The $a$-number, $p$-rank and Cartier points of genus 4 curves

We study genus $4$ curves over finite fields and two invariants of the $p$-torsion part of their Jacobians: the $a$-number ($a$) and $p$-rank ($f$). We collect and analyze statistical data of curves over ${\mathbb{F}}_p$ for $p=3,5,7,11$ and their invariants. Then, we study the existence of Cartier points, which are also related to the structure of $J[p]$. For curves with $0\le a<g$, the number of Cartier points is bounded, and it depends on $a$ and $f$.