nt.number theory – Modularity of higher genus curves

The modularity conjecture for elliptic curves over number fields is well known, and indeed, is a theorem for all elliptic curves over $mathbb{Q}$, and at least potentially, over any CM field.

What is the precise statement of the conjecture for higher genus curves? What are the modular/automorphic forms we expect to correspond to Galois representations realized in the $l$-adic cohomology of a smooth projective algebraic curve of genus $g$ > 1 over a number field $F$ via equality of $L$ functions? What is the state of progress towards the conjecture? References would be very welcome.