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Drinfeld-Lafforgue have proven function fields Langlands conjectures in type A: see https://arxiv.org/pdf/math/0212417.pdf (Laumon's survey in English), https://arxiv.org/pdf/math/0212399.pdf (LLafforgue's survey in French) and https://www.laurentlafforgue.org/math/fulltext.pdf (LLafforgue's complete proof).

Let $\sigma$ be a fixed rank $2$ irreducible $l$-adic representation of $Gal(\overline{F}/F)$, following S2 of Laumon's article. How do we construct the corresponding cuspidal automorphic representation of $GL_r(\mathbb{A})$? I'm interested in the construction (using the stack of shtukas), and not just the existence.

Please note that I'm relatively new to this construction using the stack of shtukas.

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For (1), shtukas are not needed for the construction of automorphic forms from Galois representations. Rather, this is done using the converse theorem. Shtukas are used to check the hypothesis of the converse theorem, involving automorphic forms on lower-rank groups, by converting these to Galois representations. (But shtukas aren't actually needed for this in the rank two case, as you only need the rank 1 Langlands correspondence, which is easier.)

A similar process occurs when we construct the modular form associated to an elliptic curve by taking the $L$-function of the elliptic curve, writing it as a Dirichlet series $\sum_n a_n n^{-s}$, and taking the function $\sum_n a_n q^n$ on the upper half-plane. The work in Wiles' theorem (and Taylor-Wiles and Breuil-Conrad-Diamond-Taylor) is in proving this function is a modular form, not in constructing it.

The construction of the automorphic form in the function field case is totally analogous to this - one can write down a function using an explicit Whittaker expansion, and the difficulty is in proving that this function satisfies some equivariance property.

The automorphic representation can then be constructed in the usual way from the automorphic form. Alternately, the representation can be constructed "directly" by applying the local Langlands correspondence at each place. Then again the real difficulty is in showing that this representation is indeed automorphic.

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  • $\begingroup$ Thanks, that's super-helpful! Which converse theorems are you referring to (in the sl_n case) - is there a reference you can share, for understanding the proof of these converse theorems? And if we start with a Galois representation, what's the corresponding "explicit Whittaker expansion"? $\endgroup$
    – Puraṭci Vinnani
    Commented Sep 25, 2022 at 4:48
  • $\begingroup$ @PuraṭciVinnani Probably en.wikipedia.org/wiki/Converse_theorem#References is a good place to start. For an explicit expansion, other than what's in those papers, for $GL_2$ over function fields, I wrote an explicit formula in Lemma A.1 on p. 50 here arxiv.org/pdf/1907.08098.pdf. Although it's written as a statement of the form "For each automorphic form there exists a Galois representation such that" it works equally well in the other direction since the correspondence is 1-to-1. $\endgroup$
    – Will Sawin
    Commented Sep 25, 2022 at 11:31
  • $\begingroup$ Thanks! Here's an article by JW.Cogdell about converse theorems that seems relevant (Piatetski-Shapiro’s Work on Converse Theorems). It could be worthwhile to ask a separate MO question about these converse theorems (and their extensions to VLafforgue's setting outside of type A). people.math.osu.edu/cogdell.1/PSCT-www.pdf $\endgroup$
    – Puraṭci Vinnani
    Commented Sep 28, 2022 at 1:51

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