Cartan images and $$\ell $$ ℓ -torsion points of elliptic curves with rational j-invariant
Abstract Let $$\ell $$ ℓ be an odd prime and d a positive integer. We determin...
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| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
Springer International Publishing
2021
|
| Online Access: | https://hdl.handle.net/1721.1/131476 |
| Summary: | Abstract
Let
$$\ell $$
ℓ
be an odd prime and d a positive integer. We determine when there exists a degree-d number field K and an elliptic curve E / K with
$$j(E)\in \mathbb {Q}\setminus \{0,1728\}$$
j
(
E
)
∈
Q
\
{
0
,
1728
}
for which
$$E(K)_\mathrm {tors}$$
E
(
K
)
tors
contains a point of order
$$\ell $$
ℓ
, conditionally on a conjecture of Sutherland. We likewise determine when there exists such a pair (K, E) for which the image of the associated mod-
$$\ell $$
ℓ
Galois representation is contained in a Cartan subgroup or its normalizer. We do the same under the stronger assumption that E is defined over
$$\mathbb {Q}$$
Q
. |
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