MOD l representations of arithmetic fundamental groups, I: An analog of Serre's conjecture for function fields

摘要

There is a well-known conjecture of Serre that any continuous, irreducible representation rho : G(Q) -> GL(2)((F) over bar (l)) which is odd arises from a newform. Here G(Q) is the absolute Galois group of Q, and (F) over bar (l) is all algebraic closure of the finite field (F) over tilde (l) of l elements. We formulate such a conjecture for n-dimensional mod l representations of pi(1)(X) for any positive integer n and where X is a geometrically irreducible, smooth curve over a finite field k of characteristic p (p not equal l), and we prove this conjecture in a large number of cases. In fact, a proof of all cases of the conjecture for l > 2 follows from a result announced by Gaitsgory in [G]. The methods are different.