# Action of the stabilizer group on the completed local ring

Let $$\mathcal{M}$$ be a DM stack and $$M$$ its coarse moduli space. Then we have the map $$\mathcal{M} \to M$$. Choose a geometric point $$x \in M$$ and a geometric point $$\bar{x} \in \mathcal{M}$$ which lies over $$x$$. Denote by $$\Gamma_{\bar{x}}$$ the stabilizer group at the point $$\bar{x}$$. My question is: how to understand in general the action of $$\Gamma_{\bar{x}}$$ on the completed local ring $$\hat{\mathcal{O}_{\mathcal{M}, \bar{x}}}$$?

For example, if $$\mathcal{M}$$ is the moduli stack of elliptic curves over some field $$k$$, then $$\bar{x}$$ corresponds to some elliptic curve and $$\Gamma_{\bar{x}}$$ is its group of automorphisms. Let's say $$\bar{x}$$ corresponds to $$y^2=x^3-x$$ in which case $$\Gamma_{\bar{x}} = \mathbb{Z}/6$$. I think I should find some etale covering by a scheme to be able to compute anything. For example, I can take the Legendre family and it gives me an etale map $$\operatorname{Spec} k[x][\frac{1}{x(x-1)}] \to \mathcal{M}$$. Thus, I'm interested in the action of $$\mathbb{Z}/6$$ on the completed local ring of $$k[t][\frac{1}{t(t-1)}]$$ at $$t=-1$$ which is isomorphic to $$k[[x]]$$. However, I'm struggling to figure out how to write it down.

I think I also have the map $$k[[j]] \to k[[t]]$$ given by the $$j$$-invariant which should identify $$k[[j]]$$ with the ring of invariants $$k[[t]]^{\mathbb{Z}/6}$$ however I don't see how to write down the action.

In general for any $$x \in M$$, there exists a pointed affine scheme $$(U,u)$$ and an action $$\Gamma_\bar{x}$$ on $$U$$ fixing $$u$$ and an étale map $$[U/\Gamma_\bar{x}] \xrightarrow{f} \mathcal{M}$$ such that $$f(u) = x$$. Then it suffices to compute the $$\Gamma_\bar{x}$$ on $$\mathcal{O}_{U,u}$$.

If $$\mathcal{M}$$ is presented as a global quotient $$[V/G]$$ where $$G$$ is reductive, then for every closed point $$v \in V$$, the Luna Slice Theorem allows one to construct a slice $$(U,v)$$ and an action of the stabilizer $$G_v$$ such that $$[U/G_v]$$ gives us such a chart.

This is the case for $$\overline{\mathcal{M}}_{1,1}$$ (away from characteristic $$2,3$$) which can be presented as the quotient $$[\mathbb{A}^2/\mathbb{G}_m]$$ with the action $$g(a,b) = (g^4a, g^6b)$$ and universal family $$y^2 = x^3 - ax -b$$. The point you want is $$(1,0)$$ which has stabilizer $$\mu_4$$ the group of fourth roots of unity. A Luna slice is given by $$\{(1,t)\}$$ with family $$y^2 = x^3 - x - t$$. The action (as it's induced by the restriction of the $$G$$ action on $$V$$) is simply $$t \mapsto \zeta_4^6t= \zeta_4^2t$$ and the completed local ring is $$k[[t]]$$ with this action of $$\mu_4$$.

It's not hard to see using the same method that at the point classifying $$y^2 = x^3 - 1$$, $${(t,1)}$$ us a Luna slice and $$t \mapsto \zeta_6^4t$$, and at any other point the automorphism group $$\mu_2$$ acts trivially.

If the stack isn't given to you as a global quotient I don't see any general way to answer the question. You just need to compute a versal deformation and its automorphisms. One observation that helps when $$\mathcal{M}$$ is smooth though is that in this case we can arrange it so that there is an étale map $$[U/\Gamma_\bar{x}] \to [T_x/\Gamma_\bar{x}]$$ and so the question reduces to understanding the linear action of $$\Gamma_\bar{x}$$ on the tangent space. For example for the moduli of curves, we need to understand how $$\mathrm{Aut}(C)$$ acts on $$H^1(C,T_C)$$ which is not much more explicit but at least reduces the question to a cohomology computation.

• It's a great answer! So the main idea is that for stacks of the form [X/G] it's doable and I need to reduce the problem to that case. In particular, it follows that for a general elliptic curve with Aut=Z/2 the action is trivial. However, I have a question: this presentation for the moduli stack of elliptic curves works only away of primes 2, 3. Is there any presentation as a global quotient in char. 2 or 3? (since it's basically the main example I need..) Also, do you have a reference for your last paragraph?
– iou
Commented Aug 8, 2022 at 17:16
• @iou I believe there is a global quotient presentation over $\mathrm{Spec} \mathbb{Z}$. I don't remember it off the top of my head but you can produce it by writing down the long Weierstrass equation and keeping track of the change of variables that give you isomorphic elliptic curves. I believe the group $G$ in that case is an extension of $\mathbb{G}_m$ by a unipotent group. In particular it is not reductive so I think you can't apply Luna Slice Theorem directly. I'm sure there is literature on this, and you only need a weak form of the slice theorem to produce these étale neighborhoods. Commented Aug 9, 2022 at 22:08