$\newcommand{\LT}{\mathrm{LT}}$ 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. $\newcommand{\Z}{\mathbf{Z}}$ 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?

$\DeclareMathOperator{\spf}{Spf}$ My selfish motivation for trying to understand this number-theoretic construction arises from the existence of the Lubin-Tate tower; this is the tower $\cdots\to\spf R_k\to\cdots\to \spf R_1\to \LT_n$. $\newcommand{\GL}{\mathrm{GL}}$ Each map is etale after inverting $p$, and, as mentioned above, there is an action of $\GL_n(\Z/p^k)$ on $R_k$, and the fixed points are $\LT_n$. $\newcommand{\GG}{\mathbf{G}}$ On each level, there is also an action of the Morava stabilizer group $\GG_n$.

In derived algebraic geometry, one can construct the derived Lubin-Tate space, denoted $\spf E$, 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

$\newcommand{\m}{\mathfrak{m}}$ As always, fix a prime $p$. $\newcommand{\FF}{\mathbf{F}}$ Let $(R,\m)$ be a complete local Noetherian $W(\FF_{p^n})$-algebra. Let $f(x,y)$ be a formal group law over $R$ of height $n$; we may equip $\m$ with the structure of a group given by the $f(x,y)$. $\newcommand{\dX}{\mathfrak{X}}$ Let $k$ be a positive integer.

Definition: A Drinfel’d level $p^k$-structure on $f(x,y)$ is a homomorphism of $\Z_p$-modules $\phi:(\Z/p^k)^n \to \m$ such that

where $[p](x)$ denotes the $p$-series of $f(x,y)$.

$\newcommand{\M}{\mathcal{M}}$ We can now study a generalization of the Lubin-Tate deformation problem: let $H_n$ denote the Honda formal group law over $\FF_{p^n}$ (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). $\newcommand{\cc}{\mathcal{C}}$ Let $\cc$ be the category of complete local Noetherian $W(\FF_{p^n})$-algebras, and define a functor $\M_k$ on $\cc$ by sending

Theorem (Drinfel’d): The moduli problem $\M_k$ is representable by a complete local ring $R_k$. There is a map $\M_k \to \M_{k-1}$ that displays $R_k$ as a finite flat $R_{k-1}$-module. Explicitly, if $a_1,\cdots,a_n$ form the standard basis for $(\Z/p^k)^n$ and $\phi$ is the universal level $p^{k-1}$-structure, we have an (noncanonical?) isomorphism (for $k>1$)

There is a $\GL_n(\Z/p^k)$-action on this ring, and $R_k^{\GL_n(\Z/p^k)} \simeq R_0 = \Gamma(\LT_n,\mathcal{O}_{\LT_n})$.

$\newcommand{\Eoo}{\mathbf{E}_\infty}$ Questions begin popping up when you try to write down the first few stages. Clearly the condition of being a level $p^0$-structure is vacuous, so we know, by Lubin-Tate theory, that (noncanonically)

# Shattering our dreams

Suppose now that $n=1$, to simplify the discussion. What is the ring $R_1$? This depends on the choice of the universal deformation’s formal group law, and our choice of the Honda formal group law! (If $f(x,y)$ and $g(x,y)$ are isomorphic via $h$, then $h([p]_f(x)) = [p]_g(h(x))$.) Therefore, let us pick the formal group law $x+y+xy$ over $\FF_p$, so that in our case, $R_0 = \Z_p$, and the universal deformation is $x+y+xy$, this time over $\Z_p$. The $p$-series is $[p](x) = (1+x)^p - 1$ and therefore $R_1 = \Z_p[x]/((1+x)^p - 1)/x \simeq \Z_p[y]/((y^p - 1)/(y-1)) \simeq \Z_p[\zeta_p].$

Question: Jeremy tells me that I should instead be saying $R_1 \simeq R_0[x]/(p + x^{p-1})$, and I’m unable to see why. (If we were considering the $p$-typification $g(x,y)$ of the multiplicative formal group, then you should be quotienting out by $px +_g x^p$ – but what is this, explicitly? When $p=2$ this is just $2x + x^2$, but the prime $2$ is not representative of primes in general.)

Note that there is no lift of Frobenius $\psi^p$ on $\Z_p[\zeta_p]$ – if it existed, we would have $\psi^p(x) = x^p + p\theta(x)$. When $x = \zeta_p$, we have $\psi^p(x) = \zeta_p^a$ for some integer $a$. (How do we know that $a\neq p$?) The existence of an integral lift of Frobenius then asserts that $p$ divides $(1-\zeta_p^a)$, 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 $\Eoo$-ring existed, then we would have it would be $K(1)$-local, and hence its $\pi_0$ (which is $\Z_p[\zeta_p]$) would have a lift of Frobenius, which we just showed is impossible! This is a good example that illustrates how rigid $\Eoo$-rings are in terms of the structure that exists on their homotopy groups.

Now, let $k>1$. In this case, we have $R_k\simeq R_{k-1}[x]/([p]_{R_{k-1}}[x] - \phi(1))$. Inductively, it is easy to see that $\phi(1) = \zeta_{p^{k-1}} - 1$, and since $[p]_{R_{k-1}}[x] = (1+x)^p - 1$, we know that $R_k\simeq R_{k-1}[x]/((1+x)^p - \zeta_{p^{k-1}})\simeq R_{k-1}[y]/(y^p - \zeta_{p^{k-1}})$. In other words, this is $\Z_p[\zeta_{p^k}]$, where, we remark again, $\zeta_n$ denotes any nontrivial $n$th root of unity, not necessarily primitive. It follows from the same argument as above that there is no $K(1)$-local $\Eoo$-ring with $\pi_0$ given by $R_k$.

Question: Is this true for any height and any $k$? I.e., is it true that no Drinfel’d level $p^k$-structure lifts to a $K(n)$-local $\Eoo$-ring if $k>0$ (at height $n$, of course)?

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