Let {M_{1, 1}} be the moduli stack of elliptic curves. Given a scheme {S}, maps {S \rightarrow M_{1, 1}} are given by the groupoid of elliptic curves over {S}, together with isomorphisms between them. The goal of this post is to compute {\mathrm{Pic}(M_{1, 1})} away from the primes {2, 3}. (This is done in Mumford’s paper “Picard groups of moduli problems.”)

In the previous post, we saw that {M_{1, 1}} could be described as a quotient stack. Namely, consider the scheme {B_1 = \mathrm{Spec} \mathbb{Z}[a_1, a_2, a_3, a_4, a_6]} and the Weierstrass equation

\displaystyle Y^2 Z + a_1 XYZ + a_3 YZ^2 = X^3 + a_2 X^2 Z + a_4 XZ^2 + a_6 Z^3

cutting out a subscheme {E_1 \subset \mathbb{P}^2_{B_1}}. This is a flat family of projective cubic curves over {\mathbb{P}^2_{B_1}} with a section (the point at infinity given by {[X: Y: Z] = [0 : 1 : 0]}). There is an open subscheme {B \subset B_1} over which the family {E_1 \rightarrow B_1} 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 {X \rightarrow S} 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)

\displaystyle x' = a^2 x + b, \quad y' = a^3 x + c + d, \ a \mathrm{\ invertible}

preserves the form of the equation, and these are the only transformations preserving it. In other words, the map

\displaystyle B \rightarrow M_{1, 1}

exhibits {B} as a torsor over {M_{1,1}} for the group scheme {\mathbb{G} = \mathrm{Spec} \mathbb{Z}[a^{\pm 1}, b, c, d]} with a multiplication law given by composing linear transformations. That is,

\displaystyle M_{1, 1} \simeq B/\mathbb{G};

that is, to give a map {S \rightarrow M_{1, 1}}, one has to choose an étale cover {\left\{S_\alpha\right\}} of {S} (Zariski is enough here), maps {S_\alpha \rightarrow B} inducing elliptic curves over the {S_\alpha}, and isomorphisms (coming from maps to {\mathbb{G}}) over {S_\alpha \times_S S_\beta}. (more…)