Higher modularity of elliptic curves

Abstract: Elliptic curves $E$ over the rational numbers are modular: this means there is a nonconstant map from a modular curve to $E$. When instead the coefficients of $E$ belong to a function field, it still makes sense to talk about the modularity of $E$ (and this is known), but one can also extend the idea further and ask whether $E$ is '$r$-modular' for $r=2,3\ldots$. To define this generalization, the modular curve gets replaced with Drinfeld's concept of a 'shtuka space'. The $r$-modularity of $E$ is predicted by Tate's conjecture. In joint work with Adam Logan, we give some classes of elliptic curves $E$ which are $2$- and $3$-modular.

number theory

Audience: researchers in the topic

Harvard number theory seminar