fourier exponents for the product of a Dirichlet character and a suitable power of the multiplier system~\(\nu_\theta\) of the Jacobi theta function. For many applications this leaves a gap: The equally important eta multiplier~\(\nu\) is not covered. By an ad-hoc construction one can obtain a Shimura lift, but crucial information about the level is lost.\N\NThe authors provide lifts of level~\(N\) cusp forms (orthogonal to theta series)\N\begin{align*}\N\mathrm{S}_{\lambda + \frac{1}{2}}(N, \psi \nu^r) &\longrightarrow \mathrm{S}^{\mathrm{new}\, 2,3}_{2\lambda}(6N, \psi^2, \varepsilon_{2,r,\psi}, \varepsilon_{3,r,\psi}) \quad && \text{with } \gcd(N,6) = \gcd(r,6) = 1 \text{,} \\\N\mathrm{S}_{\lambda + \frac{1}{2}}(N, \psi \nu^r) &\longrightarrow \mathrm{S}^{\mathrm{new}\, 2}_{2\lambda}(2N, \psi^2, \varepsilon_{2,r,\psi}) \quad && \text{with } \gcd(N,2) = 1, \gcd(r,6) = 3 \text{,}\N\end{align*}\Nwhere the range consists of cusp forms that are new at~\(2\) and~\(3\) or new at~\(2\), respectively, with Atkin-Lehner eigenvalues provided in parentheses:\N\begin{gather*}\N\varepsilon_{2,r,\psi} = - \psi(2) \big( \tfrac{8}{r \slash \gcd(r,3)} \big) \text{,}\quad \varepsilon_{3,r,\psi} = - \psi(3) \big( \tfrac{12}{r} \big) \text{.}\N\end{gather*}\N\NIn guiding examples the authors illustrate that as opposed to the usual Shimura lift the images of forms for different~\(N\) and~\(\psi\) can coincide. In particular, there is no corresponding variant of the Shintani lift. Towards the end they showcase an application to congruences for generating series, which leverages the refined information at the places to~\(2\) and~\(3\) to control Galois representations that appear in the generalization of an argument by Serre.