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
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.
Let denote the moduli stack of elliptic curves, so that a map is a smooth elliptic curve over . 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 -point of a scheme has automorphisms given by elements of ). Certainly elliptic curves (even over ) have nontrivial automorphism groups (for instance, the curves with -invariants and ).
Let be an elliptic curve, classified by a map . The point at infinity gives a section , called the zero section. We can complete at the zero section (which just means looking at the infinitesimal neighborhood of the additive identity on ), to get a formal group . 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 , and the choice of such an isomorphism is the choice of coordinate on this formal group. As is a formal group, we obtain a comultiplication on , and the image of under this comultiplication is the formal group law.
This gives a map of stacks . The scrupulous reader might note that we did not use the fact that is an elliptic curve, but merely the fact that had a zero section. And that’s correct – we in fact constructed a map , where classifies cubic curves (i.e., curves with a Weierstrass form) equipped with a zero section. Inside this sits as the locus where the discriminant is invertible.
Anyway, we have the following theorem.
Theorem (Hopkins-Miller): The map is flat.
To make sense of this statement, we first have to know that the map of stacks is representable. It turns out that is representable, and the inclusion of into is also clearly representable. How do we show that the map is flat? One constructs an affine cover of , and shows that the composite is flat. From last time, this is equivalent to showing that the formal group law over corresponding to this ‘‘universal elliptic curve’’ is Landweber exact.
We first need to construct a cover of . This is not too hard to do: we can first construct a cover of . 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 is an affine cover. But in fact, we know that is presented by a Hopf algebroid! This Hopf algebroid is , where (let’s write ) the ring parametrizes isomorphisms of the universal Weierstrass cubic. These isomorphisms are given by and . This means that the moduli of elliptic curves is presented by inverting the determinant, i.e., is presented by .
We now need to show that the formal group law over is Landweber exact. The maximum height of an elliptic curve is , so we need to show that form a regular sequence. First, we know that is definitely nonzero since is torsion-free. Now, consider the ring ; this is an integral domain, and hence we need to show that is nonzero. But this means that the ‘‘universal elliptic curve’’ mod has a fiber that is nonsingular, and this is true. (In fact, Marc told me that the element is the Hasse invariant.) The result will follow if we show that is not a unit in . Suppose otherwise, so that the quotient of by the maximal ideal containing is a field of characteristic , over which the reduction of the universal elliptic curve lives. The associated formal group law would be of height greater than , which is impossible.
What is the Hopkins-Miller theorem?
By the above result, we know that any flat map gives rise to a spectrum. This is exactly the situation from the previous post, where, instead of , we were considering . Just like in that case, we have a lifting theorem: this is the Hopkins-Miller theorem, which I thought I’d at least state.
Theorem (Hopkins-Miller): There is an even-periodic sheaf of -rings on the etale site of the moduli stack of elliptic curves, such that , and the formal group associated to is exactly the completion .
We can now take the global sections ; this is another -ring – not Landweber exact (just like real K-theory from before) – that is denoted and is called the spectrum of topological modular forms. It turns out that this sheaf of -rings is defined not only on , but also on its ‘‘Deligne-Mumford compactification’’ . This is the moduli stack of elliptic curves with a possible nodal singularity. The global sections is denoted , and is also called the spectrum of modular forms. This is not a connective spectrum (there is a nontrivial element of ), but we can take its connective cover to get . Here is a question to which I do not know the answer.
Question: Is there a stack with a sheaf of -rings (in the etale topology) such that ?
I’m debating whether I should now try to understand the construction of the sheaf on , or if I should take the theorem for granted. Haynes recommended reading Behrens’ The construction of , 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.