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Abstract
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For every elliptic curve
which has complex multiplication (CM) and is defined over a number field
containing the CM
field
, we prove that
the family of
-division
fields of
, with
prime, becomes
linearly disjoint over
after removing an explicit finite subfamily of fields. We then give
a necessary condition for this finite subfamily to be entangled
over , which is
always met when
.
In this case, and under the further assumption that the elliptic curve
is obtained as a
base-change from
,
we describe in detail the entanglement in the family of division fields of
.
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Keywords
elliptic curves, complex multiplication, division fields,
entanglement
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Mathematical Subject Classification
Primary: 11G05, 11G15, 14K22
Secondary: 11F80, 11S15
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Milestones
Received: 25 July 2020
Revised: 10 March 2021
Accepted: 21 August 2021
Published: 19 June 2022
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