  In this article, we construct codes with hierarchical locality using natural
geometric structures in Artin-Schreier surfaces of the form $y^p-y=f(x,z)$. Our
main theorem describes the codes, their hierarchical structure and recovery
algorithms, and gives parameters. We also develop a family of examples using
codes defined over $\mathbb{F}_{p^2}$ on the surface
$y^p-y=x^{p+1}z^2+x^2z^{p+1}$. We count the $\mathbb{F}_{p^2}$-rational points
on the surface, a topic of more general number theoretic interest, and provide
more explicit parameters a better bound on minimum distance for these codes. An
additional example and some generalizations are also considered.
