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

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.

Keywords

Cartier operator, Cartier–Manin matrix, Hasse–Witt matrix, Artin–Schreier curve, trigonal curve, $a$-number

2010 Mathematics Subject Classification

11G20, 14F40, 14H05, 15B33

Full Text (PDF format)

Received 25 November 2019

Received revised 27 September 2020

Accepted 6 October 2020

Published 4 May 2021