Let be the moduli stack of elliptic curves. Given a scheme , maps are given by the groupoid of elliptic curves over , together with isomorphisms between them. The goal of this post is to compute away from the primes . (This is done in Mumford’s paper “Picard groups of moduli problems.”)
In the previous post, we saw that could be described as a quotient stack. Namely, consider the scheme and the Weierstrass equation
cutting out a subscheme . This is a flat family of projective cubic curves over with a section (the point at infinity given by ). There is an open subscheme over which the family is smooth, i.e., consists of elliptic curves. A little effort with cohomology and Riemann-Roch allows us to show that, Zariski locally, any elliptic curve can be pulled back from one of these: that is, any elliptic curve locally admits a Weierstrass equation.
The Weierstrass equation was not unique, though; any change of parametrization (in affine coordinates here)
preserves the form of the equation, and these are the only transformations preserving it. In other words, the map
exhibits as a torsor over for the group scheme with a multiplication law given by composing linear transformations. That is,
that is, to give a map , one has to choose an étale cover of (Zariski is enough here), maps inducing elliptic curves over the , and isomorphisms (coming from maps to ) over . (more…)