By the Modularity Theorem (a.k.a. the Shimura--Taniyama--Weil Conjecture), if $E$ is an elliptic curve over $\textbf{Q}$ with conductor $N$, then there exists a “modular parametrization” $\psi: X_0(N) \to E$, a non-constant surjective morphism defined over $\textbf{Q}$. Let me call this “geometric” modularity.
It is known that this is equivalent to the existence of a holomorphic modular form of weight 2 and level $\Gamma_0(N)$ such that $L(E, \chi, s) = L(f, \chi, s)$ for all Dirichlet characters $\chi$. Indeed, over $\textbf{Q}$, such a result follows from the weaker result that $L(E, s) = L(f,s)$. Let me call this “analytic” modularity.
According to the Langlands philosophy, one expects all motivic L-functions to be automorphic. In particular, for Galois representations coming from the etale cohomology of any variety over a number field, one expects to be able to attach an automorphic representation (or some set of them, depending on the context); so we expect “analytic” modularity in some general sense. It seems like most modularity/automorphy results aim to prove this kind of statement.
My question is the following:
Does one also expect a general form of “geometric” modularity to hold?
I have not seen any such parametrizations outside of the well-known case of modular parametrizations of elliptic curves over $\textbf{Q}$, and occasionally by Shimura curves (but am less familiar with the implications, compared to the modular curve case). Since many of the nice “coincidences” in this classical modularity setting seem to break down as one passes to more general settings, and since the connection between geometric and analytic modularity seems weaker in the function field setting (e.g. the relevant automorphic forms are not functions or sections of line bundles on the Drinfeld modular curve), does one expect a general "geometric" modularity statement to hold? For example, given a higher genus curve over $\textbf{Q}$, or an elliptic curve over a number field, or a surface (say, abelian) over $\textbf{Q}$, do you expect some form of modularity to correspond to the existence of a map from some space (like a Shimura variety) to the variety in question?
I am aware that one can sometimes, say, find the appropriate motives in the cohomology of Shimura varieties (e.g. if V is a variety such that $L(V,s) = L(\pi, s)$ for some automorphic representation $\pi$, and then you may have a Galois representation attached to $\pi$ lying in the cohomology of some Shimura variety), but let me call this a case of "analytic" modularity instead of a weak form of "geometric" modularity, for the sake of the question. In other words, by "geometric" modularity, I am specifically referring to the existence of something like a morphism from, say, a Shimura variety to a variety over a number field. When I asked this question (IRL), some suggested that such a statement may be better formulated in the language of correspondences (or purely in the language of motives), but then that such an answer may be essentially the case of the variety $V$ above. I would be interested in any such answers, or in any corrections to the formulation of the question.
