Special points in the Jacobian of a hyperelliptic curve under Frobenius

Question

Good evening.

Let $H/\overline{\mathbb{F}}_q$ be a hyperelliptic curve of genus 2 with a rational Weierstrass point and let $J$ be its jacobian, if $\phi\in \text{End}(J)$ is the $q$-Frobenius.

I am trying to calculate the cardinality of the following set:

$Z := \lbrace P\in H(\overline{\mathbb{F}}_q):[P-\infty]\oplus [P^\phi - \infty]\sim 0$ or $ [R-\infty] \rbrace$ for some $R\in H(\overline{\mathbb{F}}_q)$.

Is easy to see that all the points $Q\in Twist^2(H)$ because $Q^\phi = \iota Q$, also the $\mathbb{F}_q$-rational $2$ and $3$ torsion points on the Jacobian of the form $D:= [T-\infty]$ as $2D=0$ and in case of being 3 torsion we have that $3D=[\iota P-\infty]$.

So here we have that

$\#Z \geq \#Twist^2(H)(\mathbb{F}_q) + \#W + \#T=2(q+1) - \#H(\mathbb{F}_q)+\#W+\#T$

Where $W$ are the $\mathbb{F}_q$-rational Weierstrass points on $H$ and $T$ are the three torsion points given by prime divisors.

Are there more?, I don't know if Frobenius can do more Weird stuff in different extensions.

It will be nice to know if there is something to study related to sums of elements in the same orbits by some action in abelian varieties

Thanks

Answer 1

I presume you are familiar with the Mumford representation of the divisor class group of a hyperelliptic curve $C$, which we may identify with its Jacobian $\text{Jac}\,C$ once we assume we have a rational point?

The case where $C$ has a rational Weierstrass point is described in Cantor's paper here.

http://www.ams.org/journals/mcom/1987-48-177/S0025-5718-1987-0866101-0/S0025-5718-1987-0866101-0.pdf

Since you are working over $\overline{\mathbb{F}}_q$, there is no loss of generality in assuming $C$ has a rational Weierstrass point.

The key point about the Mumford representation is that every element of $\text{Jac}\,C$ has a unique representation as a reduced divisor which can be encoded by a pair of polynomials $(v(x),u(x))$, where $v(x)$ is monic and separable, $\deg u < \deg v \le g$, and $u(x)$ divides $v^2-f(x)$, where $C:y^2=f(x)$ (I am assuming we are not in characteristic $2$, but the characteristic $2$ case is very similar). An affine point $P=(x_0,y_0)$ is in the divisor if and only if $v(x_0)=0$ and $u(x_0)=y_0$.

If a point $P$ appears in a reduced divisor then it appears only once and $-P$ does not appear. By $-P$, I mean the point $(x(P),-y(P))$.

Every divisor of the form $P-\infty$ is a reduced divisor, and if $x(P)$ $!$$= x(Q)$, then $P+Q-2\infty$ is also a reduced divisor. It follows that we cannot have $P+\infty + Q+\infty$ linearly equivalent to $0$ or $R+\infty$ except when $x(P)=x(Q)$, which implies that $Q=\pm P$. If $Q=-P$ then $Q=P$ and $P+\infty+Q+\infty = 0$, and otherwise we have $P=Q$ and $2(P+\infty) \sim R+\infty$, which implies$ R+\infty - P+\infty \sim P+\infty$. We clearly cannot have $R=P$, and if $R$ $!$$=-P$ then $R+\infty - P+\infty$ is reduced, as is $P+\infty$, a contradiction. This leaves only the case $Q=P$ and $R=-P$, in which case $P+\infty$ corresponds to a point of order $3$ in the Jacobian.

You should check the details of this argument, but I believe it implies that you only need to consider $2$- and $3$-torsion points as you suggest above. I note that this argument and does not depend on the fact that you are specifically interested in the case $Q=P^\phi$, so you don't need to worry about Frobenius doing any "weird stuff".