On the existence of curves with prescribed $a$-number

Abstract

We study the existence of Artin–Schreier curves with large $a$‑number. We show that Artin–Schreier curves with large $a$‑number can be written in certain forms and discuss their supersingularity. We also give a basis of the de Rham cohomology of Artin–Schreier curves. By computing the rank of the Hasse–Witt matrix of the curve, we also give bounds on the $a$‑number of trigonal curves of genus $5$ in small characteristic.