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As anyone who's been reading the forums closely can see, I've been averaging a question a day about Tannakian formalism for the past few days. It's quite an interesting concept!

In any case, I wish now to relate it to yet another topic of which I have only a tenuous grasp: the Langlands program. As I understood more and more about Tannakian formalism, it seemed more and more like it has something to do with Langlands. A google search confirms this. There are several sources that group these two things together. Here is a sample:



But my understanding of Langlands is weak. I'm certainly familiar with Class Field Theory, and to some limited extent with Taniyama-Shimura. I always found Langlands difficult to penetrate. But now that I know that there is a relationship between Langlands and Tannakian formalism, I am hopeful that this will give me a bird's eye view of Langlands.

So the question is: Does Tannakian formalism simplify the statement of Langlands, or at least motivate it? Does it have to do with the motivic Galois group (defined to be the group predicted from Tannakian formalism on the category of numerical motives)? How precisely is Tannakian formalism used in Langlands?

In light of these ideas, I ask a secondary question: is there a relationship between the standard conjectures and Langlands? (does one imply the other?)

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You might enjoy the following page, in which several experts kindly responded to my plea to explain a tiny amount about the Langlands programme: golem.ph.utexas.edu/category/2010/08/… . I suspect that some of the conversation will be below your level, but some of it might be useful to you. –  Tom Leinster Aug 1 '11 at 4:58
@Tom: the beginning of that discussion always amused me. Several people seem not to have taken "that's the sum total of my knowledge" very seriously... –  Qiaochu Yuan Aug 3 '11 at 0:17
I know exactly what you mean... All the same, it shows up a good side of blogs: even if I didn't understand all the answers, probably there were other readers who did. –  Tom Leinster Aug 3 '11 at 1:04

1 Answer 1

Though there are several automorphic papers discussing the Tannakian outlook (notably Ramakrishnan's article in Motives (Seattle 1991, AMS) and Arthur's A note on the Langlands group, (referred to above) there is as yet no formulation of Langlands correspondence between Galois representations and automorphic representations as an equivalence of Tannakian categories. There are (at least) two outstanding fundamental questions on the Tannakian aspects of the Langlands correspondence.

1) What is the definition of the category of automorphic representations for any number field?

here one means automorphic representations for GL, any $n \ge 0$.

2) How to endow the category in 1) with a tensor structure so as to render it Tannakian?

here the postulated Tannakian group is the "Langlands Group" which is much larger than the motivic Galois group (not all automorphic representations correspond to Galois representations..only algebraic ones do - see work of Clozel and more recent work of Buzzard-Gee).

An interesting point: Arthur's paper postulates the Langlands group as an extension of the usual Galois group by a (pro-) locally compact group whereas the motivic Galois group is an extension of the usual Galois group by a pro-algebraic group. An illustration of the difference is provided by the case of abelian motives; the Langlands group is the abelianisation of the Weil group whereas the motivic group is the Taniyama group (see references below).

But the Tannakian outlook, despite its present inaccessibility, has already made a profound impact. See Langlands paper "Ein Marchen etc" (where the Tannakian aspect was first written out with many consequences for the Taniyama group (Milne's notes)) as well as Serre's book Abelian l-adic representations for many references.

Nothing seems to be known regarding the second question. However, see (page 6 of) these comments of Langlands:

"Although there is little point in premature speculation about the form that the final theory connecting automorphic forms and motives will take, some anticipation of the possibilities has turned out to be useful. Motivic $L$-functions, in terms of which Hasse-Weil zeta functions are expressed, are introduced in a Tannakian context.


An adequate Tannakian formulation of functoriality and of the relation between automorphic representations and motives ([Cl1, Ram]) will presumably include the Tate conjecture ( [Ta] ) as an assertion of surjectivity. The Tate conjecture itself is intimately related to the Hodge conjecture whose formulation is algebro-geometrical and topological rather than arthmetical. ..."

The references here are to Clozel and Ramakrishnan's papers and then Tate's paper for the Tate conjecture.

This is just a rough answer from a novice..for a precise and detailed answer, let us wait for the experts!

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