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Is there any kind of generalization of Vélu formulae for Jacobians?

The question technically is:

Given a hyperelliptic curve $H/\overline{\mathbb{F}}_q$ and its jacobian $J_H$ of dimension $2$, if $D\in J_H$:

Is there a way to construct an isogeny $\psi:J\to{J_D}$ with kernel $nD$ for all $n\in \mathbb{Z}$ and construct the Jacobian variety $J_D$ of some other hyperelliptic curve? (maybe $J_D$ should be an abelian variety in general)

Is well known that elliptic curve isogenies have specific form, this helps to represent easily an isogeny.

What is known about isogenies between genus 2 jacobians for example?

I know that if $J$ is non simple, this could make things easier, and maybe reduce it to Vélu formulae in an $(a,b)$-isogeny, but, if $J$ is simple?

Is it possible to use the Affine part of a Jacobian of dimension 2 to reconstruct a hyperelliptic curve of genus 2 using an isogeny in Mumford representation?

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This question seems a bit confused.

If $D$ is an arbitrary point in the Jacobian then one cannot construct an isogeny with kernel the subgroup generated by $D$ -- as this subgroup is typically infinite, and the kernel of an isogeny is finite.

If however $D$ is assumed torsion, then one can certainly construct an isogeny with kernel generated by $D$ -- one can just quotient out the abelian variety by the finite group generated by $D$; this quotient exists in algebraic geometry and is another abelian variety.

However this quotient will typically not be a Jacobian if the dimension is 2 or more -- indeed in general the quotient will not admit a principal polarisation, which precludes it from being a Jacobian. This phenomenon does not occur in dimension 1, because every elliptic curve admits a principal polarization, but in higher dimension it represents an obstruction to what you want to do.

Your other questions are a bit too vague for me to say anything. Isogenies between genus 2 Jacobians can happen -- for example multiplication by a positive integer is an isogeny from a Jacobian to itself. I don't know what "the affine part" of a Jacobian is -- perhaps you think about Jacobians in a different way to me: for me a Jacobian is an abstract variety -- it can be covered by affines, but it doesn't have some sort of canonical "affine part".

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  • $\begingroup$ Dear @kevin Thank you for your answer Yes, I am assuming that the base field is $\overline{\mathbb{F}}_q$ so every point has torsion. When I talk the "Affine part", is because in genus $2$, you can find the locus of all the points of the form $P+Q-2\infty$ using that if $y^2=f(x)$ is the equation of the hyperelliptic curve, then these points can be represented by two polynomials $\langle u(t),v(t) \rangle$ such that $u\mid f-v^2$ and $\deg(v)<\deg(u)\leq 2$. So This representation for a general point I was wondering if it is useful to define explicitly an isogeny. $\endgroup$ – Eduardo R. Duarte Jan 29 '17 at 20:04
  • $\begingroup$ But you answer me more less the question with your argument of not having a polarisation. Do you know if there is some criteria for the exceptions when $J/\langle D \rangle \cong \langle J_D,\lambda \rangle$ $\endgroup$ – Eduardo R. Duarte Jan 29 '17 at 20:04
  • $\begingroup$ A hyperelliptic curve can have more than one equation so there's more than one "affine part" I guess. But I know what you mean. My gut feeling is that it would be rare for the quotient by a cyclic subgroup to still have a principal polarization but my intuition is not so good in characteristic p. Did you look in Cassels-Flynn to see if they say anything? They do many explicit calculations with Jacobians in this very form. $\endgroup$ – Kevin Buzzard Jan 29 '17 at 20:33
  • $\begingroup$ Thank you, I will check it, I will wait for someone else to give an answer some a couple of days, if no answer I will put yours. Thank you. $\endgroup$ – Eduardo R. Duarte Jan 29 '17 at 20:34

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