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.)
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?
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.