The number of curves of given genus over a field Let $k$ be a field. Let $g\geq 0$ be an integer.
I have an elementary question.
Let $N$ be the "number" of $k$-isomorphism classes of smooth projective geometrically connected curves over $k$ of genus $g$. (Note that $N$ can also be $\infty$.)
Is $N$ finite if $k$ is finite?
When is $N$ finite in general?
I'm looking for the "most elementary" answer to this question.
 A: Yes, this number is finite when the field $k$ is finite. Use some version of the canonical embedding to show that your curve can be realized in some projective space by equations bounded by some degree, and then observe that there are only finitely many polynomials with a given number of variables and of given degree over $k$. For an infinite field the number of isomorphism classes is infinite as soon as $g \geq 1$: one can consider a hyperelliptic curve defined by a choice of $2g+2$ points on $\mathbf P^1$ (considered up to the action of $\mathrm{Aut}(\mathbf P^1)$), and there are infinitely many such choices when the field is infinite.
A more sophisticated approach (when $g \geq 2$) uses the existence of a moduli space of curves of given genus. If $M_g$ is the coarse moduli space of curves of genus $g$ and $k$ is a finite field, then one has the formula
$$ \# M_g(k) = \sum_{[C]} \frac{1}{\# \mathrm{Aut}_k(C)}$$
where the sum ranges over $k$-isomorphism classes of curves $C$, and clearly $\# M_g(k)$ is finite. Unfortunately the only proof of this formula that I know uses the Grothendieck-Lefschetz trace formula on the moduli stack... 
