Let $N \ge 1$ be an integer and $\mathscr{M}_*(N)$ the stack of elliptic curves with the level $\Gamma(N), \Gamma_0(N), \Gamma_1(N)$ or $\Gamma_{\text{bal.}1}(N)$. (For its definition see Katz-Mazur.)

Then there is the coarse moduli scheme of $\mathscr{M}_*(N)$, denoted by $ Y = Y_*(N)$, over $\mathbb{Z}$, and it is a connected regular affine scheme of pure dimension 2, and is flat of finite type of relative dimension 1.

Moreover, $Y \times \mathbb{Z}[1/N]$ is smooth over $\mathbb{Z}[1/N]$.

I showed this proposition except the connectedness. I want to show it.

Here is what I've tried:

Since $Y \times \mathbb{Q} \to Y$ has the dense image, it sufficies to show that $Y \times \mathbb{C}$ is connected.
And so since the Euclid toplogy is finer than the Zariski topology, it sufficies to show that the complex manifold $Z$ induced by $Y \times \mathbb{C}$ (i.e., the closed submanifold induced by $Y(\mathbb{C}) \subseteq \mathbb{A}^N(\mathbb{C}) = \mathbb{C}^N$.) is connected.

Next,it seems that, as Riemann surfaces, $Z \cong \mathbb{H}/ \Gamma_*(N)$, at least for $* = 0, 1, \varnothing$. (Where $\mathbb{H}$ is the upper half plane.) If so, trivially $Z$ is connected, and so $Y$ is connected.

So it sufficies to show $Z \cong \mathbb{H}/ \Gamma_*(N).$

The "coarse moduli map" $\mathscr{M}_*(N) \to Y$ induces $|\mathscr{M}_*(N)(\mathbb{C})| $ $\cong Y(\mathbb{C}) \cong Z$.
On the other hand, by a fundamental proposition of modular forms, we have $|\mathscr{M}_*(N)(\mathbb{C})| \cong \mathbb{H}/\Gamma_*(N)$.

So as sets, we have $Z \cong \mathbb{H}/ \Gamma_*(N)$.

It's hard for me to show this map is holomorphic (or even it is continuous, which is enough in order to show $Z$ is connected), because I cannot understand the topology on $Z$ "moduli theoritically". (i.e., the interpretation such as "$U \subseteq Z$ is open iff elliptic curves in $U$ are such-and-such...")

P.S.
I showed other properties as follows:

First, the separated Deligne-Mumford stack $\mathscr{M}_*(N)$ is regular of dimension $2$ at every closed point, and is affine flat of relative dimension 1 over $\mathbb{Z}$ of finite type.
(By (5.1.1) of Katz-Mazur.)

Next, for $r \ge 1$, the diagram $\require{AMScd}$ \begin{CD} \mathscr{M}_{*, \varnothing} (N, r)[1/r] @>>> \mathscr{M}(r)[1/r] \\ @VVV @V{ \text{etale finite}}VV\\ \mathscr{M}_*(N)[1/r] @>{ \text{finite flat}}>> \mathscr{M}[1/r] \end{CD} is 2-Cartesian, and $\mathscr{M}(r)[1/r]$ is a regular scheme. So $\mathscr{M}_{*, \varnothing} (N, r)[1/r]$ is also a regular scheme, $T$.

It's easy to show that $(|\mathscr{M}_{*, \varnothing} (N, r)[1/r]| / \operatorname{GL}_2(\mathbb{Z}/r) )^\# \to |\mathscr{M}_*(N)[1/r]|^\#$ is isomorphism. (Where $|\mathscr{X}|$ is the functor taking $S$ to the isomoprhism classes of $\mathscr{X}(S)$, and $\mathscr{F}^\#$ is the sheafification of $\mathscr{F}$.)

So $T/ \operatorname{GL}_2(\mathbb{Z}/r)$ is the coarse moduli scheme over $\mathbb{Z}[1/r]$.
And since this is the quotient by etale finite group scheme, it inherits many properties of $T$.

Therefore, patching these schemes for relatively prime $r, r'$, we have the result.

I'm not familiar with the stack theory, so if my "proof" is wrong, please correct it.

And can I show the connectedness stack-theoritically? For example, by 4.14 of Deligne-Mumford, if $\mathscr{M}_*(N) \times \mathbb{C}$ is connected as a stack, then it seems that the coarse moduli scheme is connected as a scheme. (But I can't show the stack is connected.)

Thank you very much!