Let
be an abelian variety
over a number field
and let
be a
prime. Cohen–Lenstra–Delaunay-style heuristics predict that the Tate–Shafarevich group of
should contain an element
of order
for a positive
proportion of quadratic twists
of
. We give
a general method to prove instances of this conjecture by exploiting independent isogenies of
. For each
prime
,
there is a large class of elliptic curves for which our method shows
that a positive proportion of quadratic twists have nontrivial
-torsion
in their Tate–Shafarevich groups. In particular, when the modular curve
has infinitely
many
-rational
points, the method applies to “most” elliptic curves
having a cyclic
-isogeny. It also applies
in certain cases when
has only finitely many rational points. For example, we find an elliptic curve over
for
which a positive proportion of quadratic twists have an element of order
in
their Tate–Shafarevich groups.
The method applies to abelian varieties of arbitrary dimension,
at least in principle. As a proof of concept, we give, for each prime
, examples
of CM abelian threefolds with a positive proportion of quadratic twists having elements
of order
in their Tate–Shafarevich groups.
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