The modular parametrization of an elliptic curve over $\mathbb{Q}$ (and maybe over a number field in general?) is well-known; also for an elliptic curve over global function field with some condition (having split multiplicative reduction at some place), one also has an analogue of modular parametrization using Drinfeld modular variety in the function field case.

My main question is: Is there a version, over global function field, of modular parametrization using shtukas?

By the way, I read the survey ''Elliptic Curves and Analogies Between Number Fields and Function Fields'' by Dough Ulmer where he mentioned that there is a work by him entitled "Automorphic forms on GL2 over function fields and Gross–Zagier theorems'', but I could not find it. Has anyone read this paper and know where can I find it?


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I believe there is not, except for the fact that Drinfeld modular curves are certain subspaces of moduli spaces of shtukas.

The reason is that the classical modular parameterization is a combination of (1) the Galois representation associated to an elliptic curve appearing inside the Galois representation of a modular curve and (2) the Tate conjecture for morphisms of abelian varieties, which gives a morphism between the Jacobians of the two curves.

However, for moduli spaces of dimension > 1, no analogue of the Tate conjecture is known in general.

However, conditionally on the Tate conjecture, given any (!) variety over a function field, and any summand of its cohomology, there will be a correspondence between X x X (over the square of the base field) and a suitable moduli space of GL_n-shtukas (n the dimension of the summand) whose induced morphism on cohomology has image that summand tensor its dual. This should follow immediately from Lafforgue's theorem.


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