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](https://mathoverflow.net/questions/59088/endomorphisms-of-jacobians-of-hyperelliptic-curves-taking-exceptional-divisors-t)


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



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


