Genus 2 Curves in Small Characteristic

Abstract

We study genus 2 curves over finite fields of small characteristic. The $p$-rank $f$ of a curve induces a stratification of the coarse moduli space ${\mathcal{M}}_2$ of genus 2 curves up to isomorphism. We are interested in the size of those strata for all $f\in \{0,1,2\}$. In characteristic 2 and 3, previous results show that the supersingular $f=0$ stratum has size $q$. We show that for $q=3^r$, over ${\mathbb{F}}_q$ the non-ordinary $f=1$ and ordinary $f=2$ strata are of size $q(q-1)$ and $q^2(q-1)$, respectively. We give results found from computer calculations which suggest that these formulas hold for all $p\le 7$ and break down for $p>7$.

8 pages

References in corpus (1)

• Hasse-Witt and Cartier-Manin matrices: A warning and a request