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arXiv · 2111.07270

Genus 2 Curves in Small Characteristic

Lukas Zobernig
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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 $p \leq 7$ and break down for $p > 7$.

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Lukas Zobernig. 2021-11-14. Genus 2 Curves in Small Characteristic. https://arxiv.org/abs/2111.07270

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