Relation between motives and geometric Langlands

Question

When working over a number field (or a function field over a finite field), one predicts that the Langlands program is related to the theory of motives over this field. There are several ways I have seen this formulated:

1. Every motivic $L$-function should coincide with an automorphic $L$-function.
2. Using some Tannakian formalism, there should be a homomorphism from the Langlands group to the motivic Galois group, satisfying certain properties.
3. In some special cases, something like a Shimura variety (in the number field case) or a moduli space of shtukas (for function fields) should be related the motive via some sort of correspondence.

The last point is not a systematic prediction, but examples should be the parameterization of a modular elliptic curve over $\mathbb{Q}$, or the notion of higher modularity in https://arxiv.org/abs/2211.11149

My question is: Are any analogous relations between motives and automorphic objects expected to hold in geometric Langlands?

I do not know how to formulate points 2 and 3 in this setting (I suspect you can't, as we don't deal with Frobenius or non-derived categories of representations), but here is my attempt to formulate an analog of the first prediction

Let $X$ be a (smooth, connected) curve over $\mathbb{C}$. Let $K$ denote its function field. For an algebraic variety $A$ over $K$, we can consider the etale cohomology $H^i(A\times \overline{K}, \mathbb{C} )$. This is naturally endowed with an action of $\text{Gal}(\overline{K}/K)$. Supposing ramification is not an issue for the moment, this defines a local system $E$ on $X$. I will say $A$ is automorphic if there is a Hecke Eigensheaf $\mathcal{F}$ with eigenvalue $E$.

Is this a good notion of automorphy of a variety in the geometric setting? Does knowing "automorphy" in this sense give us any new information about the variety as happens in number theory?

Answer 1

First note that the formulation of geometric Langlands (the eigensheaf part, not the full conjectured equivalence of categories) is that associated to a local system on $X$ there exists an eigensheaf on $\operatorname{Bun}_G(X)$ which is a perverse sheaf. Both the local system and eigensheaf are objects in the derived category of sheaves.

There are multiple variants of this statement depending on what (derived) category of sheaves one works in, e.g. $D$-modules, constructible $\mathbb Q$-sheaves, constructible $\mathbb Q_\ell$-sheaves.

There are multiple notions of the category of motives, but a key property they share is they admit "realization" maps from the category of motives to the category of sheaves.

Based on this it's relatively clear what a motivic analogue of the existence of eigensheaves in geometric Langlands should be: Given a motive over $X$ (specifically a motivic local system, in particular whose realization is a local system) there should be a (derived) motive over $\operatorname{Bun}_G(X)$ whose realization should be the Hecke eigensheaf. Since there is a motivic version of geometric Satake one can demand this motive is itself an "eigenmotive" for the motivic Hecke action.

This property, of the existence of a motivic eigensheaf on $\operatorname{Bun}_G(X)$, is not really a property of the variety, but only a property of its motive.

To prove cases of this motivic statement, the obvious strategy to try is to take existing constructions of the Hecke eigensheaf and attempt to follow the same construction in the motivic world. For some constructions, like that of Frenkel, Gaitsgory, and Villonen, this seems viable: The construction is a sequence of simpler operations on sheaves such as Fourier transform and intermediate extension and one just has to make sure these operations make sense in a given category of motives. Other constructions, like that of Beilinson and Drinfeld, are specific to a given realization and do not have an obvious meaning in the category of motives.

The application of automorphy, in a geometric or motivic sense, to learning information about the original variety may be limited. The biggest application of automorphy in the classical setting is towards the holomorpy of $L$-functions associated to the variety. But this holomorphy is already known in the function field setting. The only analogue for local systems is that sufficiently high wedge powers vanish, which is obvious in each category. I believe in some categories of motives the vanishing of wedge powers is not known, but it is unlikely that automorphic methods will be the solution - rather one probably has to prove the vanishing first on the way to constructing eigensheaves, if that's possible at all.

I'm sure there is information one can draw from this but it will be of a less dramatic nature.