Let $G$ be the quasi-split unitary similitude group $GU(2, 1)$, for some choice of imaginary quadratic field $E$; and let $T = GU(1)$ be the torus $Res_{E/Q}\mathbf{G}_m$. Then there's a morphism $\theta: G \to T$ given by $g \mapsto \det(g)/\nu(g)$, where $\nu$ is the unitary multiplier. One can check that $\theta$ corresponds to a morphism of Shimura varieties, where we take the Shimura data to be the usual ones, namely $z \mapsto \mathrm{diag}(z, z, \bar{z})$ for $G$, and $z \mapsto z$ for $T$.
Both of these Shimura varieties are PEL-type, hence moduli spaces for abelian varieties over $E$-algebras: $Sh_G$ is the moduli of abelian 3-folds with $E$-multiplication and a signature condition, and $Sh_T$ is the moduli of CM elliptic curves, which are abelian 1-folds with $E$-multiplication and a signature condition. What does the map $\theta: Sh_G \to Sh_T$ "do" in terms of moduli spaces? Given an abelian 3-fold $A$ with PEL-structure, how do I make a CM elliptic curve out of it in a functorial fashion (i.e. compatible with base-change of $E$-algebras)? I'd really like it if $h^1$ of the resulting elliptic curve was canonically $(\wedge^3_E h^1(A))(-1)$.
