Elements of given order in Tate–Shafarevich groups of abelian varieties in quadratic twist families

Let A be an abelian variety over a number field F and let p be a prime. Cohen–Lenstra–Delaunay-style heuristics predict that the Tate–Shafarevich group III(As) should contain an element of order p for a positive proportion of quadratic twists As of A. We give a general method to prove instances of this conjecture by exploiting independent isogenies of A. For each prime p, there is a large class of elliptic curves for which our method shows that a positive proportion of quadratic twists have nontrivial p-torsion in their Tate–Shafarevich groups. In particular, when the modular curve X0(3p) has infinitely many F-rational points, the method applies to “most” elliptic curves E having a cyclic 3p-isogeny. It also applies in certain cases when X0(3p) has only finitely many rational points. For example, we find an elliptic curve over ℚ for which a positive proportion of quadratic twists have an element of order 5 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 p≡1(mod9), examples of CM abelian threefolds with a positive proportion of quadratic twists having elements of order p in their Tate–Shafarevich groups.