I would like to know if there is any relationship between the **motivic Galois groups** and the **Langlands program**.

Many thanks.

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I would like to know if there is any relationship between the **motivic Galois groups** and the **Langlands program**.

Many thanks.

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Take $K=\mathbb{Q}$ for simplicity, but this applies to any number field $K$, and let $L$ be the Langlands group and $\mathcal{G}$ the motivic Galois group of $\mathbb{Q}$.

Then the conjectural relation between automorphic forms and motives (the Langlands program) implies that there is an homomorphism (up to a certain conjugation) such that the diagram

$$\require{AMScd}\begin{CD} L @>{}>> \mathcal{G};\\ @VVV @VVV \\ \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) @>{}>>\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}); \end{CD}$$

is commutative.

To get a better idea of both sides of this picture it is useful to use some classic (and better understood) intermediate groups. Let $W$ be the Weil group, $\mathcal{S}$ the Serre group and $\mathcal{T}$ the Taniyama group, then we have something like this

$$L \longrightarrow W \longrightarrow \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{(automorphic side)}$$

$$\mathcal{G} \longrightarrow \mathcal{S} \longrightarrow \mathcal{T} \longrightarrow \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{(motivic side)}$$

Most of this goes back to

- Robert Langlands, "Automorphic representations, Shimura varieties and motives" (1977)

Some other relevant references are

James Milne, Kuang-yen Shih, "Langlands's Construction of the Taniyama Group" (1982)

Norbert Schappacher, "CM motives and the Taniyama group" (1994)

James Arthur, "A note on the automorphic Langlands group" (2002)

Langlands' Taniyama group is isomorphic to the Motivic galois group of the category of absolute hodge cycle CM motives over Q.Deligne, Hodge cycles, motives, and shimura varieties.$\endgroup$ – tttbase Nov 4 '16 at 14:19