I was originally going to post this later, but I realized that I’d already written this story up before. Hopefully this will be another illustrative example of a ‘‘derived stack’’, which will properly be defined in the next post.

# The moduli of elliptic curves

$\newcommand{\Mfg}{\mathcal{M}_\textbf{fg}}$ Before we begin, I just want to say that a fabulous reference for elliptic curves is Katz-Mazur’s book. I read through parts of chapter 1 and 2, and I’m in love. I’ve learnt the content of most of this post from the TMF book. $\newcommand{\Mell}{\mathcal{M}_\textbf{ell}}$

$\DeclareMathOperator{\spec}{Spec}$ Let $\Mell$ denote the moduli stack of elliptic curves, so that a map $\spec R \to \Mell$ is a smooth elliptic curve over $R$. Why is this a stack, and not a scheme? If I have understood the story correctly, one of the reasons for inventing stacks is that (closed?) points of schemes do not have nontrivial automorphisms (although geometric points certainly do, e.g. a $\overline{k}$-point of a scheme $X$ has automorphisms given by elements of $\mathrm{Gal}(\overline{k}/k)$). Certainly elliptic curves (even over $\mathbf{C}$) have nontrivial automorphism groups (for instance, the curves with $j$-invariants $0$ and $1728$).

$\DeclareMathOperator{\spf}{Spf}$ Let $C\to \spec R$ be an elliptic curve, classified by a map $\spec R\to \Mell$. The point at infinity gives a section $\spec R \xrightarrow{0} C$, called the zero section. We can complete $C$ at the zero section (which just means looking at the infinitesimal neighborhood of the additive identity on $C$), to get a formal group $\widehat{C}$. There is an explicit choice of coordinate on this formal group that gives a formal group law, but it’s tedious to work through. This simply means that $\widehat{C}\simeq \spf R[[t]]$, and the choice of such an isomorphism is the choice of coordinate on this formal group. As $\widehat{C}$ is a formal group, we obtain a comultiplication on $R[[t]]$, and the image of $t$ under this comultiplication is the formal group law.

$\newcommand{\M}{\mathcal{M}}$ This gives a map of stacks $\Mell\to \Mfg$. The scrupulous reader might note that we did not use the fact that $C$ is an elliptic curve, but merely the fact that $C$ had a zero section. And that’s correct – we in fact constructed a map $\M_\textbf{cub}\to \Mfg$, where $\M_\textbf{cub}$ classifies cubic curves (i.e., curves with a Weierstrass form) equipped with a zero section. Inside this sits $\Mell$ as the locus where the discriminant is invertible.

Anyway, we have the following theorem.

Theorem (Hopkins-Miller): The map $\Mell\to \Mfg$ is flat.

To make sense of this statement, we first have to know that the map of stacks is representable. It turns out that $\M_\textbf{cub}\to \Mfg$ is representable, and the inclusion of $\Mell$ into $\M_\textbf{cub}$ is also clearly representable. How do we show that the map $\Mell\to \Mfg$ is flat? One constructs an affine cover $\spec A$ of $\Mell$, and shows that the composite $\spec A \to \Mell \to \Mfg$ is flat. From last time, this is equivalent to showing that the formal group law over $A$ corresponding to this ‘‘universal elliptic curve’’ is Landweber exact.

$\newcommand{\Z}{\mathbf{Z}}$ We first need to construct a cover of $\Mell$. This is not too hard to do: we can first construct a cover of $\M_\textbf{cub}$. The universal Weierstrass cubic is the curve $$y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 + a_6$$ In particular, we find that $\spec \Z[a_1,\cdots,a_6]\to \M_\textbf{cub}$ is an affine cover. But in fact, we know that $\M_\textbf{cub}$ is presented by a Hopf algebroid! This Hopf algebroid is $(\Z[a_1,\cdots,a_6],\Z[a_1,\cdots,a_6][f,g,h])$, where (let’s write $A = \Z[a_1,\cdots,a_6]$) the ring $A[f,g,h]$ parametrizes isomorphisms of the universal Weierstrass cubic. These isomorphisms are given by $x\mapsto x+f$ and $y\mapsto y+gx+h$. This means that the moduli of elliptic curves is presented by inverting the determinant, i.e., $\Mell$ is presented by $(A[\Delta^{\pm 1}],A[f,g,h,\Delta^{\pm 1}])$.

We now need to show that the formal group law over $A[\Delta^{\pm 1}]$ is Landweber exact. The maximum height of an elliptic curve is $2$, so we need to show that $p,v_1,v_2$ form a regular sequence. First, we know that $p$ is definitely nonzero since $A[\Delta^{\pm 1}]$ is torsion-free. Now, consider the ring $A[\Delta^{\pm 1}]/p$; this is an integral domain, and hence we need to show that $v_1\bmod p$ is nonzero. But this means that the ‘‘universal elliptic curve’’ mod $p$ has a fiber that is nonsingular, and this is true. (In fact, Marc told me that the element $v_1$ is the Hasse invariant.) The result will follow if we show that $v_2$ is not a unit in $A[\Delta^{\pm 1}]/(p, v_1)$. Suppose otherwise, so that the quotient of $A[\Delta^{\pm 1}]$ by the maximal ideal containing $p,v_1,v_2$ is a field of characteristic $p>0$, over which the reduction of the universal elliptic curve lives. The associated formal group law would be of height greater than $2$, which is impossible.

# What is the Hopkins-Miller theorem?

$\newcommand{\Aff}{\mathrm{Aff}^\mathrm{et}}$ By the above result, we know that any flat map $\spec R \to \Mell$ gives rise to a spectrum. This is exactly the situation from the previous post, where, instead of $\Mell$, we were considering $B\Z/2$. Just like in that case, we have a lifting theorem: this is the Hopkins-Miller theorem, which I thought I’d at least state.

$\newcommand{\co}{\mathcal{O}^\mathbf{top}}$ Theorem (Hopkins-Miller): There is an even-periodic sheaf of $\mathbf{E}_\infty$-rings on the etale site $\Aff_{\Mell}$ of the moduli stack of elliptic curves, such that $\pi_0 \co(\spec R\to \Mell) \simeq R$, and the formal group associated to $\co(\spec R\xrightarrow{C} \Mell)$ is exactly the completion $\widehat{C}$.

We can now take the global sections $\Gamma(\Mell,\co)$; this is another $\mathbf{E}_\infty$-ring – not Landweber exact (just like real K-theory from before) – that is denoted $TMF$ and is called the spectrum of topological modular forms. It turns out that this sheaf of $\mathbf{E}_\infty$-rings is defined not only on $\Mell$, but also on its ‘‘Deligne-Mumford compactification’’ $\overline{\Mell}$. This is the moduli stack of elliptic curves with a possible nodal singularity. The global sections $\Gamma(\overline{\Mell},\co)$ is denoted $Tmf$, and is also called the spectrum of modular forms. This is not a connective spectrum (there is a nontrivial element of $\pi_{-21} Tmf$), but we can take its connective cover to get $tmf = Tmf\langle 0\rangle$. Here is a question to which I do not know the answer.

Question: Is there a stack $\M$ with a sheaf $\co$ of $\mathbf{E}_\infty$-rings (in the etale topology) such that $tmf = \Gamma(\M,\co)$?

I’m debating whether I should now try to understand the construction of the sheaf $\co$ on $\Mell$, or if I should take the theorem for granted. Haynes recommended reading Behrens’ The construction of $tmf$, but it seems very technical. I might leave as reading for a later time. So, next time, we will talk about Lurie’s theorem, and get into some other explicit examples of derived algebraic geometry.