Prime divisors on the Jacobian of a genus 2 curve over $\mathbb{F}_q$ under the $n$ map

Question

Let $H$ be a hyperelliptic curve geometrically irreducible of genus 2 over $\mathbb{F}_q$ with a rational point $\infty$ given by the model $y^2=f(x)$, where $f$ is monic of degree 5.

Consider the Jacobian $J_H\cong \text{Pic}^0(H)$.

Consider a point $P\in H(\mathbb{F}_q)$, $[P-\infty]\in J_H$ and $[n]\in \text{End}_{\mathbb{F}_q}(J_H)$.

a) Is there any characterization of the points $P$ such that $[n][P-\infty]=[Q-\infty]$ ? I mean, when a prime divisor under the $n$ map results in another prime divisor.

b) Additionally, when $[n][P-\infty]=[2Q-2\infty]$? Of course with $P\neq Q$ and $n\neq 2$.

I was trying to read Cantor's division polynomials papers for genus 2 curves, but is a little convoluted.

I think Cantor's division polynomials can be of help, because I could make the question "when an $n^{th}$ divison polynomial is not defined (for question 1).

There is a question related to this here Endomorphisms of Jacobians of Hyperelliptic Curves taking Exceptional Divisors to Exceptional Divisors

But one reply says that the endomorphism must be of degree one, which is not true. For example.

Let $H$ be defined by $y^2=x^5 + x + 16$ over $\mathbb{F}_{23}$ and take the point $(0,4)$ and if $n=10$ then $10[(0,4)-\infty]=[(5,15)-\infty]$. We have that $\#J_H=496=2^4 \cdot 31$, $\#H=22$ and the degree of the $n$ map is not one.

I think that if $\iota:H\hookrightarrow J_H$ is the embedding of the curve in the Jacobian, and $\theta:=\text{Im}\iota$ is a divisor in the jacobian, that is, an element of $\text{Pic}^0(J_H)$ then I am asking which elements of $\text{Supp}(\theta)$ stay in $\text{Supp}(\theta)$ or $\text{Supp}(2\theta)$ under $[n]$.

Thanks

Answer 1

If $\iota$ denotes the hyperelliptic involution, then the condition $n[P-\infty] = [Q-\infty]$ is equivalent to $nP+\iota(Q)$ linearly equivalent to $(n+1)\infty$. In a few pathological cases, where the linear system $|(n+1)\infty|$ is non-classical, every point satifies this. But in the typical case, this means that $P$ is a Weierstrass point of this linear system and those $P$'s are the zeros of a Wronskian formed with a basis for $L((n+1)\infty)$. This applies to points in $H(\bar{\mathbb{F}}_q)$. I don't think there is a clean way to distinguish the rational points satisfying this.