I'm trying to interpret things in the following terminology:

Assume the standard conjectures, the existence of the conjectural Langlands group, and anything else you wish.

I assume the following statement: the category of isomorphism classes of irreducible continuous algebraic (I added the word `algebraic' after it was pointed out that it was needed) representations $\mathcal{L}_{\mathbb{Q}}$ (the Langlands group) $\rightarrow GL_n(\mathbb{C})$ is equivalent to the category of $\mathbb{Q}$-motives with coefficients in $\overline{\mathbb{Q}}$.

In the above terminology, there are three numbers to keep track of: $n$ (of $GL_n$), dimension of the scheme that the motive comes from, and the degree of purity of the motive within that scheme (by this I mean that if we denote it by $i$, then in any Weil cohomology it would be realize as the $i^{th}$ cohomology of this scheme).

I will give two examples:

i. Classic CFT comes from $n=1$, the dimension of the scheme that the motive comes from is $0$ (because we're dealing with fields), and the degree of purity of the motive is $0$.

I imagine that when people talk about CFT in higher dimensions they mean the $n=1$ case where you allow the other two numbers to be different.

ii. Taniyama Shimura: this is the $n=2$ case, where the dimension of the scheme is $1$ and the degree of purity of the motive is $1$. Here on the Langlands side, this must correspond to newforms of weight $2$ with rational Hecke eigenvalues.

In general, one could take $n=2$, and on the Langlands side look at a general weight $k$ newform. Let $F$ be the field generated by the Hecke eigenvalues. Then the corresponding motive is coming from a $[F:\mathbb{Q}]$-dimensional variety, and the purity degree of the motive is $k-1$. If $k=2$ one can prove that the variety from which the motive is coming is an abelian variety via an argument involving Hodge structures.

Now I wish to understand complex multiplication in this context. It seems that the goal is to classify abelian extensions of quadratic fields. This seems to imply that we are in the $n=1$ case. But what are the other two numbers? Are we really looking at motives coming from $0$-dimensional schemes? This seems weird, because in the theory of complex multiplication elliptic curves are implicated.

Can one give a similar context to complex multiplication as I did to CFT and Taniyama Shimura? How would that work? What would the three numbers I described be?

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