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I hope my question is not too vague or basic to be here. I have been constructing a setting to count points on a curve, but I am stucked solving one part of my problem for some time. Now I would like to have some advise from experts. I hope someone can give me a hint, it is related to genus 2 curves and degrees of morphisms to the Kummer surface of its jacobian.

What I am doing in some sense is constructing the 2:1 map from a hyperelliptic curve $H$ to $\mathbb{P}^1$ in a little complicated way but in order to get it explicitly.

Let $H$ be a hyperelliptic curve of genus 2 over $\mathbb{F}_q$ defined by $y^2=x^5 + a_3x^3 + a_2x^2 + a_1x+a_0=f(x)$ and consider $J$ its jacobian and $Kum(J)\subset \mathbb{P}^3_{\mathbb{F}_q}$ its Kummer surface .

Let $\theta=\lbrace (P-\infty) \in J : P\in H\rbrace$ be its theta divisor.

Let $\phi,\tau,[n]\in End_{\mathbb{F}_q}(J)$ be the frobenius, the involution in the jacobian induced by the hyperelliptic involution and the $n$ map respectively.

Consider the following divisor on $J$ given by a generic point $P\in C$.

$D_n := ((P^{\phi}-\infty)+[n](P^{\tau}-\infty))$

This divisor can be thought as the Frobenius - nIdentity in $End_{\mathbb{F}_q}(J)$

Consider the map from the theta divisor (the curve in this case) to Kummer Surface

$\psi_n:\theta \to Kum(J) $

$(P-\infty) \mapsto [1:\kappa_2(D_n): \kappa_3(D_n) : \kappa_4(D_n)]$

As the Kummer identifies $\pm 1$ divisors, this is going to be a 2:1 map, and is similar to the 2:1 map from the curve $H$ to $\mathbb{P}^1$ (except for the torsion prime divisors in $J$)

I am trying to measure the degree of the map to the fourth coordinate for every $n$ i.e.:

$\hat\psi_n:\theta\to \mathbb{P}^{1}_{\mathbb{F}_q}$

$(P-\infty)\mapsto \kappa_4(D_n)$

This fourth coordinate is the one that determines the quotient by $\pm 1$ which can be found explicitly for general divisors in page five of:

http://www.math.uni-hamburg.de/home/js.mueller/general_kummer.pdf

My main problem here is that as $H$ does not form a group, I cannot explore the morphism structure, and some values for my degree function fail.

I am interested in the positive integer $deg(\psi_n)$, which is the degree of the map.

I have made some calculations and the degree seems for every $n\in \mathbb{Z}$ to behave as a quadratic function $Q(n)$ where the trace of frobenius is involved in the structure of $Q(n)$, but for some $n$ it fails, for example if the coefficient $a_0$ in the equation of $H$ is a square then $Q(2)$ and $deg(\psi_n)$ differ by 2, so maybe $\psi_n$ is not well defined at some points or I am measuring degree over a singularity, but I am not sure.

If its not a square the same happens, but for $Q(-2)$, which is expected as the twist will have in its equation the coefficient $a_0'$ as a square.

Another thing is that if $a_0=0$ I have at every even integer $n$ an error also of -2 with respect of $Q(n)$.

I have been exploring a lot of cases with/without $\mathbb{F}_q$-rational Weierstrass points but I have not been able to deduce what happen at my errors, sometimes the degree function seems to not have errors (except for n=-2,2 which always appears as an error by 2), and sometimes it has more errors, If I pic up a translation of the curve, I can make some errors disappear, and the values are the same. So I am trying to deduce which representative of the isomorphism class has less errors.

Of course for $Q(1)$ and $Q(-1)$ I get interesting values, never seem to fail which give me information of the number of points in the curve and its twist directly without passing to the Jacobian.

But well I have been trying to solve this via intersection theory or even thinking what it means for $\Psi_n:=\phi-[n]$ the set $\Psi_n^{-1}(\infty)\subset \theta$ (As I cannot talk about kernel here because $\theta$ is not an abelian variety).

So $\Psi_n^{-1}(\infty)$ is like "All the elements in $\theta$ (which is isomorphic to $H$) such that the frobenius action in them, behaves exactly as $n$ map on them"

Thanks

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The degree of ${\hat \psi}_n$ is the intersection number of the image of $\psi_n$ with the divisor on the Kummer surface defined by $\kappa_4 = 0$. Maybe the issues you are having are because the model of the surface in $\mathbb{P}^3$ is singular. It should be easier to work in the Jacobian instead. I think the calculation you are trying to do is essentially the same as the one done in my paper with Stohr (Proc LMS 52 (1986) 1-19) in the appendix.

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  • $\begingroup$ Prof. Felipe. Thanks for the answer, I have been trying to deduce the divisor of zeroes and poles od $\kappa_4$ , but also seems to be very mysterieus, for example the poles are in fact for $n=1$ the $\mathbb{F}_q$-rational points which is something nice, but the zeroes look weird, in fact this $\kappa_4$ is defined at Flynn's book on genus 2 curves in the first pages trying to fit the function ${\frac{y_1-y_2}{x_1-x_2}}^2$ in the $L(2\infty)$ space to find the projective embedding. I will check your paper, and try to think in $\kappa_4=0$, do you know the zero divisor of $\kappa_4$? $\endgroup$ – Eduardo R. Duarte Aug 7 '16 at 19:53
  • $\begingroup$ @Eduardo If indeed what you are trying to compute is the same as what we did (I didn't check) then the description of the divisor of the function, to the extent that is possible, is in the body of the paper. $\endgroup$ – Felipe Voloch Aug 7 '16 at 22:54

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