5
$\begingroup$

Let $E$ be an elliptic curve defined over a number field $F$ and $F_\infty$ be the cyclotomic $\mathbb{Z}_p$-extension of $F$. Is it true that the $p$-primary subgroup of $E$ over $F_\infty$ i.e. $E[p^\infty](F_\infty)$ is finite ?

Proofs or references are welcome.

Improve this question
$\endgroup$
1

1 Answer 1

Reset to default
4
$\begingroup$

Yes. If $E$ has potentially good reduction, this is due to H. Imai, Proc Japan Adac Math Sci 51 (1975). A non-standard proof is Theorem A.2.8 in Coates-Sujatha's "Galois cohomology of elliptic curves"

Improve this answer
$\endgroup$
2
  • $\begingroup$ Is it true if $E$ has supersingular reduction at $p$ ? $\endgroup$
    – Robert
    Commented Mar 17, 2016 at 20:09
  • 1
    $\begingroup$ Yes, they don't assume anything on $E$. It has more to do with the shape of the image of the global Galois representation on $T_pE$ rather than how this looks locally at $p$. In particular their proof splits up according to whether $E$ has cm or not. $\endgroup$
    – Chris Wuthrich
    Commented Mar 17, 2016 at 20:11

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .