One may see the modular interpretation of (points of) modular curves in the very first course on modular forms and modular curves. I am wondering if it is well-known that modular interpretation of the stalks of the structure sheaf at points of modular curves, i.e. if $Y$ is an open modular curve parameterising elliptic curves with some level structures and $x$ is a closed point, what can we say about $\mathcal{O}\_{Y,x}$?



Modular interpretation of the stalks of modular curves
I guess I'll make my comment into an answer so this question is no longer marked unanswered.


If the modular curve corresponds to a torsion-free subgroup, then the completion of the etale local ring at $x$ is the universal deformation ring of the object corresponding to $x$. In general, the completion of the etale local ring at a geometric point of a Deligne Mumford stack is the universal deformation ring at that point. The intuition for this is that the unique infinitesimal lifting property of etale morphisms exactly corresponds to the pro-representability property for universal deformation rings.


Thus as long as the etale local ring (i.e., strict henselization of the local ring) at $x$ is the same as the etale local ring of the moduli stack at $x$, then the same statement holds. For example, this is the case if $x$ does not lie above $0,1728$.


