Map between formal neighbourhoods

Question

Let $M$ be a moduli stack of elliptic curves. The $j$-invariant defines a map $M \to \mathbb{A}^1$. In the article of Fulton and Olsson they say that the map $\mathcal{O}_{\mathbb{A}^1, 1728} \to \mathcal{O}_{M, y^2=x^3-x}$ after completion looks like $k[[t]] \to k[[z]]$ where $t$ goes to $z^2$. How can I show it? It seems not really intuitive because this point $1728$ is special, namely the group of automorphisms of $y^2=x^3-x$ is $\mathbb{Z}/6$. Any ideas are greatly appreciated!

Answer 1

The elliptic curve with $j$-invariant $1728$ has $\mathbb{Z}/4\mathbb{Z}$ automorphism away from characteristic $2$ and $3$. A generic elliptic curve has automorphism group $\mathbb{Z}/2\mathbb{Z}$ so the coarse moduli map $j : M \to \mathbb{A}^1$ is ramified to order $2$ over $j = 1728$. Since both source and target are smooth curves, then we know that formally locally any map with ramifiication $2$ can be written as $t \mapsto z^2$ with appropriate choice of coordinates $z$ and $t$.

Alternatively, you can take an explicit étale cover of $M$, for example the one given by the Legendre family $y^2 = x(x-1)(x- \lambda)$, and just compute the $j$ invariant. In this case we have $$ j = 2^8\frac{(\lambda^2 - \lambda + 1)^3}{\lambda^2(\lambda -1)^2} $$ and you can directly compute that $j = 1728$ is a critical value of $j(\lambda)$ with multiplicity $2$ so after formal completion and an appropriate choice of coordinates the map has to look like $t \mapsto z^2$.