Genus 2 Curves in Small Characteristic

Fix a prime $p$, and an algebraically closed field $k$ of characteristic $p$ containing $\mathbb{F}_{p}$. Consider the additive group scheme $\mathbb{G}_{a}=\operatorname{Spec}k[X]$ and the multiplicative group scheme $\mathbb{G}_{m}=\operatorname{Spec}k[X,X^{-1}]$. The kernel of the relative Frobenius on these yields the finite group schemes $\alpha_{p}\cong\operatorname{Spec}k[X]/X^{p}$ and $\mu_{p}\cong\operatorname{Spec}k[X]/(X^{p}-1)$, respectively. The Cartier dual of $\alpha_{p}$ is itself, and the Cartier dual of $\mu_{p}$ is the constant group scheme $\mathbb{Z}/p\mathbb{Z}$.

Let $A/k$ be an abelian variety of genus $g$ and consider its $p$-torsion $A[p]$ as a group scheme. The $p$-rank of $A$ is given by $f=\dim_{\mathbb{F}_{p}}\operatorname{Hom}(\mu_{p},A[p])$. Similarly, the $a$-number of $A$ is given by $a=\dim_{k}\operatorname{Hom}(\alpha_{p},A)$. Thus, geometrically, $A[p](k)\cong(\mathbb{Z}/p\mathbb{Z})^{f}$. It holds that $0\leq f\leq g$ and $1\leq a+f\leq g$.

We define the $p$-rank and the $a$-number of a genus g curve $C$ as the corresponding invariants of its Jacobian $\operatorname{Jac}(C)$ as an abelian variety.

Assume $\operatorname{\mathrm{char}}(k)\neq 2$ and consider a genus $g$ hyperelliptic curve $C$ defined by an equation $y^{2}=f(x)$ for $f(x)\in k[x]$ of degree $2g+1$ or $2g+2$. Let $c_{i}$ denote the coefficient of $x^{i}$ in the expansion of $f(x)^{(p-1)/2}$, and define for $\ell=0,\dots,g-1$ the $g\times g$ matrix $A_{\ell}$ with entries $(A_{\ell})_{i,j}=(c_{ip-j})^{p^{\ell}}$. Following [Yui78] and heeding [AH19] we call $A_{0}$ the Cartier-Manin matrix of the curve $C$ and define the matrix $M=A_{g-1}\cdots A_{1}A_{0}$. Then we have the following lemma.

Points in the coarse moduli space $\mathcal{M}_{2}$ of genus 2 curves up to isomorphism have various different descriptions. Consider the weighted projective space $S_{k}=\mathbb{P}(2,4,6,8,10)(k)$ over an arbitrary field $k$. [Igu60] gave a set of invariants $[J_{2}:J_{4}:J_{6}:J_{8}:J_{10}]\in S_{k}$ associated to every genus 2 curve $C$ defined over $k$. Two curves are isomorphic over $k^{\text{alg}}$ if any only if their Igusa-invariants are the same in $S_{k}$. If $\operatorname{\mathrm{char}}(k)\neq 2$, we can assume that $C$ is given by a model of the form $y^{2}=f(x)$, where $f(x)\in k[x]$ is a polynomial of degree 5 or degree 6. Then $2^{12}J_{10}$ is simply the discriminant of $f(x)$.

For fields of characteristic different from 2, Cardona, Quer, Nart, and Pujolàs [CQ05, CNP05] gave the absolute “G2” invariants

Again, two curves are isomorphic over $k^{\text{alg}}$ if any only if their absolute invariants are the same. Hence, the $k$-points $\mathcal{M}_{2}(k)$ are in bijection with the set of points $(g_{1},g_{2},g_{3})\in\mathbb{A}^{3}(k)$ as above.

In most generality, a genus 2 curve is given by an equation of the form $y^{2}=f(x)$ with $f(x)=c_{6}x^{6}+\cdots c_{1}x+c_{0}$ for coefficients $c_{0},\dots,c_{6}\in k$. After homogenising, we find $F(X,Z)=c_{6}X^{6}+\cdots c_{1}XZ^{5}+c_{0}Z^{6}$. Then the group $\operatorname{GL}_{2}$ acts on $F(X,Z)$ via the following transformation: Let $M=\begin{pmatrix}\alpha&\beta\\ \gamma&\delta\end{pmatrix}\in GL_{2}$ and let

We find $F(X,Z)=c_{6}^{\prime}X^{\prime 6}+\cdots c_{1}^{\prime}X^{\prime}Z^{\prime 5}+c_{0}^{\prime}Z^{\prime 6}=G(X^{\prime},Z^{\prime})$ for new coefficients $c_{0}^{\prime},\dots,c_{6}^{\prime}$. Hence, we may call $F(X,Z)$ and $G(X^{\prime},Z^{\prime})$ to be $\operatorname{GL}_{2}$-conjugates. Let $g(x^{\prime})=c_{6}^{\prime}x^{\prime 6}+\cdots c_{1}^{\prime}x^{\prime}+c_{0}^{\prime}$ be the dehomogenisation of $G(X^{\prime},Z^{\prime})$. Then the two curves $C:y^{2}=f(x)$ and $C^{\prime}:y^{\prime 2}=g(x^{\prime})$ are isomorphic via

This is actually an if and only if relation: Two genus 2 curves $C,C^{\prime}$ are isomorphic if any only if their associated sextic forms are conjugate under $\operatorname{GL}_{2}$-action. Thus we can also study points in the moduli space $\mathcal{M}_{2}$ by considering sextic forms up to $\operatorname{GL}_{2}$-conjugacy, see [Mes91].

[MN07], and [How08] studied supersingular genus 2 curves in characteristic 2 and 3, respectively. [How08, Theorem 2.2] states that in characteristic 3, the coarse moduli space $\mathcal{S}_{2}$ of supersingular genus 2 curves is isomorphic to the affine line $\mathbb{A}^{1}$: The absolute invariants $(0,0,g_{3})$ correspond to the curve $y^{2}=x^{6}+g_{3}^{2}x^{3}+g_{3}^{3}x+g_{3}^{4}$ if $g_{3}\neq 0$ and the point $(0,0,0)$ corresponds to $y^{2}=x^{5}+1$. Hence, over the finite field $\mathbb{F}_{q}$ where $q=3^{r}$, the subspace $\mathcal{S}_{2}(\mathbb{F}_{q})$ of $\mathcal{M}_{2}(\mathbb{F}_{q})$ corresponding to isomorphism classes of supersingular genus 2 curves contains $q$ elements.

In this section we recall some theory about the size of the supersingular locus $\mathcal{S}_{2}$. [IKO86], [KO87], and [Kob75] show that the supersingular locus of $\mathcal{M}_{2}$ is a union of projective lines with the singular points of this union corresponding to the superspecial points. Denote the finite set of superspecial points by $\mathcal{SP}_{2}$.

Let $B$ be the definite quaternion algebra over $\mathbb{Q}$ with discriminant $D$ and let $\mathcal{O}$ be a maximal order of $B$. Write $D=D_{1}D_{2}$ for two positive integers $D_{1},D_{2}$ and set $L_{n}(D_{1},D_{2})$ the set of left $\mathcal{O}$-lattices in $B^{n}$ which are equivalent to $(\mathcal{O}\otimes\mathbb{Z}_{p})^{n}$ at $p$ if $p$ does not divide $D_{2}$, and otherwise to the other local equivalence class if $p$ divides $D_{2}$, see [IKO86]. Denote by $H_{n}(D_{1},D_{2})$ the number of global equivalence classes in $L_{n}(D_{1},D_{2})$. In particular, for a prime number $p$, denote by $H_{n}(p,1)$ the class number of the principal genus and by $H_{n}(1,p)$ the class number of the non-principal genus.

Let $k$ be a finite field of characteristic $p$. On the one hand, it is known that every principally polarised superspecial abelian surface $A$ over $k^{\text{alg}}$ arises from choosing a principal polarisation on a product $E\times E$ for a supersingular elliptic curve $E$. Let $B=\operatorname*{\mathrm{End}}(E)\otimes\mathbb{Q}$ the endomorphism algebra of $E$, which is the definite quaternion algebra ramified at $p$. Then the number of principally polarisations on $E\times E$ over $k^{\text{alg}}$ up to automorphisms is equal to $H_{2}(p,1)$ for $B$.

On the other hand, as a principally polarised abelian variety, either $A=E_{1}\times E_{2}$ for two supersingular elliptic curves $E_{1}$ and $E_{2}$, or $A=\operatorname{Jac}(C)$ for a superspecial genus 2 curve $C$. Hence the number of superspecial genus 2 curves over $k^{\text{alg}}$ is given by $H_{p}=H_{2}(p,1)-h_{p}(h_{p}+1)/2$, where $h_{p}=H_{1}(p,1)$ is the number of isomorphism classes of supersingular elliptic curves over $k^{\text{alg}}$, see [IKO86, Corollary 2.12 ].

The number $H_{p}$ is finite; every superspecial genus 2 curve can be defined over $\mathbb{F}_{p^{2}}$. This allows us to determine the finite contribution of superspecial genus 2 curves to $\#\mathcal{S}_{2}(k)$ when $k$ contains $\mathbb{F}_{p^{2}}$. For small primes $p$ we find \Freftable:ssing_sspec_sizes; we have also determined the number of superspecial curves defined over $\mathbb{F}_{p}$.

The number of irreducible components of the supersingular locus of $\mathcal{M}_{2}$, i.e. the aforementioned projective lines, is equal to the class number of the non-principal genus $H_{2}(1,p)$, see [KO87, Theorem 5.7 ].