Proves that ratios of Sha[4] over quadratic extensions and Sha[2] of twists can grow arbitrarily large without assuming finiteness, and that Sha[2] is bounded with vanishing twists for infinitely many D in the family y^2 = x^3 + p x assuming finiteness.
Qiu, On Quadratic Twists of Elliptic Curves and Some A pplications of a Refined Version of Yu’s Formula, Communications in Algebra, 42(12), 5050–5064, 20 14
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Behaviors of the Tate--Shafarevich group of elliptic curves under quadratic field extensions
Proves that ratios of Sha[4] over quadratic extensions and Sha[2] of twists can grow arbitrarily large without assuming finiteness, and that Sha[2] is bounded with vanishing twists for infinitely many D in the family y^2 = x^3 + p x assuming finiteness.