# Formal completion of modular curves

Let $$N\geq 4$$ be an integer coprime with $$p$$, where $$p$$ is a fixed prime number. Then we know that there exists a scheme $$Y_N$$ over $$\text{Spec}(\mathbb{Z}_p)$$ whose $$\text{Spec}(R)$$-points are elliptic curves over $$R$$ with a level $$N$$-structure, where $$R$$ is a $$\mathbb{Z}_p$$-algebra. $$Y_N$$ is the non-compactified modular curve of level $$\Gamma_1(N)$$. We also know, by representability, that there exists a universal elliptic curve $$\mathcal{E}$$ over $$Y_N$$. My question is the following. What does it happen if we now $$p$$-adically complete the picture, i.e. if we consider the formal scheme $$\mathcal{Y}_N$$ over $$\text{Spf}(\mathbb{Z}_p)$$ given by completion, does it still have a moduli interpretation? Of course we can complete also $$\mathcal{E}$$, but now it's no more an elliptic curve, it's a kind of formal elliptic curve. Is it still true that a morphism $$\text{Spf}(R)\rightarrow\mathcal{Y}_N$$ gives an elliptic curve (formal or real?) over $$R$$ with a $$N$$-level structure?

• This is a great question but might be too advanced for this site. I suggest you try your luck on MathOverflow. – David Loeffler Oct 4 '18 at 6:55
• Thank you David. By the way, I think I found a partial answer to my question. If we consider $R$ to be a Noetherian complete $\mathbb{Z}_p$-algebra maybe we can use GAGA to conclude that the base change of the universal elliptic curve over $R$ is in fact algebrizable, by the properness of $E$. What do you think? Btw, maybe I should move to Overflow, as you suggested! – Zariski93 Oct 14 '18 at 16:41
• If you want to move to MO, just ask it there. Either delete here, or add links. Both is fine. "Migration" in the technical sense is possible but not a good option in my opinion. – quid Oct 15 '18 at 12:30