To let $ N ge 1 $ be an integer and $ mathscr {M} _ * (N) $ the stack of elliptic curves with the plane $ Gamma (N), Gamma_0 (N), Gamma_1 (N) $ or $ Gamma _ { text {bal.} 1} (N) $,

(See Katz-Mazur for its definition.)

Then there is the rough module scheme of $ mathscr {M} _ * (N) $, denoted by $ Y = Y _ * (N) $, About $ mathbb {Z} $and it is a coherent regular affine scheme of pure dimension 2 and flat of the finite type of relative dimension 1.

in addition, $ Y times mathbb {Z} (1 / N) $ is smooth over $ mathbb {Z} (1 / N) $,

I've shown this sentence with the exception of connectedness.

I want to show it.

I tried the following:

Since $ Y times mathbb {Q} to Y $ has the dense picture, it is enough to show that $ Y times mathbb {C} $ Is connected.

And since the Euclid topology is finer than the Zariski topology, it is enough to show that the complex diversity $ Z $ caused by $ Y times mathbb {C} $ (i.e. the closed sub-diversity induced by $ Y ( mathbb {C}) subseteq mathbb {A} ^ N ( mathbb {C}) = mathbb {C} ^ N $.) Is connected.

Next it looks like Riemann $ Z cong mathbb {H} / Gamma _ * (N) $, At least for $ * = 0, 1, varnothing $,

(Where $ mathbb {H} $ is the upper half-plane.)

If so, trivial $ Z $ is connected, and such $ Y $ Is connected.

So it is enough to show $ Z cong mathbb {H} / Gamma _ * (N). $

The "rough module card" $ mathscr {M} _ * (N) to Y $ induced $ | mathscr {M} _ * (N) ( mathbb {C}) | $ $ cong Y ( mathbb {C}) cong Z $,

On the other hand, we have a basic set of modular shapes $ | mathscr {M} _ * (N) ( mathbb {C}) | cong mathbb {H} / Gamma _ * (N) $,

So as sets we have $ Z cong mathbb {H} / Gamma _ * (N) $,

It's hard for me to show that this map is holomorphic (or even continuous, which is enough to show) $ Z $ is connected) because I can't understand the topology on $ Z $ "Modules theoretically".

(i.e. the interpretation like "$ U subseteq Z $ is open when elliptic curves in $ U $ are so and so … ")

Thanks a lot!