2
$\begingroup$

Let $E/K$ be an elliptic curve over a field (take it as a number field or a local field if necessary). If $E[2]$ is defined over $K$, then we can associate the multiplication by $2$ isogeny to a biquadratic extension of $K(E)$. This is well-known from the 2-descent calculation. We can write down the extension very explicitly.

Can we do this for any $n$? And what happens if $E[n]$ is not defined over $K$?

Improve this question
$\endgroup$
2
  • 1
    $\begingroup$ Q1: Yes. Q2: Then it's not a Galois extension. $\endgroup$ Commented Dec 20, 2024 at 14:10
  • $\begingroup$ Thanks for the reply. Do you know some reference for Q1? $\endgroup$ Commented Dec 20, 2024 at 15:10

0

Reset to default

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.