An elliptic curve $E$ with complex multiplication by an imaginary quadratic field $F$ has $\ell$-adic Galois representations that essentially encode the class field theory of $F$ - in other words, the reciprocity in abelian extensions of $F$, or equivalently, in extensions generated by the division points of $E$. For me, "reciprocity" means a way of determining the splitting of primes in an extension.
As Shimura (c.f. A Reciprocity Law in Non-Solvable Extensions) and many others have shown, one can use modularity to deduce reciprocity laws in extensions generated by torsion points of non-CM elliptic curves.
In another direction, Lubin and Tate showed that the theory of complex multiplication can be constructed formally for $p$-adic fields in order to generate the class field theory of any $p$-adic field.
My question is: can one generalize Lubin and Tate to non-abelian extensions and get a formal analogue of the reciprocity laws given by non-CM elliptic curves?