The cubic Shimura correspondence (Q1585465)

The author writes an explicit cubic Shimura correspondence from the three-fold metaplectic cover of \(\text{GL}(2)\) to \(\text{GL}(2)\). More precisely, this is done for classical automorphic forms on \(\mathbb H^3\) transforming under a character \(\chi\) of \(\Gamma_2\) (resp. \(\Gamma)\), where \(\mathbb H^3\) is the positive upper half space \(\mathbb C\times \mathbb R^*_+\), \(\Gamma= \text{SL}_2(\mathbb Z [\omega])\), \(\Gamma_2=\text{SL}_2 (\mathbb A)\Gamma(3)\). Here \(\omega^2+ \omega+1=0\). The correspondence is given through a kernel integral, and the kernel is given explicitly in full detail. It realizes in this special case the Shimura correspondence from the cubic cover of \(\text{GL}(2)\) over \(\mathbb Q(\omega)\) to \(\text{GL}(2)\). Recall that Flicker proved the existence of the Shimura correspondence for an \(n\)-fold cover of \(\text{GL}(2)\) (locally and globally) using the trace formula. The author explains in the discussion in Section 5, how the method and arguments of the paper can be applied to other cases. For example, when applied to non-metaplectic forms, the explicit kernel becomes a theta function associated to an indefinite form.