Number Theory: Higher modularity of elliptic curves

SEMINARS, NUMBER THEORY

Speaker:

Jared Weinstein

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.