Timeline for Isogeny from kernel in higher dimensional abelian varieties
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 10, 2017 at 8:51 | vote | accept | Eduardo R. Duarte | ||
| Jan 29, 2017 at 20:34 | comment | added | Eduardo R. Duarte | 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. | |
| Jan 29, 2017 at 20:33 | comment | added | Kevin Buzzard | 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. | |
| Jan 29, 2017 at 20:04 | comment | added | Eduardo R. Duarte | 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 $J/\langle D \rangle \cong \langle J_D,\lambda \rangle$ | |
| Jan 29, 2017 at 20:04 | comment | added | Eduardo R. Duarte | 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. | |
| Jan 29, 2017 at 19:53 | history | answered | Kevin Buzzard | CC BY-SA 3.0 |