The $a$-number, $p$-rank and Cartier points of genus $4$ curves

Let $X$ be a smooth irreducible genus $g$ curve over $k$. Denote by $J_{X}$ the Jacobian variety associated to $X$. Then $J_{X}$ is an abelian variety of dimension $g$ isomorphic to $\text{Pic}^{0}_{X/k}$, equipped with a principal polarization.

The $p$-torsion part of the Jacobian, denoted by $J_{X}[p]$, is a group scheme and here we will study two invariants associated to it. The first one is the $p$-rank, defined as the integer $f$ such that $\#J_{X}[p](k)=p^{f}$. The $p$-torsion subgroup of an abelian variety of dimension $g$ has order at most $g$, hence $0\leq f\leq g$.

Suppose that $x$ is a separating variable of $k(X)/k$, then every $t\in k(X)$ can be written as

with $t_{i}\in k(X)$.

The Cartier operator is $1/p$-linear, meaning that $\mathcal{C}(a^{p}\omega_{1}+b^{p}\omega_{2})=a\mathcal{C}(\omega_{1})+b\mathcal{C}(\omega_{2})$, for $a,b$ in $k(X)$ and $\omega_{1},\omega_{2}\in\Omega_{X}^{1}$. It induces a well defined map $\mathcal{C}$ on the $k$-vector space of regular differentials $H^{0}(X,\Omega_{X}^{1})$.

The Frobenius endomorphism is $p$-linear, that is $\mathcal{F}_{X}(a\xi)=a^{p}\mathcal{F}_{X}(\xi)$ for all $a\in k$ and all $\xi\in H^{1}(X,\mathcal{O}_{X})$.

The space $H^{1}(X,\mathcal{O}_{X})$ is the dual of $H^{0}(X,\Omega_{X}^{1})$ and there is a perfect paring $<,>$ on $H^{1}(X,\mathcal{O}_{X})\times H^{0}(X,\Omega_{X}^{1})$ such that $<\mathcal{F}_{X}\xi,\omega>=<\xi,\mathcal{C}\omega>^{p}$.

When $B$ and $B^{\prime}$ are dual basis,the Cartier–Manin matrix $M$ with respect to $B$ is the transpose of the Hasse–Witt matrix $H$ with respect to $B^{\prime}$.

By Serre [13], the $p$-rank $f$ is the stable rank of the Frobenius and since this operator is $p$-linear, it implies that

where $H^{(i)}$ is the matrix obtained by raising every entry of $H$ to the $i$-th power.

The second invariant of $J_{X}[p]$ that we study here is the $a$-number, defined as the rank of the Cartier operator (see Oort [7]), i.e.,

It is known that $0\leq a+f\leq g$. Generically $f=g$, in which case $X$ is said to be ordinary. The other extreme case is when $X$ is superspecial and it occurs when $a=g$ or equivalently, when the Cartier operator is identically $0$ on $H^{0}(X,\Omega_{X}^{1})$.

In Section 4.2 we follow the work from [6] to describe how to explicitly find the Hasse–Witt matrix, when $X$ is the complete intersection of two homogeneous polynomials and use it to compute the $p$-rank and $a$-number.

In this section we review some of the main and more recent results with respect to genus $g$ curves of positive characteristic and possible values for the $a$-number and $p$-rank that occur.

Baker gives in [1] an alternative proof for Theorem 3.5, based on the existence of Cartier points (see Section 6 for definition). Let $m$ be the rank of the Cartier operator. Re ([12], Theorem 3.1 and Proposition 3.1) provides a generalization of this result to any value of $m$. The author proves that in fact $g\leq(m+1)p\frac{(p-1)}{2}+pm$ and if $X$ is also hyperelliptic then $g<\frac{p+1}{2}+mp$.

Zhou ([14], Theorem 1.1) gives a strengthening of this result for the case when $m=1$ that gives a bound for $g$ of $p+\frac{p(p-1)}{2}$. Moreover, Frei ([3], Theorem 3.1]) proved that if $g\geq p$ where $p$ is an odd prime, then there are no smooth hyperelliptic curves of genus $g$ defined over a field of characteristic $p$ with $a$-number equal to $g-1$.

There are examples, however, of curves with $a$-number $g-1$ that are non-hyperelliptic. Also Zhou finds in [15] a family of Artin–Schreier curves with these properties.

More generally, Pries [10] proves the existence of smooth curves with $a$-number $1,2$ and $3$, under certain conditions.

If $X$ is a genus $4$, smooth, irreducible and non-hyperelliptic curve, then the canonical map embeds $X$ into $\mathbb{P}^{3}_{k}$ as the intersection of the zero loci of a quadratic and a cubic homogeneous polynomial in four variables (see [5]), that we detail below.

We restrict our computations to what we will define as genus $4$ curves in standard form. These are based on equations given by Kudo and Harashita [6].

Let $X=V(F,G)$ be the complete intersection on $\mathbb{P}^{3}_{k}$ defined by homogeneous polynomials $F$ and $G$ in $k[x,y,z,w]$ of degrees $d$ and $c$, respectively. Following [5] and [1] we see that $H^{1}(X,\mathcal{O}_{X})\cong H^{3}(\mathbb{P}^{3},\mathcal{O}_{\mathbb{P}}^{3}(-c-d))$, where the basis $\mathcal{B}$ of $H^{1}(X,\mathcal{O}_{X})$ that corresponds to the coordinates $x,y,z,w$ is associated to the basis of $H^{3}(\mathbb{P}^{3},\mathcal{O}_{\mathbb{P}}^{3}(-c-d))$ given by

Using this fact, Kudo and Harashita present an algorithm to compute the Hasse–Witt of such curves, which can be generalized to compute the corresponding matrix for any complete intersection over a perfect field of positive characteristic.

This formula gives a matrix $H$ that represents the action of $\mathcal{F}_{X}$ by left matrix multiplication. Let $\mathbf{v}$ be the column vector corresponding to an element of $H^{1}(X,\mathcal{O}_{X})$ expressed in terms of the basis $\mathcal{B}$. Then the image of $\mathbf{v}$ under $\mathcal{F}_{X}$ is given by $H\cdot\mathbf{v}^{(p)}$, since $\mathcal{F}_{X}$ is $p$-linear.

In the next section, we use Proposition 4.2 together with the equations from Section 4 to compute examples of non-hyperelliptic smooth curves of genus $4$ with $a$-number $3$.

We apply the above procedure to random samples of tuples in $\mathbf{D}_{p}$, $\mathbf{N1i}_{p}$, $\mathbf{N1ii}_{p}$ and $\mathbf{N2}_{p}$ for $p\in\{3,5,7,11\}$ in order to gather statistical information. We construct the sample using the intrinsic Random function in Magma. In addition, we classify the curves by $a$-number and $p$-rank.

When we analyze the samples of curves in our data, we will often compare the number of curves with certain $p$-rank and/or $a$-number with the total of smooth curves in standard form obtained. In our search we classify the curves with $a$-number $0,1,2$ and $3$. We ignore the curves with $a=4$ because they represent a very small proportion of the curves and hence they do not affect the percentages in a significant way.

Also, our focus will be studying the occurrence of Cartier points (see Section 6), and we already know that superspecial curves have infinitely many of them (Baker [1]). Here is how we will refer to our different samples:

• •

  Total set: the set $\mathbf{D}_{p}$, $\mathbf{N1i}_{p}$, $\mathbf{N1ii}_{p}$ or $\mathbf{N2}_{p}$, depending on the case.

• •

  Sampled set: the curves of the sets above that are included in our random search.

• •

  Smooth sample: the set of cubics from the sampled set that give smooth, irreducible, genus $4$ curves (excluding superspecial curves).

• •

  Singular sample: the set of cubics from the sampled set that give curves that fail either one of the conditions for smoothness, irreducibility or genus.

If we do not specify the subcase, then Sampled set, Smooth sample and Singular sample will refer to the total samples throughout the four cases.

In this section we display the overall results from the sampling search. The sizes of our final Sampled sets by case are shown in Table 2.

In Table 3 we list the number of curves with $p$-ranks $f=0,1,2,3$ and $a$-number $a=1,2,3$, over $\mathbb{F}_{p}$ for $p\in\{3,5,7,11\}$. We include the totals for the singular samples and the ordinary curves (that is, those with $a$-number $0$). From this classification we have the following statement:

Next we discuss the results from the search for each $p$. We include the sizes of the total set, sampled set, smooth sample and singular sample, normalized by $\log_{p}$. Then we show the break down of curves with $a=1,2,3$ and specify the percentage of the sampled set and smooth sample they represent.

In order to focus on the analysis of some aspects of genus $4$ curves with $a$-number $3$, we apply the procedure above to all the tuples in $\mathbf{D}_{p},\mathbf{N1i}_{p},\mathbf{N1ii}_{p}$ and $\mathbf{N2}_{p}$, but only store the smooth, irreducible curves with $a=3$. We did this for $p=3,5$. The search is also done for $p=7$ but only for tuples in $\mathbf{D}_{7}$ and a subset of $\mathbf{N1i}_{7}$, because of the long computing times. It is important to remark that after the search, we classify the curves by $\mathbb{F}_{p}$-isomorphism classes.

In the cases $p=3,5$ we also have a complete list of all curves in standard form that have $a$-number $3$. For $p=7$ we have a subset of them: the ones in case D and those in case N1i where $(b_{1},b_{2})=(0,0)$, we denote these lists by 7(D) and 7(N1i’). The information on these curves, classified by isomorphism classes, is shown in Table 4. A first conclusion that we can draw from this data is the following:

Theorem 3.5 states that the genus of a superspecial curve in characteristic $p$ is bounded by $p(p-1)/2$. Ekedahl [2] bases the proof of this result on the fact that a curve is superspecial if and only if its Jacobian is isomorphic to the product of supersingular elliptic curves (Oort, [8]). Baker, on the other hand, presents in [1] an alternative proof which makes use of an equivalent definition: a curve is superspecial if and only if the Cartier operator annihilates $H^{0}(X,\Omega_{X}^{1})$. The other component of his proof is the existence of linear systems of dimension $1$, associated to a certain type of points on $X$, defined as Cartier points.

The following result yields Theorem 3.5 as a corollary.

From here on, we consider the Hasse–Witt matrix $H$ of $X$ to be in terms of the basis $\mathcal{B}$, unless otherwise stated.

Notice that if $k$ is not algebraically closed, then the Cartier points of $X/k$ might not be defined over $k$, but over some extension. In general we consider Cartier points as points in $X(\overline{k})$. The next lemma gives us a way to find Type $1$ points.