[tacomonstrous]

Geometric Langlands, while inspired by questions Weil was interested in, is actually answering somewhat different questions, which are not arithmetic in nature: hence the 'geometric' moniker. The actual Langlands program, which deals with number fields and hence with questions of an arithmetic flavor (meaning solving equations over the rational numbers rather than the real or complex numbers) is still very much unexplored in its full generality.

Edit: it appears that I might have spoken too soon. At the link below is a paper by Sam Rankin establishing some consequences for arithmetic questions, though over function fields (such as those formed by rational functions over finite fields).

https://gauss.math.yale.edu/~sr2532/springer.pdf

[Koshkin]

An obligatory quote on the Langlands Program from a well-known category theorist: The thing is, I’ve never succeeded in understanding the slightest thing about it.