The Picard groups of the stacks $\mathscr{Y}_0(2)$ and $\mathscr{Y}_0(3)$

Abstract

We compute the Picard group of the stack of elliptic curves equipped with a cyclic subgroup of order two, and of the stack of elliptic curves equipped with a cyclic subgroup of order three, over any base scheme on which $6$ is invertible. This generalizes a result of Fulton-Olsson, who computed the Picard group of the stack of elliptic curves (with no level structure) over a~wide variety of base schemes.