Timeline for Detect all isogenies of an elliptic curve over a given number field
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Oct 29 at 11:41 | comment | added | did | @ChrisWuthrich thank you! The last comment clarifies my confusion. | |
| Oct 22 at 11:46 | comment | added | Chris Wuthrich | If $\phi: E \to E'$ is an isogeny defined over $F$ between two curves defined over $K$, then $\phi$ is the composition of an isogeny of that degree defined over $K$ and an isomorphism from $E/E[\phi]$ to $E'$ defined over $F$. | |
| Oct 22 at 11:46 | comment | added | Chris Wuthrich | The algorithms look for all isogenies $\phi$ leaving $E$ defined over a fixed number field $K$, which is equivalent to finding their kernels $C = E[\phi]$ as subgroup of $E$ defined over $K$. The codomain $E/C$ is defined over $K$. | |
| Oct 20 at 18:27 | comment | added | did | Thank you. I have a follow-up naive question. In this algorithm, they show how to find primes $l$ such that $E/K$ has an $l$-isogeny. It seems to me that in this context the $l$-isogenies are thought to be defined over $K$, is that right? | |
| Oct 20 at 15:47 | history | edited | M.G. | CC BY-SA 4.0 |
guven -> given
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| Oct 20 at 15:32 | comment | added | Chris Wuthrich | (3) No, because of quadratic twists this isn't true even for isomorphic curves (degree 1 isogenies). | |
| Oct 20 at 15:31 | comment | added | Chris Wuthrich | My answer to this question contains links to algorithms even over number fields. That should help for (1). | |
| Oct 20 at 14:55 | history | asked | did | CC BY-SA 4.0 |