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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?

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  • $\begingroup$ I think this is exactly what people call "non-Abelian Lubin-Tate theory." $\endgroup$ – Moosbrugger Jun 19 '12 at 4:09
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    $\begingroup$ Dear Moosbrugger and Davidac, The direct analogue of passing from a CM elliptic curve to a non-CM elliptic curve would be passing from a Lubin--Tate formal group to some more general kind of formal group. This is not what happens in non-abelian LT theory. Rather, one looks at the moduli space of Lubin--Tate formal groups (and its congruence covers), and looks at its higher cohomology. (The $H^0$ recovers Lubin--Tate formal CM theory.) So this is more like using modular curves to construct non-abelian reciprocity laws than random elliptic curves. (Note that Shimura's paper is also ... $\endgroup$ – Emerton Jun 19 '12 at 4:48
  • $\begingroup$ ... about the modular elliptic curve $X_0(11)$.) Apologies to Moosbrugger, who surely knows all this; Davidac, if you want to pursue this, you could look at Carayol's paper in the Ann Arbor volumes (available on Milne's web-page) and then follow up the more recent literature (Harris and Taylor's book is probably the most significant more recent contribution). Regards, $\endgroup$ – Emerton Jun 19 '12 at 4:51
  • $\begingroup$ Dear Emerton, is not reasonable to ask if the galois representation afforded by the Tate module of a specific Lubin-Tate formal group of height 2 appears in the cohomology of some moduli space with level? If it is, wouldn't this be a reasonable direct analogue? $\endgroup$ – Dror Speiser Jun 19 '12 at 11:02

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