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.

中文翻译：

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关于具有规定的$ a $数的曲线的存在

我们研究了$ a $值较大的Artin–Schreier曲线的存在。我们证明，可以用某些形式写出具有大$ a $数的Artin–Schreier曲线，并讨论它们的奇异性。我们还为Artin-Schreier曲线的de Rham同调提供了基础。通过计算曲线的Hasse–Witt矩阵的秩，我们还给出了小特征类$ 5 $的三角曲线的$ a $数的界限。