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.



Modularity of higher genus curves
Following the suggestion of Faris I looked at [Abelian Surfaces over Totally Real Fields are Potentially Modular *by Boxer, Calegari, Gee & Pilloni*](https://arxiv.org/abs/1812.09269), whose section 1.4.1 discusses the modularity conjecure for higher genus curves and points to [On the Langlands Correspondence for Symplectic Motives *by Benedict Gross*](http://people.math.harvard.edu/%7Egross/preprints/symplectic.pdf).


Gross constructs a new form in the generic cuspidal automorphic representations of split orthogonal groups $SO\_{2g+1}$ which are conjecturally attached by global Langlands correspondence to *discrete* symplectic motives over $\mathbb{Q}$ of rank $2g$. Some of these motives are $H^1(.)(1) = H\_1(.)$ of genus $g$ curves over $\mathbb{Q}$, and more generally of polarized abelian varieties of rank $g$ over $\mathbb{Q}$ whose endomorphism rings are an order in a product of totally real fields. The $l$-adic realizations of these motives are Galois representations $G\_\mathbb{Q} \rightarrow GSp\_{2g}(\mathbb{Q}\_l)$. The Langlands dual of $GSp\_{2g}$ is $GSpin\_{2g+1}$, interpreted here as a split orthogonal group.


The (conjecturally automorphic) representation attached to such a motive is put together from explicit local representations at all places of $SO\_{2n+1}$. The explicit new form is built as restricted tensor product of local forms.


Results in the converse direction - attaching a symplectic motive to an automorphic representation of $SO\_{2n+1}$ - appear in [Potential Automorphy and Change of Weight *by Barnet-Lamb, Gee, Geraghty, and Taylor*](https://arxiv.org/abs/1010.2561).


Please feel free to strengthen this answer with more details and generalizations to other number fields.


