There is a well-known conjecture of Serre that any continuous, irreducible representation which is odd arises from a newform. Here is the absolute Galois group of , and is an algebraic closure of the finite field of of ℓ elements. We formulate such a conjecture for -dimensional mod ℓ representations of for any positive integer and where is a geometrically irreducible, smooth curve over a finite field of characteristic (), and we prove this conjecture in a large number of cases. In fact, a proof of all cases of the conjecture for follows from a result announced by Gaitsgory in [G]. The methods are different.
Gebhard Böckle. Chandrashekhar Khare. "Mod ℓ representations of arithmetic fundamental groups, I: An analog of Serre's conjecture for function fields." Duke Math. J. 129 (2) 337 - 369, 15 August 2005. https://doi.org/10.1215/S0012-7094-05-12925-8