Need for Drinfeld modules compared to elliptic curves over function field

Question

In a sense ever since they were invented that Drinfeld modules and later shtukas are the "right" objects to study and play the role of elliptic curves over function fields by virtue that they allow us to prove nice theorems (say Langlands in the function field setting). However a really naive thing is very unclear to me: what is wrong with the naive guess that such results could be achieved by studying the moduli space of elliptic curves over function fields?

Or rather why are elliptic curves over function fields not the right analogue of elliptic curves over number fields?

The two arguments I have for this are explicit class field theory and the underlying symmetric space:

• we would like the Tate module to give us an explicit construction of class field theory, and it is not clear what $\mathfrak{p}\subset \mathbb{F}_p[T]$-torsion of an elliptic curve should mean, whereas this makes perfect sense for Drinfeld modules as a "linear algebra gadget"
• on the other hand an argument could be made that the "space of lattices" is the corrcect object to study, and elliptic curves and Drinfeld modules "just happen" to be the objects allowing us to give this space the structure of an algebraic variety.

However both of these are "post-hoc" arguments, surely Drinfeld would have had another reason to know that $X_0(N)$ wouldn't do the job for function fields.

Answer 1

I am not entirely sure what the best answer to this question would look like, but let me elaborate a little on the explicit class field theory point.

Most broadly speaking, the reason why elliptic curves are useful for explicit class field theory of imaginary quadratic field $K$ is that there exist elliptic curves with complex multiplication by (orders in) $K$. In general, if we wish to mimic such constructions for other fields $K$, then we would need to have analogous objects with an endomorphism ring $K$.

Even in the context of number fields, this should tell you that elliptic curves are not enough for this purpose. One can generalize this somewhat by considering more general abelian varieties, but at best this only gives you CM fields (and you don't get full explicit CFT for those - see the Aside on page 109 of Milne's notes).

But none of that can help if $K$ is a function field - the endomorphism rings of abelian varieties, regardless of their field of definition, is going to be a characteristic zero ring. Thus one cannot hope to get explicit CFT over function fields from elliptic curve and alike. Instead you need some other kind of object on which those positive characteristic rings act. And if you recall how elliptic curves arise analytically as complex tori, you can come up with how Drinfeld modules can serve a similar purpose. This is very nicely exposed in Poonen's Introduction to Drinfeld modules.

Now, I am not familiar enough with the proofs of Langlands conjectures over function fields to provide analogous motivation for using Drinfeld modules there. However one thing I can say is that you definitely need something beyond elliptic curves if you want to go beyond the case of $GL_2$, as modular curves are Shimura varieties for $GL_2$ and it is the Hecke algebra of that group that acts on them. For larger groups you need other moduli spaces on which you have corresponding Hecke actions, and it is moduli of Drinfeld modules, or shtukas, of rank $r$ that gets an action of Hecke algebra of $GL_r$.

Edit: I've realized that the broad reason why moduli of elliptic curves couldn't work here is that while it has the action of the Hecke algebra over $GL_2$, it's the "wrong $GL_2$" - specifically, it admits an action of Hecke algebra for $GL_2$ over $\mathbb Q$, while for anything Langlands over a function field, we want the corresponding Hecke algebra over that field, as this is the algebra acting on the automorphic forms of interest. This is something that you have on the moduli of shtukas via modifications, but there is no way to make sense of the function field Hecke algebra acting on the modular curve.