One of my favorite questions is: for which $g, n, p$ is the moduli space of $n$-pointed genus $g$ curves $\mathscr{M}_{g,n, \mathbb{F}_p}$ unirational/uniruled?  Will Sawin has just posted a beautiful paper on the ArXiv answering this question in most cases, for $g=1$.  Indeed, he shows that for $n\geq p\geq 11, \mathscr{M}_{1, n, \mathbb{F}_p}$ is not uniruled... (more below the fold)