HYPERELLIPTIC CURVES WITH a-NUMBER 1 IN SMALL CHARACTERISTIC

Abstract

For every $g\ge 3$, we show there exists a hyperelliptic curve of genus $g$ with $p$-rank $g-3$ and $a$-number $1$ in characteristic $p$ when $p=3$ or $p=5$. The method of proof is to show that a generic point of the moduli space of hyperelliptic curves of genus $3$ and $p$-rank $0$ has $a$-number $1$. When $p=3$, we also show that this moduli space is irreducible.