Families of supersingular curves in characteristic 2

This paper determines normal forms of all hyperelliptic supersingular curves of genus $g$ over an algebraically closed field $F$ of characteristic $2$ for $1\leq g\leq 8$. We also show that every hyperelliptic supersingular curve of genus $9$ over $F$ has an equation $y^2-y=x^{19}+c^8x^9+c^3x$ for some $c\in\overline\mathbb{F}_2$. Consequently, the paper determines the dimensions of the open locus of hyperelliptic supersingular curves of genus $g\leq 9$ over $\overline\mathbb{F}_2$.

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Published 1 January 2002