The point of Shimura varieties, as far as I've understood it, is that for a given Shimura datum $(G,D)$, there exist models, by which I mean that for congruence subgroups $\Gamma$ there exists a Shimura variety $X(\Gamma)$ defined over some number field. Hence we get a action of the absolute galois group $G_{\mathbb{Q}}$ on $$V:=\lim_{\Gamma} H^*_{ét}(X(\Gamma),\mathbb{Q}_{\ell}).$$ However, from an adelic point of view, we also get a (continuous) action of $G(\mathbb{A}_f)$ on the shimura variety, and so in fact $V$ is a representation of $G_{\mathbb{Q}}\times G(\mathbb{A}_f)$. The point now (from a Galois representation/ Langlandian point of view) is that for every representation $\rho:G_{\mathbb{Q}}\rightarrow \mathbb{\mathbb{Q}_\ell}^\times$, we can associate a representation of $G(\mathbb{A}_f)$ as $$\text{Hom}(\rho,V).$$ The main difficulty (as far as I understand it) is now to show that we can generate sufficently many representations that way to prove Langlands Program. My question is how this argument looks like in abelian case, i.e. what happens when $G=\text{GL}_1$? The Shimura varieties of tori are relatively simple to understand, namely we know that the Shimura variety associated to $\text{GL}_1$ is of the form $$\mathbb{Q}^\times \backslash \mathbb{A}_f^\times/K$$ for an open compact subgroup $K\subset \mathbb{A}_f^\times$ and is a finite étale scheme over some number field. How do we finish the proof from there on to get Artin's reciprocity? Or do we need to restrict ourself to the local case to even be able to complete the proof?


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If $K$ is 'everything 1 mod N' for some N, then the canonical model of $\mathbf{Q}^\times_+ \backslash \mathbf{A}^\times_{\mathrm{f}} / K$ is exactly $\mu_N / \mathbf{Q}$, the scheme of $N$-th roots of unity. Any open compact $K$ will contain one of these, so $GL_1 / \mathbf{Q}$ Shimura varieties all look like quotients of $\mu_N$ for some $N$.

Hence the answer to "How do we finish the proof from there on to get Artin's reciprocity?" is "we prove the Kronecker--Weber theorem" [i.e. every abelian extension of $\mathbf{Q}$ is contained in a cyclotomic field].

Perhaps it's disappointing that Shimura varieties don't tell you how to prove Kronecker--Weber. But they do something much more important: they tell you how to generalize Kronecker--Weber, pointing you towards a much more general (mostly conjectural) picture of which Kronecker--Weber is just one small part.

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    $\begingroup$ Thank you for the nice answer. Does the situation change if we want to study $G_{\mathbb{Q}_p}^{ab}$? Harris and Taylor proved local Langlands with Shimura varieties, so is there hope for $\text{GL}_1$ in the local case? $\endgroup$ Aug 29, 2020 at 17:32
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    $\begingroup$ "Harris and Taylor proved local Langlands [for $GL_n$] with Shimura varieties" is in the same category of statements as "Michaelangelo painted the Sistine Chapel with a paintbrush". Yes, Shimura varieties for unitary groups were involved; but it's very far from that to imply that once you know the standard properties of Shimura varieties, the local Langlands conjecture is somehow obvious. $\endgroup$ Aug 30, 2020 at 8:12

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