# nt.number theory – Modularity of higher genus curves

The modularity conjecture for smooth projective algebraic curves of genus 1, aka 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.