TITLE: Existence of fine moduli space for curves and elliptic curves
QUESTION [13 upvotes]: For the moduli problem of a curve of genus $g$ with $n$ marked points, how large an $n$ is needed to ensure the existence of a fine moduli space? For this question, terminology is that of Mumford's GIT.
For the following three moduli problems, how big an $N$ is required for existence of a fine moduli space? The terminology is from the exposes of Deligne-Rapoport and Katz-Mazur, or Shimura. The first is in French, the second is too big, and the third is using old language and never mentions the modern terminology of universal elliptic curve, etc.. Therefore it is not possible for me to dig up the information myself.

i) Elliptic curves equipped with a cyclic subgroup of order $N$ -- this moduli problem corresponds to the modular group $\Gamma_0(N)$.
ii) Elliptic curves equipped with a point of order $N$ -- this moduli problem corresponds to the modular group $\Gamma_1(N)$.
ii) Elliptic curves equipped with a symplectic pairing on $N$-torsion points -- this moduli problem corresponds to the modular group $\Gamma (N)$.
References other than the above, will be appreciated.

REPLY [15 votes]: If you want to work over a base ring such as $\mathbf{Z}[1/n]$ rather than over $\mathbf{Q}$ or $\mathbf{C}$ then the relevant numerical condition is that the part of $N$ coprime to $n$ not be "too small" in the $\Gamma_1$ and full level cases.  For an extreme example, if $N$ is a $p$-power and you work over $\mathbf{Z}_{(p)}$ then you'll always have problems in characteristic $p$ at the supersingular points.
On the other hand, if you're willing to go beyond schemes and work with algebraic spaces or Deligne-Mumford or Artin stacks then these issues go away (at the expense of more technical background) in the sense that one has a reasonable "moduli space" over $\mathbf{Z}$ with nice regularity properties for all $N$ (even incorporating degenerations in the sense of generalized elliptic curves with level structure). It has better properties than a coarse moduli space (aside from perhaps not being a scheme...).