Genus 2 Curves in Small Characteristic

TL;DR

This paper investigates the sizes of stratification strata of genus 2 curves over finite fields with small characteristic, providing explicit formulas for certain cases and computational evidence for their validity.

Contribution

It derives explicit formulas for the sizes of non-ordinary and ordinary strata of genus 2 curves over finite fields of small characteristic, supported by computational data.

Findings

Supersingular stratum size is q in characteristic 2 and 3.

Non-ordinary and ordinary strata sizes are q(q-1) and q^2(q-1) for q=3^r.

Formulas hold for p ≤ 7 and break down for p > 7.

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 $p > 7$.