The Picard groups of the stacks $Y_0(2)$ and $Y_0(3)$

TL;DR

This paper calculates the Picard groups of specific moduli stacks of elliptic curves with level structures of order two and three over schemes where 6 is invertible, extending previous results to broader bases.

Contribution

It generalizes Fulton-Olsson's computation of Picard groups to stacks with level 2 and 3 structures over arbitrary base schemes where 6 is invertible.

Findings

Computed Picard groups for stacks with level 2 and 3 structures

Extended previous results to more general base schemes

Provided explicit descriptions of these Picard groups

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.