In modular curves and modular forms, there is an adelic formulation, in which smaller open subgroups of some adelic group relate to higher level structure. As we know, higher level structure corresponds to higher torsion points (when we say "higher" we mean, e.g., $E[n]$ is "higher" than $E[m]$ if $m \mid n$). In complex multiplication, higher torsion points correspond to higher ray class fields, which correspond to smaller open subgroups of the ideles of the field of complex multiplication in question! (*Since someone asked, I sketch this in more detail below)

Thus I would like to ask whether there is a formulation that combines the adelic formulation of complex multiplication with the adelic formulation of modular curves.

To simplify and thus explain the principles**, it would be best to work with elliptic curves with complex multiplication. Let $F$ be a quadratic imaginary field.

Then I have the following guess for an idea: We have an injection from the ideles of $F$ into the rational adelic points of $\mathrm{GL}_2$, and after quotienting by the rational (i.e. non-adelic) points, we still have a nice inclusion? I would guess that the resulting points at those corresponding to CM elliptic curves and their level structures, but I don't entirely understand how to prove this.

Furthermore, assuming this is true, what does this say about the Galois action? Does this allow us to construct the Galois action (i.e. rational structure) on modular curves?

(Also, to be even more precise, we should let the ideles of $F$ be represented by the adelic points of the Weil restriction of $G_m$ from $F$ to $\mathbb{Q}$).

*In the adelic formulation of modular curves, we view modular curves as quotients of the adelic points of $GL_2$. We get different modular curves, e.g. $\Gamma(N)$ instead of $\Gamma(M)$ (for $M,N$ some positive integers) by considering different open subgroups of the adeles. As is well-known, $\Gamma(N)$ parametrizes elliptic curves with "level N" structure, meaning pairs consisting of elliptic curves and $N$-torsion points. In complex multiplication, the $N$-torsion points correspond to the ray class field of level $N$, and this is, in particular, done through an adelic formulation of the main theorem of complex multiplication.

**Though I assume everything will extend to higher-dimensional Shimura varieties in an analogous way.

  • $\begingroup$ I would really appreciate, if you sketch out the correspondances indicated in the first paragraph, or at least give a reference for this. $\endgroup$ – Marc Palm May 8 '12 at 12:37
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    $\begingroup$ I don't really understand the question, but Silverman's book Advanced Topics in the Arithmetic of Elliptic Curves gives an adelic formulation of the main results on complex multiplication for elliptic curves. $\endgroup$ – David E Speyer May 8 '12 at 13:08
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    $\begingroup$ BR has it basically right. The adelic formulation gives you a map (let's call it $\eta$) from the pro-finite set $H(\mathbb{Q})∖H(\mathbb{A}_f)$ to the inverse limit of the modular curves over varying levels. The main theorem of complex multiplication can be interpreted as saying that, if you equip the pro-finite set with the $Gal(\overline{\mathbb{Q}}/K)$-action induced from the global reciprocity map for $K$, then $\eta$ is compatible with Galois actions and is therefore defined over $K$. $\endgroup$ – Keerthi Madapusi Pera May 8 '12 at 17:01
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    $\begingroup$ Shou-Wu Zhang discussed this in his graduate course on Heegner points this past semester. (The class followed math.columbia.edu/~szhang/papers/elc.pdf up through the end of chapter 4, and then went on to talk about adelic modular curves instead of the full generality of Shimura curves). You should talk to him when you're back in Princeton. $\endgroup$ – Alison Miller May 8 '12 at 22:46
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    $\begingroup$ Dear Davidac, Just to add to Keerthi's answer, you are asking for an adelic formulation of the Shimura reciprocity law, in the context of CM elliptic curves. This can be found in texts on Shimura varieties (in more generality), e.g. in Deligne's Bourbaki seminars, or in Langlands's article Ein Maerchen (highly recommended for this and many other reasons). Regards, Matthew $\endgroup$ – Emerton May 9 '12 at 3:58

This is an expansion of my comment above: You can view the modular curve of level $K$ over $\mathbb{C}$ as the double coset space $$GL_2(\mathbb{Q})\backslash\mathbb{H}^{\pm}\times GL_2(\mathbb{A}_f)/K.$$ Here, $\mathbb{H}^{\pm}$ is the union of the half planes. This set is in bijection with isogeny classes of elliptic curves over $\mathbb{C}$ with $K$-level structure, and the bijection is obtained as follows: A point $\tau\in\mathbb{H}^{\pm}$ gives us an isomorphism of vector spaces $\mathbb{R}^2\xrightarrow{\simeq}\mathbb{C}$, which in turn equips $\mathbb{Q}^2$ with a Hodge structure of weights $(0,-1),(-1,0)$. This allows us to interpret it as the $\mathbb{Q}$-homology of an elliptic curve $E_{\tau}$ (This is just a fancier way of saying that $E_{\tau}=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau)$). So, given a point $[(\tau,g)]$ in the double coset space, we attach to it the elliptic curve (up to isogeny) $E_{\tau}$, with the $K$-level structure given by the $K$-orbit of the isomorphism $$\mathbb{A}_f^2\xrightarrow{g}\mathbb{A}_f^2= H_1(E_{\tau},\mathbb{A}_f).$$

Let $\alpha$ be some generator of $F$ over $\mathbb{Q}$: it allows us to identify $F$ with $\mathbb{Q}^2$. Choose an embedding $F$ in $\mathbb{C}$, so that $\alpha$ can be viewed as an element of $\mathbb{H}$. Let $H$ be the rank $2$ torus over $\mathbb{Q}$ attached to $F$: we can view it as the sub-group of $GL_2$ that commutes with the action of $F$. We then get a map $$\eta:H(\mathbb{Q})\backslash H(\mathbb{A}_f)\rightarrow GL_2(\mathbb{Q})\backslash\mathbb{H}^{\pm}\times GL_2(\mathbb{A}_f)/K$$ $$\;\;\;\;[h]\mapsto [(\alpha,h)].$$ The image of this map consists exactly of those elliptic curves admitting CM by $F$, whose level structures admit an $F$-equivariant representative. (One subtlety is that, if $E$ is an elliptic curve in the image, then you can also look at the curve $\overline{E}$, where you twist the action of $F$ by complex conjugation. This won't show up in the image of $\eta$).

Now, we have the global reciprocity isomorphism $$Gal(\overline{F}/F)^{ab}\xrightarrow{\simeq}H(\mathbb{Q}\backslash H(\mathbb{A}_f),$$ which equips $H(\mathbb{Q})\backslash H(\mathbb{A}_f)$ with the structure of a pro-finite $Gal(\overline{F}/F)$-set. The main theorem of CM for elliptic curves says that, if we identify the right hand side of $\eta$ with the set of $\overline{\mathbb{Q}}$-points of the modular curve, then $\eta$ is in fact Galois-equivariant.

There is a slight sign issue here, since there are two possible choices for the reciprocity map (arithmetic or geometric Frobenius), but this is the general shape.

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