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Abstract
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Let
be an elliptic curve
over a number field
defined by a monic irreducible cubic polynomial
. When
is
nice at all finite
primes of , we bound
its
-Selmer rank in
terms of the
-rank
of a modified ideal class group of the field
, which we call the
seminarrow class group of
.
We then provide several sufficient conditions for
being nice at a finite prime.
As an application, when
is a real quadratic field,
is
semistable and the discriminant of
is totally negative, we frequently determine the
-Selmer rank of
by computing the
root number of
and the
-rank of the
narrow class group of
.
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Keywords
elliptic curve, 2-Selmer rank, ideal class group,
Mordell–Weil rank
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Mathematical Subject Classification
Primary: 11G05, 11G07, 11R29
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Milestones
Received: 3 January 2021
Revised: 28 September 2021
Accepted: 23 October 2021
Published: 16 October 2022
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