A tame stacky curve over a field $k$ is a geometrically connected proper smooth DM stack of dimension 1 which has a dense open substack which is a scheme, and whose automorphism group of each geometric point has order prime to the characteristic of $k$.

I've heard that the theory of stacky curves and one of M-curves are the same (e.g., Poonen, Schaefer, and Stoll - Twists of $X(7)$ and primitive solutions to $x^2+y^3=z^7$). But I essentially don't understand anything about it.

Fix a field $k$.

Let $\mathscr{X}$ be a tame stacky curve. Then the pair $(X; \{ m_P \})$ is a M-curve (see Darmon - Faltings plus epsilon, Wiles plus epsilon, and the Generalized Fermat Equation), where $X$ is its coarse moduli (which is a smooth scheme curve) and $P$ runs over the set of stacky point of $\mathscr{X}$, and $m_P$ is the order of the automorphism group of $P$.

Now consider a morphism $f : \mathscr{X} \to \mathscr{Y}$ of tame stacky curves over $k$. Then it induces a morphism of the coarse moduli schemes $X \to Y$.

**Question 1. Is this a morphism of M-curves?**

Next, conversly consider a morphism $f : X \to Y$ of the coarse moduli schemes, which is a morphism of M-curves.

**Question 2. Does this induce the morphism $\mathscr{X} \to \mathscr{Y}$?**

Glancing through the proof of the 5.3.10.a of Voight and Zureck-Brown - The canonical ring of a stacky curve and its reference Geraschenko and Satriano - A "bottom up" characterization of smooth Deligne–Mumford stacks, it seems (although I understand nothing about it) that $\mathscr{X}$ is the root stack $\sqrt{D/X}$, where $D$ is a ramification divisor. (I don't know its definition, but it seems to be $\sum m_P P$, where $P$ runs over the all stacky point of $\mathscr{X}$, and identify it with the closed point on $X$.)

So using the universal property of the root stack, it seems that the property that $f$ is a morphism of M-curves leads the morphism of stacky curves.

But I can't find a good diagram relating $X$, $Y$, $[\mathbb{A}^1/\mathbb{G}_m]$ and their ramification divisors.

Next consider an M-curve $X$.

Question 3. Does $X$ induce a tame stacky curve $\mathscr{X}$?

Similar to the question 2, it seems that the root stack of $X$ with respect to the divisor is the one, but I can't find any references.

Next, assume that $k$ is a number field, and $S$ a nice finite set of places of $K$.

**Question 4. What does the $S$-integral points of the M-curve induced by $\mathscr{X}$ correspond to?
(For the definition, see Darmon - Faltings plus epsilon, Wiles plus epsilon, and the Generalized Fermat Equation.)**

It seems for me that, fixing a "model" of $\mathscr{X}$ over $O_{k, S}$ (i.e., a smooth proper DM stack over $O_{k, S}$ whose fibres are tame stacky curves), considering the $S$-integral points are the same to considering the set of isomorphism classes of $\mathscr{X}(O_{k, S})$.

Finally, consider a morphism $f : \mathscr{X} \to \mathscr{Y}$ of tame stacky curves (over general field $k$).

**Question 5. What is the ramification index of $f$?**

I can't find any references which define the ramification indeces of a morphism of stacky curves, but I think that it is the one of a morphism $U \to V$, where $V$ is an etale covering of $\mathscr{Y}$, $U$ is of $\mathscr{X}$, and the map $U \to V$ makes the diagram commutative.

But are these concept about stacky curves and about M-curves equivalent?

Because I'm studying this field for Poonen, Schaefer, and Stoll - Twists of $X(7)$ and primitive solutions to $x^2+y^3=z^7$ the M-curve version is sufficient for me (at least now), however I want to know this correspondence.

Any help will be much appreciated!