I'm a newcomer to the geometric Langlands setting, and have mostly consulted surveys like Laumon's overview of L. Lafforgue's proof or Frenkel's recent advances survey, so apologies if this is material easily found in standard references.

I can see the utility of the sheaf-function dictionary, e.g. in the very nice toy example of unramified geometric class field theory in the Frenkel link, and also in the natural generalization to the case of Riemann surfaces. However, one thing bothers me a little: the techniques used to prove the function field Langlands conjectures (at least for $\text{GL}_n$), using trace formula methods in the cohomology of moduli spaces of shtukas, don't have any obvious relationship to this dictionary. Is there such a relationship, and is it written down somewhere in the literature?

It's a bit of a vague question; sorry, I guess I would hope for something like a perspective in which these cohomology groups somehow naturally parameterize (complexes of) $l$-adic perverse sheaves or something.

Also, as an aside, in general, is it known in general that every cuspidal automorphic function (say) for a function field corresponds to an $l$-adic sheaf on a suitable moduli space? If so, what are the techniques used to prove this?

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    $\begingroup$ I think the problem of constructing perverse sheaves on the stack of G-bundles or shtukas which recover a given eigenform is a hard problem, core to the difficulties in the geometric Langlands program. In V. Lafforgue's proof of the Langlands conjectures for function fields, he sidesteps this issue by noting that $H^0_c(\mathrm{Sht}_0, \overline{\mathbf{F_q}}, \mathbf{Q}_\ell)$ may be identified with the set of all cuspidal automorphic forms, where $\mathrm{Sht}_0$ is the stack of shtukas with no legs. Then you can build the Galois reps by looking at cohomology of IC ("constant") sheaves. $\endgroup$ – dorebell May 7 at 2:18

I don't understand the question well but the notes of Gaitsgory "FROM GEOMETRIC TO FUNCTION-THEORETIC LANGLANDS (OR HOW TO INVENT SHTUKAS)" might be useful (link), specifically Section 2.3. The abstract suggests that this is what you are looking for:

This is an informal note that explains that the classical Langlands theory over function fields can be obtained from the geometric one by taking the trace of Frobenius. The operation of taking the trace of Frobenius takes place at the categorical level, and this we deduce that the space of automorphic functions is the trace of the Frobenius on the category of automorphic sheaves.

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  • $\begingroup$ this looks very helpful, thanks I'll take a look! $\endgroup$ – xir Aug 19 '19 at 22:04

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