Loading…
Open image computations for elliptic curves over number fields
For a non-CM elliptic curve E defined over a number field K , the Galois action on its torsion points gives rise to a Galois representation ρ E : Gal ( K ¯ / K ) → GL 2 ( Z ^ ) that is unique up to isomorphism. A renowned theorem of Serre says that the image of ρ E is an open, and hence finite index...
Saved in:
| Published in: | Research in number theory 2025-03, Vol.11 (1), Article 1 |
|---|---|
| Main Author: | |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | For a non-CM elliptic curve
E
defined over a number field
K
, the Galois action on its torsion points gives rise to a Galois representation
ρ
E
:
Gal
(
K
¯
/
K
)
→
GL
2
(
Z
^
)
that is unique up to isomorphism. A renowned theorem of Serre says that the image of
ρ
E
is an open, and hence finite index, subgroup of
GL
2
(
Z
^
)
. In an earlier work of the author, an algorithm was given that computed the image of
ρ
E
up to conjugacy in
GL
2
(
Z
^
)
in the special case
K
=
Q
. A fundamental ingredient of this earlier work was the Kronecker–Weber theorem whose conclusion fails for number fields
K
≠
Q
. We shall give an overview of an analogous algorithm for a general number field and work out the required group theory. We also give some bounds on the index in Serre’s theorem for a typical elliptic curve over a fixed number field. |
|---|---|
| ISSN: | 2522-0160 2363-9555 |
| DOI: | 10.1007/s40993-024-00599-2 |