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Shtukas were defined by Drinfeld as a generalization of Drinfeld modules. While the relationship between the definitions of Drinfeld modules and shtukas is not obvious, one does have a natural bijection between Drinfeld modules and shtukas with two legs satisfying a nilpotence condition on one leg (or something like that). Since that time, a lot of research has been done involving Drinfeld modules and involving shtukas.

What I would like to understand is what the implication is of the relationship between Drinfeld modules and shtukas for this work. I don't usually see researchers working on one of the two kinds of object discuss the relation of their work to the other object. To make my question precise, let me ask:

For what problems is it crucial that we work with Drinfeld modules and not shtukas? For what problems is it crucial that we work with shtukas and not Drinfeld modules? Why?

For instance, I know that Drinfeld proved Langlands for GL_2 with Drinfeld modules, and Lafforgue proved it for GL_n with shtukas. Is it impossible to make a similar argument work with Drinfeld modules for GL_n, or is it more a matter of convenience?

In the reverse direction, the proof of Langlands for GL_2 by Drinfeld modules seems to give a slightly stronger statement than the proof by shtukas. In the Drinfeld module case, the desired Galois representation appears inside the cohomology of the 1-dimensional Drinfeld modular curve, while I think the relative moduli space of Shtukas for $GL_n$ is $2n-2$ dimensional, so $2$-dimensional in the case of $GL_2$. Because every Galois representation appearing in the cohomology of a curve appears in the cohomology of a surface, but not vice versa, Drinfeld's is (I believe) a stronger statement than Lafforgue.

I ask for a modern perspective because I want an answer informed by some of the research that has been done since Drinfeld defined shtukas, rather than a purely abstract description of the relationship.

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  • $\begingroup$ Just for completeness. A reference where the relation between shtukas, Anderson T-motives and Drinfeld modules is stablished is the following springer.com/gp/book/9783540635413 $\endgroup$ – user40276 Aug 24 '17 at 22:06
  • $\begingroup$ I think Drinfeld first proved certain cases of GL_2 (those representations being Steinberg) in his thesis using Drinfeld modules and later proved the full result for GL_2 using shtukas. So I am wondering why you said using Drinfeld modular varieties would give more representations than using moduli stacks of shtukas. $\endgroup$ – wkf Oct 19 '17 at 12:21
  • $\begingroup$ @wkf I didn't say it gave more representations - although I didn't know / didn't remember that it actually gave less, thanks. What I said is that it gives a geometrically stronger statement, i.e. the representations actually appear in the cohomology of a curve / family of curves. $\endgroup$ – Will Sawin Oct 19 '17 at 13:27

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