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Let $F$ be a global function field over a finite field, and $E$ be an elliptic curve over $F$. Much like the number field case, it is natural to study the Galois representation on the Tate module of $E$ and its associated Selmer groups and L-function. However, much like the characteristic zero setting, the theory would be much aided if we knew in some way these these curves were modular in some sense. There could be some interesting applications of knowing this, for instance in the potential study of Fermat's equation over function fields perhaps. These could also help in perhaps building Euler systems, studying questions in Iwasawa theory and formulating and studying main conjectures.

I have been studying the theory of Drinfeld modular forms, and modular curves over function fields, I wonder if there are some analogues of modularity that are known or expected in this context, or if Drinfeld modular forms are not the right objects to look at?

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    $\begingroup$ I think this is all quite well-understood (but not by me). Articles like this point to "Jacobians of Drinfeld modular curves" by Gekele and Reversat. Though often Iwasawa theory in the function field case doesn't use this as the L-functions can be described directly. $\endgroup$
    – Chris Wuthrich
    Commented Jul 26, 2022 at 10:41

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