Title: 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.... 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.