The average size of the 3-isogeny Selmer groups of elliptic curves y² = x³ + k · arXivDesk1610.05759Oct 18, 201625 pages
The average size of the 3-isogeny Selmer groups of elliptic curves y2=x3+k
Manjul Bhargava, Noam Elkies, Ari Shnidman
Abstract
The elliptic curve Ek:y2=x3+k
admits a natural 3-isogeny
φk:Ek→E−27k . We compute the average size of the
-Selmer group as
varies over the integers. Unlike previous results of Bhargava and Shankar on
-Selmer groups of elliptic curves, we show that this average can be very sensitive to congruence conditions on
; this sensitivity can be precisely controlled by the Tamagawa numbers of
and
. As consequences, we prove that the average rank of the curves
,
, is less than 1.21 and over
(resp.
) of the curves in this family have rank 0 (resp. 3-Selmer rank 1).
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