# Drinfel'd level structures

There seem to be two different definitions of Drinfel’d level structures that exist – they don’t conflict, but one’s a special case of the other. The goal of this post is to try to understand what’s going on.

(This post was modified a little after it was posted, to reflect my improved understanding gained by chatting with Jeremy Hahn.)

# Why?

My selfish motivation for trying to understand this
number-theoretic construction arises from the existence
of the *Lubin-Tate tower*; this is
the tower .
Each map is etale after inverting ,
and, as mentioned above, there is an action
of on , and the fixed points are .
On each level, there is also an action of the
Morava stabilizer group .

In derived algebraic geometry, one can construct the *derived Lubin-Tate space*, denoted , as was described in the previous post.
The motivating question behind trying to understand Drinfel’d level structures was:

**Question:** Can we lift the Lubin-Tate tower defined above to the world of derived algebraic geometry?

# The specifics

As always, fix a prime . Let be a complete local Noetherian -algebra. Let be a formal group law over of height ; we may equip with the structure of a group given by the . Let be a positive integer.

**Definition:** A *Drinfel’d level -structure* on is a
homomorphism of -modules such that

where denotes the -series of .

We can now study a generalization of the Lubin-Tate deformation problem: let denote the Honda formal group law over (this vague phrasing that I found in other papers has caused me and Marc a bunch of problems – what does this mean? Fix a particular Honda fgl; but how do I write this down? I should probably see Hazewinkel’s book). Let be the category of complete local Noetherian -algebras, and define a functor on by sending

**Theorem (Drinfel’d):** The moduli problem is representable by a complete local ring .
There is a map that displays as a finite flat -module.
Explicitly, if form the standard basis for
and is the universal level -structure,
we have an (noncanonical?) isomorphism (for )

There is a -action on this ring, and .

Questions begin popping up when you try to write down the first few stages. Clearly the condition of being a level -structure is vacuous, so we know, by Lubin-Tate theory, that (noncanonically)

# Shattering our dreams

Suppose now that , to simplify the discussion. What is the ring ? This depends on the choice of the universal deformation’s formal group law, and our choice of the Honda formal group law! (If and are isomorphic via , then .) Therefore, let us pick the formal group law over , so that in our case, , and the universal deformation is , this time over . The -series is and therefore

**Question:** Jeremy tells me that I should instead be saying
, and I’m unable to see why.
(If we were considering the -typification of the multiplicative formal group,
then you should be quotienting out by – but what is this, explicitly?
When this is just , but the prime is not representative of primes in general.)

Note that there is no lift of Frobenius on – if it existed, we would have . When , we have for some integer . (How do we know that ?) The existence of an integral lift of Frobenius then asserts that divides , which is impossible. This shatters our dreams from the motivation, namely of constructing a derived lift of the Lubin-Tate tower – indeed, if such an -ring existed, then we would have it would be -local, and hence its (which is ) would have a lift of Frobenius, which we just showed is impossible! This is a good example that illustrates how rigid -rings are in terms of the structure that exists on their homotopy groups.

Now, let .
In this case, we have .
Inductively, it is easy to see that ,
and since ,
we know that .
In other words, this is ,
where, we remark again, denotes *any* nontrivial th root of unity, not necessarily primitive.
It follows from the same argument as above that there is no -local -ring with given by .

**Question:** Is this true for any height and any ?
I.e., is it true that **no** Drinfel’d level -structure
lifts to a -local -ring if (at height , of course)?

This is an interesting project that I’m trying to work on.