# From Galois representations to automorphic forms for $\mathfrak{sl}_2$ (via Drinfeld's shtukas)

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.

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.
• @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. Sep 25, 2022 at 11:31