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…)

Let ${S}$ be a scheme. An elliptic curve over ${S}$ should be thought of as a continuously varying family of elliptic curves parametrized by ${S}$.

Definition 1 An elliptic curve over ${S}$ is a proper, flat morphism ${p: X \rightarrow S}$ whose geometric fibers are curves of genus one together with a section ${0: S \rightarrow X}$.

This is a reasonable notion of “family”: observe that a morphism ${T \rightarrow S}$ can be used to pull back elliptic curves over ${S}$. The flatness condition can be thought of as “continuity.” For an algebraically closed field, this reduces to the usual notion of an elliptic curve.

A basic property of elliptic curves over algebraically closed fields is that they imbed into ${\mathbb{P}^2}$ and are cut out by (nonsingular) Weierstrass equations of the form

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

This equation is unique up to an action of a certain four-dimensional group of transformations. The first goal is to show that, locally, the same is true for an elliptic curve over a base.  (more…)