Are there analogues of the $j$-invariant for higher genus curves?


Of course for genus $g \ge 2$ there will have to be at least $3g-3$ invariants. Are there "canonical choices" for these invariants, viewed as functions on the coarse moduli schemes? (ideally with relatively nice arithmetic properties)


I would appreciate any references on this. Googling seemed to only yield results for genus 2 curves.



Invariants of higher genus curves
For $g > 2$ you're not going to be able to write such invariants as *global* functions on the moduli space. 


In general for an algebraic variety $X$, there is a canonical morphism $a \colon X \to \mathrm{Spec}(H^0(X,\mathcal O\_X))$, the *affinization* of $X$. Points $x$ and $y$ of $X$ are separated by some global regular function if only if $a(x) \neq a(y)$. So your question can only have a positive answer if the affinization morphism is injective (so $M\_g$ is affine or quasi-affine). Now $M\_2$ is affine, but $M\_g$ for $g >2$ contains a complete curve. Since a regular function restricted to a complete subvariety is constant, this means that $a$ has positive dimensional fibers. In fact one can prove using the Satake compactification that through any two points of $M\_g$ passes a complete curve, if $g>2$. So the affinization is in fact a point.


This is also why in the $g=3$ case of David Lehavi's answer there are only such choices of invariants on the hyperelliptic locus and on the non-hyperelliptic locus separately. Both loci are affine, but their union is not.


It also seems relevant to mention that $M\_g$ is of general type for $g \gg 0$, so that there will be no rational parametrization of the set of genus $g$ curves. And it also seems relevant to mention Fenchel--Nielsen coordinates: these will give you local real-analytic coordinates around any point of the moduli stack in any genus.


