Algebra and Geometry Seminar/Number Theory Seminar
Elliptic curves over the rationals are modular. This means either of two equivalent conditions: an automorphic condition (the L-function is meromorphic, with functional equation), and a geometric condition (there is a uniformization by a modular curve). For elliptic curves over global function fields, there is a similar story, where the role of modular curves is played by Drinfeld modular curves. In this setting we can hope for a more strict version of the geometric condition, involving moduli spaces of shtukas. We'll explain this, along with some explicit descriptions of moduli spaces of shtukas over the projective line.