Let $M_g$ be the moduli space of smooth projective geometrically connected curves over a field $k$ with $g\geq 2$.

Does $M_g$ contain an elliptic curve?

The answer is no if $g=2$. In fact, $M_2$ doesn't contain any complete curves.

Probably for $g>>0$ and $k$ algebraically closed, the answer is yes.

What if $k$ is a number field?

Let $M_g$ be the moduli space of smooth projective geometrically connected curves over a field $k$ with $g\geq 2$. Note that $M_g$ is not complete.

Does $M_g$ contain an elliptic curve?

The answer is no if $g=2$. In fact, $M_2$ doesn't contain any complete curves.

Note that one can construct complete curves lying in $M_g$ for $g\geq 3$. There are explicit constructions known.

Probably for $g>>0$ and $k$ algebraically closed, the answer is yes.

What if $k$ is a number field?