Abstract
AbstractLet $$A/{\mathbb {Q}}$$
A
/
Q
be an abelian variety such that $$A({\mathbb {Q}}) =\{0_A\}$$
A
(
Q
)
=
{
0
A
}
. Let $$\ell $$
ℓ
and p be rational primes, such that A has good reduction at p, and satisfying $$\ell \equiv 1 \,(\mathrm{mod} \,p)$$
ℓ
≡
1
(
mod
p
)
and $$\ell \not \mid \# \,A({\mathbb {F}}_p)$$
ℓ
∤
#
A
(
F
p
)
. Let S be a finite set of rational primes. We show that $$(A \setminus \{0_A\})({\mathscr {O}}_{L,S}) =\varnothing $$
(
A
\
{
0
A
}
)
(
O
L
,
S
)
=
∅
for 100% of cyclic degree $$\ell $$
ℓ
fields $$L/{\mathbb {Q}}$$
L
/
Q
, when ordered by conductor, or by absolute discriminant.
Funder
Engineering and Physical Sciences Research Council
Publisher
Springer Science and Business Media LLC
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