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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.

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    $\begingroup$ Since the non-abelian aspects will not be captured anyway it's better to consider the Jacobians instead of the curves themselves. You can find some discussion in the preprint "ABELIAN SURFACES OVER TOTALLY REAL FIELDS ARE POTENTIALLY MODULAR" $\endgroup$
    – Faris
    Dec 31, 2020 at 5:39
  • $\begingroup$ For a generic curve of genus $g$ (or abelian variety of dimension $g$) we expect an automorphic representation of the group $\mathrm{GSp}_{2g}(\mathbb{A}_F)$. A refinement is to take the Mumford-Tate group of the given variety instead. $\endgroup$
    – François Brunault
    Dec 31, 2020 at 7:19
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    $\begingroup$ @FrançoisBrunault I think the correct expectation is an automorphic rep of $GSpin_{2g + 1}(\mathbf{A}_F)$, the group whose Langlands dual is $GSp_{2g}$. This is isomorphic to $GSp_{2g}$ if $g \le 2$ but not otherwise. $\endgroup$
    – David Loeffler
    Dec 31, 2020 at 8:42

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Following the suggestion of Faris I looked at Abelian Surfaces over Totally Real Fields are Potentially Modular by Boxer, Calegari, Gee & Pilloni, 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.

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.

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

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